Fixed Income Flashcards by Jorge V.C Enrique

The discount function and spot yield curve (or spot curve) represent the discount factors and spot rates for a range of maturities. The same information can be derived from each.

The spot rate represents the annualized return on a zero-coupon bond that has no default risk and no embedded options. Because a zero-coupon bond does not produce any cash flows prior to maturity, there is no reinvestment risk.

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Forward Rates and Forward Pricing Model

1- Forward Rates:
– Forward rates are interest rates agreed upon today for loans or deposits that will occur in the future. The set of these rates across different time horizons is called the forward curve.
– A forward rate for a deposit made at time “A” and maturing at time “B” is denoted as “f_A,B-A.”
– Example: If an investor plans to deposit money in 2 years for a period of 3 years, the forward rate is denoted as “f_2,3.”

2- Forward Pricing Model:
– The forward pricing model ensures that there is no arbitrage. It equates the return of an investor making a single deposit for a longer term to the return of rolling over shorter-term deposits.
– The forward price for a one-unit deposit made at time “A” and maturing at time “B” is derived using the discount factors “DF_A” and “DF_B”:
“F_A,B-A = DF_B / DF_A.”
– Example: The forward price of a one-year deposit starting in 4 years can be calculated using the ratio of discount factors for years 4 and 5.

DFn = [1 / (1 + Zn)^n]

Zn : Spot rate
n : Maturity

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The Forward Rate Model

1- Forward Price Determination:
– The forward price “F_A,B-A” is determined using the forward rate “f_A,B-A”.

2- Formula (using forward rate):
– “F_A,B-A = 1 / (1 + f_A,B-A)^(B-A)”

3- Relation to Spot Rates (Forward Rate Model):
– The forward rate can also be defined by spot rates:
– “(1 + z_B)^B = (1 + z_A)^A * (1 + f_A,B-A)^(B-A)”

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Forward Rate Model and Spot Rate Relationships

1- T-Year Spot Rate Function:
– The T-year spot rate, “z_T,” is a function of the one-year spot rate and successive one-year forward rates:
– “(1 + z_T)^T = (1 + z_1) * (1 + f_1,1) * (1 + f_2,1) * … * (1 + f_T-1,1)”

2- Geometric Mean Interpretation:
– The formula shows that the T-year spot rate is the geometric mean of the one-year spot rate and forward rates:
– “z_T = [(1 + z_1) * (1 + f_1,1) * (1 + f_2,1) * … * (1 + f_T-1,1)]^(1/T) - 1”

3- Spot and Forward Curve Relationship:
– Positive Slope (Normal Curve):
— A positively sloped spot curve means the forward curve will lie above the spot curve.
– Negative Slope (Inverted Curve):
— A negatively sloped spot curve will lie below the forward curve.
– Equal Rates:
— The spot and forward curves match only when rates are identical across all maturities.

4- Par Curve Representation:
– The par curve represents the yield to maturity on coupon-paying government bonds priced at par.
– Par curves are based on recently issued (“on-the-run”) bonds.

5- Bootstrapping Method:
– Bootstrapping is a technique used to derive the spot curve from the par curve.

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Explanation of the Spot Curve, Forward Curve, and Par Curve

1- Spot Curve:
– The spot curve represents the yields to maturity of zero-coupon bonds (bonds with no interim coupon payments) at various maturities.
– Uses:
— It is used to calculate present values of cash flows by discounting them using the specific spot rate for the cash flow’s maturity.
— Helps determine the price of any fixed-income security through discounting.
– Key Characteristics:
— Each point on the curve reflects the yield for a zero-coupon bond of a specific maturity.
— Derived using bootstrapping from the par curve when zero-coupon bonds are not available in the market.

2- Forward Curve:
– The forward curve represents the expected future interest rates for specific time periods, as implied by today’s spot rates.
– Uses:
— Helps investors predict short-term rates and guide expectations for future monetary policy.
— Useful for pricing derivatives, such as forward rate agreements (FRAs) and swaps.
– Key Characteristics:
— Calculated based on the spot curve using the no-arbitrage principle.
— Shows implied future yields, not necessarily the rates that will actually occur.
— Example: A one-year forward rate two years from now represents the interest rate expected for the period starting two years from now and ending three years from now.

3- Par Curve:
– The par curve represents the yields to maturity of coupon-paying bonds that are priced at par.
– Uses:
— Used as a reference for constructing the spot curve through bootstrapping.
— Reflects yields of “on-the-run” bonds, which are newly issued and actively traded government securities.
– Key Characteristics:
— Each point on the curve reflects the yield for a par bond (coupon-paying bond priced at face value) of a specific maturity.
— Easier to observe in the market compared to spot and forward curves, making it a common starting point for deriving other curves.

Practical Insights for Investors:
Spot Curve: Provides the purest measure of interest rates for different maturities, crucial for bond valuation.
Forward Curve: Key for making expectations about future rates and market sentiment.
Par Curve: Offers real-world yields that are observable in the market and serves as the basis for deriving the spot curve.

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Yield to maturity (YTM) is a commonly used pricing concept in bond markets. It can be thought of as a weighted average of the spot rates that are used to value a bond’s expected cash flows.

Assumes the curve is flat (dose not change)

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Understanding Yield to Maturity (YTM)

1- Definition of YTM:
– Yield to maturity is the internal rate of return (IRR) for a bond, calculated by equating the bond’s current price with the present value of its expected future cash flows (coupon payments and principal repayment).
– It is essentially the rate of return an investor can expect if they:
— Hold the bond until maturity.
— Receive all payments (coupons and principal) on time.
— Reinvest all interim cash flows at the same rate as the YTM.

2- Key Assumptions Behind YTM:
– For investors to earn the YTM, three conditions must hold:
— The bond must be held to maturity—selling it before maturity introduces reinvestment and price risk.
— The bond issuer must pay all cash flows (coupons and principal) on time and in full—default risk would affect actual returns.
— All interim cash flows must be reinvested at the YTM—deviations in reinvestment rates can alter actual returns.

3- When YTM is a Poor Estimate of Expected Return:
– Interest Rate Volatility:
— If interest rates fluctuate, reinvestment at the original YTM becomes unrealistic.
– Steep Yield Curve:
— When the yield curve is sharply sloped (positively or negatively), reinvestment assumptions implied by YTM are inaccurate.
– High Default Risk:
— If there’s a significant chance of default, YTM overestimates expected return because it assumes full payment of all cash flows.
– Embedded Options in the Bond:
— Callable bonds, putable bonds, or other options can alter cash flows, making YTM an unreliable measure of return.

4- Limitations of YTM:
– YTM implicitly assumes a flat yield curve, meaning the same interest rate applies across all maturities. This is rarely the case in real markets where yield curves can be positively sloped (normal), negatively sloped (inverted), or humped.

5- Practical Implications for Investors:
– YTM is a useful measure for comparing bonds under stable market conditions, but investors should not rely solely on it when:
— Rates are expected to change.
— There is credit risk or embedded options.
– In such cases, alternative measures like the yield to call (YTC), option-adjusted yield (OAS), or scenario analysis may provide better insights.

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Yield Curve Movement and the Forward Curve

If future spot rates evolve as predicted in today’s forward curve, forward contract prices will not change. In this scenario, all risk-free bonds will earn the current one-year spot rate over a one-year holding period, regardless of their maturity.

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Forward prices change when the spot curve deviates from what is predicted in the current forward curve. Active bond managers can try to anticipate changes in interest rates relative to those implied by current forward rates. For example, a manager who expects that future spot rates will be less than what is implied by the current forward curve should buy the forward contract because it will increase in price.

How Forward Prices Change with the Spot Curve

1- Relationship Between Forward Prices and Spot Rates:
– Forward prices are based on current spot rates and the forward curve, which predicts future interest rates.
– If actual future spot rates deviate from the predictions in the current forward curve, forward prices will adjust to reflect the new expectations.

2- Example of Forward Price Adjustment:
– Suppose the current forward curve predicts that future spot rates will rise to 5%.
– If, in reality, future spot rates are expected to rise less than 5% (say, to 4%), the forward price of a bond will increase because the implied discount rate (based on the lower spot rates) decreases, raising the bond’s value.

3- How Active Bond Managers Use This Information:
– Active bond managers analyze forward curves to anticipate whether actual future spot rates will differ from the rates implied in the forward curve.
– For Example:
— If a manager expects future spot rates to be lower than what the forward curve predicts, they will buy the forward contract now.
— As forward prices increase due to the unexpected drop in spot rates, the manager can profit from the price appreciation.

4- Summary of Strategy:
– The key idea is that if actual future interest rates (spot rates) are lower than those implied by the forward curve, forward contracts become more valuable.
– Conversely, if spot rates are higher than expected, forward contracts decrease in value.
– Managers leverage these deviations to make profitable trades based on their predictions of future rate movements.

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Components of Total Return for Fixed-Rate, Option-Free Bonds:
1- Receipt of Coupons:
– The periodic coupon payments that bondholders receive.

2- Return of Principal:
– The face value of the bond, repaid at maturity.

3- Reinvestment of Coupons:
– The additional returns earned by reinvesting the coupon payments.

4- Capital Gains/Losses on Sales Before Maturity:
– If the bond is sold before maturity, the price difference from its purchase price results in a gain or loss.

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Carry Trade:
1- General Definition:
– Borrowing in a low-yielding currency (or market) and investing in a higher-yielding currency (or market) to capture the yield difference.

2- Maturity Spread Carry Trade in Bond Markets:
– A type of carry trade based on the expectation that future interest rates will remain stable or rise less than implied by the spot curve.

3- How It Works:
– Borrow short-term at lower interest rates.
– Invest in longer-term bonds with higher yields.

4- Profitability Conditions:
– The strategy is profitable as long as short-term interest rates do not rise sharply.

5- Risk of Maturity Spread Carry Trade:
– Vulnerable to a spike in short-term interest rates, which can increase borrowing costs and lead to losses.

Riding the Yield Curve:
1- Definition:
– A trading strategy used when the yield curve is positively sloped.

2- Mechanism:
– Traders buy bonds with longer maturities and hold them as they “roll down the yield curve.”
– As time passes, the bond’s remaining maturity shortens, and its price rises because shorter-maturity bonds typically have lower yields (higher prices).

3- Profit Opportunity:
– The trader earns returns from the price increase (capital gains) as the bond rolls down the curve.
– The strategy is profitable when the yield curve maintains its current shape and slope.

4- Best Conditions for This Strategy:
– A positively sloped yield curve where the forward rates exceed spot rates.

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Key Insights:

Riding the Yield Curve: Profits from holding bonds and benefiting from price increases due to the yield curve’s slope.
Carry Trade: Profits from the yield differential between borrowing short-term and investing long-term, but with greater sensitivity to interest rate changes.

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Bond Strategy Regarding the Yield Curve

1- Key Components of the Formula:
– Left Side:
— Represents the accumulated value of a zero-coupon bond maturing at time B:
(1 + zB)^B

– Right Side:
— Combines two components:
—- The accumulated value of a zero-coupon bond maturing at time A:
(1 + zA)^A
—- The accumulated value of a (B - A)-year zero-coupon bond starting at time A, calculated using the forward rate:
(1 + fA,B-A)^(B - A)
– Formula:
(1 + zB)^B = (1 + zA)^A * (1 + fA,B-A)^(B - A)

2- Interpretation of the Formula:
– This equation demonstrates the relationship between spot rates (z) and forward rates (f).
– It shows that the forward rate reflects the implied rate for reinvesting from the shorter maturity bond to the longer maturity bond.

3- Strategy Based on Spot and Forward Rates:
– Compare the expected future spot rate with the implied forward rate:

– Case 1:
Expected future spot rate is lower than the forward rate:
— Implication: The bond is undervalued.
— Action: Buy the bond now.

– Case 2:
Expected future spot rate is higher than the forward rate:
— Implication: The bond is overvalued.
— Action: Sell the bond now.

4- Practical Application for Bond Traders:
– Evaluate the current forward curve and predict future spot rates.
– If rates deviate from the forward curve as expected, traders can exploit price changes for profit.
– This strategy requires accurate forecasting of future interest rate movements.

5- Related Formula for Comparing Spot Rates Over Time:
(1 + zB)^B / (1 + fB-1,1)^(B - 1) = (1 + z1)
– This formula shows how forward rates are linked to one-year spot rates and (B - 1)-year bonds.

Summary:

– The forward curve provides expectations of future interest rates.
– A mismatch between the expected future spot rate and the implied forward rate creates trading opportunities.
– Buy undervalued bonds when the expected spot rate is below the forward rate.
– Sell overvalued bonds when the expected spot rate is above the forward rate.

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Riding the Yield Curve / Rolling Down the Yield Curve Strategy

This is a popular yield curve trading strategy designed to take advantage of the positive slope of the yield curve. Below are the key points and steps involved:

1- Assumptions Underlying the Strategy: – The yield curve has a positive slope, meaning longer-term bonds have higher yields compared to shorter-term bonds.
– The forward curve is always above the spot curve because the forward curve incorporates expectations of future rates.
– The yield curve remains stable over the investment horizon, meaning the shape and slope of the curve do not change.
– Future spot rates are expected to be less than forward rates, which creates an opportunity for capital gains.

2- How the Strategy Works: – Traders purchase bonds with longer maturities to earn higher yields associated with longer-term instruments.
– As the bond approaches its maturity date, its yield declines (rolls down the yield curve), and the bond’s price increases. This generates capital gains in addition to coupon payments.
– By holding the bond and selling it before maturity, traders can realize returns greater than those earned by reinvesting in a series of shorter-term bonds.

3- Key Advantage: – Traders can earn an extra return by investing in longer-maturity bonds compared to continually reinvesting in shorter-term bonds.
— Return on bonds with longer maturity > Return on continually reinvesting in shorter-maturity bonds.

Practical Implications:
This strategy is effective when:
– Interest rates remain stable or decline during the holding period.
– Future spot rates are indeed lower than the forward rates implied by the current yield curve.

However, the strategy carries interest rate risk because an unexpected rise in interest rates would reduce bond prices, leading to potential capital losses.

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Swap Rate Curve

1- Definition of Swaps and Their Function:

– A swap is a derivative contract in which counterparties exchange fixed-rate interest payments for floating-rate interest payments.
– The size of the payments is based on the swap rates (for the fixed leg), the floating reference rate (e.g., LIBOR or SOFR), and the notional principal amount.
– Swaps are used for speculation and risk management by hedging interest rate exposure.

2- Key Features of Swap Rates:

– The fixed-rate leg of the swap is called the swap rate and is denoted as sT, where T is the maturity of the swap.
– The swap rate is set such that the swap has a zero value at inception, meaning the present value (PV) of fixed-rate payments equals the PV of floating-rate payments.
– The floating leg is tied to a short-term reference rate, such as LIBOR or SOFR, and resets periodically.

3- The Swap Curve:

– The swap curve represents the yield curve for swap rates across different maturities. It provides a benchmark for interest rates in swap markets.
– The swap curve differs from government bond yield curves because it is derived from swap rates, not bond yields.
– For countries with illiquid long-term government bond markets, the swap curve often serves as a more reliable benchmark for interest rates.

4- Advantages of Swaps and Their Market:

– Liquidity:
— The swap market is highly liquid because it does not require matching borrowers and lenders; it only needs counterparties willing to exchange cash flows.
— Swaps are widely used for hedging interest rate risks, which increases their liquidity.
– Benchmark Role:
— In countries with less liquid government bond markets, the swap curve serves as an essential benchmark curve for valuing fixed-income securities.
— Swap rates and government bond yields are often used together for comprehensive valuation.

Summary:
– Swaps exchange fixed-rate and floating-rate interest payments for speculation and risk management.

– The swap curve reflects the yield curve for swap rates across maturities and is a vital benchmark, especially where long-term government bonds are less liquid.

– Swap markets are highly liquid due to their effectiveness in hedging interest rate risk and the simplicity of counterparties agreeing to exchange cash flows.

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[Swap Rate Formula and Example Calculation

1- Swap Rate Formula Using Spot Rates
– The fixed swap rate “s_T” is the rate that equates the present value of fixed payments to the present value of floating payments at the inception of the swap.
– Under the no-arbitrage condition, the swap has zero net value at inception.

– Formula (based on spot rates):
— “T∑_t=1 [s_T ÷ (1 + z_t)^t] + [1 ÷ (1 + z_T)^T] = 1”

2- Swap Rate Formula Using Discount Factors
– Discount factors are derived from spot rates and simplify the present value calculation of each cash flow.
– The formula can be restated using discount factors DF_t as follows:

– Formula (using discount factors):
— “T∑_t=1 [s_T × DF_t] + DF_T = 1”

3- Example: Calculating the 3-Year Swap Rate

Fictional Scenario
– A 3-year interest rate swap is being priced.
– The fixed leg pays annually.
– The spot rates (z_t) are as follows:
– 1-year spot rate (z1) = 3.0%
– 2-year spot rate (z2) = 3.5%
– 3-year spot rate (z3) = 4.0%

Step 1: Convert spot rates to discount factors
– DF_1 = 1 ÷ (1 + 0.030)^1 = 0.97087
– DF_2 = 1 ÷ (1 + 0.035)^2 = 0.93351
– DF_3 = 1 ÷ (1 + 0.040)^3 = 0.88900

Step 2: Use the discount factor formula to solve for s_3
– Apply the formula:
— “s_3 × (DF_1 + DF_2 + DF_3) + DF_3 = 1”
— s_3 × (0.97087 + 0.93351 + 0.88900) + 0.88900 = 1
— s_3 × 2.79338 + 0.88900 = 1
— s_3 × 2.79338 = 1 - 0.88900
— s_3 × 2.79338 = 0.11100
— s_3 = 0.11100 ÷ 2.79338 ≈ 0.03974

– Result:
— “s_3 ≈ 3.974%”

Key Takeaways
– The swap rate is calculated to equate the PV of fixed payments to the PV of the floating leg at inception.
– The calculation can be performed using either spot rates or discount factors, with identical results.

List of Variables
– s_T: Swap rate for term T
– z_t: Spot rate for period t
– DF_t: Discount factor for period t = 1 ÷ (1 + z_t)^t
– T: Total term of the swap in years
– ∑: Summation operator over all payment periods]

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Importance of the Swap Curve

1- Countries Without Liquid Government Bond Markets:
– For countries where government bond markets lack liquidity for maturities beyond one year, the swap curve serves as a benchmark for interest rates.
— This provides market participants with a reliable reference for pricing and valuing financial instruments. In the US, wholesale banks tend to use the swap curve for valuation, while retail banks prefer the government spot curve.

2- Countries with Larger Private Sectors:
– In countries where the private sector is larger than government debt markets, the swap curve is used as a measure of the time value of money.
— It reflects the market-driven cost of borrowing and investing across different maturities.

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Government Bond Market vs. Swap Market in Valuation

1- Both Markets Are Very Liquid:
– In countries like the U.S., where both the government bond market and the swap market are highly liquid, the choice of benchmark depends on the nature of the business operations.

2- Wholesale Banks:
– Risk Hedging: These banks often engage in hedging activities to manage risk.
– Preference for Swap Curve: Likely to value fixed-income securities using the swap curve due to its relevance in derivative and hedging activities.

3- Retail Banks:
– Limited Swap Exposure: Retail banks typically have limited involvement in swaps.
– Preference for Government Spot Curve: They tend to use the government spot curve as a benchmark, reflecting their focus on traditional banking products like loans and deposits.

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Why the Swap Rate is Slightly Less Than the Spot Rate

1- Weighted Average of Spot Rates:
– The swap rate represents a weighted average of the spot rates.
— Weights Depend on Cash Flows: The weights are determined by the cash flow structure of the swap.

2- Concentration on Key Spot Rates:
– Most of the weight will align with the spot rate corresponding to the timing of the notional amount.
— Example: For a three-year swap, the majority of the weight is on the three-year spot rate, reflecting the critical cash flow timing.

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Key Concepts on Swap Spread, I-Spread, and Z-Spread

1- Swap Spread:
– Definition: The swap spread is the difference between the swap rate and the yield of the on-the-run government bond with the same maturity.
– Interpretation:
— Indicates credit spreads and liquidity spreads in the market.
— Highlights the risk premium of firms compared to risk-free government securities.

2- I-Spread (Interpolated Spread):
– Definition: The I-spread measures the difference between a bond’s yield and the swap rate for the same maturity.
– Comparison with Swap Spread:
— The I-spread focuses on the difference between a specific bond and the swap rate, while the swap spread compares the swap rate to a government bond yield.
– Bonds yield - [Spot rate + Swap Spread]
– [Spot rate + Swap Spread] = Swap Rate

3- Libor/Swap Curve:
– Most widely used interest rate curve due to its association with the credit risk of firms rated A1/A+ (typical for commercial banks).
– Influences on Swap Rates:
— Default risk of firms.
— Supply and demand conditions in government bond markets.

4- Advantages of Swap Markets:
– Unregulated by governments, providing cross-country comparability.
– Offers more maturities compared to government bond markets, increasing flexibility for pricing and hedging.

5- Z-Spread (Zero-Volatility Spread):
– Definition: The Z-spread is a constant basis point spread added to the implied spot yield curve so that discounted future cash flows equal the bond’s current market price.
– Key Characteristics:
— More accurate than a linearly interpolated yield, particularly when the yield curve is steep.
— Reflects credit and liquidity risks better than simple spreads.

Summary:
– Swap Spread: Measures credit and liquidity spreads by comparing the swap rate to government bond yields of the same maturity.
– I-Spread: Focuses on the difference between a bond’s yield and the swap rate for the same maturity.
– Z-Spread: Captures credit and liquidity risks by adjusting spot yields to equate discounted cash flows to the bond’s market price.
– The Libor/swap curve is preferred due to its association with A1/A+ firms and its broad applicability across maturities and countries.

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Key Concepts on Spreads as a Price Quotation Convention

3- TED Spread (Treasury-Eurodollar Spread):
– Definition: The TED spread is the difference between Libor and the yield on a T-bill with matching maturity.
— Formula: TED Spread = Libor − T-Bill Yield.
— Indicator of Credit Risk:
— A higher TED spread signals increased perceived default risk in interbank loans.
— Reflects counterparty credit risk and overall risk in the banking system.

4- Libor-OIS Spread:
– Definition: The Libor-OIS spread measures the difference between Libor and the Overnight Indexed Swap (OIS) rate.
— Characteristics:
— The OIS rate is the geometric average of a floating overnight rate during the payment period (e.g., federal funds rate for USD).
— Indicates the risk and liquidity of money market securities.
— A higher spread implies increased risk or reduced liquidity in the money markets.

Summary:
– Treasury rates and swap rates differ due to default risk, liquidity variations, and arbitrage limitations.
– The Swap Spread reflects compensation for counterparty credit risk, typically positive but occasionally negative.
– The TED Spread captures counterparty credit risk in the banking system, rising during times of increased interbank default concerns.
– The Libor-OIS Spread highlights risk and liquidity conditions in money markets, with wider spreads indicating heightened financial stress.

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Expectations Theory

1- Unbiased Expectations Theory (Pure Expectations Theory):
– Purpose: States that forward rates are unbiased predictors of future spot rates.
– Key Concept: Bonds of any maturity are perfect substitutes over any holding period. For example, the expected return of holding a seven-year bond for three years is equal to the expected return of holding a five-year bond for the same three years.
– Assumption: Investors are risk-neutral (indifferent to risk).
– Criticism: Not consistent with observed risk aversion, as most investors demand risk premiums for longer maturities.

2- Local Expectations Theory:
– Purpose: A modified version of the pure expectations theory that focuses on short time periods.
– Key Concept: All bonds, whether risk-free or risky, are expected to earn the risk-free rate of return over short holding periods.
— Over longer periods, risk premiums may exist, allowing for risk-aversion effects.
– Advantage: Applicable to both risk-free and risky bonds, making it more flexible than the pure expectations theory.
– Inconsistency in Practice:
— Empirical evidence shows that longer-dated bonds often produce higher returns than shorter-maturity bonds over short holding periods.
— This suggests investors require compensation for the illiquidity of longer-term bonds and the challenges in hedging risks for these securities.

Summary:

– The Unbiased Expectations Theory assumes forward rates are accurate predictors of future spot rates and that investors are risk-neutral.
– The Local Expectations Theory adjusts for short-term neutrality but allows for risk premiums over longer periods, making it more applicable in practice.
– Empirical evidence suggests that both theories struggle to fully explain observed market behavior, particularly for longer-maturity bonds.

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Liquidity Preference Theory

1- Purpose:
– Explains the shape of the yield curve by incorporating the idea of a risk premium (liquidity premium) for longer-term bonds.
– Suggests that investors demand additional compensation for the interest rate risk and reduced liquidity associated with lending over longer periods.

2- Key Features:
– Risk Premium: Longer maturities have higher interest rate risk, so investors require higher returns (liquidity premiums) to compensate for holding them.
– Yield Curve Implications:
— Typically leads to an upward-sloping yield curve because the risk premium increases with maturity.
— Can still produce a downward-sloping or hump-shaped yield curve if deflationary expectations dominate the market.

3- Implications for Forward Rates:
– Forward rates derived from the yield curve are upwardly biased estimators of future spot rates because they include the liquidity premium.
– This distinguishes the liquidity preference theory from the pure expectations theory, which assumes no risk premium.

Summary:

– The Liquidity Preference Theory accounts for the interest rate risk and liquidity concerns associated with longer maturities, leading to the inclusion of a liquidity premium in yields.
– This theory generally explains the upward-sloping yield curve but allows for other shapes in unique scenarios like deflation.
– Forward rates under this theory are biased upwards due to the embedded liquidity premium.

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Segmented Markets Theory

1- Purpose:
– Explains the shape of the yield curve based on supply and demand dynamics across different maturities.
– Suggests that lenders and borrowers operate in distinct maturity segments based on their specific needs and preferences, leading to independently determined interest rates for each segment.

2- Key Features:
– No Arbitrage Between Segments: Bonds of different maturities are not perfect substitutes, meaning investors are unwilling to move between segments to exploit arbitrage opportunities.
– Influence of Market Participants:
— Life insurers and pension funds drive demand for long-term bonds to match their long-term liabilities.
— Money market funds dominate demand for short-term bonds due to their liquidity requirements.
– Yield Curve Variability: The shape of the yield curve reflects imbalances in supply and demand within specific maturity segments.

3- Implications for the Yield Curve:
– High demand or limited supply in a segment reduces yields in that maturity range, while low demand or high supply increases yields.
– The theory does not inherently predict an upward or downward slope for the yield curve but provides insight into anomalies in specific sections of the curve.

Summary:

– The Segmented Markets Theory explains yield curve shapes by focusing on supply and demand dynamics within distinct maturity segments.
– It assumes that bonds of different maturities are not substitutes, so market participants’ preferences dominate pricing for each segment.
– The shape of the yield curve varies depending on which segments experience imbalances in demand or supply.

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Preferred Habitat Theory 1- Purpose: -- Expands on Segmented Markets Theory by allowing for investor flexibility. -- Explains yield curve shapes based on investors’ maturity preferences and the compensation required to leave their preferred maturity range (or "habitat"). 2- Key Features: -- Preferred Segments: Investors and institutions have a "natural" maturity range (e.g., long-term bonds for pension funds). -- Inducement to Switch: Investors may purchase bonds outside their preferred range if they receive additional compensation (e.g., higher yields). -- Market Dynamics: --- If demand for a specific segment is high and supply is low, yields in that segment will decrease. --- Investors may then move to adjacent segments if they are adequately compensated, affecting yields in those segments as well. 3- Implications for the Yield Curve: -- The theory can explain both positively sloped yield curves (investors demand a premium for long-term bonds) and negatively sloped curves (if sufficient demand exists for long-term bonds or short-term yields spike).

Summary: -- The Preferred Habitat Theory assumes that investors have natural maturity preferences but will move to other segments for sufficient compensation. -- Unlike Segmented Markets Theory, it explains how demand for specific maturities can spill over into other segments. -- The theory provides a framework for understanding both upward and downward-sloping yield curves.

Liquidity Preference Theory (Expanded Explanation) 1- Purpose of the Theory: -- Explains why the yield curve is typically upward-sloping by incorporating a liquidity premium to account for interest rate risk. -- Recognizes that investors prefer shorter maturities due to lower interest rate risk and require additional compensation (liquidity premium) to hold longer-term bonds. 2- Key Insights: -- Liquidity Premium Explained: --- Represents compensation for interest rate risk (not liquidity risk) associated with holding longer-term bonds. --- Longer-term bonds experience greater price sensitivity to interest rate changes. -- Forward Rate as a Biased Estimate: --- Forward rates are upwardly biased predictors of future spot rates because of the liquidity premium. --- Even if spot rates are expected to remain constant, the liquidity premium causes forward rates to exceed expected future spot rates. -- Yield Curve Implications: --- An upward-sloping yield curve is expected because the liquidity premium increases with maturity. --- A downward-sloping yield curve can still occur if future spot rates are expected to decline more than the liquidity premium. 3- Applications in Practice: -- Investors require enticement (via higher yields) to purchase bonds with maturities beyond their investment horizons. -- The liquidity premium becomes a key factor in the pricing of longer-term bonds.

Summary: -- Liquidity Preference Theory predicts a positively sloped yield curve, even if short-term spot rates are not expected to change, due to the liquidity premium required for interest rate risk. -- Forward rates are biased estimates of future spot rates because they include this liquidity premium. -- While typically explaining upward-sloping curves, the theory accommodates downward-sloping yield curves when future interest rate declines outweigh the premium.

A Bond’s Exposure to Yield Curve Movement Shaping risk is the sensitivity of a bond’s price to the changing shape of the yield curve. Yield curves rarely shift in a purely parallel fashion, so the shape is constantly changing. Such changes are particularly important for the valuation of bonds with embedded options.

Factors Affecting the Shape of the Yield Curve Yield curve factor models aim to explain historical interest rate movements by identifying the key factors that influence the curve's shape. The Litterman and Scheinkman three-factor model highlights the following factors: 1- Level (Parallel Shifts): -- Description: Represents parallel upward or downward shifts in the entire yield curve. -- Impact: --- Negative for all bonds because of the inverse relationship between yields and bond prices. --- Example: A level coefficient of -0.5 means the bond's price will decline by 0.5% for a 1 standard deviation increase in the yield curve level. -- Importance: The level factor is the dominant component of yield curve movements, explaining most variations. 2- Steepness (Nonparallel Shifts): -- Description: Captures differences in changes between short-term and long-term rates. --- Example: Short-term rates rise more than long-term rates, causing the curve to become steeper. -- Impact: Nonparallel shifts are generally less significant compared to parallel shifts (level factor). -- Relevance: Changes in steepness often reflect monetary policy actions or expectations for future growth and inflation. 3- Curvature (Twists): -- Description: Reflects changes where short-term rates increase while long-term rates fall, creating a twist in the curve. -- Impact: --- Negative for intermediate maturities (e.g., 5- to 10-year bonds). --- Positive for short-term and long-term maturities. -- Relevance: Often linked to changes in market sentiment or unique economic conditions (e.g., flight to safety in long-term bonds).

Summary: -- Level: Describes parallel shifts and has the greatest explanatory power. -- Steepness: Captures nonparallel shifts (e.g., short-term rates rising faster than long-term rates). -- Curvature: Represents twists in the yield curve, typically with intermediate maturities falling while short and long-term rates rise. -- Together, these factors offer a comprehensive understanding of how and why the yield curve changes over time.

Yield Volatility Interest rate volatility plays a significant role in fixed-income securities valuation and risk management, particularly for securities with embedded options. Below are the key aspects: 1- Definition and Impact: -- Description: Yield volatility reflects fluctuations in interest rates, influencing the value of bonds and their embedded options (e.g., callable or putable bonds). -- Relevance: --- Higher volatility increases the value of embedded options, as uncertainty raises the potential benefit of exercising options. --- Critical in risk management for forecasting potential changes in bond values due to rate movements. 2- Volatility Curve (Term Structure): -- Definition: The volatility curve shows how interest rate volatility varies across maturities. -- Shape: Typically downward sloping, indicating: --- Short-term rates: Tend to be more volatile due to higher sensitivity to monetary policy changes. --- Long-term rates: Are relatively stable because they are influenced by slower-moving factors such as long-term growth and inflation expectations. 3- Factors Driving Volatility: -- Short-term rates: Primarily influenced by monetary policy uncertainty. --- Example: Market speculation on central bank interest rate changes increases short-term rate volatility. -- Long-term rates: Driven by uncertainty about real economic growth and inflation expectations. --- Example: Unpredictable long-term inflation trends can cause volatility in long-term rates. 4- Duration Effect on Prices: -- Despite the relative stability of long-term rates, prices of long-term bonds are generally more volatile than short-term bonds due to the duration effect (greater sensitivity to interest rate changes). --- Example: A small change in long-term interest rates can significantly affect the price of a long-duration bond.

Summary: -- Yield volatility is crucial for understanding the behavior of fixed-income securities, particularly those with embedded options. -- The volatility curve is typically downward sloping, as short-term rates are more volatile than long-term rates. -- Short-term volatility reflects monetary policy uncertainty, while long-term volatility is driven by economic growth and inflation expectations. -- Longer-term bonds, while less volatile in rates, exhibit higher price volatility due to the duration effect.

Managing Yield Curve Risks Using Key Rate Duration Yield curve risk refers to the potential impact of unanticipated changes in the yield curve on a portfolio’s value. Key rate durations and factor models can be used to manage this risk. 1- Key Rate Duration: -- Measures the sensitivity of a bond’s value to changes in specific points on the yield curve. -- The bond’s effective duration is calculated as the sum of its key rate durations across all maturities. -- For a portfolio of bonds with maturities at 1-year, 5-year, and 10-year, the expected percentage change in value can be approximated as: ΔP ÷ P ≈ -D1 Δr1 - D5 Δr5 - D10 Δr10 Where: --- ΔP ÷ P: Percentage change in portfolio value. --- D1, D5, D10: Key rate durations for 1-year, 5-year, and 10-year maturities. --- Δr1, Δr5, Δr10: Changes in interest rates for 1-year, 5-year, and 10-year maturities. 2- Factor Models for Yield Curve Movements: -- Yield curve movements can be decomposed into: --- Level: Parallel shifts. --- Steepness: Nonparallel changes where short-term rates increase more than long-term rates. --- Curvature: "Twists" where short-term rates and long-term rates move in opposite directions. For a portfolio with maturities at 1-year, 5-year, and 10-year: --- D_Level: D1 + D5 + D10 --- D_Steepness: -D1 + D10 --- D_Curvature: D1 - D5 + D10

Summary: -- Key rate durations measure sensitivity to specific maturities, enabling precise management of yield curve risk. -- Factor models break yield curve changes into components (level, steepness, curvature) for additional insights. -- These tools help portfolio managers identify risks and make informed decisions to mitigate exposure.

List of Duration Measures Below is a detailed breakdown of key duration measures, their uses, advantages, and disadvantages. 1- Effective Duration: -- Usage: --- Measures the sensitivity of a bond’s price to a small parallel shift in the yield curve. --- Commonly used for bonds with embedded options. -- Advantages: --- Captures the effects of changes in interest rates and option-related convexity. --- Applicable to option-embedded securities. -- Disadvantages: --- Assumes parallel yield curve shifts, which may not always hold true. --- Requires complex option-adjusted calculations. 2- Key Rate Duration: -- Usage: --- Measures the sensitivity of a bond’s price to changes in specific maturities (key rates) on the yield curve. --- Useful for managing yield curve risk. -- Advantages: --- Identifies vulnerabilities to non-parallel shifts in the yield curve. --- Provides precision in managing risk for portfolios with varying maturities. -- Disadvantages: --- Complex and requires detailed yield curve data. --- Not useful for securities that are evenly impacted by yield curve shifts. 3- Modified Duration: -- Usage: --- Measures the percentage change in a bond’s price for a 1% change in yield. --- Often used for plain vanilla fixed-income securities. -- Advantages: --- Straightforward and widely understood. --- Applicable to bonds without embedded options. -- Disadvantages: --- Assumes constant cash flows, so unsuitable for callable/option-embedded bonds. --- Does not capture convexity effects. 4- Macaulay Duration: -- Usage: --- Measures the weighted-average time to receive a bond’s cash flows. --- Primarily used for immunization strategies in portfolio management. -- Advantages: --- Provides a time-weighted measure of bond maturity. --- Useful for matching duration to liability timelines. -- Disadvantages: --- Assumes a flat yield curve and constant cash flows. --- Less relevant for actively managed portfolios. 5- Portfolio Duration: -- Usage: --- Measures the weighted-average duration of a portfolio of bonds. --- Useful for assessing overall interest rate risk of a portfolio. -- Advantages: --- Aggregates interest rate sensitivity across multiple bonds. --- Useful for diversified portfolios. -- Disadvantages: --- Ignores interaction effects between bonds. --- Assumes parallel shifts in the yield curve. 6- Convexity-Adjusted Duration: -- Usage: --- Adjusts effective or modified duration to account for convexity (nonlinear price changes). --- Used when interest rate changes are large. -- Advantages: --- Improves accuracy for non-linear price-yield relationships. --- Essential for large interest rate shifts. -- Disadvantages: --- More complex to calculate. --- May overstate effects for smaller interest rate changes.

Key Insights: -- Effective Duration is ideal for option-embedded bonds. -- Key Rate Duration is best for assessing yield curve risk. -- Modified Duration is the simplest measure for vanilla bonds. -- Convexity Adjustment is crucial for large rate shifts.

Factors Affecting Spot and Forward Rate Curves 1- Short- and Intermediate-Term Rates: -- Primary Influences: --- Monetary Policy: Central banks' decisions on interest rates directly impact short- and intermediate-term rates. --- Macroeconomic Factors: GDP growth and inflation also play a role, but to a lesser extent. 2- Long-Term Rates: -- Primary Influences: --- Inflation: Approximately two-thirds of the variation in long-term rates is driven by inflation expectations. --- Monetary Policy: Residual variation in long-term rates is attributed to central bank actions and monetary policies. 3- Additional Factors Influencing Bond Yields: -- Fiscal Policy: --- Governments issuing more debt to finance budget deficits cause bond yields to rise as supply increases. --- Restrictive fiscal policy reduces new government debt issuance, lowering yields. -- Maturity Structure: --- An increase in the supply of longer-dated bonds raises yields and enhances excess returns for longer maturities. -- Investor Demand: --- Domestic demand (e.g., from pension funds or insurance companies) lowers yields, particularly in the long-term segment. --- Foreign investors influence yields indirectly through exchange rate management and international transactions.

Summary: -- Short- and intermediate-term rates are mainly influenced by monetary policy, with macroeconomic factors like GDP growth and inflation having secondary effects. -- Long-term rates are primarily driven by inflation expectations and, to a lesser extent, monetary policy. -- Fiscal policy, maturity structure, and investor demand also play critical roles in shaping bond yields and rate curves.

Active Bond Investment Strategies and Yield Curve Dynamics 1- Passive vs. Active Investors: -- Passive Investors: Accept the forward rates implied by the yield curve and simply roll bonds over as they mature. -- Active Investors: Develop independent views of interest rate movements and market conditions, seeking to profit from discrepancies between their expectations and market consensus. --- Derivative Usage: Active investors can use derivatives, such as futures contracts, to express their views while minimizing portfolio turnover. 2- Bond Risk Premium (Term Premium): -- Definition: The additional yield required to hold a long-term, default-free bond above the short-term risk-free rate. -- Characteristics: --- It is forward-looking and based on government bond yields. --- It excludes illiquidity or credit risk premiums. 3- Strategies Based on Interest Rate Expectations: -- Parallel Shifts in Rates: --- Downward Shift: Increase portfolio duration to benefit from rising bond prices. --- Upward Shift: Decrease portfolio duration to minimize losses from falling bond prices. --- Key Note: Parallel shifts are rare; more nuanced strategies are typically required. -- Non-Parallel Shifts: --- Steepening Yield Curve: Focus on shorter maturities as long-term yields rise faster than short-term yields. --- Flattening Yield Curve: Extend portfolio duration as long-term yields fall relative to short-term yields.

Summary: -- Passive investors accept forward rates and manage portfolios with minimal turnover. -- Active investors seek to profit from discrepancies between market consensus and their own expectations, often using derivatives to express their views. -- The bond risk premium explains why yields are typically upward-sloping, reflecting compensation for holding longer-maturity bonds. -- Portfolio strategies vary based on expected interest rate movements, with duration adjustments playing a critical role.

Yield Curve Movements and Investment Strategies 1- Bullish Steepening -- Definition: Short-term rates fall faster than long-term rates. -- Cause: Central banks loosen monetary policy to stimulate an economy operating below potential. -- Investment Strategy: --- Position: Short long-term bonds and buy short-term bonds. --- Objective: Profit from the widening spread between short- and long-term yields. 2- Bearish Steepening -- Definition: Long-term rates rise faster than short-term rates. -- Cause: Often occurs when inflation expectations increase, pushing long-term rates higher relative to short-term rates. -- Investment Strategy: --- Position: Buy long-term bonds and short short-term bonds. --- Objective: Benefit from the widening gap as long-term rates increase more than short-term rates. 3- Bullish Flattening -- Definition: Long-term rates fall faster than short-term rates. -- Cause: Observed after market turmoil due to a "flight to quality" where investors move into safe government bonds. -- Investment Strategy: --- Position: Buy long-term bonds and short short-term bonds. --- Alternative: For investors unable to short, shift from a bullet portfolio (concentrated on a single maturity) to a barbell portfolio (combining short- and long-dated bonds). --- Objective: Profit from narrowing spreads between short- and long-term yields.

4- Bearish Flattening -- Definition: Short-term rates rise faster than long-term rates. -- Cause: Typically occurs during an economic expansion when central banks tighten monetary policy to control inflation. -- Investment Strategy: --- Position: Short short-term bonds and hold long-term bonds. --- Objective: Take advantage of narrowing spreads between short- and long-term rates. 5- Bullish Parallel Shift -- Definition: Rates across all maturities decline simultaneously. -- Cause: A decrease in inflation expectations or significant monetary policy easing. -- Investment Strategy: --- Position: Increase portfolio duration by buying long-term bonds. --- Objective: Maximize gains from falling rates as longer-duration bonds are more sensitive to interest rate changes. 6- Bearish Parallel Shift -- Definition: Rates across all maturities increase simultaneously. -- Cause: A rise in inflation expectations or significant monetary policy tightening. -- Investment Strategy: --- Position: Reduce portfolio duration by holding short-term bonds. --- Objective: Minimize losses from rising rates as shorter-duration bonds are less sensitive to interest rate changes. 7- Bullish Twist -- Definition: Short-term rates fall while long-term rates rise. -- Cause: Often associated with monetary policy easing accompanied by rising inflation expectations in the long run. -- Investment Strategy: --- Position: Buy short-term bonds and short long-term bonds. --- Objective: Benefit from declining short-term rates and rising long-term rates. 8- Bearish Twist -- Definition: Short-term rates rise while long-term rates fall. -- Cause: Often observed during stagflation or when the economy faces both weak growth and inflationary pressures. -- Investment Strategy: --- Position: Short short-term bonds and buy long-term bonds. --- Objective: Profit from short-term rates increasing more than long-term rates declining.

Duration-Neutral Positioning -- Definition: Structuring a portfolio to eliminate exposure to changes in the overall level of the yield curve while focusing solely on steepening or flattening effects. -- Benefit: Isolates yield curve slope changes from shifts in the curve's level, providing a clearer risk-return profile.

Yield Curve Movement Strategies 1- Overall Interest Rates Expected to Fall -- Action: Extend portfolio duration. -- Reasoning: Longer-duration bonds are more sensitive to interest rate changes, allowing investors to benefit more from falling rates. -- Objective: Maximize gains from a declining interest rate environment. 2- Overall Interest Rates Expected to Rise -- Action: Reduce portfolio duration. -- Reasoning: Shorter-duration bonds are less sensitive to rising interest rates, minimizing potential losses. -- Objective: Protect the portfolio from adverse price movements due to rising rates. 3- Yield Curve Expected to Steepen -- Action: --- Sell: Long-term bonds. --- Buy: Short-term bonds. -- Reasoning: Steepening implies long-term rates increase relative to short-term rates. Selling long-term bonds minimizes exposure to rate increases, while buying short-term bonds captures relative stability. -- Objective: Profit from the divergence between short- and long-term rates. 4- Yield Curve Expected to Flatten -- Action: --- Sell: Short-term bonds. --- Buy: Long-term bonds. -- Reasoning: Flattening implies short-term rates rise or long-term rates fall. Selling short-term bonds avoids potential losses from rising rates, while buying long-term bonds captures gains from falling rates. -- Objective: Benefit from the convergence of short- and long-term yields.

1. Banks Preferred Rate: Swap rates. Reason: Swap rates reflect the credit risk of A-rated financial institutions, aligning better with the risk profile of banks. They are also more liquid and globally comparable, making them a suitable benchmark for bank-issued bonds. 2. Corporate Issuers Preferred Rate: Interpolated spreads (I-spreads) over swap rates. Reason: I-spreads reflect credit risk relative to swaps, making them more accurate for pricing corporate bonds with varying levels of credit risk. 3. Municipal Bond Issuers Preferred Rate: Municipal bond indices or taxable municipal spreads. Reason: These issuers often rely on tax-advantaged pricing benchmarks, such as municipal yield curves, to reflect tax-adjusted returns for investors. 4. Emerging Market Governments Preferred Rate: Sovereign credit spreads over swaps or global bond indices. Reason: Emerging market governments often issue bonds with higher credit risk than developed market governments, so swap-based spreads or sovereign-specific indices better reflect the associated risk. 5. Supranational Organizations (e.g., World Bank) Preferred Rate: LIBOR or SOFR-based curves. Reason: Supranational entities often have AAA ratings, and they issue bonds in highly liquid markets. They may prefer floating rates or other benchmark curves that align with global funding conventions. 6. High-Yield Bond Issuers Preferred Rate: High-yield bond indices or spreads relative to corporate indices. Reason: These issuers often require benchmarks that reflect their lower credit ratings and higher default risks, which are not captured by government or swap rates. 7. Securitized Debt Issuers (e.g., Mortgage-Backed Securities) Preferred Rate: Asset-specific benchmarks such as OAS (option-adjusted spread). Reason: Securitized debt pricing depends on cash flow structures and embedded options, requiring benchmarks that adjust for prepayment or extension risk.

6.2 The Arbitrage Free Valuation Framework

-- Explain what is meant by arbitrage-free valuation of a fixed-income instrument. -- Calculate the arbitrage-free value of an option-free, fixed-rate coupon bond. -- Describe a binomial interest rate tree framework. -- Describe the process of calibrating a binomial interest rate tree to match a specific term structure. -- Describe the backward induction valuation methodology and calculate the value of a fixed-income instrument given its cash flow at each node. -- Compare pricing using the zero-coupon yield curve with pricing using an arbitrage-free binomial lattice. -- Describe pathwise valuation in a binomial interest rate framework and calculate the value of a fixed-income instrument given its cash flows along each path. -- Describe a Monte Carlo forward-rate simulation and its application. -- Describe term structure models and how they are used.

Arbitrage-Free Valuation: Key Points -- Definition: Arbitrage-free valuation calculates the value of securities under the assumption that no arbitrage opportunities exist, ensuring the market prices reflect all available information. -- Principle of No Arbitrage: In efficient markets, arbitrage opportunities may arise temporarily but are eliminated quickly as prices adjust. Arbitrage allows investors to earn riskless profits with no net investment, which violates market efficiency. -- Valuation Framework: --- The value of a financial asset equals the present value of its cash flows. --- Risk-free assets: Discount cash flows at the risk-free rate. --- Riskier assets: Use higher discount rates reflecting the risk premium. -- Yield Curve Impact: --- If the yield curve is flat, all cash flows are discounted at the same rate. --- If the yield curve is not flat, each cash flow is discounted using the appropriate spot rate, treating cash flows as individual zero-coupon bonds. --- This ensures consistency with arbitrage-free pricing principles by reflecting the time value of money and differing rates for various maturities.

Law of One Price The law of one price states two goods that are perfect substitutes should have the same price. If different prices existed, the trader could buy the cheaper one and sell the more expensive one to lock in a risk-free profit based on the price differential. However, for two assets to be perfect substitutes, it is not sufficient that they offer identical cash flows. They must also be equal in terms of risk.

Arbitrage Opportunity: -- Definition: Arbitrage opportunities arise when an investor can achieve a riskless profit with zero net investment. -- Types of Arbitrage: --- 1. Violation of Value Additivity: ---- Occurs when the value of the whole differs from the sum of its parts. ---- Example: If a bond’s individual cash flows (valued as separate zero-coupon bonds) sum to $1,000 but the bond is priced at $950, an arbitrage opportunity exists. --- 2. Violation of Dominance: ---- Occurs when a risk-free future payoff is available at a zero or negative price today. ---- Example: If a security guaranteeing a $1,000 risk-free payment is priced at $950, investors can arbitrage by purchasing the security and receiving a certain profit at maturity. -- Market Impact: Investors will exploit these arbitrage opportunities until prices adjust, restoring efficiency in well-functioning markets.

Lognormal Model and Approximation of Interest Rates The lognormal random walk model provides an efficient way to approximate future interest rates by utilizing implied forward rates. This model ensures the interest rate tree aligns with the forward rates implied by the benchmark yield curve. Key Concepts: 1- Midpoint Approximation in Nodes: -- The one-year forward rate for a given period serves as the best estimate for the midpoint between adjacent nodes in that period. -- Example: --- At Time 1, the forward rate approximates the midpoint between i1,H (upper node rate) and i1,L (lower node rate). --- Similarly, at Time 2, the forward rate approximates the midpoint for i2,HL. -- However, actual calibrated rates may not match the midpoint exactly. 2- Simplified Notation for the Binomial Lattice: -- Rates at each node can be represented in terms of the one-year forward rate at a given time (i_T). -- Each node's interest rate is adjusted for the number of standard deviations (σ) away from the midpoint. -- Formula: --- For an upward movement: i_T * e^(nσ). --- For a downward movement: i_T * e^(-nσ). Restated Binomial Lattice (Simplified): Using the notation i_T to represent the one-year forward rate at time T, the tree can be expressed as: Time 0: -- i_0. Time 1: -- Upper Node: i_1 * e^(σ). -- Lower Node: i_1 * e^(-σ). Time 2: -- Upper Node: i_2 * e^(2σ). -- Lower Node: i_2 * e^(-2σ). Time 3: -- Upper Node: i_3 * e^(3σ). -- Lower Node: i_3 * e^(-3σ). This restated lattice maintains the flexibility of the binomial tree while simplifying the interpretation of each node’s rates in terms of the forward rate and its deviations.

Key Insight: The lognormal model ensures consistency with the implied forward curve while accounting for volatility through the standard deviation factor (σ). This ensures a robust framework for valuing bonds with embedded options.

With the lognormal distribution, the standard deviation of the one-year rate is the product of the current one-year rate and the assumed level of volatility. This means interest rate movements are larger when interest rates are high. This also means negative interest rates are not possible.

Volatility can be measured with two methods. One method measures historical interest rate volatility. Another method calculates the volatility that is implied by the current market prices of interest rate derivatives.

Determining the Value of a Bond at a Node 1- Process Overview: -- To value a bond in the binomial tree, the backward induction valuation method is used. -- This involves starting at the bond’s known cash flows at maturity and working backward to determine its value at each node. 2- Node Value Representation: -- At each node: --- VH represents the bond’s value if the forward rate increases. --- VL represents the bond’s value if the forward rate decreases. -- The model assumes equal probabilities of an increase or decrease in rates (50% probability for each outcome). 3- Calculation at Each Node: -- The value at any node is determined by calculating the present value of the bond’s probability-weighted future values using the appropriate one-period rates. -- Formula for node valuation: NodeValue = 0.5 * [ [( VH + C) ÷ (1 + i)] + [ (VL + C) ÷ (1 + i)] ]. or = [C + 0.5 * (VH + VL) / 1+ i ] 4- Purpose: -- This backward-looking process ensures that the bond’s present value is consistent with the binomial interest rate tree, reflecting possible future rate paths.

Recall that a binomial interest rate tree reflects the current benchmark yield curve, an assumed level of interest rate volatility, and a stochastic process that determines the interest rate movements. If market prices are assumed to be correct (i.e., no arbitrage opportunities exist), then a binomial interest rate tree is calibrated by choosing rates that are consistent with current benchmark yields and bond prices.

Explanation: Calibration of a Binomial Interest Rate Tree 1- Key Components of the Binomial Interest Rate Tree: -- The binomial interest rate tree models interest rate movements based on: --- The current benchmark yield curve: Reflects observed market interest rates for different maturities. --- Assumed interest rate volatility: Captures the uncertainty in future interest rate changes. --- A stochastic process: Governs the random movement of rates (upward or downward at each node). 2- Calibration Process: -- The tree is calibrated under the assumption of no arbitrage, meaning that market prices are correct and free from arbitrage opportunities. -- Rates are chosen such that: --- The tree matches the current benchmark yields (to reflect observed market interest rates). --- Bond prices calculated using the tree align with market prices of those bonds. 3- Purpose of Calibration: -- Ensures consistency between the binomial tree, observed market conditions, and no-arbitrage pricing. -- Enables the accurate valuation of bonds and other fixed-income securities based on the tree.

[Three-Year Binomial Tree: Step-by-Step Example for Bond Valuation 1- Objective and Structure -- The goal is to construct and calibrate a 3-year binomial interest rate tree consistent with the observed par curve, and use it to price a bond. -- Assumptions: -- 1-year par rate = 2.500% -- 2-year par rate = 2.750% -- 3-year par rate = 3.000% -- Volatility (σ) = 20% -- Face value = 100 -- Annual coupon = 3.000% 2- Step 1: Time 0 Rate -- Initial short-term rate: i0 = 2.500% = 0.025 3- Step 2: Time 1 Rates Using Lognormal Walk -- Assume i1_H = 3.605% = 0.03605 -- Then i1_L = i1_H × e^(-2σ) = 0.03605 × e^(-0.40) ≈ 2.417% 4- Step 3: Time 2 Rates Using Lognormal Walk -- Assume i2_M = 3.401% = 0.03401 -- i2_H = 0.03401 × e^(0.40) ≈ 5.074% -- i2_L = 0.03401 × e^(-0.40) ≈ 2.280% 5- Step 4: Price the 3-Year Par Bond Using Backward Induction -- At Time 3: all nodes = 103 -- At Time 2: --- PV_H = 103 ÷ (1 + 0.05074) ≈ 98.048 --- PV_M = 103 ÷ (1 + 0.03401) ≈ 99.598 --- PV_L = 103 ÷ (1 + 0.02280) ≈ 100.669 -- At Time 1: --- At i1_H: (98.048 + 99.598)/2 + 3 = 100.823 → PV = 100.823 ÷ 1.03605 ≈ 97.31 --- At i1_L: (99.598 + 100.669)/2 + 3 = 101.634 → PV = 101.634 ÷ 1.02417 ≈ 99.22 -- At Time 0: (97.31 + 99.22)/2 + 3 = 100.765 → PV = 100.765 ÷ 1.025 ≈ 98.34 -- Final bond price ≈ **98.34** → slightly below par, adjust calibration if needed. 6- Step 5: Comparing Tree Rates with Implied Forward Rates -- **1-Year Forward Rate One Year From Now (f1):** --- Formula: "f1 = [(1 + z2)^2 ÷ (1 + z1)] - 1" --- z1 = 0.025 (1-year par rate), z2 = 0.0275 (2-year par rate) --- f1 ≈ [(1.0275)^2 ÷ 1.025] - 1 ≈ 2.987% -- Tree’s average at Time 1: (3.605% + 2.417%) ÷ 2 = 3.011% → **very close** -- **1-Year Forward Rate Two Years From Now (f2):** --- Formula: "f2 = [(1 + z3)^3 ÷ (1 + z2)^2] - 1" --- z2 = 0.0275, z3 = 0.03 --- f2 ≈ [(1.03)^3 ÷ (1.0275)^2] - 1 ≈ 3.5254% -- Tree middle node at Time 2: 3.401% → **closely aligned** -- Conclusion: The tree approximates the implied forward rates well, confirming correct calibration.

Key Takeaways -- A binomial interest rate tree is calibrated to match par rates and estimate future rate paths. -- Lognormal rate movements define the branching structure. -- Forward rates from the par curve help validate node consistency. -- Backward induction is used to value bonds at each node. -- Once calibrated, the tree can be extended or used for option-adjusted pricing. List of Variables -- i0: Initial rate -- i1_H, i1_L: Rates at Time 1 (high/low) -- i2_H, i2_M, i2_L: Rates at Time 2 -- σ: Volatility -- f1, f2: Implied 1-year forward rates (1 and 2 years ahead) -- z1, z2, z3: 1-, 2-, and 3-year par rates -- PV: Present value -- DF: Discount factor = 1 ÷ (1 + rate)^t -- ∑: Summation operator -- T: Total number of years (3)]

Steps to Calibrate a Two-Period Arbitrage-Free Binomial Tree 1- Start with i0: -- Use the current one-year rate from the benchmark par curve. 2- Make a Guess for i1_L: -- Begin with an initial guess for the lower node rate at Time 1 (i1_L). 3- Calculate i1_H: -- Determine the upper node rate at Time 1 (i1_H) using the lognormal relationship: "i1_H = i1_L × e^(2σ)." 4- Calculate the Value of the Two-Year Bond: -- Use the binomial tree to compute the value of the two-year benchmark bond that pays a coupon matching the two-year par rate. --- If the bond’s value is less than the market price, reduce the guess for i1_L to increase the bond value. --- If the bond’s value is greater than the market price, increase the guess for i1_L to lower the bond value. 5- Adjust i1_L: -- Modify the initial guess for i1_L until the value of the two-year bond equals its market price. 6- Make a Guess for i2_HL: -- Use a similar iterative approach to estimate i2_HL at Time 2. 7- Calculate i2_LL and i2_HH: -- Apply the lognormal relationships to compute the lower node (i2_LL) and upper node (i2_HH) rates at Time 2: -- "i2_LL = i2_HL × e^(-2σ)." -- "i2_HH = i2_HL × e^(2σ)." 8- Use the Expanded Tree to Value the Three-Year Bond: -- Utilize the expanded tree to calculate the value of the three-year benchmark bond. 9- Adjust i2_LL: -- Refine i2_LL until the value of the three-year bond equals its market price.

[Two-Year Binomial Tree: Step-by-Step Example for Bond Valuation 1- Objective and Structure -- Build and calibrate a **2-year binomial interest rate tree** that aligns with a given par yield curve, and use it to **price a 2-year bond**. -- Assumptions: -- 1-year par rate = 2.500% -- 2-year par rate = 2.750% -- Volatility (σ) = 20% -- Face value = 100 -- Annual coupon = 2.750% (to match 2-year par rate) 2- Step 1: Time 0 Rate -- From the 1-year par rate: i0 = 2.500% = 0.025 3- Step 2: Time 1 Rates Using Lognormal Walk -- Let’s assume i1_H = 3.0075% = 0.030075 (estimated from the implied 1-year forward rate) -- Then: --- i1_L = i1_H × e^(-2σ) = 0.030075 × e^(-0.40) ≈ 0.030075 × 0.6703 ≈ **2.015%** -- Time 1 nodes: --- i1_H = 3.0075% --- i1_L = 2.015% 4- Step 3: Price the 2-Year Par Bond Using Backward Induction -- Bond pays 2.750 annually and 100 principal at maturity. **Step 3.1: At Time 2 Nodes (Maturity)** -- Value at all nodes = 100 + 2.750 = 102.750 **Step 3.2: At Time 1 Nodes** -- Discount back from Time 2 using rates at Time 1: -- At i1_H = 3.0075%: --- PV = 102.750 ÷ (1 + 0.030075) ≈ 102.750 ÷ 1.030075 ≈ 99.757 -- At i1_L = 2.015%: --- PV = 102.750 ÷ (1 + 0.02015) ≈ 102.750 ÷ 1.02015 ≈ 100.708 **Step 3.3: At Time 0** -- Value = [0.5×99.757 + 0.5×100.708 + 2.750] ÷ (1 + 0.025) -- Average future value = (99.757 + 100.708) ÷ 2 = 100.233 -- Total cash flow = 100.233 + 2.750 = 102.983 -- Present value = 102.983 ÷ 1.025 ≈ **100.47** -- Result: Bond value ≈ **100.47** → Slightly above par → calibration is close. 5- Step 4: Compare Tree Rates with Implied Forward Rate -- **1-Year Forward Rate in One Year (f1):** --- Formula: "f1 = [(1 + z2)^2 ÷ (1 + z1)] - 1" --- z1 = 2.500% = 0.025 --- z2 = 2.750% = 0.0275 --- f1 = [(1.0275)^2 ÷ 1.025] - 1 = (1.05556 ÷ 1.025) - 1 ≈ **2.987%** -- Tree average at Time 1: --- (3.0075% + 2.015%) ÷ 2 = **2.511%** --- Slightly below the implied forward rate → further calibration of i1_H could improve accuracy.

Key Takeaways -- A 2-year binomial tree is simpler to build and calibrate compared to longer trees. -- Rates at Time 1 are estimated using forward rates implied by the par curve, then adjusted using volatility assumptions. -- Backward induction is used to value the bond starting from maturity. -- Tree-derived rates should align closely with implied forward rates to ensure accurate calibration. List of Variables -- i0: Initial short rate at Time 0 -- i1_H, i1_L: Interest rates at Time 1 (high and low nodes) -- σ: Interest rate volatility -- z1, z2: 1-year and 2-year par rates -- f1: 1-year forward rate one year from now -- PV: Present value -- DF: Discount factor = 1 ÷ (1 + rate)^t -- ∑: Summation operator -- T: Number of years (2)]

Pathwise Valuation for Option-Free Bonds Using a Binomial Tree (Correctly Following The Rules) Steps for Pathwise Valuation: Specify All Paths Through the Tree: Each path represents a sequence of interest rate movements over the bond’s life. In a three-time-period tree, there are 4 paths: -- Path 1: Up to 3.605%, up to 5.074%. -- Path 2: Up to 3.605%, down to 3.401%. -- Path 3: Down to 2.417%, up to 3.401%. -- Path 4: Down to 2.417%, down to 2.280%. Calculating Present Value for Path 1: -- Step 1: Discount the cash flow of 1,032 at Year 3 using the forward rate of 5.074%: PresentValue = 1,032 ÷ (1 + 0.05074) = 982.16 -- Step 2: Add the Year 2 coupon of 32, and discount the sum using the forward rate of 3.605%: PresentValue = (982.16 + 32) ÷ (1 + 0.03605) = 978.88 -- Step 3: Add the Year 1 coupon of 32, and discount the sum using the forward rate of 2.500%: PresentValue = (978.88 + 32) ÷ (1 + 0.025) = 986.22 Averaging Results Across All Paths: -- Repeat the above process for all four paths. -- The average of all present values matches the bond's arbitrage-free value of 1,005.68.

Monte Carlo Method for Valuation 1- Definition and Purpose: -- The Monte Carlo method simulates numerous possible interest rate paths by selecting them randomly. -- It is particularly useful for valuing securities with path-dependent cash flows, such as mortgage-backed securities, where prepayments depend on the sequence of interest rates rather than the current rate alone. 2- Key Features: -- Arbitrage-Free Calibration: --- Like binomial trees, the Monte Carlo method becomes arbitrage-free by: ---- Making an interest rate volatility assumption. ---- Calibrating the paths to the current benchmark term structure. --- A drift-adjusted model adds a constant drift term to short-term rates, ensuring the benchmark bond's value aligns with its market price. -- Path-Dependence: --- Captures cash flow behaviors that depend on the sequence of interest rate changes, not just the terminal value at a specific node. -- Mean Reversion: --- Can simulate mean-reverting interest rates by setting upper and lower boundaries. --- Produces rates that cluster around the implied forward rates derived from the current yield curve. 3- Statistical Accuracy vs. Input Quality: -- Increasing Modeled Paths: --- Increases statistical accuracy, reducing sampling errors in simulated outputs. --- Does not guarantee closer alignment with the security's intrinsic value unless inputs are accurate and realistic. --- Quality of the model and assumptions (e.g., volatility, drift, and yield curve structure) significantly impacts output reliability. 4- Applications: -- Mortgage-Backed Securities: --- Models prepayment behavior, which depends on the path of interest rate movements. -- Complex Derivatives: --- Useful for pricing derivatives with path-dependent payoffs (e.g., Asian options).

❓ What’s the confusion? Billingsley says: "More Monte Carlo paths = more accuracy = better estimate of fundamental value." This sounds logical at first, but it’s not fully correct. Here's why: ✅ What Increasing Paths Actually Does Increasing the number of Monte Carlo paths (simulations) reduces random error (sampling variability). This makes your estimate more statistically stable. BUT: That only improves the precision of the simulation, not the accuracy of your model’s assumptions or inputs. 🔁 In short: More paths = more precise output, not necessarily more correct output. 🚫 Why This Doesn't Guarantee a Better Fundamental Value Estimate Monte Carlo simulation is just a tool. It gives results based on: Your model assumptions (e.g., geometric Brownian motion, risk-neutral framework) Your input data (e.g., volatility, drift, correlations) If the model is wrong or the inputs are off — even millions of simulations won’t help. You’re just more precisely wrong. 📌 Garbage in, garbage out — but more neatly averaged.

Interest Rate Factors The simplest term structure models rely entirely on the one-period rate. This single factor determines the entire term structure. These one-factor models assume that all rates move in the same direction over a given interval, although they may not necessarily move by the same amount. More complex multi-factor models allow analysts to incorporate additional assumptions about, for example, the slope of the yield curve.

Interest Rate Process 1- Definition: -- Interest rate processes are stochastic models used to describe the dynamics of interest rate movements, assuming rates evolve randomly over time. -- These processes are continuous and can be applied to frameworks like binomial lattice models. 2- General Formula for the Process: -- The process consists of two components: --- A drift term that defines the path assuming zero volatility. --- A dispersion term that introduces volatility into the model. -- General formula: "dr = θ_t * dt + σ_t * dZ" --- Where: ---- dr: Change in the short-term interest rate (𝒾). ---- θ_t * dt: Drift term, which can be constant or mean-reverting. ---- σ_t * dZ: Dispersion term, incorporating randomness through a Weiner process (Z). 3- Explanation of Terms: -- The drift term (θ_t * dt): --- Represents the expected deterministic component of rate movements. --- May vary based on assumptions like mean reversion. -- The dispersion term (σ_t * dZ): --- Adds randomness and reflects the volatility of interest rates. --- Z: A normally distributed Weiner process allowing for the possibility of negative interest rates. 4- Applications: -- Used for pricing fixed-income securities with embedded options (e.g., callable/putable bonds). -- Also applied in valuing derivatives tied to interest rates. 5- Model Preferences: -- Analysts may select models based on objectives: --- Equilibrium models: Focus on theoretical relationships (e.g., term structure of interest rates). --- Arbitrage-free models: Ensure consistency with current market data (e.g., yield curves). -- Differences in models stem from how they structure the drift and volatility components.

Class of Models for Interest Rate Dynamics 1- Overview: -- Interest rate models are categorized into arbitrage-free models and equilibrium models, each with distinct features, assumptions, and use cases. 2- Arbitrage-Free Models: -- Definition: --- Assume that bond prices and the term structure implied by those prices are correct, meaning no arbitrage opportunities exist. -- Key Features: --- Parameterized to align with current prices and rates. --- Grounded in market prices, making them suitable for applications requiring consistency with current market conditions (e.g., hedging). -- Advantages: --- Provide accurate pricing for derivatives and securities based on current market data. -- Disadvantages: --- Computationally intensive due to the higher number of parameters required. --- Outputs rely on risk-neutral probabilities, which may not align with real-world probabilities. 3- Equilibrium Models: -- Definition: --- Use fundamental economic variables to model the term structure of interest rates. --- Derive equilibrium prices for bonds and interest rate options based on these variables. -- Key Features: --- Parameter values may deviate from current market prices, making them less suitable for static decisions (e.g., hedging). --- Allow for multiple possible future interest rate paths, making them ideal for dynamic applications. -- Well-Known Examples: --- Cox-Ingersoll-Ross (CIR) Model: ---- Incorporates mean reversion and avoids negative interest rates. --- Vasicek Model: ---- Allows for negative interest rates, but also assumes mean reversion. -- Advantages: --- Simpler than arbitrage-free models. --- Suitable for dynamic analyses where future uncertainty matters. -- Disadvantages: --- Not parameterized to current market prices, reducing their accuracy for pricing based on existing data.

Equilibrium Models: Cox-Ingersoll-Ross (CIR) Model 1- Overview: -- The Cox-Ingersoll-Ross (CIR) model is a one-factor equilibrium model used to describe the dynamics of the short-term interest rate. -- The model assumes that short-term rates are mean-reverting to a long-term average and incorporates stochastic elements to account for randomness. 2- Components of the CIR Model: -- Deterministic Component (Drift Term): --- Ensures that interest rates revert to a long-run mean value (θ). --- Described as: k(θ - rt), where: ---- θ = Long-run mean of the short-term rate. ---- rt = Current short-term rate. ---- k = Speed of mean reversion (how quickly rt moves toward θ). -- Stochastic Component (Random Risk Term): --- Accounts for random interest rate movements. --- Described as: σ√rt dZ, where: ---- σ = Volatility of the short-term rate. ---- √rt = Ensures volatility is proportional to the square root of the short-term rate, avoiding negative rates. ---- dZ = A normally distributed random variable. 3- General Formula: -- CIR Formula: "dr_t = k(θ - r_t) dt + σ√r_t dZ" Where: --- dr_t = Expected change in the short-term rate in the next period. --- dt = Number of periods over which the rate change occurs. 4- Key Assumptions: -- Interest rates are mean-reverting, meaning they eventually return to the long-term average (θ). -- Volatility is positively correlated with the level of the short-term rate, making large rate swings less likely when rates are low. -- No negative interest rates: Since the volatility term is proportional to √rt, rates cannot drop below zero. 5- Advantages: -- Realistic modeling of interest rates by incorporating mean reversion. -- Avoids negative rates, making it suitable for many practical applications. 6- Disadvantages: -- Complexity: Requires multiple parameters (k, θ, σ) that can make calibration challenging. -- Assumes a single-factor process, which may not fully explain multi-dimensional changes in the yield curve. 7- Application: -- The CIR model is widely used in pricing interest rate derivatives and fixed-income securities due to its theoretical rigor and practical assumptions.

Equilibrium Models: Vasicek Model 1- Overview: -- The Vasicek model is another one-factor equilibrium model used to describe the dynamics of short-term interest rates. -- Like the CIR model, it assumes that interest rates are mean-reverting to a long-term average. -- However, the Vasicek model differs by assuming constant volatility, which can theoretically allow for negative interest rates. 2- Formula: -- Vasicek Formula: "dr_t = k(θ - r_t) dt + σ dZ" Where: --- dr_t = Expected change (differential) in the short-term rate in the next period. --- θ = Long-term mean of the short-term rate. --- r_t = Current short-term rate. --- k = Speed of mean reversion (rate at which r_t moves toward θ). --- σ = Constant volatility of the short-term rate. --- dZ = Normally distributed random variable representing stochastic changes. --- dt = Time period over which the rate change occurs. 3- Key Differences Compared to CIR Model: -- Volatility: In the Vasicek model, volatility (σ) is constant, while the CIR model links volatility to the square root of the short-term rate. -- Negative Rates: Unlike the CIR model, the Vasicek model can theoretically allow for negative interest rates due to its linear volatility assumption. 4- Assumptions: -- Interest rates are mean-reverting, eventually moving back toward the long-term average (θ). -- Volatility is constant and does not depend on the level of interest rates. 5- Advantages: -- Simplicity: The Vasicek model’s constant volatility makes it mathematically simpler and easier to implement. -- Useful for theoretical insights into interest rate dynamics. 6- Disadvantages: -- The possibility of negative interest rates is unrealistic in many practical scenarios. -- May not accurately reflect the real-world relationship between volatility and the level of interest rates. 7- Applications: -- The Vasicek model is commonly used in academic research and theoretical modeling. -- It provides a foundation for more complex models, including those that adjust for its limitations (e.g., allowing for positive-only rates).

Arbitrage-Free Models: Ho-Lee Model 1- Overview: -- The Ho-Lee model is an arbitrage-free model that uses a binomial lattice framework to simulate changes in the short-term interest rate. -- At each node in the lattice, the short-term interest rate can either increase or decrease by equal probabilities. -- This model ensures that the generated term structure is consistent with current market prices of bonds and the observed term structure. 2- Formula: -- Ho-Lee Formula: "dr_t = θ_t dt + σ dZ" Where: --- dr_t = Expected change (differential) in the short-term rate in the next period. --- θ_t = Time-dependent drift term ensuring arbitrage-free valuation. --- σ = Constant volatility of the short-term rate. --- dZ = Normally distributed random variable representing stochastic changes. --- dt = Time period over which the rate change occurs. 3- Key Features: -- Arbitrage-Free: Ensures consistency with the observed market prices of bonds and the term structure. -- Time-Dependent Drift: The drift term (θ_t) adjusts over time to ensure that the model generates correct term structures and avoids arbitrage opportunities. -- Constant Volatility: Assumes σ is constant, simplifying calculations. 4- Advantages: -- Ensures that interest rate simulations match current bond prices and the yield curve. -- Simple implementation using a binomial tree structure. -- Provides a normal symmetrical distribution of possible future interest rates. 5- Disadvantages: -- The assumption of constant volatility may not reflect real-world conditions where interest rate volatility varies with the level of rates. -- Negative Interest Rates: The model can produce negative rates due to its linear drift and volatility assumptions. 6- Applications: -- The Ho-Lee model is primarily used for pricing and valuing fixed-income securities, including bonds and derivatives, in markets where arbitrage-free valuation is critical. -- It serves as a foundation for more complex arbitrage-free models.

Arbitrage-Free Models: Kalotay-Williams-Fabozzi (KWF) Model 1- Overview: -- The KWF model, like the Ho-Lee model, is an arbitrage-free model assuming constant volatility and no mean reversion in interest rates. -- However, the KWF model focuses on the logarithm of the short-term interest rate rather than the rate itself. -- This ensures that the short-term interest rate is lognormally distributed, which prevents the generation of negative interest rates. 2- Formula: -- KWF Formula: " dln(r_t) = θ_t dt + σ dZ " Where: --- dln(r_t) = Change in the natural logarithm of the short-term rate. --- r_t = Short-term rate, assumed to be lognormally distributed. --- θ_t = Time-dependent drift term ensuring arbitrage-free valuation. --- σ = Constant volatility of the short-term rate. --- dZ = Normally distributed random variable representing stochastic changes. --- dt = Time period over which the rate change occurs. 3- Key Features: -- Lognormal Distribution: The logarithm of the short-term rate (ln(r_t)) is modeled, ensuring that the rate remains positive and avoids unrealistic negative rates. -- Arbitrage-Free: The drift term (θ_t) adjusts to match current bond prices and the term structure. -- Constant Volatility: Assumes that volatility (σ) is constant over time. -- No Mean Reversion: Rates are not expected to revert to a long-term average value. 4- Advantages: -- The model avoids generating negative interest rates due to its lognormal distribution assumption. -- Ensures arbitrage-free valuation consistent with current market prices and yield curves. -- Suitable for applications where rates are expected to remain positive. 5- Disadvantages: -- The assumption of constant volatility may not align with real-world interest rate dynamics. -- No Mean Reversion: This limitation makes the model less applicable in environments where rates are expected to revert to historical averages. 6- Applications: -- Used for pricing and valuation of fixed-income securities in arbitrage-free contexts. -- Suitable for scenarios where maintaining positive interest rates is a critical assumption.

Modern Models The models that have been covered to this point use a single factor, the short-term interest rate, to model the entire term structure. More recent multi-factor models are more sophisticated, using data from observed market rates as well as the volatilities implied by option prices. The Gauss+ model, for example, uses short-, medium-, and long-term rates to produce a humped-shaped volatility curve. Changes in short-term rates are not assumed to be random, which is consistent with central banks manipulating rates at this end of the curve to achieve their monetary policy objectives. Medium- and long-term rates are more volatile in comparison.

Models Allowing Negative Rates 1- Ho-Lee Model (AB Model): -- The model assumes constant volatility and a normal distribution for future rates, meaning interest rates can fall below zero. 2- Vasicek Model: -- Due to its assumption of constant volatility and lack of restrictions on rate levels, the model allows for the possibility of negative interest rates.

Models That Do Not Allow Negative Rates 1- Cox-Ingersoll-Ross (CIR) Model: -- Rates are mean-reverting with volatility proportional to the square root of the current rate. This ensures that as rates approach zero, volatility decreases, preventing negative rates. 2- Kalotay-Williams-Fabozzi (KWF) Model (AB Model): -- Models the logarithm of the short-term rate (ln(r_t)), ensuring rates remain lognormally distributed and always positive.

[Interest Rate Models: Equilibrium and Arbitrage-Free Frameworks] 1- Cox-Ingersoll-Ross (CIR) Model -- One-factor equilibrium model with mean-reverting and strictly positive short-term interest rates. -- Formula: "dr_t = k(θ - r_t) dt + σ√r_t dZ" -- 1- Deterministic Component: --- Term: k(θ - r_t), ensures mean reversion toward long-run average θ. -- 2- Stochastic Component: --- Term: σ√r_t dZ, volatility scales with √r_t to prevent negative rates. -- 3- Use and Evaluation: --- Useful for pricing bonds and interest rate derivatives with non-negative rate assumptions. --- Its structure reflects economic realism, particularly mean reversion and volatility scaling. --- However, it is harder to calibrate due to parameter complexity and lacks multi-factor dynamics. 2- Vasicek Model -- One-factor equilibrium model with constant volatility and possible negative rates. -- Formula: "dr_t = k(θ - r_t) dt + σ dZ" -- 1- Differences from CIR: --- Volatility is constant and independent of r_t; allows negative rates. -- 2- Use and Evaluation: --- Preferred in academic and theoretical contexts for its mathematical simplicity. --- Allows closed-form solutions for bond pricing, making it analytically attractive. --- Less realistic due to constant volatility and the possibility of negative interest rates.

3- Ho-Lee Model -- Arbitrage-free model calibrated to the current yield curve using time-dependent drift. -- Formula: "dr_t = θ_t dt + σ dZ" -- 1- Key Features: --- θ_t fits the term structure; volatility is constant; binomial tree-compatible. -- 2- Use and Evaluation: --- Used to value bonds and interest rate derivatives in arbitrage-free environments. --- Matches current bond prices exactly and is easy to implement. --- However, it does not prevent negative rates and assumes constant volatility. 4- Kalotay-Williams-Fabozzi (KWF) Model -- Arbitrage-free model ensuring strictly positive short-term rates by modeling the log of rates. -- Formula: "dln(r_t) = θ_t dt + σ dZ" -- 1- Key Features: --- Log-normal specification ensures positivity; arbitrage-free calibration. -- 2- Use and Evaluation: --- Applied in settings where interest rates must remain positive, such as fixed-income pricing. --- It prevents arbitrage and fits the observed yield curve effectively. --- The downside is it lacks mean reversion and uses a fixed volatility assumption. 5- Modern Multi-Factor Models -- Models with multiple stochastic drivers (e.g., Gauss+ with short-, medium-, and long-term factors). -- 1- Key Features: --- Captures complex curve movements; accommodates policy impacts; hump-shaped volatility possible. -- 2- Use and Evaluation: --- Suited for sophisticated pricing, yield curve modeling, and interest rate risk analysis. --- Provides higher realism by integrating multiple term structure drivers. --- More complex to implement and calibrate but offers improved empirical accuracy.

Written Solution: The standard deviation of the one-year rate is calculated as the product of the volatility and the one-year rate. Formula: Standard deviation = Volatility × One-year rate Calculation: 0.15 × 0.0135 = 0.002025 This value can be expressed as either: 0.2025% 20.25 basis points (bps)

Explanation: Concept: The best estimate for the middle rate at Time 2 (Y) in a binomial lognormal interest rate tree is the one-year forward rate two years into the future, denoted as F(2,1). This forward rate is implied by the two-year and three-year spot rates provided in Exhibit 1. Calculation: Formula for F(2,1): F(2,1) = [(1 + S3)^3 ÷ (1 + S2)^2] - 1, where: --- S3 = 5.0% (three-year spot rate), --- S2 = 4.0% (two-year spot rate). Substitution: F(2,1) = [(1.05)^3 ÷ (1.04)^2] - 1. Result: F(2,1) = 7.029%. Interpretation: Based on the given data, the middle rate (Y) in Exhibit 2 is closest to 7.03%. Note: While this is the best estimate using the provided information, a fully calibrated binomial tree (with a 20% volatility assumption) may produce a slightly different value for Y, closer to 6.8%. For exam purposes, precise calibration using software tools (e.g., Excel) is not required.

In a binomial interest rate tree, an increase in the assumed volatility causes the rates at adjacent nodes to spread further apart. This happens because the rates at each node differ by a factor of e^(2σ), where σ represents volatility. A higher volatility leads to lower rates at the bottom nodes and higher rates at the top nodes, increasing the dispersion of possible rates. For Node Z in Exhibit 2, as volatility increases, the rate at Z decreases because it represents the lower bound of the possible rates. For example, if the middle rate (Y) at Time 2 is calibrated to 6.8%, then Z, calculated as Y × e^(-2σ), would decrease further as σ increases. This explains why, with higher volatility, Z is most likely to decrease.

6.3 Valuation & Analysis: Bonds with Embeded Options

-- Describe fixed-income securities with embedded options. -- Explain the relationships between the values of a callable or putable bond, the underlying option-free (straight) bond, and the embedded option. -- Describe how the arbitrage-free framework can be used to value a bond with embedded options. -- Explain how interest rate volatility affects the value of a callable or putable bond. -- Explain how changes in the level and shape of the yield curve affect the value of a callable or putable bond. -- Calculate the value of a callable or putable bond from an interest rate tree. -- Explain the calculation and use of option-adjusted spreads. -- Explain how interest rate volatility affects option-adjusted spreads. -- Calculate and interpret effective duration of a callable or putable bond. -- Compare effective durations of callable, putable, and straight bonds. -- Describe the use of one-sided durations and key rate durations to evaluate the interest rate sensitivity of bonds with embedded options. -- Compare effective convexities of callable, putable, and straight bonds. -- Calculate the value of a capped or floored floating-rate bond. -- Describe defining features of a convertible bond. -- Calculate and interpret the components of a convertible bond’s value. -- Describe how a convertible bond is valued in an arbitrage-free framework. -- Compare the risk–return characteristics of a convertible bond with the risk–return characteristics of a straight bond and of the underlying common stock.

Embedded options are contingency provisions that are written into a bond’s indenture.

For example, a call option allows the bond issuer to benefit from falling interest rates by refinancing at a lower rate. A put option allows the bondholder to benefit from rising rates. Because they are part of the bond contract, embedded options cannot be traded separately like exchange-traded options contracts. The underlying bond has specific features, such as the coupon rate and payment structure.

Summary: Call Options and Callable Bonds 1- Definition of Callable Bonds: Bonds with an embedded call option give the issuer the right to redeem the bond before maturity. Issuers exercise this option when: -- Interest rates decrease. -- Issuer’s credit quality improves. 2- Historical Context: Historically, most U.S. corporate bonds were callable after a set time (e.g., 10 years). Call prices typically declined over time, reaching par near maturity. 3- Modern Callable Bonds: Most investment-grade bonds are non-refundable, preventing refinancing for a lower rate. Make-whole provisions may compensate investors for price changes due to interest rate movements, limiting issuer advantages. 4- Call Protection Period: Callable bonds usually have a protection period (e.g., 10 years on a 20-year bond), during which the issuer cannot call the bond. 5- Call Option Styles: European-style: Callable only at the end of the protection period. American-style: Callable anytime after the protection period. Bermudan-style: Callable on specific scheduled dates after the protection period. 6- Common Issuers of Callable Bonds: Government-sponsored entities (e.g., Fannie Mae, Freddie Mac) issue callable bonds, usually at par with Bermudan-style options. Most municipal bonds in the U.S. are callable at par after the 10th year.

Issuer's Motivation: The call option benefits the issuer by allowing flexibility to reduce financing costs when market conditions are favorable.

Simple Embeded Options - Put Options and Extension Options 1- Putable Bond Definition: A bond with an embedded put option gives investors the right to sell the bond back to the issuer before maturity, usually at par. -- Investors exercise this option when: --- Interest rates rise. --- The proceeds can be reinvested in higher-yielding bonds. 2- Characteristics of Putable Bonds: -- Typically include a protection period during which the put option cannot be exercised. -- Most embedded put options are European-style, with some Bermudan-style options. -- American-style put options do not exist. 3- Extension Options: An extension option allows the holder of an extendible bond to extend its maturity, potentially at a new coupon rate. -- Operates in the opposite direction of a put option, allowing investors to extend the bond's life rather than shorten it. 4- Comparing Putable and Extendible Bonds: These two structures can often be used interchangeably. -- Example: --- A 3-year bond with the option to extend to Year 5 is similar to a 5-year bond with a put option exercisable after Year 3.

The ability to exercise an embedded call option may also be limited by the option's style: 1- European-style call options can only be called on the date the protection period expires. 2- American-style call options can be called any time after the protection period has expired. 3- Bermudan-style call options can be exercised on scheduled dates after the protection period has expired. Most bonds issued by government-sponsored entities in the United States (e.g., Fannie Mae, Freddie Mac) are callable. Usually, the call price is par, and the style is Bermudan. Most municipal bonds issued in the United States are callable at par after the 10th year.

Complex Embedded Options 1- Overview of Complex Embedded Options: Most bonds are either callable or putable, but some feature unique or combined options, offering flexibility to issuers and investors. -- Example: A bond can be both callable and putable. 2- Convertible Bonds: Allow the investor to convert the bond into common shares of the issuing company. -- Convertible bonds are typically callable, giving issuers flexibility to force conversion under favorable conditions. 3- Contingent Options: Options tied to specific events unique to the investor. -- Estate Put: Allows the bondholder's estate to sell the bond back to the issuer upon the bondholder's death. --- Valuation of this option depends on the investor's life expectancy. 4- Sinking Fund Bonds: Require issuers to set aside funds to retire the bond, reducing credit risk. -- Some sinking fund bonds include an acceleration provision, akin to a call option, allowing issuers to repurchase additional bonds beyond the specified sinking fund amount. -- Others feature a delivery option, permitting issuers to deliver bonds to the trustee instead of cash. --- This benefits issuers when interest rates rise and bond prices fall.

Embedded Options and Straight Bonds 1- Embedded Options Characteristics: Embedded options cannot be traded separately as they are attached to the underlying bond. 2- Specific Features of the Underlying Bond: The bond's key characteristics include: -- Issuer. -- Issue date. -- Maturity date. -- Coupon rate. -- Currency denomination. 3- Straight Bond Definition: The underlying option-free bond is referred to as the straight bond.

Relationships Between the Values of a Callable or Putable Bond, Straight Bond, and Embedded Option 1- Value of a Bond with an Embedded Option: The value of a bond with an embedded option is determined by adjusting the value of a straight (option-free) bond for the arbitrage-free value of the embedded option. -- Adding an option affects the bond's value: --- A call option decreases the bond's value to investors since it benefits the issuer. --- A put option increases the bond's value to investors as it provides them additional flexibility. 2- Key Equations: -- Value of callable bond = Value of straight bond - Value of issuer call option. -- Value of putable bond = Value of straight bond + Value of investor put option. 3- Rearranging the Equations: -- These formulas can also be used to calculate the value of embedded options: --- The call option reduces a bond's value because it is owned by the issuer. --- The put option adds value for investors, enhancing their control over the bond.

Valuation of a Callable Bond at Zero Volatility 1- Impact of Zero Volatility: -- The value of an embedded call option is heavily influenced by the volatility of interest rates. --- Greater volatility increases the value of the call option. -- Assuming zero volatility is unrealistic but provides a basic framework for extending to non-zero volatility scenarios. 2- Valuation Process Under Zero Volatility: -- A callable bond is valued by discounting its expected future cash flows at maturity and working back to the present. -- This approach is similar to the arbitrage-free method used to value option-free bonds. 3- Key Difference for Callable Bonds: -- The cash flows in any given year depend on whether the embedded call option is exercised. -- Forward rates are used to assess whether the option will be exercised at the start of each period, beginning with the cash flows at maturity.

The same concept is used to value putable bonds except that the option will be exercised if the bond's value falls below the exercise price.

[Two-Year Binomial Tree: Callable Bond Valuation Example 1- Objective and Structure -- Build and calibrate a **2-year binomial interest rate tree** and use it to **value a callable bond**. -- The callable bond allows the issuer to redeem the bond **at par (100)** at **any coupon date** (annually) before maturity. -- Use backward induction to price the bond and reflect the **issuer’s right to call** the bond. -- Assumptions: -- Par yield curve: --- 1-year par rate = 2.500% --- 2-year par rate = 2.750% -- Volatility (σ) = 20% -- Face value = 100 -- Annual coupon = 2.750% (to match 2-year par rate) -- **Call price = 100** -- Callable at **end of Year 1 and Year 2** 2- Step 1: Time 0 Rate -- i0 = 2.500% = 0.025 3- Step 2: Time 1 Rates Using Lognormal Walk -- Assume: --- i1_H = 3.0075% = 0.030075 (forward rate estimate) --- i1_L = i1_H × e^(-2σ) ≈ 0.030075 × 0.6703 ≈ **2.015%** -- Time 1 Nodes: --- i1_H = 3.0075% --- i1_L = 2.015% 4- Step 3: Value at Time 2 (Maturity) -- Cash flows = 100 (face) + 2.750 (final coupon) = **102.750** -- Value at all Time 2 nodes = **102.750** 5- Step 4: Value at Time 1 with Call Option -- Use backward induction. Value = min(continuation value, call price) -- At i1_H = 3.0075%: --- PV = 102.750 ÷ 1.030075 ≈ **99.757** --- Add coupon: 99.757 + 2.750 = **102.507** --- Compare to call price = 100 → **issuer will call** --- Value at node = **min(102.507, 100) = 100** -- At i1_L = 2.015%: --- PV = 102.750 ÷ 1.02015 ≈ **100.708** --- Add coupon: 100.708 + 2.750 = **103.458** --- Compare to call price = 100 → **issuer will call** --- Value at node = **min(103.458, 100) = 100** 6- Step 5: Value at Time 0 -- Expected value = (0.5 × 100 + 0.5 × 100) = 100 -- Add coupon = 100 + 2.750 = 102.750 -- PV = 102.750 ÷ (1 + 0.025) = 102.750 ÷ 1.025 = **100.24** -- Final value of callable bond ≈ **100.24** 7- Call Option Mechanics -- The issuer can **call the bond at par** at the end of **Year 1** or **Year 2**. -- At each node, the value is: --- "min(call price, continuation value)" -- In this case, since rates rise slightly, the bond is called at both Time 1 nodes.

[Two-Year Binomial Tree: Putable Bond Valuation Example 1- Objective and Structure -- Build and calibrate a **2-year binomial interest rate tree** and use it to **value a putable bond**. -- A putable bond gives the **bondholder** the right to sell the bond **back to the issuer at par (100)** at specified dates. -- The valuation reflects the **bondholder’s right to put** the bond when continuation value is below the put price. -- Assumptions: -- Par yield curve: --- 1-year par rate = 2.500% --- 2-year par rate = 2.750% -- Volatility (σ) = 20% -- Face value = 100 -- Annual coupon = 2.750% (to match 2-year par rate) -- **Put price = 100** -- Putable at **end of Year 1 and Year 2** 2- Step 1: Time 0 Rate -- i0 = 2.500% = 0.025 3- Step 2: Time 1 Rates Using Lognormal Walk -- Assume: --- i1_H = 3.0075% = 0.030075 (from implied forward rate) --- i1_L = i1_H × e^(-2σ) ≈ 0.030075 × 0.6703 ≈ **2.015%** -- Time 1 Nodes: --- i1_H = 3.0075% --- i1_L = 2.015% 4- Step 3: Value at Time 2 (Maturity) -- Final cash flow = 100 (face) + 2.750 (coupon) = **102.750** -- Value at all Time 2 nodes = **102.750** -- Since the put option cannot be exercised after maturity, no adjustment is needed. 5- Step 4: Value at Time 1 with Put Option -- Value = **max(continuation value, put price)** -- At i1_H = 3.0075%: --- PV = 102.750 ÷ 1.030075 ≈ **99.757** --- Add coupon: 99.757 + 2.750 = **102.507** --- Compare to put price = 100 → **bondholder will not exercise put** --- Value = **max(102.507, 100) = 102.507** -- At i1_L = 2.015%: --- PV = 102.750 ÷ 1.02015 ≈ **100.708** --- Add coupon: 100.708 + 2.750 = **103.458** --- Compare to put price = 100 → **bondholder will not exercise put** --- Value = **max(103.458, 100) = 103.458** 6- Step 5: Value at Time 0 -- Expected value = (0.5 × 102.507 + 0.5 × 103.458) = **102.9825** -- Add coupon = 102.9825 + 2.750 = 105.7325 -- PV = 105.7325 ÷ 1.025 ≈ **103.65** -- Final value of putable bond ≈ **103.65** 7- Put Option Mechanics -- The **bondholder** can put the bond at **par (100)** at the end of **Year 1 or Year 2**. -- At each node, the bond’s value is: --- "max(continuation value, put price)" -- The option protects the holder against interest rate increases that lower bond prices.

Interest Rate Volatility 1- Impact of Interest Rate Volatility on Option Value: -- The value of an embedded option increases as interest rate volatility rises. --- Greater volatility provides more opportunities for the option to be exercised. 2- Effect on Callable and Putable Bonds: -- Callable Bonds: --- The value of a callable bond decreases as the value of the embedded call option increases. --- This is because the call option is owned by the issuer, reducing the bond’s appeal to investors. -- Putable Bonds: --- The value of a putable bond increases as the value of the embedded put option rises. --- This is because the put option benefits the bondholder, enhancing the bond's appeal. 3- Straight Bonds: -- The value of straight (option-free) bonds is unaffected by changes in interest rate volatility because they contain no embedded options. 4- Key Conclusion: -- Rising interest rate volatility: --- Reduces the value of callable bonds. --- Increases the value of putable bonds.

Level and Shape of the Yield Curve 1- Effect on the Value of a Callable Bond: -- Relationship to Interest Rates: --- Callable bonds have limited upside potential when interest rates fall. --- Increases in the value of the straight bond are offset by the rising value of the embedded call option, as issuers are more likely to repurchase bonds in a low-interest-rate environment. -- Impact of the Yield Curve: --- The value of a call option increases as the yield curve flattens or becomes inverted, making it more likely that the option will be exercised. --- The value of the embedded call option is lowest when the yield curve is upward-sloping because implied forward rates are high, reducing the probability of the call option being exercised.

2- Effect on the Value of a Putable Bond: -- Relationship to Interest Rates: --- Rising interest rates reduce the value of straight bonds, but putable bonds lose less value because the embedded put option becomes more valuable. --- The decrease in the value of the straight bond is at least partially offset by the increase in the value of the put option. -- Impact of the Yield Curve: --- A put option and, by extension, a putable bond are more valuable when the yield curve is upward-sloping. --- This relationship is the opposite of what is observed for callable bonds.

[Binomial Tree Valuation of Callable and Putable Bonds: Step-by-Step Guide 1- Objective and Structure -- This guide explains how to **value callable and putable bonds** using a **binomial interest rate tree**. -- The key principle: --- For **callable bonds**, value = **min(continuation value, call price)** (issuer's right). --- For **putable bonds**, value = **max(continuation value, put price)** (investor's right). -- We use **backward induction**, starting from maturity. -- Assumptions for example: --- Face value = 100 --- Annual coupon = 4 --- Call/Put price = 100 --- Tree has 2 periods (Time 0, 1, 2) --- Interest rates at all nodes = 2.000% = 0.02 (simplified) 2- Valuing a Callable Bond Using Backward Induction **Step 1: Value at Maturity (Time 2)** -- Each node’s value = face value + final coupon = 100 + 4 = **104** -- Since the bond is **callable**, issuer can call at 100 → cap values: --- V_T2_H = min(104, 100) = **100** --- V_T2_L = min(104, 100) = **100** **Step 2: Value at Time 1 Nodes** -- Use expected discounted value of Time 2 nodes: --- V_node = 0.5 × [ (V_up + C) ÷ (1 + r_up) + (V_down + C) ÷ (1 + r_down) ] -- Calculation: --- V_1 = 0.5 × [ (100 + 4)/1.02 + (100 + 4)/1.02 ] = 0.5 × [2 × 104 ÷ 1.02] --- V_1 = 0.5 × (208 ÷ 1.02) ≈ 0.5 × 203.9216 ≈ **101.96** -- Callable → cap at 100 → V_1 = **min(101.96, 100) = 100** **Step 3: Value at Time 0** -- Again use the expected discounted value: --- V_0 = 0.5 × [ (100 + 4)/1.02 + (100 + 4)/1.02 ] --- Same as above → V_0 ≈ **101.96** -- Callable → cap at 100 → Final callable bond value = **100** 3- Valuing a Putable Bond Using Backward Induction **Step 1: Value at Maturity (Time 2)** -- Final payoff = 100 + 4 = **104** -- Since the **put option cannot be exercised at maturity**, V_T2 = **104** **Step 2: Value at Time 1** -- Use the same discounting as before: --- V_1 = 0.5 × (104 ÷ 1.02 + 104 ÷ 1.02) = 0.5 × (208 ÷ 1.02) ≈ **101.96** -- Putable → value = **max(101.96, 100) = 101.96** **Step 3: Value at Time 0** -- V_0 = 0.5 × (101.96 + 4)/1.02 + (101.96 + 4)/1.02 --- Total = 0.5 × (105.96 + 105.96)/1.02 = 0.5 × 211.92 ÷ 1.02 ≈ 103.86 -- Putable → value = **max(103.86, 100) = 103.86**

[Binomial Tree Valuation of a Putable Bond: Step-by-Step Guide 1- Objective and Structure -- This guide explains how to **value a putable bond** using a **2-period binomial interest rate tree**. -- The bondholder has the **right to sell the bond back to the issuer at par (put price = 100)** on specified dates. -- At each node, we take the **maximum of the continuation value and the put price**. -- Assumptions: --- Face value = 100 --- Annual coupon = 4 --- Put price = 100 --- Tree has 2 periods (Time 0, 1, 2) --- Interest rates: r_up = r_down = 2.000% = 0.02 (simplified for illustration) 2- Step 1: Value at Maturity (Time 2) -- Final cash flow = 100 + 4 = **104** -- No put at maturity → value at all Time 2 nodes = **104** 3- Step 2: Value at Time 1 with Put Option -- Formula: --- "V_node = 0.5 × [ (V_up + C) ÷ (1 + r_up) + (V_down + C) ÷ (1 + r_down) ]" -- Given: --- V_up = 100 --- V_down = 100.97 --- C = 4 --- r = 0.02 -- Compute: --- V_1 = 0.5 × [ (100 + 4) ÷ 1.02 + (100.97 + 4) ÷ 1.02 ] --- V_1 = 0.5 × [104 ÷ 1.02 + 104.97 ÷ 1.02] --- V_1 = 0.5 × [101.96 + 102.90] = 0.5 × 204.86 = **102.43** -- Compare to put price = 100 → **max(102.43, 100) = 102.43** -- Put is **not exercised**, so value = **102.43** 4- Step 3: Value at Time 0 -- Use same approach: --- V_0 = 0.5 × [104 ÷ 1.02 + 104.97 ÷ 1.02] --- V_0 = 0.5 × [101.96 + 102.90] = **102.43** -- Compare to put price = 100 → **max(102.43, 100) = 102.43** -- Final value of the putable bond = **102.43**

Valuation of Risky Bonds 1- Assumption of Zero Default Risk: -- The assumption of zero default risk is reasonable for sovereign bonds denominated in the issuing government's domestic currency. -- Most other bonds, however, are subject to default risk. 2- Adjusting the Discount Rate: -- The most common method to value bonds with default risk is by adjusting the discount rate. --- Investors add premiums to the discount rate to account for risks such as: ---- Default risk. ---- Reduced liquidity. --- These premiums increase the discount rate, thereby reducing the valuation of risky bonds compared to equivalent sovereign bonds. 3- Alternative Approach: Explicit Modeling: -- Investors may explicitly model default probabilities and expected recovery rates in bond valuation. --- This approach integrates: ---- The likelihood of default. ---- The expected recovery rate (amount recovered if the bond defaults). --- These inputs can be inferred from market prices, such as credit default swaps (CDS). 4- Key Difference in Approaches: -- Adjusting the discount rate simplifies the valuation process but indirectly accounts for default risk. -- Explicit modeling provides a more granular approach, directly incorporating probabilities and recovery rates into bond valuation.

Option-Adjusted Spread (OAS) 1- Purpose of OAS and Z-Spread: -- The Z-spread is a constant spread added to one-year forward rates from a benchmark yield curve to value option-free risky bonds. -- For risky bonds with embedded options, the OAS is a constant spread similar to the Z-spread but adjusts for the value of the embedded options. --- For option-free bonds, the Z-spread equals the OAS under the assumption of zero volatility. 2- Valuing Bonds Using OAS: -- To calculate a bond's OAS: --- Start with the one-year forward rates derived from the benchmark yield curve. --- Add a constant spread (OAS) to all rates and calculate the bond's value using the arbitrage-free valuation method. --- Adjust the spread iteratively until the calculated value matches the bond's current market price. 3- How OAS Removes Option Uncertainty: -- The OAS adjusts for whether an embedded option is exercised at each node in the valuation tree. -- This removes uncertainty related to option risk, making the OAS a measure of relative value for comparing bonds with and without embedded options. 4- Interpretation of OAS: -- Bonds with higher OAS compared to similar bonds are undervalued and may represent a buying opportunity. -- Bonds with lower OAS are relatively overvalued. 5- Key Advantages of OAS: -- OAS accounts for both credit risk and embedded options. -- It allows for direct comparison of bonds regardless of whether they have embedded options or not.

Effect of Interest Rate Volatility on Option-Adjusted Spread (OAS) 1- Impact of Volatility on OAS: -- All else equal, increasing the volatility assumption reduces a bond's OAS. -- This relationship assumes that the callable bond's price remains constant while the volatility level changes. 2- Key Relationships: -- The value of a callable bond = Value of an option-free bond - Value of the embedded call option. -- Increasing volatility widens the distribution of rates in the binomial tree: --- High-rate nodes: Rates increase further, reducing the callable bond's value. --- Low-rate nodes: Rates decrease further, but the callable bond's value does not change since the bond will still be called at the exercise price. 3- Effect on Callable Bond Pricing: -- At higher assumed volatility, the embedded call option becomes more valuable to the issuer. --- This reduces the callable bond's value to investors. -- If the callable bond's price is held constant, the OAS calculation adjusts as follows: --- The OAS assumes that increases in the value of the call option are offset by increases in the value of the option-free bond. --- A lower spread (OAS) is required to align the option-adjusted valuation with the benchmark bond's price. 4- Conclusion: -- The OAS for a callable bond decreases as the assumed interest rate volatility increases. -- This reflects the rising value of the embedded call option, which reduces the bond's attractiveness to investors.

[Option-Adjusted Spread (OAS) Valuation Using a Binomial Tree: Step-by-Step Example 1- Objective and Structure -- This example demonstrates how to **value a putable bond** using a **binomial interest rate tree** adjusted for the **Option-Adjusted Spread (OAS)**. -- OAS reflects the constant spread added to each forward rate to match the bond's market price. -- The bond is **putable at par (100)** at the end of each year. -- Assumptions: --- Bond maturity = 2 years --- Coupon = 4% annually --- Face value = 100 --- OAS = 50 basis points = 0.50% --- Benchmark forward rates: ---- Time 0 = 2.0% ---- Time 1 upward node = 5.0% ---- Time 1 downward node = 3.0% 2- Step 1: Adjust the Tree for the OAS -- Add OAS to each forward rate to construct the **adjusted rate tree**: --- Time 0 rate = 2.0% + 0.50% = **2.5%** --- Time 1 upward node = 5.0% + 0.50% = **5.5%** --- Time 1 downward node = 3.0% + 0.50% = **3.5%** 3- Step 2: Value at Maturity (Time 2) -- Final payoff = 100 (par) + 4 (coupon) = **104** -- Since it's the final node, no put option is applied. -- Value at all Time 2 nodes = **104** 4- Step 3: Value at Time 1 with Put Option -- Use the backward induction formula: --- "V_node = 0.5 × [ (V_up + C) ÷ (1 + r_up) + (V_down + C) ÷ (1 + r_down) ]" **Time 1 Upward Node (r = 5.5% = 0.055):** -- V_node = 0.5 × [ (104 ÷ 1.055) + (104 ÷ 1.055) ] -- V_node = 0.5 × [2 × 98.58] = 0.5 × 197.16 = **98.58** -- Compare to put price = 100 → Value = **max(98.58, 100) = 100** (put exercised) **Time 1 Downward Node (r = 3.5% = 0.035):** -- V_node = 0.5 × [ (104 ÷ 1.035) + (104 ÷ 1.035) ] -- V_node = 0.5 × [2 × 100.48] = 0.5 × 200.96 = **100.48** -- Compare to put price = 100 → Value = **max(100.48, 100) = 100.48** (put not exercised) 5- Step 4: Value at Time 0 -- Use adjusted Time 0 rate = 2.5% = 0.025 -- V_0 = 0.5 × [ (100 + 4) ÷ 1.055 + (100 + 4) ÷ 1.055 ] = **100.48** (upward) -- and -- V_0 = 0.5 × [ (100 + 4) ÷ 1.035 + (100 + 4) ÷ 1.035 ] = **98.58** (downward) -- Then: --- V_0 = 0.5 × [ (100.48 + 4) ÷ 1.025 + (98.58 + 4) ÷ 1.025 ] --- V_0 = 0.5 × [104.48 ÷ 1.025 + 102.58 ÷ 1.025 ] --- V_0 = 0.5 × [101.95 + 100.05] = 0.5 × 202.00 = **101.00** 6- Final Result -- After accounting for the 50 bps OAS and adjusting the binomial tree: --- The **value of the putable bond = 101.00**

Key Takeaways -- OAS is added to each node rate to reflect credit and option-adjusted risk premium. -- A binomial tree is then used to value the bond with embedded options. -- The **put option sets a floor**, while the OAS ensures consistency with market pricing. List of Variables -- C: Annual coupon = 4 -- Face value: 100 -- Put price: 100 -- OAS: Option-adjusted spread = 0.50% -- r_t: Node-specific interest rates including OAS -- V_node: Bond value at each node -- V_up, V_down: Values from upper and lower future nodes -- PV: Present value using backward induction -- max(a, b): Used to apply the put option logic -- T: Time to maturity = 2 years]

Duration Duration measures the sensitivity of a bond’s full price to changes in its yield to maturity (i.e., yield duration) or changes in benchmark interest rates (i.e., curve duration). Yield duration measures, such as modified duration, can only be used with option-free bonds.

To monitor the sensitivity of bonds with embedded options, it is necessary to use curve duration measures.

Effective Duration 1- Definition: Effective duration, also known as option-adjusted duration, measures the percentage change in a bond’s price for a 100 bps parallel shift of the benchmark yield curve. -- It is a curve duration measure, meaning it accounts for changes in interest rate levels and is used for bonds with embedded options, but it is also applicable to option-free bonds. 2- Formula for Effective Duration: Name of Formula: Effective duration formula. Formula: "Effectiveduration = [(PV_minus - PV_plus) ÷ (2 * ΔCurve * PV_0)]" Where: -- PV_minus: Bond price after the benchmark yield curve shifts down by ΔCurve. -- PV_plus: Bond price after the benchmark yield curve shifts up by ΔCurve. -- PV_0: Initial bond price before any curve shift. -- ΔCurve: Change in the benchmark yield curve (expressed as a decimal). 3- Steps to Calculate Effective Duration: 1- Determine the initial price (PV_0): -- Calculate the bond’s implied OAS for the given initial price and assume an appropriate level of interest rate volatility. 2- Shift the benchmark yield curve down (ΔCurve): -- Generate a new interest rate tree and calculate PV_minus using the OAS from Step 1. 3- Shift the benchmark yield curve up (ΔCurve): -- Generate a new interest rate tree and calculate PV_plus using the same OAS from Step 1. 4- Apply the effective duration formula: -- Use the values for PV_minus, PV_plus, PV_0, and ΔCurve in the formula to calculate the bond’s effective duration.

Key Takeaways: -- Effective duration accounts for interest rate sensitivity and optionality, making it suitable for valuing bonds with embedded options. -- It is more comprehensive than modified duration, as it adjusts for potential changes in cash flows caused by the embedded options.

Effective Duration: Callable, Putable, and Other Bonds 1- Definition of Effective Duration: Effective duration measures a bond’s sensitivity to interest rate changes, accounting for embedded options and curve shifts. It is particularly important for bonds with optionality, as their cash flows can change due to exercised options. 2- Effective Duration by Type of Bond: -- 1- Cash: --- Effective duration is 0, meaning cash is completely insensitive to changes in interest rates. -- 2- Zero-Coupon Bond: --- Effective duration is approximately equal to the bond’s maturity, as the cash flow consists of a single payment at maturity. -- 3- Fixed-Rate Bond: --- Effective duration is less than maturity, as cash flows from periodic coupon payments reduce the bond’s sensitivity to interest rate changes. -- 4- Callable Bond: --- Effective duration is less than or equal to the duration of a comparable straight bond. ---- When interest rates are high (option is out-of-the-money), the callable bond behaves like a straight bond. ---- As interest rates decrease, the callable bond’s effective duration falls, as the bond is more likely to be called. -- 5- Putable Bond: --- Effective duration is less than or equal to the duration of a comparable straight bond. ---- When interest rates are low (option is out-of-the-money), the putable bond behaves like a straight bond. ---- As interest rates increase, the put option becomes more valuable, and the bond’s effective duration decreases significantly compared to a straight bond. -- 6- Floating-Rate Bond (Floater): --- Effective duration is approximately equal to the time (in years) until the next reset date. ---- Floaters reset their coupon rates periodically, so their prices remain relatively stable even in volatile interest rate environments.

Key Takeaways: -- Callable Bonds: Effective duration decreases as interest rates fall because of the higher likelihood of the bond being called by the issuer. -- Putable Bonds: Effective duration decreases as interest rates rise because bondholders are more likely to exercise the put option. -- Straight Bonds: Maintain higher sensitivity to yield curve shifts, as there are no embedded options affecting cash flows. -- Other Types: Cash has zero duration, zero-coupon bonds’ duration equals maturity, and floating-rate bonds have durations tied to their next reset date.

Effective durations are normally calculated by averaging the changes from shifting the yield curve up and down. However, this can be a misleading indicator for bonds with embedded options because their price sensitivity is asymmetric. Callable bonds are typically more sensitive to rate increases than decreases, while putable bonds tend to be more sensitive to rate decreases than increases. One-sided durations often better capture the interest rate sensitivity of callable or putable bonds.

Key Rate Durations 1- Definition of Key Rate Durations: Key rate durations, also known as partial durations, measure a bond's sensitivity to changes in specific maturities on the benchmark yield curve. -- Unlike effective duration, which assumes parallel shifts in the yield curve, key rate durations capture shaping risk—the risk of the yield curve becoming steeper or flatter. 2- Calculation of Key Rate Durations: -- Key rate durations are calculated similarly to effective duration but focus on a single rate while holding all other rates constant. -- A key rate duration is essentially the effective duration for a particular maturity point on the yield curve. 3- Key Rate Durations of Option-Free Bonds: -- 1- Par Bonds: --- An option-free bond trading at par is only sensitive to its maturity point. --- Key rate durations for all other maturities are zero. --- Example: A 10-year, 5% coupon bond with a flat 5% yield curve has a non-zero key rate duration only at the 10-year rate. -- 2- Premium Bonds: --- Bonds trading at a premium are more sensitive to interim maturities because of their higher coupon payments. --- Example: A 10-year, 10% coupon bond will have positive key rate durations for rates before its maturity. -- 3- Zero-Coupon Bonds: --- Zero-coupon bonds are most sensitive to their maturity point. --- Interim key rate durations are typically negative. --- Example: A 10-year zero-coupon bond will be even more sensitive to the 10-year rate than a coupon bond. 4- Key Rate Durations of Bonds with Embedded Options: -- Embedded options introduce additional sensitivities: --- A callable or putable bond is sensitive to both its time to maturity and its time to exercise. --- As the probability of option exercise increases, the bond’s key rate durations shift toward the time to exercise. -- Example: --- A 30-year callable bond, callable in 10 years, will have key rate durations increasingly resembling those of a 10-year option-free bond as interest rates fall and the likelihood of calling increases.

Key Takeaways: -- Key rate durations provide a detailed view of how a bond reacts to non-parallel shifts in the yield curve. -- Option-free bonds are primarily sensitive to their time to maturity, with variations for par, premium, and zero-coupon bonds. -- Bonds with embedded options are sensitive to both their maturity and their exercise time, with sensitivities shifting as exercise probabilities change.

Key Rate Durations of Bonds with Embedded Options 1- Key Rate Durations and Embedded Options: -- For bonds with embedded options (e.g., callable or putable bonds), key rate durations depend on both: --- Time to maturity. --- Time to exercise of the embedded option. 2- Sensitivity Based on Likelihood of Option Exercise: -- 1- When the option is unlikely to be exercised: --- The key rate duration corresponding to the bond’s maturity is the most significant. -- 2- When the option is likely to be exercised: --- The key rate duration corresponding to the exercise time becomes more important. 3- Example: 30-Year Callable Bonds (Callable in 10 Years) The table illustrates key rate durations for 30-year callable bonds at a 4% flat yield curve with 15% interest rate volatility: -- 1- Low-Coupon Bonds (e.g., 2%): --- Key rate duration is heavily weighted at the 30-year point because these bonds are unlikely to be called unless interest rates drop significantly. -- 2- High-Coupon Bonds (e.g., 10%): --- Key rate duration shifts toward the 10-year point because these bonds are more likely to be called as rates fall.

Key Takeaways: -- Key rate durations help identify the most sensitive maturities based on a bond’s structure and embedded options. -- Low-coupon bonds are more sensitive to their full maturity, while high-coupon bonds shift sensitivity toward the call date as the probability of exercise increases.

Effective Convexity 1- Limitations of Duration: -- A bond’s duration provides a linear approximation of price changes for a given change in interest rates. -- This works reasonably well for small yield curve shifts, but becomes less accurate for larger interest rate changes. --- Overestimation occurs for price decreases when yields rise. --- Underestimation occurs for price increases when yields fall. 2- Definition of Effective Convexity: Effective convexity measures the curvature of the price/yield relationship, accounting for changes in a bond’s duration as interest rates fluctuate. -- It enhances the accuracy of price change estimates for larger interest rate movements. 3- Formula for Effective Convexity: Name of Formula: Effective convexity formula. Formula: "Effectiveconvexity = [(PV_minus + PV_plus - (2 * PV_0)) ÷ ((ΔCurve)^2 * PV_0)]" Where: -- PV_minus = Bond price after the benchmark yield curve shifts down by ΔCurve. -- PV_plus = Bond price after the benchmark yield curve shifts up by ΔCurve. -- PV_0 = Initial bond price. -- ΔCurve = Change in the benchmark yield curve (expressed as a decimal).

Key Takeaways: -- Effective convexity captures the non-linear relationship between bond price and yield, improving accuracy for larger rate changes. -- Duration estimates alone are insufficient for pricing bonds accurately when significant shifts in interest rates occur.

[Determining Moneyness of Embedded Options 1- Key Concept -- Moneyness is defined by the relationship between current market interest rates and the bond’s coupon rate relative to its exercise price. 2- Moneyness Criteria -- 1- Call Option (issuer right to redeem at exercise price): --- In-the-money: Market rate < coupon rate --- At-the-money: Market rate = coupon rate --- Out-of-the-money: Market rate > coupon rate -- 2- Put Option (holder right to sell at exercise price): --- In-the-money: Market rate > coupon rate --- At-the-money: Market rate = coupon rate --- Out-of-the-money: Market rate < coupon rate 3- Example: Negative Convexity and Moneyness -- Effective convexity = –748 → indicates call option is in-the-money. -- In this case, price sensitivity to falling rates decreases due to higher likelihood of the bond being called, causing negative convexity.

Key Takeaways: -- For callable bonds, falling interest rates increase the likelihood that the option will be exercised, driving the bond in-the-money. -- For putable bonds, rising interest rates increase the likelihood of exercise, driving the bond in-the-money. -- Negative convexity is a clear signal of an in-the-money call option.

Convexity of Option-Free, Callable, and Putable Bonds 1- Option-Free Bonds: -- Option-free bonds exhibit low positive convexity, meaning that the price increase from falling rates is greater than the price decrease from rising rates by an equivalent amount. 2- Callable Bonds: -- Callable bonds display negative convexity when interest rates are low because the bond is more likely to be called. --- As a result, price increases from falling rates are limited, as the issuer will exercise the call option. -- When interest rates are high and the likelihood of exercising the call option is low, callable bonds have positive convexity similar to that of an option-free bond. 3- Putable Bonds: -- Putable bonds have the highest convexity when interest rates are high because the bondholder is more likely to exercise the put option. -- These bonds exhibit positive convexity across all interest rate environments. 4- Key Observations from the Price/Yield Curve: -- Callable bonds’ prices approach the price of straight bonds when yields increase, as the call option becomes unlikely to be exercised. -- Putable bonds’ prices approach the price of straight bonds when yields decrease, as the put option becomes unlikely to be exercised.

Key Takeaways: -- Option-free bonds consistently exhibit positive convexity. -- Callable bonds can exhibit negative convexity when rates are low, while maintaining positive convexity when rates are high. -- Putable bonds always maintain positive convexity, with the highest convexity occurring at high rates.

- The price of a callable bond is always less than or equal to the price of an equivalent straight bond.

- The price of a Put bond is always more than or equal to the price of an equivalent straight bond.

Callable Bond Price-Yield Relationship 1- Price-Yield Relationship Overview: -- Callable bonds exhibit unique price-yield characteristics due to the embedded call option, which limits price appreciation when yields fall. -- Compared to straight bonds, callable bonds demonstrate negative convexity at lower yields and positive convexity at higher yields. 2- Convexity and Interest Rates: -- 1- At High Interest Rates: --- The likelihood of the call option being exercised is low. --- Callable bonds behave similarly to straight bonds, with positive convexity and prices increasing as rates decrease. -- 2- At Low Interest Rates: --- The likelihood of the call option being exercised is high. --- Callable bonds exhibit negative convexity, limiting price increases because issuers are likely to call the bond and refinance at lower rates. --- The price of the callable bond approaches the call price. 3- Effective Duration and Price Changes: -- Effective duration provides a linear approximation for price changes, which is useful for small yield movements. --- However, it underestimates price increases when yields rise and overestimates price increases when yields fall, especially for callable bonds. -- The actual price change of a callable bond accounts for curvature and is lower than the effective duration estimate for large yield decreases.

Options in floating-rate bonds (floaters) are exercised automatically. A cap or floor will take effect if the coupon rate rises or falls below the specified threshold. In the absence of caps or floors, a floating-rate bond will trade very close to its par value.

Valuation of Floating-Rate Bonds (Floaters) 1- General Characteristics of Floaters: -- Floating-rate bonds adjust coupon payments automatically based on changes in interest rates. -- Caps and floors limit the coupon rate by setting maximum (cap) or minimum (floor) thresholds: --- Without caps or floors, floaters trade near par value. 2- Valuation of a Capped Floater: -- A capped floater includes an embedded cap that protects the issuer from rising interest rates by capping coupon payments. -- The cap reduces the value for investors who own the floater. Formula: "Value of capped floater = Value of straight bond - Value of embedded cap" -- Coupon payments are set in arrears, meaning the rate is determined at the end of the coupon period. 3- Valuation of a Floored Floater: -- A floored floater includes an embedded floor that protects the investor from falling interest rates by ensuring a minimum coupon payment. -- The floor increases the value for investors who own the floater. Formula: "Value of floored floater = Value of straight bond + Value of embedded floor" -- A floor establishes a minimum coupon payment, opposite to the cap's role in limiting the maximum payment.

Valuation of a Capped Floater 1- Purpose of a Capped Floater: -- A capped floater protects the issuer against rising interest rates by limiting the coupon payments. -- It is viewed as an issuer option, even though the issuer does not need to actively decide to exercise it.

2- Coupon Payment Structure: -- Coupon rates are set at the beginning of the period and paid at the end of the period. -- This ensures coupon payments are known in advance for the upcoming payment period.

Valuation of a Floored Floater 1- Purpose of a Floored Floater: -- A floored floater protects the investor against falling interest rates by ensuring a minimum coupon payment. -- It is viewed as an investor option, increasing the bond’s value for the holder, as the floor guarantees a minimum return.

Defining Features of a Convertible Bond 1- Convertible Bond Characteristics: -- Convertible bonds are hybrid securities that combine features of debt and equity. -- Investors have the right to convert the bond into the issuer’s common stock during the conversion period at the conversion price, both of which are pre-determined. 2- Key Components of Convertible Bonds: -- Structure: A convertible bond is equivalent to a straight bond plus a call option on the issuer's stock. --- Investors accept a lower coupon rate in exchange for the upside potential of the conversion option, benefiting the issuer. --- If conversion occurs, the issuer avoids repaying the principal, though shareholder dilution may occur. -- Conversion Ratio: --- The number of common shares a bondholder receives upon conversion. --- Adjustments may occur for: ---- Stock splits. ---- Large dividend payments (e.g., the conversion price may decrease to compensate bondholders for missed dividends). 3- Protections and Options in Convertible Bonds: -- Change-of-Control Protection: --- Bondholders may be reluctant to lend to a new entity after a merger or acquisition. --- Protections in the bond prospectus may include: ---- A put option allowing bondholders to sell the bond back to the issuer. ---- A downward adjustment to the conversion price. -- Put Options in Convertible Bonds: --- Hard Put: Requires the issuer to redeem the bond in cash. --- Soft Put: Allows redemption in cash, common stock, or subordinated notes. -- Call Options in Convertible Bonds: --- Many convertible bonds are callable, often including a call premium and a lockout period. --- Issuers use call options to force conversion when the share price exceeds the conversion price, allowing them to stop paying coupons.

Key Takeaways: -- Convertible bonds offer a blend of debt-like security and equity-like upside. -- Features like conversion ratios, change-of-control protections, and embedded options provide flexibility for both issuers and investors. -- Callable convertible bonds enable issuers to manage interest costs by encouraging conversion under favorable market conditions.

Analysis of a Convertible Bond 1- Conversion Value: -- The conversion value is the value of the bond if it is converted into shares at the current market price. Formula: "Conversion value = Underlying share price × Conversion ratio" 2- Minimum Value of a Convertible Bond: -- The minimum value is the greater of: -- 1- The conversion value. -- 2- The straight bond value (value of an equivalent bond without conversion options). -- The minimum value serves as a floor value, which adjusts with interest rates and credit spreads. 3- Market Conversion Price, Market Conversion Premium, and Market Conversion Premium Ratio: -- Market Conversion Price: --- Represents the cost to investors for converting the bond into shares. Formula: "Market conversion price = Convertible bond price ÷ Conversion ratio" -- Market Conversion Premium per Share: --- The premium investors pay over the underlying share price for conversion. Formula: "Market conversion premium per share = Market conversion price - Underlying share price" -- Market Conversion Premium Ratio: --- The premium expressed as a percentage of the share price. Formula: "Market conversion premium ratio = Market conversion premium per share ÷ Underlying share price" -- Reason for Premium: Investors accept this premium because the bond provides downside protection (floor value) and upside potential through conversion. 4- Downside Risk and Upside Potential: -- The downside risk of a convertible bond is measured by its straight bond value. -- All else equal, a bond is relatively overvalued if priced at a higher premium over its straight value. Formula: "Premium over straight value = (Convertible bond price ÷ Straight value) - 1" -- Limitations: --- The straight bond value is not fixed, making this measure imperfect. --- The underlying stock must perform well enough to justify the premium investors pay for the bond.

[Convertible Bond Valuation and Premium Metrics 1- Conversion Value Formula: "Conversionvalue = Shareprice × Conversionratio" 2- Minimum Value of Convertible Bond Formula: "Minimumvalue = max(Conversionvalue, Straightbondvalue)" 3- Market-Based Premium Metrics -- Market Conversion Price: "Marketconversionprice = Convertiblebondprice ÷ Conversionratio" -- Market Conversion Premium per Share: "Marketconversionpremiumpershare = Marketconversionprice - Shareprice" -- Market Conversion Premium Ratio: "Marketconversionpremiumratio = Marketconversionpremiumpershare ÷ Shareprice" 4- Downside Risk and Relative Valuation -- Premium Over Straight Value: "Premiumoverstraightvalue = (Convertiblebondprice ÷ Straightbondvalue) - 1"]

Conversion Period: -- The conversion period defines when investors are allowed to convert their bonds into shares. -- It typically begins after issuance and ends before or at bond maturity. -- Timeline: --- Conversion period begins: Specified start date after issuance. --- Conversion period ends: At or before the bond's maturity.

Call Options in Convertible Bonds 1- Purpose of a Call Option in Convertible Bonds: -- Convertible bonds often include a call option allowing the issuer to redeem the bond early. -- This option is typically exercised when: --- Interest rates have fallen, enabling refinancing at lower costs. --- The issuer's credit quality has improved, reducing borrowing costs. -- An issuer may also call the bond strategically to: --- Prevent dilution of existing shareholders if the share price is expected to rise significantly. 2- Features of the Call Option: -- Protection Period: --- A specific timeframe during which the issuer cannot call the bond, providing initial protection to investors. -- Call Price: --- Usually set at a premium to the par value of the bond. 3- Forced Conversion: -- Forced conversion occurs when the share price exceeds the conversion price, prompting issuers to threaten to call the bond. -- Investors are incentivized to convert their bonds into shares rather than accept redemption. -- Benefits to the Issuer: --- Strengthens the capital structure by reducing leverage. --- Eliminates ongoing coupon payments.

Minimum Value of a Convertible Bond 1- Definition of Minimum Value: -- The minimum value of a convertible bond is the greater of: --- The conversion value, calculated as the product of the underlying share price and the conversion ratio. --- The value of the underlying option-free bond, determined by discounting future bond cash flows at the appropriate rate, assuming no conversion option. -- This ensures the convertible bond price avoids arbitrage opportunities. 2- Factors Affecting the Minimum Value: -- Underlying Share Price: Higher share prices increase the conversion value. -- Interest Rates: Changes in interest rates impact the value of the underlying bond. -- Credit Spreads: Wider credit spreads reduce the value of the bond component.

[Market Conversion Metrics for Convertible Bonds 1- Market Conversion Premium per Share -- Measures the premium paid over the stock’s current market price when acquiring shares via a convertible bond. -- Formula: "Market conversion premium per share = Market conversion price - Share price" 2- Market Conversion Price -- Reflects the effective price per share paid if the bond is converted at current market value. -- Formula: "Market conversion price = Convertible bond price ÷ Conversion ratio"]

Key Features and Differences Between Convertible Bonds and Call Options 1- Convertible Bond Characteristics: -- Convertible bonds include a straight bond floor that protects investors against a drop in the share price. -- The investor benefits from limited downside risk (due to the bond floor) and unlimited upside potential (from conversion rights). -- The conversion premium represents the additional cost of the bond compared to the value of the embedded call option. 2- Key Differences Between Convertible Bonds and Call Options: -- Call Option: --- Premium paid for the option provides downside risk protection, which is known and fixed. -- Convertible Bond: --- Downside risk is estimated because the straight bond value is not fixed and can fluctuate based on: ---- 1- Changes in interest rates. ---- 2- Deterioration in the issuer's credit quality.

Convertible Bond Valuation 1- Complexity of Convertible Bonds: -- Convertible bonds are hybrid securities that combine elements of bonds, stocks, and options. -- Additional features, such as issuer or investor options, increase their complexity and make them more suitable for professional or institutional investors. 2- Factors to Consider in Valuation: -- Interest rate environment. -- Credit risk. -- Factors affecting the issuer's stock price. 3- Valuation Formula for Convertible Bonds: -- Basic Formula: "Convertible bond value = Straight bond value + Call option on the issuer's stock value." -- Expanded Formula (Including Additional Features): "Convertible bond value = Straight bond value + Call option on the issuer's stock value − Issuer call option value + Investor put option value."

Comparison of Risk-Return Characteristics of a Convertible Bond, Straight Bond, and Underlying Stock 1- Behavior of Convertible Bonds Relative to Stock Price: -- When the stock price is low, the convertible bond behaves more like a bond because conversion is unlikely. -- When the stock price is high, the convertible bond behaves more like a stock because conversion becomes likely. 2- Key Observations from the Graph: -- The straight bond value represents the floor value for the convertible bond when the stock price is very low. -- As the stock price increases, the conversion value determines the upside potential. -- The convertible bond value is always greater than or equal to both the straight bond value and the conversion value. 3- Protection and Premium of Convertible Bonds: -- The convertible bond provides downside protection due to its straight bond value, which prevents it from dropping as much as the stock price during poor performance. -- Even at a high stock price, the convertible bond maintains a slight premium over the conversion value due to its bond features.

Items to Consider When Valuing Convertible Bonds 1- Issuer’s Financial Ability: -- Assess the ability of the issuer to service the debt and repay the principal. 2- Terms of Issuance: -- Evaluate factors such as collateral, covenants, and contingent provisions included in the bond terms. 3- Interest Rate Environment: -- Analyze the prevailing interest rate levels and their potential effect on the bond's valuation. 4- Common Stock Factors: -- Consider aspects like dividend payments and the stock's future growth potential.

Minimum Value of a Convertible Bond : max [ Conversion Value, Value of the underlying option free bond]

Value of the Underlying Option-Free Bond: -- Definition: The value of the bond without the conversion feature (i.e., if it were a traditional straight bond). -- How to Find It: -- Discount the bond’s future cash flows (coupons and face value) at an appropriate discount rate, typically reflecting market interest rates and the issuer’s credit risk. -- Formula: Value of the Option-Free Bond = Present Value of Coupons + Present Value of Principal.

A convertible bond that is both callable and putable can be synthetically constructed by combining a long position in an option-free bond with the following options positions: A long position in a call option on the issuer's common stock A short position in a call option on the underlying option-free bond A long position in a put option on the underlying option-free bond

[Market Conversion Premium Ratio Calculation 1- Given Data (15 September 20X4) -- Convertible bond price = $121,000 -- Conversion ratio = 20,000 shares -- Underlying share price = $5.25 2- Step-by-Step Calculation **a) Market Conversion Price** Formula: "Market conversion price = Convertible bond price ÷ Conversion ratio" Calculation: "Market conversion price = 121,000 ÷ 20,000 = 6.05" **b) Market Conversion Premium per Share** Formula: "Market conversion premium per share = Market conversion price - Share price" Calculation: "Market conversion premium per share = 6.05 - 5.25 = 0.80" **c) Market Conversion Premium Ratio** Formula: "Market conversion premium ratio = (Market conversion premium per share ÷ Share price) × 100" Calculation: "Market conversion premium ratio = (0.80 ÷ 5.25) × 100 ≈ 15.24%" Answer: The market conversion premium ratio is closest to **15% (Answer: B).**]

[Premium of Convertible Bond Over Straight Bond 1- Given Data -- Convertible bond price = $121,000 -- Coupon = $4,000 annually -- Par value = $100,000 (maturity in 4 years) -- Discount rate = 2.5% -- Maturity = 4 years 2- Step-by-Step Calculation **a) Straight Bond Value** Formula: "Straight bond value = [C ÷ (1 + r)^1] + [C ÷ (1 + r)^2] + [C ÷ (1 + r)^3] + [(C + Par) ÷ (1 + r)^4]" Substitution: "Straight bond value = [4,000 ÷ 1.025] + [4,000 ÷ (1.025^2)] + [4,000 ÷ (1.025^3)] + [104,000 ÷ (1.025^4)] ≈ 105,643" **b) Premium Over Straight Bond** Formula: "Premium over straight bond = [(Convertible bond price ÷ Straight bond value) - 1] × 100" Substitution: "Premium over straight bond = [(121,000 ÷ 105,643) - 1] × 100 ≈ 14.54%" Answer: The premium at which the convertible bond is trading over its straight value is **14.5% (Answer: A).**]

1- A convertible bond that is both callable and putable can be synthetically constructed by combining the following positions: -- 1- A long position in an option-free bond. -- 2- A long position in a call option on the issuer's common stock. -- 3- A short position in a call option on the underlying option-free bond. -- 4- A long position in a put option on the underlying option-free bond. 2- Explanation of the positions: -- 1- Long position in the option-free bond: This provides the base value of the synthetic convertible bond, ensuring a fixed-income component. -- 2- Long position in a call option on the issuer's common stock: This represents the conversion feature, allowing the bondholder to convert the bond into equity at a predetermined price. -- 3- Short position in the call option on the bond: This reflects the callable feature, giving the issuer the right to redeem the bond early, typically when interest rates drop or the issuer's credit quality improves. -- 4- Long position in a put option on the bond: This represents the putable feature, giving the bondholder the right to sell the bond back to the issuer at a predetermined price, typically providing downside protection.

[Effect of Declining Volatility on Bond Values Answer: **B) Bond B only** 1- Bond B (Callable Bond) -- Callable bond value = Option-free bond value **−** Call option value -- A decrease in interest rate volatility **reduces the value of the embedded call option**, which increases the overall value of Bond B. -- Therefore, **Bond B's value increases** in Scenario 1. 2- Bond A (Putable Bond) -- Putable bond value = Option-free bond value **+** Put option value -- A decrease in volatility **reduces the value of the embedded put option**, lowering the overall value of Bond A. -- Bond A becomes **less attractive** in Scenario 1. 3- Bond C (Option-Free Bond) -- Contains **no embedded options**. -- Its value is **unaffected by volatility** and changes only with interest rate levels. Conclusion: **Only Bond B increases in value** as a result of reduced interest rate volatility.

[Explanation: Impact of Rising Interest Rates on Duration 1- Scenario 3 Overview -- The yield curve shifts **upward** (interest rates rise). This impacts bonds with embedded options differently from option-free bonds. 2- Bond A (Putable Bond) -- Embedded put option becomes **more valuable** as rates rise above the exercise price. -- Investors are **more likely to exercise the put**, capping downside and reducing interest rate sensitivity. -- Result: **Effective duration decreases**. 3- Bond B (Callable Bond) -- Higher rates make it **less likely** for issuers to call the bond, since refinancing is unattractive. -- With the call option becoming less relevant, the bond behaves more like an option-free bond. -- Result: **Effective duration increases**. 4- Bond C (Option-Free Bond) -- No embedded options, so duration is affected **only by rate changes**, not option behavior. -- Result: Duration reflects **standard sensitivity** to rising rates.

6.4 Credit Analysis Models

-- Explain expected exposure, the loss given default, the probability of default, and the credit valuation adjustment. -- Explain credit scores and credit ratings. -- Calculate the expected return on a bond given transition in its credit rating. -- Explain structural and reduced-form models of corporate credit risk, including assumptions, strengths, and weaknesses. -- Calculate the value of a bond and its credit spread, given assumptions about the credit risk parameters. -- Interpret changes in a credit spread. -- Explain the determinants of the term structure of credit spreads and interpret a term structure of credit spreads. -- Compare the credit analysis required for securitized debt to the credit analysis of corporate debt.

Modeling Credit Risk 1- Definition of Credit Risk: -- Credit risk is measured by the credit spread, defined as the yield difference between a corporate bond and a government bond of the same maturity. -- It compensates investors for taking on default risk, which involves the likelihood of default and the loss incurred if default occurs (expected loss). 2- Assumptions in Measuring Credit Risk Using Credit Spreads: -- The corporate and government bonds being compared are assumed to have the same liquidity and tax treatment. -- Government bonds tend to be more liquid and may have preferential tax treatments, partially explaining the credit spread. 3- Key Factors in Modeling Credit Risk: -- 1- Expected Exposure to Default Loss: --- Represents the maximum loss an investor could incur if a default occurs. --- This includes both principal and coupon payments. --- The exposure depends on the market value of the bond, which changes with interest rates. --- Example: For a 1-year bond with a 5% coupon and $100 par value, the exposure to default loss would be $105 (par value + coupon). -- 2- Recovery Rate: --- Refers to the percentage of the bond’s value recovered in case of default. --- Loss severity is calculated as "100% - Recovery Rate." --- Example: A 40% recovery rate implies a loss severity of 60%. --- The loss given default equals the loss severity multiplied by the exposure to default. -- 3- Probability of Default: --- Measures the likelihood that the bond issuer will fail to meet contractual obligations. --- If risk-neutral probabilities are used, cash flows are discounted at the risk-free rate. --- If actual default probabilities are used, cash flows are discounted at risk-adjusted rates. --- Risk-neutral probabilities tend to be higher than actual default probabilities because they incorporate a risk premium for uncertainty regarding default timing.

Key Takeaways: -- Credit risk modeling combines exposure to default, recovery rate, and default probability to estimate potential losses. -- The credit spread reflects compensation for these risks but is also influenced by liquidity and tax factors.

[Credit Value Adjustment (CVA) 1- Definition -- CVA is the **present value of expected credit losses** due to issuer default over the bond’s life. -- It adjusts the bond’s value or yield to reflect **credit risk**. 2- Formula for Present Value of Expected Losses (PVEL) -- Name of Formula: Present value of expected loss. -- Formula: "PVEL = Discount factor × (Exposure × Loss severity) × Probability of default" 3- Explanation of Variables -- Discount factor: Present value factor based on time and interest rates. -- Exposure: Bond value at risk in case of default. -- Loss severity: "100% − Recovery rate" -- Probability of default: Likelihood the issuer will default within a given time horizon.

Key Takeaways: -- CVA accounts for the time value of money and expected losses over the bond’s life. -- It provides a comprehensive way to quantify credit risk and adjust bond valuations accordingly.

The components of the PVEL formula, in its various forms, can be illustrated using the example of a 3-year, zero-coupon bond with the following assumptions The government yield curve is flat at 4% The recovery rate is 40% At the end of each year, there is a 1% probability of default

Credit Value Adjustment (CVA) Components: 1- Discount Factors (DF): -- The discount factors are calculated using the spot curve, which is flat at 4%, to determine the present value of expected losses for each year: --- DF1 = 1 ÷ (1.04) = 0.96154. --- DF2 = 1 ÷ (1.04)^2 = 0.92456. --- DF3 = 1 ÷ (1.04)^3 = 0.88900. 2- Exposure: -- The exposure at the beginning of each year reflects the bond's potential loss due to default, discounted at the risk-free rate: --- At the beginning of Year 3, the exposure is the bond's face value: 100. --- At the beginning of Year 2: Exposure = 100 ÷ (1.04) = 96.154. --- At the beginning of Year 1 (present day): Exposure = 96.154 ÷ (1.04) = 92.456. -- Note: This exposure differs from the present value of the bond’s cash flows assuming no default risk, which is 100 ÷ (1.04)^3 = 88.900. The difference accounts for the assumption that default can only occur at the end of each year. 3- Loss Given Default (LGD): -- LGD is derived as the product of the bond's exposure and its loss severity. Here, the loss severity is 60%, calculated as 100% minus the 40% recovery rate: --- LGD1 = 92.456 × 60% = 55.473. --- LGD2 = 96.154 × 60% = 57.692. --- LGD3 = 100.000 × 60% = 60.000.

Probability of Default and Survival 1- Introduction to the Theory: -- The Probability of Default (POD) in a given year is the conditional probability that the bond defaults at the end of the year, assuming it has not already defaulted. This is driven by the hazard rate, which is 1% for this example. -- The Probability of Survival (POS) is the likelihood that the bond does not default up to a specific point in time. It is helpful to calculate POS at the beginning of a year (e.g., POS_BY1) and at the end of a year (e.g., POS_EY1). 2- Step-by-Step Thought Process: -- Year 1 Calculations: --- At the beginning of Year 1, the bond has not defaulted, so POS_BY1 = 100%. --- The POS_EY1 (end of Year 1) is determined as POS_BY1 - (POS_BY1 × Hazard Rate): - POS_EY1 = 100% - (100% × 1%) = 99.0%. -- Year 2 Calculations: --- The POS_BY2 (beginning of Year 2) equals POS_EY1: - POS_BY2 = 99.0%. --- POD2 is calculated as the product of the hazard rate and POS_BY2: - POD2 = 1.00% × 99.0% = 0.99%. --- The POS_EY2 (end of Year 2) is determined as POS_BY2 - POD2: - POS_EY2 = 99.0% - 0.99% = 98.01%. -- Year 3 Calculations: --- The POS_BY3 (beginning of Year 3) equals POS_EY2: - POS_BY3 = 98.01%. --- POD3 is calculated as the product of the hazard rate and POS_BY3: - POD3 = 1.00% × 98.01% = 0.9801%. --- The POS_EY3 (end of Year 3) is determined as POS_BY3 - POD3: - POS_EY3 = 98.01% - 0.9801% = 97.03%. -- General Formula for Survival to the End of Year t: --- POS_EYt = (100% - Hazard Rate)^t. For Year 3: - POS_EY3 = (100% - 1%)^3 = 97.03%. 3- Extending the Credit Value Adjustment (CVA) Formula: -- Using these probabilities, the Present Value of Expected Losses (PVEL) can be expressed as: - PVEL = DF × Expected Loss. - Expected Loss = LGD × POD. - LGD = Exposure × Loss Severity. -- Substituting all components: - PVEL = DF × (Exposure × Loss Severity) × (POS_BYt × Hazard Rate).

Credit Value Adjustment (CVA) Calculation and Application 1- CVA Calculation: -- The CVA represents the sum of the present values (PV) of expected losses over the bond's lifetime. -- Formula: CVA = ∑ [PV of expected loss_t], for t = 1 to 3. -- Inputs Required: --- Exposure (E): The bond’s exposure at the beginning of each year, as previously calculated. --- Loss Severity (LS): Assumed to be 60%. --- Probability of Survival (POS_BY): The likelihood of survival at the beginning of the year. --- Hazard Rate (HR): Assumed to be 1%. --- Discount Factor (DF): Based on the spot curve flat at 4%. --- PVEL (Present Value of Expected Loss): Calculated as DF × E × LS × (POS_BY × HR). -- Detailed CVA Calculation: --- Year 1: --- PVEL_1 = 0.96154 × 92.456 × 60% × (100% × 1%) = 0.533. --- Year 2: --- PVEL_2 = 0.92456 × 96.154 × 60% × (99% × 1%) = 0.528. --- Year 3: --- PVEL_3 = 0.88900 × 100.000 × 60% × (98.01% × 1%) = 0.523. --- Total CVA = 0.533 + 0.528 + 0.523 = 1.584. 2- CVA Application: -- Step 1: Calculate the bond's fair value, which adjusts for the credit risk by subtracting CVA from the default-free value. --- Formula: Fair value = Default-free value - CVA. --- Fair value = 88.900 - 1.584 = 87.316. -- Step 2: Determine the bond's credit-adjusted yield using the fair value. --- Formula: Fair value = 100 ÷ (1 + Yield)^3. --- 87.316 = 100 ÷ (1 + Yield)^3 → Yield = 4.63%. -- Step 3: Calculate the credit spread as the difference between the credit-adjusted yield and the risk-free rate. --- Formula: Credit spread = Yield - Risk-free rate. --- Credit spread = 4.63% - 4.00% = 0.63%.

Key Takeaways: -- The CVA quantifies the credit risk by incorporating probabilities of default, loss severity, and exposures into a present value calculation. -- The credit-adjusted fair value, yield, and credit spread reflect the bond's pricing and yield considering the default risk. -- Assuming no default, investors will earn the adjusted yield to maturity.

ESG Factors and Default Risk 1- Impact of ESG Factors on Default Risk: -- ESG factors are integral to evaluating default risk due to their influence on a company's operations and reputation: --- Companies with weak governance (e.g., poor internal controls) are more prone to fraudulent practices, increasing financial instability. --- Negative social factors, such as unethical practices, may lead to consumer boycotts and reduced revenues. 2- Bonds Linked to ESG Factors: -- Bonds explicitly tied to ESG initiatives are increasingly prevalent: --- Climate (Green) Bonds: ---- Raise funds for environmental projects. ---- May offer preferential tax treatment to attract investors. --- Catastrophe Bonds (Cat Bonds): ---- Provide high coupon payments but carry significant risk, as investors may lose their entire principal if specific catastrophic events occur. ---- Example: Pandemic bonds issued in 2017 provided cash flows for over two years before becoming worthless during the COVID-19 pandemic. ---- These bonds resemble insurance policies more than traditional fixed-income securities.

Formula for POS_BYt 1- Definition: The probability of survival to the beginning of Year t (POS_BYt) is calculated iteratively by adjusting for survival probabilities in prior years and subtracting the probability of default in earlier periods. -- Formula: POS_BYt = POS_BYt-1 - (POS_BYt-1 × Hazard rate) -- Where: --- POS_BYt-1: Probability of survival to the beginning of the prior year (t-1). --- Hazard rate: Conditional probability of default in a given year. 2- Key Steps to Derive POS_BYt: -- 1- Year 1 (t = 1): --- At the beginning of Year 1, the probability of survival is 100%: POS_BY1 = 100%. -- 2- Year 2 (t = 2): --- The probability of survival to the beginning of Year 2 adjusts for the hazard rate in Year 1: --- POS_BY2 = POS_BY1 - (POS_BY1 × Hazard rate) --- Substituting values: POS_BY2 = 100% - (100% × 1%) = 99%. -- 3- Year 3 (t = 3): --- The probability of survival to the beginning of Year 3 adjusts for the hazard rate in Year 2: --- POS_BY3 = POS_BY2 - (POS_BY2 × Hazard rate) --- Substituting values: POS_BY3 = 99% - (99% × 1%) = 98.01%. 3- Difference Between POS_BYt and End-of-Year Survival: -- POS_EYt (Probability of Survival to the End of Year t): Calculated as (100% - Hazard rate)^t, representing the cumulative probability of survival to the end of Year t. -- POS_BYt (Probability of Survival to the Beginning of Year t): Tracks survival probabilities at the start of each year, incorporating year-by-year adjustments for default risk.

-- The probability of surviving until the end of year t is the same as the probability of surviving to the beginning of year t+1. -- The survival probability reflects the likelihood that no default occurs up to a specific point in time, as shown in the timeline: --- From the start of year t to the beginning of year t+1, the survival probability accounts for the absence of default through year t.

Credit Scores and Ratings 1- Credit Scores: Used primarily for small businesses and individuals. -- Credit scores are determined based on various observed factors, including payment history, debt burden, length of credit history, types of credit used, and recent credit searches. -- Scores are often specific to a country’s credit system. Some systems consider only negative information, while others incorporate broader factors. -- 1- Example in the United States: --- The FICO score is a widely used credit risk measure, ranging from 300 to 850. --- A lower score indicates greater credit risk. --- Factors such as ethnicity, gender, and marital status have no impact on FICO scores. 2- Credit Ratings: Ordinal measures used for corporations and government entities. -- Ratings are issued by agencies such as Moody’s, Standard & Poor’s, and Fitch Ratings. -- These ratings primarily focus on the probability of default, with historical data showing that defaults on investment-grade bonds are rare. 3- Notching: Adjustments to reflect differences in loss given default for specific securities. -- Companies typically have an issuer rating based on their senior unsecured debt. -- Ratings of specific securities are adjusted depending on their structure: --- Secured debt may have a higher rating than the issuer rating. --- Subordinated debt may have a lower rating than the issuer rating. 4- Outlooks: Agencies may assign outlooks to issuers as either positive, stable, or negative, reflecting expected changes in credit quality. 5- Transition Matrices: Tools used to assess the probability of a rating change. -- Transition matrices illustrate the likelihood of moving from one rating to another over a one-year period. -- They show that issuers are highly likely to maintain the same rating, but downgrades occur more frequently than upgrades. 6- Credit Spread Migration: -- Downgrades tend to increase credit spreads more significantly than upgrades reduce them. -- This asymmetry reduces expected returns on risky bonds, as issuers face higher penalties for worsening credit quality compared to the benefits of improving creditworthiness.

Key Takeaways -- Credit scores assess individuals and small businesses, while credit ratings evaluate corporations and government entities. -- Credit ratings incorporate notching to adjust for differences in securities' loss given default. -- Transition matrices highlight the stability of credit ratings and the higher likelihood of downgrades compared to upgrades. -- Credit spread migration typically reduces the expected return on risky bonds.

1- Seniority Ranking Overview: The seniority ranking categorizes debt based on the priority of claims in the event of default: -- Secured Debt: Includes First Lien/Mortgage, Senior Secured, and Junior Secured debts. --- Secured debt provides bondholders with direct claims on specific assets or associated cash flows, resulting in lower loss given default (LGD). -- Unsecured Debt: Includes Senior Unsecured, Senior Subordinated, Subordinated, and Junior Subordinated debts. --- Unsecured debt gives bondholders general claims on assets and cash flows, leading to higher LGD compared to secured debt. The hierarchy ensures that secured debt is repaid first, followed by unsecured debt in descending order of seniority. 2- Notching: Issuer ratings and notching are related to seniority: -- Senior Unsecured Debt: Forms the basis for the issuer rating. -- Notching Down: Debt instruments lower in seniority, such as subordinated debt, receive lower credit ratings than the issuer rating due to their higher credit risk and weaker protection. This notching process reflects differences in loss given default across debt types. For example, subordinated debt has higher risk and lower recovery prospects, resulting in a lower credit rating

Seniority Ranking -- Secured Debt -- -- 1- : First Lien/Mortgage. -- 2- Senior Secured: Same as above with slightly less priority. -- 3- Junior Secured: Secured claim but subordinate to higher-priority secured debt. -- Unsecured Debt -- -- 4- Senior Unsecured: General claim on assets with higher priority than subordinated debt. -- 5- Senior Subordinated: Subordinated debt but with higher priority than other subordinated debts. -- 6- Subordinated: General claim on assets with lower priority than senior subordinated debt. -- 7- Junior Subordinated: Lowest priority claim among all unsecured debts.

Credit Analysis Models 1- Structural Models: These models explain why defaults occur based on the structure of a company’s balance sheet. -- Definition: Structural models, also called company-value models, assume a company will default if the value of its liabilities exceeds the value of its assets. -- Key Features: --- Based on the insight that the probability of default depends on the company's balance sheet structure. --- Default probability is modeled using option-pricing techniques, such as the Black-Scholes-Merton model, which requires inputs like historical equity price volatility and a default barrier based on liabilities. --- Use probability distributions to estimate the likelihood that future asset values will fall below liability values, triggering default. -- Challenges: --- Assume all assets are actively traded, which is often unrealistic. --- Estimating the true value of a company’s assets and liabilities is difficult. 2- Reduced-Form Models: These models focus on when defaults are likely to occur, not why they happen. -- Definition: Reduced-form models, also known as intensity-based or stochastic default rate models, emerged in the 1990s and treat defaults as random external events. -- Key Features: --- Avoid the assumption of actively traded assets. --- Model the timing of defaults using a Poisson stochastic process. --- Use observable inputs like company-specific variables (e.g., leverage ratio) or macroeconomic factors (e.g., unemployment rate). -- Advantages: --- Reflect the current stage of the business cycle due to reliance on observable data. --- Easier to implement than structural models. -- Limitations: --- Do not explain the economic reasons for defaults. --- Assume defaults are a total surprise, which is inconsistent with the warning signals typically seen, such as credit rating downgrades.

Key Differences Between Structural and Reduced-Form Models -- Structural models focus on understanding the reasons for default based on asset-liability structure, whereas reduced-form models emphasize the timing of defaults without explaining why they occur. -- Structural models rely on difficult-to-measure inputs (e.g., asset values), while reduced-form models use observable data that reflect current economic conditions. -- Structural models incorporate economic reasoning but are harder to implement, whereas reduced-form models are easier to apply but make unrealistic assumptions about the randomness of defaults.

✅ Short’s Statement Summary: "The probability of default used in my model reflects both the likely timing of a default and the uncertainty over the timing." This is correct — and here’s why: 🔍 Reduced-Form Models: Key Characteristics Default is treated as a random, exogenous event — not something determined by firm value (unlike structural models). They do not explain why default happens. Instead, they model when default might occur — statistically — using a hazard rate or intensity function. 🧠 What is the Hazard Rate? It’s the instantaneous probability that default will happen at a certain time given it hasn’t happened yet. Think of it as: “What are the chances default occurs now, given survival so far?” This allows the model to capture: The likely timing of default (e.g., Year 2 more likely than Year 1). Uncertainty in timing (i.e., even if Year 2 is most likely, default could still happen earlier or later). 🧪 What the Curriculum Says: From the image and direct curriculum quote: "Reduced-form models aim to explain statistically when [default occurs]... modeled using a Poisson stochastic process." So yes — they capture: The probability of default at each time point (timing), And the randomness/uncertainty (via stochastic processes). 🎯 So Why Is the Confusion Happening? Some interpret “default is a surprise” as meaning you can’t predict timing at all. But in context, “surprise” means: The model doesn’t rely on internal firm fundamentals (like debt vs. asset value) to trigger default. It’s exogenous — probabilistic, not deterministic. Still, it models the distribution of timing, so you can say: Default is likely in Year 2 (timing), But not guaranteed (uncertainty). ✅ Final Verdict: Short’s statement is fully consistent with the curriculum and reduced-form theory: ✔️ It reflects both the likely timing (hazard rate function) ✔️ And the uncertainty (stochastic nature of the model) Hence, Answer A (“Yes”) is correct.

Structural Models Framework 1- Structural Model Representation: Structural models use probability distributions for asset values to estimate the likelihood of default, which occurs when asset values fall below liability values. -- Balance Sheet Equation: At time T, the balance sheet equation can be expressed as: --- A(T) = D(T) + E(T). --- Where: ---- A(T): Market value of assets at time T. ---- D(T): Market value of debt at time T. ---- E(T): Market value of equity at time T. -- Equity and Debt Valuation: --- Equity value: E(T) = max[A(T) - K, 0]. --- Debt value: D(T) = A(T) - max[A(T) - K, 0]. --- Where: ---- K: Maturity value (face value) of debt. 2- Key Implications: -- Equity as a Call Option: --- Equity holders have a call option on the assets with a strike price of K (the face value of the debt). --- The value of equity cannot be negative due to limited liability. -- Debt as a Written Call Option: --- Debt holders effectively write a call option to equity holders. --- If A(T) > K, debt holders receive the face value (K), while equity holders receive the residual value [A(T) - K]. --- If A(T) < K, debt holders receive the total asset value (A(T)), as equity value becomes zero. 3- Alternative Representation with Put-Call Parity: Debt and equity can also be represented using a combination of long positions and options. -- Equity: --- E(T) = A(T) - K + max[K - A(T), 0]. --- Represents a long position in assets, a long put option, and a short bond. -- Debt: --- D(T) = K - max[K - A(T), 0]. --- Represents a combination of a long bond and a short put option written by shareholders. 4- Use Cases and Limitations: -- Structural Models: --- Require detailed information about assets and liabilities, often accessible only to internal management, bankers, or credit rating agencies. --- Best suited for internal risk management, bank credit risk measures, and credit ratings. -- Reduced-Form Models: --- Utilize publicly available data, making them more accessible. --- Focus on modeling default timing rather than understanding default causes.

Structural models use equity price volatility, which is estimated from historical data, as an input.

Comparison of Structural and Reduced-Form Models 1- Structural Models -- Advantages: --- Provide insight into the nature of credit risk. -- Disadvantages: --- Difficult to implement. 2- Reduced-Form Models -- Advantages: --- Able to use observable variables, including historical data. --- Examples include company-specific variables and macroeconomic variables. -- Disadvantages: --- Do not explain the economic reasons for default. --- Assume defaults come as a surprise.

Put-Call Parity Combinations 1- Standard Put-Call Parity Relationship: -- Formula: Stock + Put option = Bond + Call option. 2- Rearranging for the Call Option: -- Formula: Call option = Stock - Bond + Put option. 3- Rearranging for the Put Option: -- Formula: Put option = Bond + Call option - Stock. 4- Rearranging for the Stock: -- Formula: Stock = Bond + Call option - Put option. 5- Rearranging for the Bond: -- Formula: Bond = Stock + Put option - Call option.

Application of Put-Call Parity to Equity and Debt 1- Formula for Equity: -- Equity is analogous to a call option on the company’s assets, where the strike price equals the value of debt. -- Formula: Equity = max [A(T) - K, 0]. --- Where: ---- A(T): Value of the company's assets at time T. ---- K: Face value of debt at maturity. Alternatively, using put-call parity: -- Formula: Equity = A(T) - K + max [K - A(T), 0]. --- This represents equity as the residual value of assets after accounting for debt, plus the protective put option for equity holders in case the asset value falls below debt. 2- Formula for Debt: -- Debt is a combination of a short put option written by shareholders and a bond. -- Formula: Debt = K - max [K - A(T), 0]. --- This represents debt as the face value (K) minus the short put option that accounts for the possibility of default.

Valuing Risky Bonds in an Arbitrage-Free Framework 1- Calculate the Discount Factors from the Par Curve -- Use the bond prices and coupon payments of benchmark government bonds to derive the discount factors for each maturity. -- Formula for Year 1: "100 = (100 + coupon1) × DF1 → DF1 = 100 ÷ (100 + coupon1)" -- For subsequent years, solve iteratively: --- Formula for Year 2: "100 = (coupon2 × DF1) + [(100 + coupon2) × DF2]" --- Formula for Year 3: "100 = (coupon3 × DF1) + (coupon3 × DF2) + [(100 + coupon3) × DF3]". 2- Derive Spot and Forward Rates -- From the discount factors, calculate: --- Spot Rate (st): "st = [(1 ÷ DFt)^(1/t)] - 1". --- Forward Rate (ft): "ft = DFt ÷ DFt+1 - 1". 3- Create a Binomial Interest Rate Tree -- Use 1-year forward rates as the starting point. Adjust for a specified level of interest rate volatility. -- At each node, rates can increase or decrease with equal probability. --- For example: "f1u = f1 + volatility" and "f1d = f1 - volatility". -- Calibrate the tree so it accurately reflects bond values under the risk-free assumption. -- Note: The process of creating an interest rate tree is iterative and requires a computer program, so you will not be required to compute the rates in the binomial tree on the exam. 4- Calculate Bond Values Under the Binomial Tree -- Use backward induction to calculate bond values at each node, factoring in coupon payments and the expected value at the end of the period. -- Formula for Node Value (V): --- At time T: "V = 100 + coupon". --- At time T-1: "V = 0.5 × [(VH ÷ (1 + f1u)) + (VL ÷ (1 + f1d))] + coupon". 5- Estimate the Expected Exposure -- Expected exposure at each node depends on: --- Bond value. --- Probability of reaching the node. --- Coupon payment. -- Formula: "Expected Exposure = (Probability1 × NodeValue1) + (Probability2 × NodeValue2) + Coupon". 6- Calculate the CVA (Credit Valuation Adjustment) -- Incorporate probabilities of survival (POS) and hazard rates: --- POS Year t: "POS_BYt = POS_BYt-1 - (POS_BYt-1 × Hazard Rate)". -- Use the following formula for CVA: --- CVA = ∑[DFt × (Exposure × Loss Severity × (POS_BYt × Hazard Rate))]. 7- Determine the Fair Value of the Risky Bond -- Subtract the CVA from the value assuming no default risk (VND). -- Formula: "Fair Value = VND - CVA".

Value Risky Floating-Rate Notes 1- Coupon Variability -- Coupons would vary at each node based on changes in the index rate. 2- Coupon Rate Determination -- Coupon rates are based on the index rate at the previous node. --- Interest rates are set at the beginning of the period. 3- Relative Stability of Bond Values -- Bond values remain relatively stable as coupon rates and discount rates move in the same direction. 4- Discounting of Payments -- Higher coupon payments are discounted at higher interest rates, maintaining the stability of the note's value.

If a binomial tree has been correctly calibrated, changing the interest rate volatility normally will have no impact on a corporate bond's VND and only a minimal impact on its CVA and fair value. The volatility assumption is more important for bonds with embedded options, such as callable bonds.

Relationships Between Par, Spot, and Forward Rates 1- From Par Rates to Discount Factors -- Use the par rates and bond prices to calculate discount factors (DF) for each maturity. -- Formula for Year 1: "Price = (Par + Coupon1) × DF1 → DF1 = Price ÷ (Par + Coupon1)". -- Formula for Year 2: "Price = (Coupon2 × DF1) + [(Par + Coupon2) × DF2] → Solve for DF2". -- Formula for Year 3: "Price = (Coupon3 × DF1) + (Coupon3 × DF2) + [(Par + Coupon3) × DF3] → Solve for DF3". 2- From Discount Factors to Spot Rates -- Spot rates (st) can be derived from the discount factors. -- Formula for Year t: "st = [(1 ÷ DFt)^(1/t)] - 1". --- Year 1: "s1 = [(1 ÷ DF1)] - 1". --- Year 2: "s2 = [(1 ÷ DF2)^(1/2)] - 1". --- Year 3: "s3 = [(1 ÷ DF3)^(1/3)] - 1". 3- From Discount Factors to Forward Rates -- Forward rates (ft) can be calculated from discount factors. -- Formula for Year t: "ft = (DFt ÷ DFt+1) - 1". --- Between Year 1 and Year 2: "f1 = DF1 ÷ DF2 - 1". --- Between Year 2 and Year 3: "f2 = DF2 ÷ DF3 - 1". 4- From Spot Rates to Discount Factors -- Discount factors can be calculated directly from spot rates. -- Formula for Year t: "DFt = 1 ÷ [(1 + st)^t]". --- Year 1: "DF1 = 1 ÷ (1 + s1)". --- Year 2: "DF2 = 1 ÷ [(1 + s2)^2]". --- Year 3: "DF3 = 1 ÷ [(1 + s3)^3]". 5- From Forward Rates to Spot Rates -- Combine forward rates to calculate spot rates. -- Formula: "(1 + st)^t = [(1 + f1) × (1 + f2) × ... × (1 + ft)]". --- Example for Year 3: "(1 + s3)^3 = (1 + f1) × (1 + f2) × (1 + f3)". 6- From Forward Rates to Discount Factors -- Discount factors can also be derived using forward rates. -- Formula for Year t: "DFt = DFt-1 ÷ (1 + ft-1)". --- Example: "DF2 = DF1 ÷ (1 + f1)". 7- From Par Rates to Spot Rates -- Use the iterative relationship between par rates and spot rates. -- Formula for Year t: "Price = (Coupon × ∑DFt) + (Par × DFt) → Use to solve for st via DFt".

Credit Spreads and Yield Components 1- Components of a Corporate Bond's Yield -- Corporate bond yields consist of a benchmark yield and a credit spread. --- Benchmark Yield Components: ---- Expected real interest rate. ---- Expected inflation. ---- Compensation for inflation uncertainty. --- Credit Spread Components: ---- Expected losses on defaults. ---- Liquidity premiums. ---- Taxation effects (if applicable). ---- Compensation for the uncertainty of default losses. 2- Credit Valuation Adjustment (CVA) -- CVA is a widely used method to model counterparty credit risk, funding costs, liquidity, and taxation effects in derivative valuation. 3- Implied Default Probabilities -- Using an observed credit spread and an assumed recovery rate, implied default probabilities can be calculated through trial and error. -- Implied default probabilities are typically higher than historical default rates, especially for investment-grade bonds, because: --- Investors also factor in credit migration (downgrades) that increase spreads and reduce valuations. --- Investors demand compensation for differences in liquidity and tax treatment. 4- Relationship Between Recovery Rates, Default Probabilities, and Credit Spreads -- All else equal: --- Higher default probabilities lead to higher credit spreads. --- Lower recovery rate assumptions lead to higher credit spreads. 5- Notching and Credit Spreads -- Issuers may employ notching to enhance bond ratings and reduce credit spreads.

Term Structure of Credit Spreads 1- Definition and Usefulness: The term structure of credit spreads represents an issuer's credit spread over the benchmark curve across different maturities. -- It is based on the yield curve, which is the series of rates on government bonds of various maturities. -- Applications: --- Issuers use it to optimize financing decisions. --- Investors utilize it for trading decisions. 2- Drivers of the Term Structure: -- Credit Quality: --- High-quality bonds have flat or slightly upward-sloping credit term structures due to limited potential for credit migration. --- Lower-quality bonds exhibit a steeper positive slope because of greater sensitivity to the credit cycle. -- Financial Conditions: --- Credit spreads narrow during periods of economic growth (lower defaults). --- They widen during economic downturns as default risks increase. -- Market Supply and Demand: --- Greater liquidity narrows credit spreads. --- However, if new debt issuance increases perceived credit risk, spreads may widen. -- Company-Specific Factors: --- Fundamentals such as leverage ratios and cash flows influence credit spreads. --- Increased equity volatility typically puts upward pressure on spreads. 3- Credit Spread Measurement: -- Credit spreads are measured relative to an appropriate benchmark rate, typically government bond yields. -- Linear interpolation may be required for corporate bonds with intermediate maturities. 4- Analysis Considerations: -- Bonds included in the analysis should have similar credit characteristics, such as unsecured general obligations without embedded options. -- The shape of the credit curve reflects market expectations of default risk over various horizons: --- An upward-sloping curve indicates increasing long-term default risk. --- A downward-sloping curve may occur for high-yield issuers when credit quality is expected to improve.

Key Takeaways -- Bonds likely to default trade closer to recovery rates rather than based on credit spreads. -- The credit curve's slope provides insights into investor expectations for default risks and issuer credit quality over time.

Securitized and Covered Debt 1- Securitized Debt: Definition and Benefits -- Issuers finance specific assets or receivables by issuing securitized debt, which is backed by the underlying assets as collateral. -- Benefits for issuers: --- Increases debt capacity. --- Reduces financing costs. -- Collateral examples: Auto loans, credit card receivables, and mortgages. 2- Investor Advantages -- Securitized debt often offers higher returns compared to securities with similar ratings, providing incentives for investors to accept the added complexity. 3- Structured Finance Products -- Classification: --- Homogeneous products have similar characteristics across individual obligations (e.g., auto loans). --- Heterogeneous structures require individual asset analysis. -- Granularity: --- High granularity portfolios (many creditors): Summary statistics suffice for analysis. --- Low granularity portfolios (few creditors): Require detailed, loan-by-loan credit analysis. 4- Servicing and Origination -- Investors depend on the originator/servicer to: --- Maintain asset standards. --- Address delinquencies. --- Collect cash flows to service ABS debt. --- Maintainning proper documentation 5- Secured Debt Structure -- Key features: --- Obligors are often Special Purpose Entities (SPEs), which are bankruptcy-remote from the originator. --- Structural enhancements: ---- Over-collateralization. ---- Credit tranching (tiering cash flows to match investors' risk preferences). -- Early payment triggers may be used if servicers fail to make timely deposits. 6- Covered Bonds -- Origin: Dating back to 18th-century Germany, these bonds give investors dual recourse: --- Claim on the originating issuer. --- Claim on the underlying asset pool. -- Benefits: Covered bonds often receive higher ratings than the issuer due to dual recourse. -- Redemption Regimes: --- Hard-Bullet: Immediate default and payment acceleration upon missed payments. --- Soft-Bullet: Default is delayed, with a new maturity date typically a year later. --- Conditional Pass-Through: Converts to pass-through securities after the original maturity if payments are missed.

Key Takeaways -- Securitized debt and covered bonds provide investors with higher yields and varying levels of security through structured enhancements or dual recourse protection. -- The choice of structure (granularity, tranching, and redemption regimes) determines the level of complexity and credit risk for both issuers and investors.

Bonds with higher credit ratings are less likely to experience either upgrades or downgrades over a one-year period.

6.5 Credit Default Swaps

-- Describe credit default swaps (CDS), single-name and index CDS, and the parameters that define a given CDS product. -- Describe credit events and settlement protocols with respect to CDS. -- Explain the principles underlying and factors that influence the market’s pricing of CDS. -- Describe the use of CDS to manage credit exposures and to express views regarding changes in the shape and/or level of the credit curve. -- Describe the use of CDS to take advantage of valuation disparities among separate markets, such as bonds, loans, equities, and equity-linked instruments.

Credit Default Swaps (CDS) 1- Definition: A CDS is a financial derivative that provides protection against default or market-perceived credit risk, similar to insurance against credit events. The underlying of a CDS is the credit quality of the borrower, and the contract transfers credit risk from the protection buyer to the protection seller. 2- Contract Structure: -- The credit protection buyer: --- Pays periodic premiums (like insurance premiums). --- Receives compensation for losses if a pre-defined credit event occurs. -- The credit protection seller: --- Receives periodic premiums. --- Promises to compensate the buyer for credit losses if the specified event occurs. 3- Purpose of CDS: -- CDS are most commonly used to hedge credit risk, but they also serve other purposes: -- Managing portfolio leverage. -- Accessing maturities unavailable in the bond market. -- Obtaining credit risk exposure without taking on interest rate risk. -- Enhancing portfolio liquidity compared to holding corporate bonds. 4- Structure and Analogy: -- CDS contracts are structured similarly to put options: --- The buyer pays premiums and expects compensation if a credit event occurs. --- The seller accepts premiums and bears the obligation to compensate the buyer in case of a credit event. 5- Key Characteristics: -- CDS contracts require clear definitions of credit events (e.g., default) to avoid disputes. -- Industry bodies like ISDA have worked to standardize contract language, but ambiguities and disagreements can still arise. 6- Applications: -- Most CDS contracts cover corporate debt but may also include: --- Sovereign government debt. --- Municipal bonds. --- Portfolios of mortgages or other debt securities. 7- Risks: -- CDS contracts do not fully eliminate credit risk because: --- A reference entity might default in a way that does not qualify as a default under the CDS agreement. --- The protection seller (typically large financial institutions) could default on their commitment to provide compensation.

Key Takeaways: -- CDS contracts transfer credit risk from buyer to seller but involve risks related to definitions of credit events and the financial health of the seller. -- They are primarily used for hedging and liquidity purposes, offering an alternative to direct exposure to corporate or sovereign debt.

CDS transactions are executed in the over-the-counter (OTC) market. Information about CDS trades is reported to the Depository Trust and Clearinghouse Corporation (DTCC), which is based in the United States and provides services such as post-trade clearing and settlement. Details of CDS trades are maintained in the DTCC's Trade Information Warehouse. The International Swaps and Derivatives Association's Determinations Committee is responsible for determining whether a credit event occurs.

Types of CDS 1- Single-name CDS: The most common type of CDS, accounting for 60% of the credit derivatives market. -- Reference Obligation: Covers a specific debt obligation issued by a single reference entity (e.g., a corporation). --- Typically, the reference obligation is a senior unsecured debt issue, but other debts with the same or higher priority are also covered. -- Cheapest-to-Deliver Obligation: Compensation to the credit protection buyer is based on the lowest-cost debt with the same seniority as the reference obligation. --- Example: If a two-year senior unsecured bond trades at 30% of par, it can be delivered for compensation, even if the reference obligation is a 10-year senior unsecured bond trading at 40% of par. 2- Index CDS: A contract covering a portfolio of reference obligations. -- Purpose: Allows investors to take credit risk exposure to a combination of borrowers. -- Correlation Impact: The cost of credit protection increases with higher perceived correlation among issuers in the index. -- More details about these contracts are typically covered in advanced discussions. 3- Tranche CDS: A specialized form of index CDS. -- Structure: Allows investors to limit credit protection to a pre-specified level of losses within the portfolio. -- Market Share: Represents a relatively small portion of the CDS market compared to single-name and index CDS.

CDS Contract Terms 1- Notional Amount: Represents the amount of credit protection under the CDS. -- This amount may be up to the full issue size for a single contract. -- The total notional amount across all contracts can exceed the issue size of the reference obligation. 2- Period of Coverage: Specifies the CDS duration. -- Maturities typically range from 1 to 10 years. 3- CDS Spread: The regular premium paid by the credit protection buyer to the seller. -- Analogous to the credit spread on a bond. -- Standardized coupon rates: --- 1- 1% for investment-grade companies. --- 2- 5% for high-yield companies. -- Adjusting CDS Spread with Upfront Payments: --- If the actual credit spread exceeds the standard rate, the buyer pays an upfront premium to the seller. --- If the actual spread is less, the seller pays an upfront premium to the buyer.

Key Takeaways -- ISDA Master Agreements standardize CDS contract terms. -- CDS spreads reflect credit risk and are adjusted using upfront payments for specific cases.

CDS Market Dynamics 1- Changes in CDS Market Value: -- The market value of a CDS fluctuates with the reference entity's credit quality. -- If the reference entity’s credit quality decreases: --- The credit protection buyer gains. --- The credit protection seller loses. 2- Terminology for CDS Positions: -- Unlike other markets, CDS positions require specific terminology due to their structure. -- Descriptions differ for single-name CDS and CDS index contracts: -- 1- Single-Name CDS: --- The credit protection buyer has short credit exposure (benefits from a deterioration in credit quality). --- The credit protection seller has long credit exposure (suffers losses if credit quality worsens). -- 2- CDS Index: --- The credit protection buyer has long credit exposure (gains from improvements in credit quality). --- The credit protection seller has short credit exposure (loses if credit quality deteriorates). 3- Avoiding Confusion: -- To prevent misunderstandings, it is recommended to consistently use the terms credit protection buyer and credit protection seller rather than "long" or "short."

Key Takeaways -- The value of a CDS is inversely related to the reference entity's credit quality. -- Position terminology differs between single-name CDS and CDS index contracts, emphasizing the importance of clear labeling.

Credit and Succession Events 1- Types of Credit Events: Payments from credit protection sellers are triggered by predefined credit events. There are three broad categories: -- Bankruptcy: --- A legal procedure where creditors are forced to defer claims while the defaulting company works with creditors and courts to develop a repayment plan. --- Bankruptcy may lead to liquidation or reorganization. -- Failure to Pay: --- Occurs when the borrower misses a principal or interest payment. --- A missed payment is not considered a credit event if it is made within the specified grace period. -- Restructuring: --- For restructuring to qualify as a credit event, it must be forced by creditors. --- This is a less common credit event in the United States, where bankruptcy is typically preferred. 2- Defining Credit Events in CDS Contracts: -- Credit events should be clearly specified in the CDS contract terms to avoid ambiguity. -- Final decisions on whether a credit event has occurred are made by the ISDA's Declaration Committee. 3- Succession Events: -- Succession events, such as mergers, divestitures, and spinoffs, may require modifications to existing CDS contracts. -- These decisions are also made by the ISDA's Declaration Committee.

Key Takeaways -- Credit events include bankruptcy, failure to pay, and restructuring, and they must be clearly defined in CDS contracts. -- Succession events may necessitate adjustments to CDS terms, with the ISDA Declaration Committee serving as the deciding authority.

Settlement Protocols for CDS 1- Overview: Credit Default Swaps (CDS) allow for two types of settlement when a credit event occurs—physical settlement and cash settlement. These are triggered 30 days after a credit event is declared by ISDA's Determination Committee. 2- Physical Settlement: -- The credit protection buyer delivers debt securities to the credit protection seller in exchange for their full face value. -- This type of settlement is less common. 3- Cash Settlement: -- The credit protection seller pays a cash amount equal to the loss suffered by the credit protection buyer. -- This approach allows credit protection buyers to receive payouts quickly after a credit event is declared, bypassing the lengthy bankruptcy process that actual lenders may face. 4- Payout from a CDS Contract: -- The payout is based on the Loss Given Default (LGD), which is the complement of the recovery rate. -- Formula for LGD: "LGD = 1 - Recovery rate". -- Payout Amount Formula: "Payout amount = LGD × Notional amount". 5- Importance of Cheapest-to-Deliver (CTD) Bond in Cash Settlements: -- In a CDS auction, bids from banks and dealers determine the recovery rate using the CTD bond. -- Investors holding bonds trading above the CTD bond will benefit more from cash settlement. -- Conversely, holders of the CTD bond remain indifferent to whether the settlement is physical or cash.

Key Takeaways: -- Settlement options provide flexibility for protection buyers. -- Cash settlement typically accelerates the payout process for credit protection buyers.

CDS Index Products 1- Overview -- CDS indexes provide credit protection for a portfolio of reference obligations. They trigger payouts when any of the underlying obligations experiences a credit event. 2- Structure and Weighting -- CDS indexes are equally weighted. If an index has 100 components, a single default leads to a settlement equal to 1% of the notional value. -- After a default, the defaulted component is removed from the index, and the notional amount is reduced accordingly. 3- Classification -- Index CDS are categorized by geographic region and credit quality. -- 1- Geographic regions include North America, Europe, and Asia-Pacific. -- 2- Credit quality classifications include investment-grade and high-yield. 4- Updates and Liquidity -- The latest iteration of an index is referred to as the on-the-run series. Older versions are known as off-the-run. -- On-the-run CDS indexes are typically more liquid, with five major indexes accounting for more than 90% of CDS market activity. 5- Rolling Feature -- Investors can transition ("roll") from an off-the-run series to the current on-the-run series when a new series is introduced.

Key Takeaways -- CDS indexes provide an efficient means of gaining exposure to or hedging against credit risk across a diversified portfolio. -- Their equal weighting and liquidity make them more accessible and practical than single-name CDS for many market participants.

Market Characteristics 1- Risk Exposure in Banking -- Banks primarily face two types of risk: interest rate risk and credit risk. -- 1- Interest rate risk is commonly managed through duration-based strategies. -- 2- Credit risk has traditionally been addressed with internal methods such as credit limits, collateral requirements, and ongoing monitoring. 2- Emergence of CDS Contracts -- Credit default swaps (CDS) were introduced in the mid-1990s, enabling banks to transfer credit risk to other parties. -- This innovation allowed banks to significantly expand lending activity, contributing to economic growth. -- CDS contracts have proven particularly effective in the bond market due to the standardized terms of bonds. 3- Market Evolution -- Initially, CDS contracts became the most popular type of credit derivative among investors. -- By December 2007, the gross notional value of CDS contracts peaked at nearly $60 trillion. -- Following the 2008 financial crisis, during which institutions like AIG required bailouts, the market contracted significantly. -- By June 2012, the gross notional value of CDS contracts had fallen to less than half of the 2007 peak and continued declining, reaching $7.9 trillion by December 2019, with a market value of $199 billion. 4- Regulatory Changes -- The CDS market has seen increased regulation since the 2008 financial crisis. -- 1- As of 2010, only approximately 10% of CDS contracts were centrally cleared. -- 2- Today, most CDS transactions are processed through authorized clearinghouses that facilitate settlement, collect payments, and enforce margin requirements.

Key Takeaways -- CDS contracts revolutionized risk management by enabling the transfer of credit risk, thereby facilitating increased lending and economic growth. -- Despite their initial popularity, the CDS market has contracted and become more regulated following the 2008 financial crisis. -- Modern regulatory frameworks, including central clearing and margin requirements, have enhanced the stability of the CDS market.

Settlement of CDS Contracts 1- Challenges in the Recovery Process -- The actual recovery process for defaulted debt can take a long time, delaying settlements. -- CDS parties prefer not to wait for the resolution of the recovery process to settle the contract. 2- Auction-Based Settlement -- A payout ratio is determined through an auction process conducted by major banks and dealers. -- The auction identifies the cheapest-to-deliver defaulted debt. 3- Agreement Among CDS Parties -- All CDS parties agree to accept the payout ratio determined by the auction, ensuring a quicker settlement process.

Key Takeaways -- The auction-based process expedites CDS settlements by bypassing the potentially lengthy recovery process. -- The use of the cheapest-to-deliver obligation aligns with market standards and facilitates efficient settlement.

Markit Indices and CDS Index Features 1- Classification of Markit Indices -- Markit indices are classified by: --- Region: Geographical distinctions. --- Credit Quality: Two primary categories. ---- Investment-grade: Quoted with spreads. ---- High-yield: Quoted with prices. -- Both categories have standardized coupons and equally weight the entities in the index. 2- Series Updates -- A new series is introduced for each index every six months. --- On-the-run series: The latest series. --- Off-the-run series: Older series. -- Investors can roll their positions from the off-the-run series to the on-the-run series to maintain exposure to the latest index. 3- Default Treatment in Indices -- When an entity in the index defaults: --- It is removed from the index. --- It is settled like a single-name CDS, with the settlement based on its proportion in the index.

Components of a CDS Contract 1- Protection Leg -- Definition: The contingent payments made by the protection seller if a credit event occurs. -- Value: Equal to the sum of the present value of expected losses for the reference obligation. -- Relation to CVA: Represents the amount that eliminates the reference obligation’s credit risk. 2- Premium Leg -- Definition: The series of periodic payments made by the protection buyer. -- Value: Equal to the sum of the present value of these payments, discounted at the risk-free rate. 3- Upfront Payment Calculation From the perspective of the protection buyer, the upfront payment is calculated as: Formula: "Upfront payment = PV of protection leg − PV of premium leg". -- Positive Value: Paid by the credit protection buyer. -- Negative Value: Paid by the credit protection seller.

Key Takeaway: -- The protection leg mitigates credit risk, while the premium leg reflects the cost of this protection. Upfront payments balance these components at the initiation of a CDS contract.

The Credit Curve 1- Definition The credit curve represents the range of credit spreads over various maturities. It is analogous to the term structure of interest rates, showing how credit spreads change with time. -- A positively sloped credit curve implies an increasing probability of default in later years. 2- CDS Spread Approximation The CDS credit spread reflects the return investors demand for taking on credit risk. It can be approximated using the formula: Formula: "CDS spread ≈ Loss given default × Probability of default". Since Loss given default (LGD) = (1 − Recovery rate), the formula becomes: Formula: "CDS spread ≈ (1 − Recovery rate) × Probability of default". 3- Market Reference Rates -- Historically, the London Interbank Offered Rate (Libor) served as the reference rate for CDS contracts. -- As Libor is phased out, market reference rates are based on what commercial banks charge each other for interbank loans and may include some credit risk.

CDS Pricing Conventions 1- Overview CDS spreads are preferred over prices because they are easier to compare across investments. CDS spreads are standardized at: -- 1% for investment-grade companies. -- 5% for high-yield companies. The present value of a CDS credit spread comprises two components: Formula: "Present value of credit spread = Upfront premium + Present value of fixed coupon". 2- Upfront Premium Calculation The fixed coupon refers to the periodic premium paid by the credit protection buyer, which differs from coupon payments made by bond issuers. The upfront premium is calculated using the following approximation: Formula: "Upfront premium ≈ (Credit spread − Fixed coupon) × CDS Duration". -- CDS Duration represents the CDS's effective duration, reflecting contingent coupon payments similar to bonds with embedded options. 3- Percentage Upfront Premium The upfront premium can also be expressed as a percentage of par value: Formula: "Upfront premium (%) = 100 − CDS price per 100 par". 4- Credit Spread Approximation The credit spread in percentage terms can be derived by rearranging the upfront premium formula: Formula: "Credit spread ≈ (Upfront premium ÷ Duration) + Fixed coupon".

Key Takeaways: -- The upfront premium accounts for differences between the credit spread and fixed coupon. -- CDS spreads simplify comparisons between contracts of varying maturities and risk levels.

Valuation Changes in CDS During Their Lives 1- Overview The value of a CDS contract fluctuates throughout its life due to factors such as: -- Duration: Time to maturity and sensitivity to changes in spread. -- Default probability: The likelihood of the reference entity defaulting. -- Expected loss given default: The percentage of exposure not recoverable in the event of default. 2- Formula for Valuation Changes From the protection buyer’s perspective, the change in the value of a CDS is calculated as: Formula: "% Change in CDS price = Change in spread in bps × Duration".

Key Takeaways: -- A larger duration amplifies the impact of spread changes on the CDS price. -- Positive spread changes increase the value of the CDS for the protection buyer, while negative spread changes reduce it.

Monetizing Gains and Losses 1- Monetization Process -- Gains or losses from a CDS contract can be realized by entering into offsetting contracts. -- The buyer of the original CDS becomes the seller of the new offsetting CDS. -- The offsetting CDS does not require the same counterparty as the original contract. 2- Outcomes Based on Credit Events -- No Credit Event: --- The CDS seller gains as no payout is made. --- The CDS buyer loses, having paid premiums. --- The CDS buyer benefits by having protection over the CDS term despite the loss.

Key Takeaway: -- Offsetting contracts allow CDS participants to manage or realize the value of their positions without waiting for the contract to expire.

Example: Premiums and Credit Spreads 1- Concept of Credit Spread The credit spread reflects the compensation investors demand for taking on credit risk. It is calculated as: "Credit spread = (Upfront premium ÷ Duration) + Fixed coupon." In the given example: -- Upfront premium = -3% (paid by the credit protection seller). -- Duration = 5 years. -- Fixed coupon = 1% (for an investment-grade issuer). Calculation: Credit spread = (-3 ÷ 5) + 1 = -0.6% + 1% = 0.4%. This means the CDS contract reflects a 0.4% credit spread, and the seller pays an upfront premium because the fixed coupon exceeds the actual credit spread. 2- Why Does the Seller Pay an Upfront Premium? -- In the CDS market, fixed coupons for investment-grade companies are standardized at 1%. -- When the issuer's credit spread is less than 1%, the credit protection seller compensates the buyer for the difference by paying an upfront premium. Explanation in Visuals: -- The buyer essentially "overpays" by paying a higher fixed coupon (1%) than the credit spread (0.4%). -- The seller offsets this overpayment by paying the upfront premium at the start of the contract. 3- If the Issuer’s Credit Spread Exceeds 1% -- The credit protection buyer would make an upfront payment to the seller to account for the spread exceeding the fixed 1% coupon rate. -- Example: If the credit spread were 1.5%, the buyer would pay an upfront premium to balance the difference.

Key Takeaways -- Credit spread is derived from both upfront premium and fixed coupon, using the formula: "Credit spread = (Upfront premium ÷ Duration) + Fixed coupon." -- The direction of the upfront premium depends on whether the issuer’s credit spread is greater or less than the standardized fixed coupon (1% for investment-grade issuers). -- Upfront premiums ensure fair pricing of CDS contracts when the fixed coupon does not match the issuer’s actual credit spread.

Change in Value of a CDS 1- Formula for Change in CDS Value The change in the value of a CDS can be estimated using the following formula: "% Change in CDS price ≈ Change in spread in bps × Duration." -- Change in spread in bps: Reflects the movement in the CDS credit spread (basis points). -- Duration: Represents the sensitivity of the CDS value to changes in credit spreads, similar to the modified duration concept in bonds. 2- Comparison to Bond Pricing The formula for CDS value changes is analogous to the price sensitivity of bonds: "% Change in the price of a bond ≈ Change in interest rates × Modified duration." -- Parallel Concepts: -- The change in spread in bps in CDS pricing corresponds to the change in interest rates in bond pricing. -- CDS duration serves a similar role as the bond's modified duration.

Managing Credit Exposures 1- Reducing Credit Risk Exposure -- Lenders often prefer CDS to reduce credit exposure instead of selling loans, especially for short risk horizons. -- Selling loans may not be practical or optimal, making CDS a more flexible solution. 2- Role of CDS Sellers -- Dealers are the primary sellers of CDS contracts, aiming to profit by providing credit protection. -- Dealers manage the credit risk from selling CDS contracts by entering offsetting positions, often with other dealers. -- Advanced risk models are used by dealers to balance their exposure, although this is outside the CFA curriculum. 3- Naked CDS Positions -- A naked CDS position occurs when a party purchases credit protection without holding the reference obligation. -- These positions represent a speculative bet on the deterioration of the reference entity's credit quality. -- Regulatory Concerns: --- Europe: Trading naked CDS on sovereign debt has been banned following the 2008 crisis. --- Supporters: Argue naked CDS enhances market liquidity and draws comparisons to short stock sales or long put options. 4- Common CDS Trading Strategies -- Long/Short Trading: --- A strategy involving a long position in one CDS and a short position in another. --- Objective: Profit from relative changes in credit quality between reference entities. -- Curve Trading: --- Involves buying a CDS for one maturity and selling a CDS with a different maturity for the same reference entity. --- This is a bet on credit curve changes. 5- Examples of Curve Trading If an investor believes the credit curve will steepen (long-term credit risk increases relative to short-term credit risk): -- Bullish Short-Term vs. Bearish Long-Term: -- 1- Buy a long-term single-name CDS. -- 2- Sell a long-term CDS index. -- 3- Sell a short-term single-name CDS. -- 4- Buy a short-term CDS index. By contrast, curve-flattening trades (opposite of the above) reflect bearish short-term and bullish long-term expectations for the reference entity.

Key Takeaways -- Lenders benefit from CDS by reducing credit exposure without selling loans, while dealers profit by managing credit risk through offsetting positions. -- Naked CDS positions add liquidity but are controversial due to regulatory concerns. -- Long/Short and Curve Trading allow investors to express views on relative credit quality and credit curve changes.

Basis Trades and Exploiting Mispricing 1- Theory of Credit Risk and Yield -- In theory, the credit spread of a bond should equal the credit spread of a CDS. -- Investors may identify mispricing using advanced models (outside the CFA curriculum scope). 2- Basis Trades -- Basis trades exploit differences in credit spreads between the bond market and CDS market. 3- Negative Basis -- A negative basis occurs when the bond market implies a higher credit risk premium than the CDS market. -- Strategy: --- 1- Purchase credit protection using a CDS. --- 2- Take a long position in the underlying bond. -- Objective: Profit from the eventual convergence of credit risk premiums. 4- CDS Index Arbitrage -- Arbitrage opportunities exist if the cost of assembling the components of a CDS index is lower than the cost of the index itself.

Key Takeaways -- Basis trades allow investors to profit from temporary mispricing between the bond and CDS markets. -- Negative basis trades combine long bond positions with CDS protection purchases. -- Arbitrage can be achieved if the sum of individual CDS components is less than the CDS index.

[Example of a Basis Trade 1- Given Data -- Bond yield = 7.00% -- Bond maturity = 5 years -- CDS spread (5-year) = 2.75% -- Investor’s cost of borrowing = 4.00% 2- Step-by-Step Calculation **a) Yield Spread on the Bond** Formula: "Yield spread = Bond yield − Cost of borrowing" Calculation: "Yield spread = 7.00% − 4.00% = 3.00%" **b) Net Basis (Hedged Yield)** Formula: "Net basis = Yield spread − CDS spread" Calculation: "Net basis = 3.00% − 2.75% = 0.25%" 3- Investor Strategy -- **Buy the bond** to earn the 7.00% yield -- **Buy CDS protection** at 2.75% to hedge credit risk -- **Finance the bond** at 4.00% borrowing cost -- Result: Net hedged return = **0.25%**, with credit risk neutralized via CDS.

[Example of a Negative Basis Trade 1- Given Data -- Bond yield = 6.50% -- Cost of borrowing = 4.00% -- CDS spread = 2.75% -- Yield spread = 6.50% − 4.00% = **2.50%** 2- Implication -- The **yield spread (2.50%) is less than the CDS spread (2.75%)**. -- Result: Buying the bond and CDS protection leads to a **net loss of 0.25%**, as the cost of hedging exceeds the bond's excess return. 3- Investor Strategy for Negative Basis -- If the investor expects the negative basis to **narrow or normalize**, they can execute a **negative basis trade**: **Strategy:** -- **Short the bond** to avoid the lower yield. -- **Sell CDS protection** to earn the higher CDS spread. **Net basis:** "Net basis = CDS spread − Yield spread = 2.75% − 2.50% = 0.25%" -- The investor earns **0.25% net profit** from the convergence trade, profiting as the bond underperforms relative to the CDS.

Key Takeaways -- A positive basis (bond yield spread > CDS spread) encourages buying the bond and CDS protection. -- A negative basis (bond yield spread < CDS spread) encourages shorting the bond and selling CDS protection. -- Both strategies rely on the expectation that credit spreads will eventually converge.

Curve Trading with CDS 1- Definition -- Curve trading involves taking positions in CDS contracts with different maturities to profit from expected changes in the shape of the credit curve (relationship between credit spreads and maturities). 2- Strategies Based on Credit Curve Expectations A. If the Credit Curve is Expected to Steepen: -- This means long-term credit risk will increase relative to short-term credit risk, implying the company’s short-term outlook is better than its long-term prospects. -- Strategy: --- Take a Short Position in a Long-Term CDS: --- Sell credit protection for long-term maturities, betting that long-term spreads will widen. --- Take a Long Position in a Short-Term CDS: --- Buy credit protection for short-term maturities, betting that short-term spreads will remain stable or narrow. -- Result: Profit if long-term spreads widen more than short-term spreads. B. If the Credit Curve is Expected to Flatten: -- This means short-term credit risk will increase relative to long-term credit risk, implying the company’s short-term outlook is worse than its long-term prospects. -- Strategy: --- Take a Long Position in a Long-Term CDS: --- Buy credit protection for long-term maturities, betting that long-term spreads will remain stable or narrow. --- Take a Short Position in a Short-Term CDS: --- Sell credit protection for short-term maturities, betting that short-term spreads will widen. -- Result: Profit if short-term spreads widen more than long-term spreads.

Key Takeaways -- Steepening Trades (Bullish Short-Term Outlook): Buy short-term CDS protection and sell long-term CDS protection. -- Flattening Trades (Bearish Short-Term Outlook): Sell short-term CDS protection and buy long-term CDS protection. -- These trades rely on the investor's expectations of relative changes in credit spreads across maturities.

[Impact of a Downward Shift in the Credit Curve on CDS Positions 1- Interpreting a Downward Credit Curve Shift -- A downward shift means **credit spreads fall across all maturities**, signaling lower perceived credit risk. -- For CDS contracts: --- **Long CDS positions (buyers of protection)** lose value as spreads fall. --- **Short CDS positions (sellers of protection)** gain value by having locked in higher premiums. 2- Profit Opportunity from Short CDS -- Short CDS positions benefit from falling spreads: --- The protection seller receives a fixed premium while the cost to unwind the position drops. -- Example: --- If CDS was sold at a 5% spread and now trades at 4%, the **value of the short CDS position increases**. 3- Why CDS C Is Optimal -- The forecast indicates a **1% decline** in credit spreads across the curve. -- **CDS C has a 10-year maturity**, implying the **highest spread duration** among all CDS. -- Longer duration = **greater price sensitivity** to spread changes. -- Thus, a **short position in CDS C** offers the **largest potential gain** from the downward shift. 4- Optimal Strategy -- Take a **short position in CDS C only** to maximize profit from the anticipated spread compression. -- Avoid diluting the exposure by combining with other CDS positions, as all spreads are shifting equally.

Key Takeaways: -- A downward shift in the credit curve benefits short CDS positions because the cost of protection decreases. -- The longer the CDS's duration, the more sensitive it is to changes in spreads, offering greater profit potential from a short position.

Coverage of a CDS with Bond C as the Reference Obligation 1- Key Concept of CDS Coverage A CDS contract covers not only the referenced obligation but also other debts of the reference entity with equal or higher seniority. This ensures that protection is extended to obligations that have a similar or superior claim in case of a default. 2- Bond C's Seniority In this case, Bond C has a secured seniority ranking. -- This means that a CDS with Bond C as the reference obligation will provide protection for any debt issued by the reference entity with equal or higher seniority than Bond C. -- Bond A, which also has a senior secured ranking, qualifies for coverage under the same CDS contract. 3- Exclusion of Other Bonds -- Bond B: Excluded because it is senior unsecured and has a lower seniority ranking compared to Bond C. -- Bond D: Excluded because it is subordinated, which ranks lower than both Bond C and Bond A. Final Answer: The CDS with Bond C as the reference obligation covers Bond A and Bond C only.

Impact of Credit Curve Changes on CDS Value 1- Premia Paid in a CDS Contract -- The premium leg of a CDS contract, which is the periodic payment made by the protection buyer, is fixed at the contract's initiation. -- This means that even if the credit curve shifts downward, the periodic premiums remain unchanged. -- Therefore, changes in the credit curve do not directly affect the premia paid during the life of the CDS. 2- Effect on CDS Value -- The market value of the CDS contract is highly sensitive to changes in the credit curve. -- A downward shift in the credit curve (e.g., credit spreads narrowing by 0.2%) reflects an improvement in the credit quality of the reference entity. -- Consequently, the value of credit protection decreases because the likelihood of default has reduced, making the contract less valuable to the protection buyer. 3- Monetizing Losses with a Downward Shift -- If a protection buyer, like Denman, has taken a long CDS position (short credit exposure) and the credit curve shifts downward, the CDS becomes less valuable. -- If Denman unwinds his position by entering into an offsetting CDS contract (selling protection), he would lock in a loss. --- This is because he initially locked in a higher premium rate than what the market is now offering. 4- Summary -- Changes in the credit curve influence the value of the CDS rather than the premiums paid, which are fixed. -- A downward shift in the credit curve decreases the value of a CDS contract for the protection buyer, resulting in a loss if unwound. -- Conversely, a protection seller benefits from this change due to a reduced risk of default.

Cash Payment from CDS Counterparty After Bankruptcy Declaration 1- CDS Terms and Physical Delivery -- Denman’s CDS contract specifies physical delivery of bonds. -- In this case, Denman delivers bonds to the counterparty in exchange for their face value ($25 million). -- This eliminates the impact of market prices on the settlement, as the payment matches the notional value of the CDS contract. 2- Role of Seniority in CDS Coverage -- The CDS linked to Bond C covers Bond A and Bond C, as both are senior secured obligations of ITR. -- Bond B and Bond D are excluded, as their seniority ranking is lower than that of the reference obligation (Bond C). 3- Cheapest-to-Deliver (CTD) Obligation -- For CDS contracts with cash settlement, the payout is based on the cheapest-to-deliver obligation. -- Bond C trades at 55% of par, meaning the payout ratio is 45% (100% - 55%). -- If the CDS was cash-settled, the counterparty would have paid Denman $11.25 million (45% of $25 million). --- Denman would have retained the bonds, now worth $13.75 million (55% of par). 4- Physical Delivery Settlement -- Since Denman’s CDS specifies physical delivery, the counterparty pays $25 million in exchange for the bonds’ face value. -- This avoids determining the market value or payout ratio, simplifying the settlement.

Key Takeaway -- In a physical delivery CDS, the face value of the reference obligation determines the cash payment, independent of the market price of the bonds. -- This ensures full recovery for the protection buyer, regardless of bond prices post-default.

[CDS Case: Evaluating Scenarios for Potential Credit Events CDS Contract Terms -- CDS type: 5-year contract on BPM’s secured senior bond -- CDS spread = 3.2% -- CDS duration = 4.4 years -- Reference obligation duration = 4.1 years -- Coupon: 4.6%, payable 31 October with a **45-day grace period** Scenario Analysis 1- **Scenario 1: BPM Spins Off Assets** -- BPM divests 30% of assets and revenue to a new entity -- Result: **Not a credit event** -- May be considered a **succession event**; ISDA may split the CDS between BPM and the spin-off entity -- No missed payment or restructuring involved 2- **Scenario 2: Missed Payment and Late Recovery** -- BPM misses 31 October payment, repays on 30 November -- Result: **Not a credit event** -- Payment made **within the 45-day grace period**, so no failure-to-pay is triggered 3- **Scenario 3: Voluntary Restructuring** -- BPM and creditors agree to modify debt terms -- Result: **Potential credit event** -- A **restructuring credit event** occurs only if changes are deemed coercive by ISDA (e.g., forced haircut or maturity extension) 4- **Scenario 4: Municipal Moratorium** -- A local government suspends municipal bond issuance -- Result: **Not a credit event** -- Municipal bonds are unrelated to BPM’s corporate debt and the CDS reference obligation is unaffected Conclusion -- Only **Scenario 3** has potential to trigger a CDS credit event, contingent on ISDA’s judgment of creditor coercion. -- Scenarios 1, 2, and 4 **do not meet the criteria** for triggering a CDS payout.

Key Takeaways -- A credit event in a CDS requires specific triggers: bankruptcy, failure to pay (beyond the grace period), or forced restructuring. -- In this case: --- Scenario 2 and Scenario 3 have potential relevance, but only Scenario 3 could be deemed a credit event if restructuring is involuntary. --- Scenarios 1 and 4 are not credit events under CDS definitions. --- ISDA's Role: Determination of credit and succession events depends on their official ruling, underscoring the uncertainty inherent in CDS settlements.

[Calculating the Upfront Premium for BPM’s CDS 1- Formula Name of formula: Upfront premium formula Formula: "Upfront premium = (Credit spread − Fixed coupon) × CDS duration" 2- Given Data -- Credit spread = 3.2% -- Fixed coupon = 1.0% (standard for investment-grade CDS) -- CDS duration = 4.1 years 3- Calculation "Upfront premium = (3.2% − 1.0%) × 4.1 = 2.2% × 4.1 = 9.02%" Notes -- Use **CDS duration**, not the reference bond duration. -- The **fixed coupon** depends on credit quality: 1% for investment-grade, 5% for high-yield. -- The upfront premium is paid **once** to reconcile the difference between the fixed coupon and the market-implied spread.]

[Scenario 1: Classification of the BPM Spin-Off 1- Scenario Overview -- BPM spins off a division representing ~30% of its assets and revenues into a new entity with separate common shares. -- This introduces uncertainty about which entity is responsible for the reference obligation under existing CDS contracts. 2- Classification and Analysis **Succession Event (Correct Classification)** -- Defined as a significant change in the reference entity's structure (e.g., merger, spin-off). -- The spin-off alters BPM’s structure and creates ambiguity over CDS reference obligation responsibility. -- Result: **This is a succession event.** **Credit Event (Incorrect)** -- Applies to default-like scenarios (e.g., bankruptcy, failure to pay, restructuring). -- BPM remains solvent with no payment disruption. -- Result: **Not a credit event.** **Restructuring (Incorrect)** -- Involves coercive changes to debt terms (e.g., reduced principal, maturity extension). -- No debt terms are altered in this scenario. -- Result: **Not a restructuring.** 3- Conclusion -- Scenario 1 qualifies as a **succession event**, not a credit or restructuring event, due to the corporate spin-off's impact on the CDS reference entity structure.

[Scenario 2: Why This Is Not a Credit Event 1- Scenario Summary -- BPM misses its 4.6% annual interest payment due on 31 October. -- The payment is made **in full within the 45-day contractual grace period**. 2- Credit Event Criteria: Failure to Pay -- A **failure to pay** credit event occurs only if: --- The issuer **does not make payment within the grace period**, and --- The **full contractual amount remains overdue** after the deadline. -- In Scenario 2: --- The bond contract includes a **45-day grace period**. --- BPM makes the payment **within this window**, by 15 December. --- Result: **No credit event is triggered**. 3- Importance of Grace Periods -- The **grace period is contractually defined** and protects issuers from technical default due to minor delays. -- If no grace period were defined, the missed 31 October payment **could** be a credit event. Conclusion -- Scenario 2 **does not meet the criteria** for a failure-to-pay credit event because the payment was made **within the permitted grace period**.

Fixed Income Flashcards

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