CFA L2 Fixed Income Flashcards
(213 cards)
1
Q
Spot rate/Spot price
A
The price quoted for immediate settlement on an interest rate, commodity, security, or a currency. It is the YTM on zero-coupon bonds.
Calculation for zero-coupon: Pt = 1 ÷ (1 + S_T)^T
Calculation for coupon bond: Vt = PMT ÷ (1 + S_T)^T
- Ex: $100 bond set to mature in 1 year. No coupons or any other payments during that period. The spot rate is the PV of this bond.
- Pt = price of $1 par zero-coupon bond
- S_T= Spot rate at time T
2
Q
Spot curve
A
Yield curve for spot rates. The spot rate is on the y-axis and maturity is on the x-axis.
3
Q
Forward rate
A
An annualized interest rate for a financial transaction that is agreed upon today, but starts at another time (j), and has a maturity of k.
Calculation: F(j,k) = 1 ÷ [ 1 + f(j,k) ]^k
- f(j,k) = the forward RATE applicable on a k-year loan starting in j years.
- the forward PRICE of a $1 par zero-coupon bond maturing at time j+k delivered at time j.
4
Q
Forward rate example:
The rate for a 5 year loan is 6% and the rate for a 2 year loan is 4%. What is the forward rate for a 3 year loan beginning 2 years from now?
A
J = 2
K = 3
f(2,3) = [ (1.06)^5 ÷ (1.04)^2 ]^(1/3) - 1 = 7.35%
OR
[ (6% * 5) - (4% * 2) ÷ 3 ] = 7.33% → this is an approximation
5
Q
Yield-to-maturity (YTM)
A
The total return anticipated on a bond if the bond is held until it matures.
- For a zero-coupon bond, the spot rate = YTM. For a coupon bond, the two ARE NOT equal.
6
Q
Spot rate example:
Compute the price and YTM of a 3 year 4% annual pay, $1,000 par bond given the following spot rate curves: S1 = 5%, S2=6%, and S3 = 7%.
A
[ 40 ÷ (1 + .05)^1 ] + [ 40 ÷ (1 + .06)^2 ] + [ (1000 + 40) ÷ (1 + .07)^3 ] = $922.64
YTM can be found by using a calculator:
N = 3 ; I/Y = x ; PV = 922.64 ; PMT = 40 ; FV = $1,000
7
Q
When will the return on a bond equal a bond’s yield?
A
When the bond is HTM, when all payments (coupons and principal) are made in full and on time, and all coupons are reinvested at the original YTM.
- If the yield curve is not flat, the coupon payments will not be reinvested at the YTM and the expected return will differ from the yield.
8
Q
Forward pricing model
A
Values forward contracts based on arbitrage-free pricing. This model equates buying a long-maturity zero-coupon bond to entering into a forward contract to buy a zero-coupon bond that matures at the same time
Formula: F(j,k) = P(j+k) ÷ Pj
9
Q
Forward pricing example:
Calculate the forward price two years from now for a $1 par, zero-coupon, three-year bond given the following spot rates:
The two-year spot rate, S2 = 4%.
The five-year spot rate, S5 = 6%.
A
P(j) = P(2) = [ 1 ÷ (1 + .04)^2 ] = 0.9246
P(j+k) = P(2+3) = P(5) = [ 1 ÷ (1 + .06)^5 ] = 0.7473
F(j,k) = 0.7473 ÷ 0.9246 = 0.8082
…
So, $0.8082 is the price agreed to today, to pay in two years, for a three-year bond that will pay $1 at maturity.
10
Q
The forward rate model
A
The forward rate model tells us that the forward rate should make investors indifferent between buying a long-maturity zero-coupon bond versus buying a shorter-maturity zero-coupon bond and reinvesting the principal.
Formula: [ 1 + S(j+k) ]^(j+k) = (1 + S_j)^j * [ 1 + f(j,k) ]^k
OR
[ 1 + f(j,k) ]^k = [1 + S(j+k) ]^(j+k) ÷ (1 + Sj)^j
This equation suggests that the forward rate f(2,3) should make investors indifferent between buying a five-year zero-coupon bond versus buying a two-year zero-coupon bond and at maturity reinvesting the principal for three additional years.
11
Q
Par rate
A
The YTM of a bond trading at par
12
Q
Bootstrapping
A
A process that calculates spot rates from the par rate curve. Bootstrapping involves using the output of one step as an input to the next step.
13
Q
Bootstrapping example:
Given:
Maturity 1 = 1% par rate
Maturity 2 = 1.25% par rate
Maturity 3 = 1.50% par rate
Compute the 2 year spot rate
A
100 = (1.25 ÷ 1.01) + ((100 + 1.25) ÷ (1 + r2)^2)
100 = 1.2376 + (101.25 ÷ (1 + r2)^2)
98.7624 = 101.25 ÷ (1 + r2)^2
(1 + r2)^2 = 1.025187723
1 + r2 = 1.012515542
r2 = 0.012515542
This shows how zero coupon rates can be derived from the par curve
14
Q
True or false: If the spot rate curve is flat, the forward rate curve is flat and lies equally with it. If the spot rate curve is upward sloping, the forward rate curve is also upward sloping and lies above it. And if the spot rate curve is inverted, the forward rate curve is inverted and lies below it?
A
True
15
Q
True or false: The spot rate for a long-maturity security will equal the geometric mean of the one period spot rate and a series of one-year forward rates?
A
True
- Active bond portfolio managers will try to outperform the market by predicting how the future spot rates will differ from those predicted by the current forward curve.
- If the future spot rates are below the current forward rates, the portfolio manager wil see a greater return than the one-year Rf.
16
Q
Forward price evolution
A
If the future spot rates evolve as the forward curve predicted, the forward price will remain unchanged. Therefore, a change in the forward price indicates that the future spot rate(s) did not conform to the forward curve. When spot rates turn out to be lower than implied by the forward curve, the forward price will increase and vice versa.
17
Q
Maturity matching
A
A bond investment strategy that’s purchasing bonds that have a maturity equal to the investor’s investment horizon.
18
Q
Riding the yield curve/rolling down the yield curve
A
A bond investment strategy. When a yield curve is upward sloping, investors seeking superior returns may pursue this strategy. This strategy involves an investor purchasing bonds w/ longer maturities than their investment horizons. In an upward-sloping yield curve, shorter maturity bonds have lower yields than longer maturity bonds. As the bond approaches maturity (rolls down the yield curve), it is valued using successively lower yields and, therefore, at successively higher prices.
If the yield curve remains unchanged over the investment horizon, riding the yield curve strategy will produce higher returns than a simple maturity matching strategy, increasing the total return of a bond portfolio. The greater the difference between the forward rate and the spot rate, and the longer the maturity of the bond, the higher the total return.
This strategy increases IRR
19
Q
Holding period return formula
A
(Closing value ÷ Beginning value) - 1
20
Q
Holding period return example:
Given benchmark spot rates:
Maturity 1 = 3% spot rate
Maturity 2 = 4% spot rate
Maturity 3 = 5% spot rate
Expected spot rates:
Year 1 = 5.01%
Year 2 = 6.01%
Calculate the one-year holding period return of a 1-year zero coupon bond, 2-year zero coupon bond, and a 3-year zero coupon bond.
A
(1): ($1 ÷ 1.03) = 0.9709
Holding period return = ($1 ÷ 0.9709) -1 = 3%
(2): ($1 ÷ (1.04)^2) = 0.924556213
($1 ÷ 1.0501) = 0.9523
Holding period return = 0.9523 ÷ 0.924556213 = 3%
(3) = ($1 ÷ (1.05)^3) = 0.8638375985
($1 ÷ 1.0601^2) = 0.8898285399
Holding period return = 0.8898285399 ÷ 0.8638375985 = 3%
21
Q
Interest rate swap
A
A forward contract where one party makes payments based on a fixed rate while the counterparty makes payments based on a floating rate. The fixed rate in an interest rate swap is called the swap rate.
22
Q
Swap rate curve
A
A curve that plots how different swap rates are at different maturities.
- DOES NOT indicate credit risk.
- Can be used to indicate premium for time value of money.
23
Q
Why do market participants use the swap rate curve?
A
- Swap rates reflect the credit risk of commercial banks rather than the credit risk of governments.
- The swap market is not regulated by any government, which makes swap rates in different countries more comparable.
- The swap curve typically has yield quotes at many maturities, while the U.S. government bond yield curve has only a small number of maturities.
- Swap rate curves are not affected by technical market factors that affect the yields on government bonds.
24
Q
Swap spread
A
The amount the fixed-rate side of an interest rate swap exceeds the yield of a government bond w/ the same maturity.
Formula: Swap fixed rate - treasury yield
- Swap spreads are almost always positive, reflecting the lower credit risk of governments compared to the credit risk of surveyed banks that determines the swap rate.
25
Spot rate curve example:
Given:
Maturity 1 = 3% spot rate
Maturity 2 = 4% spot rate
Maturity 3 = 5% spot rate
Compute the swap fixed rate for years 1, 2, and 3.
(1) [ SFR1 ÷ (1 + S1) ] + [ 1 ÷ (1 + S1) ] = 1
[ SFR1 ÷ (1.03) ] + [1 ÷ 1.03 ] = 1
[ SFR1 ÷ (1.03) ] = 0.9709
SFR1 = 3%
(2) [ SFR2 ÷ (1 + S1) ] + [ SFR2 ÷ (1 + S2)^2 ] + [ 1 ÷ (1 + S2)^2 ] = 1
[ SFR2 ÷ (1.03) ] + [SFR2 ÷ (1.04)^2 ] + [1 ÷ (1.04)^2 ] = 1
26
True or false: Retail banks are more likely to use the swap rate curve as a benchmark than the government spot curve?
False, retail banks are likely to use as the government spot rate as a benchmark. However, wholesale banks are likely to use the swap rate curve as a benchmark.
27
Interpolated spread (I-spread)
A type of short-term interest rate spread. The amount by which the yield on the risky bond exceeds the swap rate for the same maturity. If a swap rate isn't available for a specific maturity, the missing swap rate can be estimated from the swap rate curve using linear interpolation.
Formula: Yield on the risky bond - swap rate
## Footnote
* While a bond's yield reflects time value as well as compensation for credit and liquidity risk, I-spread only reflects compensation for credit and liquidity risks. The higher the I-spread, the higher the compensation for liquidity and credit risk.
28
Interpolated rate calculation
rate for lower bound + [ ((# of years for interpolated rate − # of years for lower bound) * [ higher bound rate − lower bound rate)) ÷ (# of years for upper bound − # of years for lower bound) ]
29
I-Spread example:
a 6% bond is current yielding 2.5% and matures in 1.6 years. Given the swap curve:
tenor-0.5 = 1%
tenor-1 = 1.25%
tenor-1.5 = 1.35%
tenor-2 = 1.50%
Compute the I-spread
Interpolated rate = 0.0135 + [ ((1.6 - 1.5)(.015 - .0135)) ÷ (2 - 1.5) ] = 1.38%
I-spread = 2.5% - 1.38% = 0.97%
30
Z-spread/Zero volatility spread
The spread that, when added to each spot rate on the default-free spot curve, makes the present value of a bond's cash flows equal to the bond's market price. Therefore, the Z-spread is a spread over the entire spot rate curve.
Formula: Market price of risky bond = (CF1 ÷ (1 + r1 + z)) + (CF1 ÷ (1 + r2 + z)^2) ...
| Z-spread MUST be used for bonds WITHOUT embedded options.
## Footnote
* Zero-volatility spread is a commonly used measure of relative value for MBS and ABS. However, it only considers one path of interest rates, while OAS considers every spot rate along every interest rate path.
31
TED spread
A short-term interest rate spread. The amount by which the 3-month MRR exceeds the interest rate on short-term U.S. government debt (3-month T-bill) of the same maturity. The TED spread is seen as an indication of the credit and liquidity risk in the banking sector. A rising TED spread indicates that market participants believe banks are increasingly likely to default on loans and that risk-free T-bills are becoming more valuable in comparison. The TED spread captures the risk in the banking system more accurately than the 10-year swap spread.
32
MRR OIS Spread
The amount by which the MRR rate (which includes some credit risk) exceeds the OIS rate (which includes only minimal credit risk) and also indicates the level of credit and liquidity risk in the banking system.
OIS stands for overnight indexed swap and represents interest rate on unsecured overnight lending between banks. The OIS rate roughly reflects the federal funds rate and includes minimal counterparty credit risk.
33
True or false: The swap spread of a default free bond should provide an indication of the bond's illiquidity—or, alternatively, that the bond is mispriced?
True
34
Theories that explain why a yield curve takes a particular shape:
1. Unbiased expectations theory
2. Local expectations theory
3. Liquidity preference theory
4. Segmented markets theory
5. Preferred Habitat Theory
35
Unbiased expectations theory/pure expectations theory
This theory states that investors' expectations determine the shape of the interest rate term structure and that forward rates are solely a function of expected future spot rates and that every maturity strategy has the same expected return over a given investment horizon. The underlying principle behind this theory is risk neutrality: Investors DON'T DEMAND A RISK PREMIUM for maturity strategies that differ from their investment horizon.
| Long-term interest rates = the mean of future expected short-term rates
## Footnote
* If the yield curve is upward sloping, the short-term rates are expected to rise.
36
Local Expectations Theory
Similar to the unbiased expectations theory but only holds the risk-neutrality assumption for short periods. OVER LONG PERIODS OF TIME, RISK PREMIUMS SHOULD EXIST. Over short periods, every bond should earn the Rf.
## Footnote
* This theory doesn't hold because the short-holding-period returns of long-maturity bonds can be shown to be higher than short-holding-period returns on short-maturity bonds due to liquidity premiums and hedging concerns.
37
Liquidity preference theory
This theory states that forward rates reflect investors' expectations of future spot rates, plus a liquidity premium to compensate investors for exposure to interest rate risk. The liquidity premium is positively related to maturity (ex: a 25-year bond should have a larger liquidity premium than a five-year bond).
## Footnote
* States that the forward rate is biased because it includes a liquidity premium.
* Under this theory, the yield curve may take any shape. Even after adding the premium to a steep downward sloping yield curve the result will still be downward sloping.
* An upward slowing yield curve may be due to future expectations if rising rates or the liquidity premium- we don't know.
38
Segmented markets theory
This theory states that yields are not determined by liquidity premiums and expected spot rates, but rather, the shape of the yield curve is determined by the preferences of borrowers and lenders, which drives supply/demand for loans of different maturities. The theory suggests that the yield at each maturity is determined independently of the yields at other maturities.
## Footnote
* The segmented markets theory proposes that market participants have strong preferences for specific maturities.
* Under segmented markets theory, investors in one maturity segment of the market will not move into any other maturity segments.
39
Preferred Habitat Theory
This theory states that forward rates represent expected future spot rates plus a premium, but it does not support the view that this premium is directly related to maturity. Borrowers require cost savings (lower yields) and lenders require a yield premium (higher yields) to move out of their preferred habitats (preferred maturities).
## Footnote
* The preferred habitat theory proposes that market participants have strong preferences for specific maturities, however risk premiums can incentivize investors to change maturities to take advantage of certain opportunities.
* Unlike the liquidity preference theory, under the preferred habitat theory a 10-year bond might have a higher or lower risk premium than the 25-year bond.
40
Yield curve risk
Risk to the value of a bond portfolio due to unexpected shifts (parallel vs non-parallel) in the yield curve.
41
Parallel shifts vs nonparallel shifts
Parallel shifts = changes in the yield curve where the yield of every maturity goes up or down by the same proportion.
Nonparallel shift (steepness change) = Long-term interest rates increase while short-term rates decrease or vice versa.
Nonparallel shift (curvature change) = Increasing curvature means short- and long-term interest rates increase while intermediate rates do not change.
42
How to measure sensitivity to yield curve risk?
There are three main approaches:
1. Effective duration
2. Key rate duration
3. A three-factor model that decomposes changes in the yield curve into changes in level, steepness, and curvature. This is a regression model. The three factors will be estimated using past data.
## Footnote
* Changes in the shape of yield curve is explained by (in order of importance): level, steepness and curvature.
43
Effective duration
Effective duration is used mostly for bonds w/ embedded options and measures the price sensitivity to a 1% change in rates. Measures price sensitivity to small PARALLEL shifts in the yield curve. This measure of duration takes into account the fact that expected CFs will fluctuate as rates change and is, therefore, a measure of risk.
Calculation: ED = ((BV _ -△Y) - (BV _ △Y)) ÷ (2 * BV0 * △Y)
| Does not work w/ non-parallel shifts in the yield curve.
## Footnote
* Parallel shifts explain more than 75% of the variation in bond portfolio returns.
* The larger the coupon, the lower the duration.
44
Shaping risk
Changes in portfolio value due to changes in the shape of the benchmark yield curve.
## Footnote
* Effective rate DOES NOT measure shaping risk.
45
Key rate duration
Measures the sensitivity of the value of a security or bond portfolio to changes in the yield at a particular maturity only. It's the approximate % change in the value of a bond portfolio in response to a 100 basis point change in the corresponding par rate (YTM of a bond that trades at par), holding all other par rates constant.
## Footnote
* More precise than effective duration.
* Can be used for non-parallel shifts.
* Captures shaping risk.
46
3 factor model
The 3rd approach is to measure yield curve risk is to run a regression w/ 3 independent variables: level ΔXL (a parallel increase/decrease in int. rates), steepness ΔXS, and curvature ΔXC.
47
True or false: Effective duration = the sum of the key rate durations?
True
48
True or false: Empirical data shows that long-term rates are more volatile than short-term rates?
False
## Footnote
* Volatility at the long-maturity end is thought to be associated with uncertainty regarding the real economy and inflation, while volatility at the short-maturity end reflects risks regarding monetary policy.
* **Two-thirds of the variation in short and intermediate-term yields is explained by monetary policy, and the remaining is explained by the other factors. In contrast, inflation explains two-thirds of the variation in long-term yields, with the remaining mostly explained by monetary policy.**
49
True or false: Embedded options make bonds less sensitive to interest rate fluctuations?
False, more sensitive
50
Bond risk premium/term premium/duration premium
The excess return over the one-year Rf earned by investors for investing in long-term government bonds.
Formula: Er - Rf
## Footnote
* Er = expected return on a long-term government bond.
51
Bearish flattening vs bullish steepening
Bearish flattening hapens when central banks raise the short-term rates. Oppositely, bullish steepening happens when central banks lower the short-term rates.
52
Factors besides monetary policy and inflation that affect bond prices?
* Fiscal policy: Expansionary policy increases yields and vice versa.
* Maturity structure: The government's choice of maturity when issuing new securities affects the supply (and yield) of bonds in those maturity segments. An increase in offerings in a specific segment of the market increases the supply and increases the yield in that segment (the market segmentation theory).
* Investor demand: Domestic and foreign investor demand and preferences for specific maturity segments affect the yield in that segment. The more demand, the lower the yield.
53
Bullet portfolio vs barbell portfolio
Bullet portfolio: A bond portfolio concentrated in a single maturity.
Barbell portfolio: A portfolio with short and long maturities.
54
Bond duration
Measures how long it takes, in years, for an investor to be repaid a bond’s price by the bond’s total CFs.
## Footnote
* PMs will increase duration in anticipation of a fall in rates, and decrease duration in anticipation of a rise in rates.
* PMs will rotate out of bullet structuring and into barbell structuring in expecation of a bullish flattening of the yield curve.
* If PMs expect a curvature change, for example, where 10 year yields increase while other yields remain the same, PMs will use a bullet strategy.
55
What are the two types of arbitrage?
1. Value additivity: When the value of the whole differs from the sum of the values of parts
2. Dominance: When one asset trades at a lower price than another asset with identical characteristics
56
Arbitrage-free valuation
Values securities such that no market participant can earn an arbitrage profit in a trade involving that security. Arbitrage-free valuation of a fixed-rate, option-free bond entails discounting each of the bond's future CFs using the corresponding spot rate (how we normally value a bond).
## Footnote
* Arbitrage-free valuation gives us the price of a bond where we cannot carry out a value additivity strategy or dominance strategy.
57
Stripping/reconstitution
A five-year, 5% Treasury bond should be worth the same as a portfolio of its coupon and principal strips. If the portfolio of strips is trading for less than an intact bond, one can purchase the strips, combine them (reconstituting), and sell them as a bond. Similarly, if the bond is worth less than its component parts, one could purchase the bond, break it into a portfolio of strips (stripping), and sell those components.
58
Arbitrage example:
Security A - current price = $99 ; payoff in one year = $100
Security B- current price = $990 ; payoff in one year = $1,010
Security C- current price = $100 ; payoff in one year = $102
Security D- current price = $100 ; payoff in one year = $103
Securities A and B are identical in every respect other than as noted. Similarly, securities C and D are identical in every other respect. Find the arbitrage opportunities.
(1) Short 10 units of secrity A ($990 inflow) and long 1 unit of security B. This leads to 0 outflow currently. Then, in one year, there will be a $1,000 outflow from security A but a $1,010 inflow from security B, so there will be a $10 profit.
(2) Short 1 unit of Security C and long 1 unit of security D. This lead to 0 outflow currently. Then, in one year, there will be a $102 outflow from security C but a $103 inflow from security D.
59
Binomial interest rate tree
A model to value a bond that allows both rates and the underlying cash flows to vary. This model assumes that interest rates have an equal probability of taking one of two possible values in the next period. This model is a lognormal random walk w/ two properties: higher volatility at higher rates and non-negative interest rates.
Binomial interest rate trees begin w/ a root (i0), which is the current 1-period spot rate. Then, it moves to two nodes: An upper node (1,U) and a lower node (1,L). Then, each of those nodes will have two nodes of their own and so on. To value each node: take the corresponding node * e^(2σ)
## Footnote
* If the binomial lattice is correctly calibrated, it should give the same value for an option-free bond as using the par curve used to calibrate the tree.
60
Backward induction
The process of valuing a bond using a binomial interest rate tree. For a bond that has N compounding periods, the current value of the bond is determined by computing the bond's possible values at Period N and working backwards to Node 0. The value of a bond at a given node in a binomial tree is *the average of the present values of the two possible values from the next period.*
61
Backwards induction example:
A $100 par 7% annual coupon bond has two years to maturity. The discount rate for i1U is 7.1826%, the rate for i1L is 5.3210%, and the rate for i0 is 4.5749%. Fill in the tree and calculate the value of the bond today.
i1U = (($100 + $7) ÷ (1.07826)) = $99.83
i1L = (($100 + 7) ÷ (1.053210)) = $101.59
i0 = [ (($99.83 + $101.59) ÷ 2) + $7 ] ÷ 1.045749 = $103
62
3 rules of building binomial interest rate trees:
1. The interest rate tree should generate arbitrage-free values for the benchmark security (the value of bonds produced by the interest rate tree must be equal to their market price).
2. Adjacent forward rates must be two standard deviations apart.
3. The middle forward rate in a period is approximately equal to the implied (from the benchmark spot rate curve) one-period forward rate for that period.
63
Example: Calibrating a binomial interest rate tree to match a specific term structure.
Firm A generates a binomial interest rate tree consistent with a set data and an assumed volatility of 20%. The data is as follows:
Maturity 1 - par rate = 3% - spot rate = 3%
Maturity 2 - par rate = 4% - spot rate = 4.020%
Maturity 3 - par rate = 5% - spot rate = 5.069%
Given that the forward rates at i0 = 3% and i1U = 5.7883%, calculate:
(1): The forward rate at i1L
(1): i1L = i1U * e^(-2σ) = .057883 * e^(-2 * 0.20) = 0.0388 = 3.88%
(2): We must use bootstrapping:
64
Pathwise valuation
Discounting a bond’s CFs for each likely interest rate path and calculating the average of these values across all the paths. It is an alternative method to the backward induction approach.
Number of paths = 2^(n-1)
65
Pathwise valuation example:
Firm A wants to value a 3-year 3% annual-pay bond that is option-free and $100 par. The forward rates are as follows:
i0 = 3%
i1U = 5.7883%
i1L = 3.8800%
i2UU = 10.7383%
i2UL/i2LU = 7.1981%
i2LL = 4.8250%
Compute the value at i0 using pathwise valuation
There are 4 paths:
Path 1: [ (3 ÷ (1.03)) + (3 ÷ (1.03 * 1.057883)) + (103 ÷ (1.03 * 1.057883 * 1.107383)) ] = $91.03
Path 2: [ (3 ÷ (1.03)) + (3 ÷ (1.03 * 1.057883)) + (103 ÷ (1.03 * 1.057883 * 1.071981)) ] = $93.85
Path 3: [ (3 ÷ (1.03)) + (3 ÷ (1.03 * 1.0388)) + (103 ÷ (1.03 * 1.0388 * 1.071981)) = $95.52
Path 4: [ (3 ÷ (1.03)) + (3 ÷ (1.03 * 1.0388)) + (103 ÷ (1.03 * 1.0388 * 1.048250)) = $97.53
($91.04 + $93.85 + $95.52 + $97.53) ÷ 4 = $94.49
66
True or false: Prepayment risk is affected not only by the level of interest rate at a particular point in time, but also by the path rates took to get there?
True
67
True or false: An important assumption of the binomial valuation process is that the value of the CFs at a given point in time is dependent on the path that interest rates followed up to that point?
False, the important assumption is it's indepedent.
## Footnote
* Because of this assumption, we cannot use a binomial model to value MBSs where prepayment risk lurks.
68
Monte Carlo forward-rate simulation
This method randomly generates a large # of interest rate paths, using a model that incorporates a volatility assumption and an assumed probability distribution. Similar to pathwise valuation, the value of the bond is the average of the diffferent values from the various paths. The simulated paths should be calibrated so benchmark interest rate paths value benchmark securities at their market price (i.e., arbitrage-free valuation). This calibration process results in a **drift adjusted** model.
| W/ this method, the underlying CFs can be path DEPENDENT.
69
Term structure
The relationship between credit spreads and maturity
## Footnote
* The credit spread is inversely related to the recovery rate and positively related to the probability of default.
70
Term structure models
Models that analyze the statistical properties of interest rate movements. The two major types of term structure models are:
1. Equilibrium term structure models
2. Arbitrage-free models
71
Equilibrium term structure models
Models that attempt to model the term structure using fundamental economic variables that are thought to determine interest rates. Two main types are:
1. Cox-Ingersoll-Ross model
2. Vasicek Model
72
Cox-Ingersoll-Ross (CIR) Model
Assumes the economy has a natural long-run interest rate (b) that the short-term rate (r) converges to at a speed of (a). Interest rate volatility varies with r and is not constant. Produces nonnegative rates only.
Formula: drt = (a(b - r) * dt) + (σ * (√rdz))
## Footnote
* drt = change in the short-term interest rate
* a = speed of mean reversion.
* b= long-run value of short-term interest rate
* r = the short-term interest rate
* dt = a small increase in time
* σ = volatility
* dz = a small random walk movement
73
Vasicek Model
This model suggests that interest rates are mean reverting to some long-run value.
Formula: drt = a(b - r) * dt + (σ * dz)
- b = mean reverting level
- r = int. rate
- a = speed- how fast rates are reverting to mean level. Between 0 and 1.
- d = ∆ (dt is ∆ in time, dr is ∆ in rates, and dz is ∆ in noise. z is from a standard normal dist.)
- σ = assumed level of volatility
## Footnote
* The difference between this and the CIR model is that volatility in this model does not increase as the level of interest rates increases
* The main disadvantage of the Vasicek model is that the model does not force interest rates to be nonnegative
74
Arbitrage-free models
These models assume markets price bonds correctly.
75
Ho-Lee model
An arbitrage-free model that is calibrated by using market prices to find the θ that generates the current term structure. This model assumes that changes in the yield curve are consistent with a no-arbitrage condition. The model assumes constant volatility and a constant drift. This model is derived from the concepts of the Black-Scholes model. The interest rates generated from this model can be used to determine the prices of zero-coupon bonds and the spot curve.
Formula: drt = (θt * dt) + (σ * dzt)
## Footnote
* θt = a time-dependent drift term
* This model can have negative rates
76
Kalotay-Williams-Fabozzi (KWF) Model
An arbitrage-free model that assumes no volatility and no drift and also does not assume mean reversion. However, the KWF model assumes that the short-term rate is lognormally distributed.
Formula: (d * ln(rt)) = (θt * dt) + (σ * dzt)
77
Gauss+ Model
A multifactor model that incorporates short, medium, and long-term rates, where the long-term rate is designed to be mean reverting and depends on macroeconomic variables. Medium-term rates revert to the long-term rate, while the short-term rate depends on the central bank's the short-term rate, and does not have a random component.
78
True or false: Increasing the # of paths used in the pathwise valuation model increases the statistical accuracy of the estimated value and produces a value closer to the true fundamental value of the security?
False, the larger the # of paths, the more accurate the value in a statistical sense. However, whether the value is closer to the true fundamental value depends on the accuracy of the model inputs.
79
True or false: One of the drawbacks of the Vasicek and CIR models is that the model prices generated by these models generally do not coincide with observed market prices?
True
80
Advantages of embedded options for issuers:
* Manage IRR
* Ability to issue bonds at an attractive coupon rate
81
Protection period for callable bonds
A period during which the bond cannot be called
82
European-style vs American-style vs Bermudan-style
European-style: When the option can only be exercised on a single day immediately after the protection period.
American-style: When the option can be exercised at any time after the protection period.
Bermudan-style: When the option can be exercised at fixed dates after the protection period.
83
Extendible bond
A bond that allows the investor to extend the maturity of the bond. An extendible bond can be evaluated as a putable bond with longer maturity
84
Estate put
A bond which includes a provision that allows the heirs of an investor to put the bond back to the issuer upon the death of the investor. The value of this contingent put option is inversely related to the investor's life expectancy; the shorter the life expectancy, the higher the value.
85
Sinking fund provision
A bond that requires the issuer to set aside funds periodically to to retire the bond. This provision reduces the credit risk of the bond.
## Footnote
* These bonds typically have several other issuer options embedded (ex: call options).
86
How to value a callable bond
Value of callable bond = Value of a straight (option-free) bond - the value of the call.
87
How to value a putable bond
Value of putable bond = Value of straight bond + value of put
88
Using binomial trees to value callable and putable bonds
Call rule: When valuing a callable bond, the value at any node where the bond is callable must be either the price where the issuer will call the bond (call price) OR the computed value if the bond is not called, whichever is lower.
Put rule: When valuing a putable bond, the value used at any node corresponding to a put date must be either the price at which the investor will put the bond or the computed value if the bond is not put, whichever is higher.
## Footnote
* We still use backward induction but this time it's just a bit different.
89
True or false: The value at each node in a binomial tree is the PV of the two possible values from the next period?
False, it's the average of the PVs of the two possible values from the next period.
90
How does interest rate volatility affect the value of a callable or putable bond?
***Option values*** are positively related to the volatility of their underlying, so when interest rate volatility increases, the values of both call and put options increase. When interest rate volatility increases, the model value (NOT the MV) of a callable bond decreases and the model value of a putable bond increases.
## Footnote
* The value of a straight bond is affected by changes in the level of interest rates but is unaffected by changes in the volatility of interest rates.
91
How do changes in the level of interest rates affect callable and putable bonds?
As interest rates decline, the value of a callable bond rises less rapidly than the value of an otherwise-equivalent straight bond. Similarly, as interest rates increase, the value of a putable bond falls less rapidly than the value of an otherwise-equivalent straight bond. Call option value is inversely related to the level of interest rates, while put option value varies directly with the level of interest rates.
## Footnote
* Market price of a bond IS NOT affected. Market price is determined by supply/demand. What we're talking about here is the MODEL PRICE.
92
How does the shape of the yield curve affect callable and putable bonds?
The value of a call option will be lower for an upward sloping yield curve. As an upward-sloping yield curve becomes flatter, the call option value increases. Oppositely, put options are more valuable when the yield curve is upward-sloping and become less valuable as the curve flattens.
93
True or false: If risk-free rates are used to discount cash flows of a credit risky corporate bond, the calculated value will be too low?
False, too high
94
How to use backwards induction for bonds w/ risk?
We must consider the **option adjusted spread (OAS)**. A constant spread must be added to all one-period rates in the tree so that the model's price equals the market price of the risky bond.
| Calculation of OAS done by computers- not needed for exam
## Footnote
* Bonds w/ similar credit risk should have the same OAS. ***If the OAS for a bond is higher than the OAS of its peers, it is considered to be undervalued***
* OAS accounts for compensation for credit and liquidity risk after the optionality has been removed
95
Relationship between interest rate volatility and OAS
Keeping market price of a bond constant, higher volatility results in a lower OAS for a callable bond and a higher OAS for a putable bond.
## Footnote
* Volatility doesn't affect market price, only MODEL price.
96
Modified duration
Measures a bond's price sensitivity to interest rate changes, assuming that the bond's CFs do not change as interest rates change.
| Not useful for bonds w/ embedded options.
## Footnote
* Macaulay duration is a measure of the bond's price sensitivity to changes in time to maturity, whereas modified duration is a measure of the bond's price sensitivty to changes in yields.
97
Convexity
The curvature in the relationship between bond prices and interest rates. It reflects the rate at which the duration of a bond changes as interest rates change.
98
Effective convexity
A measure of the sensitivity of duration to changes in interest rates
Calculation: [ (BV _ -△Y) + (BV _ △Y) - (2 * BV0) ] ÷ (BV0 * (△Y) ^2)
## Footnote
* △Y = change in required yield
* BV _ -△Y = estimated price if yield decreased by △Y
* BV _ △Y = estimated prie if yield increased by △Y
* BV0 = initial observed bond price
99
How to estimate price of a bond if yield increases/decreases by Y
1. Using backward induction, calculate OAS
2.Shock yields by △Y, add OAS from #1 and calculate BV _ △Y and BV _ -△Y
100
Compare effective durations of callable, putable, and straight bonds.
* Effective duration (callable) ≤ effective duration (straight).
* Effective duration (putable) ≤ effective duration (straight).
* Effective duration (zero-coupon) ≈ maturity of the bond.
* Effective duration of fixed-rate coupon bond < maturity of the bond.
* Effective duration of floater ≈ time (in years) to next reset.
101
Macaulay duration
The length of time taken by an investor to recover the money they invested in a bond through coupons and principal repayment.
| NOT A PRICE SENSATIVITY METRIC!
102
One-sided durations
These durations are better for bonds w/ embedded options. One-sided durations only apply when interest rates rise OR when interest rates fall (not both). Used for callable bonds or putable bonds that are at-the-money or in-the-money.
103
True or false: When the underlying option is at-the-money or near-the-money, callable bonds will have lower one-sided down-duration than one-sided up-duration: the price change of a callable when rates fall is smaller than the price change for an equal increase in rates?
True
104
True or false: A near-the-money putable bond will have smaller one-sided down-duration than one-sided up-duration.
False, a near-the-money putable bond will have larger one-sided down-duration than one-sided up-duration.
105
True or false: Callable bonds with low coupon rates will most likely not be called and hence their maturity–matched rate is their most critical rate (and has the highest key rate duration). As the coupon rate increases, a callable bond is more likely to be called and the time-to-exercise rate will start dominating the time-to-maturity rate?
True
106
True or false: Putable bonds with high coupon rates are unlikely to be put and are most sensitive to its maturity-matched rate. As the coupon rate decreases, a putable bond is more likely to be put and the time-to-exercise rate will start dominating the time-to-maturity rate?
True
107
Compare effective convexities of callable, putable, and straight bonds.
Straight bonds have positive effective convexity: the increase in the value is higher when rates fall than the decrease in value when rates increase by an equal amount.
Putable bonds also exhibit positive convexity
Callable bonds show positive convexity when rates are high. When the underlying call option is near the money, its effective convexity turns negative.
108
Capped floater
Contains an issuer option that prevents the coupon rate from rising above a specified maximum rate known as the cap. Capped floaters ***must trade below par.***
Value of capped floater = value of a straight floater - value of the embedded cap
109
Floored floater
Contains an investor option that will not let the coupon rate fall below a certain point. Floored floaters must trade above par.
Value of a floored floater = value of straight floater + value of the embedded floor
110
Example of valuing capped and floored floaters:
Firm A is looking to value a $100 par, two-year, floating-rate note that pays MRR (set in arrears). The underlying bond has the same credit quality as reflected in the swap curve. The forward rate in i0 = 4.5749% ; i1U = 7.1826% ; i1L = 5.3210%.
(1) Compute the value of the floater, assuming that it is an option-free bond.
(2) Compute the value of the floater, assuming that it is capped at a rate of 6%. Also compute the value of the embedded cap.
(3) Compute the value of the floater, assuming that it is floored at a rate of 5%. Also compute the value of the embedded floor.
(1) An option-free bond with a coupon rate equal to the required rate of return will be worth par value. Hence, the straight value of the floater is $100.
(2) Since we know i0 is 4.5749%, we know that the coupon in i1U and i1L is $4.57. And since i1U is 7.1826%, the coupon at maturity would normally be $7.18 but since we have the cap it's $6 OR if we go the other path it'll be $5.32. Now we can value the price of i1U (($100 + $6) ÷ (1.071826) = $98.90) and i1L (($100 + $5.32) ÷ (1.0532) = $100). Then, we can calculate i0.
(3) Like #2, we know that the coupon at maturity will either be $7.18 or $5.32. However, at period 1 we must know that the floor is $5, so that's what the coupon will be. Then, we can calculate the prices from there.
111
True or false: When an upward sloping yield curve flattens, call options decrease in value while put options increase in value?
False, when an upward sloping yield curve flattens, call options increase in value while put options decrease in value.
112
True or false: Straight bonds generally have higher effective durations than bonds with embedded options?
True
113
True or false: In a rising rate scenario, a call option is likely to be called and a put option is unlikely to be put?
False, in a rising rate scenario, a call option is unlikely to be called (duration will increase) but put options are likely to be called (duration will decrease).
114
Conversion ratio
The # of common shares in which a convertible bond can be exchanged.
Calculation: Par value ÷ conversion price
115
Conversion price
The effective price per share when converting
Calculation: Market price of bond ÷ Conversion ratio
116
Contingent put option
Allows the bondholder to put the bond back to the issuer in the case of some extreme event (ex: merger/acquisition).
117
Conversion value
The value of the common stock into which the bond can be converted.
Calculation: Mark price of stock after conversion * conversion ratio
118
Straight value/investment value
The value of the bond if it were not convertible
119
Minimum value of convertible bond
The greater of a convertible bond's conversion value or its straight value.
120
Minimum value example:
Firm A has a convertible bond with a 7% coupon that is currently selling at $985 with a conversion ratio of 25 and a straight value of $950. Suppose that the value of Firm A's common stock is currently $35 per share, and that it pays $1 per share in dividends annually. What is this bond's minimum value?
Straight value = $950
Conversion value = 25 * 35 = $875
minimum value = $950
121
Market conversion price/Conversion parity price
The price that the convertible bondholder would effectively pay for the stock if they bought the bond and immediately converted it
Calculation: market price of convertible bond ÷ conversion ratio
122
Market conversion premium per share
The premium that investors are willing to pay for the opportunity to profit should the market price of the stock rise above the market conversion price.
Formula: Market conversion price - stock's market price
## Footnote
* The conversion ratio times the price per share of common stock is a lower limit on the bond's price.
123
Market conversion premium ratio
Market conversion premium per share ÷ market price of common stock
124
Premium over straight value
(Market price of convertible bond ÷ straight value) - 1
## Footnote
* The greater the premium over straight value, the less attractive the convertible bond.
* The disadvantage to the ratio is that the straight value is not constant- it varies w/ changes in interest rates and with the credit spread of a bond.
125
True or false: Investing in a convertible bond w/o embedded options is equal to buying an option-free bond AND a call option on an amount of the common stock equal to the conversion ratio?
True, the value of a convertible bond w/o embedded options = straight value + value of call option on stock
126
Callable convertible bond value
Straight value of bond + value of call option on stock - value of call option on bond
127
True or false: Convertible bond investors must be concerned with credit risk, call risk, interest rate risk, and liquidity risk?
True
128
Busted convertible
When the stock price associated w/ a convertible bond is trading so low that the bond is basically a straight bond.
129
Common stock equivalent
When the stock associated w/ a convertible bond is so high it's basically an equity security.
130
True or false: A decrease in interest rate volatility will decrease the value of the embedded short call on the bond, and thus increase the value of the convertible bond?
True
131
Expected exposure
The amt of money a bondholder could lose at any point in time BEFORE factoring in any recovery
Calculation: Exposure_t ÷ (1 + Rf) + coupon payment
## Footnote
* When using backwards induction, we can calculate expected exposure by taking the sum of the (node probability * PV) at each period (ex: (node probability * i1U) + (node probability * i1L) ...)
132
Recovery rate
The % recovered in the event of a default. It's the opposite of loss severity (ex: if recovery rate= 40%, loss severity = 60%).
## Footnote
*
133
Loss Severity
The % loss in the event of a default.
134
Loss given default (LGD)
Loss severity * exposure OR (1 - recovery rate)
135
Hazard rate
The conditional probability of default given that default has not previously occurred.
136
Probability of survival
(1 - hazard rate)^t
- Given that the hazard rate is constant
137
Probability of default
Hazard rate * Probability of survival_(t-1)
- In the first year, probability of default = hazard rate
- After the first year, probability of default < hazard rate
## Footnote
* This is the unconditional probability of default, whereas the hazard rate is the conditional probability of default given that default hasn't happened before.
* CVA and credit spreads are positively related to probability of default.
138
Expected loss
Loss given default * probability of default
## Footnote
* If a question asks which bond has the most/least amt of credit risk, it's asking you to calculate expected losses of each.
139
Credit valuation adjustment (CVA)
A measure of credit risk. The sum of the PV of expected loss for each period.
Formula: CVA = price of risk-free bond - price of risky bond
140
Example:
A 3-year, $100 par, zero-coupon corporate bond has a hazard rate of 2% per year. It's recovery rate is 60%, and the benchmark rate curve is flat at 3%. Calculate the expected exposure, probability of survival, probability of default, loss given default, CVA, and the credit spread.
Expected exposure in year 3 = $100 (par). year 2 = $100 ÷ (1.03) = $97.09. year 3 = $100 ÷ (1.03)^2.
Probability of survival in year 1 = (1 - 0.02) = 98%. Year 2 - (1 - 0.02)^2 = 96%. year 3 = (1 - 0.02)^3 = 94%
Probability of default year 1= (.02 * .98) = 0.0196. year 2 = (.02 * .96) = 0.0192. year 3 = (.02 * .94) = 0.0188
Loss given default = (1 - .6) = 0.4
CVA: Since expected loss = LGD * probability of default
- year 1 = 40 * 0.0196 = 0.784
- year 2 = 40 * 0.0192 = 0.768
- year 3 = 40 * 0.0188 = 0.752
- Discount factor is the PV of $1
- year 1 = 1 ÷ (1.03) =
- year 2 = 1 ÷ (1.03)^2 =
- year 3 = 1 ÷ (1.03)^3 =
- Pv of expected loss = DF * expected loss
- CVA = sum of the three PV of expected loss
Value of credit risky bond = Benchmark bond (100 ÷ 1.03^3) - CVA
**Credit spread = YTM risky - YTM risk-free** (use calculator)
141
Risk neutral probability of default
The probability of default implied in the current market price.
142
True or false: Analysts must also consider environmental, social, and governance factors when evaluating the default risk of the company?
True
143
Example:
A 3-year, $100 par, zero-coupon corporate bond has a hazard rate of 2% per year. It's recovery rate is 60%, and the benchmark rate curve is flat at 3% and the CVA is $2.15. Exposures at the end of years 1, 2, and 3 are $94.26, $97.09, and $100, respectively. Calculate the IRR on the investment if the bond defaults in each of the 3 years, as well as if it doesn't default.
Year 1: If default: recovery rate * exposure = 0.6 * $94.26 = $56.56 = FV. PV = $100 ÷ (1.03)^3 = $91.51 - $2.15 = $89.36... N = 1 ; I/Y = x ; PV = $89.36 ; PMT = 0 ; FV = $56.56
Year 2: If default: FV = 0.6 * $97.09 = $58.25. N = 2 ; I/Y = x ; PV = $89.36 ; PMT = 0 ; FV = $58.25.
Year 3: If default: FV = 0.6 * $100 = $60 ; N = 3 ; I/Y = X ; PV = $89.36 ; FV = $60
No default: N = 3 ; I/Y = x ; PV = $89.36 ; PMT = 0 ; FV = $100
144
True or false: Credit risk cannot be evaluated based on the expected loss?
False
145
Notching
When credit rating agencies simply lower the rating by one or more levels for more subordinate debt of the issuer. Notching accounts for LGD differences between different classes of debt by the same issuer (higher LGD for issues with lower seniority).
146
True or false: Higher-rated bonds trade at higher spreads relative to their benchmark rates?
False, higher-rated bonds trade at lower spreads relative to their benchmark rates
147
Credit migration
A change in the credit rating
148
How to calculate the expected return on a bond given transition in its credit rating?
%△Price = - (modified duration) * △ in spread
## Footnote
* Remember, a credit spread might be quoted in bps. If so, we have multiply by (1 ÷ 10,000). If quoted in %, we can leave it.
149
Structural models of corporate credit risk
These models focus on a firm's b/s and defines a mechanism for default (why default occurs). This model is focused on the principals of the black scholes model (assumes Rf and volatility are constant). Essentially, this model says that common shareholders have a call option on the firm's assets w/ a strike price equal to the firm's debt. If at the maturity of the debt, the value of the company's assets is higher than the face value of debt, shareholders will exercise their call option to acquire the assets (and then pay off the debt and keep the residual). On the other hand, if the value of the company's assets is less than the face value of debt, the shareholders will let the option expire worthless (i.e., default on the debt), leaving the company's assets to the debt investors.
Value of equity (shareholder) = max(0 , A_T - K)
Value of debt (bondholder) = A_T - value of equity OR min(A_T , K)
An alternative approach views equity investors as long the net assets of a company and long a put option, allowing them to sell at K.
Value of put option = max(0, K - A_T) OR CVA
Value of risky debt = value of Rf debt - value of put option
Value of equity = A - K + max(0, K - A_T)
| One of the assumptions of this model is that assets are perfectly traded
## Footnote
* A_T = Value of a company's assets at time T
* K = face value of debt
* Under structural model the put option value = value of risk-free bond – value of the risky bond = CVA.
* Structural models do not account for the impact of interest rate risk of the value of a company’s assets.
150
Advantages and disadvantages of structural models
Advantages:
* Structural models provide an economic rationale for default
* Structural models utilize option pricing models to value risky debt
Disadvantages:
* Complex balance sheets cannot be modeled.
* Off b/s items cause outputs of this model to be inaccurate
* One of the key assumptions of the structural model is that the assets of the company are traded in the market. This restrictive assumption makes the structural model impractical. Most companies use their assets to function, so they're not trading them.
* Structural models do not account for the impact of interest rate risk of the value of a company’s assets.
151
Reduced form models
Unlike structural models, reduced form models do not explain why a default occurs, but rather when one occurs. Under this model, a default is a randomly occurring *exogenous* event.
## Footnote
* Reduced form and structural models can be used for ABSs as long as they take into account the complex structure of the ABS.
152
Default intensity
The probability of default over the next (small) time period.
| Estimated using regression models.
153
Advantages and disadvantages of reduced form models:
Advantages:
* Do not assume that assets of a firm trade
* Default intensity can change when the structure of a firm changes or when the economy changes.
* The probability of default and recovery rate are not constant
Disadvantages:
* Do not explain why default occurs
* Default is treated as a random event (unrealistic most of the time)
## Footnote
* Both reduced form and structural models can be used for ABSs.
154
True or false: For a risky bond's market price, its risk neutral probability of default is negatively correlated w/ the assumed recovery rate?
False, positively correlated. Think about it, if you are able to recover more in the event of a default, the bond is going to cost more.
155
Value given no default (VND)
The value of a risky bond, assuming it does not default. This is the price of the risk-free bond. Recall, CVA = price of risk-free bond (*VND*) - price of risky bond
156
True or false: Credit spreads include compensation for default, liquidity, and taxation risks relative to the benchmark?
True
## Footnote
* Adjustment to the price for all these risk factors together is known as the CVA
157
True or false: A benchmark yield should be equal to the real risk-free rate plus expected inflation, as well as a risk-premium for uncertainty in future inflation?
True
158
True or false: To create a spread curve, all of the bonds whose spreads are used should have different characteristics?
False, similar characteristics. If different characteristics would distort the credit spread and not be an accurate reflection of the term structure.
159
Determinents of term structure of credit spreads
1. Credit quality: lower-rated sectors tend to have steeper spread curves
2. Financial conditions: Spreads narrow during economic expansions and widen during cyclical downturns. During boom times, benchmark yields tend to be higher while credit spreads tend to be narrower.
3. Market demand/supply: Less liquid bonds have higher spreads.
4. Equity market volatility: The higher the volatility, the wider the spread.
## Footnote
* Credit spreads are positively correlated to the probability of default and loss severity.
* Credit spread curves of the highest rated bond sectors tend to be flat or SLIGHTLY upward sloping.
160
Example:
Given the benchmark par rate in year 1 = 0.75%, year 3 = 1.25%, year 5 = 1.75%, and year 10 = 2.25%. Also, a AAA bond's CVA in year 1 = $1.79, year 3 = $2.12, year 5 = $3.24, year 10 = $5.84. Using a typical 4% coupon AAA bond for each maturity category, the credit spread curve is most likely upward sloping, downward sloping, or flat?
VND in year 1: N = 1 ; I/Y = 0.75 ; PV = x ; PMT = 4 ; FV = 100 --- PV = 103.23
VND in year 3: PV = 108.05, year 5: PV = 110.68 , year 10: PV = 115.52
Value of risky bond in year 1 = 103.23 - 1.79 = 101.44 , year 3 = 105.93 , year 3 = 107.44, year 10 = 109.68
Risky yield: year 1: N = 1 ; I/Y = X ; PV = 103.23 , PMT = 4 , FV = 100 --- I/Y = 2.53%. Year 2: I/Y = 1.95%, year 5: I/Y = 2.40%, and year 10: I/Y = 2.88%.
Credit spread year 1 = (2.53% - 0.75%) = 1.78%, year 3 = 0.70%, year 5 = 0.65%, and year 10 = 0.63%
Since the credit spread is going down as maturity increases, the credit spread curve is inverted.
161
Credit default swap (CDS)
A form of insurance in the form of a contract between two parties where the credit protection buyer purchases protection from the credit protection seller against losses on a default. When there's a credit event, the swap will be settled in cash or by physical delivery.
W/ physical delivery, the seller receives the devalued reference obligation from the purchaser and the purchaser receives the par value from the seller.
W/ cash delivery, the payout is the payout ratio * notional principal.
- Payout ratio = 1 - recovery rate
162
CDS spread
The premium a credit protection buyer pays a credit protection seller. The CDS spread is usually a fixed payment: 1% for investment-grade securities, 5% for high-yield securities. The CDS spread and actual credit spread will likely be different.
163
True or false: The credit protection buyer is long the credit risk in a CDS?
False, the credit buyer is short the credit risk and the CDS while the credit seller is long the credit risk and the CDS.
164
Single name CDS vs Index CDS
Single name CDS: A CDS for a specific purchaser. A payoff on a single-name CDS is based on the cheapest-to-deliver (CTD) obligation w/ the same maturity.
Index CDS: Covers multiple issuers. This allows purchasers to assume exposure to credit risk of multiple companies.
165
Single-name CDS payout example:
Firm A is a protection buyer in a $10MM notional principal senior CDS of Firm B. There is a credit event where firm B defaults, and the market prices of firm B's bonds after the credit event are as follows:
- Bond A, a subordinated unsecured debenture trading at 15% of par.
- Bond B, a 5-year senior unsecured debenture trading at 25% of par
- Bond C, a 3-year senior unsecured debenture trading at 30% of par
What will the payoff be on the CDS?
First of all, a CDS will never be based on a subordinated bond. Since we know the payoff is based on the cheapest-to-deliver (CTD), the payoff will be based on bond B.
Payoff = $10MM - (25% * $10MM) = $7.5MM
166
True or false: The PV of the difference between the standardized coupon rate and the credit spread on the reference obligation is paid upfront by one of the parties to the contract?
True, for example, a CDS w/ a credit spread of 75bps would require a premium payout 100bps (CDS coupon rate) by the protection buyer. To compensation the buyer for the higher-than-market premium, the protection seller would then pay upfront to the buyer the PV of 25bps.
167
The International Swaps and Derivatives Association (ISDA)
The unofficial governing body of derivative transactions.
168
ISDA Master Agreement
Where the ISDA publishes standardized contract terms and conventions.
169
Reference obligations vs reference entity of a CDS
A reference obligation is what the CDS is insuring. The issuer of the reference obligation is the reference entity. The CDS pays off when the reference entity defaults on the reference obligation OR any other issue that is pari passu.
170
Index CDS payout example:
Firm A is a protection buyer in a 5 year $100MM notional principal CDS for index B, which contains 125 entities. One of the index constituents, firm B, defaults on its bonds that trade at 30% of par after default.
(1) What will be the payoff on the CDS?
(2) What will be the notional principal of the CDS after default?
(1) Notional principal = $100MM ÷ 125 = $800,000. Firm A will receive a payout of $800,000 - (0.3 * $800,000) = $560,000
(2) $100MM - $800,000 = $9.2MM
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Types of credit events specified in CDS agreements?
- Bankruptcy: Allows the defaulting party to work w/ creditors (under supervision of a bankruptcy court) to avoid full liquidation.
- Failure to Pay: When an issuer misses a payment w/o filing for formal bankruptcy
- Restructuring: When the issuer forces creditors to accept terms that are different than those specified in the original issue. (In the U.S., bankruptcy is more common)
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Determinations Committee (DC)
A 15 member group of the ISDA that declares when a credit event occurs. At least 12/15 (supermajority) of the members must agree.
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True or false: The pricing of an index CDS is dependent on the correlation of default (credit correlation) among the entities in the index. The higher the correlation of default among index constituents, the higher the spread on the index CDS?
True
174
What factors affect the price of a CDS?
- Probability of default: Higher PD = higher CDS spread. This value fluctuates and usually gets larger over time.
- Loss given default: High LGD = higher CDS spread
- Coupon rate on a swap
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How to estimate single-period CDS premium
(1 - recovery rate) * probability of default
OR
(upfront premium % ÷ CDS duration) + CDS coupon
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Premium leg
The payments made by the protection buyer to the seller in a CDS
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Protection leg
When the protection seller pays the protection buyer in the event of a default. The difference between the premium leg and protection leg determines the upfront payment.
Upfront payment = PV(protection leg) - PV(premium leg)
OR
upfront premium ≈ (CDS spread - CDS coupon) * duration of CDS
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How to calculate initial CDS price
Notional principal - upfront premium
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How to calculate profit for the protection buyer
Change in CDS spread * CDS duration * notional principal
180
Monetizing the gain/losses
Capturing the gains/losses on an exiting in-the-money/out-the-money CDS.
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Credit curve
The relationship between credit spreads for different bonds issued by an entity, and the bonds' maturities. The credit curve is similar to the term structure of interest rates. If the longer maturity bonds have a higher credit spread compared to shorter maturity bonds, the credit curve will be upward sloping. However, if the hazard rate is constant, the credit curve will be flat.
## Footnote
* Bond portfolio managers can manage spreads by predicting the creidt curve. In anticipation of declining credit spreads, a portfolio manager may increase credit exposure by selling CDSs.
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Naked CDS
When an investor w/ no underlying exposure purchases or sells a CDS
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Long/short CDS trade
When an investor purchases protection on one reference entity while simultaneously selling protection on another reference entity. The investor is betting that the difference in credit spreads between the two reference entities will change to the investor's advantage.
184
Curve trade
A type of long/short trade where the investor is buying and selling protection on the same reference entity but with a different maturity.
## Footnote
* An investor that expects an upward-sloping credit curve to flatten should buy a short-maturity CDS and sell protection in a long-maturity CDS
* A **curve steepening** trade is when the investor believes that the short-term outlook is safer than the long-term, the investor will sell the short-term CDS and buy a long-maturity CDS.
185
True or false: Investors cannot earn aribtrage in the CDS market?
False
186
Basis trade
When investors try to profit off of differences in credit spreads between bond markets and the CDS market.
187
Collateralized debt obligtions (CDOs)
A type of derivative based on a pool of underlying debt instruments
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Synthetic vs cash CDO
Synthetic CDO: Comprised of CDSs rather than debt securities.
Cash CDO: Comprised of debt securities
## Footnote
* Both types have similar credit exposure
* If the synthetic CDO can be created at a cost lower than that of the cash CDO, investors can buy the synthetic CDO and sell the cash CDO, engaging in a profitable arbitrage.
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Securitized debt
Financing of specific assets w/o financing the entire b/s. Secured debt is usually financed via a bankruptcy-remote SPE, which allows for higher leverage and lower costs for the issuer. Investors also benefit from greater diversification, more stable cash flows and a higher risk premium relative to similarly-rated general obligation bonds due to higher complexity.
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Components of Credit Analysis of Secured Debt
1. Collateral pool: **granularity** means the transparency of the assets. The higher the granularity, the more assets. **Homogenity** means the similar of the assets in the collateral pool.
2. Servicer quality: A servicer is a middleman between the lender and the borrower the will collect and distribute payments.
3. Structure: Determines the tranching or other management of credit and other risks in a collateral pool. One key structural element is credit enhancement, which may be internal or external.
## Footnote
* W/ regads to the servicer, investors face operational and counterparty risk
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Covered bond
Senior, secured bonds issued by financial institutions backed by a collateral pool as well as by the issuer. These bonds have recourse rights.
## Footnote
* Common forms of collateral include commerical and residential mortgages and public sector loans.
192
True or false: Short-term granular and homogenous structured finance vehicles and medium-term granular and homogenous structured finance vehicles are most appropriately evaluated using a statistcal-based approach?
False, short-term granular and homogenous structured finance vehicles are best evaluated using a statistical-based approach while medium-term use a portfolio-based approach.
193
True or false: Long-term spot rates can be viewed as geometric averages of short-term forward rates?
True
194
What are the two properties of the binomial interest rate model?
1. Non-negative interest rates
2. Higher volatility at higher rates
195
Why can't a binomial model be used to value MBSs?
A binomial model or any other model that uses the backward induction method cannot be used to value an MBS because the cash flows for the MBS are dependent upon the path that interest rates have followed.
196
True or false: The OAS is the constant interest rate spread added to all rates in a binomial tree so that model price = market price?
True
## Footnote
* OAS is interpreted as the average spread over the Treasury spot rate curve
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Duration
The % change in bond price given a 1% parallel shift in the yield curve
| * Only works w/ parallel shifts and typically needs to be a small shift.
198
True or false: Convertible options are sensitive to interest rate volatility?
False
199
Pros and cons of investing in convertible bonds
Pros: There is price appreciation resulting from an increase in the value of the common stock. Also, if stock prices fall, convertible bonds limit downside risk.
Cons: The bond will underperform investing directly in the stock.
## Footnote
* If the stock price remains stable, the return on the bond may exceed the stock returns due to the coupon payments received from the bond.
200
Which of the following statements about how interest rate volatility affects the value bond is most accurate? When interest rate volatility increases, the value of a:
A. Straight bond decreases
B. Callable bond decreases
C. Putable bond decreases
B. Option values are positively related to the volatility of the underlying. Thus, when interest rate volatility increases, the values of both call and put options increase. When interest rate volatility increases, the value of a callable bond (where the investor is short the call option) decreases and the value of a putable bond (where the investor is long the put option) increases. The value of a straight bond is unaffected by changes in the volatility of interest rate, though value is affected by changes in the level of interest rate.
201
True or false: A call option will always have a positive value prior to expiration?
True
202
True or false: Negative convexity applies to convertible bonds?
False
203
True or false: During periods of market turmoil, a flight to safety may reduce long-term government bond yields resulting in a bullish flattening of the yield curve?
True
204
Soft put
A bond with an embedded soft put is redeemable through the issuance of cash, subordinated notes, common stock, or any combination of these three securities. In contrast, a bond with a hard put is only redeemable using cash.
205
How is effective convexity calculated using the binomial model?
Apply parallel shifts to the yield curve and use these curves to compute new forward rates in the interest rate tree. The resulting bond values are then used to compute the effective convexity.
206
Wall now turns his attention to the value of the embedded call option. How does the value of the embedded call option react to an increase in interest rates? The value of the embedded call is most likely to:
A. Increase
B. Decrease
C. Stay the same
B. There are two different effects that an increase in interest rate will cause in this situation. The first (and primary) impact stems from the relationship between interest rates and bond values: when interest rates increase, bond values decrease. Since the underlying asset to the option (the bond) decreases in value, the option will decrease in value also. The second (and much smaller) effect stems from the fact that when interest rates are higher, call option prices are generally higher because holding a call (rather than the underlying) is more attractive when interest rates are high. However, this secondary effect is likely to be smaller than the impact of the change in bond value.
207
Using the structural model, the value of the put option on the assets of the company is equal to:
A. the value of the call option on assets of the company.
B. value of the risky bond minus value of the risk-free bond.
C. credit valuation adjustment of the bond.
C. Under structural model the put option value = value of risk-free bond – value of the risky bond = CVA.
208
True or false: If you underestimate the value of a bond, you will also underestimate the OAS of the bond?
True
209
True or false: Holding the company's equity is economically equivalent to owning an American call option on the company's assets?
False, holding the company's equity is economically equivalent to owning a European call option on the company's assets.
210
True or false: Given the market price of a credit risky bond, the estimated risk-neutral probabilities of default and recovery rates are negatively correlated?
False, given the market price of a credit risky bond, the estimated RISK-NEUTRAL probabilities of default and recovery rates are positively correlated.
211
Which rates are option free bonds sensitive to vs callable bonds?
Option free bonds: Sensitive to the par rate that matches the bond's maturty
Callable bond: Sensitive to the par rate that matches the bond's maturity and the par rate that matches the year where the bond is callable
212
True or false: The maximum amount an investor would to pay to remove the credit risk is the the expected loss.?
False, the maximum amount an investor would to pay to remove the credit risk is the present value of the expected loss.
213
True or false: the estimated risk neutral probabilities of default and recovery rates are positively correlated?
True
