Fixed Income Flashcards
Describe the relationship between spot rates and forward rates.
Forward rates are implied future spot rates. The relationship is defined by $ (1 + z_B)^B = (1 + z_A)^A (1 + f_{A,B-A})^{B-A} $. If the spot curve is upward sloping, the forward curve is above the spot curve. If downward sloping, the forward curve is below.
How can zero-coupon rates (spot rates) be obtained from a par curve?
Using bootstrapping. Start with the shortest maturity (par rate = spot rate). Then, use the known shorter-term spot rates (and their discount factors) along with the par rate for the next maturity to solve for the unknown spot rate (discount factor) for that maturity, ensuring the par bond is priced at 100. Repeat for successive maturities. Formula: $ DF_N = \frac{1 - C_N(\sum_{i=1}^{N-1} DF_i)}{1 + C_N} $
What is the strategy of “rolling down the yield curve”?
Assuming an upward-sloping and stable yield curve, buying a bond with a maturity longer than the investment horizon. As time passes, the bond’s maturity shortens (“rolls down”), and its yield is expected to fall (price rise), generating capital gains in addition to coupon income.
What is the swap rate curve, and why is it used?
A yield curve based on the fixed rates of interest rate swaps (often based on MRR like SOFR) across different maturities. Used by market participants (especially banks) as a benchmark for funding costs and pricing derivatives, reflecting bank credit risk rather than government risk.
What are the traditional theories of the term structure?
Unbiased Expectations: Forward rates are unbiased predictors of future spot rates; assumes risk neutrality. Local Expectations: Expected return over short periods equals risk-free rate; allows for risk aversion over longer periods. Liquidity Preference: Long-term rates include a liquidity premium for bearing interest rate risk; forward rates are upwardly biased predictors. Segmented Markets: Rates determined by supply/demand within distinct maturity segments. Preferred Habitat: Like segmented, but investors will shift maturities if compensated sufficiently.
How can yield curve risk be decomposed and managed?
Yield curve movements can be decomposed into factors: Level (parallel shift), Steepness (twist), and Curvature (butterfly). Risk managed using Duration (for level risk) and Key Rate Durations (for sensitivity to specific maturity points, capturing steepness/curvature risks).
What is meant by arbitrage-free valuation of fixed income?
Valuing a bond by discounting its cash flows using multiple discount rates derived from a benchmark yield curve (typically spot rates) such that the model price equals the market price, preventing arbitrage opportunities. Ensures value additivity and dominance principles hold.
How is a binomial interest rate tree calibrated?
The tree is built such that: 1) Adjacent forward rates are consistent with assumed volatility ($ r_{uh} = r_{ul} e^{2\sigma} $). 2) The values of benchmark bonds calculated using the tree match their market prices (arbitrage-free). This is often done using backward induction, adjusting rates iteratively.
How is backward induction used to value bonds in a binomial tree?
Start at the final maturity nodes (value = par + coupon). At each preceding node, calculate the expected present value of the bond by averaging the values from the two subsequent nodes (plus coupon at the current node) and discounting back one period using the forward rate at the current node. Check for embedded options (e.g., call/put) at each node and adjust value accordingly (Value cannot exceed call price or be less than put price).
Contrast equilibrium vs. arbitrage-free term structure models.
Equilibrium Models (e.g., CIR, Vasicek): Start with assumptions about economic variables, derive term structure endogenously. May not perfectly fit current market curve. Arbitrage-Free Models (e.g., Ho-Lee, KWF): Calibrated to exactly fit the current term structure, ensuring no arbitrage. Drift term often time-dependent.
How are callable and putable bonds valued relative to straight bonds?
Value(Callable Bond) = Value(Straight Bond) - Value(Call Option). Value(Putable Bond) = Value(Straight Bond) + Value(Put Option). The options adjust the value based on the issuer’s right (call) or investor’s right (put).
How does interest rate volatility affect the value of callable and putable bonds?
Higher volatility increases the value of embedded options. Therefore, higher volatility decreases the value of a callable bond (call option more valuable to issuer) and increases the value of a putable bond (put option more valuable to investor).
What is the Option-Adjusted Spread (OAS)?
The constant spread added to all forward rates in an arbitrage-free binomial tree that makes the model value of a bond with embedded options equal to its market price. Represents the spread after removing the influence of the option.
How does interest rate volatility affect OAS?
For a callable bond, higher volatility implies a lower OAS (as the option value subtracted is larger). For a putable bond, higher volatility implies a higher OAS (as the option value added is larger).
Compare Effective Duration and Convexity for callable, putable, and straight bonds.
Duration: Callable bonds have lower effective duration than straight bonds when rates fall (call risk limits price upside). Putable bonds have lower effective duration than straight bonds when rates rise (put floor limits price downside). Convexity: Straight & Putable bonds always have positive convexity. Callable bonds can exhibit negative convexity when rates are low (near call price) as price appreciation is capped.
How is a convertible bond valued?
Value = Max(Conversion Value, Value of Straight Bond) + Value of Conversion Option. Can be valued in an arbitrage-free framework considering the equity option component. Minimum value is Max(Conversion Value, Straight Bond Value).
Define the components of credit risk modeling: Expected Exposure, POD, LGD, CVA.
Expected Exposure: Amount potentially lost before recovery if default occurs. Probability of Default (POD): Likelihood issuer fails to meet obligations. Loss Given Default (LGD): Percentage of exposure lost after recovery (LGD = Exposure * (1 - Recovery Rate)). Credit Valuation Adjustment (CVA): PV of expected credit losses; difference between risk-free bond value and risky bond value (VRF−VRisky).
Contrast Structural and Reduced-Form credit models.
Structural: Models default based on issuer’s balance sheet structure (value of assets relative to liabilities). Default occurs if asset value falls below debt threshold. Views equity as call option on assets. Reduced-Form: Models default statistically based on observed market data (credit spreads, ratings) without modeling company structure. Uses hazard rates/default intensities.
What determines the term structure of credit spreads?
Economic conditions (wider in recessions), market liquidity, issuer-specific factors, supply/demand for debt. Lower-rated bonds typically have steeper credit curves.
What is a Credit Default Swap (CDS)?
A derivative where a protection buyer pays periodic premiums to a protection seller, who compensates the buyer if a predefined credit event (e.g., bankruptcy, failure to pay) occurs for a reference entity/obligation.
How are CDS contracts settled?
Cash Settlement: Protection buyer receives Notional * (1 - Recovery Rate), keeps the bond. Physical Settlement: Protection buyer delivers the defaulted bond (or cheapest-to-deliver) to seller and receives Notional. Recovery rate determined by auction for cash settlement.
How is the upfront premium on a CDS calculated?
Required when the CDS Spread (market price) differs from the standard CDS Coupon (e.g., 1% for IG, 5% for HY). Upfront Premium ≈ (CDS Spread - CDS Coupon) * CDS Duration.
Describe common CDS trading strategies (Curve Trades, Basis Trades).
Curve Trade: Long CDS at one maturity, short CDS at another maturity on the same entity to bet on changes in the shape of the credit curve (e.g., steepening/flattening). Basis Trade: Exploits difference between CDS spread and bond yield spread (e.g., buy bond + buy CDS protection if bond spread > CDS spread, hoping for convergence).
