Fixed Income Flashcards
(50 cards)
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.
Unbiased Expectations
Forward rates are unbiased predictors of future spot rates; assumes risk neutrality.
Local Expectations
Local Expectations: Expected return over short periods equals risk-free rate; allows for risk aversion over longer periods.
Liquidity Preference
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.
Cox-Ingersoll-Ross Model
𝑑𝑟𝑡=𝑘(𝜃−𝑟𝑡)𝑑𝑡 +𝜎√(𝑟𝑡)d𝑍
Assumes interest rates mean reverting
k modulates the rate of mean reversion
Variance of rate changes depends on rate itself, so helps prevent rate becoming negative
Vasicek Model
𝑑𝑟𝑡=𝑘(𝜃−𝑟𝑡)𝑑𝑡 +𝜎d𝑍
Interest rates calculated with constant volatility
With this its theoretically possible to have negative interest rates
Ho-Lee Model
𝑑𝑟𝑡=𝜃𝑡𝑑𝑡+𝜎𝑑𝑍
Arbitrage free model calibrated to market data using binomial lattice
No mean reversion assumption
Drift term 𝜃 is time dependent
Had constant volatility
Kalotay-Williams-Fabozzi Model
𝑑ln(𝑟𝑡)=𝜃𝑡𝑑𝑡+𝜎𝑑𝑍
Differential equation describes the log of the short rate, meaning the short rate is log normally distributed
Interest rate options influenced by tails of distribution
Can’t be negative
Gauss+ Model
Incorporates short, medium, long term rates
Long-term is mean reverting and reflects macro parameters
Medium term reverts to long-run rate
Short-term impacted by central bank policy
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.
OAS formula
OAS_call = Z-spread - call option
OAS_put = Z-spread + put option
Volatility decreases the OAS for call options, increases for put options
Bond effective 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).
