Fixed Income

Question 1

Describe relationships among spot rates, forward rates, yield to maturity, expected and realized returns on bonds, and the shape of the yield curve.

Answer 1

The spot rate for a particular maturity is equal to a geometric average of the one-period spot rate and a series of one-period forward rates.

When the spot curve is flat, forward rates will equal spot rates and yields. When the spot curve is upward sloping (downward sloping), forward rate curves will be above (below) the spot curve and the yield for a maturity of T will be less than (greater than) the spot rate ST .

The forward pricing model values forward contracts by using an arbitrage-free framework that equates buying a zero-coupon bond to entering into a forward contract to buy a zero-coupon bond in the future that matures at the same time.

The forward rate model tells us that the investors will be indifferent between buying a long-maturity zero-coupon bond versus buying a shorter-maturity zero-coupon bond and reinvesting the principal at the locked in forward rate .

Question 2

Explain how a bond’s exposure to each of the factors driving the yield curve can be measured and how these exposures can be used to manage yield curve risks.

Answer 2

We can measure a bond’s exposures to the factors driving the yield curve in a number of ways:

Effective duration—Measures the sensitivity of a bond’s price to parallel shifts in the benchmark yield curve.

Key rate duration—Measures bond price sensitivity to a change in a specific spot rate keeping everything else constant.

Sensitivity to parallel, steepness, and curvature movements—Measures sensitivity to three distinct categories of changes in the shape of the benchmark yield curve.

Question 3

Describe short-term interest rate spreads used to gauge economy-wide credit risk and liquidity risk.

Answer 3

The Z-spread is the spread that when added to each spot rate on the yield curve makes the present value of a bond’s cash flows equal to the bond’s market price. The Z refers to zero volatility—a reference to the fact that the Z-spread assumes interest rate volatility is zero. Z-spread is not appropriate to use to value bonds with embedded options.

TED spreads

TED = T-bill + ED (“ED” is the ticker symbol for the Eurodollar futures contract)

TED spread = (three-month LIBOR rate) − (three-month T-bill rate)

The TED spread is used as an indication of the overall level of credit risk in the economy.

LIBOR-OIS spread

The LIBOR-OIS spread is the amount by which the LIBOR rate (which includes some credit risk) exceeds the overnight indexed swap (OIS) rate (which includes only minimal credit risk).

Question 4

Explain the maturity structure of yield volatilities and their effect on price volatility.

Answer 4

The maturity structure of yield volatilities indicates the level of yield volatilities at different maturities. This term structure thus provides an indication of yield curve risk. The volatility term structure usually indicates that short-term rates (which are linked to uncertainty over monetary policy) are more volatile than long-term rates (which are driven by uncertainty related to the real economy and inflation). Fixed income instruments with embedded options can be especially sensitive to interest rate volatility.

Question 5

Describe how zero-coupon rates (spot rates) may be obtained from the par curve by bootstrapping.

Answer 5

By using a process called bootstrapping, spot rates (i.e., zero-coupon rates) can be derived from the par curve iteratively—one spot rate at a time.

Question 6

Explain how key economic factors are used to establish a view on benchmark rates, spreads, and yield curve changes.

Answer 6

Inflation forecasts, GDP growth, and monetary policy affect bond yields. Bond risk premium is the excess return (over the one-year risk-free rate) from investing in longer-term government bonds. Additionally, fiscal policy, maturity structure, and investor demand all affect yields.

In expectation of a rise (fall) in rates, investors will lower (extend) the duration of their bond portfolios. An investor will rotate out of a bullet portfolio and into a barbell portfolio in expectation of a bullish flattening of the yield curve.

Question 7

Describe the strategy of rolling down the yield curve.

Answer 7

When the yield curve is upward sloping, bond managers may use the strategy of “rolling down the yield curve” to chase above-market returns. By holding long-maturity rather than short-maturity bonds, the manager earns an excess return as the bond “rolls down the yield curve” (i.e., approaches maturity and increases in price). As long as the yield curve remains upward sloping and the spot rates continue to be lower than previously implied by their corresponding forward rates, this strategy will add to the return of a bond portfolio.

Question 8

Describe the assumptions concerning the evolution of spot rates in relation to forward rates implicit in active bond portfolio management.

Answer 8

If spot rates evolve as predicted by forward rates, bonds of all maturities will realize a one-period return equal to the one-period spot rate and the forward price will remain unchanged.

Active bond portfolio management is built on the presumption that the current forward curve may not accurately predict future spot rates. Managers attempt to outperform the market by making predictions about how spot rates will change relative to the rates suggested by forward rate curves.

If an investor believes that future spot rates will be lower than corresponding forward rates, then the investor will purchase bonds (at a presumably attractive price) because the market appears to be discounting future cash flows at “too high” of a discount rate.

Question 9

Explain the swap rate curve and why and how market participants use it in valuation.

Answer 9

The swap rate curve provides a benchmark measure of interest rates. It is similar to the yield curve except that the rates used represent the interest rates of the fixed-rate leg in an interest rate swap.

Market participants prefer the swap rate curve as a benchmark interest rate curve rather than a government bond yield curve for the following reasons:

Swap rates reflect the credit risk of commercial banks rather than that of governments.
The swap market is not regulated by any government.
The swap curve typically has yield quotes at many maturities.

Question 10

Explain traditional theories of the term structure of interest rates and describe the implications of each theory for forward rates and the shape of the yield curve.

Answer 10

There are several traditional theories that attempt to explain the term structure of interest rates:

Unbiased expectations theory—Forward rates are an unbiased predictor of future spot rates. Also known as the pure expectations theory.

Local expectations theory—Bond maturity does not influence returns for short holding periods.

Liquidity preference theory—Investors demand a liquidity premium that is positively related to a bond’s maturity.

Segmented markets theory—The shape of the yield curve is the result of the interactions of supply and demand for funds in different market (i.e., maturity) segments.

Preferred habitat theory—Similar to the segmented markets theory, but recognizes that market participants will deviate from their preferred maturity habitat if compensated adequately.

Question 11

Calculate and interpret the swap spread for a given maturity.

Answer 11

We define swap spread as the additional interest rate paid by the fixed-rate payer of an interest rate swap over the rate of the “on-the-run” government bond of the same maturity.

swap spread = (swap rate) − (Treasury bond yield)

Investors use the swap spread to separate the time value portion of a bond’s yield from the risk premia for credit and liquidity risk. The higher the swap spread, the higher the compensation for liquidity and credit risk.

For a default-free bond, the swap spread provides an indication of (1) the bond’s liquidity and/or (2) possible mispricing.

Question 12

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.

Answer 12

In the pathwise valuation approach, the value of the bond is simply the average of the values of the bond at each path. For a n-period binomial tree, there are 2(n-1) possible paths.

Question 13

Describe a binomial interest rate tree framework.

Answer 13

The binomial interest rate tree framework is a lognormal model with two equally likely outcomes for one-period forward rates at each node. A volatility assumption drives the spread of the nodes in the tree.

Question 14

Describe the process of calibrating a binomial interest rate tree to match a specific term structure.

Answer 14

A binomial interest rate tree is calibrated such that (1) the values of benchmark bonds using the tree are equal to the bonds’ market prices, (2) adjacent forward rates at any nodal period are two standard deviations apart and (3) the midpoint for each nodal period is approximately equal to the implied one-period forward rate for that period.

Question 15

Describe term structure models and how they are used.

Answer 15

Two major classes of term structure models are as follows:

Equilibrium term structure models: Attempt to model the term structure using fundamental economtextc variables that are thought to determtextne interest rates.

Arbitrage-free models: Begin with observed market prices and the assumption that securities are correctly priced.

Question 16

Describe the Equilibrium term models and how they are used.

Answer 16

Cox-Ingersoll-Ross 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 non-negative rates only.

Vasicek model: Simtextlar to the CIR model, but assumes that the interest rate volatility level is constant and independent of the level of short-term interest rates. Can produce negative rates.

Question 17

Describe the Arbitrage-free models and how they are used.

Answer 17

Ho-Lee model: Calibrated by using market prices to find the time-dependent drift term θt that generates the current term structure. Assumes that short-term rates are normally distributed with a constant volatility. Produces a normal distribution of rates and rates can be negative.

Kalotay–Williams–Fabozzi (KWF) model: Simtextlar to the Ho-Lee model, but assumes that short rate has a lognormal distribution.

Question 18

Describe the backward induction valuation methodology and calculate the value of a fixed-income instrument given its cash flow at each node.

Answer 18

Backward induction is the process of valuing a bond using a binomial interest rate tree. The term backward is used because in order to determine the value of a bond at Node 0, we need to know the values that the bond can take on at nodal period 1, and so on.

Question 19

Compare pricing using the zero-coupon yield curve with pricing using an arbitrage-free binomial lattice.

Answer 19

Valuation of bonds using a zero-coupon yield curve (also known as the spot rate curve) is suitable for option-free bonds. However, for bonds with embedded options where the value of the option varies with outcome of unknown forward rates, a model that allows for variability of forward rates is necessary. One such model is the binomial interest rate tree framework.

Question 20

Explain what is meant by arbitrage-free valuation of a fixed-income instrument.

Answer 20

Arbitrage-free valuation leads to a security value such that no market participant can earn an arbitrage profit in a trade involving that security. In other words, the valuation is consistent with the value additivity principle and without dominance of any security relative to others in the market.

Question 21

Calculate the arbitrage-free value of an option-free, fixed-rate coupon bond.

Answer 21

Arbitrage-free valuation of fixed-rate, option-free bonds entails discounting each of the bond’s future cash flows at its corresponding spot rate.

Question 22

Describe a Monte Carlo forward-rate simulation and its application.

Answer 22

The Monte Carlo simulation method uses pathwise valuation and a large number of randomly generated simulated paths. Mortgage-backed securities have path-dependent cash flows on account of the embedded prepayment option. The Monte Carlo simulation method should be used for valuing MBS as the binomial tree backward induction process is inappropriate for securities with path-dependent cash flows.

Question 23

Explain the relationships between the values of a callable or putable bond, the underlying option-free (straight) bond, and the embedded option.

Answer 23

Value of option embedded in a callable or putable bond:

Vcall = Vstraight − Vcallable

Vput = Vputable − Vstraight

Question 24

Explain how interest rate volatility affects the value of a callable or putable bond.

Answer 24

When interest rate volatility increases, the value of both call and put options on bonds increase. As volatility increases, the value of a callable bond decreases (remember that the investor is short the call option) and the value of a putable bond increases (remember that the investor is long the put option).

Question 25

Describe fixed-income securities with embedded options.

Answer 25

Bonds with embedded options allow issuers to manage their interest rate risk or issue bonds at attractive coupon rates. The embedded options can be simple call or put options, or more complex options such as provisions for sinking fund, estate puts, et cetera.

Question 26

Calculate and interpret the components of a convertible bond’s value.

Answer 26

The value of a bond with embedded options is determined as the value of the straight bond plus (minus) the value of options that the investor is long (short).

callable and putable convertible bond value	
= straight value of bond
\+ value of call option on stock
– value of call option on bond
\+ value of put option on bond

Question 27

Calculate the value of a capped or floored floating-rate bond.

Answer 27

The owner of a convertible bond can exchange the bond for the common shares of the issuer. A convertible bond includes an embedded call option giving the bondholder the right to buy the common stock of the issuer.

Question 28

Calculate the value of a capped or floored floating-rate bond.

Answer 28

The owner of a convertible bond can exchange the bond for the common shares of the issuer. A convertible bond includes an embedded call option giving the bondholder the right to buy the common stock of the issuer.

Question 29

Compare effective durations of callable, putable, and straight bonds.

Answer 29

effective duration (callable) ≤ effective duration (straight)

effective duration (putable) ≤ effective duration (straight)

Question 30

Describe how a convertible bond is valued in an arbitrage-free framework.

Answer 30

The major benefit from investing in convertible bonds is the price appreciation resulting from an increase in the value of the common stock.

The main drawback of investing in a convertible bond versus investing directly in the stock is that when the stock price rises, the bond will underperform the stock because of the conversion premium of the bond.

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.

If the stock price falls, the straight value of the bond limits downside risk (assuming bond yields remain stable).

Question 31

Explain the calculation and use of option-adjusted spreads.

Answer 31

The option adjusted spread (OAS) is the constant spread added to each forward rate in a benchmark binomial interest rate tree, such that the sum of the present values of a credit risky bond’s cash flows equals its market price.

Question 32

Calculate the value of a callable or putable bond from an interest rate tree.

Answer 32

A backwards induction process is used in a binomial interest rate tree framework for valuing a callable (or putable) bond. In the binomial tree, we use one-period forward rates for each period. For valuing a callable (putable) bond, the value used at any node corresponding to a call (put) date must be either the price at which the issuer will call (investor will put) the bond, or the computed value if the bond is not called (put)—whichever is lower (higher).

Question 33

Describe defining features of a convertible bond.

Answer 33

The conversion ratio is the number of common shares for which a convertible bond can be exchanged.

conversion value = market price of stock × conversion ratio

market conversion price = market price of convertible bond / conversion ratio

market conversion premium per share = market conversion price − market price

The minimum value at which a convertible bond trades is its straight value or its conversion value, whichever is greater.

Question 34

Compare the risk–return characteristics of a convertible bond with the risk–return characteristics of a straight bond and of the underlying common stock.

Answer 34

The major benefit from investing in convertible bonds is the price appreciation resulting from an increase in the value of the common stock.

The main drawback of investing in a convertible bond versus investing directly in the stock is that when the stock price rises, the bond will underperform the stock because of the conversion premium of the bond.

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.

If the stock price falls, the straight value of the bond limits downside risk (assuming bond yields remain stable).

Question 35

Explain how changes in the level and shape of the yield curve affect the value of a callable or putable bond.

Answer 35

The short call in a callable bond limits the investor’s upside when rates decrease, while the long put in a putable bond hedges the investor against rate increases.

The value of the call option will be lower in an environment with an upward-sloping yield curve as the probability of the option going in the money is low. A call option gains value when the upward-sloping yield curve flattens. A put option will have a higher probability of going in the money when the yield curve is upward sloping; the option loses value if the upward-sloping yield curve flattens.

Question 36

Explain how interest rate volatility affects option-adjusted spreads.

Answer 36

Binomial trees generated under an assumption of high volatility will lead to higher values for a call option and a corresponding lower value for a callable bond. Under a high volatility assumption, we would already have a lower computed value for the callable bond, and hence, the additional spread (i.e., the OAS) needed to force the discounted value to equal the market price will be lower.

When an analyst uses a lower-than-actual (higher-than-actual) level of volatility, the computed OAS for a callable bond will be too high (low) and the bond will be erroneously classified as underpriced (overpriced).

Similarly, when the analyst uses a lower-than-actual (higher-than-actual) level of volatility, the computed OAS for a putable bond will be too low (high) and the bond will be erroneously classified as overpriced (underpriced).

Question 37

Describe how the arbitrage-free framework can be used to value a bond with embedded options.

Answer 37

To value a callable or a putable bond, the backward induction process and a binomial interest rate tree framework is used. The benchmark binomial interest rate tree is calibrated to ensure that it values benchmark bonds correctly (i.e., that it generates prices equal to their market prices).

Question 38

Compare effective convexities of callable, putable, and straight bonds.

Answer 38

Straight and putable bonds exhibit positive convexity throughout. Callable bonds also exhibit positive convexity when rates are high. However, at lower rates, callable bonds exhibit negative convexity.

Question 39

Describe the use of one-sided durations and key rate durations to evaluate the interest rate sensitivity of bonds with embedded options.

Answer 39

For bonds with embedded options, one-sided durations—durations when interest rates rise versus when they fall—are better at capturing interest rate sensitivity than the more common effective duration. When the underlying option is at (or near) the money, callable (putable) bonds will have lower (higher) one-sided down-duration than one-sided up-duration.

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.

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.

Question 40

Explain structural and reduced-form models of corporate credit risk, including assumptions, strengths, and weaknesses.

Answer 40

Structural models of corporate credit risk are based on the structure of a company’s balance sheet and rely on insights provided by option pricing theory. Structural models consider equity as a call option on company assets. Alternately, an investment in a risky bond can be viewed as equivalent to purchasing a risk-free bond and writing a put option on company assets.

Unlike the structural model, reduced-form models do not explain why default occurs. Instead, they statistically model when default occurs. Default under a reduced-form model is a randomly occurring exogenous variable.

Question 41

Interpret changes in a credit spread.

Answer 41

Credit spreads change as investors’ perceptions change about the future probability of default and recovery rates. Expectations of impending recessions lead to expectations of higher default and lower recovery rates.

Question 42

Explain credit scores and credit ratings.

Answer 42

Credit scoring is used for small businesses and individuals. Credit ratings are issued for corporate debt, asset-backed securities, and government and quasi-government debt. Credit scores and ratings are ordinal rankings (higher = better). Notching different issues of the same issuer based on seniority ensures that credit ratings incorporate the probability of default as well as loss given default.

Question 43

Calculate the expected return on a bond given transition in its credit rating.

Answer 43

The change in the price of a bond resulting from credit migration depends on the modified duration of the bond and the change in spread.

Δ%P = –(modified duration of the bond) × (Δ spread)

Question 44

Compare the credit analysis required for securitized debt to the credit analysis of corporate debt.

Answer 44

Credit analysis of ABS involves the analysis of the collateral pool, servicer quality, and the structure of secured debt (i.e., distribution waterfall).

Question 45

Calculate the value of a bond and its credit spread, given assumptions about the credit risk parameters.

Answer 45

credit spread on a risky bond = YTM of risky bond − YTM of benchmark

The value of a risky bond, assuming it does not default, is its value given no default (VND). VND is calculated using the risk-free rate to value the risky bond.

A backward induction procedure using a risky bond’s cash flows and a benchmark interest-rate tree can also be used to calculate the bond’s VND.

Question 46

Explain expected exposure, the loss given default, the probability of default, and the credit valuation adjustment.

Answer 46

Expected exposure is the amount of money a bond investor in a credit risky bond stands to lose at a point in time before any recovery is factored in. Loss given default is equal to loss severity multiplied by exposure. Credit valuation adjustment (CVA) is the sum of the present values of expected losses for each period.

Question 47

Explain the determinants of the term structure of credit spreads and interpret a term structure of credit spreads.

Answer 47

The term structure of credit spreads shows the relationship between credit spreads and maturity. Term structure depends on credit quality, financial conditions, demand and supply in the bond market, and expected volatility in the equity markets.

Question 48

Explain the principles underlying and factors that influence the market’s pricing of CDS.

Answer 48

In a naked CDS, an investor with no exposure to the underlying purchases protection in the CDS market.

In a long/short trade, an investor purchases protection on one reference entity while selling protection on another reference entity.

A curve trade is a type of long/short trade where the investor is buying and selling protection on the same reference entity but with different maturities. An investor who believes the short-term outlook for the reference entity is better than the long-term outlook can use a curve-steepening trade (buying protection in a long-term CDS and selling protection in a short-term CDS) to profit if the credit curve steepens. Conversely, an investor who is bearish about the reference entity’s prospects in the short term will enter into a curve-flattening trade.

Question 49

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.

Answer 49

A basis trade is an attempt to exploit the difference in credit spreads between bond markets and the CDS market. Basis trades rely on the idea that such mispricings will be temporary and that disparity should eventually disappear after it is recognized.

If a synthetic CDO can be created at a cost lower than that of the equivalent cash CDO, investors can buy the synthetic CDO and sell the cash CDO, producing a profitable arbitrage.

Question 50

Describe credit events and settlement protocols with respect to CDS.

Answer 50

The factors that influence the pricing of CDS (i.e., CDS spread) include the probability of default, the loss given default, and the coupon rate on the swap. The CDS spread is higher for a higher probability of default and for a higher loss given default. The conditional probability of default (i.e., the probability of default given that default has not already occurred) is called the hazard rate.

(expected loss)t = (hazard rate)t × (loss given default)t

The upfront premium on a CDS can be computed as:

upfront payment (by protection buyer) = PV(protection leg) − PV(premium leg)

Or approximately:

upfront premium ≈ (CDS spread − CDS coupon) × duration

The change in value for a CDS after inception can be approximated by the change in spread multiplied by the duration of the CDS.

profit for protection buyer ≈ change in spread × duration × notional principal

profit for protection buyer (%) ≈ change in spread (%) × duration

Question 51

Describe the use of CDS to take advantage of valuation disparities among separate markets, such as bonds, loans, equities, and equity-linked instruments.

Answer 51

A default is defined as occurrence of a credit event. Common types of credit events specified in CDS agreements include bankruptcy, failure to pay, and restructuring.

When there is a credit event, the swap will be settled in cash or by physical delivery.

Question 52

Describe credit default swaps (CDS), single-name and index CDS, and the parameters that define a given CDS product.

Answer 52

A credit default swap (CDS) is essentially an insurance contract wherein upon occurrence of a credit event, the credit protection buyer gets compensated by the credit protection seller. To obtain this coverage, the protection buyer pays the seller a premium called the CDS spread. In the case of a single-name CDS, the reference obligation is the fixed income security on which the swap is written. An index CDS covers an equally-weighted combination of borrowers.