Fixed Income

Question 1

What a spot rate and how does the spot rate curve look like when it is upward sloping, downward sloping, and flat?

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, forward rate curves will be above the spot curve and the yield for a maturity of T will be less than the spot rate ST .

When the spot curve is downward sloping, forward rate curves will be below the spot curve and the yield for a maturity of T will be greater than the spot rate ST .

Question 2

What is the forward pricing model?

Answer 2

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.

Question 3

What is the forward rate model?

Answer 3

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 4

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

Answer 4

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 5

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

Answer 5

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 because the market appears to be discounting future cash flows at “too high” of a discount rate.

Question 6

Describe the strategy of rolling down the yield curve.

Answer 6

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” (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 7

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

Answer 7

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 becasue:

• 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 8

Calculate and interpret the swap spread for a given maturity.

Answer 8

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.

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 9

What are the 3 short-term interest rate spreads used to gauge economy-wide credit risk and liquidity risk?

Answer 9

The 3 spreads used to used to gauge economy-wide credit risk and liquidity risk are

1. The Z-spread
2. TED spreads
3. LIBOR-OIS spread

Question 10

What is the Z-spread?

Answer 10

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.

Question 11

What is the TED Spread?

Answer 11

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.

Question 12

What is the LIBOR-OIS spread?

Answer 12

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 13

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 13

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 14

What are the 3-ways you can measure a bond’s exposures to the factors driving the yield curve?

Answer 14

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 15

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

Answer 15

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 16

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

Answer 16

What economic factors affect bond yields

• Inflation forecasts
• GDP growth
• Monetary policy: More/less supply of any maturity will lower/raise yields
• Bond risk premium is the excess return (over the one-year risk-free rate) from investing in longer-term government bonds
• Fiscal policy: larger/longer deficits = more supply = higher yields
• maturity structure
• investor demand

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 17

What happens when rates rise or falls?

Answer 17

Bull Steepener: Rates Fall, Curve Steepens, Bullet; Drop rates recession

Bull Flattener: Rates Fall, Curve Flatter, Barbell; Flight to qualilty or QE

Bear Steepener: Rates Rise, Curve Steepens, Bullet; Rising LT Inflation

Bear Flattener: Rates Rise, Curve Flatter, Barbell; Raise Rates for Inflation

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 18

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

Answer 18

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 19

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

Answer 19

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 20

Describe a binomial interest rate tree framework.

Answer 20

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 21

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

Answer 21

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
3. The midpoint for each nodal period is approximately equal to the implied one-period forward rate for that period

Question 22

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

Answer 22

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 23

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

Answer 23

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 24

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 24

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 25

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

Answer 25

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 26

Describe term structure models and how they are used.

Answer 26

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. “Mean Reverting”

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

Question 27

Describe the Equilibrium term models and how they are used.

Answer 27

Cox-Ingersoll-Ross model:

• Mean reverting
• Negative rates not possible
• Interest rate volatility varies with √rt and is not constant.

Vasicek model:

• Mean reverting
• Negative rates possible
• Constant volatility

Question 28

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

Answer 28

Ho-Lee model:

• Time dependent drift term
• Not mean reverting
• Negative rates possible
• Constant volatility

Kalotay–Williams–Fabozzi (KWF) model:

• Time dependent drift term
• Not mean reverting
• Log of rt –> negative rates not possible
• Interest rate volatility varies

Question 29

Describe fixed-income securities with embedded options.

Answer 29

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 30

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

Answer 30

Value of option embedded in a callable or putable bond:

Vcall = Vstraight − Vcallable

Vput = Vputable − Vstraight

Question 31

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

Answer 31

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 32

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

Answer 32

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 33

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

Answer 33

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 34

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

Answer 34

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 35

Explain the calculation and use of option-adjusted spreads.

Answer 35

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 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

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

Answer 37

effective duration (callable) ≤ effective duration (straight)

effective duration (putable) ≤ effective duration (straight)

Question 38

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

Answer 38

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 39

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

Answer 39

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 40

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

Answer 40

A capped floater contains an issuer option that prevents the coupon rate on a floater from rising above a specified maximum “cap rate”.

Value of a capped floater = value of a straight floater - value of the embedded cap

A related floating-rate bond is the floored floater where the coupon rate will not fall below a specified minimum “the floor”.

Value of a floored floater = value of a straight floater + value of the embedded floor.

Question 41

Describe defining features of a convertible bond.

Answer 41

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 42

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

Answer 42

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 43

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

Answer 43

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 44

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 44

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 45

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

Answer 45

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 46

Explain credit scores and credit ratings.

Answer 46

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 47

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

Answer 47

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 48

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

Answer 48

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/model. They try to explain why default occurs.

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.

Question 49

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

Answer 49

They statistically model when default occurs. Default under a reduced-form model is a randomly occurring exogenous variable. These models use ratios and macroeconomic factos

Question 50

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

Answer 50

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 51

Interpret changes in a credit spread.

Answer 51

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 52

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

Answer 52

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
• expected volatility in the equity markets

Question 53

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

Answer 53

Credit analysis of ABS involves the analysis of:

• the collateral pool
• servicer quality
• the structure of secured debt (distribution waterfall)

Question 54

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

Answer 54

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.

Question 55

Describe credit events and settlement protocols with respect to CDS.

Answer 55

A default is defined as occurrence of a credit event.

Common types of credit events specified in CDS agreements include:

• bankruptcy
• failure to pay
• restructuring

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

Question 56

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

Answer 56

The factors that influence the pricing of CDS (CDS spread) include:

• the probability of default
• the loss given default
• 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 hazard rate:“the conditional probability of default” probability of default given that default has not already occurred

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

Question 57

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 57

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.

Question 58

What is a curve trade for a CDS

Answer 58

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 59

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

Answer 59

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.