Fixed Income

Question 1

Spot Rates

Answer 1

Spot rates are the annualized market interest rates for a single payment to be received in the future. Generally, we use spot rates for government securities (risk-free) to generate the spot rate curve. We sometimes refer to spot rates as zero-coupon rates.

Question 2

Forward Rate

Answer 2

A forward rate is an interest rate agreed today for a loan to be made at some future date.

Question 3

Expected Return will be equal to the bond’s yield only when all three of the following are true

Answer 3

• The bond is held to maturity
• All payments (coupon and principal) are made on time and in full
• All coupons are reinvested at the original YTM

Question 4

Par Rate

Answer 4

A par rate is the yield to maturity of a bond trading at par. Par rates for bonds with different maturities make up the par rate curve or simply the par curve. By definition, the par rate will be equal to the coupon rate on the bond. Generally, par curve refers to the par rates for government or benchmark bonds.

Question 5

Maturity matching

Answer 5

Purchasing bonds that have a maturity equal to the investor’s investment horizon.

Question 6

Swap rates are preferred over government bond yield because

Answer 6

• 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,
• Wholesale banks that manage interest rate risk with swap contracts are more likely to use swap curves to value their assets and liabilities. Retail banks are more likely to use a government bond yield curve.

Question 7

Swap spread

Answer 7

Swap spread refers to the amount by which the swap rate exceeds the yield of a government bond with the same maturity.

swap spread_t = swap rate_t − Treasury yield_t

The LIBOR swap curve is arguably the most commonly used interest rate curve. This rate curve roughly reflects the default risk of a commercial bank.​

Question 8

I Spread

Answer 8

The I-spread for a credit-risky bond is the amount by which the yield on the risky bond exceeds the swap rate for the same maturity.

Interpolated rate = 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)

Question 9

Z spread

Answer 9

The Z-spread is 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.

Question 10

TED Spread

Answer 10

The “TED” in “TED spread” is an acronym that combines the “T” in “T-bill” with “ED” (the ticker symbol for the Eurodollar futures contract).

Conceptually, the TED spread is the amount by which the interest rate on loans between banks (formally, three-month LIBOR) exceeds the interest rate on short-term U.S. government debt (three-month T-bills).

TED spread = (3-month LIBOR rate) − (3-month T-bill rate)

TED spread is seen as an indication of the risk of interbank loans and a measure of counterparty risk.

Question 11

LIBOR OI spread

Answer 11

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

OIS stands for overnight indexed swap. The OIS rate roughly reflects the federal funds rate and includes minimal counterparty risk.

LIBOR-OIS spread is a useful measure of credit risk and an indication of the overall wellbeing of the banking system. A low LIBOR-OIS spread is a sign of high market liquidity while a high LIBOR-OIS spread is a sign that banks are unwilling to lend due to concerns about creditworthiness.

Question 12

Unbiased expectations theory

Answer 12

• Investors’ expectations determine the shape of the interest rate term structure. The underlying principle behind the pure expectations theory is risk neutrality: Investors don’t demand a risk premium for maturity strategies that differ from their investment horizon.
• 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.
• Implications for the shape of the yield curve
  
  • If the yield curve is upward sloping, short-term rates are expected to rise.
  • If the curve is downward sloping, short-term rates are expected to fall.
  • A flat yield curve implies that the market expects short-term rates to remain constant.

Question 13

Local expectations theory

Answer 13

• Similar to the unbiased expectations theory with one major difference: the local expectations theory preserves the risk-neutrality assumption only for short holding periods. In other words, over longer periods, risk premiums should exist.
• This implies that over short time periods, every bond (even long-maturity risky bonds) should earn the risk-free rate.
• Can be shown not to 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.

Question 14

Liquidity preference theory

Answer 14

• Proposing that forward rates reflect investors’ expectations of future spot rates, plus a liquidity premium to compensate investors for exposure to interest rate risk.
• Liquidity premium is positively related to maturity.
• Implication of yield curve:
  
  • Upward sloping:
    
    • The market expects future interest rates to rise or
    • Rates remain constant (or even fall), but the addition of the liquidity premium results in a positive slope.
  • Downward sloping: steeply falling short-term rates

Question 15

Segmented markets theory and Preferred habitat theory

Answer 15

• 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 16

Effective duration

Answer 16

• Effective duration measures price sensitivity to small parallel shifts in the yield curve
• It is not an accurate measure of interest rate sensitivity to non-parallel shifts in the yield curve

Question 17

Shaping risk

Answer 17

Shaping risk refers to changes in portfolio value due to changes in the shape of the benchmark yield curve.

Question 18

Sensitivity to Parallel, Steepness and Curvature Movements

Answer 18

All yield curve movements can be described using a combination of one or more of these movements:

• Level (ΔxL) − A parallel increase or decrease of interest rates.
• Steepness (ΔxS) − Long-term interest rates increase while short-term rates decrease.
• Curvature (ΔxC) − Increasing curvature means short- and long-term interest rates increase while intermediate rates do not change.

Question 19

Term structure of interest rate volatility

Answer 19

The term structure of interest rate volatility is the graph of yield volatility versus maturity.

Short-term interest rates are generally more volatile than are long-term rates.

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.

Question 20

Equilibrium vs arbitrage-free term structure

Answer 20

• Equilibrium term structure models are factor models that seek to describe the dynamics of the term structure by using fundamental economic variables that are assumed to affect interest rates.
• Arbitrage-free term structure models use observed market prices of a reference set of financial instruments, assumed to be correctly priced, to model the market yield curve.
• Equilibrium term structure models require fewer parameters to be estimated relative to arbitrage-free models, and arbitrage-free models allow for time-varying parameters. Consequently, arbitrage-free models can model the market yield curve more precisely than equilibrium models.

Question 21

Arbitrage-free valuation

Answer 21

Arbitrage-free valuation methods value securities such that no market participant can earn an arbitrage profit in a trade involving that security. An arbitrage transaction involves no initial cash outlay but a positive riskless profit (cash flow) at some point in the future.

Question 22

2 types of arbitrage opportunities

Answer 22

• Value additivity: value of whole differs from the sum of the values of parts. arbitrage profits can be earned by stripping or reconstitution
• Dominance: one asset trades at a lower price than another asset with identical characteristics

Question 23

Binomial interest rate tree

Answer 23

• The binomial interest rate tree framework is a lognormal random walk model with two desirable properties:
  
  • Higher volatility at higher rates
  • Non-negative interest rates
• Interest rate trees should generate arbitrage-free values for the benchmark security.
• Adjacent forward rates (for the same period) are two standard deviations apart (calculated as e^2σ)
• The middle forward rate (or mid-point in case of even number of rates) in a period is approximately equal to the implied (from the benchmark spot rate curve) one-period forward rate for that period.
• Important assumption: cash flows are not path dependent. As such, cannot be used to value mortgage-backed securities.

Question 24

Backward induction

Answer 24

Backward induction refers to the process of valuing a bond using a binomial interest rate tree. Because the probabilities of an up move and a down move are both 50%, 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. The appropriate discount rate is the forward rate associated with the node.

Question 25

Simple Options

Answer 25

• Callable bonds
  
  • European-style: the option can only be exercised on a single day immediately after the lockout period
  • American-style: the option can be exercised at any time after the lockout period
  • Bermudan-style: the option can be exercised at fixed dates after the lockout period
• Putable bonds
  
  • Extendible bond: allows the investor to extend the maturity of the bond. can be evaluated as a putable bond with longer maturity.

Question 26

Estate Put

Answer 26

An estate put 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.

Question 27

Sinking fund bonds

Answer 27

Sinking fund bonds (sinkers) which require the issuer to set aside funds periodically to retire the bond (a sinking fund). This provision reduces the credit risk of the bond. Sinkers typically have several related issuer options (e.g., call provisions, acceleration provisions, and delivery options).

Question 28

Option Adjusted Spread (OAS)

Answer 28

OAS is used by analysts in relative valuation; bonds with 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 and hence an attractive investment (i.e., it offers a higher compensation for a given level of risk). Conversely, bonds with low OAS (relative to peers) are considered to be overvalued.

Question 29

Modified Duration

Answer 29

• Modified duration measures a bond’s price sensitivity to interest rate changes, assuming that the bond’s cash flows do not change as interest rates change
• The standard measure of convexity can be used to improve price changes estimated from modified duration.
• Effective duration and effective convexity should be used for bonds with embedded options, because these measures take into account how changes in interest rates may alter cash flows.

Question 30

Effective duration in Practice

Answer 30

• 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.

Question 31

One-sided durations

Answer 31

For bonds with embedded options, one-sided durations - durations that apply only when interest rates rise (or, alternatively, only when rates fall) - are better at capturing interest rate sensitivity than simple effective duration.

Question 32

Key Rate Duration Generalization

Answer 32

• If an option-free bond is trading at par, the bond’s maturity rate is the only rate that affects the bond’s value. Its maturity key rate duration is the same as its effective duration, and all other rate durations are zero.
• For an option-free bond not trading at par, the maturity-matched rate is still the most important rate.
• A bond with a low (or zero) coupon rate may have negative key rate durations for horizons other than the bond’s maturity.
• A callable bond with a low coupon rate is unlikely to be called; hence the bond’s maturity-matched rate is the most critical rate.
• All else equal, higher coupon bonds are more likely to be called and therefore the time-to-exercise rate will tend to dominate the time-to-maturity rate.
• Putable bonds with high coupon rates are unlikely to be put, and thus are most sensitive to their maturity-matched rates.
• All else equal, lower coupon bonds are more likely to be put, and therefore the time-to-exercise rate will tend to dominate the time-to-maturity rate.

Question 33

Capped floater

Answer 33

A capped floater effectively contains an issuer option that prevents the coupon rate from rising above a specified maximum rate known as the cap.

value of a capped floater = value of a “straight” floater − value of the embedded cap

Question 34

Floored floater

Answer 34

A related floating rate bond is the floored floater, which has a coupon rate that will not fall below a specified minimum rate known as the floor.

value of a floored floater = value of a “straight” floater + value of the embedded floor

Question 35

Minimum value of a convertible bond

Answer 35

The minimum value of a convertible bond is the greater of its conversion value or its straight value. This must be the case, or arbitrage opportunities would be possible.

• The conversion value of a convertible bond is the value of the common stock into which the bond can be converted.
• The straight value, or investment value, of a convertible bond is the value of the bond if it were not convertible—the present value of the bond’s cash flows discounted at the return required on a comparable option-free issue.

Question 36

Expected exposure

Answer 36

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. A bond’s expected exposure changes over time.

Question 38

Recovery rate

Answer 38

Recovery rate is the percentage recovered in the event of a default. Recovery rate is the opposite of loss severity

Question 39

Loss given default

Answer 39

(LGD) is equal to loss severity multiplied by exposure.

Question 40

Probability of default and probability of survival

Answer 40

• Probability of default is the likelihood of default occurring in a given year. The initial probability of default is also known as the hazard rate.
• Probability of survival is 1 minus the cumulative conditional probability of default.

Question 41

Credit valuation adjustment

Answer 41

Credit valuation adjustment (CVA) is the sum of the present value of the expected loss for each period. CVA is the monetary value of the credit risk in present value terms; it is the difference in value between a risk-free bond and an otherwise identical risky bond

Question 42

Risk neutral probability of default

Answer 42

• Probability of default implied in the current market price.
• Given the market price, the estimated risk neutral probabilities of default and recovery rates are positively correlated.

Question 43

FICO Scores

Answer 43

FICO is a well-known credit scoring model used in the US. FICO scores are higher for those with

• longer credit histories (age of oldest account),
• absence of delinquencies,
• lower utilization (outstanding balance divided by available line),
• fewer credit inquires, and
• a wider variety of types of credit used.

Question 44

Structural models pro and cons

Answer 44

Advantages of structural models:

1. Structural models provide an economic rationale for default (i.e., AT < K) and explain why default occurs.
2. Structural models utilize option pricing models to value risky debt.

Disadvantages of structural models:

1. Because structural models assume a simple balance sheet structure, complex balance sheets cannot be modeled. Additionally, when companies have off-balance sheet debt, the default barrier under structural models (K) would be inaccurate and hence the estimated outputs of the model will be inaccurate.
2. 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.

Question 45

Reduced Form Models pros and cons

Answer 45

Advantages of reduced form models:

1. RF models do not assume that the assets of a company trade.
2. Default intensity is allowed to vary as company fundamentals change, as well as when the state of the economy changes.

Disadvantages of reduced form models:

1. RF models do not explain why default occurs.
2. Under the RF models, default is treated as a random event (i.e., a surprise), but in reality, default is rarely a surprise (it is often preceded by several downgrades).

Question 46

Determinants of Term Structure of Credit Spreads

Answer 46

• Credit quality
• Financial conditions
• Market demand and supply
• Equity market volatility

Question 47

Credit default swap (CDS)

Answer 47

AA credit default swap (CDS) is a contract between two parties in which one party purchases protection from another party against losses from the default of a borrower. To obtain this coverage, the protection buyer pays the seller a premium called the CDS spread. The protection seller is assuming (i.e., long) credit risk, while the protection buyer is short credit risk. Note that the CDS does not provide protection against market-wide interest rate risk, only against credit risk. The contract is written on a face value of protection called the notional principal (or “notional”).