Fixed Income

Question 1

CA - PV of Expected Loss

Present Value of Expect Loss

Answer 1

Question 2

CA - PV of Expected Loss

Credit Analysis

PV of Expected Loss

Question 3

EO - Callable Bond

Embedded Options

Callable bond

(explain, formula)

Answer 3

V_call = V_straight - V_callable

Question 4

CDS - Credit Default Swaps

Credit Default Swaps (CDS)

• Expected Loss
• Upfront payment (buyer)
• Upfront premium
• Profit for protection buyer%

Answer 4

Expected Loss_t = (hazard rate)_t x (loss given default)_t

Upfront payment = (PV_{protection leg}) - (PV_{premium leg})

_fs1Upfront premium (%) ≈ (CDS_spread - CDS_coupon) x CDS_duration

_fs2 Profit (protection buyer) ≈ change in spread (bps) x CDS_duration x nottional principal

Think of

CDS spread like mkt price for insurance

CDS coupon like contract variable

Question 5

• Forward price @ (t=j) of zero-coupon bond maturing (j+k)
• Forward pricing model
• Forward rate model

Answer 5

Fwd price (t = j) of zero bond maturing @ (j + k)

F_j,k = ¹ / [1 + f(j,k)] ^k

Fwd pricing model

P_{(j + k)} = P_jF_(j,k) Therefore: F_(j,k) = P(j + k)

^P_j

Fwd rate (Spot) model 
[1 + S<sub>(j + k) </sub>]<sup>(j + k)</sup> = (1 + S<sub>j</sub>)<sup>j </sup>**x** [1 + f(j,k)]<sup>k</sup>

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

If spot curve flat, forward rates = spot rates.

If spot curve is upward sloping (downward sloping), forward rate curves will be above (below) the spot curve, and the yield for maturity of T-bill will be less than (greater than) the spot rate ST

Evolution of Spot Rates in Relation to Forward Rates

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 evolve relative to the rates suggested by forward rate curves.

Question 6

Putable bond

Answer 6

Question 7

CA - Reduced Form Model

Credit Analysis

Reduced Form Model

(credit analysis)

• assumptions
• strengths/weaknesses

Answer 7

Assumptions

• ZeroCpn included in Debt, BUT its not the only debt like Structural!
• Risk-free Rate is stochastic (randomly determined)
• State of Economy is stochastic depending on nonconstant macroeconomic variables.
  
  • ​hence business cycles change and probability of default is not contact

​​Strengths

• inputs estimated w/ historical data
• credit risk fluctuates w/ business cycles
• company’s B/S details not required

Weaknesses

• past market conditions market conditions may not reflect future.
• estimates derived from historical data may be inappropriate.

Question 8

Convertible Bonds (know terminology)

• Conversion ratio
• Market conversion price
• Conversion value
• Straight value
• Minimum value of convertible bond
• Market conversion premium
• Market conversion premium ratio
• Premium over straight value

Answer 8

• Conversion ratio: #shares per bond
• Market conversion (effective) price ​​

​Market Price Bond / Conversion ratio

• Conversion value: (Post-conv share price)(conv ratio)
  
  • CV = P₀ x CR
• Straight value: PV of CFs if not converted (usually given)
• Min value of convertible bond: Greater of conversion value or straight value
  
  • ​most likely to show up on exam
• Mkt conversion premium: (Mkt conv price) - (mkt stock price)
  
  • MCP/sh = (PV₀ / CR) - P₀
• Market conversion premium ratio:
  
  • (Mkt conv premium) / (mkt price)
• Premium over straight value:
  
  • (MV of bond / straight value) -1

Mkt conv price is effective price per share when converting.

Question 9

Convexity

Answer 9

• Positive convexity for putable or straight bond
• Negative convexity for callable bond

Question 10

price of a T-period Zero-coupon bond

Answer 10

Question 11

Spread formulas and definitions

• Swap spread
• TED spread
• LIBOR-OIS Spread
• Z-spread

Answer 11

Swap spread_t = swap rate_t - Treasury yield_t

• additional interest rate paid by fixed-rate payer of interest rate swap over the rate of the “on-the-run” government bond of same maturity.

TED spread = (3-month LIBOR) - (3-month T-Bill)

• Difference between rates on interbank loans and short-term U.S. govt debt (“T-bills”). TED is T-Bill and ED, the ticker symbol for the Eurodollar futures contract.
• Indicates overall level of credit risk in the economy.

LIBOR-OIS spread = LIBOR rate - “overnight indexed swap” rate

• is the amount by which the LIBOR rate (which includes credit risk) exceeds the overnight indexed swap (OIS) rate (which includes only minimal credit risk). The LIBOR-OIS spread is a useful measure of credit risk and provides an indication of the overall well-being of the banking system.

Z-spread = spot rate_t - (PV of CFs = bond’s mkt price)

• Zero-volatility spread (assume rate volatility zero)
• Not appropriate for embedded options (use Z for straight only)
• –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.

OAS

• option adjusted spread
• if straight, then OAS and Z-spread are equal

Credit spread => ☺make fun of.

• simple and likely not used in level 2.
• = YTM credit-risky ZeroCpn - YTM risk-free ZeroCpn

Question 12

Traditional Theories (5) of the

Term Structure of Interest Rates

Answer 12

1. Unbiased expectations theory—Forward rates are an unbiased predictor of future spot rates. Also known as the pure expectations theory.
2. Local expectations theory—Preserves the risk-neutrality assumption only for short holding periods, while 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.
3. Liquidity preference theory—Investors demand a liquidity premium that is positively related to a bond’s maturity.
4. 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.
5. 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 13

Effective Duration

• (relationships in call/putable, zero, fixed, floater)
• formula

Answer 13

Effective durations:

• duration_{callable/putable} ≤ effective duration_straight
• duration_(zero) ≈ bond maturity
• duration_{fixed rate} < bond maturity
• effective duration of floater ≈ time in years to next reset

Question 14

Key Rate Durations

(no need to calculate. Know relationships!)

Answer 14

Duration assumes parallel interest rate shift (short, interm, long all move the same). Problem is that usually rate shifts have curvature. So….

Rate duration is what happens to change in value of bond if we change just one spot-rate (par rates)

Key rate duration takes multiple rate durations to analyze impact on bond value.

• Option-FREE bond = maturity match rate most important. key rate highest b/c biggest CF at maturity.

Measures bond price sensitivity to a change in a specific par rate, keeping everything else constant.

Question 15

Level and Shape of Yield Curve

• Impact of rates up/down on calls and puts?
• Yield curve steepens?
• Yield curve flattens?

Answer 15

• Upside on callable limited when rates decline
• Downside on putable limited when rates increase
• Steepened curve makes call option unlikely to be exercised and has low value while put option is valuable
• Flattened curve makes call option become more valuable while put option losses value.

Question 16

Modern Term Structure Models:

1. Equilibrium term structure models
2. Arbitrage-free models

Answer 16

1. Equilibrium term structure models

• Cox-Ingersoll-Ross (CIR) model:
  
  • _​_dr = a(b−r)dt + σ √(rdz)
  • Assumes the economy has a natural long-run interest rate (b) that the short- term rate (r) converges to.
• Vasicek model:
  
  • dr = a(b – r)dt + σdz
  • Similar to the CIR model but assumes that interest rate volatility level is independent of the level of short-term interest rates.

2. Arbitrage-free models—Begins with observed market prices and the assumption that securities are correctly priced.

• Ho-Lee model: dr_t = θtd_t + σdz_t
• This model is calibrated by using market prices to find the time-dependant drift term θt that generates the current term structure.

Question 17

One-Sided Durations

Answer 17

Callable bonds → lower one-sided down-duration (rates ↓ not price!) than one-sided up-duation

Putable bonds → HIGHER one-sided down-duration than one-sided up-duration

Question 18

Structural Model

(credit analysis for probability of default)

• Assumptions
• Strengths
• Problems

Answer 18

Assumptions

1) Traded assets within firm
* (b/c need to know volatility)
2) Simple B/S
* ZeroCpn Debt_{(only one-class)​} + Equity

Strengths

• Option pricing theory used for 1. probability of default and 2. loss given default
• Inputs can be estimated through callibration
  
  • [E(R), volatility_(constant), T, Assets like call option, Debt like R_{f (constant)} - put option]

Problems

1. Guess work (callibration w/ 5 variables!)
2. Silly B/S (balance sheet = bullshit)
3. No Business cycle considerations

Question 19

Volatility impact on Embedded Options

• Straight bond value impact if volatility increases?
• Embedded bond value impact if volatility increase?
• OAS impact?

Answer 19

Straight bond impact from volatility?

• None! Straights are impacted by rate changes!

Embedded impact of increased volatility?

• V_{call option} ↑, V_{<span>put option</span>} ↑, V_{callable bond} ↓, V_{<span>putable bond</span>} ↑

Volatility (σ) is an assumption!

↑ σ = ↓ lower OAS for callable bonds

↑ σ = ↑ higher OAS for puttable bonds

Assumed σ impacts calculated OAS (which isn’t observable so different data providers can have very different OAS). If σ is wrong, then OAS will be wrong.

Question 20

Why swap rate curve preferred as benchmnark interest rate curve?

• Swap rates reflect?
• Swap market regulation?
• Swap curve quotes?

Answer 20

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 governments.
• The swap market is not regulated by any government.
• The swap curve typically has yield quotes at many maturities.

Institutions like wholesale banks are familiar with swaps and, as a result, often use swap curves (rather than other interest rate benchmarks) to value their assets and liabilities.

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

• 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.

Question 21

Factors driving Yield Curve?

Answer 21

1. Effective duration—Measures the sensitivity of a bond’s price to parallel shifts in the benchmark yield curve.
2. Key rate duration—Measures bond price sensitivity to a change in a specific par rate, keeping everything else constant.
3. Sensitivity to parallel, steepness, and curvature movements—Measures sensitivity to three distinct categories of changes in the shape of benchmark yield curve