CFA 2018 Level 2 Fixed Income

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Fixed Income

Level 2 -- 2017

Instructor: Feng

Brief Introduction

Topic weight:

Study Session 1-2 Ethics & Professional Standards 10 -15% Study Session 3 Quantitative Methods 5 -10% Study Session 4 Economics 5 -10% Study Session 5-6 Financial Reporting and Analysis 15 -20% Study Session 7-8 Corporate Finance 5 -15% Study Session 9-11 Equity Investment 15 -25% Study Session 12-13 Fixed Income 10 -20% Study Session 14 Derivatives 5 -15% Study Session 15 Alternative Investments 5 -10% Study Session 16-17 Portfolio Management 5 -10% Weights: 100%

Brief Introduction

Content:

Ø Study Session 12: Valuation Concepts

• Reading 35: The Term Structure and Interest Rate Dynamics

• Reading 36: The Arbitrage-Free Valuation Framework

Brief Introduction

Content:

Ø Study Session 13: Topics in Fixed Income Analysis

• Reading 37: Valuation and Analysis: Bonds with Embedded Options

• Reading 38: Credit Analysis Models

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Brief Introduction

考纲对比:

Ø 与2016年相比，2017年的考纲没有变化。

Brief Introduction

Forward Rate Models & Forward Pricing

Models

Tasks:

Ø Describe relationships among spot rates and forward rates, and the shape of the yield curve;

Ø Describe the forward pricing and forward rate models, and calculate forward and spot prices and rates using those models.

Spot rate

Ø The rate of interest on a security that makes a single payment at a future point in time (a.k.a zero-coupon rate or zero rate).

üS(j): annualized spot rate for time j.

Ø Spot curve: the term structure of spot rates, the graph of the spot rate versus maturity.

t=0 t=j t=j+k

S(j)

$1 [1+S(j)]j

Forward Rate Models & Forward Pricing Models

Forward rate

Ø Forward rate is the rate of interest set today for a single-payment security to be issued at a future date.

üf(j, k): annualized k-year forward rate starting at time j. Ø Forward curve: the term structure of forward rates, the

graph of the forward rate versus maturity.

t=0 t=j t=j+k

f(j, k)

$1 [1+f(j, k)]k

Forward rate models

Ø Forward rate models show how forward rates can be

extrapolated from spot rates.





j k





j





k

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Example

Ø The spot rates for three hypothetical zero-coupon bonds with maturities of one, two, and three years are 9%, 10% and 11%. Calculate the forward rate for a one-year zero issued two years from today.

Answer:

(1 + 0.11)3_{= (1 + 0.10)}2_×_{[1 + f(2,1)]}

f(2,1) = 13.03%

Spot curve vs. forward curve

Ø If yield curves are upward sloping, forward curves lie above the spot curve.

ü The later the initiation date, the higher the forward curve.

Yield curves at Jul. 31st_{, 2013}

Spot curve vs. forward curve (Cont.)

Ø If yield curves are downward sloping, forward curves lie below the spot curve.

üThe later the initiation date, the lower the forward curves.

Yield curves at Dec. 31st_{, 2006}

Discount factor

Ø The price of a risk-free single-unit payment at time j:

 

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Forward price

Ø The price at time j years from today for a zero-coupon bond with maturity j+k years and unit principal.





Forward pricing models

Ø The forward pricing model describes the valuation of forward contracts.

Example

Ø Consider a 2-year loan beginning in 1 year. The 1-year spot rate is 7% and the 3-year spot rate is 9%. Calculate the forward price of a 2-year bond to be issued in 1 year.

Answer:

F(1,2) = 0.7722 ÷ 0.9346 = 0.8262.

Ø Importance: ☆☆☆ Ø Content:

• Spot rates & forward rates;

• Spot curve and forward curve;

• Forward rate models & forward pricing models. Ø Exam tips:

• 常考点1：两个模型的运用，计算题;

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Yield and Spread

Tasks:

Ø Describe relationships among YTM, expected and realized returns on bonds;

Ø Describe bootstrapping;

Ø Explain swap rate curve and calculate swap spread; Ø Describe Z-spread, TED and Libor–OIS spreads.

Yield-to-Maturity (YTM)

Ø The internal rate of return on the cash flows of a fixed-rate bond.

üYTM is the same as the spot rate for zero-coupon bonds; üFor coupon bonds, if the spot curve is not flat, the YTM

will not be the same as the spot rate; üYTM is some weighted average of spot rates.

n Yield and Spread

Yield-to-Maturity (Cont.)

Ø Three critical assumptions for YTM:

üThe investor hold the bond until maturity;

üThe issuer makes full and timely coupon and principal payments;

• The bond is option-free and there is no default risk. üThe investor is able to reinvest coupon payments at

YTM.

Yield and Spread

Expected return vs. Realized Return

Ø Expected return is the ex-ante return that a bondholder expects to earn.

üThe YTM is the expected return only if all the three critical assumptions for YTM are met.

Ø Realized return is the actual return on the bond during the time an investor holds the bond.

üIt is based on actual reinvestment rates and the yield curve at the end of the holding period.

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Bootstrapping

Ø Using the output of one step as the input of next step. üSpot rates may be obtained from the par curve by

bootstrapping.

• Par rate is the YTM of a bond trading at par;

• Par curve is the term structure of par rates: the graph of par rate versus maturity.

Yield and Spread

Example

Ø If the YTMs for annual pay bonds trading at par with maturity of 1 year, 2 years and 3 years are 1%, 1.25%, and 1.5% respectively, compute the 1-year, 2-year, and 3-year spot rates.

Yield and Spread

Answer:

Ø Step 1: S1=1%;

Ø Step 2: value the 2-year bond using spot rates: 100=1.25/(1+1%)+101.25/(1+S2)2

=> S2=1.252%;

Ø Step 3: value the 3-year bond using spot rates: 100=1.5/(1+1%)+1.5/(1+1.252%)2_+101.5/(1+S₃₎3

=> S3=1.51%. Yield and Spread

Swap rate curve

Ø Swap rate: the interest rate for the fixed-rate leg of an interest rate swap.

Ø Swap rate curve (swap curve): the yield curve of swap rate.

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Swap rate curve (Cont.)

Ø Investors prefer the swap rate curve rather than government spot curve in bond valuation because: üSwap rates reflect the credit risk of commercial banks

rather than governments.

üThe swap is not regulated by any government, which

makes swap rates in different countries more comparable.

üThe swap curve typically has yield quotes at many maturities.

Yield and Spread

Swap spread

Ø The amount by which the swap rate exceeds the rate of

the “on-the-run” government security with the same maturity.

üSwap spread = Swap rate - Government security rate

üLibor swap curve is the most widely used yield curve because it reflects the default risk of most commercial banks.

Yield and Spread

Z-spread

Ø The constant spread that would need to be added to the

implied spot curve so that the discounted cash flows of a bond are equal to its current market price.

  Yield and Spread

TED spread

Ø The spread of Libor over the yield on a T-bill with matching maturity.

üTED spread is an indicator of perceived credit risk in the general economy.

• An increase/decrease in the TED spread is a sign that lenders believe the risk of default on interbank loans is increasing/decreasing.

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Libor-OIS spread

Ø The spread of Libor over the overnight indexed swap (OIS) rate.

üAn OIS is an interest rate swap in which the periodic floating rate is equal to the geometric average of an overnight index rate over every day of the payment period.

üConsidered an indicator of the risk and liquidity of money market securities.

Yield and Spread

Ø Importance: ☆☆ Ø Content:

• YTM, expected return and realized return;

• Bootstrapping;

• Swap curve and swap spread;

• Z-spread, TED and Libor–OIS spreads. Ø Exam tips:

• 常考点：spreads的定义，概念题。 Summary

Traditional Term Structure Theories

Tasks:

Ø Explain traditional theories of the term structure of interest rates;

Ø Describe the implications of each theory for forward rates and the shape of the yield curve.

Traditional theories of term structure

Ø Present the underlying economic factors that affect the shape of the yield curve.

üPure expectations theory

üLocal Expectations Theory üLiquidity preference theory

üSegmented market theory

üPreferred habitat theory

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Unbiased (pure) expectation theory

Ø This theory suggests that the forward rate is an unbiased predictor of the future spot rate.

üUnderlying assumption: risk neutrality;

• This assumption is a significant shortcoming. üUnder this theory, bonds of any maturity are perfect

substitutes for one another.

Traditional Term Structure Theories

Unbiased expectation theory (Cont.)

Ø Under the unbiased expectations theory:

üIf yield curve is upward sloping, short-term rates are expected to rise;

üIf yield curve is downward sloping, short-term rates are expected to fall;

üIf yield curve is flat, short-term rates are expected to remain constant.

Local expectation theory

Ø This theory assume risk-neutrality only in the short term while incorporate uncertainty in the long term. üUnder this theory, the expected return for every bond

over short time periods is the same.

• But it is often observed that short-holding-period returns on long-dated bonds do exceed those on short-dated bonds.

Liquidity preference theory

Ø This theory asserts that liquidity premiums exist to compensate investors for the added interest rate risk they face when lending long term.

üForward rate = expected spot rate + liquidity premium

• Liquidity premium increase with maturity.

üGiven an expectation of unchanging short-term spot

rates, this theory predicts an upward-sloping yield curve.

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Segmented markets theory

Ø Under this theory, each maturity sector can be thought of as a segmented market in which yield is determined by supply of and demand for loan, and independent from the yields in other maturity segments.

üConsistent with a world where there are asset/liability management constraints;

üYields are not a reflection of expected spot rates or liquidity premiums.

Preferred habitat theory

Ø Similar to the segmented markets theory, but contends

that if the premium is large enough, investor will deviate from their preferred maturities or habitats.

üPremium is not directly related to maturity;

üBased on the realistic notion that investors will accept additional risk in return for additional expected returns.

• 5 traditional term structure theories. Ø Exam tips:

• 常考点：五种理论的辨识和比较，概念题。 Summary

Modern term structure models

Tasks:

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Modern term structure models

Ø Provide quantitatively precise descriptions of how interest rates evolve.

üEquilibrium term structure models

• The Cox-Ingersoll-Ross model

• The Vasicek Model

üArbitrage free models

• The Ho-Lee Model

Modern Term Structure Models

Equilibrium term structure models

Ø Describe the dynamics of the term structure using fundamental economic variables that are assumed to affect interest rates.

üCan be one-factor or multi-factor models.

Equilibrium term structure models (Cont.)

Ø In the modeling process, restrictions are imposed that allow for the derivation of equilibrium prices for bonds and interest rate options.

üRequire the specification of a drift term and the assumption of a functional form for interest rate

volatility.

Cox-Ingersoll-Ross (CIR) model

Ø CIR model is based on the idea that individuals must determine his/her optimal trade-off between

consumption today and investing and consuming at a later time.

üHigher interest rate → more investing → more capital supply → lower interest rate; and vice versa. üUltimately, interest rates will reach a market

equilibrium rate at which no one needs to borrow or lend.

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CIR model (Cont.)

Ø Mathematically, the CIR model is as follows:

Where:

üdr: a infinitely small change in short-term interest rate;

üdt: a infinitely small increase in time;

üdz: an infinitely small movement in a “random walk”.





d r = a b - r d t + σ rd z Modern Term Structure Models

üb: long-run value of the short-term interest rate; ür: the short-term interest rate;

üa: a positive parameter for speed of mean reversion adjustment, and the higher/lower, the quicker/slower; üσ: interest rate volatility.

Ø The CIR model consist of tow terms:

üDrift term: ; which means the interest rate is

mean-revert to long term value (b) with speed presented by parameter (a), also named deterministic term;

üStochastic term: ; also named volatility term.

• in the stochastic term forces interest rate (r) to be

non-negative, and assets that higher interest rate will lead to higher volatility.





a b - r dt

σ rdz r

Vasicek model

Ø Similar to the CIR model, the Vasicek model captures mean reversion:

üDisadvantages of Vasicek model:

• Does not force interest rates to be non-negative;

• Volatility remains constant over the period of analysis.





d r = a b - r d t + σ d z

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Arbitrage free models

Ø This models assume bonds trading in the market are

correctly priced (arbitrage free).

Ø The modeling process that determines the term

structure is such that the bonds valuation process generates their market prices.

Ho-Lee model

Ø The model assumes that the yield curve moves in a way

that is consistent with a no-arbitrage condition.

üθ: a time-dependant drift term.

t t t

d r = θ d t + σ d z Modern Term Structure Models

Ø Importance: ☆ Ø Content:

• 3 modern term structure models.

Ø Exam tips:

• 不是考试重点。 Summary

Active Bond Portfolio Management

Tasks:

Ø Describe the strategy of riding the yield curve; Ø Explain the measurement of yield curve risk; Ø Explain the maturity structure of yield volatilities and

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Forward rates vs. future spot rates

Ø If future spot rates actually evolve as implied by the current forward curve, the return of bonds of varying tenor over a same period is always the same.

üIf future spot rates is f(j,k), Sj is always the same.

Active Bond Portfolio Management

Example

Ø Assume that one-year spot rate (S1) is 9% and two-year

spot rate (S2) is 10%, then the implied forward rate f(1,1)

is 11.01%.

Ø The return of year zero-coupon bond over the

one-year holding period is 9%.





Example (Cont.)

Ø The price of a two-year zero-coupon bond now (P0) is:

Ø If actual one-year spot rate one year later is current forward rate f(1,1), the price of the above bond one year later (P1) is:

Example (Cont.)

Ø So, the return of two-year zero-coupon bond over the

one-year holding period is also 9%.

90 08 82 64. .  1 9%

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Forward rates vs. future spot rates (Cont.)

Ø If future spot rates turn out to be different from implied forward rates, the return of bonds of varying tenor over a same period will differ.

üIf future spot rates are expected to be lower/higher, investor would perceive the bond to be

undervalued/overvalued.

Ø An active portfolio manager may outperform the overall

market if the manager can predict how future spot rates will differ from the current implied forward rates.

Riding the yield curve

Ø When a yield curve is upward sloping and the trader believes that the yield curve will not change its level and shape, then buying bonds with a maturity longer than the investment horizon would provide a total return greater than the return on a maturity-matching strategy. üThe bond is valued at successively lower yields and

higher prices, and then sold before maturity to realize a higher return.

Example

Ø The following table shows a upward-sloping yield curve and the prices of a 3% annual-pay coupon bond with $100 par value and different maturities.

Maturity Yield Price 5 3 100 15 4 88.88 25 5 71.81 30 5.5 63.67

Example (Cont.)

Ø Assuming an investor with investment horizon of 5 years purchase the bond with maturity of 5 years (maturity-matching strategy):

üAnnual coupon payment: 3%;

üCapital gain: none.

Ø If the investor purchase the bond with maturity of 30 years and sell it after 5 years (riding the yield curve):

üAnnual coupon payment: 3%;

üCapital gain: $71.81 - $63.67 = $8.14.

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Summary of duration (review):

ØYield duration (price sensitivity to YTM)

ØCurve duration (price sensitivity to benchmark yield curve)

ØKey rate duration (sensitivity to yield at specific maturity)

•Macaulay duration •Modified duration •Money duration

üPVBP (DV01) •Uncertain future cash flow

•No well-defined IRR (YTM)

•Effective duration •Non-parallel shift

Managing yield curve risk

Ø Yield curve risk: risk to portfolio value arising from unanticipated changes in the yield curve. Ø Measurement of yield curve risk

üEffective duration üKey rate duration

üLevel, steepness, and curvature

Effective duration

Ø Measures the bond price sensitivity to a small parallel shift in a benchmark yield curve (△Curve).

- +

0

P - P EffDur =_{2 (ΔCurve) P}_ _

Key rate duration

Ø Measures the bond price sensitivity to a small change in a benchmark yield curve at a specific maturity segment. üShaping risk: bond price sensitivity to changes in the

shape of the benchmark yield curve.

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Level, steepness, and curvature

Ø The yield curve movements can be decomposed into

parallel movement (ΔXL), steepness movement (ΔXS),

and curvature movements (ΔXC). Active Bond Portfolio Management

Level, steepness, and curvature (Cont.)

Ø The proportional change in portfolio value resulted from yield curve movement can be modeled as:

üDL, DS, and DC as the sensitivities of portfolio value to

small changes in the level, steepness, and curvature, respectively.

L L S S C C

ΔP _{-D ΔX - D ΔX - D ΔX}

P ≈

Maturity structure of yield volatility

Ø A representation of the yield volatility of a zero-coupon bond for every maturity of security.

üTypically, short-term rates are more volatile than long-term rates.

• Short-term volatility is most strongly linked to uncertainty regarding monetary policy;

• Long-term volatility is most strongly linked to uncertainty regarding the real economy and inflation.

• Forward rate vs. future spot rate, and strategy of riding the yield curve;

• Measurement of yield curve risk;

• Maturity structure of yield volatility. Ø Exam tips:

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Arbitrage-Free Valuation

Tasks:

Ø Explain arbitrage-free valuation of a fixed-income instrument;

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

The law of one price

Ø Two goods that are perfect substitutes must sell for the same current price in the absence of transaction costs. üOtherwise, buying the lower and selling the higher will

earn a riskless profit, and make the two prices converge.

Arbitrage

Ø Trades that earn riskless profits without any net investment of money.

Arbitrage-Free Valuation

Arbitrage opportunities

Ø There are two types of arbitrage opportunities: üValue additivity: the value of the whole must equals

the sum of the values of the parts.

• Stripping: separate the bond’s individual cash flows and trade them as zero-coupon securities;

• Reconstitution: recombine the individual zero coupon securities and reproduce the underlying coupon bond. üDominance: a financial asset with a risk-free payoff in

the future must have a positive price today.

Arbitrage-free valuation

Ø An approach to security valuation that determines security values that are consistent with the absence of an arbitrage opportunity.

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Arbitrage-free valuation (Cont.)

Ø Viewing any bond as a package of zero-coupon bonds;

Ø Using spot rates that correspond to the maturities of the zero-coupon bonds to calculate the bond price.

Zn: spot rate for period n.

üIf market price is different with the calculated price, there is an arbitrage opportunity of “value additivity”.

n t

t n

t = 1

P M T F

P = +

( 1 + Z )t ( 1 + Z )n



Example:

Ø Suppose that the one-year spot rate is 2%, the two-year spot rate is 3%, and the three-year spot rate is 4%. Then, the price of a three-year bond that makes a 5% annual coupon payment is:



5



1



5



2



1 0 5



3

P = + + = 1 0 2 .9 6 1 .0 2 1 .0 3 1 .0 4

• Arbitrage opportunities and arbitrage-free valuation. Ø Exam tips:

• 常考点：用spot rates给债券定价。 Summary

Binomial interest rate tree

Tasks:

Ø Describe a binomial interest rate tree framework; Ø Describe the process of calibrating a binomial interest

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Binomial interest rate tree

Ø The arbitrage-free valuation of bond with spot rates is not available for bonds with embedded option, and we need a valuation approach with binomial interest rate tree.

üFor bonds with embedded options, changes in future

interest rates impact the likelihood the option will be exercised and in so doing impact the cash flows.

Binomial Interest Rate Tree

Binomial interest rate tree (Cont.)

Ø A interest rate model that assumes interest rates at any point of time (node) have an equal probability of taking one of two possible values in the next period, an upper path (U) and a lower path (L).

üThe interest rates at each node are one-period forward rates corresponding to the nodal period.

Ø E.g.: interest rate i2,LU at node 2 is the rate that will occur

if initial interest i0 at node 0 follows the lower path to

node 1, and then follows the upper path to node 2. ü

• σ: standard deviation of interest rate.

i0 Binomial Interest Rate Tree

Ø The binomial interest rate tree framework is a lognormal random walk (lognormal tree) that insures two appealing properties:

üNon-negativity of interest rates; ühigher volatility at higher interest rates.

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Ø When calibrating a binomial interest tree, we follow two rules below:

üChoose interest rates fitting to the current yield curve so that the model produces arbitrage-free values for the benchmark bonds;

üAdjacent forward rates (for the same period) are two standard deviations apart.

• Framework and calibration of binomial interest rate tree.

Ø Exam tips:

• 一般不直接出题，但是后面重要知识点的基础。 Summary

Valuing Option-Free Bonds

Tasks:

Ø Describe the backward induction valuation and

calculate the value of a bond;

Ø Describe pathwise valuation and calculate the value of a bond.

Backward induction valuation

Ø Value bond by moving backward from last period to time

zero.

Ø Bond value at any node:

üAverage PV of two possible values from next period; üDiscount rate is the one-period forward rate at that

node.

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Example

Ø Using the interest rate tree below, calculate the value for a 2-year, annual-pay bond with a coupon rate of 7%.

8% 3%

5%

Year 0 Year 1

Valuing Option-free Bonds

Answer

Ø Step 1: calculate the bond value for up-node at year 1:

1,U 1 100 + 7 100 + 7

Answer (Cont.)

Ø Step 2: calculate the bond value for down-node at year 1:

$99.07 Valuing Option-free Bonds

Answer (Cont.)

Ø Step 3: calculate the bond value at year 0:

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Pathwise valuation

Ø Calculates the present value of a bond for each possible interest rate path and takes the average of these values across paths.

üFor a binomial interest rate tree with n periods, there will be 2(n-1)_{unique paths.}

Example

Ø Using the interest rate tree below, calculate the value for a 3-year, annual-pay bond with a coupon rate of 3%.

5.778%

Path Year 1 Year 2 Year 3 Value 1 3% 5.778% 10.738% 2 3% 5.778% 7.198% 3 3% 3.88% 7.198% 4 3% 3.88% 4.825%

Answer

Ø Step 1: find all the interest rate paths according to the interest rate tree.

Path Year 1 Year 2 Year 3 Value 1 3% 5.778% 10.738% 91.03 2 3% 5.778% 7.198% 93.85 1.03 1.03 1.05578 1.03 1.05578 1.10738  

Answer (Cont.)

Ø Step 2: calculate the bond values for each path and take the values of these values.

ü

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• Backward induction valuation;

• Pathwise valuation. Ø Exam tips:

• 一般不直接出题，但是后面重要知识点的基础。 Summary

Bonds with Embedded Option

Tasks:

Ø Describe fixed-income securities with embedded options;

Ø Explain how interest rate volatility, level and shape of the yield curve affect the value of callable or putable bond.

Basics of bonds with embedded option

Ø Embedded options refers to contingency provisions in the

bond’s indenture.

üCallable bonds: allows the issuer to benefit from lower interest rates by retiring the bond issue early; üPutable bonds: allows the bondholder to benefit from

higher interest rates by putting back the bonds to the issuer early.

• Extendible bond: allows the bondholder to keep the bond for a number of years after maturity.

Bonds with Embedded Option

Basics of bonds with embedded option (Cont.)

üEstate put: allows the heirs of an investor to put the bond back to the issuer upon the death of the investor; üSinking fund bonds (sinkers): require the issuer to set

aside funds periodically to retire the bond.

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Basics of bonds with embedded option (Cont.)

Ø For a callable bond, the investor is long the straight bond but short the call option on the bond.

ü Vcallalbe = Vstraight – Vcall Bonds with Embedded Option

Bonds with embedded option (Cont.)

Ø For a putable bond, the investor has a long position in both the straight bond and the put option on the bond. ü Vputalbe = Vstraight + Vput

Interest rate volatility vs. bond value

Ø The value of any embedded option, regardless of the type of option, increases with interest rate volatility. Thus: üAs interest rate volatility increases, the value of the

callable bond decreases;

• Note: recall Vcallalbe = Vstraight – Vcall

üAs interest rate volatility increases, the value of the putable bond increases.

• Note: recall Vputalbe = Vstraight + Vput Bonds with Embedded Option

Level of yield curve vs. callable bond value

Ø As interest rates decline, the value of the callable bond rises less rapidly than the value of the straight bond, limiting the upside potential for the investor.

üCall option value increases as interest rate decline, the rise of the straight bond value is partially offset by the increase in the value of the call option;

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Level of yield curve vs. putable bond value

Ø As interest rates rise, the value of the putable bond falls less rapidly than the value of the straight bond. üPut option value increases as interest rates rise, the

decline of the straight bond value is partially offset by the increase in the value of the put option;

• Note: recall the price-yield curve of putable bond. Bonds with Embedded Option

Shape of yield curve vs. callable bond value

Ø As the yield curve moves from being upward sloping, to

flat, to downward sloping, the value of the call option increases, and the value of the callable bond decreases.

üThe one-period forward rates become lower and the

opportunities to call increase.

Shape of yield curve vs. putable bond value

Ø As the yield curve moves from being upward sloping, to

flat, to downward sloping, the value of the put option decreases, and the value of the putable bond decreases.

üThe one-period forward rates become lower and the

opportunities to put declines.

• Basic types of bonds with embedded option;

• Components of the value of bonds with embedded

option;

• Factors on bond value with embedded option.

Ø Exam tips:

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Valuing Bonds with Embedded Option

Tasks:

Ø Describe how the arbitrage-free framework can be used to value a bond with embedded options; Ø Calculate the value of a callable or putable bond from

an interest rate tree.

Valuing bonds with embedded option

Ø The basic process to value a bond with embedded option

is similar to the valuation of straight bond; Ø However, the following two points are different:

üOnly binomial interest rate tree model is applicable, valuation with spot rates is non-available any more;

Valuing Bonds with Embedded Option

Valuing bonds with embedded option (Cont.)

üNeed to check at each node to determine whether the

embedded option will be exercised or not.

• Call rule: the value of callable bond is the lower of the call price and the calculated price if the bond is not called;

• Put rule: the value of putable bond is the higher of the put price and the calculated price if the bond is not put.

Example of callable bond

Ø Using the interest rate tree below, calculate the value for a 2-year, 7% annual-pay bond, which has a par value of $100 and callable at $100 at the end of year 1.

8% 3%

5% Year 0 Year 1

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Answer

Ø Step 1: determine the bond value for up-node at year 1: üCallable bond value = Min(100, 99.07) = $99.07

$99.07

Answer (Cont.)

Ø Step 2: determine the bond value for down-node at year 1:

üCallable bond value = Min(100, 101.9) = $100

$99.07

Answer (Cont.)

Ø Step 3: determine the bond value at year 0:

üCallable bond value = $103.43

$99.07

Example of putable bond

Ø Using the interest rate tree below, calculate the value for a 2-year, 7% annual-pay bond, which has a par value of $100 and putable at $100 at the end of year 1.

8% 3%

5% Year 0 Year 1

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Answer

Ø Step 1: determine the bond value for up-node at year 1:

üPutable bond value = Max(100, 99.07) = $100

$99.07

Answer (Cont.)

Ø Step 2: determine the bond value for down-node at year 1:

üPutable bond value = Min(100, 101.9) = $101.9

$100

Answer (Cont.)

Ø Step 3: determine the bond value at year 0:

üPutable bond value = $104.8

$100

• Calculation of a callable or putable bond value from an interest rate tree.

Ø Exam tips:

• 常考点：利用binomial interest rate tree 给含权债券定价，概念题或计算题都有可能考到。

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Valuing Risky Bonds with Embedded

Option

Tasks:

Ø Explain the calculation and use of option-adjusted spreads;

Ø Explain how interest rate volatility affects option-adjusted spreads.

Valuing risky callable and putable bonds

Ø Review: Z-spread and option-adjusted spread (OAS): üZ-spread: a constant yield spread added to government

spot curve that make PV of bond CFs equal to market price.

üOAS: yield spread that remove the influence of embedded option.

Valuing Risky Bonds with Embedded Option

Rates

Valuing risky callable and putable bonds (Cont.)

Ø Review: graph presentation of Z-spread and OAS:

Ø When valuing risky bond with the interest rate tree

generated from government spot curve, the model does

not produce arbitrage-free price (typically higher than market price).

Ø Option-adjusted spread (OAS): the constant spread that, when added to all the one-period forward rates on the interest rate tree, makes the model price of the bond equal to its market price.

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Example

Ø If the market price of callable bond in the previous example is $102.71, the OAS will be 50 bps.

$98.62

Ø OAS is often used as a measure of value relative to the benchmark.

üAn OAS lower than that for a comparable bond indicates that the bond is likely overpriced. üAn OAS larger than that for a comparable bond

indicates that the bond is likely underpriced. üAn OAS close to that for a comparable bond indicates

that the bond is likely fairly priced.

Interest rate volatility vs. OAS

Ø As interest rate volatility increases, the OAS for the

callable bond decreases, and vice versa.

üHigher volatility → higher call option value → lower price for benchmark callable bond → lower OAS. Ø As interest rate volatility increases, the OAS for the

putable bond increases, and vice versa.

üHigher volatility → higher put option value → higher price for benchmark putable bond → higher OAS.

Monte Carlo forward-rate simulation

Ø Path dependency: cash flow to be received in a particular period depends on the path followed to reach its current level as well as the current level itself.

üE.g.: level of prepayments for MBS is interest rate path dependent.

üBinomial tree backward induction assumes cash flows

are not path dependent, therefore it cannot value the securities with such cash flow.

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Monte Carlo forward-rate simulation (Cont.)

Ø Monte Carlo forward-rate simulation involves randomly generating a large number of interest rate paths. üA key feature of the Monte Carlo method is that the

underlying cash flows can be path dependent.

• Calculation and application of OAS;

• Interest rate volatility vs. OAS;

• Monte Carlo forward-rate simulation.

Ø Exam tips:

• 常考点：OAS的基本概念与应用，概念题。 Summary

Interest Rate Risk of Bonds with

Embedded Option

Tasks:

Ø Calculate and interpret effective duration of a callable or putable bond;

Ø Describe one-sided durations and key rate durations; Ø Compare effective convexities of callable, putable,

and straight bonds.

Effective duration

Ø The sensitivity of the bond’s price to a 100 bps parallel shift of the benchmark yield curve, typically the government par curve, assuming no change in the bond’s credit spread.

Interest Rate Risk of Bonds with Embedded Option

-PV - -PV Effective duration

Curve PV 

   ₀

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Effective duration (Cont.)

Ø The process to calculate the PV- and PV+:

üStep 1: calculate the implied OAS to the benchmark yield curve according to the market price (PV0); üStep 2: shift the benchmark yield curve down by

(△Curve), generate a new interest rate tree, and then calculate the PV- using the OAS calculated in step 1; Interest Rate Risk of Bonds with Embedded Option

üStep 3: shift the benchmark yield curve up by (△Curve), and calculate the PV+ with similar method;

üStep 4: calculate the bond’s effective duration using the formula.

Ø Both call and put option will potentially make the bond’s life shorter, so the effective duration of both callable and putable bond will be less or equal to that of identical straight bond.

Ø The interest rate movements have little impact on effective duration of straight bond, but have much influence on that of callable and putable bonds. üDecrease of interest rate will decrease the effective

duration of callable bond;

üIncrease of interest rate will decrease the effective duration of putable bond.

• Note: recall the price-yield curves.

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Ø The effective durations may be misleading when it is calculated by averaging the changes resulting from shifting the benchmark yield curve up and down by the same amount.

üThe price sensitivity of bonds with embedded options is

not symmetrical when the embedded option is in the money.

• The callable bond price has limited upside potential;

• The putable bond price has limited downside

potential.

One-side duration

Ø The effective durations when interest rates go up or

down.

ü One-side up-duration: durations that apply only when interest rates rise.

üOne-side down-duration: durations that apply only when interest rates fall.

Key rate duration/partial duration

Ø The sensitivity of the bond’s price to changes in specific maturities on the benchmark yield curve.

üHelp to identify the “shaping risk” for bonds.

• Shaping risk: the sensitivity of bond’s price to changes in the shape of the yield curve (e.g., steepening and flattening).

Effective convexity

Ø The sensitivity of duration to changes in interest rates.

Ø Straight bond: always exhibits low positive convexity; Ø Callable bond:

üWhen interest rates are high, callable and straight bond have similar positive convexity;

üWhen the call option is in the money, the effective convexity of the callable bond turns negative.

- 

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Effective convexity (Cont.)

Ø Putable bond:

ü Always have positive convexity;

üWhen the put option is in the money, the effective convexity of the putable bond is higher than straight bond.

• Note: recall the price-yield curves.

• Effective duration, one-side duration, key rate duration;

• Effective convexity. Ø Exam tips:

• 常考点：effective duration的计算与比较，计算题或概念题。 Summary

Convertible Bonds

Tasks:

Ø Describe defining features of a convertible bond; Ø Calculate and interpret the components of a

convertible bond’s value;

Ø Compare risk-return characteristics of a convertible bond with straight bond and common stock.

Defining features of convertible bonds

Ø Convertible bond: a hybrid security that gives

bondholders the right to convert their debt into equity during a pre-determined period (conversion period) at a pre-determined price (conversion price).

üConversion option: a call option on the issuer’s common stock.

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Defining features of convertible bonds(Cont.)

Ø Conversion ratio: the number of common shares that the bondholder receives from converting the bonds into shares.

üConversion price = Issue price/Conversion ratio

• Typically, conversion bonds are issued at par value.

Convertible Bonds

Analysis of convertible bonds

Ø Conversion value: the value of the bond if it is converted to common shares.

üConversion value = Market share price ×_Conversion

ratio

üConversion ratio is fixed but the market share price is floating, so the conversion value is floating. Ø Straight value: the value of the bond if it were not

convertible.

üThe PV of the cash flows

Ø Minimum value: Max(conversion value, straight value)

Convertible Bonds

Analysis of convertible bonds (Cont.)

Ø Market conversion price: the price that investors effectively pay for the underlying common share if they buy the convertible bond and then convert it into shares. üMarket conversion price

= Convertible bond price/Conversion ratio üMarket conversion premium per share

= Market conversion price - Market share price üMarket conversion premium ratio

= Market conversion premium per share/Market share price

Convertible Bonds

Example

Ø A convertible bond of company A, issued at $900 with

face value of $1000 and coupon rate of 5 percent, can be converted to 10 common shares, which is traded at $80 per share. Now, the convertible bond is traded at $850. Calculate the conversion price, conversion value, market conversion price, market conversion premium per share, market conversion premium ratio.

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Answer

Ø Conversion price = $900/10 = $90;

Ø Conversion value = $80*10 = $800;

Ø Market conversion price = $850/10 =$85;

Ø Market conversion premium per share = $85 - $80 = $5;

Ø Market conversion premium ratio = $5/$80 = 6.25%;

Convertible Bonds

Downside risk

Ø Straight value can be used as a benchmark of the downside risk of a convertible bond.

üPremium over straight value

= (Convertible bond price/Straight price) - 1

Upside potential

Ø Depends primarily on the prospects of the underlying

common share.

Convertible Bonds

Valuation equivalence

Ø The convertible bond can be valued as the equivalence

of its component combination:

üConvertible bond = Straight bond + Call option on stock üCallable convertible bond value

= Straight bond + Call option on stock - Call option on bond

üPutable convertible bond

= Straight bond + call option on stock + put option on bond

Convertible Bonds

Risk-return characteristics of convertible bonds

Ø When the underlying share price is well below the conversion price, the convertible bond exhibits mostly

bond risk-return characteristics. üStraight value acts as a downside limit. Ø When the underlying share price is well above the

conversion price, the convertible bond exhibits mostly

stock risk-return characteristics.

Ø In between these two extremes, the convertible bond

trades like a hybrid instrument.

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• Defining features of a convertible bond;

• Components of a convertible bond’s value;

• Comparison of convertible bond with straight bond and common stock.

Ø Exam tips:

• 常考点：概念题。 Summary

Measures of Credit Risk

Tasks:

Ø Explain probability of default, loss given default, expected loss, and present value of the expected loss; Ø Calculate and interpret the present value of the

expected loss on a bond;

Ø Compare the credit analysis ABS and corporate debt.

Measures of credit risk

Ø Probability of default: the probability that the bond will default before maturity.

Ø Loss given default: the amount of the remaining coupon and principal payments lost in the event of default. üRecovery rate: the percentage of the position received

or recovered in default.

• Loss given default (%) = 100% - Recovery rate (%)

Measures of Credit Risk

Measures of credit risk (Cont.)

Ø Expected loss: probability of default ×_{loss given default}

Ø Present value of expected loss (PVEL): the largest price one would be willing to pay to a third party (e.g., an insurer) to entirely remove the credit risk of holding the bond.

üIt involves two modifications to the expected loss:

• Adjust the probabilities to account for the risk of the cash flows (risk-neutral probability);

• Include the time value of money in the calculation.

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Ø Credit spread: the yield difference between default-free and credit risky zero-coupon bonds (spot rates). üIn practice, the “true” credit spread will consist of both

the expected percentage loss and a liquidity risk premium.

üTerm structure of credit spread: the relationship of credit spreads to maturity.

Ø Present value of expected loss (PVEL) is also the difference between the value of a credit-risky bond and an otherwise identical risk-free bond.

üPVEL can be estimated from credit spread on a risky bond.

Example

Ø A 6% annual-pay bond, with par value of $1,000, is maturing in 3 years. Assume all rates are continuously compounded, using the information in the following table, calculate the present value of expected loss for the payment due in 2 years and the bond.

Time (yrs) Risk-free rate (Rf) Credit spread

1 2% 1%

2 3% 2%

3 4% 2%

Answer

Ø The PVEL for payment due in years is: $56.56 - $54.42 = $2.14

Ø The PVEL for the bond is: $1057.7 - $ 1002.7 = $ 55

Time

(yrs) Rf _spreadCredit Total _yield Cash _flow PV(Rf) PV(Risky) ΔPV

1 2% 1% 3% 60 58.82 58.25 0.57

2 3% 2% 5% 60 56.56 54.42 2.14

3 4% 2% 6% 1060 942.34 890 52.34

Total 1057.7 1002.7 55

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ABS vs. Corporate debt

Ø In the case of a corporate bond, when the issuer defaults, the cash flows cease and there is a terminal cash flow; Ø ABS do not default, but they can lose value as the SPE’s

pool of securitized assets incurs defaults.

üThe credit risk metric of probability of loss is applied rather than probability of default.

• Probability of default, loss given default, expected loss;

• PVEL and its calculation;

• ABS vs. corporate debt. Ø Exam tips:

• 不是重要考点。 Summary

Credit Models

Tasks:

Ø Explain credit scoring and credit ratings;

Ø Explain structural models and reduced form models of corporate credit risk, and their assumptions, strengths, and weaknesses.

Credit models

Ø Traditional credit models üCredit scoring üCredit rating

Ø Newer credit models for corporate credit risk

üStructural models

üReduced form models

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Credit scoring

Ø Credit scoring ranks a borrower’s credit riskiness. üProvide an ordinal ranking of a borrower’s credit risk;

• Do not tell the degree to which the credit risk differs among different ranks.

üUsed for small businesses and individuals.

Credit Models

Credit rating

Ø Credit ratings rank the credit risk of a company, government (sovereign), quasi-government, or ABS.

Credit Models

Credit rating (Cont.)

Ø

Credit Models

Ø Strengths of credit ratings

üProvide a simple statistic that summarizes a complex credit analysis of a potential borrower;

üTend to be stable over time, reducing debt market volatility.

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Ø Weaknesses of credit ratings

üThe stability reduces the correspondence to a debt’s default probability, and make the rating lag the market; üDo not depend on business cycle, while default

probability does;

üThe issuer-pays model for compensating credit-rating agencies has a potential conflict of interest that may distort the accuracy of credit ratings.

Credit Models

Structural models

Ø Structural models were originated to understand the economics of a company’s liabilities and build on the insights of option pricing theory.

üThey are called structural models because they are based on the structure of a company’s balance sheet.

Credit Models

Structural models (Cont.)

Ø Assumptions of structural models:

üCompany’s assets (A) are traded in a frictionless arbitrage-free market;

üThe value of the company’s assets has a lognormal distribution;

üThe risk-free interest rate (r) is constant over time;

üThe company has a simple balance sheet structure with

only one class of simple zero-coupon debt D(K,T).

Credit Models

Ø Call option analogy for equity: holding the company’s equity is economically equivalent to owning a European call option on the company’s assets (A) with strike price K (face value of zero-coupon debt) and maturity T (maturity of debt), because they have the same payoff at maturity T.

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üAt maturity, exercise the call option if A > K, the value of both equity and call option is (A-K);

üAt maturity, do not exercise the call option if A < K, the value of both equity and call option is zero.

Credit Models

Ø Debt option analogy: owning the company’s debt is economically equivalent to owning a riskless bond, and simultaneously selling a European put option on the assets (A) of the company with strike price K (face value of zero-coupon debt) and maturity T.

Credit Models

üAt maturity, exercise the put option if A < K, the value of both positions is A;

üAt maturity, do not exercise the put option if A > K, the value of both positions is K.

• Vrisky debt = Vrisk-free debt – Vput option Credit Models

Ø Under structural models, debt value can be computed as:

wherein:

• r: continuously compounded risk-free rate;

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Ø Strengths:

üProvide an option analogy for understanding a company’s default probability and recovery rate; üCan be estimated using only current market prices. Ø Weaknesses:

üModel assumptions of simple balance sheet and traded

assets are not realistic;

üEstimation procedures do not consider business cycle.

Credit Models

Reduced form models

Ø Assumptions of reduced form models:

üThe company’s zero-coupon bond trades in frictionless markets that are arbitrage free;

üRisk-free interest rate (r) is stochastic;

üEconomy and recovery rate are stochastic, probability of default is not constant and varies with the state of economy;

üWhether a particular company actually defaults depends only on company-specific consideration.

Credit Models

Reduced form models (Cont.)

Ø Under reduced form models, debt value can be

computed as:

wherein:

• K = face value of debt;

• = Expectation operation using risk-neutral probabilities;

Reduced form models (Cont.)

Ø Strengths:

üModel inputs are observable, historical estimation procedures can be used;

üCredit risk allowed to fluctuate with business cycle; üDo not require specification of the company’s BS

structure. Ø Weaknesses:

üUnless the model has been formulated and back tested

properly, the hazard rate estimation may not be valid.

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• Credit scoring and credit rating;

• Structural models;

• Reduced form models.

Ø Exam tips:

• 常考点：Structural models 和 Reduced form models的建模思路与优缺点，概念题。

Summary

Basic Concepts of CDS

Tasks:

Ø Describe credit default swaps (CDS), and the parameters that define a given CDS product; Ø Describe credit events and settlement protocols with

respect to CDS.

Credit derivative

Ø A derivative instrument in which the underlying is the credit quality of a borrower.

Ø Four types of credit derivative:

üTotal return swaps

üCredit spread options üCredit-linked notes üCredit default swaps (CDS) Basic Concepts of CDS

Definition

Ø A derivative contract between two parties, a credit protection buyer and credit protection seller, in which the

buyer makes a series of cash payments to the seller and

receives a promise of compensation for credit losses

resulting from the default -that is, a pre-defined credit event - of a third party.

Protection

buyer Protection seller

Periodic premium if no default Compensation if

default occurs

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Basic features of CDS

Ø Notional amount/principal: the amount of protection being purchased.

Ø CDS spread (%): the periodic premium that the buyer of a CDS pays to the seller for protection against credit risk. Ø CDS coupon rate (%): the periodic premium that the

buyer actually pays to the seller. Typically, for standardization:

ü1% for a CDS on an investment-grade company or

index;

ü5% for a CDS on a high-yield company or index.

Basic Concepts of CDS

Basic features of CDS (Cont.)

Ø Upfront payment/upfront premium: the differential between the credit spread and the standard coupon rate that converted to a present value basis.

üA credit spread more than the standard coupon rate will result in a cash upfront payment from the protection buyer to the seller;

üA credit spread less than the standard rate would result in a cash upfront payment from the protection seller to buyer.

Single name CDS

Ø A CDS on one specific borrower (reference entity). üReference obligation: a particular debt instrument

issued by the borrower that is the designated instrument being covered by CDS.

• Usually a senior unsecured obligation (senior CDS);

• Any debt obligation issued by the borrower that is pari passu (ranked equivalently in priority of claims) or higher relative to the reference obligation is covered.

Single name CDS (Cont.)

Ø The payoff of the CDS is determined by the cheapest-to-deliver obligation.

üCheapest-to-deliver obligation: the debt instrument that can be purchased and delivered at the lowest cost but has the same seniority as the reference obligation.

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Index CDS

Ø A CDS that allows participants to take positions on the credit risk of a combination of borrowers.

üThe notional principle is the sum of the protection on all the borrowers;

üCredit correlation is a key determinant of its value.

• The more correlated the defaults, the more costly it is to purchase the index CDS.

Credit events

Ø The outcome that triggers a payment from the credit protection seller to the buyer:

üBankruptcy: allows the defaulting party to work with creditors under the supervision of the court so as to avoid full liquidation;

üFailure to pay: a borrower does not make a scheduled payment of principal or interest on any outstanding obligations after a grace period, without a formal bankruptcy filing.

Credit events (Cont.)

üRestructuring: the issuer forces its creditors to accept terms that are different than those specified in the original issue.

• Reduction or deferral of principal or interest;

• Change in seniority or priority of an obligation;

• Change in the currency in which principal or interest is scheduled to be paid.

Settlement protocols

Ø Physical settlement: actual delivery of the debt instrument in exchange for a payment by the credit protection seller of the notional amount of the contract.

Protection

Reference obligation

Notional amount

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Settlement protocols (Cont.)

Ø Cash settlement: the credit protection seller pays cash to the credit protection buyer.

üPayout amount = Payout ratio (%) × Notional amount üPayout ratio (%) = 1 - recovery rate (%)

Protection

Cash payment

• Basic features of CDS;

• Single name CDS and index CDS;

• Credit events and settlement protocols. Ø Exam tips:

• 不是考试重点。 Summary

Pricing and Application of CDS

Tasks:

Ø Explain the factors that influence the pricing of CDS; Ø Describe the use of CDS to manage credit exposures

and to take advantage of valuation disparities.

Pricing of CDS

Ø Pricing means determining the CDS spread or upfront

payment given a particular coupon rate for a contract. Ø On a present value basis, the sum of the value of CDS spread, CDS coupon, and the upfront premium should be zero.

üUpfront premium (%) by buyer

≈ (CDS spread - CDS coupon)×Duration of CDS

• The lower the CDS coupon rate, the higher the upfront premium.

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Pricing of CDS (Cont.)

Ø Factors that influence the market’s pricing of CDS: üThe higher the probability of default, the higher the

CDS spread;

• Hazard rate: the probability of default given that it has not already occurred.

üThe higher the loss given default, the higher the CDS spread;

• (Expected loss)t = (Hazard rate)t×(Loss given default)t Pricing and Application of CDS

Profit of CDS

Ø The value of a CDS may change if the credit quality of the reference entity or the credit risk premium in the overall market change.

üProfit for protection buyer (%) ≈ Change in spread (%)×Duration of CDS üProfit for protection buyer ($)

≈ Change in spread (%)×Duration of CDS× Notional amount($)

Pricing and Application of CDS

Application of CDS

Ø Credit curve: the term structure of credit spread, or the credit spreads for a range of maturities of a company’s debt .

üUpward-sloping credit curves imply a greater likelihood of default in later years;

üDownward-sloping credit curves imply a greater probability of default in the earlier years.

• Downward-sloping curves are less common.

Application of CDS (Cont.)

Ø Managing credit exposure: the taking on or shedding of credit risk in light of changing expectations and/or valuation disparities.

Ø Valuation disparity: the focus is on differences in the pricing of credit risk in the CDS market relative to that of the underlying bonds.

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Managing credit exposure

Ø Adjustment of credit exposure: increasing/decreasing credit exposure by selling/buying CDS if having assumed too little/much credit risk;

Ø Naked CDS: buying or selling credit protection without credit exposure to the reference entity;

Ø Long/short trade: taking a long position in one CDS and a short position in another.

üA bet that the credit position of one entity will improve relative to that of another.

Managing credit exposure (Cont.)

Ø Curve trade: buying a CDS of one maturity and selling a CDS on the same reference entity with a different maturity.

üIf an investor believes that the credit curve will become steeper, he can buy a long-term CDS and sell a short-term CDS;

üIf an investor believes that the credit curve will become flatter, he can buy a short-term CDS and sell a long-term CDS.

Valuation disparity

Ø Basis trade: exploit the difference of credit spread between bond market and CDS market.

üThe general idea behind is that any such mispricing will disappear when the market recognizes the disparity.

Valuation disparity (Cont.)

Ø Arbitrage trade: buy the cheaper and sell the more expensive.

üif the cost of the index is not equivalent to the aggregate cost of the index components;

üIf a synthetic CDO is not equivalent to the actual CDO.

• Synthetic CDO

= Default-free security - CDS (protection seller)

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• Pricing and profit of CDS;

• Use CDS to manage credit exposure and take

advantage of valuation disparities. Ø Exam tips: