Fixed Income
Level 2  2017
Instructor: Feng
Brief Introduction
Topic weight:
Study Session 12 Ethics & Professional Standards 10 15% Study Session 3 Quantitative Methods 5 10% Study Session 4 Economics 5 10% Study Session 56 Financial Reporting and Analysis 15 20% Study Session 78 Corporate Finance 5 15% Study Session 911 Equity Investment 15 25% Study Session 1213 Fixed Income 10 20% Study Session 14 Derivatives 5 15% Study Session 15 Alternative Investments 5 10% Study Session 1617 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 ArbitrageFree 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
Brief Introduction
考纲对比:
Ø 与2016年相比，2017年的考纲没有变化。
Brief Introduction
推荐阅读:
Ø 固定收益分析
• Fabozzi, F. J 著
• ISBN: 9787565406508
• 东北财经出版社
Brief Introduction
学习建议:
Ø 本门课程难度比较大，要着重理解和总结；
Ø 逻辑递进关系比较强，要把每个知识点学懂了再继续学；
Ø 听课与做题相结合，但并不建议“刷题”；
Ø 最重要的，认真、仔细的听课。
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 zerocoupon 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 singlepayment security to be issued at a future date.
üf(j, k): annualized kyear 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 & Forward Pricing Models
Forward rate models
Ø Forward rate models show how forward rates can be
extrapolated from spot rates.
j k
j
kExample
Ø The spot rates for three hypothetical zerocoupon bonds with maturities of one, two, and three years are 9%, 10% and 11%. Calculate the forward rate for a oneyear zero issued two years from today.
Answer:
(1 + 0.11)3_{ = (1 + 0.10)}2_{×}_{[1 + f(2,1)]}
f(2,1) = 13.03%
Forward Rate Models & Forward Pricing Models
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 }
Forward Rate Models & Forward Pricing Models
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 }
Forward Rate Models & Forward Pricing Models
Discount factor
Ø The price of a riskfree singleunit payment at time j:
Forward price
Ø The price at time j years from today for a zerocoupon bond with maturity j+k years and unit principal.
Forward Rate Models & Forward Pricing Models
Forward pricing models
Ø The forward pricing model describes the valuation of forward contracts.
Forward Rate Models & Forward Pricing Models
Example
Ø Consider a 2year loan beginning in 1 year. The 1year spot rate is 7% and the 3year spot rate is 9%. Calculate the forward price of a 2year bond to be issued in 1 year.
Answer:
F(1,2) = 0.7722 ÷ 0.9346 = 0.8262.
Forward Rate Models & Forward Pricing Models
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;
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 Zspread, TED and Libor–OIS spreads.
YieldtoMaturity (YTM)
Ø The internal rate of return on the cash flows of a fixedrate bond.
üYTM is the same as the spot rate for zerocoupon 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
YieldtoMaturity (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 optionfree 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 exante 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.
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 1year, 2year, and 3year spot rates.
Yield and Spread
Answer:
Ø Step 1: S1=1%;
Ø Step 2: value the 2year bond using spot rates: 100=1.25/(1+1%)+101.25/(1+S2)2
=> S2=1.252%;
Ø Step 3: value the 3year bond using spot rates: 100=1.5/(1+1%)+1.5/(1+1.252%)2_{+101.5/(1+S}_{3}_{)}3_{ }
=> S3=1.51%. Yield and Spread
Swap rate curve
Ø Swap rate: the interest rate for the fixedrate leg of an interest rate swap.
Ø Swap rate curve (swap curve): the yield curve of swap rate.
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 “ontherun” 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
Zspread
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 Tbill 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.
LiborOIS 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;
• Zspread, 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
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, shortterm rates are expected to rise;
üIf yield curve is downward sloping, shortterm rates are expected to fall;
üIf yield curve is flat, shortterm rates are expected to remain constant.
Traditional Term Structure Theories
Local expectation theory
Ø This theory assume riskneutrality 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 shortholdingperiod returns on longdated bonds do exceed those on shortdated bonds.
Traditional Term Structure Theories
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 shortterm spot
rates, this theory predicts an upwardsloping yield curve.
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.
Traditional Term Structure Theories
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.
Traditional Term Structure Theories
Ø Importance: ☆☆☆ Ø Content:
• 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:
Modern term structure models
Ø Provide quantitatively precise descriptions of how interest rates evolve.
üEquilibrium term structure models
• The CoxIngersollRoss model
• The Vasicek Model
üArbitrage free models
• The HoLee 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 onefactor or multifactor models.
Modern Term Structure 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.
Modern Term Structure Models
CoxIngersollRoss (CIR) model
Ø CIR model is based on the idea that individuals must determine his/her optimal tradeoff 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.
CIR model (Cont.)
Ø Mathematically, the CIR model is as follows:
Where:
üdr: a infinitely small change in shortterm 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
CIR model (Cont.)
üb: longrun value of the shortterm interest rate; ür: the shortterm interest rate;
üa: a positive parameter for speed of mean reversion adjustment, and the higher/lower, the quicker/slower; üσ: interest rate volatility.
Modern Term Structure Models
CIR model (Cont.)
Ø The CIR model consist of tow terms:
üDrift term: ; which means the interest rate is
meanrevert 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
nonnegative, and assets that higher interest rate will lead to higher volatility.
a b  r dt
σ rdz r
Modern Term Structure Models
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 nonnegative;
• Volatility remains constant over the period of analysis.
d r = a b  r d t + σ d z
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.
Modern Term Structure Models
HoLee model
Ø The model assumes that the yield curve moves in a way
that is consistent with a noarbitrage condition.
üθ: a timedependant 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
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 oneyear spot rate (S1) is 9% and twoyear
spot rate (S2) is 10%, then the implied forward rate f(1,1)
is 11.01%.
Ø The return of year zerocoupon bond over the
oneyear holding period is 9%.
Active Bond Portfolio Management
Example (Cont.)
Ø The price of a twoyear zerocoupon bond now (P0) is:
Ø If actual oneyear spot rate one year later is current forward rate f(1,1), the price of the above bond one year later (P1) is:
Active Bond Portfolio Management
Example (Cont.)
Ø So, the return of twoyear zerocoupon bond over the
oneyear holding period is also 9%.
90 08 82 64. . 1 9%
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.
Active Bond Portfolio Management
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 maturitymatching strategy. üThe bond is valued at successively lower yields and
higher prices, and then sold before maturity to realize a higher return.
Active Bond Portfolio Management
Example
Ø The following table shows a upwardsloping yield curve and the prices of a 3% annualpay 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
Active Bond Portfolio Management
Example (Cont.)
Ø Assuming an investor with investment horizon of 5 years purchase the bond with maturity of 5 years (maturitymatching 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.
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 welldefined IRR (YTM)
•Effective duration •Nonparallel shift
Active Bond Portfolio Management
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
Active Bond Portfolio Management
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}_{} _{}
Active Bond Portfolio Management
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.
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 ≈
Active Bond Portfolio Management
Maturity structure of yield volatility
Ø A representation of the yield volatility of a zerocoupon bond for every maturity of security.
üTypically, shortterm rates are more volatile than longterm rates.
• Shortterm volatility is most strongly linked to uncertainty regarding monetary policy;
• Longterm volatility is most strongly linked to uncertainty regarding the real economy and inflation.
Active Bond Portfolio Management
Ø Importance: ☆☆ Ø Content:
• 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:
ArbitrageFree Valuation
Tasks:
Ø Explain arbitragefree valuation of a fixedincome instrument;
Ø Calculate the arbitragefree value of an optionfree, fixedrate 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.
ArbitrageFree 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 zerocoupon securities;
• Reconstitution: recombine the individual zero coupon securities and reproduce the underlying coupon bond. üDominance: a financial asset with a riskfree payoff in
the future must have a positive price today.
ArbitrageFree Valuation
Arbitragefree valuation
Ø An approach to security valuation that determines security values that are consistent with the absence of an arbitrage opportunity.
Arbitragefree valuation (Cont.)
Ø Viewing any bond as a package of zerocoupon bonds;
Ø Using spot rates that correspond to the maturities of the zerocoupon 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
ArbitrageFree Valuation
Example:
Ø Suppose that the oneyear spot rate is 2%, the twoyear spot rate is 3%, and the threeyear spot rate is 4%. Then, the price of a threeyear bond that makes a 5% annual coupon payment is:
5
1
5
2
1 0 5
3P = + + = 1 0 2 .9 6 1 .0 2 1 .0 3 1 .0 4
ArbitrageFree Valuation
Ø Importance: ☆☆ Ø Content:
• Arbitrage opportunities and arbitragefree 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
Binomial interest rate tree
Ø The arbitragefree 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 oneperiod forward rates corresponding to the nodal period.
Binomial Interest Rate Tree
Binomial interest rate tree (Cont.)
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
Binomial interest rate tree (Cont.)
Ø The binomial interest rate tree framework is a lognormal random walk (lognormal tree) that insures two appealing properties:
üNonnegativity of interest rates; ühigher volatility at higher interest rates.
Binomial interest rate tree (Cont.)
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 arbitragefree values for the benchmark bonds;
üAdjacent forward rates (for the same period) are two standard deviations apart.
Binomial Interest Rate Tree
Ø Importance: ☆☆ Ø Content:
• 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 OptionFree 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 oneperiod forward rate at that
node.
Example
Ø Using the interest rate tree below, calculate the value for a 2year, annualpay bond with a coupon rate of 7%.
8% 3%
5%
Year 0 Year 1
Valuing Optionfree Bonds
Answer
Ø Step 1: calculate the bond value for upnode at year 1:
1,U 1 100 + 7 100 + 7
Valuing Optionfree Bonds
Answer (Cont.)
Ø Step 2: calculate the bond value for downnode at year 1:
$99.07 Valuing Optionfree Bonds
Answer (Cont.)
Ø Step 3: calculate the bond value at year 0:
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(n1)_{ unique paths.}
Valuing Optionfree Bonds
Example
Ø Using the interest rate tree below, calculate the value for a 3year, annualpay bond with a coupon rate of 3%.
5.778%
Valuing Optionfree Bonds
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.
Valuing Optionfree Bonds
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
Ø Importance: ☆☆ Ø Content:
• 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 fixedincome 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.
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
Bonds with Embedded Option
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;
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 priceyield 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 oneperiod forward rates become lower and the
opportunities to call increase.
Bonds with Embedded Option
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 oneperiod forward rates become lower and the
opportunities to put declines.
Bonds with Embedded Option
Ø Importance: ☆☆☆ Ø Content:
• 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:
Valuing Bonds with Embedded Option
Tasks:
Ø Describe how the arbitragefree 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 nonavailable 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.
Valuing Bonds with Embedded Option
Example of callable bond
Ø Using the interest rate tree below, calculate the value for a 2year, 7% annualpay 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
Answer
Ø Step 1: determine the bond value for upnode at year 1: üCallable bond value = Min(100, 99.07) = $99.07
$99.07
Valuing Bonds with Embedded Option
Answer (Cont.)
Ø Step 2: determine the bond value for downnode at year 1:
üCallable bond value = Min(100, 101.9) = $100
$99.07
Valuing Bonds with Embedded Option
Answer (Cont.)
Ø Step 3: determine the bond value at year 0:
üCallable bond value = $103.43
$99.07
Valuing Bonds with Embedded Option
Example of putable bond
Ø Using the interest rate tree below, calculate the value for a 2year, 7% annualpay 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
Answer
Ø Step 1: determine the bond value for upnode at year 1:
üPutable bond value = Max(100, 99.07) = $100
$99.07
Valuing Bonds with Embedded Option
Answer (Cont.)
Ø Step 2: determine the bond value for downnode at year 1:
üPutable bond value = Min(100, 101.9) = $101.9
$100
Valuing Bonds with Embedded Option
Answer (Cont.)
Ø Step 3: determine the bond value at year 0:
üPutable bond value = $104.8
$100
Valuing Bonds with Embedded Option
Ø Importance: ☆☆☆ Ø Content:
• 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。
Valuing Risky Bonds with Embedded
Option
Tasks:
Ø Explain the calculation and use of optionadjusted spreads;
Ø Explain how interest rate volatility affects optionadjusted spreads.
Valuing risky callable and putable bonds
Ø Review: Zspread and optionadjusted spread (OAS): üZspread: 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 Zspread and OAS:
Valuing Bonds with Embedded Option
Valuing risky callable and putable bonds (Cont.)
Ø When valuing risky bond with the interest rate tree
generated from government spot curve, the model does
not produce arbitragefree price (typically higher than market price).
Ø Optionadjusted spread (OAS): the constant spread that, when added to all the oneperiod forward rates on the interest rate tree, makes the model price of the bond equal to its market price.
Example
Ø If the market price of callable bond in the previous example is $102.71, the OAS will be 50 bps.
$98.62
Valuing Bonds with Embedded Option
Valuing risky callable and putable bonds (Cont.)
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.
Valuing Bonds with Embedded Option
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.
Valuing Bonds with Embedded Option
Monte Carlo forwardrate 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.
Monte Carlo forwardrate simulation (Cont.)
Ø Monte Carlo forwardrate 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.
Valuing Bonds with Embedded Option
Ø Importance: ☆☆ Ø Content:
• Calculation and application of OAS;
• Interest rate volatility vs. OAS;
• Monte Carlo forwardrate 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 onesided 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}
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
Effective duration (Cont.)
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.
Interest Rate Risk of Bonds with Embedded Option
Effective duration (Cont.)
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.
Interest Rate Risk of Bonds with Embedded Option
Effective duration (Cont.)
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 priceyield curves.
Effective duration (Cont.)
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.
Interest Rate Risk of Bonds with Embedded Option
Oneside duration
Ø The effective durations when interest rates go up or
down.
ü Oneside upduration: durations that apply only when interest rates rise.
üOneside downduration: durations that apply only when interest rates fall.
Interest Rate Risk of Bonds with Embedded Option
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).
Interest Rate Risk of Bonds with Embedded Option
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.

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 priceyield curves.
Interest Rate Risk of Bonds with Embedded Option
Ø Importance: ☆☆ Ø Content:
• Effective duration, oneside 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 riskreturn 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 predetermined period (conversion period) at a predetermined price (conversion price).
üConversion option: a call option on the issuer’s common stock.
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.
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
Riskreturn characteristics of convertible bonds
Ø When the underlying share price is well below the conversion price, the convertible bond exhibits mostly
bond riskreturn 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 riskreturn characteristics.
Ø In between these two extremes, the convertible bond
trades like a hybrid instrument.
Ø Importance: ☆☆ Ø Content:
• 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 (riskneutral probability);
• Include the time value of money in the calculation.
Measures of credit risk (Cont.)
Ø Credit spread: the yield difference between defaultfree and credit risky zerocoupon 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.
Measures of Credit Risk
Measures of credit risk (Cont.)
Ø Present value of expected loss (PVEL) is also the difference between the value of a creditrisky bond and an otherwise identical riskfree bond.
üPVEL can be estimated from credit spread on a risky bond.
Measures of Credit Risk
Example
Ø A 6% annualpay 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) Riskfree rate (Rf) Credit spread
1 2% 1%
2 3% 2%
3 4% 2%
Measures of Credit Risk
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 _{spread}Credit 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
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.
Measures of Credit Risk
Ø Importance: ☆ Ø Content:
• 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
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), quasigovernment, or ABS.
Credit Models
Credit rating (Cont.)
Ø
Credit Models
Credit rating (Cont.)
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.
Credit rating (Cont.)
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 issuerpays model for compensating creditrating 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 arbitragefree market;
üThe value of the company’s assets has a lognormal distribution;
üThe riskfree interest rate (r) is constant over time;
üThe company has a simple balance sheet structure with
only one class of simple zerocoupon debt D(K,T).
Credit Models
Structural models (Cont.)
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 zerocoupon debt) and maturity T (maturity of debt), because they have the same payoff at maturity T.
Structural models (Cont.)
üAt maturity, exercise the call option if A > K, the value of both equity and call option is (AK);
üAt maturity, do not exercise the call option if A < K, the value of both equity and call option is zero.
Credit Models
Structural models (Cont.)
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 zerocoupon debt) and maturity T.
Credit Models
Structural models (Cont.)
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 = Vriskfree debt – Vput option Credit Models
Structural models (Cont.)
Ø Under structural models, debt value can be computed as:
wherein:
• r: continuously compounded riskfree rate;
Structural models (Cont.)
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 zerocoupon bond trades in frictionless markets that are arbitrage free;
üRiskfree 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 companyspecific 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 riskneutral 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.
Ø Importance: ☆☆ Ø Content:
• 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 üCreditlinked 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 predefined credit event  of a third party.
Protection
buyer Protection seller
Periodic premium if no default Compensation if
default occurs
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 investmentgrade company or
index;
ü5% for a CDS on a highyield 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.
Basic Concepts of CDS
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.
Basic Concepts of CDS
Single name CDS (Cont.)
Ø The payoff of the CDS is determined by the cheapesttodeliver obligation.
üCheapesttodeliver obligation: the debt instrument that can be purchased and delivered at the lowest cost but has the same seniority as the reference obligation.
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.
Basic Concepts of 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.
Basic Concepts of CDS
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.
Basic Concepts of CDS
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
buyer Protection seller
Reference obligation
Notional amount
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
buyer Protection seller
Cash payment
Basic Concepts of CDS
Ø Importance: ☆ Ø Content:
• 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.
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 .
üUpwardsloping credit curves imply a greater likelihood of default in later years;
üDownwardsloping credit curves imply a greater probability of default in the earlier years.
• Downwardsloping curves are less common.
Pricing and Application of CDS
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.
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.
Pricing and Application of CDS
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 longterm CDS and sell a shortterm CDS;
üIf an investor believes that the credit curve will become flatter, he can buy a shortterm CDS and sell a longterm CDS.
Pricing and Application of 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.
Pricing and Application of CDS
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
= Defaultfree security  CDS (protection seller)
Ø Importance: ☆ Ø Content:
• Pricing and profit of CDS;
• Use CDS to manage credit exposure and take
advantage of valuation disparities. Ø Exam tips: