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Session NO. Content Weightings
Study Session 1-2 Ethics & Professional Standards 10-15
Study Session 3 Quantitative Methods 5-10
Study Session 4 Economic Analysis 5-10
Study Session 5-6 Financial Statement Analysis 15-20
Study Session 7-8 Corporate Finance 5-15
Study Session 9-11 Equity Analysis 15-25
Study Session 12-13 Fixed Income Analysis 10-20
Fixed Income Analysis
• R35 Term Structure and Interest Rates Dynamics • R36 The arbitrage-free
SS13: Topics in Fixed Income Analysis
• R37 Valuation and analysis: Bonds with Embedded
1. Benchmark curve • Spot curve
• Forward curve • Par curve
• YTM, spot rate and return on bond • The swap rate curve
• The swap spread • I-spread
• Z-spread • TED spread
3. Traditional theories of the term structure of interest rates
• Local expectation theory • Liquidity preference theory • Preferred habitat theory • Segmented markets theory 4. Modern term structure models
• Equilibrium term structure models • Arbitrage-free model
5. Yield curve factor models • level、steepness、curvature • Interest rate volatility
A spot interest rate (spot rate) is a rate of interest on a security that makes a single payment at a future point in time.
Discount factor: the discount factor, P(T)
Spot yield curve (spot curve): the spot rate, r(T), for a range of maturities in years T > 0
The annualized return on an option-free and default-risk-free zero-coupon bond (zero for short) with a single payment of principal at maturity.
The shape and level of the spot yield curve are dynamic
The yield to maturity(YTM) or the yield of a zero-coupon bond with maturity T is the spot interest rate for a maturity of T.
A Forward rate is an interest rate that is determined today for a loan that will be initiated in a future time period.
reinvestment rate that would make an investor indifferent between buying an eight-year zero-coupon bond or investing in a seven-year zero-coupon bond and at maturity reinvesting the proceeds for one year. In this sense, the forward rate can be viewed as a type of breakeven interest rate.
Forward curve :The term structure of forward rates for a loan made on a specific initiation date.
Forward rates model (the relationship between spot rate and forward rate)：
( * )
( *, )
F(T*,T)= 1 1+ f (T*,T)
éë ùû T
F(T*,T)=the forward price of a $1 par zero-coupon bond maturing at time T *+T delivered at time T *
The spot rates for three hypothetical zero-coupon bonds (zeros) with maturities of one, two, and three years are given in the following table.
Calculate the forward rate for a one-year zero issued one year from today, f(1,1)
Calculate the forward rate for a one-year zero issued two years from today, f(2,1).
Calculate the forward rate for a two-year zero issued one year from today, f(1,2).
Based on your answers to 1 and 3, describe the relationship
Maturity(T) 1 2 3
f(1,1) is calculated as follows
f(2,1) is calculated as follows
f(1,2) is calculated as follows
The upward-sloping zero-coupon yield curve is associated with an upward-sloping forward curve
Relationship between spot rate and forward rate:
When the spot curve is upward sloping, the forward curve will lie above the spot curve.
When the spot curve is downward sloping, the forward curve will lie below the spot curve.
Yield curve shapes:
In developed markets, yield curves are most commonly upward sloping with diminishing marginal increases in yield for identical changes in maturity; that is, the yield curve "flattens" at longer maturities.
Because nominal yields incorporate a premium for expected
inflation, an upward-sloping yield curve is generally interpreted as reflecting a market expectation of increasing or at least level future inflation (associated with relatively strong economic growth).
An inverted yield curve is somewhat uncommon. Such a term structure may reflect a market expectation of declining future inflation rates
(because a nominal yield incorporates a premium for expected inflation) from a relatively high current level.
Expectations of declining economic activity may be one reason that inflation might be anticipated to decline
a downward-sloping yield curve has frequently been observed before recessions.
A flat yield curve typically occurs briefly in the transition from an upward-sloping to a downward-sloping yield curve, or vice versa.
A humped yield curve, which is relatively rare, occurs when
Forward pricing model:
Describes the valuation of forward contracts.
The no-arbitrage argument that is used to derive the model is frequently used in modern financial theory
Tradable securities with identical cash flow payments must have the same price. Otherwise, traders would be able to generate risk-free arbitrage profits.
Applying this argument to value a forward contract
Forward contract price that delivers a T-year-maturity bond at time T* using forward pricing model
F T T
Calculate the forward price two years from now for a $1 par, zero-coupon, three-year bond given the following spot rates.
The two-year spot rate, S2 = 4%.
The five-year spot rate, S5 = 6%.
Calculate discount factors Pi and P(i+k).
The forward price of a three-year bond in two years is represented as
0.7473 / 0.9246
Yield curve movement and the forward curve：
Forward contract price remains unchanged as long as future spot rates evolve as predicted by today's forward curve.
a change in the forward price reflects a deviation of the spot curve from that predicted by today's forward curve
if a trader expects that the future spot rate will be lower than what is predicted by the prevailing forward rate, the forward contract value is expected to increase. The trader would buy the forward contract.
if the trader expects the future spot rate to be higher than what is predicted by the existing forward rate, then the forward
Riding the yield curve or rolling down the yield curve
Under this strategy, an investor will purchase bonds with maturities longer than his investment horizon.
In an upward-sloping yield curve, shorter maturity bonds have lower yields than longer maturity bonds.
As the bond approaches maturity (i.e., rolls down the yield curve), it is valued using successively lower yields and, therefore, at
successively higher prices.
The par curve represents the yields to maturity on coupon-paying government bonds, priced at par, over a range of maturities.
recently issued ("on the run") bonds are typically used to create the par curve because new issues are typically priced at or close to par.
The zero-coupon rates are determined by using the par yields and solving for the zero-coupon rates one by one, in order from earliest to latest maturities, via a process of forward substitution known as
One-year par rate = 5%, Two-year par rate = 5.97%, Three-year par rate = 6.91%, Four-year par rate = 7.81%. From these we can
bootstrap zero-coupon rates.
Two year zero coupon rate:
Three year zero coupon rate
Four year zero coupon rate
0.0691 0.0691 1 0.0691
Relationship between YTM and Spot rate
The YTM of these bonds with maturity T would not be the same as the spot rate at T.
most bonds outstanding have coupon payments and many have various options, such as a call provision.
The YTM of the bond should be some weighted average of spot rates used in the valuation of the bond.
Because the principle of no arbitrage shows that a bond’s value is
Compute the price and yield to maturity of a three-year, 4% annual-pay, $1,000 face value bond given the following spot rate curve: S1 = 5%, S2 = 6%, and S3 = 7%.
Calculate the price of the bond using the spot rate curve:
Calculate the yield to maturity (y3):
N = 3; PV = -922.64; PMT = 40; FV = 1,000; CPT I/Y = 6.94
40 40 1040
YTM and the expected return on a bond
The expected rate of return is the return one anticipates earning on an investment.
The YTM is the expected rate of return for a bond that is held until its maturity, assuming that all coupon and principal payments are made in full when due and that coupons are reinvested at the original YTM.
YTM is not the expected return on a bond in general.
The assumption regarding reinvestment of coupons at the original yield to maturity typically does not hold
The YTM can provide a poor estimate of expected return if
(1) interest rates are volatile;
(2) the yield curve is steeply sloped, either upward or downward;
(3) there is significant risk of default;
or (4) the bond has one or more embedded options (e.g., put, call, or conversion).
Realized return on a bond
The realized return is the actual return on the bond during the time an investor holds the bond.
It is based on actual reinvestment rates and the yield curve at the end of the holding period.
Swap rate: the interest rate for the fixed-rate leg of an interest rate swap
The level of the swap rate is such that the swap has zero value at the initiation of the swap agreement.
The yield curve of swap rates is called the swap rate curve(swap curve) Because it is based on so-called par swaps, in which the fixed rates are
The swap market is a highly liquid market for two reasons.
First, unlike bonds, a swap does not have multiple borrowers or lenders, only counterparties who exchange cash flows.
offer significant flexibility and customization in the swap contract's design.
swaps provide one of the most efficient ways to hedge interest rate risk.
Function of swap curve:
The swap curve is a necessary market benchmark for interest rates.
Many countries do not have a liquid government bond market with maturities longer than one year.
swap curve is a far more relevant measure of the time value of money than is the government's cost of borrowing.
Swap rate curves VS government spot curves
The choice of a benchmark for the time value of money often depends on the business operations of the institution using the benchmark in the United States where there is both an active Treasury security market and a swap market.
Wholesale banks frequently use the swap curve to value assets and liabilities because these organizations hedge many items on their balance sheet with swaps.
Retail banks with little exposure to the swap market are more likely to use the government spot curve as their benchmark.
The preference for swap rate curve as benchmark
Swap rates reflect the credit risk of commercial banks rather than governments
The swap market is not regulated by any government
More comparable swap rates in different countries
Determining swap rate
The right side: the value of the floating leg, which is 1 at origination
The swap rate is determined by equating the value of the fixed leg, on the left-hand side to the value of the floating rate.
suppose a government spot curve implies the following discount factors
P(1)=0.9524, P(2)=0.8900, P(3)=0.8163, P(4)=0.7350
Determine the swap rate curve based on this information
Similarly, For T=3, s(3)=6.91%; For T=4, s(4)=7.81%
Swap spread: spread paid by the fixed-rate payer of an interest rate swap over the rate of the on-the- run(most recently issued) government security with the same maturity with the swap.
Swap spreadt = swap ratet - Treasure yieldt
For example, if the fixed rate of a five-year fixed-for-float Libor swap is 2.00% and the five-year Treasury is yielding 1.70%, the swap spread is 2.00%-1.7%=0.30%, or 30 bps
The Treasury rate can differ from the swap rate for the same term for several reasons.
Unlike the cash flows from US Treasury bonds, the cash flows from swaps are subject to much higher default risk.
Market liquidity for any specific maturity may differ.
For example, some parts of the term structure of interest rates may be more actively traded with swaps than with Treasury bonds.
I-spreads: amount the yield on the risky bond exceed the swap rate for the same security
The missing swap rate can be estimated from the swap rate curve using linear interpolation when the swap rate for a specific swap curve in not available.
6% Z bond are currently yield 2.35% and mature in 1.6 years. Compute the I-spread from the provided swap curve
I-spread only reflects compensation for credit and liquidity risks. Tenor Swap rate
1.6 year swap rate=1.5 year swap rate+ 1.38% 0.50
Z-spread: spread when added to each spot rate on the default-free spot curve makes the present value of a bond’s cash flows equal to the
bond’s market price, a spread over the entire spot rate curve
Zero volatility: assumption of zero interest rate volatility.
one-year spot rate is 4% and the two-year spot rate is 5%. The market price of a two-year bond with annual coupon payments of 8% is
$104.12. The Z-spread is the spread that balances the following equation:
A. 80bps B. 90bps C. 100bps
TED spread：an indicator of perceived credit risk in the general
economy. TED is an acronym formed from US T-bill and ED, the ticker symbol for the eurodollar futures contract.
The TED spread is calculated as the difference between Libor and the yield on a T-bill of matching maturity.
An increase (decrease) in the TED spread is a sign that lenders believe the risk of default on interbank loans is increasing
as it relates to the swap market, the TED spread can also be thought of as a measure of counterparty risk.
Compared with the 10-year swap spread, the TED spread more
Libor–OIS spread: the difference between Libor and the overnight indexed swap (OIS) rate
Useful measure of credit risk
Indication of the overall wellbeing of the banking system
Pure (Unbiased ) Expectations Theory Description:
The pure (unbiased) expectations theory suggests that forward rates are solely a function of expected future spot rates
long-term interest rates equal the mean of future expected short-term rates.
Every maturity strategy has the same expected return over a given investment horizon. an investor should earn the same return by investing in a five-year bond or by investing in a three-year bond and then a two-year bond after the three-year bond matures.
The underlying principle behind the pure expectations theory is risk neutrality: Investors don't demand a risk premium for maturity
The implications for the shape of the yield curve under the expectations theory are:
If the yield curve is upward sloping, short-term rates are expected to rise
If the curve is downward-sloping, short-term rates are expected to fall
Shortcomings——fail to consider the riskiness of bond investing
Price risk: the uncertainty associated with the future price of a bond that may be sold prior to its maturity
suppose the 1-year spot rate 5% and the 2-year spot rate is 7%. Under the pure expectations theory, the 1-year forward rate in one year must be 9%
Because investing for two years at 7% yields approximately the same annual return as investing for the first year at 5% and the second year at 9%.
In other words, the 2-year rate of 7% is the average of the
Local Expectation Theory
Similar to the unbiased expectations theory with one major difference: the local expectations theory preserves the risk-neutrality assumption only for short holding periods.
In other words, over longer periods, risk premiums should exist.
This implies that over short time periods, every bond(even long-term maturity risk bonds)should earn the risk-free rate
The local expectations theory can be shown not hold because the short-holding period return of long-maturity bonds can be shown to be
Liquidity preference theory
The liquidity theory says that forward rates are biased estimates of the
market’s expectation of future rates because they include a liquidity
A positive-sloping yield curve may indicate that: The market expects future interest rates to rise; or
that rates are expected to remain constant, but the addition of the liquidity premium results in positive slope.
Segmented Market Theory: yields are not determined by liquidity premiums and expected spot rates.
The shape of the yield curve is determined by the preferences of
borrowers and lenders, which drives the balance between supply and demand for loans of different maturities.
Yield at each maturity is determined independently of the yields at other maturities
Supposes that various market participants only deal in securities of a particular maturity because they are prevented from operating at different maturities
Preferred habitat theory
The preferred habitat theory also proposes that forward rates represent expected future spot rates plus a premium, but it does not support
the view that this premium is directly related to maturity.
The preferred habitat theory suggests that the existence of an imbalance between the supply and demand for funds in a given maturity range will induce lenders and borrowers to shift from their preferred habitats (maturity range) to one that has the opposite imbalance
To entice investors to do so, the investors must be offered an incentive to compensate for the exposure to price and/or reinvestment rate risk in the less-than-preferred habitat.
This theory can be used to explain almost any yield curve shape.
Subtle difference between preferred habitat theory and liquidity theory
The premium is a positive or negative risk premium related to
supply and demand for funds at various maturities, nor necessarily a liquidity premium
Description Shapes Pure expectations
forward rates are solely a function of
expected future spot rates
Upward slope: rise Downward slope:
Flat yield: remain
rates or rates remain but liquidity
premium added related to supply
and demand for
Modern term structure models：provide quantitatively precise descriptions of how interest rates evolve.
Equilibrium Term Structure Models：Models that seek to describe the dynamics of the term structure using fundamental economic variables that are assumed to affect interest rates
Restrictions are imposed that allow for the derivation of equilibrium prices for bonds and interest rate options.
Require the specification of a drift term (explained later) and the assumption of a functional form for interest rate volatility
The best-known equilibrium models are the Cox—Ingersoll—Ross modeland the Vasicek model
Arbitrage-free model: assuming that bonds trading in the market are correctly priced, and the model is calibrated to value such bonds
consistent with their market price(arbitrage-free).
Characteristics of Equilibrium Term Structure Models They are one factor or multifactor models.
One-factor models assume that a single observable factor
(sometimes called a state variable) drives all yield curve movements
Both the Vasicek and CIR models assume a single factor, the short-term interest rate, r.
They make assumptions about the behavior of factors
They are, in general, more sparing with respect to the number of
The CIR model assumes that every individual has to make consumption and investment decisions with their limited capital.
The individual must determine his or her optimal trade-off assuming that he or she can borrow and lend in the capital market.
Ultimately, interest rates will reach a market equilibrium rate at which no one needs to borrow or lend.
The CIR model can explain interest rate movements in terms of an
individual's preferences for investment and consumption as well as the risks and returns of the productive processes of the economy.
The model shows how the short-term interest rate is related to the risks facing the productive processes of the economy.
Assuming that an individual requires a term premium on the long-term rate, the model shows that the short-long-term rate can delong-termine the entire term structure of interest rates and the valuation of
The first part of this expression is a drift term, the second part is the random component.
a(b-r) forces the interest rate to mean-revert toward the long-run value b, at a speed determined by parameter a.
Volatility increases with interest rate. The amount of period-over-period fluctuation in rates in high at high interest rates
Suggest that interest rates are mean reverting to some long-run value
Difference from CIR model:
Non interest rate( r ) term appears in the second term
Volatility in this model does not increase as the level of interest rates increase
Disadvantage: the model does not force interest rates to be non-negative
Advantage over equilibrium model: the ability to calibrate arbitrage-free models to match current market prices.
The Ho-Lee Model
t t t
Three factors explain historical treasury security returns.
Changes in the level of interest rates (parallel shifts in the yield curve), contributing 77% of the observed variation in total returns for all
Changes in the slope or steepness of the yield curve, contributing 17% Changes in the curvature of the yield curve , contributing 3%
Parallel shift 77% Duration/convexity
Key rate duration
The level movement refers to an upward or downward shift in the yield curve.
The steepness movement refersto a non-parallel shift in the yield curve when either short-term rates change more than term rates or long-term rates change more than short-long-term rates.
The curvature movement is a reference to movement in three segments of the yield curve: the short-term and long-term segments rise while the
A movement of the yield curve in which the short rate decreases by 150 bps and the long rate decreases by 50 bps would best be
described as a:
A. flattening of the yield curve resulting from changes in level and steepness.
B. steepening of the yield curve resulting from changes in level and steepness.
C. steepening of the yield curve resulting from changes in steepness and curvature.
Yield curve risk: risk to portfolio value arising from unanticipated changes in the yield curve, can be managed on the basis of several measures of
sensitivity to yield curve movements
Managing yield curve risk
Effective duration: measures the sensitivity of a bond's price to a small parallel shift in a benchmark yield curve
Address risk associated with parallel yield curve changes
key rate duration: measures a bond's sensitivity to a small change in a benchmark yield curve at a specific maturity segment
Key rate duration
Duration is an adequate measure of bond price risk only for small paralleled shifts in the yield curve.
Example: non-parallel shift Bond
(zero coupon) Weight D1 D2 D3 D4
Decompose the risk into sensitivity to the following three categories of yield curve movement:
Level ( ): A parallel increase or decrease of interest rate
Steepness( ): Long-term interest rates increase while short-term rates decrease
Curvature( ): Increasing curvature means short- and long-term interest rates while intermediate rates do not change
Interest rate volatility is a key concern for bond manager because
interest rate volatility drives price volatility in a fixed income portfolio. Important for securities with embedded options, which are especially
sensitive to volatility.
Term structure of interest rate volatilities: a representation of the yield volatility of a zero-coupon bond for every maturity of security
Interest rate volatility is not the same for all interest rates along the yield curve. Short-term interest rates are generally more volatile than are
On the basis of the typical assumption of a lognormal model, the
1. Arbitrage opportunity
2. Introduction of Arbitrage free valuation 3. Interest rate trees and arbitrage-free
• Binomial interest rate tree • Option-free Bond valuation • Pathwise valuation
Arbitrage-free valuation: an approach to security valuation that
determines security values that are consistent with the absence of arbitrage opportunities.
Arbitrage opportunities are opportunities for trades that earn riskless profits without any net investment of money.
Arbitrage opportunities arise as a result of violations of the law of one price.
The law of one price states that two goods that are perfect substitutes must sell for the same current price in the absence of transaction costs.
Well functioning market complies with principle of no arbitrage.
There are two types of arbitrage opportunities
Any fixed-income security should be thought of as a package or portfolio of zero-coupon bonds using the arbitrage-free approach.
Stripping: The market mechanism for US Treasuries that enables this approach is the dealer's ability to separate the bond's individual cash flows and trade them as zero-coupon securities.
Reconstitution: dealers can recombine the appropriate individual zero-coupon securities and reproduce the underlying zero-coupon Treasury
A fundamental principle of valuation is that the value of any financial asset is equal to the present value of its expected future cash flows.
This principle holds for any financial asset from zero-coupon bonds to interest rate swaps. Thus, the valuation of a financial asset involves the following three steps:
Step 1 Estimate the future cash flows.
Step 2 Determine the appropriate discount rate or discount rates that should be used to discount the cash flows.
Example: the arbitrage-free valuation of an option-free bond
Sam Givens, fixed income analyst at GBO Bank is interested in
valuing a three-year, 3% annual pay, €100 par bond with the same liquidity and risk as the benchmark. What is the value of the bond using the benchmark par curve provided below?
Binomial interest rate tree construction
The set of possible interest rate paths that are used to value bonds with binomial model over multiple periods
Assumes that interest rates have an equal probability of taking one of two possible values in the next period.
Determine interest rate in interest rate trees
Estimation of volatility
historical interest rate volatility
volatility is calculated by using data from the recent past with the assumption that what has happened recently is indicative of the future
Observed market prices of interest rate derivatives
assumed volatility of the one-year rate
i the lower one-year forward rate one year from now at Time 1, and
i the higher one-year forward rate one year from now at Time 1
Construction of a binomial interest rate tree
Interest rate tree is generated using specialized computer software in practice.
The interest rate tree should generate arbitrage-free values for the benchmark security. The value of bonds produced by the interest rate tree must be equal to their market price
Adjacent forward rates are two standard deviation apart
Knowing one forward rate allows us to compute the other forward rates for the period in the tree
Forward rate C is the middle rate for Period 3 and hence the best estimate for that rate is the one-year forward rate in two years f(2,1). Using the spot rates, we can bootstrap the forward rate:
option free;2 years; annual coupon rate of 7%
100 7.0 99.830
(99.830 7) (101.594 7) / (1 4.5749%) $102.999
Zero-coupon yield curve: Each known future cash flow is discounted at the underlying spot rate (also known as the zero-coupon yield)
Comparison between zero-coupon yield curve and binomial tree Value bonds with embedded options, the rates need to be allowed to
The future cash flows are uncertain as they depend on whether the embedded option will be in the money
The value of the option depends on uncertain future interest rates, the underlying cash flows are also dependent on the same future interest rates.
Samuel Favre wants to value the same three-year, 3% annual-pay
treasury bond. The interest rate tree is the same as before but this time, Favre wants to use a pathwise valuation approach.
One-period Forward Rate in year
Compute the value of the $100 par option-free bond.
0 1 2
3% 5.7883% 10.7383%
The value of the bond in Path 1 is computed as:
Path Year1 Year 2 Year 3 Value
1 SUU 3% 5.7883% 10.7383% $91.03
2 SUL 3% 5.7883% 7.1981% $93.85
3 SLU 3% 3.8800% 7.1981% $95.52
4 SLL 3% 3.8800% 4.8250% $97.55
The implications for valuation models
An important assumption of the binomial valuation process is that the value of the cash flows at a given point in time is independent of the path that interest rates followed up to that points.
Cash flows for MBS are dependent on the path that interest rates
follow and can not be properly valued with the binomial model or any other model that employs the backward induction methodology.
Path dependency in passthrough securities
Cash flows for pass-through securities are a function of prepayment rates, which are affected by interest rates in the past. Two sources of path dependency:
If mortgage rates decrease, prepayment rates will increase at the beginning as home owners refinance their mortgage, the
prepayments will slow as more refinancing push the mortgage rats higher. (prepayment burnout)
Monte Carlo forward-rate simulation: involves randomly generating a large number of interest rate paths, using a model that incorporates a volatility assumption and an assumed probability distribution.
A key feature of the Monte Carlo method is that the underlying cash flows can be path dependent.
The value of the bond: average of values from the various paths.
The calibration process entails adding(subtracting) a constant to all
rates when the value obtained from the simulated paths is too high(too low) relative to market prices, resulting in a drift adjusted model.
A Monte Carlo simulation may impose upper and lower bounds on interest rates as part of the generating the simulated paths.
These bounds are based on the notion of mean reversion
Steps in the valuation of an MBS using the Monte Carlo simulation model:
Step 1: Simulate interest rate paths (e.g., 1,000 different paths) and cash flows using assumptions concerning benchmark rates, rate volatility, refinancing spreads, and prepayment rates. Non-agency MBS also require assumptions regarding default and recovery rates.
Step 2: Calculate the present value of the cash flows along each of the 1,000 interest rate paths.
Monte Carlo simulation model
Simulate CFs on each of the interest rate paths, Ct(n) Simulate spot rate,
Calculate the PV[Path(n)], , Generate 1-month future interest rate, ft(n)
Generate mortgage refinancing rate, rt(n)
Select the number of interest rate path, representative paths
Determine the theoretical value, theory value = average PV[Path(i)]
1. Fixed-income securities with embedded options
2. Valuation of callable and putable bonds • Relationship between embedded bond
and option-free bond
• Valuation of callable or putable bond using binomial tree
• OAS calculation and explanation • Interest rate volatility
• Effect of changes in the level and shape of the yield curve
3. Valuation of a capped or floored floating-rate bond
• Ratchet bonds
Callable bonds give the issuer the option to call back the bond; the investor is short the call option.
European-style option: the option can only be exercised on a single day immediately after the lockout period
American-style option: the option can be exercised at any time after the lockout period
Bermudan-style option: the option can be exercised at fixed dates after the lockout period
Putable bonds allow the investor to put back the bond to the issuer prior to maturity. The investor is long the underlying put option.
Extendible bond: allow the investor to extend the maturity of the bond
An estate put which includes a provision that allows the heirs of an investor to put the bond back to the issuer upon the death of the
investor. The value of this contingent put option is inversely related to the investor's life expectancy; the shorter the life expectancy, the higher the value.
Sinking fund bonds (sinkers) which require the issuer to set aside funds periodically to retire the bond (a sinking fund). This provision reduces the credit risk of the bond. Sinkers typically have several related issuer options.
(99.830 7) (100 7) / (1 4.5749%) 102.238
(100 7) (101.594 7) / (1 4.5749%) 103.081
Valuing a callable bond with four years to maturity, a coupon rate of 6.5%, and with a call price schedule (10% volatility assumed)
Firstly, judge call or put
They are American options.
Callable bond (option-embedded）- cannot calculate the value of call 3 year maturity; option expires in
2 years; Strike price is 100; Annual coupon rate is 7%
Call option on European bond – could calculate the value of call directly. 3 year maturity; option expires
in 2 years; Strike price is 99; Annual coupon rate is 7%
Call option on American bond – could calculate the value of call directly. Annual coupon rate is 7%
The procedure to value a bond with an embedded option in the presence of interest rate volatility
Generate a tree of interest rates based on the given yield curve and interest rate volatility assumptions.
At each node of the tree, determine whether the embedded options will be exercised.
Apply the backward induction valuation methodology to calculate the bond's present value. This methodology involves starting at maturity and working back from right to left to find the bond's present value.
The rate in the up state (Ru) is given by
Effect of volatility on the arbitrage-free value of an option
Interest rate volatility effect on the value of a callable or putable bond:
The values of call and put options increase when interest rate volatility increases
The value of a callable bond decreases
The value of a putable bond increases
The value of a straight bond is unaffected by changes in the volatility of interest rate
Benchmark interest rates:
US treasury securities
A specific sector of the bond market with a certain credit rating higher than the issue valued
A specific issuer
Z-spread Assume the interest rate volatility is zero
Option cost =Z-spread – OAS
The constant spread added to all one-period rates in the tree such that the
O A S
Credit risk Liquidity risk Option risk
Treasury (risk free)
A $100-par, three-year, 6% annual-pay ABC Inc. callable bond trades at $99.95. The underlying call option is a Bermudan-style option that can be exercised in two or three years at par.
Benchmark interest rate tree assuming volatility of 20% is provided below.
Bonds with similar credit risk should have the same OAS. If the OAS for a bond is higher than OAS of its peers, it is considered to be
undervalued and hence an attractive investment (i.e., it offers a higher compensation for a given level of risk). Conversely, bonds with low OAS (relative to peers) are considered to be overvalued.
An analyst makes the following spread estimates relative to U.S. Treasuries for a callable corporate bonds:
Z-spread relative to the Treasury yield curve is 225 basis points
OAS relative to the Treasury yield curve is 190 basis points The analyst also determine that the Z-spread over Treasuries on
comparable option-free bonds in the market is 210 basis points.
Determine whether the bond is overvalued, undervalued, or properly valued
The required OAS in this case is the Z-spread on comparable free bonds (because Z-spread is equal to OAS for option-free bonds), which is 210 basis points. This bond is overvalued,
because its OAS of 190 basis points is less than the required OAS. It
Treasury benchmark Bond sector
Credit risk Credit risk
Liquidity risk Liquidity risk Liquidity risk
Option risk Option risk Option risk
OAS Credit risk Credit risk
Liquidity risk Liquidity risk Liquidity risk
OAS﹥0 Over if actual
Undervalued if OAS
Over if actual
Undervalued if OAS
OAS＝0 Over Over Fairly priced
Interest rate volatility and OAS:
The computed OAS for a callable bond decreases when the assumed level of volatility in an interest rate tree increase
Higher volatility of the benchmark rates used in a binomial tree, the computed value of a callable bond will be lower, therefore closer to its actual market price. The constant spread needs to be added to the benchmark rates to correctly price the bond is therefore lower
Lower-than-actual(lower-than-actual) level of volatility, the callable
bond will appear to have higher-than-actual(lower-than-actual) level of OAS, erroneously classified as underprice(overpriced)
Higher-than-actual(lower-than-actual) level of volatility, the computed OAS for a putable bond will be too high(low), the bond will be
erroneously classified as underprice(overpriced)
Assumed level of
Level of interest rates
Interest rate declines: the value of a callable bond rise less rapidly than the value of an otherwise-equivalent straight bond
Interest rate increase: the value of a putable bond falls less rapidly than the value of an otherwise-equivalent straight bond
Shape of the yield curve Call option
Interest rate decline, the value of an embedded call option increases
Value of a call option will be lower for an upward sloping yield curve
A higher interest rate scenario limits the probability of the call option being in the money.
As an upward-sloping yield curve becomes flatter, the call option value increases.
Interest rate increases, the value of a put option increases
Put option value will be lower as an upward-sloping yield curve flattens
Effective duration and effective convexity
Modified duration and convexity are not useful for bonds with embedded options, because the cash flows from these bonds may change if the option is exercise.
To overcome this problem, effective duration and convexity should be used because these measures take into account how changes in interest rates may alter cash flows.
V is originally 908; Yield changes 50bps; when yield increases, V equals to 866.80; when yield decreases, V equals to 952.30
Using the binomial model to compute effective duration and convexity. The procedure for calculating the value of V+ is as follows:
Step 1 Given assumptions about the benchmark interest rates, interest rate volatility, and any calls and /or puts, calculate the OAS for the isue using the current market price and the binomial model.
Step 2 Shift the on-the-run yield curve up by a small number of basis points (Δy).
Step 3 Construct a binomial interest rate tree based on the new yield curve in Step 2.
Step 4 Add the OAS from step 1 to each of the one-year rates in the
interest rate tree to get a ―modified‖ tree.
Comparison of effective durations among callable, putable and straight bonds
The effective duration of straight bonds is relatively unaffected by changes in interest rates
An increase(decrease) in rates would decrease the effective duration of a putable(callable) bond
Effective duration callable
effective duration straight
Effective duration putable
effective duration straight
Effective duration zero coupon
maturity of the bond
Effective duration of fixed
Comparison among effective convexities of callable, putable and straight bonds
Straight bonds have positive effective convexity
The increase in the value of an option-free bond is higher when rates fall than the decrease in value when rates increase by an equal amount
Callable bonds are unlikely to be called and will exhibit positive convexity when rates are high
The effective convexity turns negative when the underlying call option is near the money
The upside potential of the bond’s price is limited due to the
call(while the downside is not protected)
Effective durations: normally calculated by averaging the changes
resulting from shifting the benchmark yield curve up and down by the same amount.
This calculation works well for option-free bonds
In the presence of embedded options, the results can be misleading.
The problem is that when the embedded option is in the money, the price of the bond has limited upside potential if the bond is callable or limited downside potential if the bond is putable.
The price sensitivity of bonds with embedded options is not
One-sided durations: durations that apply only when interest rates go up (or, alternatively, only when rates go down.
better at capturing the interest rate sensitivity of a callable or putable bond than the (two-sided) effective duration
When the underlying option is at (or near) money, callable bonds will have lower one-sided down-duration than one-sided up-duration; the price change of a callable when rates fall is smaller than the price
change for an equal increase in rates.
Key rate duration: shifting any par rate has an effect on the value of the bond
For an option-free bond, the maturity matched rate is the most important rate.
Trading at par, the bond’s maturity matched rate is the only rate that
affects the bond’s value. Its maturity key rate duration is the same as
its effective duration, and all other key rate durations are zero.
Bonds with low(or zero) coupon rate may have negative key rate durations for horizons other than its maturity
Callable bonds with low coupon rate are unlikely to be called, hence, their maturity-matched rate is their most critical rate.
As the coupon rate increases, a callable bond is more likely to be called and the time-to-exercise rate will start dominating the time-to-maturity rate
Putable bonds with high coupon rates are unlikely to be put and are most sensitive to their maturity-matched rates.
Floating rate bond(floater): pays a coupon that adjusts every period based on an underlying reference rate
Options in floating-rate bonds (floaters) are exercised automatically depending on the course of interest rates—that is, if the coupon rate rises or falls below the threshold, the cap or floor
The capped floater protects the issuer against rising interest rates and is thus an issuer option
Value of capped floater= Value of ‘straight’ bond – Value of embedded cap
The floor floater protects the investor against declining and thus offers protection from falling interest rates
Value of floored floater= Value of ‘straight’ bond + Value of embedded floor
Standard backward induction methodology is applied in a binomial interest rate tree to value a capped or floored floater
Susan Albright works as a fixed income analyst with Zedone Banks NA. She has been asked to value a $100 par, two-year, floating rate note that pays LIBOR set in arrears. The underlying bond has the same credit quality as reflected in the LIBOR swap curve. Albright has constructed the following two-year binomial LIBOR tree:
1. The value of the floater, assuming that it is an option-free bond 2. The value of the floater, assuming that it is capped with a cap rate
of 6%. Also compute the value of the embedded cap
3. The value of the floater, assuming that it is floored with a floor rate of 5%. Also compute the value of the embedded floor.
One-period forward rate
Year 0 Year 1
Correct Answer 1 :
An option-free bond with a coupon rate equal to the required rate of return will be worth par value. Hence the straight value of the floater is $100.
The value of the capped floater is $99.47 as shown below:
Note that when the option is not in the money, the floater is valued at par
The upper node in the year 2 shows the exercise of the cap (the coupon is capped at $6 instead of $7.18)
The year 0 value is the average of the year 1 values (including their adjusted coupons) discounted for one period. The year 1
coupon don’t require any adjustment as the coupon rate is
below the cap rate.
The value of the embedded cap=$100-$99.47=$0.53
Correct Answer 2 :
Ratchet bonds are floating-rate bonds with both issuer and investor options.
As with conventional floater, the coupon is reset periodically according to a formula based on a reference rate and a credit spread.
A capped floater protects the issuer against rising interest rates.
Ratchet bonds offer extreme protection: At the time of reset, the coupon can only decline; it can never exceed the existing level. So,
over time, the coupon "ratchets down‖.
At issuance, the coupon of a ratchet bond is much higher than that of a standard floater.
The initial coupon is set well above the issuer's long-term
A capped floater protects the issuer against rising interest rates.
A ratchet bond is similar to a conventional callable bond: When the bond is called, the investor must purchase a replacement in the prevailing lower rate environment.
Ratchet bonds also contain investor options.
Whenever a coupon is reset, the investor has the right to put the bonds back to the issuer at par.
The embedded option is called a "contingent put" because the right to put is available to the investor only if the coupon is reset.
The coupon reset formula of ratchet bonds is designed to assure that the market price at the time of reset is above par, provided that the issuer's credit quality does not deteriorate.
A convertible bond includes an embedded call option. The option is slightly different from the embedded option in a callable bond.
The convertible bondholder owns the call option, the issuer owns the call option in a callable bond.
The holder has the right to buy shares with a bond that changes in value, not with cash at a fixed exercise price.
Conversion ratio is the number of common shares for which a convertible bond can be exchanged.
Conversion price is the bond issue price divided by the conversion ratio.
(some investors refer to it as the stated conversion price.)
Market conversion price is the market price of bond divided by the conversion ratio
The market conversion premium per share = market conversion price – market price Market conversion premium ratio
= market conversion premium per share/market price of common stock
The minimum value of a convertible bond must be the greater of its conversion value or its straight value
market price of convertible bond
Premium over straight value
the price of straight bond
Effects of embedded option in convertible bond
Stock price volatility↓=>Value of the call option on the
stock↓=>convertible bond value ↓
Stock price volatility ↑ =>Value of the call option on the stock ↑
=>convertible bond value ↑
Comparisons between convertible bonds and stocks
Stock price↓=>the returns on convertible bonds exceed those of the
Reason: The convertible bond’s price has a floor =its straight bond
Stock price↑=>the bond will underperform Reason: conversion premium.
Stock price remains stable=>the returns on convertible bonds exceed those of the stock
Comparison between underlying stock and risk-return characteristics of the convertible bond
When the stock’s price falls, the returns on convertible bonds exceed
those of the stock because the convertible bond’s price has a floor equal
to its straight bond value
When the stock’s price rises, the bond will underperform because of the
This is the main drawback of investing in convertible bonds versus investing directly in the stock
If the stock’s price remains stable, the return on a convertible bond may
exceed the stock return due to the coupon payments received from the bond, assuming no change in interest rates or credit risk of the issuer.
Fixed-income equivalent(busted convertible): the price of the
common stock associated with a convertible issue is so low that it has
little or no effect on the convertible’s market price and the bond trades
as though it is a straight bond
Issuer: Heavy Element Inc.
Issue Date: 15 September 2010
Maturity Date: 15 September 2015
Interest: 3.75% payable annually
Issue Size: $100,000,000
Issue Price: $1,000 at par
Conversion Ratio: 23.26
Convertible Bond Price on 16 September 2012: $1,230
Share Price on 16 September 2012: $52
1. The conversion price is closest to:
2. The conversion value on 16 September 2012 is closest to:
4. The risk—return characteristics of the convertible bond on 16 September 2012 most likely resemble that of:
A. a busted convertible.
B. Heavy Element's common stock.
C. a bond of Heavy Element that is identical to the convertible bond but without the conversion option.
5. As a result of favorable economic conditions, credit spreads for the chemical industry narrow, resulting in lower interest rates for the debt of companies such as Heavy Element. All else being equal, the price of Heavy Element's convertible bond will most likely:
A. decrease significantly.
6. Suppose that on 16 September 2012, the convertible bond is available in the secondary market at a price of $1,050. An
arbitrageur can make a risk-free profit by:
A. buying the underlying common stock and shorting the convertible bond.
B. buying the convertible bond, exercising the conversion option, and selling the shares resulting from the conversion.
7. A few months have passed. Because of chemical spills in lake water at the site of a competing facility, the government has
introduced very costly environmental legislation. As a result, share prices of almost all publicly traded chemical companies, including Heavy Element, have decreased sharply. Heavy Element's share price is now $28. Now, the risk–return characteristics of the convertible bond most likely resemble that of:
A. a bond.
B. a hybrid instrument.
Valuation of convertible bond
Most convertible bonds are callable, giving the issuer the right to call the issue prior to maturity.
Noncallable/nonputable convertible bond value
= Straight value of bond
+ Value of the call option on the stock
Callable convertible bond value
= Straight value of bond
+ Value of the call option on the stock
- Value of the call option on the bond
Callable and putable convertible bond value
= Straight value of bond
+ Value of the call option on the stock
- Value of the call option on the bond
The outputs from the use of specialized software system to derive the binomial-tree model must meet following minimum qualifications:
Check that the put–call parity holds
Value(C) – Value(P) = PV(Forward price of bond on exercise date ) –PV (Exercise price)
Check that the valve of the underlying option free bond does not depend on interest rate volatility.
1. Measures of Credit risk：PD、LGD、EL、
2. Two models for credit risk：
• Traditional model：Credit scoring & Credit ratings
• Structure model & Reduced form model