Assignment 9

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Given that an upward-sloping yield curve implies a forward rate higher than the spot, or current, yield to maturity, we ask next what can account for that higher forward rate. The challenge is that there always are two possible answers to this question. Recall that the forward rate can be related to the expected future short rate according to:

fn!=!E(rn )!+!Liquidity premium (15.8)

where the liquidity premium might be necessary to induce investors to hold bonds of maturities that do not correspond to their preferred investment horizons.

By the way, the liquidity premium need not be positive, although that is the position generally taken by advocates of the liquidity premium hypothesis. We showed previously that if most investors have long-term horizons, the liquidity premium in principle could be negative.

In any case, Equation 15.8 shows that there are two reasons that the forward rate could be high. Either investors expect rising interest rates, meaning that E(rn) is high, or they require a large premium for holding longer-term bonds. Although it is tempting to infer from a rising yield curve that investors believe that interest rates will eventually increase, this does not necessarily follow. Indeed, Panel A in Figure 15.4 provides a simple counterexample. There, the short rate is expected to stay at 5% forever. Yet there is a constant 1% liquidity premium so that all forward rates are 6%. The result is that the yield curve continually rises, starting at a level of 5% for 1-year bonds, but eventually approaching 6% for long-term bonds as more and more forward rates at 6% are averaged into the yields to maturity.

Therefore, while expectations of increases in future interest rates can result in a ris- ing yield curve, the converse is not true: A rising yield curve does not in and of itself imply expectations of higher future interest rates. Potential liquidity premiums confound any simple attempt to extract expectations from the term structure. But estimating the market’s expectations is crucial because only by comparing your own expectations to those reflected in market prices can you determine whether you are relatively bullish or bearish on interest rates.

One very rough approach to deriving expected future spot rates is to assume that liquidity premiums are constant. An estimate of that premium can be subtracted from the forward rate to obtain the market’s expected interest rate. For example, again making use of the example plotted in Panel A of Figure 15.4, the researcher would estimate from historical data that a typical liquidity premium in this economy is 1%. After calculating the forward rate from the yield curve to be 6%, the expectation of the future spot rate would be determined to be 5%.

This approach has little to recommend it for two reasons. First, it is next to impos- sible to obtain precise estimates of a liquidity premium. The general approach to doing so would be to compare forward rates and eventually realized future short rates and to calculate the average difference between the two. However, the deviations between the two values can be quite large and unpredictable because of unanticipated economic events that affect the realized short rate. The data are too noisy to calculate a reliable estimate of the expected premium. Second, there is no reason to believe that the liquidity premium should be constant. Figure 15.5 shows the rate of return variability of prices of long-term Treasury bonds since 1971. Interest rate risk fluctuated dramatically during the period. So we should expect risk premiums on various maturity bonds to fluctuate, and empirical evidence suggests that liquidity premiums do in fact fluctuate over time.

Still, very steep yield curves are interpreted by many market professionals as warn- ing signs of impending rate increases. In fact, the yield curve is a good predictor of the

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C H A P T E R !" The Term Structure of Interest Rates #$%

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KEY TERMSterm structure of interest rates yield curve bond stripping bond reconstitution

pure yield curve on-the-run yield curve spot rate short rate

forward interest rate liquidity premium expectations hypothesis liquidity preference theory

Forward rate of interest: 1!+!fn!=! (1!+!yn)n ____________

(1!+!yn " 1)n " 1

Yield to maturity given sequence of forward rates: 1 + yn = [ (1 + r1) (1 + f2) (1 + f3)!#!#!#!(1 + fn) ] 1/n

Liquidity premium = Forward rate " Expected short rate

KEY EQUATIONS

liquidity premium remained reasonably stable over time. However, both empirical and theoretical considerations cast doubt on the constancy of that premium.

6. Forward rates are market interest rates in the important sense that commitments to forward (i.e., deferred) borrowing or lending arrangements can be made at these rates.

1. What is the relationship between forward rates and the market’s expectation of future short rates? Explain in the context of both the expectations hypothesis and the liquidity preference theory of the term structure of interest rates.

2. Under the expectations hypothesis, if the yield curve is upward-sloping, the market must expect an increase in short-term interest rates. True/false/uncertain? Why?

3. Under the liquidity preference theory, if inflation is expected to be falling over the next few years, long-term interest rates will be higher than short-term rates. True/false/uncertain? Why?

4. If the liquidity preference hypothesis is true, what shape should the term structure curve have in a period where interest rates are expected to be constant? a. Upward sloping. b. Downward sloping. c. Flat.

5. Which of the following is true according to the pure expectations theory? Forward rates: a. Exclusively represent expected future short rates. b. Are biased estimates of market expectations. c. Always overestimate future short rates.

6. Assuming the pure expectations theory is correct, an upward-sloping yield curve implies: a. Interest rates are expected to increase in the future. b. Longer-term bonds are riskier than short-term bonds. c. Interest rates are expected to decline in the future.

7. The following is a list of prices for zero-coupon bonds of various maturities.! a. Calculate the yield to maturity for a bond with a maturity of (i) one year; (ii) two years;

(iii) three years; (iv) four years. b. Calculate the forward rate for (i) the second year; (ii) the third year; (iii) the fourth year.

Maturity (years) Price of Bond

! $"#$.#% & '"'.#( $ '#(.)& # ("&.!)

PROBLEM SETS

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!"" P A R T I V Fixed-Income Securities

8. a. Assuming that the expectations hypothesis is valid, compute the expected price of the 4-year bond in Problem 7 at the end of (i) the first year; (ii) the second year; (iii) the third year; (iv) the fourth year.

b. What is the rate of return of the bond in years 1, 2, 3, and 4? Conclude that the expected return equals the forward rate for each year.

9. Consider the following $1,000 par value zero-coupon bonds: Bond Years to Maturity YTM(%)

A ! "% B # $ C % $." D & '

According to the expectations hypothesis, what is the market’s expectation of the yield curve one year from now? Specifically, what are the expected values of next year’s yields on bonds with maturities of (a) one year? (b) two years? (c) three years?

10. The term structure for zero-coupon bonds is currently: Maturity (years) YTM (%)

! &% # "( % $(

Next year at this time, you expect it to be: Maturity (years) YTM (%)

! "% # $ % '

a. What do you expect the rate of return to be over the coming year on a 3-year zero-coupon bond?

b. Under the expectations theory, what yields to maturity does the market expect to observe on 1- and 2-year zeros at the end of the year?

c. Is the market’s expectation of the return on the 3-year bond greater or less than yours? 11. The yield to maturity on 1-year zero-coupon bonds is currently 7%; the YTM on 2-year zeros is

8%. The Treasury plans to issue a 2-year maturity coupon bond, paying coupons once per year with a coupon rate of 9%. The face value of the bond is $100. a. At what price will the bond sell? b. What will the yield to maturity on the bond be? c. If the expectations theory of the yield curve is correct, what is the market expectation of the

price for which the bond will sell next year? d. Recalculate your answer to part (c) if you believe in the liquidity preference theory and you

believe that the liquidity premium is 1%. 12. Below is a list of prices for zero-coupon bonds of various maturities.

Maturity (years) Price of $#,$$$ Par Bond

(zero-coupon)

! $)&%.&* # +'%."# % +!$.%'

a. An 8.5% coupon $1,000 par bond pays an annual coupon and will mature in three years. What should the yield to maturity on the bond be?

b. If at the end of the first year the yield curve flattens out at 8%, what will be the 1-year holding-period return on the coupon bond?

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13. Prices of zero-coupon bonds reveal the following pattern of forward rates: Year Forward Rate

! "% # $ % &

In addition to the zero-coupon bond, investors also may purchase a 3-year bond making annual payments of $60 with par value $1,000. a. What is the price of the coupon bond? b. What is the yield to maturity of the coupon bond? c. Under the expectations hypothesis, what is the expected realized compound yield of the

coupon bond? d. If you forecast that the yield curve in one year will be flat at 7%, what is your forecast for the

expected rate of return on the coupon bond for the 1-year holding period? 14. You observe the following term structure:

& Effective Annual YTM

!-year zero-coupon bond '.!% #-year zero-coupon bond '.# %-year zero-coupon bond '.% (-year zero-coupon bond '.(

a. If you believe that the term structure next year will be the same as today’s, calculate the return on (i) the 1-year zero and (ii) the 4-year zero.

b. Which bond provides a greater expected 1-year return? c. Redo your answers to parts (a) and (b) if you believe in the expectations hypothesis.

15. The yield to maturity (YTM) on 1-year zero-coupon bonds is 5%, and the YTM on 2-year zeros is 6%. The YTM on 2-year-maturity coupon bonds with coupon rates of 12% (paid annually) is 5.8%. a. What arbitrage opportunity is available for an investment banking firm? b. What is the profit on the activity?

16. Suppose that a 1-year zero-coupon bond with face value $100 currently sells at $94.34, while a 2-year zero sells at $84.99. You are considering the purchase of a 2-year-maturity bond making annual coupon payments. The face value of the bond is $100, and the coupon rate is 12% per year. a. What is the yield to maturity of the 2-year zero? b. What is the yield to maturity of the 2-year coupon bond? c. What is the forward rate for the second year? d. According to the expectations hypothesis, what are (i) the expected price of the coupon bond

at the end of the first year and (ii) the expected holding-period return on the coupon bond over the first year?

e. Will the expected rate of return be higher or lower if you accept the liquidity preference hypothesis?

17. The current yield curve for default-free zero-coupon bonds is as follows: Maturity (years) YTM (%)

! !)% # !! % !#

a. What are the implied 1-year forward rates? b. Assume that the pure expectations hypothesis of the term structure is correct. If market

expectations are accurate, what will be the yield to maturity on 1-year zero-coupon bonds next year?

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!"# P A R T I V Fixed-Income Securities

c. What about the yield on 2-year zeros? d. If you purchase a 2-year zero-coupon bond now, what is the expected total rate of return over

the next year?!(Hint: Compute the current and expected future prices.) Ignore taxes. e. What is the expected total rate of return over the next year on a 3-year zero-coupon bond? f. What should be the current price of a 3-year maturity bond with a 12% coupon rate paid

annually? g. If you purchased the coupon bond at the price you computed in part (f ), what would your

total expected rate of return be over the next year (coupon plus price change)? Ignore taxes.

18. Suppose that the prices of zero-coupon bonds with various maturities are given in the following table. The face value of each bond is $1,000.

Maturity (years) Price

! $"#$."% # &$%.%" % '&#."# ( '!$.)) $ *$).))

a. Calculate the forward rate of interest for each year. b. How could you construct a 1-year forward loan beginning in year 3? Confirm that the rate on

that loan equals the forward rate. c. Repeat part (b) for a 1-year forward loan beginning in year 4.

19. Use the data from Problem 18. Suppose that you want to construct a 2-year maturity forward loan commencing in 3 years. a. Suppose that you buy today one 3-year maturity zero-coupon bond. How many 5-year matu-

rity zeros would you have to sell to make your initial cash flow equal to zero? b. What are the cash flows on this strategy in each year? c. What is the effective 2-year interest rate on the effective 3-year-ahead forward loan? d. Confirm that the effective 2-year forward interest rate equals (1 + f4) " (1 + f5) # 1. You

therefore can interpret the 2-year loan rate as a 2-year forward rate for the last two years. Alternatively, show that the effective 2-year forward rate equals

(1!+!y5) 5 ________

(1!+!y3)3 !#!1

1. Briefly explain why bonds of different maturities have different yields in terms of the expecta- tions and liquidity preference hypotheses. Briefly describe the implications of each hypothesis when the yield curve is (1) upward-sloping and (2) downward-sloping.

2. Which one of the following statements about the term structure of interest rates is true? a. The expectations hypothesis indicates a flat yield curve if anticipated future short-term rates

exceed current short-term rates. b. The expectations hypothesis contends that the long-term rate is equal to the anticipated short-

term rate. c. The liquidity premium theory indicates that, all else being equal, longer maturities will have

lower yields. d. The liquidity preference theory contends that lenders prefer to buy securities at the short end

of the yield curve.

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C H A P T E R !" The Term Structure of Interest Rates #$!

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3. The following table shows yields to maturity of zero-coupon Treasury securities. Term to Maturity (years) Yield to Maturity (%)

! ".#$% % &.#$ " #.$$ & #.#$ # '.$$

!$ '.'$

a. Calculate the forward 1-year rate of interest for year 3. b. Describe the conditions under which the calculated forward rate would be an unbiased

estimate of the 1-year spot rate of interest for that year. c. Assume that a few months earlier, the forward 1-year rate of interest for that year had been sig-

nificantly higher than it is now. What factors could account for the decline in the forward rate? 4. The 6-month Treasury bill spot rate is 4%, and the 1-year Treasury bill spot rate is 5%. What is the

implied 6-month forward rate for six months from now? 5. The tables below show, respectively, the characteristics of two annual-pay bonds from the same

issuer with the same priority in the event of default, and spot interest rates. Neither bond’s price is consistent with the spot rates. Using the information in these tables, recommend either bond A or bond B for purchase.

Bond Characteristics

Bond A Bond B

Coupons Annual Annual Maturity " years " years Coupon rate !$% '% Yield to maturity !$.'#% !$.(#% Price )*.&$ **."&

Spot Interest Rates

Term (years) Spot Rates (zero-coupon)

! #% % * " !!

6. Sandra Kapple is a fixed-income portfolio manager who works with large institutional clients. Kapple is meeting with Maria VanHusen, consultant to the Star Hospital Pension Plan, to discuss management of the fund’s approximately $100 million Treasury bond portfolio. The current U.S. Treasury yield curve is given in the following table. VanHusen states, “Given the large differen- tial between 2- and 10-year yields, the portfolio would be expected to experience a higher return over a 10-year horizon by buying 10-year Treasuries, rather than buying 2-year Treasuries and reinvesting the proceeds into 2-year T-bonds at each maturity date.”

Maturity Yield Maturity Yield

! year %.$$% ' years &.!#% % %.)$ ( &."$ " ".#$ * &.&# & ".*$ ) &.'$ # &.$$ !$ &.($

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!"# P A R T I V Fixed-Income Securities

a. Indicate whether VanHusen’s conclusion is correct, based on the pure expectations hypothesis. b. VanHusen discusses with Kapple alternative theories of the term structure of interest rates and

gives her the following information about the U.S. Treasury market: Maturity (years) ! " # $ % & ' ( )* Liquidity premium (%) *.$$ *.$$ *.%$ *.&$ *.(* ).)* ).!* ).$* ).%*

Use this additional information and the liquidity preference theory to determine what the slope of the yield curve implies about the direction of future expected short-term interest rates.

7. A portfolio manager at Superior Trust Company is structuring a fixed-income portfolio to meet the objectives of a client. The portfolio manager compares coupon U.S. Treasuries with zero-coupon stripped U.S. Treasuries and observes a significant yield advantage for the stripped bonds:

Term Coupon

U.S. Treasuries Zero-Coupon Stripped

U.S. Treasuries

" years $.$*% $.'*% & %.&$+ &.!$+

)* &.!$+ &.%*+ "* &.&$+ '.!*+

Briefly discuss why zero-coupon stripped U.S. Treasuries could have higher yields to maturity than coupon U.S. Treasuries with the same final maturity.

8. The shape of the U.S. Treasury yield curve appears to reflect two expected Federal Reserve reductions in the federal funds rate. The current short-term interest rate is 5%. The first reduction of approximately 50 basis points (bp) is expected six months from now and the second reduction of approximately 50 bp is expected one year from now. The current U.S. Treasury term premiums are 10 bp per year for each of the next three years (out through the 3-year benchmark).

However, the market also believes that the Federal Reserve reductions will be reversed in a single 100-bp increase in the federal funds rate 2 1 ⁄ 2 years from now. You expect liquidity premi- ums to remain 10 bp per year for each of the next three years (out through the 3-year benchmark).

Describe or draw the shape of the Treasury yield curve out through the 3-year benchmark. Which term structure theory supports the shape of the U.S. Treasury yield curve you’ve described?

9. U.S. Treasuries represent a significant holding in many pension portfolios. You decide to analyze the yield curve for U.S. Treasury notes. a. Using the data in the table below, calculate the 5-year spot and forward rates assuming annual

compounding. Show your calculations. U.S. Treasury Note Yield Curve Data

Years to Maturity Par Coupon

Yield to Maturity Calculated Spot Rates

Calculated Forward Rates

) $.** $.** $.** ! $.!* $.!) $.#! " %.** %.*$ &.&$ # &.** &.)% )*.$% $ &.** ? ?

b. Define and describe each of the following three concepts: i. Short rate ii. Spot rate iii. Forward rate

Explain how these concepts are related. c. You are considering the purchase of a zero-coupon U.S. Treasury note with four years to

maturity. On the basis of the above yield-curve analysis, calculate both the expected yield to maturity and the price for the security. Show your calculations.

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C H A P T E R !" The Term Structure of Interest Rates #$%

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10. The spot rates of interest for five U.S. Treasury securities are shown in the following exhibit. Assume all securities pay interest annually.

Spot Rates of Interest

Term to Maturity Spot Rate of Interest

! year !".##% $ !$.## " !!.## % !#.## & '.##

a. Compute the 2-year implied forward rate for a deferred loan beginning in three years. b. Compute the price of a 5-year annual-pay Treasury security with a coupon rate of 9%

by using the information in the exhibit.

E&INVESTMENTS EXERCISES Go to'stockcharts.com/freecharts/yieldcurve.php where you will find a dynamic or “living” yield curve, a moving picture of the yield curve over time. Hit the Animate button to start the demon- stration. Is the yield curve usually upward- or downward-sloping? What about today’s yield curve? How much does the slope of the curve vary? Which varies more: short-term or long-term rates? Can you explain why this might be the case?

SOLUTIONS TO CONCEPT CHECKS 1. The price of the 3-year bond paying a $40 coupon is

40 ____ 1.05

!+! 40 _____ 1.062

!+! 1040 _____ 1.073

!=!38.095!+!35.600!+!848.950!=!$922.65

At this price, the yield to maturity is 6.945% [n = 3; PV = (")922.65; FV = 1,000; PMT = 40]. This bond’s yield to maturity is closer to that of the 3-year zero-coupon bond than is the yield to maturity of the 10% coupon bond in Example 15.1. This makes sense: This bond’s coupon rate is lower than that of the bond in Example 15.1. A greater fraction of its value is tied up in the final payment in the third year, and so it is not surprising that its yield is closer to that of a pure 3-year zero-coupon security.

2. We compare two investment strategies in a manner similar to Example 15.2:

Buy!and!hold!4-year!zero!=!Buy!3-year!zero;!roll!proceeds!into!1-year!bond (1!+!y4)4!=!(1!+!y3)3!#!(1!+!r4)

1.084!=!1.073!#!(1!+!r4)

which implies that r4 = 1.084/1.073 " 1 = .11056 = 11.056%. Now we confirm that the yield on the 4-year zero reflects the geometric average of the discount factors for the next 3 years:

1!+!y4!=![(1!+!r1)!#!(1!+!r2)!#!(1!+!r3)!#!(1!+!r4)]1/4

1.08!=![1.05!#!1.0701!#!1.09025!#!1.11056]1/4

3. The 3-year bond can be bought today for $1,000/1.073 = $816.30. Next year, it will have a remaining maturity of two years. The short rate in year 2 will be 7.01% and the short rate in year 3

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!"! P A R T I V Fixed-Income Securities

will be 9.025%. Therefore, the bond’s yield to maturity next year will be related to these short rates according to

(1!+!y2)2!=!1.0701!"!1.09025!=!1.1667

and its price next year will be $1,000/(1 + y2)2 = $1,000/1.1667 = $857.12. The 1-year holding- period rate of return is therefore ($857.12 # $816.30)/$816.30 = .05, or 5%.

4. The n-period spot rate is the yield to maturity on a zero-coupon bond with a maturity of n periods. The short rate for period n is the one-period interest rate that will prevail in period n. Finally, the forward rate for period n is the short rate that would satisfy a “break-even condition” equating the total returns on two n-period investment strategies. The first strategy is an investment in an n-period zero-coupon bond; the second is an investment in an n # 1 period zero-coupon bond “rolled over” into an investment in a one-period zero. Spot rates and forward rates are observable today, but because interest rates evolve with uncertainty, future short rates are not. In the special case in which there is no uncertainty in future interest rates, the forward rate calculated from the yield curve would equal the short rate that will prevail in that period.

5. 7% # 1% = 6%. 6. The risk premium will be zero. 7. If issuers prefer to issue long-term bonds, they will be willing to accept higher expected interest

costs on long bonds over short bonds. This willingness combines with investors’ demands for higher rates on long-term bonds to reinforce the tendency toward a positive liquidity premium.

8. In general, from Equation 15.5, (1 + yn)n = (1 + yn#1)n#1 " (1 + fn). In this case, (1 + y4)4 = (1.07)3 " (1 + f4). If f4 = .07, then (1 + y4)4 = (1.07)4 and y4 = .07. If f4 is greater than .07, then y4 also will be greater, and conversely if f4 is less than .07, then y4 will be as well.

9. The 3-year yield to maturity is ( 1,000 ______ 816.30 ) 1/3

!#!1!=!.07!=!7.0%

The forward rate for the third year is therefore

f3!=! (1!+!y3)3 ________ (1!+!y2)2

!#!1!=! 1.07 3 _____

1.062 !#!1!=!.0903!=!9.03%

(Alternatively, note that the ratio of the price of the 2-year zero to the price of the 3-year zero is 1 + f3 = 1.0903.) To construct the synthetic loan, buy one 2-year maturity zero, and sell 1.0903 3-year maturity zeros. Your initial cash flow is zero, your cash flow at time 2 is +$1,000, and your cash flow at time 3 is #$1,090.30, which corresponds to the cash flows on a 1-year forward loan commencing at time 2 with an interest rate of 9.03%.

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Calculate the yield to maturity of the coupon bond in Example 15.1, and you may be sur- prised. Its yield to maturity is 6.88%; so while its maturity matches that of the 3-year zero in Table 15.1, its yield is a bit lower.1 This reflects the fact that the 3-year coupon bond may usefully be thought of as a portfolio of three implicit zero-coupon bonds, one corresponding to each cash flow. The yield on the coupon bond is then an amalgam of the yields on each of the three components of the “portfolio.” Think about what this means: If their coupon rates differ, bonds of the same maturity generally will not have the same yield to maturity.

What then do we mean by “the” yield curve? In fact, in practice, traders refer to sev- eral yield curves. The pure yield curve refers to the curve for stripped, or zero-coupon,

payment from a whole Treasury bond as a separate cash flow. For example, a 1-year maturity T-bond paying semiannual coupons can be split into a 6-month maturity zero (by selling the first coupon payment as a stand-alone security) and a 12-month zero (cor- responding to payment of final coupon and principal). Treasury stripping suggests exactly how to value a coupon bond. If each cash flow can be (and in practice often is) sold off as a separate security, then the value of the whole bond should equal the total value of its cash flows bought piece by piece in the STRIPS market.

What if it weren’t? Then there would be easy profits to be made. For example, if invest- ment bankers ever noticed a bond selling for less than the amount at which the sum of its parts could be sold, they would buy the bond, strip it into stand-alone zero-coupon securi- ties, sell off the stripped cash flows, and profit by the price difference. If the bond were selling for more than the sum of the values of its individual cash flows, they would run the process in reverse: buy the individual zero-coupon securities in the STRIPS market, reconstitute (i.e., reassemble) the cash flows into a coupon bond, and sell the whole bond for more than the cost of the pieces. Both bond stripping and bond reconstitution offer opportunities for arbitrage—the exploitation of mispricing among two or more securities to clear a riskless economic profit. Any violation of the Law of One Price, that identical cash flow bundles must sell for identical prices, gives rise to arbitrage opportunities.

To value each stripped cash flow, we simply look up its appropriate discount rate in The Wall Street Journal. Because each coupon payment matures at a different time, we discount by using the yield appropriate to its particular maturity—this is the yield on a Treasury strip maturing at the time of that cash flow. We can illustrate with an example.

Suppose the yields on stripped Treasuries are as given in Table !".!, and we wish to value a !#% coupon bond with a maturity of three years. For simplicity, assume the bond makes its payments annually. Then the first cash flow, the $!## coupon paid at the end of the first year, is discounted at "%; the second cash flow, the $!## coupon at the end of the second year, is discounted for two years at $%; and the final cash flow consisting of the final coupon plus par value, or $!,!##, is discounted for three years at %%. The value of the coupon bond is therefore

!##

____ !.#"

!+! !##

____ !.#$&

!+! !,!##

____ !.#%'

!=!(".&')!+!)(.###!+!)(%.(&)!=!$!,#)&.!%

Example 15.1 Valuing Coupon Bonds

1Remember that the yield to maturity of a coupon bond is the single interest rate at which the present value of cash flows equals market price. To calculate the coupon bond’s yield to maturity on your calculator or spreadsheet, set n = 3;!price = "1,082.17; future value = 1,000;!payment = 100. Then compute the interest rate.

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