Principles of Corporate Finance, 9e

bre05108_ch04_059-084.indd Page 59 7/24/07 7:39:45 PM user1

/Volumes/102/MHIL014/mhbre/bre9ch04%0

4

CHAPTER FOUR

VALUING BONDS INVESTMENT IN NEW plant and equipment requires

money—often a lot of money. Sometimes firms can retain and accumulate earnings to cover the cost of investment, but often they need to raise extra cash from investors. If they choose not to sell additional shares of common stock, the cash has to come from borrowing. If cash is needed for only for a short while, they may borrow from a bank. If they need cash for long-term investments, they generally issue bonds, which are simply longterm loans. Companies are not the only bond issuers. Municipalities also raise money by selling bonds. So do national governments. There is always some risk that a company or municipality will not be able to come up with the cash to repay its bonds, but investors in government issues can be confident that the promised payments will be made in full and on time.1 This chapter focuses on how government bonds are valued and on the interest rates that governments pay when they borrow. The markets for these bonds are huge. The aggregate principal amount of outstanding U.S. Treasury securi-

ties in mid-2006 was about $8.4 trillion. 2 The corresponding amounts for Germany and the U.K. were about €1.1 trillion and £.4 trillion, respectively. The markets are also sophisticated. Bond traders make massive trades motivated by tiny price discrepancies. The interest rates on governments bonds are benchmarks for all interest rates. Companies can’t borrow at the same low interest rates as governments, but when government rates go up or down, corporate rates follow more or less proportionally. Therefore financial managers had better understand how the government rates are determined and what happens when they change. Government bonds pay a schedule of cash flows representing interest and repayment of principal. There is no uncertainty about either the amounts or timing. So valuation of government bonds should be simple, just a matter of discounting at the riskfree interest rate, right? Wrong: There’s not one risk-free interest rate, but dozens, depending on maturity, and you will find that bond traders may refer to “spot interest rates” or “yields to maturity,” which are not the same thing. (continued )

1

This is true only if the government bond is issued in the country’s own currency. When governments borrow in another country’s currency, investors cannot be absolutely sure of repayment. 2 This includes $3.6 trillion held by government bodies.


60


PART ONE Value

This book is not for bond traders, but if you are to be involved in managing the company’s debt, you will have to get beyond the mechanics of discounting. Professional financial managers understand the bond pages in the financial press and know what bond dealers mean when they quote spot rates or yields to maturity. They realize why

4.1

short-term rates are usually lower (but sometimes higher) than long-term rates and why the longestterm bond prices are most sensitive to fluctuations in interest rates. They can distinguish real (inflationadjusted) interest rates and nominal (money) rates and anticipate how future inflation can affect interest rates. We cover all these topics in this chapter.

USING THE PRESENT VALUE FORMULA TO VALUE BONDS If you own a bond, you are entitled to a fixed set of cash payoffs: Each year until the bond matures, you collect an interest payment; then at maturity, you also get back the face value of the bond, which is called the principal. Therefore, when the bond matures, you receive both the principal and interest.

A Short Trip to Germany to Value a Government Bond We will start our discussion of bond values with a visit to Germany, where the government issues long-term bonds known as “bunds” (short for Bundesanleihen). These bonds pay interest and principal in euros (€s). For example, suppose that in July 2006 you decided to buy €100 face value of the 5% bund maturing in July 2012. Each year until 2012 you are entitled to an interest payment of .05 100 €5. This amount is the bond’s coupon.3 When the bond matures in 2012 the government pays you the final €5 interest, plus the €100 face value. Your first coupon payment is in one year’s time in July 2007. So the cash flows from owning the bonds are as follows: Cash Flows (€) 2007

2008

2009

2010

2011

2012

€5

€5

€5

€5

€5

€105

What is the present value of these payoffs? To determine that, you need to look at the return offered by similar securities. In July 2006 other medium-term German government bonds offered a return of about 3.8%. That is what you were giving up when you bought the 5% bonds. Therefore to value the 5% bonds, you must discount the cash flows at 3.8%: PV

5 5 5 5 5 105 €106.33 1.038 1.0382 1.0383 1.0384 1.0385 1.0386

3 Bonds used to come with coupons attached, which had to be clipped off and presented to the issuer to obtain the interest payments. This is still the case with bearer bonds, where the only evidence of indebtedness is the bond itself. In many parts of the world bearer bonds are still issued and are popular with investors who would rather remain anonymous. The alternative is to issue registered bonds, in which case the identity of the bond’s owner is recorded and the coupon payments are sent automatically. Bunds are registered bonds.



CHAPTER 4 Valuing Bonds Bond prices are usually expressed as a percentage of face value. Thus we can say that your 5% bund is worth 106.33%. You may have noticed a shortcut way to value this bond. Your purchase is like a package of two investments. The first investment pays off the six annual coupon payments of €5 each and the second pays off the €100 face value at maturity. Therefore, you can use the annuity formula to value the coupon payments and add on the present value of the final payment: PV(bond) PV(coupon payments) PV(final payment) (coupon 6-year annuity factor) (final payment discount factor)

⎡ 1 ⎤ 100 1 5⎢_ __6 _6 26.38 79.95 €106.33 ⎣ .038 .038(1.038) ⎦ (1.038) Any bond can be valued as a package of an annuity (the coupon payments) and a single repayment (the repayment of the face value). Rather than asking the value of the bond, we could have phrased our question the other way around: If the price of the bond is 106.33%, what return do investors expect? In that case, you need to find the value of y that solves the following equation: 106.33

5 5 5 5 5 105 1y 11 y2 2 11 y2 3 11 y2 4 11 y2 5 11 y2 6

The rate y is called the bond’s yield to maturity. The yield to maturity on our bond is 3.8%. If you buy the bond at 106.33% and hold it to maturity, you will earn a return of 3.8% over the six years. That figure reflects both the regular interest payment that you receive and the fact that you are paying more for the bond today (€106.33) than you will receive back at maturity (€100). The only general procedure for calculating the yield to maturity is trial and error. You guess at an interest rate and calculate the present value of the bond’s payments. If the present value is greater than the actual price, your discount rate must have been too low, and you need to try a higher rate. The more practical solution is to use a spreadsheet program or a specially programmed calculator to calculate the yield.

Back to the United States: Semiannual Coupons and Bond Prices Just like the German government, the U.S. Treasury periodically raises money by auctioning new issues of bonds. Some of these issues do not mature for 30 years; others, known as notes, mature in 10 years or less. The government also issues short-term loans that mature in less than a year. These are known as Treasury bills. We will look at an example of a U.S. government note. In 2004 the Treasury issued 4.0% notes maturing in 2009. Treasury bonds have a face value of $1,000 so, if you own the 4s of 2009, the Treasury gives you back $1,000 when the bond matures. You can also look forward to a regular interest payment, but, in contrast to our German bond, interest on Treasury bonds is paid semiannually.4 Thus, the 4s of 2009 provide a coupon payment of 4.0兾2 2.0% of face value every six months. Once issued, Treasury bonds are widely traded through a network of dealers and the prices at which you can buy or sell the bonds are shown each day in the 4

The frequency of interest payments varies from country to country. For example, most euro bonds pay interest annually, while bonds in the U.K., Canada, and Japan, generally pay interest semiannually.

61

bre05108_ch04_059-084.indd Page 62 7/25/07 9:03:59 AM elhi

62

PART ONE Value

F I G U R E 4.1 Treasury bond quotes from The Wall Street Journal, June 2006. Source: The Wall Street Journal, June 2006 © Dow Jones, Inc.

financial press. Figure 4.1 is an excerpt from the bond quotation page of The Wall Street Journal. Look at the entry for our 4.0% Treasury bond maturing in June 2009. The asked price of 97:11 is the price you need to pay to buy the bond from a dealer. This price is quoted in 32nds rather than decimals. Thus a price of 97:11 means that each bond costs 97 and 11兾32, or 97.34375% of face value. The face value of the bond is $1,000, so each bond costs $973.4375.5 The bid price is the price investors receive if they sell the bond to a dealer. The dealer earns her living by charging a spread between the bid and the asked price. Notice that the spread for the 4% bonds is only 1兾32, or about .03%, of the bond’s value. The next column in Figure 4.1 shows the change in price since the previous day. The price of the 4.0% bonds has fallen by 1兾32. Finally, the column “Ask Yld” shows the ask yield to maturity. Because interest is semiannual, yields on U.S. bonds are The quoted bond price is known as the flat (or clean) price. The price that the bond buyer actually pays (sometimes called the full or dirty price) is equal to the flat price plus the interest that the seller has already earned on the bond since the last interest payment. The precise method for calculating this accrued interest varies from one type of bond to another. You need to use the flat price to calculate the yield. 5



CHAPTER 4 Valuing Bonds usually quoted as semiannually compounded yields. Thus, if you buy the 4.0% bond at the asked price and hold it to maturity, you will earn a semiannually compounded return of 4.96%. This is equivalent to a yield over six months of 4.96兾2 2.48%. We can now repeat the present value calculations that we did for the German government bond. We just need to recognize that bonds in the United States have a face value of $1,000, that their coupons are paid semiannually, and that the quoted yield is a semiannually compounded rate. Here are the cash flows from the 4s of 2009: Cash Flows ($) Dec 2006

Jun 2007

Dec 2007

Jun 2008

Dec 2008

June 2009

$20

$20

$20

$20

$20

$1,020

If investors demand a semiannual return of 2.48% for investing in three-year bonds, then the present value of these cash flows is PV

20 20 20 20 20 1020 $973.54 1.0248 1.02482 1.02483 1.02485 1.02486 1.02484

Each bond is worth $973.54, or 97.35% of face value (the slight difference from the figure in The Wall Street Journal is simply due to rounding error).

4.2

HOW BOND PRICES VARY WITH INTEREST RATES

As interest rates change, so do bond prices. For example, suppose that investors demanded a yield of 3% on three-year Treasury bonds. What would be the price of the 4s of 2009? Just repeat the last calculation with a six-month yield of 1.5%: PV

20 20 20 20 20 1020 $1,028.49 1.015 1.0152 1.0153 1.0155 1.0156 1.0154

or 102.85% of face value. The lower interest rate results in a higher bond price. The solid line in Figure 4.2 shows the value of our 4% bond for different interest rates. You can see that as yields fall, bond prices rise. When the yield is equal to the bond’s coupon (4%), the bond sells for exactly its face value. When the yield is higher than 4%, the bond sells at a discount to face value. When the yield is lower, the bond sells at a premium. Bond investors cross their fingers that market interest rates will fall, so that the price of their securities will rise. If they are unlucky and the interest rates jump up, the value of their investment declines. Any such change in interest rates is likely to have only a modest effect on the value of near-term cash flows, but it will have a much greater effect on the value of distant cash flows. Thus the price of long-term bonds is affected more by changing interest rates than is the price of short-term bonds.

Duration and Bond Volatility But what do we mean by the phrases “long-term” and “short-term” bonds? A coupon bond that matures in year 30 makes payments in each of years 1 through 30. Therefore, it is somewhat misleading to describe the bond as a 30-year bond; the average time to each cash flow is less than 30 years.

63


64


PART ONE Value

F I G U R E 4.2

Bond price, % 115

The value of the threeyear 4% bond falls as interest rates rise.

110 105 100

3-year 4% bond

95 90 85 80 0

Year 1 2 3

Ct 100 100 1,100

1

2

3

4 5 6 Interest rate, %

Proportion of Total Value [PV(Ct)/V ] 0.084 0.080 0.836 1.000

PV(Ct) at 5% 95.24 90.70 950.22 V = 1,136.16

7

8

9

10

Proportion of Total Value x Time 0.084 0.160 2.509 Duration = 2.753 years

T A B L E 4.1 The first four columns show that the cash flow in year 3 accounts for less than 84% of the present value of the three-year 10s. The final column shows how to calculate a weighted average of the times to each cash flow. This average is the bond’s duration.

x

e cel Visit us at www.mhhe.com/bma9e.

Consider a simple three-year bond that pays interest of 10% once a year. The first three columns of Table 4.1 calculate the present value (V ) of this bond assuming a yield to maturity of 5%. The total value of the bond is $1,136.16. The fourth column shows the contribution of each payment to the bond’s value. Notice that the cash flow in year 3 accounts for less than 84% of the value. The remaining 16% comes from the earlier cash flows. Bond analysts often use the term duration to describe the average time to each payment. If we call the total value of the bond V, then duration is calculated as follows:6 Duration ⫽

6

[1 ⫻ PV1C1 2] V

⫹

[2 ⫻ PV1C2 2] V

This measure is also known as Macaulay duration after its inventor.

⫹

[3 ⫻ PV1C3 2] V

⫹...



CHAPTER 4 Valuing Bonds The final column of Table 4.1 shows that for our three-year 10% bond, Duration 11 .0842 12 .0802 13 .8362 2.753 years The bond’s maturity is three years but the weighted average time to each cash flow is only 2.753 years. Suppose we take another three-year bond. This time the coupon payment is 4%. The maturity is the same as that of the 10% bond, but the first two years’ coupon payments account for a smaller fraction of the value. In this sense the bond is a longer bond. The duration of the three-year 4% bonds is 2.884 years. Consider now what happens to the price of the 10% and 4% bonds as interest rates change: 3-year 10% bond

Yield falls .5% Yield rises .5% Difference

3-year 4% bond

New Price

Change

New Price

Change

1151.19 1121.41

ⴙ1.32% ⴚ1.30 2.62

986.26 959.53

ⴙ1.39% ⴚ1.36 2.75

A 1 percentage-point variation in yield causes the price of the 10s to change by 2.62%. We can say that the 10s have a volatility of 2.62%, while the 4s have a volatility of 2.75%. Notice that the 4% bonds have the greater volatility and that they also have the longer duration. In fact, a bond’s volatility is directly related to its duration: 7 Volatility 1% 2

duration 1 yield

In the case of the 10s, Volatility 1% 2

2.753 2.62 1.05

Figure 4.3 shows how changing interest rates affect the prices of a 3-year 4% bond and a 30-year 4% bond. Each bond’s volatility is simply the slope of the line relating the bond price to the interest rate. The 30-year bond has a much longer duration than the 3-year bond and is correspondingly more volatile. This shows up in the steeper curve in Figure 4.3. Notice also that the bond’s volatility changes as the interest rate changes. Volatility is higher at lower interest rates (the curve is steeper), and it is lower at higher rates (the curve is flatter).8

A Cautionary Note Bond volatility measures the effect on bond prices of a shift in interest rates. For example, we calculated that the three-year 10s had a volatility of 2.62. This means

7 8

For this reason volatility is also called modified duration. Bond investors refer to this feature as the bond’s convexity.

65

bre05108_ch04_059-084.indd Page 66 8/17/07 7:56:19 PM user

66


PART ONE Value

F I G U R E 4.3

Bond price, %

Plots of the prices of 3-year and 30-year 4% bonds. Notice that prices of the long bonds are more sensitive to changes in the interest rate than those of short bonds. Each bond’s volatility is the slope of the curve relating the bond price to the interest rate.

x

e cel

250 200 30-year 4% bond

150 100

3-year 4% bond

50 0 0

1

2

3

4 5 6 Interest rate, %

Visit us at www.mhhe.com/bma9e.

F I G U R E 4.4

7

8

9

10

Yield, %

Short- and long-term interest rates do not always move in parallel. Between September 1992 and April 2000 U.S. short-term rates rose sharply while long-term rates declined.

7.5 7 6.5 6 5.5 5 4.5 4 3.5

September 1992 April 2000

23

5

7

10

30 Bond maturity, years

that a 1 percentage-point change in interest rates leads to a 2.62% change in the bond price: Change in bond price ⫽ 2.62 ⫻ change in interest rates We will show in Chapter 27 how this volatility measure can help firms to understand how they may be affected by interest rate changes and how they can protect themselves against these risks. If the yields on all bonds moved in precise lockstep, then the volatility measure would capture exactly the effect of interest rate changes on bond prices. However, Figure 4.4 illustrates that short- and long-term interest rates do not always move in perfect unison. Between 1992 and 2000 short-term interest rates nearly doubled while long-term rates declined. As a result, the term structure, which initially sloped steeply upward, shifted to a downward slope. Because short- and long-term yields do not move in parallel, a single measure of volatility



CHAPTER 4 Valuing Bonds cannot be the whole story, and managers may need to worry not just about the risks of an overall change in interest rates but also about shifts in the shape of the term structure.

4.3

THE TERM STRUCTURE OF INTEREST RATES

We now need to look more carefully at the relationship between short- and longterm rates of interest. Consider a simple loan that pays $1 at time 1. The present value of this loan is PV

1 1 r1

Thus we discount the cash flow at r1, today’s rate for a one-period loan. This rate is often called today’s one-period spot rate. If we have a loan that pays $1 at both time 1 and time 2, present value is PV

1 1 1 r1 11 r2 2 2

This is identical to the calculations that we performed at the start of Chapter 3 where we valued a series of risk-free cash flows. The first period’s cash flow is discounted at today’s one-period spot rate and the second period’s flow is discounted at today’s two-period spot rate. The series of spot rates r1, r2, etc., is one way of expressing the term structure of interest rates.

The Yield to Maturity and the Term Structure Rather than discounting each of the payments at a different rate of interest, we could instead find a single rate that would produce the same present value. That is what we did in Section 4.1 when we calculated the yield to maturity on the German and U.S. government bonds. In the case of our simple two-year loan, we could write the present value in terms of the yield to maturity as PV

1 1 1y 11 y2 2

Financial managers who want a quick, summary measure of interest rates look in the financial press at the yield to maturity on government bonds. Or they may refer to the yield curve, which summarizes how bond yields vary with the bond’s maturity. Thus managers may make broad generalizations such as “If we take out a five-year loan, we will have to pay an interest rate (i.e., yield) of 5%.” Throughout this book, we too will use the yield to maturity to summarize the return required by bond investors. But you also need to understand the measure’s limitations when the spot rates r1, r2, etc., are not equal. The yield to maturity resembles an average of these different spot rates and, like any average, it may hide some useful information. If you wish to understand why different bonds sell at different prices, you may need to dig deeper and look at the separate interest rates for one-year cash flows, two-year cash flows, and so on. In other words, you may need to look at the spot rates of interest.

67


68


PART ONE Value Example Here is an example where comparing the yields of two bonds is potentially misleading. It is 2009. You are contemplating an investment in U.S. Treasuries and come across the following quotations for two bonds: Bond

Price as % of Face Value

Yield to Maturity

85.211 105.429

8.78% 8.62

5s of 2014 10s of 2014

Does the higher yield on the 5s of 2014 mean that they are a better buy? The only way to know for sure is to use the spot rates of interest to calculate the bonds’ present values. This is done in Table 4.2, assuming for simplicity annual coupon payments. The important assumption in Table 4.2 is that long-term interest rates are higher than short-term interest rates. We have assumed that the one-year interest rate is r1 .05, the two year rate is r2 .06, and so on. When each year’s cash flow is discounted at the rate appropriate for that year, we see that each bond’s present value is equal to the quoted price. Thus each bond is fairly priced. If both bonds are fairly priced, why do the 5s have a higher yield? Because for each dollar that you invest in the 5s, you receive relatively little cash inflow in the first four years and a relatively high cash inflow in the final year. Therefore, although the two bonds have identical maturity dates, the 5s provide a greater proportion of their cash flows in 2014. In this sense the 5s are a longer-term investment than the 10s. Their higher yield to maturity just reflects the fact that longterm interest rates are higher than short-term rates. Notice the reason that the yield to maturity in this example is misleading. When the yield is calculated, the same rate is used to discount all payments on the bond. But in our example bondholders demand different rates of return (r1, r2, etc.) for cash flows that occur at different dates. Since the cash flows on the two bonds are also not identical, the bonds have different yields to maturity, and the yield to maturity on the 5s of 2014 offers only a rough guide to the appropriate yield on the 10s of 2014.

Present Value Calculations 5s of 2014 Year

Spot Interest Rate

2010 2011 2012 2013 2014

r1 .05 r2 .06 r3 .07 r4 .08 r5 .09

Cash Flow $

50 50 50 50 1050 Totals

10s of 2014 PV

Cash Flow

$ 47.62 44.50 40.81 36.75 682.43 $852.11

$ 100 100 100 100 1100

PV $

95.24 89.00 81.63 73.50 714.92 $1,054.29

T A B L E 4.2 Calculating present values of two bonds when long-term interest rates are higher than short-term rates.



CHAPTER 4 Valuing Bonds

F I G U R E 4.5

6 ay -1

5 M

ay -1

4 M

ay -1

3 M

ay -1

2 M

M

ay -1

1

0

ay -1 M

M

ay -1

9 ay -0

8

Spot rates on U.S. Treasury strips, June 2006.

M

ay -0 M

M

ay -0

7

Spot rate, % 5.5 5.3 5.1 4.9 4.7 4.5 4.3 4.1 3.9 3.7 3.5

Years

Measuring the Term Structure You can think of the spot rate, rt, as the rate of interest on a bond that makes a single payment at time t. Such bonds do exist. They are known as stripped bonds, or strips. On request the Treasury will split a normal coupon bond into a package of mini-bonds, each of which makes just one cash payment. Thus, our 5% bonds of 2014 could be exchanged for five coupon strips each paying $50 and a principal strip paying $1,000. The prices of strips are shown each day in the financial press. For example, in June 2006 a 10-year strip cost $609.06, and it will make a single payment of $1,000 in the summer of 2016. Hence the 10-year spot rate was (1000兾609.06)1兾10 1 .0508, or 5.08%.9 In Figure 4.5 we have used the prices of strips with different maturities to plot the term structure of spot rates from 1 to 10 years. You can see that investors required a somewhat higher interest rate for lending for 10 years rather than 1.

4.4

EXPLAINING THE TERM STRUCTURE

The term structure that we showed in Figure 4.5 was upward-sloping. In other words, long-term rates of interest were higher than short-term rates. This is the more common pattern, but sometimes it is the other way around, with short rates higher than long rates. Why do we get these shifts in term structure? Let us look at a simple example. Suppose that the one-year spot rate (r1) is 5% and the two-year spot rate is higher at r2 6%. If you invest in a one-year Treasury strip, you would earn the one-year spot rate, so that by the end of the year each dollar that you invested would have grown to $(1 r1) $1.05. If instead you were prepared to invest for two years, you would earn the two-year spot rate of r2, and by the end of the two years each dollar would have grown to $(1 r2)2 $1.062 $1.1236. By 9 This is an annually compounded rate. The yields quoted by U.S. bond dealers are semiannually compounded rates.

69


70


PART ONE Value

F I G U R E 4.6

(a ) The future value of $1 invested in a two-year loan

An investor can invest either in a twoyear loan (a) or in two successive oneyear loans (b). The expectations theory says that in equilibrium the expected payoffs from these two strategies must be equal. In other words, the forward rate, f2, must equal the expected spot rate, 1r2.

Period 0

Period 2 2

(1 + r2) = (1 + r1) (1 + f2)

(b ) The future value of $1 invested in two successive one-year loans Period 0

Period 1 (1 + r1)

Period 2 (1 + 1r2)

keeping your money invested for that second year, your savings grow from $1.05 to $1.1236, an increase of 7.01%. This extra 7.01% that you earn by keeping your money invested for two years rather than one is termed the forward interest rate or f2. Notice how we calculated the forward rate. When you invest for one year, each dollar grows to $(1 r1). When you invest for two years, each dollar grows to $(1 r2)2. Therefore, the extra return that you earn for that second year is f2 (1 r2)2兾(1 r1) 1. In our example, f2 11 r2 2 2兾11 r1 2 1 11 .062 2兾11 .052 1 .0701, or 7.01% If you twist this equation around, you can obtain an expression for the two-year spot rate, r2, in terms of the one-year spot rate, r1, and the forward rate, f2: 11 r2 2 2 11 r1 211 f2 2 In other words, you can think of the two-year investment as earning the one-year spot rate for the first year and the extra return, or forward rate, for the second year.

The Expectations Theory Would you be happy to earn an extra 7% for investing for two years rather than one? The answer depends on how you expect interest rates to change over the coming year. Suppose, for example, that you were confident that interest rates would rise sharply, so that at the end of the year the one-year rate would be 8%. In that case, rather than investing in a two-year bond and earning an extra 7% for the second year, you would do better to invest in a one-year bond and, when that matured, to reinvest the cash for a further year at 8%. If other investors shared your view, no one would be prepared to hold the two-year bond and its price would fall. It would stop falling only when the extra return from holding the two-year bond equalled the expected future one-year rate. Let us call this expected rate 1r2—that is, the spot rate at year 1 on a loan maturing at the end of year 2.10 Figure 4.6 shows that at that point investors would earn the same expected return from investing in a two-year loan as from investing in two successive one-year loans.

10

Be careful to distinguish 1r2 from r2, the spot interest rate on a bond held from time 0 to time 2. The quantity 1r2 is a one-year spot rate established at time 1.



CHAPTER 4 Valuing Bonds This is known as the expectations theory of the term structure. It states that in equilibrium the forward interest rate f2 must equal the expected one-year spot rate r . The expectations theory implies that the only reason for an upward-sloping 1 2 term structure is that investors expect short-term interest rates to rise; the only reason for a declining term structure is that investors expect short-term rates to fall.11 The expectations theory also implies that investing in a succession of short-term bonds gives exactly the same expected return as investing in long-term bonds. If short-term interest rates are significantly lower than long-term rates, it is tempting to borrow short-term rather than long-term. The expectations theory implies that such naïve strategies won’t work. If short-term rates are lower than long-term rates, then investors must be expecting interest rates to rise. When the term structure is upward-sloping, you are likely to make money by borrowing short only if investors are overestimating future increases in interest rates. Even on a casual glance the expectations theory does not seem to be the complete explanation of term structure. For example, if we look back over the period 1900–2006, we find that the return on long-term U.S. Treasury bonds was on average about 1.2 percentage points higher than the return on short-term Treasury bills.12 Perhaps short-term interest rates stayed lower than investors expected, but it seems more likely that investors wanted some extra return for holding long bonds and that on average they got it. If so, the expectations theory is wrong. These days the expectations theory has few strict adherents, but most economists believe that expectations about future interest rates have an important effect on term structure. For example, you often hear market commentators remark that the forward interest rate over the next few months is higher than the current spot rate and conclude that the market is expecting the Fed to raise interest rates. There is quite a bit of evidence for this type of reasoning. Suppose that every month from 1950 to 2005 you used the three-month forward rate of interest to predict the change in the corresponding spot rate over these three months. You would have found on average that the steeper the term structure, the more the spot rate rose. It looks as if the expectations theory has some truth to it even if it is not the whole truth.

Introducing Risk What does the expectations theory leave out? The most obvious answer is “risk.” If you are confident about the future level of interest rates, you will simply choose the strategy that offers the highest return. But, if you are not sure of your forecasts, you may well opt for a less risky strategy even if it means giving up some return. Remember that the prices of long-duration bonds are more volatile than those of short-term bonds. A sharp increase in interest rates can easily knock 30% or 40% off the price of long-term bonds. For some investors this extra volatility may not be a concern. For example, pension funds and life insurance companies with longterm liabilities may prefer to lock in future returns by investing in long-term bonds. However, the volatility of long-term bonds does create extra risk for investors who do not have such long-term obligations. These investors will be prepared to hold long bonds only if they offer the compensation of a higher return. In this case

11 This follows from our example. If the one-year spot rate, r1, exceeds the two-year spot rate, r2, then r1 also exceeds the forward rate, f2. If the forward rate equals the expected spot rate, 1r2, then r1 must also exceed 1r2. 12 Treasury bills are short-term government debts with a maximum maturity of six months. We describe Treasury bills in Chapter 30.

71


72


PART ONE Value the forward rate must be higher than the expected spot rate and the term structure will be upward-sloping more often than not. Of course, if future spot rates are expected to fall, the term structure could be downward-sloping and still reward investors for lending long. But the additional reward for risk offered by long bonds would result in a less dramatic downward slope.

Inflation and Term Structure There is one other thing that you need to think about when comparing the risk of different bonds. Although the cash flows on every U.S. Treasury bond are certain, you can’t be sure what that money will buy. That depends on the rate of inflation. Suppose you are saving for your retirement. Which of the following strategies is the more risky? Invest in a succession of one-year Treasury bonds or invest in a 20-year bond? If you buy the 20-year bond, you know what money you will have at the end of the period but you will be making a long-term bet on inflation. Inflation may seem benign now but who knows what it will be like in 10 or 20 years? This uncertainty about inflation may make it more risky for you to fix today the rates at which you will lend in the distant future. You can reduce this uncertainty by investing in successive short-term bonds. You do not know the interest rate at which you will be able to reinvest your money at the end of each year, but at least you know that it will incorporate the latest information about inflation in the coming year. So, if inflation takes off, it is likely that you will be able to reinvest your money at a higher interest rate. Here then we have another reason that long-term bonds may offer a risk premium. If inflation creates an additional source of risk for long-term lenders, borrowers must offer some extra incentive if they want investors to lend long. That is why we often see a steeply upward-sloping term structure when inflation is particularly uncertain.

4.5

REAL AND NOMINAL RATES OF INTEREST It is now time to review more carefully the relation between inflation and interest rates. Suppose you invest $1,000 in a one-year bond that makes a single payment of $1,100 at the end of the year. Your cash flow is certain, but the government makes no promises about what that money will buy. If the prices of goods and services increase by more than 10%, you will have lost ground in terms of the goods that you can buy. Several indexes are used to track the general level of prices. The best known is the Consumer Price Index, or CPI, which measures the number of dollars that it takes to pay for a typical family’s purchases. The change in the CPI from one year to the next measures the rate of inflation. Figure 4.7 shows the rate of inflation in the United States since 1900. Inflation touched a peak at the end of World War I, when it reached 21%. This figure, however, pales into insignificance compared with inflation in Germany in 1923, which was more than 20,000,000,000% a year (or about 5% per day). Of course prices do not always rise. For example, in recent years Japan and Hong Kong have both faced a problem of deflation. The United States experienced severe deflation in the Great Depression when prices fell by 24% in three years.




F I G U R E 4.7

25

Annual rates of inflation in the United States from 1900–2006.

Annual inflation, %

20 15

Source: E. Dimson, P. R. Marsh, and M. Staunton, Triumph of the Optimists: 101 Years of Investment Returns (Princeton, NJ: Princeton University Press, 2002), with updates provided by the authors. Reprinted by permission of Princeton University Press.

10 5 0 5 10 15 1900 1912 1924 1936 1948 1960 1972 1984 1996 2006

F I G U R E 4.8

12

Average rates of inflation in 17 countries from 1900–2006.

Average inflation, %

10 8 6 4 2

UK So Irel ut an G h d er Af m r an Av ica y er (e x 19 age 22 / Be 23) lg iu m Sp ai Fr n an ce Ja pa n Ita ly

N

Sw

itz e et rlan he d rla nd s US Ca A na Sw da ed N en or w Au ay st r De alia nm ar k

0

The average inflation rate in the United States between 1900 and 2006 was 3.1%. As you can see from Figure 4.8, among major economies the U.S. has been almost top of the class in holding inflation in check. Those countries that have been torn by war have generally experienced much higher inflation. For example, in Italy and Japan inflation since 1900 has averaged about 11% a year. Economists sometimes talk about current, or nominal, dollars versus constant, or real, dollars. For example, the nominal cash flow from your one-year bond is $1,100. But suppose prices of goods rise over the year by 6%; then each dollar will buy you 6% fewer goods next year than it does today. So at the end of the year $1,100 will buy the same quantity of goods as 1,100兾1.06 $1,037.74 today. The nominal payoff on the bond is $1,100, but the real payoff is only $1,037.74.

Source: E. Dimson, P. R. Marsh, and M. Staunton, Triumph of the Optimists: 101 Years of Investment Returns (Princeton, NJ: Princeton University Press, 2002), with updates provided by the authors. Reprinted by permission of Princeton University Press.

73


74


PART ONE Value The general formula for converting nominal cash flows at a future period t to real cash flows is Real cash flowt ⫽

nominal cash flowt 11 ⫹ inflation rate2 t

For example, if you were to invest that $1,000 in a 20-year bond with a 10% coupon, then your final year’s payment would still be $1,100, but with an inflation rate of 6% a year, the real value of that payoff would be 1,100兾1.0620 ⫽ $342.99. When a bond dealer says that your bond yields 10%, she is quoting a nominal interest rate. That rate tells you how rapidly your money will grow: Invest Current Dollars

Receive Period-1 Dollars →

1,000

1,100

Result 10% nominal rate of return

However, with an inflation rate of 6%, you are only 3.774% better off at the end of the year than at the start: Invest Current Dollars 1,000

Expected Real Value of Period-1 Dollars →

1,037.74

Result 3.774% expected real rate of return

Thus, we could say, “The bank account offers a 10% nominal rate of return,” or “It offers a 3.774% expected real rate of return.” The formula for calculating the real rate of return is 1 ⫹ rreal ⫽ 11 ⫹ rnominal 2兾11 ⫹ inflation rate2 In our example,13 1.03774 ⫽ 1.10兾1.06

Indexed Bonds and the Real Rate of Interest Most bonds are like our U.S. Treasury bonds; they promise you a fixed nominal rate of interest. The real interest rate that you receive is uncertain and depends on inflation. If the inflation rate turns out to be higher than you expected, the real return on your bonds will be lower than forecasted. You can nail down a real return; you do so by buying an indexed bond whose payments are linked to inflation. Indexed bonds have been around in many other countries for decades, but they were almost unknown in the United States until 1997 when the U.S. Treasury began to issue inflation-indexed bonds known as TIPS (Treasury Inflation-Protected Securities).14 A common rule of thumb states that rreal ⫽ rnominal − inflation rate. In our example this gives rreal ⫽ .10 ⫺ .06 ⫽ .04, or 4%. This is not a bad approximation to the true real interest rate of 3.774%. But there are countries where inflation is large (sometimes 100% or more). In such cases it pays to use the full formula. 14 Indexed bonds were not completely unknown in the United States before 1997. For example, in 1780 American Revolutionary soldiers were compensated with indexed bonds that paid the value of “five bushels of corn, 68 pounds and four-seventh parts of a pound of beef, ten pounds of sheep’s wool, and sixteen pounds of sole leather.” 13



CHAPTER 4 Valuing Bonds The real cash flows on TIPS are fixed, but the nominal cash flows (interest and principal) increase as the Consumer Price Index increases. For example, suppose that the U.S. Treasury issues 3% 20-year TIPS at a price of 100. If during the first year the Consumer Price Index rises by (say) 10%, then the coupon payment on the bond would increase by 10% to (1.1 3) 3.3%. And the final payment of principal would also increase in the same proportion to (1.1 100) 110%. Thus, an investor who buys the bond at the issue price and holds it to maturity can be assured of a real yield of 3%. As we write this in the summer of 2006, long-term TIPS offer a yield of about 2.3%. This yield is a real yield: It measures the extra goods your investment will allow you to buy. The 2.3% yield on TIPS is about 2.8% less than the nominal yield on nominal Treasury bonds. If the annual inflation rate proves to be higher than 2.8%, you will earn a higher return by holding long-term TIPS; if the inflation rate is less than 2.8%, you will be better off with nominal bonds. The real yield that investors demand depends on people’s willingness to save (the supply of capital)15 and the opportunities for productive investment by governments and businesses (the demand for capital). For example, suppose that investment opportunities generally improve. Firms have more good projects, so they are willing to invest more than previously at the current interest rate. Therefore, the rate has to rise to induce individuals to save the additional amount that firms want to invest.16 Conversely, if investment opportunities deteriorate, there will be a fall in the real interest rate. This implies that the required real rate of interest depends on real phenomena. A high aggregate willingness to save may be associated with high aggregate wealth (because wealthy people usually save more), an uneven distribution of wealth (an even distribution would mean fewer rich people who do most of the saving), and a high proportion of middle-aged people (the young don’t need to save and the old don’t want to —“You can’t take it with you”). Correspondingly a high propensity to invest may be associated with a high level of industrial activity or major technological advances. Real interest rates do change but they do so gradually. We can see this by looking at the U.K. where the government has issued indexed bonds since 1982. The (maroon) line in Figure 4.9 shows that the (real) yield on these bonds has fluctuated within a relatively narrow range, while the yield on nominal government bonds (the blue line) has declined dramatically.

Inflation and Nominal Interest Rates How does the inflation outlook affect the nominal rate of interest? Here is how the economist Irving Fisher answered the question. Suppose that consumers are equally happy with 100 apples today or 105 apples in a year’s time. In this case the real or “apple” interest rate is 5%. If the price of apples is constant at (say) $1 each, then we will be equally happy to receive $100 today or $105 at the end of the year. That extra $5 will allow us to buy 5% more apples at the end of the year than we could buy today. 15

Some of this saving may be done indirectly. For example, if you hold 100 shares of IBM stock, and IBM plows back $1.00 a share, IBM is saving $100 on your behalf. The government may also oblige you to save by raising taxes to invest in roads, hospitals, and so on. 16 We assume that investors save more as interest rates rise. It doesn’t have to be that way; here is an extreme example of how a higher interest rate could mean less saving. Suppose that 20 years hence you will need $50,000 at current prices for your children’s college expenses. How much will you have to set aside today to cover this obligation? The answer is the present value of a real expenditure of $50,000 after 20 years, or 50,000兾(1 real interest rate)20. The higher the real interest rate, the lower the present value and the less you have to set aside.

75


PART ONE Value

F I G U R E 4.9

14 12 Interest rate, %

The maroon line shows the real yield on long-term indexed bonds issued by the U.K. government. The blue line shows the yield on long-term nominal bonds. Notice that the real yield has been much more stable than the nominal yield.

10

10-year nominal interest rate

8 6 4 2

10-year real interest rate

Jan-06

Jan-04

Jan-02

Jan-00

Jan-98

Jan-96

Jan-94

Jan-92

Jan-90

Jan-88

Jan-84

0 Jan-86

76


But suppose now that the apple price is expected to increase by 10% to $1.10 each. In that case we would not be happy to give up $100 today for the promise of $105 next year. To buy 105 apples in a year’s time, we will need to receive 1.10 $105 $115.50. In other words, the nominal rate of interest must increase by the expected rate of inflation to 15.50%. This is Fisher’s theory: A change in the expected inflation rate will cause the same proportionate change in the nominal interest rate; it has no effect on the required real interest rate. The formula relating the nominal interest rate and expected inflation is 1 rnominal 11 rreal 211 i2 where rreal is the real interest rate that consumers require and i is the expected inflation rate. In our example, the prospect of inflation causes 1 rnominal to rise to 1.05 1.10 1.155. Nominal interest rates cannot be negative; if they were, everyone would prefer to hold cash, which pays zero interest. But what about real rates? For example, is it possible for the money rate of interest to be 5% and the expected rate of inflation to be 10%, thus giving a negative real interest rate? If this happens, you may be able to make money in the following way: You borrow $100 at an interest rate of 5% and use the money to buy apples. You store the apples and sell them at the end of the year for $110, which leaves you enough to pay off your loan plus $5 for yourself. Since easy ways to make money are rare, we can conclude that if it doesn’t cost anything to store goods, the money rate of interest can’t be less than the expected rise in prices. But many goods are even more expensive to store than apples, and others can’t be stored at all (you can’t store haircuts, for example). For these goods the money interest rate can be less than the expected price rise.

How Well Does Fisher’s Theory Explain Interest Rates? Not all economists would agree with Fisher that the real rate of interest is unaffected by the inflation rate. For example, if changes in prices are associated with changes in the level of industrial activity, then in inflationary conditions I might want more or less than 105 apples in a year’s time to compensate me for the loss of 100 today. We wish we could show you the past behavior of interest rates and expected inflation. Instead we have done the next best thing and plotted in Figure 4.10 the




F I G U R E 4.10

(a) U.S. 20

The return on Treasury bills and the rate of inflation in the U.S., Japan, and Germany, 1953–2006.

15

Source: E. Dimson, P. R. Marsh, and M. Staunton, Triumph of the Optimists: 101 Years of Investment Returns (Princeton, NJ.: Princeton University Press, 2002), with updates provided by the authors. Reprinted by permission of Princeton University Press.

Treasury bill return

%

10 5

Inflation

0

2001

2005

2001

2005

1997

1993

1989

1985

1981

1977

1973

1969

1965

1961

1957

1953

5

Year (b) Japan 25 20

%

15 10

Treasury bill return

5

Inflation

0 1997

1993

1989

1985

1981

1977

1973

1969

1965

1961

1957

1953

5

Year (c) Germany 15 Treasury bill return

5 Inflation

0

Year

2005

2001

1997

1993

1989

1985

1981

1977

1973

1969

1965

1961

1957

5 1953

%

10

77


78


PART ONE Value return on Treasury bills (short-term government debt) against actual inflation for the U.S., Japan, and Germany. Notice that since 1953 the return on Treasury bills has generally been a little above the rate of inflation. Investors in each country earned an average real return of between 1% and 2% during this period. Look now at the relationship between the rate of inflation and the Treasury bill rate. Figure 4.10 shows that investors have for the most part demanded a higher rate of interest when inflation has been high.17 So it looks as if Fisher’s theory provides at least a useful rule of thumb for financial managers. If the expected inflation rate changes, it is a good bet that there will be a corresponding change in the interest rate.

Visit us at www.mhhe.com/bma9e

17

SUMMARY

The principal exception occurred in Japan in 1973–74, when rapid monetary growth was followed by the oil crisis.

Bonds are simply long-term loans. If you own a bond, you are entitled to a regular interest (or coupon) payment and at maturity you get back the bond’s face value (or principal). In the United States bond interest is generally paid every six months but in other countries the interest may be paid annually. The value of any bond is equal to the cash payments discounted at the spot rates of interest. For example, the value of a 10-year bond with a 5% coupon paid annually equals PV 1% of face value2

5 5 105 ... 2 1 r1 11 r2 2 11 r10 2 10

Bond dealers commonly use the yield to maturity on a bond to summarize its prospective return. To calculate the yield to maturity on the 10-year 5s, you need to solve for y in the following equation: Bond price

5 5 105 ... 1y 11 y2 2 11 y2 10

The yield to maturity, y, is like an average of the spot interest rates, r1, r2, etc. Like most averages it can be a useful summary measure, but it can also hide a lot of useful information. If you want to dig deeper, we suggest that you refer to yields on stripped bonds as measures of the spot rates of interest.18 A bond’s maturity tells you when you receive your final payment, but it is also useful to know the average time to each payment. This is called the bond’s duration. Duration is important because there is a direct relationship between the duration of a bond and its volatility. A change in interest rates has a greater effect on the price of a bond with a longer duration. The one-period spot rate, r1, may be very different from the two-period spot rate, r2. In other words, investors may want a different annual rate of return for lending for one year rather than for two years. Why is this? The expectations theory

18

In Chapter 27 we will explain how bond investors also use interest rates on swaps to measure the term structure.





says that bonds are priced so that an investor who holds a succession of short bonds can expect the same return as another investor who holds a long bond. The expectations theory predicts that r2 will exceed r1 only if next period’s one-year interest rate is expected to rise. The expectations theory cannot be a complete explanation of term structure if investors are worried about risk. Long bonds may be a safe haven for investors with long-term fixed liabilities. But other investors may not like the extra volatility of long-term bonds or may be concerned that a sudden burst of inflation may largely wipe out the real value of these bonds. Such investors will be prepared to hold long-term bonds only if they offer the compensation of a higher rate of interest. Bonds promise a fixed money payment but the real interest rate that they provide depends on inflation. The real interest rate that investors require is determined by the demand for capital and the supply of savings. The demand for capital comes from governments and firms that want to invest in new projects. The supply of savings comes from individuals who are willing to consume tomorrow rather than today. The equilibrium interest rate is the rate that produces a balance between the demand and supply. The best-known theory about the effect of inflation on interest rates was suggested by Irving Fisher. He argued that the nominal, or money, rate of interest is equal to the required real rate plus the expected (and unrelated) rate of inflation. If the expected inflation rate increases by 1%, so too will the money rate of interest. During the past 50 years Fisher’s simple theory has not done a bad job of explaining changes in short-term interest rates in the United States, Japan, and Germany. When you buy a U.S. Treasury bond, you can be confident that you will get your money back. When you lend to a company, you face the risk that it will go belly-up and will not be able to repay its bonds. Therefore, companies need to compensate investors with the promise of a higher rate of interest. In this chapter we sidestepped the issue of default risk, but in Chapter 24 we will explain how investors measure the probability of default and factor it into the price of the firm’s bonds.

79

A general text on debt markets is: S. Sundaresan, Fixed Income Markets and Their Derivatives, 2nd ed. (Cincinnati, OH: SouthWestern Publishing, 2001). Schaefer’s paper is a good review of duration and how it is used to hedge fixed liabilities: S. M. Schaefer, “Immunisation and Duration: A Review of Theory, Performance and Application,” in J. M. Stern and D. H. Chew, Jr., The Revolution in Corporate Finance (Oxford: Basil Blackwell, 1986).

FURTHER READING

Log on to www.smartmoney.com and find The Living Yield Curve, which provides a moving picture of the term structure. How does today’s yield curve compare with the average? Do short-term interest rates move more or less than long-term rates? Why do you think this is the case?

WEB PROJECT


80


PART ONE Value

CONCEPT REVIEW QUESTIONS

1. Fill in the blanks: The market value of a bond is the present value of its ________ and ________ payments. (page 60) 2. What is meant by a bond’s yield to maturity and how is it calculated? (page 61) 3. If interest rates rise, do bond prices rise or fall? (page 63) For a complete listing of your chapter Concept Review Questions, please visit us at www.mhhe.com/bma9e

1. A 10-year bond is issued with a face value of $1,000, paying interest of $60 a year. If market yields increase shortly after the T-bond is issued, what happens to the bond’s

QUIZ

a. Coupon rate? b. Price?


c. Yield to maturity? 2. A bond with a coupon rate of 8% is selling at a price of 97%. Is the bond’s yield to maturity more or less than 8%?

x


x


3. In August 2006 Treasury 12.5s of 2014 offered a semiannually compounded yield of 8.669%. Recognizing that coupons are paid semiannually, calculate the bond’s price. 4. Here are the prices of three bonds with 10-year maturities: Bond Coupon (%)

Price (%)

2 4 8

81.62 98.39 133.42

If coupons are paid annually, which bond offered the highest yield to maturity? Which had the lowest? Which bonds had the longest and shortest durations? 5. a. What is the formula for the value of a two-year, 5% bond in terms of spot rates? b. What is the formula for its value in terms of yield to maturity? c. If the two-year spot rate is higher than the one-year rate, is the yield to maturity greater or less than the two-year spot rate? d. In each of the following sentences choose the correct term from within the parentheses: • “The (yield-to-maturity兾spot-rate) formula discounts all cash flows from one bond at the same rate even though they occur at different points in time.” • “The (yield-to-maturity兾spot-rate) formula discounts all cash flows received at the same point in time at the same rate even though the cash flows may come from different bonds.” 6. Construct some simple examples to illustrate your answers to the following: a. If interest rates rise, do bond prices rise or fall? b. If the bond yield is greater than the coupon, is the price of the bond greater or less than 100? c. If the price of a bond exceeds 100, is the yield greater or less than the coupon? d. Do high-coupon bonds sell at higher or lower prices than low-coupon bonds? e. If interest rates change, does the price of high-coupon bonds change proportionately more than that of low-coupon bonds?

bre05108_ch04_059-084.indd Page 81 7/27/07 3:48:39 AM epg1



81

7. The following table shows the prices of a sample of U.S. Treasury strips in August 2006. Each strip makes a single payment of $1,000 at maturity. a. Calculate the annually compounded, spot interest rate for each year. b. Is the term structure upward- or downward-sloping or flat? c. Would you expect the yield on a coupon bond maturing in August 2010 to be higher or lower than the yield on the 2010 strip? d. Calculate the annually compounded, one-year forward rate of interest for August 2008. Now do the same for August 2009. Price (%)

August 2007 August 2008 August 2009 August 2010

95.53 91.07 86.2 81.08

8. a. An 8%, five-year bond yields 6%. If the yield remains unchanged, what will be its price one year hence? Assume annual coupon payments. b. What is the total return to an investor who held the bond over this year? c. What can you deduce about the relationship between the bond return over a particular period and the yields to maturity at the start and end of that period? 9. True or false? Explain. a. Longer-maturity bonds necessarily have longer durations. b. The longer a bond’s duration, the lower its volatility. c. Other things equal, the lower the bond coupon, the higher its volatility. d. If interest rates rise, bond durations rise also. 10. Calculate the durations and volatilities of securities A, B, and C. Their cash flows are shown below. The interest rate is 8%.

A B C

Period 1

Period 2

Period 3

40 20 10

40 20 10

40 120 110

11. a. Suppose that the one-year spot rate of interest at time 0 is 1% and the two-year spot rate is 3%. What is the forward rate of interest for year 2? b. What does the expectations theory of term structure say about the relationship between the forward rate and the one-year spot rate at time 1? c. Over a very long period of time, the term structure in the United States has been on average upward-sloping. Is this evidence for or against the expectations theory? d. If longer-term bonds are more risky than short-term bonds, what can you deduce about the relationship between the forward rate and the one-year spot rate at time 1? e. If you have to meet long-term liabilities (college tuition for your children, for example), is it safer to invest in long- or short-term bonds? Assume inflation is predictable. f. If inflation is very uncertain and you have to meet long-term real liabilities, is it safer to invest in long- or short-term bonds?

x



Maturity


82


PART ONE Value

PRACTICE QUESTIONS

12. A 10-year German government bond (bund) has a face value of €100 and an annual coupon rate of 5%. Assume that the interest rate (in euros) is equal to 6% per year. What is the bond’s PV? 13. Look again at Problem 12. Suppose that the German bund paid interest semiannually like a U.S. bond. (The bond would pay .025 100 €2.5 every six months.) What is the PV in this case?

x


14. A 10-year U.S. Treasury bond with a face value of $10,000 pays a coupon of 5.5% (2.75% of face value every six months). The semiannually compounded interest rate is 5.2% (a six-month discount rate of 5.2兾2 2.6%). a. What is the present value of the bond? b. Generate a graph or table showing how the bond’s present value changes for semiannually compounded interest rates between 1% and 15%.

x

e cel



x


x


x


15. Suppose that five-year government bonds are selling on a yield of 4%. Value a five-year bond with a 6% coupon. Start by assuming that the bond is issued by a continental European government and makes annual coupon payments. Then rework your answer assuming that the bond is issued by the U.S. Treasury, that the bond pays semiannual coupons, and the yield refers to a semiannually compounded rate. 16. Refer again to Problem 15. How would the bond value in each case change if interest rates fall to 3%? 17. A six-year government bond makes annual coupon payments of 5% and offers a yield of 3% annually compounded. Suppose that one year later the bond still yields 3%. What return has the bondholder earned over the 12-month period? Now suppose that the bond yields 2% at the end of the year. What return would the bondholder earn in this case? 18. A 6% six-year bond yields 12% and a 10% six-year bond yields 8%. Calculate the sixyear spot rate. Assume annual coupon payments. Hint: What would be your cash flows if you bought 1.2 10% bonds? 19. Is the yield on high-coupon bonds more likely to be higher than that on low-coupon bonds when the term structure is upward-sloping or when it is downward-sloping? Explain. 20. The one-year spot rate is r1 6%, and the forward rate for a one-year loan maturing in year 2 is f2 6.4%. Similarly, f3 7.1%, f4 7.3%, and f5 8.2%. What are the spot rates r2, r3, r4, and r5? If the expectations hypothesis holds, what can you say about expected future interest rates? 21. Suppose your company will receive $100 million at t 4 but must make a $107 million payment at t 5. Assume the spot and forward rates from Problem 20. Show how the company can lock in the interest rate at which it will invest at t 4. Will the $100 million, invested at this locked-in rate, be sufficient to cover the $107 million liability?

x


22. Use the rates from Problem 20 one more time. Consider the following bonds, each with a five-year maturity. Calculate the yield to maturity for each. Which is the better investment (or are they equally attractive)? Each has $1,000 face value and pays coupons annually. Coupon (%)

Price (%)

5 7 12

92.07 100.31 120.92




83

x

e cel

23. You have estimated spot rates as follows:


Year

Spot Rate

1 2 3 4 5

r1 5.00% r2 5.40% r3 5.70% r4 5.90% r5 6.00%

a. What are the discount factors for each date (that is, the present value of $1 paid in year t)? b. What are the forward rates for each period? c. Calculate the PV of the following bonds assuming annual coupons: i. 5%, two-year bond.


ii. 5%, five-year bond. iii. 10%, five-year bond. d. Explain intuitively why the yield to maturity on the 10% bond is less than that on the 5% bond. e. What should be the yield to maturity on a five-year zero-coupon bond? f. Show that the correct yield to maturity on a five-year annuity is 5.75%. g. Explain intuitively why the yield on the five-year bonds described in part (c) must lie between the yield on a five-year zero-coupon bond and a five-year annuity. 24. Look at the spot interest rates shown in Problem 23. Suppose that someone told you that the six-year spot interest rate was 4.80%. Why would you not believe him? How could you make money if he was right? What is the minimum sensible value for the six-year spot rate? 25. Look again at the spot interest rates shown in Problem 23. What can you deduce about the one-year spot interest rate in four years if a. The expectations theory of term structure is right? b. Investing in long-term bonds carries additional risks? 26. Look up prices of 10 U.S. Treasury bonds with different coupons and different maturities. Calculate how their prices would change if their yields to maturity increased by 1 percentage point. Are long- or short-term bonds most affected by the change in yields? Are high- or low-coupon bonds most affected? 27. In Section 4.2 we stated that the duration of the three-year 4% bond was 2.884 years. Construct a table like Table 4.1 to show that this is so. 28. Find the “live” spreadsheet for Table 4.1 on this book’s Web site, www.mhhe.com/ bma9e. Show how duration and volatility change if (a) the bond’s coupon is 8% of face value and (b) the bond’s yield is 6%. Explain your finding. 29. The formula for the duration of a perpetual bond that makes an equal payment each year in perpetuity is (1 yield)兾yield. If bonds yield 5%, which has the longer duration—a perpetual bond or a 15-year zero-coupon bond? What if the yield is 10%? 30. You have just been fired as CEO. As consolation the board of directors gives you a five-year consulting contract at $150,000 per year. What is the duration of this contract if your personal borrowing rate is 9%? Use duration to calculate the change in the contract’s present value for a .5% increase in your borrowing rate.

x


x


x


x



84

PART ONE Value

CHALLENGE QUESTIONS

x


x




31. Write a spreadsheet program to construct a series of bond tables that show the present value of a bond given the coupon rate, maturity, and yield to maturity. Assume that coupon payments are semiannual and yields are compounded semiannually. 32. Find the arbitrage opportunity (opportunities?). Assume for simplicity that coupons are paid annually. In each case the face value of the bond is $1,000. Bond

Maturity (years)

Coupon ($)

Price ($)

A B C D E F G

3 4 4 4 3 3 2

zero 50 120 100 140 70 zero

751.30 842.30 1,065.28 980.57 1,120.12 1,001.62 834.00

33. The duration of a bond that makes an equal payment each year in perpetuity is (1 yield)兾yield. Prove it. 34. What is the duration of a common stock whose dividends are expected to grow at a constant rate in perpetuity?

x


35. a. What spot and forward rates are embedded in the following Treasury bonds? The price of one-year (zero-coupon) Treasury bills is 93.46%. Assume for simplicity that bonds make only annual payments. Hint: Can you devise a mixture of long and short positions in these bonds that gives a cash payoff only in year 2? In year 3? Coupon (%)

Maturity (years)

Price (%)

4 8

2 3

94.92 103.64

b. A three-year bond with a 4% coupon is selling at 95.00%. Is there a profit opportunity here? If so, how would you take advantage of it?