Brealey−Meyers: Principles of Corporate Finance, Seventh Edition VII. Debt Financing 24. Valuing Debt © The McGraw−Hill Companies, 2003 CHAPTER TWENTY-FOUR 666 VALUING DEBT Brealey−Meyers: Principles of Corporate Finance, Seventh Edition VII. Debt Financing 24. Valuing Debt © The McGraw−Hill Companies, 2003 HOW DO YOU estimate the present value of a company’s bonds? The answer is simple: You take the cash flows and discount them at the opportunity cost of capital. Therefore, if a bond produces cash flows of C dollars per year for N years and is then repaid at its face value ($1,000), the present value is where are the appropriate discount rates for the cash flows to be received by the bond owners in periods That is correct as far as it goes but it does not tell us anything about what determines the discount rates. For example, • In 1945 U.S. Treasury bills offered a return of .4 percent: At their 1981 peak they offered a re- turn of over 17 percent. Why does the same security offer radically different yields at different times? • In mid-2001 the U.S. Treasury could borrow for one year at an interest rate of 3.4 percent, but it had to pay nearly 6 percent for a 30-year loan. Why do bonds maturing at different dates offer dif- ferent rates of interest? In other words, why is there a term structure of interest rates? • In mid-2001 the United States government could issue long-term bonds at a rate of nearly 6 percent. But even the most blue-chip corporate issuers had to pay at least 50 basis points (.5 percent) more on their long-term borrowing. What explains the premium that firms have to pay? These questions lead to deep issues that will keep economists simmering for years. But we can give general answers and at the same time present some fundamental ideas. Why should the financial manager care about these ideas? Who needs to know how bonds are priced as long as the bond market is active and efficient? Efficient markets protect the ignorant trader. If it is necessary to know whether the price is right for a proposed bond issue, you can check the prices of similar bonds. There is no need to worry about the historical behavior of interest rates, about the term structure, or about the other issues discussed in this chapter. We do not believe that ignorance is desirable even when it is harmless. At least you ought to be able to read the bond tables in The Wall Street Journal and talk to investment bankers about the prices of recently issued bonds. More important, you will encounter many problems of bond pricing where there are no similar instruments already traded. How do you evaluate a private placement with a custom-tailored repayment schedule? How about financial leases? In Chapter 26 we will see that they are essentially debt contracts, but often extremely complicated ones, for which traded bonds are not close substitutes. Many companies, notably banks and insurance firms, are exposed to the risk of interest rate fluctuations. To control their exposure, these companies need to understand how interest rates change. 1 You will find that the terms, concepts, and facts presented in this chapter are essential to the analysis of these and other practical problems. We start the chapter with our first question: Why does the general level of interest rates change over time? Next we turn to the relationship between short- and long-term interest rates. We consider three issues: • Each period’s cash flow on a bond potentially needs to be discounted at a different interest rate, but bond investors often calculate the yield to maturity as a summary measure of the interest rate on the bond. We first explain how these measures are related. continued 1, 2, . . . , N. r 1 , r 2 , . . . , r N PV ϭ C 1 ϩ r 1 ϩ C 11 ϩ r 2 2 2 ϩ … ϩ C 11 ϩ r N 2 N ϩ $1,000 11 ϩ r N 2 N 667 1 We discuss in Chapter 27 how firms protect themselves against interest rate risk. Brealey−Meyers: Principles of Corporate Finance, Seventh Edition VII. Debt Financing 24. Valuing Debt © The McGraw−Hill Companies, 2003 Indexed Bonds and the Real Rate of Interest In Chapter 3 we drew the distinction between the real and nominal rate of interest. Most bonds promise a fixed nominal rate of interest. The real interest rate that you receive depends on the inflation rate. For example, if a one-year bond promises you a return of 10 percent and the expected inflation rate is 4 percent, the expected real return on your bond is , or 5.8 percent. Since future inflation rates are uncertain, the real return on a bond is also uncertain. For example, if in- flation turns out to be higher than the expected 4 percent, the real return will be lower than 5.8 percent. 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 coun- tries 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). 2 The real cash flows on TIPs are fixed, but the nominal cash flows (interest and principal) are increased as the Consumer Price Index increases. For example, sup- pose that the U.S. Treasury issues 3 percent 20-year TIPs at a price of 100. If during the first year the Consumer Price Index rises by (say) 10 percent, then the coupon payment on the bond would be increased by 10 percent to percent. And the final payment of principal would also be increased in the same proportion to percent. 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 percent. As we write this in the summer of 2001, long-term TIPs offer a yield of 3.46 per- cent. This yield is a real yield: It measures how much extra goods your investment would allow you to buy. The 3.46 percent yield on TIPs was about 2.3 percent less than on nominal Treasury bonds. If the annual inflation rate proves to be higher than 2.3 percent, you will earn a higher return by holding long-term TIPs; if the in- flation rate is lower than 2.3 percent, the reverse will be true. What determines the real interest rate that investors demand? The classical econ- omist’s answer to this question is summed up in the title of Irving Fisher’s great book: The Theory of Interest: As Determined by Impatience to Spend Income and Opportu- nity to Invest It. 3 The real interest rate, according to Fisher, is the price which equates 11.1 ϫ 1002ϭ 110 11.1 ϫ 32ϭ 3.3 1.10/1.04 Ϫ 1 ϭ .058 668 PART VII Debt Financing • Second, we show why a change in interest rates has a greater impact on the price of long-term loans than on short-term loans. • Finally, we look at some theories that explain why short- and long-term interest rates differ. To close the chapter we shift the focus to corporate bonds and examine the risk of default and its ef- fect on bond prices. 24.1 REAL AND NOMINAL RATES OF INTEREST 2 In 1988 Franklin Savings Association had issued a 20-year bond whose interest (but not principal) was tied to the rate of inflation. Since then a trickle of companies has also issued indexed bonds. 3 August M. Kelley, New York, 1965; originally published in 1930. Brealey−Meyers: Principles of Corporate Finance, Seventh Edition VII. Debt Financing 24. Valuing Debt © The McGraw−Hill Companies, 2003 the supply and demand for capital. The supply depends on people’s willingness to save. 4 The demand depends on the opportunities for productive investment. For example, suppose that investment opportunities generally improve. Firms have more good projects, so they are willing to invest more than previously at any interest rate. Therefore, the rate has to rise to induce individuals to save the addi- tional amount that firms want to invest. 5 Conversely, if investment opportunities deteriorate, there will be a fall in the real interest rate. Fisher’s theory emphasizes that the required real rate of interest depends on real phenomena. A high aggregate willingness to save may be associated with high ag- gregate wealth (because wealthy people usually save more), an uneven distribu- tion 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 activ- ity or major technological advances. Real interest rates do change but they do so gradually. We can see this by look- ing at the UK, where the government has issued indexed bonds since 1982. The col- ored line in Figure 24.1 shows that the (real) yield on these bonds has fluctuated within a relatively narrow range, while the yield on nominal government bonds has declined dramatically. Inflation and Nominal Interest Rates Now let us see what Irving Fisher had to say about inflation and interest rates. Sup- pose 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 percent. Suppose also CHAPTER 24 Valuing Debt 669 4 Some of this saving is done indirectly. For example, if you hold 100 shares of GM stock, and GM re- tains earnings of $1 per share, GM is saving $100 on your behalf. 5 We assume that investors save more as interest rates rise. It doesn’t have to be that way; here is an ex- ample 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 to- day to cover this obligation? The answer is the present value of a real expenditure of $50,000 after 20 years, or . The higher the real interest rate, the lower the present value and the less you have to set aside. 50,000/11 ϩ real interest rate2 20 Dec. 83 Dec. 84 Dec. 85 Dec. 86 Dec. 87 Dec. 88 Dec. 89 Dec. 90 Dec. 91 Dec. 92 Dec. 93 Dec. 94 Dec. 95 Dec. 96 Dec. 97 Dec. 98 Dec. 99 Dec. 00 0 2 4 6 8 10 12 14 Percent Real yield on UK indexed bonds Yield on UK nominal bonds FIGURE 24.1 The burgundy line shows the real yield on long-term indexed bonds issued by the UK government. The blue line shows the yield on UK government long-term nominal bonds. Notice that the real yield has been much more stable than the nominal yield. Brealey−Meyers: Principles of Corporate Finance, Seventh Edition VII. Debt Financing 24. Valuing Debt © The McGraw−Hill Companies, 2003 that I know the price of apples will increase over the year by 10 percent. Then I will part with $100 today if I am repaid $115 at the end of the year. That $115 is needed to buy me 5 percent more apples than I can get for my $100 today. In other words, the nominal, or “money,” rate of interest must equal the required real, or “apple,” rate plus the prospective rate of inflation. 6 A change of 1 percent in the expected in- flation rate produces a change of 1 percent in the nominal interest rate. That is Fisher’s theory: A change in the expected inflation rate will cause the same change in the nominal interest rate; it has no effect on the required real interest rate. 7 Nominal interest rates cannot be negative; if they were, everyone would prefer to hold cash, which pays zero interest. 8 But what about real rates? For example, is it possible for the money rate of interest to be 5 percent and the expected rate of in- flation to be 10 percent, 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 in- terest rate of 5 percent and you 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 unaf- fected 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 in- flation. Instead we have done the next best thing and plotted in Figure 24.2 the re- turn on U.S. Treasury bills against actual inflation. Notice that between 1926 and 1981 the return on Treasury bills was below the inflation rate about as often as it 670 PART VII Debt Financing 6 We oversimplify. If apples cost $1.00 apiece today and $1.10 next year, you need next year to buy 105 apples. The money rate of interest is 15.5 percent, not 15. Remember, the exact for- mula relating real and money rates is where i is the expected inflation rate. Thus In our example, the money rate should be When we said the money rate should be 15 percent, we ignored the cross-product term i . This is a common rule of thumb because the cross-product term is usually small. But there are countries where i is large (sometimes 100 percent or more). In such cases it pays to use the full formula. 7 The apple example was taken from R. Roll, “Interest Rates on Monetary Assets and Commodity Price Index Changes,” Journal of Finance 27 (May 1972), pp. 251–278. 8 There seems to be an exception to almost every statement. In late 1998 concern about the solvency of some Japanese banks led to a large volume of yen deposits with Western banks. Some of these banks charged their customers interest on these deposits; the nominal interest rate was negative. 1r real 2 r money ϭ .05 ϩ .10 ϩ .101.052ϭ .155 r money ϭ r real ϩ i ϩ i1r real 2 1 ϩ r money ϭ 11 ϩ r real 2 11 ϩ i2 1.10 ϫ 105 ϭ $115.50 Brealey−Meyers: Principles of Corporate Finance, Seventh Edition VII. Debt Financing 24. Valuing Debt © The McGraw−Hill Companies, 2003 was above. The average real interest rate during this period was a mere 0.1 percent. Since 1981 the return on bills has been significantly higher than the rate of infla- tion, so that investors have earned a positive real return on their savings. Fisher’s theory states that changes in anticipated inflation produce correspon- ding changes in the rate of interest. But Figure 24.2 offers little evidence of this in the 1930s and 1940s. During this period, the return on Treasury bills scarcely changed even though the inflation rate fluctuated sharply. Either these changes in inflation were unanticipated or Fisher’s theory was wrong. Since the early 1950s, there appears to have been a closer relationship between interest rates and infla- tion in the United States. 9 Thus, for today’s financial managers Fisher’s theory pro- vides a useful rule of thumb. If the expected inflation rate changes, it is a good bet that there will be a corresponding change in the interest rate. CHAPTER 24 Valuing Debt 671 1926 1931 1936 1941 1946 1951 1956 1961 1966 1971 1976 1981 1986 1991 1996 -15 -10 -5 0 Year 5 10 15 20 Percent Inflation Treasury bill return FIGURE 24.2 The return on U.S. Treasury bills and the rate of inflation, 1926–2000. Source: Ibbotson Associates, Inc., Chicago, 2001. 9 This probably reflects government policy, which before 1951 stabilized nominal interest rates. The 1951 “accord” between the Treasury and the Federal Reserve System permitted more flexible nominal inter- est rates after 1951. 24.2 TERM STRUCTURE AND YIELDS TO MATURITY We turn now to the relationship between short- and long-term rates of interest. Suppose that we have a simple loan that pays $1 at time 1. The present value of this loan is Thus we discount the cash flow at , the rate appropriate for a one-period loan. This rate, which is fixed today, 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 ϭ 1 1 ϩ r 1 ϩ 1 11 ϩ r 2 2 2 r 1 PV ϭ 1 1 ϩ r 1 Brealey−Meyers: Principles of Corporate Finance, Seventh Edition VII. Debt Financing 24. Valuing Debt © The McGraw−Hill Companies, 2003 Thus 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 , etc., is one way of expressing the term structure of interest rates. Yield to Maturity Rather than discounting each of the payments at a different rate of interest, we could find a single rate of discount that would produce the same present value. Such a rate is known as the yield to maturity, though it is in fact no more than our old acquain- tance, the internal rate of return (IRR), masquerading under another name. If we call the yield to maturity y, we can write the present value of the two-year loan as All you need to calculate y is the price of a bond, its annual payment, and its ma- turity. You can then rapidly work out the yield with the aid of a preprogrammed calculator. The yield to maturity is unambiguous and easy to calculate. It is also the stock- in-trade of any bond dealer. By now, however, you should have learned to treat any internal rate of return with suspicion. 10 The more closely we examine the yield to maturity, the less informative it is seen to be. Here is an example. Example. It is 2003. You are contemplating an investment in U.S. Treasury bonds and come across the following quotations for two bonds: 11 PV ϭ 1 1 ϩ y ϩ 1 11 ϩ y2 2 r 1 , r 2 672 PART VII Debt Financing 10 See Section 5.3. 11 The quoted bond price is known as the flat (or clean) price. The price that the bond buyer pays (some- times called the 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 date. You need to use the flat price to calculate yields to maturity. Yield to Maturity Bond Price (IRR) 5s of ‘08 85.21% 8.78% 10s of ‘08 105.43 8.62 The phrase “5s of ‘08” refers to a bond maturing in 2008, paying annual interest of 5 percent of the bond’s face value. The interest payment is called the coupon payment. In continental Europe coupons are usually paid annually; in the United States they are usually paid every six months, so the 5s of ‘08 would pay 2.5 percent of face value every six months. To simplify the arithmetic, we will pretend throughout this chap- ter that all coupon payments are annual. When the bonds mature in 2008, bond- holders receive the bond’s face value in addition to the final interest payment. The price of each bond is quoted as a percent of face value. Therefore, if face value is $1,000, you would have to pay $852.11 to buy the bond and your yield would be 8.78 percent. Letting 2003 be , 2004 be , etc., we have the fol- lowing discounted-cash-flow calculation: t ϭ 1t ϭ 0 Cash Flows Bond C 0 C 1 C 2 C 3 C 4 C 5 Yield 5s of ‘08 Ϫ852.11 ϩ50 ϩ50 ϩ50 ϩ50 ϩ1,050 8.78% 10s of ‘08 Ϫ1,054.29 ϩ100 ϩ100 ϩ100 ϩ100 ϩ1,100 8.62 Brealey−Meyers: Principles of Corporate Finance, Seventh Edition VII. Debt Financing 24. Valuing Debt © The McGraw−Hill Companies, 2003 Although the two bonds mature at the same date, they presumably were issued at different times—the 5s when interest rates were low and the 10s when interest rates were high. Are the 5s of ‘08 a better buy? Is the market making a mistake by pricing these two issues at different yields? The only way you will know for sure is to calculate the bonds’ present values by using spot rates of interest: for 2004, for 2005, etc. This is done in Table 24.1. The important assumption in Table 24.1 is that long-term interest rates are higher than short-term interest rates. We have assumed that the one-year interest rate is , the two-year rate is , and so on. When each year’s cash flow is discounted at the rate appropriate to that year, we see that each bond’s present value is exactly equal to the quoted price. Thus each bond is 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 identi- cal maturity dates, the 5s provide a greater proportion of their cash flows in 2008. In this sense the 5s are a longer-term investment than the 10s. Their higher yield to maturity just reflects the fact that long-term interest rates are higher than short- term rates. Notice why the yield to maturity can be misleading. When the yield is calculated, the same rate is used to discount all payments on the bond. But in our example bond- holders actually demanded different rates of return ( , etc.) for cash flows that oc- curred at different times. Since the cash flows on the two bonds were not identical, the bonds had different yields to maturity. Therefore, the yield to maturity on the 5s of ‘08 offered only a rough guide to the appropriate yield on the 10s of ‘08. 12 Measuring the Term Structure Financial managers who just want a quick, summary measure of interest rates look in the financial press at the yields to maturity on government bonds. Thus managers will make broad generalizations such as “If we borrow money today, we will have to pay an interest rate of 8 percent.” But if you wish to understand why different r 1 , r 2 r 2 ϭ .06r 1 ϭ .05 r 2 r 1 CHAPTER 24 Valuing Debt 673 Present Value Calculations 5s of ‘08 10s of ‘08 Interest Period Rate C t PV at r t C t PV at r t t ϭ 1 r 1 ϭ .05 $ 50 $ 47.62 $ 100 $ 95.24 t ϭ 2 r 2 ϭ .06 50 44.50 100 89.00 t ϭ 3 r 3 ϭ .07 50 40.81 100 81.63 t ϭ 4 r 4 ϭ .08 50 36.75 100 73.50 t ϭ 5 r 5 ϭ .09 1,050 682.43 1,100 714.92 Totals $852.11 $1,054.29 TABLE 24.1 Calculating present value of two bonds when long-term interest rates are higher than short-term rates. 12 For a good analysis of the relationship between the yield to maturity and spot interest rates, see S. M. Schaefer, “The Problem with Redemption Yields,” Financial Analysts Journal 33 (July–August 1977), pp. 59–67. Brealey−Meyers: Principles of Corporate Finance, Seventh Edition VII. Debt Financing 24. Valuing Debt © The McGraw−Hill Companies, 2003 bonds sell at different prices, you must dig deeper and look at the separate interest rates for one-year cash flows, for two-year cash flows, and so on. In other words, you must look at the spot rates of interest. To find the spot interest rate, you need the price of a bond that simply makes one future payment. Fortunately, such bonds do exist. They are known as stripped bonds or strips. Strips originated in 1982 when several investment bankers came up with a novel idea. They bought U.S. Treasury bonds and reissued their own separate mini-bonds, each of which made only one payment. The idea proved to be popu- lar with investors, who welcomed the opportunity to buy the mini-bonds rather than the complete package. If you’ve got a smart idea, you can be sure that others will soon clamber onto your bandwagon. It was therefore not long before the Trea- sury issued its own mini-bonds. 13 The prices of these bonds are shown each day in the daily press. For example, in the summer of 2001, a strip maturing in May 2021 cost $316.55 and 20 years later will give the investors a single payment of $1,000. Thus the 20-year spot rate was , or 5.92 percent. 14 In Figure 24.3 we have used the prices of strips with different maturities to plot the term structure of spot rates from 1 to 24 years. You can see that investors re- quired an interest rate of 3.4 percent from a bond that made a payment only at the end of one year and a rate of 5.8 percent from a bond that paid off only in year 2025. 11000/316.552 1/20 Ϫ 1 ϭ .0592 674 PART VII Debt Financing 13 The Treasury continued to auction coupon bonds in the normal way, but investors could exchange them at the Federal Reserve Bank for stripped bonds. 14 This is an annually compounded rate. The yields quoted by investment dealers are semiannually compounded rates. 0 2001 2006 2011 Year 2016 2021 2026 1 2 3 4 5 6 7 Spot rate, percent FIGURE 24.3 Spot rates on U.S. Treasury strips, June 2001. 24.3 HOW INTEREST RATE CHANGES AFFECT BOND PRICES Duration and Bond Volatility In Chapter 7 we reviewed the historical performance of different security classes. We showed that since 1926 long-term government bonds have provided a higher average return than short-term bills, but have also been more variable. The stan- Brealey−Meyers: Principles of Corporate Finance, Seventh Edition VII. Debt Financing 24. Valuing Debt © The McGraw−Hill Companies, 2003 dard deviation of annual returns on a portfolio of long-term bonds was 9.4 percent compared with a standard deviation of 3.2 percent for bills. Figure 24.4 illustrates why long-term bonds are more variable. Each line shows how the price of a 5-percent bond changes with the level of interest rates. You can see that the price of a longer-term bond is more sensitive to interest rate fluctua- tions than that of a shorter bond. But what do we mean by long-term and short-term bonds? It is obvious in the case of strips that make payments in only one year. However, a coupon bond that matures in year 10 makes payments in each of years 1 through 10. Therefore, it is somewhat misleading to describe the bond as a 10-year bond; the average time to each cash flow is less than 10 years. Consider the Treasury 6 7/8s of 2006. In mid-2001 these bonds had a present value of 108.57 percent of face value and yielded 4.9 percent. The third and fourth columns in Table 24.2 show where this present value comes from. Notice that the cash flow in year 5 accounts for only 77.5 percent of the bond’s value. The remain- ing 22.5 percent of the value comes from the earlier cash flows. CHAPTER 24 Valuing Debt 675 50 70 90 110 130 150 170 190 210 230 250 0 1 2 3 4 5 6 7 8 9 10 Interest rate, percent Bond price, percent 30-year 5 percent bond 10-year 5 percent bond 1-year 5 percent bond FIGURE 24.4 How bond prices change as interest rates change. Note that longer-term bonds are more sensitive to interest rate changes. Proportion of Total Value Proportion of Year C t PV(C t ) at 4.9% [PV(C t )/V] Total Value ؋ Time 1 68.75 65.54 0.060 0.060 2 68.75 62.48 0.058 0.115 3 68.75 59.56 0.055 0.165 4 68.75 56.78 0.052 0.209 5 1068.75 841.39 0.775 3.875 V ϭ 1085.74 1.000 Duration ϭ 4.424 years TABLE 24.2 The first four columns show that the cash flow in year 5 accounts for only 77.5 percent of the present value of the 6 7/8s of 2006. The final column shows how to calculate a weighted average of the time to each cash flow. This average is the bond’s duration. [...]... in the price of each bond would be exactly proportional to the bond’s duration For example, the price of a long-term bond with a duration of 20 years would always rise or fall twice as much as the price of a medium-term bond with a duration of 10 years However, Figure 24. 6 illustrates that short- and long-term interest rates do Brealey−Meyers: Principles of Corporate Finance, Seventh Edition 678 VII... value of a two-year, 5 percent 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? Brealey−Meyers: Principles of Corporate Finance, Seventh Edition VII Debt Financing © The McGraw−Hill Companies, 2003 24 Valuing Debt CHAPTER 24. .. advantage of the risk- Brealey−Meyers: Principles of Corporate Finance, Seventh Edition VII Debt Financing 24 Valuing Debt © The McGraw−Hill Companies, 2003 CHAPTER 24 Valuing Debt 699 neutral trick that we used to value options Pretend that investors are risk-neutral Now answer the following questions: a Suppose that the price of the short bond is 98 and the price of the medium is 83 What is the price of. .. of a risk-free loan and the common stock that would exactly replicate the payoffs from the option That allowed us to Brealey−Meyers: Principles of Corporate Finance, Seventh Edition VII Debt Financing © The McGraw−Hill Companies, 2003 24 Valuing Debt CHAPTER 24 Valuing Debt price the option given the price of the risk-free loan and the share Here we value a bond by constructing a portfolio of two or... around, you 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 ϩ r2 2 2 ϭ 11 ϩ r1 2 ϫ 11 ϩ f2 2 Brealey−Meyers: Principles of Corporate Finance, Seventh Edition VII Debt Financing 24 Valuing Debt © The McGraw−Hill Companies, 2003 CHAPTER 24 Valuing Debt (a ) The future value of $1 invested in a two-year loan Period 0 Period 2 (1 + r2)2... the one-year spot rate of interest at time 0 is 1 percent and the two-year spot rate is 3 percent What is the forward rate of interest for year 2? Visit us at www.mhhe.com/bm7e 6 The following table shows the prices of a sample of strips of UK gilts (government bonds) in December 1998 Each strip makes a single payment of £100 at maturity Brealey−Meyers: Principles of Corporate Finance, Seventh Edition. .. Fundamental Principles of Economic Theory, 2nd ed., Oxford University Press, Oxford, 1946 For a theoretical development, see R Roll, The Behavior of Interest Rates: An Application of the Efficient-Market Model to U.S Treasury Bills, Basic Books, Inc., New York, 1970 Brealey−Meyers: Principles of Corporate Finance, Seventh Edition VII Debt Financing 24 Valuing Debt © The McGraw−Hill Companies, 2003 CHAPTER 24. .. the value of a 10-year bond with a 5 percent coupon equals PV1percent of face value2 ϭ 5 105 5 ϩ ϩ … ϩ 2 1 ϩ r1 11 ϩ r2 2 11 ϩ r10 2 10 Brealey−Meyers: Principles of Corporate Finance, Seventh Edition VII Debt Financing © The McGraw−Hill Companies, 2003 24 Valuing Debt CHAPTER 24 Valuing Debt 693 Bond dealers generally look at the yield to maturity on a bond This is simply the internal rate of return... package of 10 sequential puts Brealey−Meyers: Principles of Corporate Finance, Seventh Edition VII Debt Financing © The McGraw−Hill Companies, 2003 24 Valuing Debt CHAPTER 24 Valuing Debt differentials of this magnitude simply in terms of default risk.29 So what is going on here? One possibility is that companies are paying too much for their debt, but it seems likely that the high yields on corporate. .. examples of models that use no-arbitrage conditions to model the term structure are O Vasicek, “An Equilibrium Characterization of the Term Structure,” Journal of Financial Economics 5 (November 1977), pp 177–188; and J C Cox, J E Ingersoll, and S A Ross, “A Theory of the Term Structure of Interest Rates,” Econometrica 53 (May 1985), pp 385–407 683 Brealey−Meyers: Principles of Corporate Finance, Seventh Edition . Brealey−Meyers: Principles of Corporate Finance, Seventh Edition VII. Debt Financing 24. Valuing Debt © The McGraw−Hill Companies, 2003 CHAPTER TWENTY-FOUR 666 VALUING DEBT Brealey−Meyers:. long-term government bonds have provided a higher average return than short-term bills, but have also been more variable. The stan- Brealey−Meyers: Principles of Corporate Finance, Seventh Edition VII $1.035 1r 2 21r 1 2 Brealey−Meyers: Principles of Corporate Finance, Seventh Edition VII. Debt Financing 24. Valuing Debt © The McGraw−Hill Companies, 2003 In other words, you can think of the two-year