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 stock, the cash has to come from borrowing. If cash is needed for only a short while, firms may borrow from a bank. If they need cash for long-term investments, they generally issue bonds, which are simply long-term loans.

Companies are not the only bond issuers. Municipalities also raise money by selling bonds. So do national governments. We start our analysis of the bond market by looking at the valuation of government bonds and at the interest rate that the government pays when it borrows. Do not confuse this interest rate with the cost of capital for a corporation. The projects that companies undertake are almost invariably risky, and investors demand higher prospective returns from these projects than from safe government bonds. (In Chapter 7, we start to look at the additional returns that investors demand from risky assets.)

The markets for government bonds are huge. At the start of 2018, investors held $14.8 trillion of U.S. government securities, and U.S. government agencies held a further $5.7 trillion. The bond markets are also sophisticated. Professional bond traders make massive trades that are often motivated by tiny price discrepancies. This book is not for them, but if you are to be involved in managing the company’s debt, you will need to get beyond the simple mechanics of bond valuation. Qualified 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 short-term rates are usually lower (but sometimes higher) than long-term rates and why the longest-term bond prices are most sensitive to fluctuations in interest rates. They can distinguish real (inflation-adjusted) interest rates and nominal (money) rates and anticipate how future inflation can affect interest rates. We cover all these topics in this chapter.

Companies can’t borrow at the same low interest rates as governments. The interest rates on government bonds are benchmarks for all interest rates, however. When government interest 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.

Corporate bonds are more complex securities than government bonds. It is more likely that a corporation may be unable to come up with the money to pay its debts, so investors have to worry about default risk. Corporate bonds are also less liquid than government bonds: They are not as easy to buy or sell, particularly in large quantities or on short notice. These complications affect the “spread” of corporate bond rates over interest rates on government bonds of similar maturities.

This chapter only introduces corporate debt. We take a more detailed look in Chapters 23 and 24.

3-1Using the Present Value Formula to Value Bonds

If you own a bond, you are entitled to a fixed set of cash payoffs. Every year until the bond matures, you collect regular interest payments. At maturity, when you get the final interest payment, you also get back the face value of the bond, which is called the bond’s principal.

A Short Trip to Paris to Value a Government Bond

Why are we going to Paris, apart from the cafés, restaurants, and sophisticated nightlife? Because we want to start with the simplest type of bond, one that makes payments just once a year.

French government bonds, known as OATs (short for Obligations Assimilables du Trésor), pay interest and principal in euros (€). Suppose that in October 2017 you decide to buy €100 face value of the 6.00% OAT maturing in October 2025. Each year until the bond matures, you are entitled to an interest payment of .06 × 100 = €6.00. This amount is the bond’s coupon.¹ When the bond matures in 2025, the government pays you the final €6.00 interest, plus the principal payment of €100. Your first coupon payment is in one year’s time, in October 2018. So the cash payments from the bond are as follows:

What is the present value of these payments? It depends on the opportunity cost of capital, which in this case equals the rate of return offered by other government debt issues denominated in euros. In October 2017, other medium-term French government bonds offered a return of just .3%. That is what you were giving up when you bought the 6.00% OATs. Therefore, to value the 6.00% OATs, you must discount the cash flows at .3%:

Bond prices are usually expressed as a percentage of face value. Thus the price of your 6.00% OAT was quoted as 144.99%.

You may have noticed a shortcut way to value this bond. Your OAT amounts to a package of two investments. The first investment gets the eight annual coupon payments of €6.00 each. The second gets the €100 face value at maturity. You can use the annuity formula from Chapter 2 to value the coupon payments and then add on the present value of the final payment.

Thus, the bond can be valued as a package of an annuity (the coupon payments) and a single, final payment (the repayment of principal).²

We just used the .3% interest rate to calculate the present value of the OAT. Now we turn the valuation around: If the price of the OAT is 144.99%, what is the interest rate? What return do investors get if they buy the bond and hold it to maturity? To answer this question, you need to find the value of the variable y that solves the following equation:

The rate of return y is called the bond’s yield to maturity. In this case, we already know that the present value of the bond is 144.99% at a .3% discount rate, so the yield to maturity must be .3%. If you buy the bond at 144.99% and hold it to maturity, you will earn a return of .3% per year.

Why is the yield to maturity less than the 6.00% coupon payment? Because you are paying €144.99 for a bond with a face value of only €100. You lose the difference of €44.99 if you hold the bond to maturity. On the other hand, you get eight annual cash payments of €6.00. (The immediate, current yield on your investment is 6.00/144.99 = .0414, or 4.14%.) The yield to maturity blends the return from the coupon payments with the declining value of the bond over its remaining life.

Let’s generalize. A bond, such as our OAT, that is priced above its face value is said to sell at a premium. Investors who buy a bond at a premium face a capital loss over the life of the bond, so the yield to maturity on these bonds is always less than the current yield. A bond that is priced below face value sells at a discount. Investors in discount bonds look forward to a capital gain over the life of the bond, so the yield to maturity on a discount bond is greater than the current yield.

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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. At the end of this chapter, you will find a box that lists the Excel function for calculating yield to maturity plus several other useful functions for bond analysts.

Back to the United States: Semiannual Coupons and Bond Prices

Just like the French government, the U.S. Treasury raises money by regular auctions of new bond issues. Some of these issues do not mature for 20 or 30 years; others, known as notes, mature in 10 years or less. The Treasury also issues short-term debt maturing in a year or less. These short-term securities are known as Treasury bills. Treasury bonds, notes, and bills are traded in the fixed-income market.

Let’s look at an example of a U.S. government bond. In 1991, the Treasury issued 8.00% bonds maturing in 2021. These bonds are called “the 8s of 2021.” Treasury bonds and notes have face values of $1,000, so if you own the 8s of 2021, the Treasury will give you back $1,000 at maturity. You can also look forward to a regular coupon, but in contrast to our French bond, coupons on Treasury bonds and notes are paid semiannually.³ Thus, the 8s of 2021 provide a coupon payment of 8/2 = 4% of face value every six months.

You can’t buy Treasury bonds, notes, or bills on the stock exchange. They are traded by a network of bond dealers, who quote prices at which they are prepared to buy and sell. For example, suppose that you decide to buy the 8s of 2021. You phone a broker, who checks the current price on her screen. If you are happy to go ahead with the purchase, your broker contacts a bond dealer and the trade is done.

The prices at which you can buy or sell Treasury notes and bonds are shown each day in the financial press and on the web. In November 2017, there were 310 different Treasury notes and bonds. Table 3.1 shows the prices of just a small sample of them. Look at the entry for our 8.00s of 2021. The asked price, 123.41, is the price you need to pay to buy the bond from a dealer. This means that the 8.00% bond costs 123.41% of face value. The face value of the bond is $1,000, so each bond costs $1,234.10.⁴

Maturity	Coupon	Asked Price (%)	Yield to Maturity (%)
November 2020	2.625	102.38	1.807
November 2021	8.00	123.41	1.893
November 2022	1.625	98.05	2.036
November 2023	2.75	103.48	2.128
November 2024	7.50	134.46	2.166
November 2025	2.25	99.80	2.278
November 2026	6.50	134.28	2.267
November 2027	6.125	133.99	2.300
November 2028	5.25	128.58	2.344

TABLE 3.1 Treasury bond quotes, November 2017

Source: The Wall Street Journal website, www.wsj.com.

The final column in the table shows the yield to maturity. Because interest is semiannual, annual yields on U.S. bonds are usually quoted as twice the semiannual yields. Thus, if you buy the 8.00% bond at the asked price and hold it to maturity, every six months you earn a return of 1.893/2 = .947%.

You can now repeat the present value calculations that we did for the French government bond. You just need to recognize that bonds in the U.S. 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 payments for the 8.00s of 2021:

If investors demand a return of .947% every six months, then the present value of these cash flows is

Each bond is worth $1,234.10, or 123.41% of face value.

Again we could turn the valuation around: Given the price, what’s the yield to maturity? Try it, and you’ll find (no surprise) that the semiannual rate of return that you can earn over the eight remaining half-year periods is .00947. Take care to remember that the yield is reported as an annual rate, calculated as 2 × .00947 = .01893, or 1.893%. If you see a reported yield to maturity of R%, you have to remember to use y = R/2% as the semiannual rate for discounting cash flows received every six months.

3-2How Bond Prices Vary with Interest Rates

Figure 3.1 plots the yield to maturity on 10-year U.S. Treasury bonds⁵ from 1900 to 2017. Notice how much the rate fluctuates. For example, interest rates climbed sharply after 1979 when Paul Volcker, the new chairman of the Fed, instituted a policy of tight money to rein in inflation. Within two years the interest rate on 10-year government bonds rose from 9% to a midyear peak of 15.8%. Contrast this with the summer of 2016, when long-term Treasury bonds offered a measly 1.4% rate of interest.

FIGURE 3.1 The interest rate on 10-year U.S. Treasury bonds

As interest rates change, so do bond prices. For example, suppose that investors demanded a semiannual return of 2% on the 8s of 2021, rather than the .947% semiannual return we used previously. In that case the price would be

The higher interest rate results in a lower price.

Bond prices and interest rates must move in opposite directions. The yield to maturity, our measure of the interest rate on a bond, is defined as the discount rate that explains the bond price. When bond prices fall, interest rates (i.e., yields to maturity) must rise. When interest rates rise, bond prices must fall. We recall a hapless TV pundit who intoned, “The recent decline in long-term interest rates suggests that long-term bond prices may rise over the next week or two.” Of course, the bond prices had already gone up. We are confident that you won’t make the pundit’s mistake.

The brown line in Figure 3.2 shows the value of our 8% bond for different interest rates. As the yield to maturity falls, the bond price increases. When the annual yield is equal to the bond’s annual coupon rate (8%), the bond sells for exactly its face value. When the yield is higher than 8%, the bond sells at a discount to face value. When the yield is lower than 8%, the bond sells at a premium.

FIGURE 3.2 Plot of bond prices as a function of the interest rate. The price of long-term bonds is more sensitive to changes in the interest rate than is the price of short-term bonds.

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 interest rates jump up, the value of their investment declines.

EXAMPLE 3.1 Changes in Interest Rates and Bond Returns

On May 15, 2008, the U.S. Treasury sold $9 billion of 4.375% bonds maturing in February 2038. The bonds were issued at a price of 96.38% and offered a yield to maturity of 4.60%. This was the return to anyone buying at the issue price and holding the bonds to maturity. In the months following the issue, the financial crisis reached its peak. Lehman Brothers filed for bankruptcy with assets of $691 billion, and the government poured money into rescuing Fannie Mae, Freddie Mac, AIG, and a host of banks. As investors rushed to the safety of Treasury bonds, prices soared. By mid-December, the price of the 4.375s of 2038 had reached 138.05% of face value, and the yield had fallen to 2.5%. Anyone fortunate enough to have bought the bond at the issue price would have made a capital gain of $1,380.50 − $963.80 = $416.70. In addition, on August 15 the bond made its first coupon payment of $21.875 (this is the semiannual payment on the 4.375% coupon bond with a face value of $1,000). Our lucky investor would, therefore, have earned a seven-month rate of return of 45.5%:

Suddenly, government bonds did not seem quite so boring as before.

A change in interest rates has only a modest impact on the value of near-term cash flows but a much greater impact on the value of distant cash flows. Thus the price of long-term bonds is affected more by changing interest rates than the price of short-term bonds. For example, compare the two curves in Figure 3.2. The brown line shows how the price of the four-year 8% bond varies with the interest rate. The blue line shows how the price of a 30-year 8% bond varies. You can see that the 30-year bond is much more sensitive to interest rate fluctuations than the four-year bond.

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Duration and Volatility

Changes in interest rates have a greater impact on the prices of long-term bonds than on those of short-term bonds. But what do we mean by “long term” and “short term”? A coupon bond that matures in year 30 makes payments in each of years 1 through 30. It’s misleading to describe the bond as a 30-year bond; the average time to each cash payment is less than 30 years.

EXAMPLE 3.2 Which Is the Longer-Term Bond?

Table 3.2 calculates the prices of two seven-year bonds. We assume annual coupon payments and a yield to maturity of 4% per year. Take a look at the time pattern of each bond’s cash payments and review how the prices are calculated:

TABLE 3.2 A comparison of the cash flows and prices of two bonds. Price is calculated assuming annual coupon payments and a yield to maturity of 4%.

Note: Both securities mature at the end of year 7.

Which of these two bonds is the longer-term investment? They both have the same final maturity, of course. But the timing of the bonds’ cash payments is not the same. In the case of the 3s, the average time to each cash flow is longer, because a higher proportion of the cash flows occurs at maturity, when the face value is paid off.

Suppose now that the yield to maturity on each bond falls to 3%. Which bond would you most like to own? The 3s, of course. Since they have the longer effective life, they should benefit most from a fall in yields. Table 3.3 confirms that this is indeed the case:.

	Yield = 4%	Yield = 3%
Coupon	Price ($)	Price ($)	Change in Price (%)
3%	$ 939.98	$1,000.00	+6.4%
9%	1,300.10	1,373.82	+5.7

TABLE 3.3 The effect of a 1% fall in yield on the prices of two seven-year bonds

The 9s have the shorter average life and therefore a shift in interest rates has a more muted effect on the price. That much is clear. However, it would be useful to have a precise measure of the average life, one that could be used to predict the exposure of each bond’s price to fluctuations in interest rates. There is such a measure, and it is called duration or Macaulay duration after its founder.

Duration is the weighted average of the times to each of the cash payments. The times are the future years 1, 2, 3, etc., extending to the final maturity date, which we call T. The weight for each year is the present value of the cash flow received at that time divided by the total present value of the bond.

Table 3.4 shows how to compute duration for the 9% seven-year bonds, assuming annual payments. First, we value each of the coupon payments of $90 and the final payment of coupon plus face value of $1,090. Of course, the present values of these payments add up to the bond price of $1,300.10. Then we calculate the fraction of the price accounted for by each cash flow and multiply each fraction by the year of the cash flow. The results sum across to a duration of 5.69 years.

TABLE 3.4 Calculating the duration of the 9% seven-year bonds. The yield to maturity is 4% a year.

We leave it to you to calculate durations for the 3% bonds in Table 3.2. You will find that duration increases to 6.40 years.

Table 3.5 shows the duration in November 2017 of the sample of U.S. Treasury bonds set out in Table 3.1. Notice that, in all cases, the duration is shorter than the number of years to maturity and that the difference is most marked in the case of high coupon bonds.

Years to Maturity	Coupon	Yield to Maturity (%)	Duration (years)
3	2.625	1.807	2.906
4	8.00	1.893	3.559
5	1.625	2.036	4.820
6	2.75	2.128	5.581
7	7.50	2.166	5.797
8	2.25	2.278	7.365
9	6.50	2.267	7.274
10	6.125	2.300	7.993
11	5.25	2.344	8.831

TABLE 3.5 The duration in November 2017 of the Treasury bonds shown in Table 3.1. Duration calculations recognize that coupon payments are semiannual.

We mentioned that investors and financial managers track duration because it measures how bond prices change when interest rates change. For this purpose, it’s best to use modified duration or volatility, which is just duration divided by one plus the yield to maturity:

Modified duration measures the percentage change in bond price for a 1 percentage-point change in yield.⁶ Let’s try out this formula for our seven-year 9% bond in Table 3.3. The bond’s modified duration is duration/(1 + yield) = 5.69/1.04 = 5.47. This means that a 1% change in the yield to maturity should change the bond price by 5.47%.

Let’s check that prediction. Suppose the yield to maturity either increases or declines by .5%:

Yield to Maturity (%)	Price ($)	Change (%)
4.5%	$1265.17	−2.687%
4.0	1300.10	—
3.5	1336.30	+2.784

USEFUL SPREADSHEET FUNCTIONS

Valuing Bonds

Spreadsheet programs such as Excel provide built-in functions to solve for a variety of bond valuation problems. You can find these functions by pressing fx on the Excel toolbar. If you then click on the function that you wish to use, Excel will ask you for the inputs that it needs. At the bottom left of the function box, there is a Help facility with an example of how the function is used.

Here is a list of useful functions for valuing bonds, together with some points to remember when entering data:

· PRICE: The price of a bond given its yield to maturity.

· YLD: The yield to maturity of a bond given its price.

· DURATION: The duration of a bond.

· MDURATION: The modified duration (or volatility) of a bond.

Note:

· You can enter all the inputs in these functions directly as numbers or as the addresses of cells that contain the numbers.

Source: Microsoft Excel.

· You must enter the yield and coupon as decimal values, for example, for 3% you would enter .03.

· Settlement is the date that payment for the security is made. Maturity is the maturity date. You can enter these dates directly using the Excel date function; for example, you would enter 15 Feb 2009 as DATE(2009,02,15). Alternatively, you can enter these dates in a cell and then enter the cell address in the function.

· In the functions for PRICE and YLD you need to scroll down in the function box to enter the frequency of coupon payments. Enter 1 for annual payments or 2 for semiannual.

· The functions for PRICE and YLD ask for an entry for “basis.” We suggest you leave this blank. (See the Help facility for an explanation.)

Spreadsheet Questions

The following questions provide an opportunity to practice each of these functions.

1. (PRICE) In February 2009, Treasury 8.5s of 2020 yielded 3.2976%. What was their price? If the yield rose to 4%, what would happen to the price?

2. (YLD) On the same day, Treasury 3.5s of 2018 were priced at 107.46875%. What was their yield to maturity? Suppose that the price was 110.0%. What would happen to the yield?

3. (DURATION) What was the duration of the Treasury 8.5s? How would duration change if the yield rose to 4%? Can you explain why?

4. (MDURATION) What was the modified duration of the Treasury 8.5s? How would modified duration differ if the coupon were only 7.5%?

The total percentage difference between price at yields of 4.5% and 3.5% is 2.687 + 2.784 = 5.47%. Thus, a 1% change in interest rates means a 5.47% change in bond price, just as predicted.⁷

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The modified duration for the 3% bond in Table 3.3 is 6.40/1.04 = 6.15%. In other words, a 1% change in yield to maturity results in a 6.15% change in the bond’s price.

You can see why duration (or modified duration) is a handy measure of interest-rate risk.⁸ For example, the mini-case at the end of Chapter 26 looks at how financial managers can use the measure to protect the pension plan against unexpected changes in interest rates.

3-3The Term Structure of Interest Rates

When we explained in Chapter 2 how to calculate present values, we used the same discount rate to calculate the value of each period’s cash flow. This discount rate was the bond’s yield to maturity y. For many purposes, using a single discount rate is a perfectly acceptable approximation, but there are also occasions when you need to recognize that short-term interest rates may be different from long-term rates. In this case, it may be worth discounting each cash flow at a different rate.

The relationship between short- and long-term interest rates is called the term structure of interest rates. Look, for example, at Figure 3.3, which shows the term structure in two different years. Notice that in the later year, the term structure sloped downward; long-term interest rates were lower than short-term rates. In the earlier year, the pattern was reversed and long-term bonds offered a much higher interest rate than short-term bonds. You now need to learn how to measure the term structure and understand why long- and short-term rates often differ.

FIGURE 3.3 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.

Consider a simple loan that pays $1 at the end of one year. To find the present value of this loan you need to discount the cash flow by the one-year rate of interest, r₁:

PV = 1/(1 + r₁)

This rate, r₁, is called the one-year spot rate. To find the present value of a loan that pays $1 at the end of two years, you need to discount by the two-year spot rate, r₂:

PV = 1/(1 + r₂)²

The first year’s cash flow is discounted at today’s one-year spot rate, and the second year’s flow is discounted at today’s two-year spot rate. The series of spot rates r₁, r₂, . . . , r_t, . . . traces out the term structure of interest rates.

Now suppose you have to value $1 paid at the end of years 1 and 2. If the spot rates are different—say, r₁ = 3% and r₂ = 4%—then we need two discount rates to calculate present value:

Once we know that PV = 1.895, we can go on to calculate a single discount rate that would give the same answer. That is, we could calculate the yield to maturity by solving for y in the following equation:

This gives a yield to maturity of 3.66%. Once we have the yield, we could use it to value other two-year annuities. But we can’t get the yield to maturity until we know the price. The price is determined by the spot interest rates for dates 1 and 2. Spot rates come first. Yields to maturity come later, after bond prices are set. That is why professionals often identify spot interest rates and discount each cash flow at the spot rate for the date when the cash flow is received.

Spot Rates, Bond Prices, and the Law of One Price

The law of one price states that the same commodity must sell at the same price in a well-functioning market. Therefore, all safe cash payments delivered on the same date must be discounted at the same spot rate.

Table 3.6 illustrates how the law of one price applies to government bonds. It lists three government bonds, which we assume make annual coupon payments. All the bonds have the same coupon but they have different maturities. The shortest (bond A) matures in two years and the longest (bond C) in four.

TABLE 3.6 The law of one price applied to government bonds

Spot rates and discount factors are given at the top of each column. The law of one price says that investors place the same value on a risk-free dollar regardless of whether it is provided by bond A, B, or C. You can check that the law holds in the table.

Each bond is priced by adding the present values of each of its cash flows. Once total PV is calculated, we have the bond price. Only then can the yield to maturity be calculated.

Notice how the yield to maturity increases as bond maturity increases. The yields increase with maturity because the term structure of spot rates is upward-sloping. Yields to maturity are complex averages of spot rates. For example, you can see that the yield on the four-year bond (5.81%) lies between the one- and four-year spot rates (3% and 6%).

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Try It! Table 3.5: Spot rates, forward rates, and arbitrage

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Financial managers who want a quick, summary measure of interest rates bypass spot interest rates and look in the financial press at yields to maturity. They may refer to the yield curve, which plots yields to maturity, instead of referring to the term structure, which plots spot rates. They may use the yield to maturity on one bond to value another bond with roughly the same coupon and maturity. They may speak with a broad brush and say, “Ampersand Bank will charge us 6% on a three-year loan,” referring to a 6% yield to maturity.

Throughout this text, we too use the yield to maturity to summarize the return required by bond investors. But you also need to understand the measure’s limitations when spot rates are not equal.

Measuring the Term Structure

You can think of the spot rate, r_t, as the rate of interest on a bond that makes a single payment at time t. Such simple bonds do exist. They are known as stripped bonds, or strips. On request the U.S. Treasury will split a normal coupon bond into a package of mini-bonds, each of which makes just one cash payment. Our 8% bonds of 2021 could be exchanged for eight semiannual coupon strips, each paying $40, and a principal strip paying $1,000. In November 2017, this package of coupon strips would have cost $307.58 and the principal strip would have cost $927.53, making a total cost of $1,235.11, just a little more than it cost to buy one 8% bond. The similarity should be no surprise. Because the two investments provide identical cash payments, they must sell for very close to the same price.

We can use the prices of strips to measure the term structure of interest rates. For example, in November 2017, a 10-year strip cost $793.37. In return, investors could look forward to a single payment of $1,000 in November 2027. Thus, investors were prepared to pay $.79337 for the promise of $1 at the end of 10 years. The 10-year discount factor was DF₁₀ = 1/(1 + r₁₀)¹⁰ = .79337, and the 10-year spot rate was r₁₀ = (1/.79337)^.10 − 1 = .0234, or 2.34%. In Figure 3.4, we use the prices of strips with different maturities to plot the term structure of spot rates from 1 to 25 years. You can see that in 2017 investors required a higher interest rate for lending for 25 years rather than for 1.

FIGURE 3.4 Spot rates on U.S. Treasury strips, December 2017

Why the Discount Factor Declines as Futurity Increases—and a Digression on Money Machines

In Chapter 2, we saw that the longer you have to wait for your money, the less is its present value. In other words, the two-year discount factor DF₂ = 1/(1 + r₂)² is less than the one-year discount factor DF₁ = 1/(1 + r₁). But is this necessarily the case when there can be a different spot interest rate for each period?

Suppose that the one-year spot rate of interest is r₁ = 20% and the two-year spot rate is r₂ = 7%. In this case, the one-year discount factor is DF₁ = 1/1.20 = .833 and the two-year discount factor is DF₂ = 1/1.07² = .873. Apparently, a dollar received the day after tomorrow is not necessarily worth less than a dollar received tomorrow.

But there is something wrong with this example. Anyone who could borrow and invest at these interest rates could become a millionaire overnight. Let us see how such a “money machine” would work. Suppose the first person to spot the opportunity is Hermione Kraft. Ms. Kraft first buys a one-year Treasury strip for .833 × $1,000 = $833. Now she notices that there is a way to earn an immediate surefire profit on this investment. She reasons as follows. Next year the strip will pay off $1,000 that can be reinvested for a further year. Although she does not know what interest rates will be at that time, she does know that she can always put the money in a checking account and be certain of having $1,000 at the end of year 2. Her next step, therefore, is to go to her bank and borrow the present value of this $1,000. At 7% interest the present value is PV = 1000/(1.07)² = $873.

So Ms. Kraft borrows $873, invests $830, and walks away with a profit of $43. If that does not sound like very much, notice that by borrowing more and investing more she can make much larger profits. For example, if she borrows $21,778,584 and invests $20,778,584, she would become a millionaire.⁹

Of course this story is completely fanciful. Such an opportunity would not last long in well-functioning capital markets. Any bank that allowed you to borrow for two years at 7% when the one-year interest rate was 20% would soon be wiped out by a rush of small investors hoping to become millionaires and a rush of millionaires hoping to become billionaires. There are, however, two lessons to our story. The first is that a dollar tomorrow cannot be worth less than a dollar the day after tomorrow. In other words, the value of a dollar received at the end of one year (DF₁) cannot be less than the value of a dollar received at the end of two years (DF₂). There must be some extra gain from lending for two periods rather than one: (1 + r₂)² cannot be less than 1 + r₁.

Our second lesson is a more general one and can be summed up by this precept: “There is no such thing as a surefire money machine.” The technical term for money machine is arbitrage. In well-functioning markets, where the costs of buying and selling are low, arbitrage opportunities are eliminated almost instantaneously by investors who try to take advantage of them.

Later in the book we invoke the absence of arbitrage opportunities to prove several useful properties about security prices. That is, we make statements like, “The prices of securities X and Y must be in the following relationship—otherwise there would be potential arbitrage profits and capital markets would not be in equilibrium.”

3-4Explaining the Term Structure

The term structure that we showed in Figure 3.4 was upward-sloping. Long-term rates of interest in December 2017 were more than 2.5%; short-term rates were about 1.8%. Why then didn’t everyone rush to buy long-term bonds? Who were the (foolish?) investors who put their money into the short end of the term structure?

Suppose that you held a portfolio of one-year U.S. Treasuries in December 2017. Here are three possible reasons you might decide to hold on to them, despite their low rate of return:

1. You believe that short-term interest rates will be higher in the future.

2. You worry about the greater exposure of long-term bonds to changes in interest rates.

3. You worry about the risk of higher future inflation.

We review each of these reasons now.

Expectations Theory of the Term Structure

Recall that you own a portfolio of one-year Treasuries. A year from now, when these Treasuries mature, you can reinvest the proceeds for another one-year period and enjoy whatever interest rate the bond market offers then. The interest rate for the second year may be high enough to offset a low return in the first year. You often see an upward-sloping term structure when future interest rates are expected to rise.

EXAMPLE 3.3 Expectations and the Term Structure

Suppose that the one-year interest rate, r₁, is 5%, and the two-year rate, r₂, is 7%. If you invest $100 for one year, your investment grows to 100 × 1.05 = $105; if you invest for two years, it grows to 100 × 1.07² = $114.49. The extra return that you earn for that second year is 1.07²/1.05 − 1 = .090, or 9.0%.¹⁰

Would you be happy to earn that extra 9% for investing for two years rather than one? The answer depends on how you expect interest rates to change over the coming year. If you are confident that in 12 months’ time one-year bonds will yield more than 9.0%, you would do better to invest in a one-year bond and, when that matured, reinvest the cash for the next year at the higher rate. If you forecast that the future one-year rate is exactly 9.0%, then you will be indifferent between buying a two-year bond or investing for one year and then rolling the investment forward at next year’s short-term interest rate.

If everyone is thinking as you just did, then the two-year interest rate has to adjust so that everyone is equally happy to invest for one year or two. Thus the two-year rate will incorporate both today’s one-year rate and the consensus forecast of next year’s one-year rate.

We have just illustrated (in Example 3.3) the expectations theory of the term structure. It states that in well-functioning bond markets investment in a series of short-maturity bonds must offer the same expected return as an investment in a single long-maturity bond. Only if that is the case would investors be prepared to hold both short- and long-maturity bonds.

The expectations theory implies that the only reason for an upward-sloping 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.

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 at 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 1900–2017, we find that the return on long-term U.S. Treasury bonds was on average 1.5 percentage points higher than the return on short-term Treasury bills. 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 only a first step.

These days, the expectations theory has few strict adherents. Nevertheless, most economists believe that expectations about future interest rates have an important effect on the term structure. For example, you often hear market commentators remark that since the six-month interest rate is higher than the three-month rate, the market must be expecting the Federal Reserve Board to raise interest rates.

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 prices of short-duration bonds. A sharp increase in interest rates can knock 30% or 40% off the price of long-term bonds.

For some investors, this extra volatility of long-duration bonds may not be a concern. For example, pension funds and life insurance companies have fixed long-term liabilities and 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, the term structure will be upward-sloping more often than not. Of course, if interest 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.

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Arbitrage models of term structure

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Inflation and Term Structure

Suppose you are saving for your retirement 20 years from now. Which of the following strategies is more risky? Invest in a succession of one-year Treasuries, rolled over annually, or invest once in 20-year strips? The answer depends on how confident you are about future inflation.

If you buy the 20-year strips, you know exactly how much money you will have at year 20, but you don’t know what that money will buy. Inflation may seem benign now, but who knows what it will be in 10 or 15 years? This uncertainty about inflation may make it uncomfortably risky for you to lock in one 20-year interest rate by buying the strips.

You can reduce exposure to inflation risk by investing short-term and rolling over the investment. You do not know future short-term interest rates, but you do know that future interest rates will adapt to inflation. If inflation takes off, you will probably be able to roll over your investment at higher interest rates.

If inflation is an important source of risk for long-term investors, 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.

3-5Real 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 lose ground in terms of purchasing power.

Several indexes are used to track the general level of prices. The best known is the consumer price index (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.

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The German hyperinflation

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Figure 3.5 shows the rate of inflation in the U.S. since 1900. Inflation touched a peak at the end of World War I, when it reached 21%. However, this figure pales into insignificance compared with the hyperinflation in Zimbabwe in 2008. Prices there rose so fast that a Z$50 trillion bill was barely enough to buy a loaf of bread.

FIGURE 3.5 Annual rates of inflation in the United States from 1900–2017

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.

Prices can fall as well as rise. The U.S. experienced severe deflation in the Great Depression, when prices fell by 24% in three years. In Japan, which has experienced persistent deflation, prices in 2017 were only marginally higher than they had been 20 years earlier.

The average U.S. inflation rate from 1900 to 2017 was 3.0%. As you can see from Figure 3.6, among major economies, the United States has been almost top of the class in holding inflation in check. Countries torn by war have generally experienced much higher inflation. For example, in Italy and Japan, inflation since 1900 has averaged around 10% a year.

FIGURE 3.6 Average rates of inflation in 20 countries from 1900–2017

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.

Economists and financial managers refer to current, or nominal, dollars versus constant, or real, dollars. For example, the nominal cash flow from your one-year bond is $1,100. But if prices rise over the year by 6%, then each dollar will buy you 6% less next year than it does today. So at the end of the year, $1,100 has the same purchasing power as 1,100/1.06 = $1,037.74 today. The nominal payoff on the bond is $1,100, but the real payoff is only $1,037.74.

The formula for converting nominal cash flows in a future period t to real cash flows today is

For example, suppose you invest in a 20-year Treasury strip, but prices over that period grow by 6% per year. The strip pays $1,000 in year 20, but the real value of that payoff is only 1,000/1.06²⁰ = $311.80. In this example, the purchasing power of $1 today declines to just over $.31 after 20 years.

These examples show you how to get from nominal to real cash flows. The journey from nominal to real interest rates is similar. 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—say, over one year:

Invest Current Dollars	Receive Dollars in Year 1	Result
$1,000 →	$1,100	10% nominal rate of return

However, with an expected inflation rate of 6%, you are only 3.774% better off at the end of the year than at the start:

Invest Current Dollars	Expected Real Value of Dollars in Year 1	Result
$1,000 →	$1,037.74 (=1,100/1.06)	3.774% expected real rate of return

Thus, we could say, “The bond 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 + r_real = (1 + r nominal ) / (1 + inflation rate)

In our example, 1 + r_real = 1.10/1.06 = 1.03774. A common rule of thumb states that r_real = rnominal − inflation rate. In our example, this gives r_real = .10 − .06 = .04, or 4%. This is not a bad approximation to the true real interest rate of 3.774%. But when inflation is high, it pays to use the full formula.

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.

You can nail down a real return, however. You do so by buying an indexed bond that makes cash payments 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).¹¹

The real cash flows on TIPS are fixed, but the nominal cash flows (interest and principal) increase as the CPI increases.¹² For example, suppose that the U.S. Treasury issues 3% 20-year TIPS at a price equal to its face value of $1,000. If during the first year the CPI rises by 10%, then the coupon payment on the bond increases by 10% from $30 to 30 × 1.10 = $33. The amount that you will be paid at maturity also increases to $1,000 × 1.10 = $1,100. The purchasing power of the coupon and face value remain constant at $33/1.10 = $30 and $1,100/1.10 = $1,000. Thus, an investor who buys the bond at the issue price earns a surefire real interest rate of 3%.

Long-term TIPS offered a yield of about .5% in December 2017.¹³ This is a real yield to maturity. It measures the extra goods and services your investment will allow you to buy. The .5% yield on TIPS was about 1.9% less than the nominal yield on ordinary Treasury bonds. If the annual inflation rate turns out to be higher than 1.9%, investors will earn a higher return by holding long-term TIPS; if the inflation rate turns out to be less than 1.9%, they would have been better off with nominal bonds.

What Determines the Real Rate of Interest?

The real rate of interest depends on people’s willingness to save (the supply of capital)¹⁴ 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 real interest rate. Therefore, the rate has to rise to induce individuals to save the additional amount that firms want to invest.¹⁵ Conversely, if investment opportunities deteriorate, there will be a fall in the real interest rate.

Short- and medium-term real interest rates are affected by the monetary policy of central banks. For example, sometimes central banks keep short-term nominal interest rates low despite significant inflation. The resulting real rates can be negative. However, nominal interest rates cannot be negative except in fairly rare circumstances because investors can simply hold cash. Cash always pays zero interest, which is better than negative interest.

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How to make a certain loss

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For many years, real interest rates were much more stable than nominal rates, but as you can see from Figure 3.7, both nominal and real interest rates have plummeted since the financial crash.

FIGURE 3.7 The green line shows the real yield on long-term TIPS issued by the U.S. government. The brown line shows the yield on long-term nominal bonds.

Inflation and Nominal Interest Rates

How does the inflation outlook affect the nominal rate of interest? Here is how economist Irving Fisher answered the question. Suppose that consumers are equally happy with 100 apples today or 103 apples in a year’s time. In this case, the real or “apple” interest rate is 3%. If the price of apples is constant at (say) $1 each, then we will be equally happy to receive $100 today or $103 at the end of the year. That extra $3 will allow us to buy 3% more apples at the end of the year than we could buy today.

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The yield on TIPs

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But suppose now that the apple price is expected to increase by 5% to $1.05 each. In that case, we would not be happy to give up $100 today for the promise of $103 next year. To buy 103 apples in a year’s time, we will need to receive 1.05 × $103 = $108.15. In other words, the nominal rate of interest must increase by the expected rate of inflation to 8.15%.

This is Fisher’s theory: A change in the expected inflation rate causes 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 + r_nominal = (1 + r_real ) (1 + i)

where r_real is the real interest rate that consumers require and i is the expected inflation rate. In our example, the prospect of inflation causes 1 + r_nominal to rise to 1.03 × 1.05 = 1.0815.

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 103 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 3.8 the return on Treasury bills (short-term government debt) against actual inflation for the United States, U.K., 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.

FIGURE 3.8 The return on Treasury bills and the rate of inflation in the U.K., U.S., and Germany, 1953–2017

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.

Look now at the relationship between the rate of inflation and the Treasury bill rate. Figure 3.8 shows that investors have for the most part demanded a higher rate of interest when inflation was high. So it looks as if Fisher’s theory provides 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 nominal interest rate. In other words, a strategy of rolling over short-term investments affords some protection against uncertain inflation.

3-6The Risk of Default

Corporate Bonds and Default Risk

Look at Table 3.7, which shows the yields to maturity on a sample of corporate bonds. Notice that the bonds all mature in 2023, but their yields to maturity differ dramatically. With a yield of 13.0%, the bonds of Quorum Health appeared to offer a mouth-watering rate of return. However, the company had been operating at a loss and had substantial debts. Investors foresaw that there was a good chance that the company would default and that they would not get their money back. Thus, the payments promised to the bondholders represent a best-case scenario: The firm will never pay more than the promised cash flows, but in hard times, it may pay less. This risk applies in some measure to all corporate bonds, but investors were much more confident that Johnson & Johnson would be able to service its debt and this is reflected in the low yield on its bonds.

TABLE 3.7 Prices and yields of a sample of corporate bonds

Source: Bond transactions reported on FINRA’s TRACE service: http://finra-markets.morningstar.com/BondCenter/Screener.jsp.

The safety of most corporate bonds can be judged from bond ratings provided by Moody’s, Standard & Poor’s (S&P), and Fitch. Table 3.8 lists the possible bond ratings in declining order of quality. For example, the bonds that receive the highest Standard & Poor’s rating are known as AAA (or “triple A”) bonds. Then come AA (double A), A, BBB bonds, and so on. Bonds rated BBB and above are called investment grade, while those with a rating of BB or below are referred to as speculative grade, high-yield, or junk bonds. Notice that the bonds in the first four rows of Table 3.7 are all investment-grade bonds; the rest are junk bonds.

Moody’s	Standard & Poor’s and Fitch
Investment-grade bonds
Aaa	AAA
Aa	AA
A	A
Baa	BBB
Junk bonds
Ba	BB
B	B
Caa	CCC
Ca	CC
C	C

TABLE 3.8 Key to bond ratings. The highest-quality bonds rated Baa/BBB or above are investment grade.

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Bond rating definitions

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It is rare for highly rated bonds to default. However, when an investment-grade bond does go under, the shock waves are felt in all major financial centers. For example, in May 2001, World-Com sold $11.8 billion of bonds with an investment-grade rating. About one year later, WorldCom filed for bankruptcy, and its bondholders lost more than 80% of their investment. For these bondholders, the agencies that had assigned investment-grade ratings were not the flavor of the month.

Because of the risk of default, yields on corporate bonds are higher than those of government bonds. For example, Figure 3.9 shows the yield spread of corporate bonds against U.S. Treasuries. Notice that the spreads widen as safety falls off. Notice also how spreads vary over time. On average, the spread on Baa bonds has been about 2%, but it widened to more than 6% during the credit crunch of 2007 to 2009.¹⁶

FIGURE 3.9 Yield spreads between corporate and 10-year Treasury bonds, January 1953–January 2018

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Some sovereign defaults

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Sovereign Bonds and Default Risk

When investors buy corporate bonds or a bank lends to a company, they worry about the possibility of default and spend considerable time and effort assessing differences in credit risk. By contrast, when the U.S. government issues dollar bonds, investors can generally be confident that those bonds will be repaid in full and on time. Of course, bondholders don’t know what that money will be worth. Governments have a nasty habit of reducing the real value of their debts by inflationary policies.

FINANCE IN PRACTICE

A Game of Political Chicken

In 2010, the U.S. Congress set a ceiling of $14.3 trillion on the amount that the federal government could borrow. However, government spending was fast outrunning revenues, and unless Congress voted to increase the debt ceiling, the U.S. government forecasted that by August 2, 2011, it would run out of cash to pay its bills. It would then face a stark choice between drastic cuts in government spending or defaulting on its debt. Treasury Secretary Tim Geithner warned that “failure to raise the limit would precipitate a default by the United States. Default would effectively impose a significant and long-lasting tax on all Americans and all American businesses and could lead to the loss of millions of American jobs. Even a very short-term or limited default would have catastrophic economic consequences that would last for decades.”

Although there was general agreement that any increase in the debt ceiling should be accompanied by a deal to reduce the deficit, there was little meeting of minds as to how this should be achieved. Few observers believed that the United States would actually default on its debt, but as the dispute dragged on, the unthinkable became thinkable. Negotiations went down to the wire. On August 2, the day that the country was fore-casted to run out of borrowing power, President Obama finally signed the Budget Control Act that increased the debt ceiling by $900 billion. Two days later, Standard & Poor’s downgraded the long-term credit rating of the U.S. government from AAA to AA.

Although sovereign debt is generally less risky than corporate debt, we should not leave you with the impression that it is always safe even in money terms. Indeed, as the nearby Finance in Practice box explains, even in the United States investors in government bonds have had their slightly scary moments. Countries do occasionally default on their debts and, when they do so, the effects can be catastrophic. We will look briefly at three circumstances in which countries may default.

Foreign Currency Debt Most government bond defaults have occurred when a foreign government borrows dollars. In these cases investors worry that in some future crisis the government may run out of taxing capacity and may not be able to come up with enough dollars to repay the debt. This worry shows up in bond prices and yields to maturity. For example, in 2001 the Argentinian government defaulted on $82 billion of debt. As the prospect of default loomed, the price of Argentinian bonds slumped, and the promised yield climbed to more than 40 percentage points above the yield on U.S. Treasuries. Argentina has plenty of company. Since 1970, there have been more than 100 occasions that sovereign governments have defaulted on their foreign currency bonds.¹⁷

Own Currency Debt If a government borrows in its own currency, there is less likelihood of default. After all, the government can always print the money needed to repay the bonds. Very occasionally, governments have chosen to default on their domestic debt rather than create the money to pay it off. That was the case in Russia in the summer of 1998, when political instability combined with a slump in oil prices, declining government revenues, and pressure on the exchange rate. By August yields on government ruble bonds had reached 200% and it no longer made sense for Russia to create the money to service its debt. That month the government devalued the ruble and defaulted on its domestic ruble debt.

Eurozone Debt The 19 countries in the eurozone do not even have the option of printing money to service their domestic debts; they have given up control over their own money supply to the European Central Bank. This was to pose a major problem for the Greek government, which had amassed a massive €330 billion (or about $440 billion) of debt. In May 2010, other eurozone governments and the International Monetary Fund (IMF) rushed to Greece’s aid, but investors were unconvinced that their assistance would be sufficient, and the yield on 10-year Greek government debt climbed to nearly 27%. In 2012, in return for a further bailout package, investors in Greek bonds were obliged to accept a write-down of some $100 billion in the value of their bonds. It was the largest ever sovereign default. However, the difficulties for Greece were by no means over. The rescue package required it to adopt an austerity policy that resulted in a sharp fall in national income and considerable hardship for the Greek people. The country was also unable to escape from its large debt burden. It missed a payment on its debt to the IMF in 2015, and two years later, despite a further bailout, was back in crisis mode.

The sovereign debt crisis in Europe was not confined to Greece. Cyprus also delayed repayment of its bonds, and the Irish and Portuguese government debt was down-rated to junk level. Investors joked that, instead of offering a risk-free return, eurozone government bonds just offered a return-free risk.

SUMMARY

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 U.S., coupons are normally paid every six months, but in other countries they may be paid annually.

The value of any bond is equal to its cash payments discounted at the spot rates of interest. For example, the present value of a 10-year bond with a 5% coupon paid annually equals

This calculation uses a different spot rate of interest for each period. A plot of spot rates by maturity shows the term structure of interest rates.

Spot interest rates are most conveniently calculated from the prices of strips, which are bonds that make a single payment of face value at maturity, with zero coupons along the way. The price of a strip maturing at a future date t reveals the discount factor and spot rate for cash flows at that date. All other safe cash payments on that date are valued at that same spot rate.

Investors and financial managers 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:

The yield to maturity discounts all cash payments at the same rate, even if spot rates differ. It is, therefore, rather like an average of the different spot rates.

A bond’s maturity tells you the date of its 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 the exposure of its price to changes in interest rates. A change in interest rates has a greater effect on the price of long-duration bonds.

The term structure of interest rates is upward-sloping more often than not. This means that long-term spot rates are higher than short-term spot rates. But it does not mean that investing long is more profitable than investing short. The expectations theory of the term structure tells us 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 an upward-sloping term structure only when future short-term interest rates are 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. These investors will be prepared to hold long-term bonds only if they offer the compensation of a higher rate of interest.

Bonds promise fixed nominal cash payments, but the real interest rate that they provide depends on inflation. 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 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.

When you buy a U.S. Treasury bond, you can be fairly 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. Defaults are rare for companies with investment-grade bond ratings, but investors worry nevertheless. Companies need to compensate investors for default risk by promising to pay higher rates of interest.

FURTHER READING

Some good general texts on fixed income markets are:

F. J. Fabozzi and S. V. Mann, Handbook of Fixed Income Securities, 8th ed. (New York: McGraw-Hill, 2012).

S. Sundaresan, Fixed Income Markets and Their Derivatives, 4th ed. (San Diego, CA: Academic Press, 2014).

B. Tuckman, Fixed Income Securities: Tools for Today’s Markets(New York: Wiley, 2002).

P. Veronesi, Fixed Income Securities: Valuation, Risk, and Risk Management (New York: Wiley, 2010).

PROBLEM SETS

 Select problems are available in McGraw-Hill’s Connect. Answers to questions with an “*” are found in the Appendix.

1. Bond prices and yields* A 10-year bond is issued with a face value of $1,000, paying interest of $60 a year. If interest rates increase shortly after the bond is issued, what happens to the bond’s

a. Coupon rate?

b. Price?

c. Yield to maturity?

2. Bond prices and yields The following statements are true. Explain why.

a. If a bond’s coupon rate is higher than its yield to maturity, then the bond will sell for more than face value.

b. If a bond’s coupon rate is lower than its yield to maturity, then the bond’s price will increase over its remaining maturity.

3. Bond prices and yields 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 to maturity 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 to maturity 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, do the prices of high-coupon bonds change proportionately more than that of low-coupon bonds?

4. Bond prices and yields* A 10-year German government bond (bund) has a face value of €100 and a coupon rate of 5% paid annually. Assume that the interest rate (in euros) is equal to 6% per year. What is the bond’s PV?

5. Bond prices and yields In November 2017, Treasury 4¾s of 2041 offered a semiannually compounded yield to maturity of 2.6%. Recognizing that coupons are paid semiannually, calculate the bond’s price.

6. Bond prices and yields A 10-year U.S. Treasury bond with a face value of $1,000 pays a coupon of 5.5% (2.75% of face value every six months). The reported yield to maturity is 5.2% (a six-month discount rate of 5.2/2 = 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%.

7. Bond prices and yields Choose 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? (Assume annual coupon payments.)

8. Bond returns If a bond’s yield to maturity does not change, the return on the bond each year will be equal to the yield to maturity. Confirm this with a simple example of a four-year bond selling at a premium to face value. Now do the same for a four-year bond selling at a discount. For convenience, assume annual coupon payments.

9. Bond returns*

a. An 8%, five-year bond yields 6%. If this yield to maturity remains unchanged, what will be its price one year hence? Assume annual coupon payments and a face value of $1,000.

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?

10. Bond returns 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 did the bondholder earn in this case?

11. Duration* 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.

12. Duration Here are the prices of three bonds with 10-year maturities:

Bond Coupon (%)	Price (%)
2%	81.62%
4	98.39
8	133.42

13. 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?

14. Duration Calculate the durations and volatilities of securities A, B, and C. Their cash flows are shown below. The interest rate is 8%.

	Period 1	Period 2	Period 3
A	40	40	40
B	20	20	120
C	10	10	110

15. Duration Calculate durations and modified durations for the 3% bonds in Table 3.2. You can follow the procedure set out in Table 3.4 for the 9% coupon bonds. Confirm that modified duration closely predicts the impact of a 1% change in interest rates on the bond prices.

16. Duration Find the spreadsheet for Table 3.4 in Connect. 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.

17. Duration The formula for the duration of a perpetual bond that makes an equal payment each year in perpetuity is (1 + yield)/yield. If each bond yields 5%, which has the longer duration—a perpetual bond or a 15-year zero-coupon bond? What if the yield is 10%?

18. Spot interest rates and yields Which comes first in the market for U.S. Treasury bonds:

a. Spot interest rates or yields to maturity?

b. Bond prices or yields to maturity?

19. Spot interest rates and yields Look again at Table 3.6. Suppose that spot interest rates all change to 4%—a “flat” term structure of interest rates.

a. What is the new yield to maturity for each bond in the table?

b. Recalculate the price of bond A.

20. Spot interest rates and yields Look again at Table 3.6. Suppose the spot interest rates change to the following downward-sloping term structure: r₁ = 4.6%, r₂ = 4.4%, r₃ = 4.2%, and r₄ = 4.0%. Recalculate discount factors, bond prices, and yields to maturity for each of the bonds listed in the table.

21. Spot interest rates and yields Look at the spot interest rates shown in Problem 19. Suppose that someone told you that the five-year spot interest rate was 2.5%. Why would you not believe him? How could you make money if he was right? What is the minimum sensible value for the five-year spot rate?

22. Spot interest rates and yields Assume annual coupons.

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?

23. Spot interest rates and yields A 6% six-year bond yields 12% and a 10% six-year bond yields 8%. Calculate the six-year spot rate. Assume annual coupon payments. (Hint: What would be your cash flows if you bought 1.2 10% bonds?)

24. Spot interest rates and yields 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.

25. Spot interest rates and yields* You have estimated spot rates as follows:

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. Calculate the PV of the following bonds assuming annual coupons and face values of $1,000: (i) 5%, two-year bond; (ii) 5%, five-year bond; and (iii) 10%, five-year bond.

26. Look again at the bonds in part (b) of Problem 24.

a. Explain intuitively why the yield to maturity on the 10% bond is less than that on the 5% bond.

b. What should be the yield to maturity on a five-year zero-coupon bond?

c. Show that the correct yield to maturity on a five-year annuity is 5.75%.

d. Explain intuitively why the yield on the five-year bonds described in part (b) of Problem 24 must lie between the yield on a five-year zero-coupon bond and a five-year annuity.

27. Measuring term structure* The following table shows the prices of a sample of Narnian Treasury strips in December 2018. 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 December 2023 to be higher or lower than the yield on the 2020 strip?

Maturity	Price (%)
December 2020	90.703%
December 2021	85.892
December 2022	81.491
December 2023	77.243

28. Term-structure theories The one-year spot interest rate is r₁ = 5% and the two-year rate is r₂ = 6%. If the expectations theory is correct, what is the expected one-year interest rate in one year’s time?

29. Term-structure theories Look again at the spot interest rates shown in Problem 24. What can you deduce about the one-year spot interest rate in three years if:

a. The expectations theory of term structure is right?

b. Investing in long-term bonds carries additional risks?

30. Real interest rates The two-year interest rate is 10% and the expected annual inflation rate is 5%.

a. What is the expected real interest rate?

b. If the expected rate of inflation suddenly rises to 7%, what does Fisher’s theory say about how the real interest rate will change? What about the nominal rate?

31. Nominal and real returns* Suppose that you buy a two-year 8% bond at its face value.

a. What will be your total nominal return over the two years if inflation is 3% in the first year and 5% in the second? What will be your total real return?

b. Now suppose that the bond is a TIPS. What will be your total two-year real and nominal returns?

32. Bond ratings A bond’s credit rating provides a guide to its price. In the fall of 2017 Aaa bonds yielded 3.6% and Baa bonds yielded 4.3%. If some bad news causes a 10% five-year bond to be unexpectedly downrated from Aaa to Baa, what would be the effect on the bond price? (Assume annual coupons.)

CHALLENGE

32. Bond prices and yields 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.

33. Price and spot interest rates Find the arbitrage opportunity(ies). 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	3	0	751.30
B	4	50	842.30
C	4	120	1,065.28
D	4	100	980.57
E	3	140	1,120.12
F	3	70	1,001.62
G	2	0	834.00

34. Duration The duration of a bond that makes an equal payment each year in perpetuity is (1 + yield)/yield. Prove it.

35. Prices and spot interest rates What spot interest rates are implied by the following Treasury bonds? Assume for simplicity that the bonds pay annual coupons. The price of a one-year strip is 97.56%, and the price of a four-year strip is 87.48%.

Maturity (years)	Coupon	Price (%)
5	2	92.89
5	3	97.43
3	5	105.42

36. Prices and spot interest rates Look one more time at Table 3.6.

a. Suppose you knew the bond prices but not the spot interest rates. Explain how you would calculate the spot rates. (Hint: You have four unknown spot rates, so you need four equations.)

b. Suppose that you could buy bond C in large quantities at $1,040 rather than at its equilibrium price of $1,076.20. Show how you could make a zillion dollars without taking on any risk.

FINANCE ON THE WEB

The websites of The Wall Street Journal (www.wsj.com) and the Financial Times (www.ft.com) are wonderful sources of market data. You should become familiar with them.

1. Use www.wsj.com to answer the following questions:

a. Find the prices of coupon strips. Use these prices to plot the term structure. If the expectations theory is correct, what is the expected one-year interest rate three years hence?

b. Find a three- or four-year bond and construct a package of coupon and principal strips that provides the same cash flows. The law of one price predicts that the cost of the package should be very close to that of the bond. Is it?

c. Find a long-term Treasury bond with a low coupon and calculate its duration. Now find another bond with a similar maturity and a higher coupon. Which has the longer duration?

d. Look up the yields on 10-year nominal Treasury bonds and on TIPS. If you are confident that inflation will average 2% a year, which bond will provide the higher real return?

2. Bond transactions are reported on FINRA’s TRACE service, which was the source of the data for Table 3.7. Use the Advanced Search facility in TRACE to find bond prices for Johnson & Johnson (JNJ), Walmart (WMT), Disney (DIS), SunTrust Banks (STI), and U.S. Steel (X). If possible, exclude callable issues that the company can buy back. What has happened to the yields of these companies’ bonds? (You will find that bonds issued by the same company may have very different yields, so you will need to use your best judgment to answer this question.)

¹Bonds used to come with a coupon 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 ownership is the bond certificate. In many parts of the world, bearer bonds are still issued and are popular with investors who would rather remain anonymous. The alternative is registered bonds, where the identity of the bond’s owner is recorded and the coupon payments are sent automatically. OATS are registered bonds.

²You could also value a seven-year annuity of €6.00 plus a final payment of €106.00.

³The frequency of interest payments varies from country to country. For example, most bonds issued by eurozone governments pay interest annually (Italy and Malta are exceptions), while most bonds in the U.K., Canada, and Japan pay interest semiannually.

⁴The 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. Always use the flat price to calculate the yield.

⁵ From this point forward, we will just say “bonds” and not distinguish notes from bonds unless we are referring to a specific security. Note also that bonds with long maturities end up with short maturities when they approach the final payment date. Thus, you will encounter 30-year bonds trading 20 years later at the same prices as new 10-year notes (assuming equal coupons).

⁶In other words, the derivative of the bond price with respect to a change in yield to maturity is dPV/dy = −duration/(1 + y) = −modified duration.

⁷If you look back at Figure 3.2, you will see that the plot of price against yield is not a straight line. This means that modified duration is a good predictor of the effect of interest rate changes only for small moves in interest rates.

⁸For simplicity, we assumed that the two Treasury bonds paid annual coupons. Calculating Macaulay duration for a bond with semiannual coupons is no different except that there are twice as many cash flows. To calculate modified duration with semiannual coupons you need to divide Macaulay duration by the semiannual yield to maturity.

⁹We exaggerate Ms. Kraft’s profits. There are always costs to financial transactions, though they may be very small. For example, Ms. Kraft could use her investment in the one-year strip as security for the bank loan, but the bank would need to charge more than 7% on the loan to cover its costs.

¹⁰The extra return for lending for one more year is termed the forward rate of interest. In our example, the forward rate is 9.0%. In Ms. Kraft’s arbitrage example, the forward interest rate was negative. In real life, forward interest rates can’t be negative. At the lowest they are zero.

¹¹Indexed 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.”

¹²The reverse happens if there is deflation. In this case, the coupon payment and principal amount are adjusted downward. However, the U.S. government guarantees that when the bond matures, it will not pay less than its original nominal face value.

¹³In 2012, the yield on short-term TIPs was negative. You were actually losing in real terms.

¹⁴Some of this saving is done indirectly. For example, if you hold 100 shares of IBM stock, and IBM retains and reinvests earnings of $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, etc.

¹⁵We assume that investors save more as interest rates rise. It doesn’t have to be that way. Suppose that 20 years hence you will need $50,000 in today’s dollars for your children’s college tuition. 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/(1 + real interest rate)²⁰ . The higher the real interest rate, the lower the present value and the less you have to set aside.

¹⁶Corporate bonds are also less liquid than Treasuries: They are more difficult and expensive to trade, particularly in large quantities or on short notice. Many investors value liquidity and will demand a higher interest rate on a less liquid bond. Lack of liquidity accounts for some of the spread between yields on corporate and Treasury bonds.

¹⁷Occasionally, defaults have been a case of “won’t pay” rather than “can’t pay.” For example, in 2008 Ecuador’s president announced that his country would disavow $3.9 billion of “illegal” debts contracted by earlier regimes. In dealing with international lenders, he said, “We are up against real monsters.”