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Read 10.1 -10.3 on bond characteristics, pricing, and Yields Read 13.3 Dividend Discount Models Read 15.1 - 15.3 on Options. The main things to take from the reading are the terminology and understanding the payoff and profit diagrams. Read 17.1 The Futures Contract Read 4.1 – 4.6 on mutua...

Read 10.1 -10.3 on bond characteristics, pricing, and Yields Read 13.3 Dividend Discount Models Read 15.1 - 15.3 on Options. The main things to take from the reading are the terminology and understanding the payoff and profit diagrams. Read 17.1 The Futures Contract Read 4.1 – 4.6 on mutual funds Read 12.2 Domestic Macroeconomy Read 12.6 Business Cycles Read 8.1 Random Walks and Efficient Markets Learning Objectives LO 10-1 Explain the general terms of a bond contract and how bond prices are quoted in the financial press. LO 10-2 Compute a bond’s price given its yield to maturity, and compute its yield to maturity given its price. LO 10-3 Calculate how bond prices will change over time for a given interest rate projection. LO 10-4 Describe call, convertibility, and sinking fund provisions, and analyze how these provisions affect a bond’s price and yield to maturity. LO 10-5 Identify the determinants of bond safety and rating and how credit risk is reflected in bond yields and the prices of credit default swaps. LO 10-6 Calculate several measures of bond return, and demonstrate how these measures may be affected by taxes. LO 10-7 Analyze the factors likely to affect the shape of the yield curve at any time, and impute forward rates from the yield curve. In the previous chapters on risk and return relationships, we have treated securities at a high level of abstraction. We have assumed implicitly that a prior, detailed analysis of each security already has been performed and that its risk and return features have been assessed. We turn now to specific analyses of particular security markets. We examine valuation principles, determinants of risk and return, and portfolio strategies commonly used within and across the various markets. We begin by analyzing debt securities, which are claims on a specified periodic stream of cash flows. Debt securities are often called fixed-income securities because they promise either a fixed stream of income or one determined according to a specified formula. These securities have the advantage of being relatively easy to understand because the payment formulas are specified in advance. Uncertainty about cash flows is minimal as long as the issuer of the security is sufficiently creditworthy. That makes these securities a page 285convenient starting point for our analysis of the universe of potential investment vehicles. The bond is the basic debt security, and this chapter starts with an overview of bond markets, including Treasury, corporate, and international bonds. We turn next to pricing, showing how bond prices are set in accordance with market interest rates and why they change with those rates. Given this background, we can compare myriad measures of bond returns such as yield to maturity, yield to call, holding-period return, and realized compound rate of return. We show how bond prices evolve over time, discuss tax rules that apply to debt securities, and show how to calculate after-tax returns. Next, we consider the impact of default or credit risk on bond pricing and look at the determinants of credit risk and the default premium built into bond yields. Finally, we turn to the term structure of interest rates, the relationship between yield to maturity and time to maturity. 10.1 BOND CHARACTERISTICS A bond is a security that is issued in connection with a borrowing arrangement. The borrower issues (i.e., sells) a bond to the lender for some amount of cash; the bond is in effect the “IOU” of the borrower. The issuer agrees to make specified payments to the bondholder on specified dates. In a typical coupon bond, the issuer makes semiannual payments of interest for the life of the bond. These are called coupon payments because, in precomputer days, most bonds had coupons that investors would clip off and present to claim the interest payment. When the bond matures, the issuer retires the debt by paying the bond’s par value (or equivalently, its face value). The coupon rate determines the interest payment: The annual payment equals the coupon rate times the bond’s par value. The coupon rate, maturity date, and par value of the bond are part of the bond indenture, which is the contract between the issuer and the bondholder. bond A security that obligates the issuer to make specified payments to the holder over a period of time. par value, face value The payment to the bondholder at the maturity of the bond. coupon rate A bond’s annual interest payment per dollar of par value. To illustrate, a bond with a par value of $1,000 and a coupon rate of 8% might be sold for $1,000. The issuer then pays the bondholder 8% of $1,000, or $80 per year, for the stated life of the bond, say, 30 years. The $80 payment typically comes in two semiannual installments of $40 each. At the end of the bond’s life, the issuer also pays the $1,000 par value to the bondholder. Bonds usually are issued with coupon rates set just high enough to induce investors to pay par value to buy the bond. Sometimes, however, zero-coupon bonds are issued that make no coupon payments. In this case, investors receive par value at the maturity date but receive no interest payments until then: The bond has a coupon rate of zero. These bonds are issued at prices considerably below par value, and the investor’s return comes solely from the difference between issue price and the payment of par value at maturity. We will return to these bonds below. zero-coupon bond A bond paying no coupons that sells at a discount and provides only a payment of par value at maturity. Treasury Bonds and Notes Figure 10.1 is an excerpt from the listing of Treasury issues from the The Wall Street Journal Online. Treasury notes are issued with original maturities between 1 and 10 years, while Treasury bonds are issued with maturities ranging from 10 to 30 years. Both bonds and notes may be purchased directly from the Treasury in denominations of only $100, but denominations of $1,000 are far more common. Both make semiannual coupon payments. FIGURE 10.1 Prices and yields of U.S. Treasury bonds, November 15, 2019 Source: The Wall Street Journal Online, November 15, 2019. The highlighted issue in Figure 10.1 matures on November 15, 2025. Its coupon rate is 2.25%. Par value is $1,000; thus, the bond pays interest of $22.50 per year in two semiannual payments of $11.25. Payments are made in May and November of each year. Although bonds are typically sold in denominations of $1,000 par value, the bid and ask prices1 page 286are quoted as a percentage of par value. Therefore, the ask price is 103.04% of par value, or $1,030.40. The last column, labeled “Asked Yield,” is the bond’s yield to maturity based on the ask price. The yield to maturity is often interpreted as a measure of the average rate of return to an investor who purchases the bond for the asked price and holds it until its maturity date. We will have much to say about yield to maturity below. ACCRUED INTEREST AND QUOTED BOND PRICES The prices you see quoted online or in the financial pages are not the prices investors actually pay for the bond. This is because the quoted price does not include the interest that accrues between coupon payment dates. If a bond is purchased between coupon payments, the buyer must pay the seller for accrued interest, the prorated share of the upcoming semiannual coupon. For example, if 30 days have passed since the last coupon payment, and there are 182 days in the semiannual coupon period, the seller is entitled to a payment of accrued interest of of the semiannual coupon. The sale, or invoice, price of the bond, which is the amount the buyer actually pays, would equal the stated price plus the accrued interest. In general, the formula for the accrued interest between two dates is EXAMPLE 10.1 Accrued Interest Suppose that the coupon rate is 8%. Then the semiannual coupon payment is $40. Because 30 days have passed since the last coupon payment, the accrued interest on the bond is If the quoted price of the bond is $990, then the invoice price will be $990 + $6.59 = $996.59. The practice of quoting bond prices net of accrued interest explains why the price of a maturing bond is listed at $1,000 rather than $1,000 plus one coupon payment. A purchaser of an 8% coupon bond one day before the bond’s maturity would receive $1,040 on the following day and so should be willing to pay $1,040. But $40 of that total payment constitutes the accrued interest for the preceding half-year period. The bond price is quoted net of accrued interest and thus appears as $1,000.2 page 287 Corporate Bonds Like the government, corporations borrow money by issuing bonds. Figure 10.2 is a sample of corporate bond listings. Although some bonds trade electronically on the NYSE Bonds platform, and some electronic trading platforms now allow participants to trade bonds directly, most bonds still trade on a traditional over-the-counter market in a network of bond dealers linked by a computer quotation system. In part, this is due to the lack of uniformity of bond issues. While most public firms have issued only one class of common stock, they may have dozens of bonds differing by maturity, coupon rate, seniority, and so on. Therefore, the bond market can be quite “thin,” in that there are few investors interested in trading a particular issue at any particular time. FIGURE 10.2 Listing of corporate bonds Source: FINRA (Financial Industry Regulatory Authority), October 10, 2019. The bond listings in Figure 10.2 include the coupon, maturity, price, and yield to maturity of each bond. The “Rating” column is the estimation of bond safety given by two major bond rating agencies, Moody’s and Standard & Poor’s. Bonds with A ratings are safer than those rated B or below. As a general rule, safer bonds with higher ratings promise lower yields to maturity. We will return to this topic toward the end of the chapter. CALL PROVISIONS ON CORPORATE BONDS Some corporate bonds are issued with call provisions, allowing the issuer to repurchase the bond at a specified call price before the maturity date. For example, if a company issues a bond with a high coupon rate when market interest rates are high, and interest rates later fall, the firm might like to retire the high-coupon debt and issue new bonds at a lower coupon rate to reduce interest payments. The proceeds from the new bond issue are used to pay for the repurchase of the existing higher-coupon bonds at the call price. This is called refunding. Callable bonds typically come with a period of call protection, an initial time during which the bonds are not callable. Such bonds are referred to as deferred callable bonds. callable bonds Bonds that may be repurchased by the issuer at a specified call price during the call period. The option to call the bond is valuable to the firm, allowing it to buy back the bonds and refinance at lower interest rates when market rates fall. Of course, the firm’s benefit is the bondholder’s burden. When bonds are called, investors must forfeit them for the call price, thereby giving up the prospect of an attractive lending rate on their original investment. As compensation for this risk, callable bonds are issued with higher coupons and promised yields to maturity than noncallable bonds. CONCEPT c h e c k 10.1 Suppose that Verizon issues two bonds with identical coupon rates and maturity dates. One bond is callable, however, while the other is not. Which bond will sell at a higher price? page 288 CONVERTIBLE BONDS Convertible bonds give bondholders an option to exchange each bond for a specified number of shares of common stock of the firm. The conversion ratio gives the number of shares for which each bond may be exchanged. Suppose a convertible bond is issued at par value of $1,000 and is convertible into 40 shares of a firm’s stock. The current stock price is $20 per share, so the option to convert is not profitable now. Should the stock price later rise to $30, however, each bond may be converted profitably into $1,200 worth of stock. The market conversion value is the current value of the shares for which the bonds may be exchanged. At the $20 stock price, for example, the bond’s conversion value is $800. The conversion premium is the excess of the bond price over its conversion value. If the bond were selling currently for $950, its premium would be $150. convertible bond A bond with an option allowing the bondholder to exchange the bond for a specified number of shares of common stock in the firm. Convertible bondholders benefit from price appreciation of the company’s stock. Not surprisingly, this benefit comes at a price; convertible bonds offer lower coupon rates and stated or promised yields to maturity than nonconvertible bonds. At the same time, the actual return on the convertible bond may exceed the stated yield to maturity if the option to convert becomes profitable. We discuss convertible and callable bonds further in Chapter 15. PUTTABLE BONDS Whereas a callable bond gives the issuer the option to extend or retire the bond at the call date, an extendable or put bond gives this option to the bondholder. If the bond’s coupon rate exceeds current market yields, for instance, the bondholder will choose to extend the bond’s life. If the bond’s coupon rate is too low, it will be optimal not to extend; the bondholder instead reclaims principal, which can be invested at current yields. put bond A bond that the holder may choose either to exchange for par value at some date or to extend for a given number of years. FLOATING-RATE BONDS Floating-rate bonds make interest payments tied to some measure of current market rates. For example, the rate might be adjusted annually to the current T-bill rate plus 2%. If the one-year T-bill rate at the adjustment date is 4%, the bond’s coupon rate over the next year would then be 6%. Therefore, the bond always pays approximately current market rates. floating-rate bonds Bonds with coupon rates periodically reset according to a specified market rate. The major risk in floaters has to do with changing credit conditions. The yield spread is fixed over the life of the security, which may be many years. If the financial health of the firm deteriorates, then investors will demand a greater yield premium than is offered by the security, and the price of the bond will fall. While the coupon rate on floaters adjusts to changes in the general level of market interest rates, it does not adjust to changes in financial condition. Preferred Stock Although preferred stock strictly speaking is considered to be equity, it often is included in the fixed-income universe. This is because, like bonds, preferred stock promises to pay a specified cash flow stream. However, unlike bonds, the failure to pay the promised dividend does not result in corporate bankruptcy. Instead, the dividends owed simply cumulate, and the common stockholders may not receive any dividends until the preferred stockholders have been paid in full. In the event of bankruptcy, the claim of preferred stockholders to the firm’s assets has lower priority than that of bondholders but higher priority than that of common stockholders. Preferred stock usually pays a fixed dividend. Therefore, it is in effect a perpetuity, providing a level cash flow indefinitely. In contrast, floating-rate preferred stock is much like floating-rate bonds. The dividend rate is linked to a measure of current market interest rates and is adjusted at regular intervals. Unlike interest payments on bonds, dividends on preferred stock are not considered tax-deductible expenses. This reduces their attractiveness as a source of capital to issuing firms. On the other hand, there is an offsetting tax advantage to preferred stock. When one corporation buys the preferred stock of another, it pays taxes on only 50% of the dividends received. For example, if the firm’s tax bracket is 21%, and it receives $10,000 in preferred-dividend payments, it pays taxes on only $5,000: Total taxes owed are .21 × $5,000 = $1,050. page 289The firm’s effective tax rate on preferred dividends is therefore only .50 × 21% = 10.5%. Given this tax rule, it is not surprising that most preferred stock is held by corporations. Preferred stock rarely gives its holders full voting privileges in the firm. However, if the preferred dividend is skipped, the preferred stockholders will then be provided some voting power. Other Domestic Issuers There are, of course, several issuers of bonds in addition to the Treasury and private corporations. For example, state and local governments issue municipal bonds. Their outstanding feature is that their interest payments are tax-free. We examined municipal bonds, the value of the tax exemption, and the equivalent taxable yield of these bonds in Chapter 2. Government agencies, such as the Federal Home Loan Bank Board, the Farm Credit agencies, and the mortgage pass-through agencies Ginnie Mae, Fannie Mae, and Freddie Mac also issue considerable amounts of bonds. These too were reviewed in Chapter 2. International Bonds International bonds are commonly divided into two categories: foreign bonds and Eurobonds. Foreign bonds are issued by a borrower from a country other than the one in which the bond is sold. The bond is denominated in the currency of the country in which it is marketed. For example, a dollar-denominated bond issued in the U.S. by a German firm is considered a foreign bond. These bonds are given colorful names based on the countries in which they are marketed. Foreign bonds sold in the U.S. are called Yankee bonds. Yen-denominated bonds sold in Japan by non-Japanese issuers are called Samurai bonds. British-pound-denominated foreign bonds sold in the U.K. are called bulldog bonds. In contrast to foreign bonds, Eurobonds are denominated in one currency, usually that of the issuer, but sold in other national markets. For example, the Eurodollar market refers to dollar-denominated bonds sold outside the U.S. (not just in Europe). Because the Eurodollar market falls outside U.S. jurisdiction, these bonds are not regulated by U.S. federal agencies. Similarly, Euroyen bonds are yen-denominated bonds selling outside Japan, Eurosterling bonds are pound-denominated bonds selling outside the U.K., and so on. Innovation in the Bond Market Issuers constantly develop innovative bonds with unusual features; these issues illustrate that bond design can be extremely flexible. Here are examples of some novel bonds. They should give you a sense of the potential variety in security design. MATURITY Bonds conventionally are issued with maturities up to 30 years, but there is nothing sacrosanct about that limit. In recent years, Japan, the U.K., and Austria have issued bonds with maturities ranging from 50 to 100 years. In the 18th century, the British government issued bonds called consols with infinite maturity (thus making these bonds perpetuities). These, however, have been redeemed, and thus no longer trade. INVERSE FLOATERS These are similar to the floating-rate bonds we described earlier, except that the coupon rate on these bonds falls when the general level of interest rates rises. Investors in these bonds suffer doubly when rates rise. Not only does the present value of each dollar of cash flow from the bond fall, but the level of those cash flows falls as well. (Of course, investors in these bonds benefit doubly when rates fall.) ASSET-BACKED BONDS Miramax has issued bonds with coupon rates tied to the financial performance of its films. Tesla has issued bonds with payments backed by revenue generated by leases of its Model S and Model X cars. Some shale companies have created bonds that will be paid off using the revenue generated by oil and gas wells. These are examples of asset-backed securities. The income from a specified group of assets is used to service the debt. More conventional asset-backed securities are mortgage-backed securities or securities backed by auto or credit card loans, as we discussed in Chapter 2. page 290 PAY-IN-KIND BONDS Issuers of pay-in-kind bonds may choose to pay interest either in cash or in additional bonds. If the issuer is short on cash, it will likely choose to pay with new bonds rather than scarce cash. CATASTROPHE BONDS Oriental Land Co., which manages Tokyo Disneyland, has issued bonds with a final payment that depends on whether there has been an earthquake near the park. FIFA (the Fédération Internationale de Football Association) once issued catastrophe bonds with payments that would have been halted if terrorism had forced the cancellation of the World Cup. In 2017, the World Bank issued “pandemic bonds” in which investors forfeit their principal if any of six deadly viruses such as Ebola reach a specified contagion level. The bonds raise money for the World Bank’s efforts to fight potential pandemics, but they relieve the Bank of its payment burden if it is overwhelmed with expenses from disease outbreaks. These bonds are a way to transfer “catastrophe risk” from insurance companies to the capital markets. Investors in these bonds receive compensation in the form of higher coupon rates for taking on the risk. But in the event of a catastrophe, the bondholders will lose all or part of their investments. “Disaster” can be defined either by total insured losses or by specific criteria such as wind speed in a hurricane, Richter level in an earthquake, or contagion level as in the pandemic bonds. Issuance of catastrophe bonds has grown in recent years as insurers have sought ways to spread their risks across a wider spectrum of the capital market. More than $35 billion of various catastrophe bonds were outstanding in 2019. INDEXED BONDS Indexed bonds make payments that are tied to a general price index or the price of a particular commodity. For example, Mexico has issued bonds with payments that depend on the price of oil. Some bonds are indexed to the general price level. The U.S. Treasury started issuing such inflation-indexed bonds in January 1997. They are called Treasury Inflation Protected Securities (TIPS). By tying the par value of the bond to the general level of prices, coupon payments, as well as the final repayment of par value, on these bonds increase in direct proportion to the consumer price index. Therefore, the interest rate on these bonds is a risk-free real rate. To illustrate how TIPS work, consider a newly issued bond with a three-year maturity, par value of $1,000, and a coupon rate of 4%. For simplicity, we will assume the bond makes annual coupon payments. Assume that inflation turns out to be 2%, 3%, and 1% in the next three years. Table 10.1 shows how the bond cash flows will be calculated. The first payment comes at the end of the first year, at t = 1. Because inflation over the year was 2%, the par value of the bond increases from $1,000 to $1,020. The coupon payment is therefore .04 × $1,020 = $40.80. Notice that principal value increases by the inflation rate, and because the coupon payments are 4% of principal, they too increase in proportion to the general price level. Therefore, the cash flows paid by the bond are fixed in real terms. When the bond matures, the investor receives a final coupon payment of $42.44 plus the (price-level-indexed) repayment of principal, $1,061.11.3 Table 10.1 Principal and interest payments for a Treasury Inflation Protected Security page 291 The nominal rate of return on the bond in the first year is The real rate of return is precisely the 4% real yield on the bond: One can show in a similar manner (see Problem 19 in the end-of-chapter questions) that the rate of return in each of the three years is 4% as long as the real yield on the bond remains constant. If real yields do change, then there will be capital gains or losses on the bond. In early 2020, the real yield on TIPS bonds with 30-year maturity was around 0.4%. 10.2 BOND PRICING A bond’s coupon and principal repayments all occur months or years in the future. Therefore, the price an investor is willing to pay for them depends on the value of dollars to be received in the future compared to dollars in hand today. This “present value” calculation depends in turn on market interest rates. As we saw in Chapter 5, the nominal risk-free interest rate equals the sum of (1) a real risk-free rate of return and (2) a premium above the real rate to compensate for expected inflation. In addition, because most bonds are not riskless, the discount rate will embody an additional premium that reflects bond-specific characteristics such as default risk, liquidity, tax attributes, call risk, and so on. We simplify for now by assuming there is one interest rate that is appropriate for discounting cash flows of any maturity, but we can relax this assumption easily. In practice, there may be different discount rates for cash flows accruing in different periods. For the time being, however, we ignore this refinement. To value a security, we discount its expected cash flows by the appropriate discount rate. Bond cash flows consist of coupon payments until the maturity date plus the final payment of par value. Therefore, If we call the maturity date T and call the discount rate r, the bond value can be written as (10.1) The summation sign in Equation 10.1 directs us to add the present value of each coupon payment; each coupon is discounted based on the time until it will be paid. The first term on the right-hand side of Equation 10.1 is the present value of an annuity. The second term is the present value of a single amount, the final payment of the bond’s par value. You may recall from an introductory finance class that the present value of a $1 annuity that lasts for T periods when the interest rate equals r is We call this expression the T-period annuity factor for an interest rate of r.4 Similarly, we call page 292the PV factor, that is, the present value of a single payment of $1 to be received in T periods. Therefore, we can write the price of the bond as (10.2) EXAMPLE 10.2 Bond Pricing We discussed earlier an 8% coupon, 30-year-maturity bond with par value of $1,000 paying 60 semiannual coupon payments of $40 each. Suppose that the interest rate is 8% annually, or r = 4% per six-month period. Then the value of the bond can be written as It is easy to confirm that the present value of the bond’s 60 semiannual coupon payments of $40 each is $904.94 and that the $1,000 final payment of par value has a present value of $95.06 for a total bond value of $1,000. You can calculate this value directly from Equation 10.2, perform these calculations on any financial calculator (see Example 10.3), use a spreadsheet (see Column F of Spreadsheet 10.1), or a set of present value tables. In this example, the coupon rate equals the market interest rate, and the bond price equals par value. If the interest rate were not equal to the bond’s coupon rate, the bond would not sell at par. For example, if the interest rate rises to 10% (5% per six months), the bond’s price will fall by $189.29, to $810.71, as follows: At a higher discount rate, the present value of the payments is lower. Therefore, bond prices fall as market interest rates rise. This illustrates a crucial general rule in bond valuation.5 Bond prices are tedious to calculate without a spreadsheet or financial calculator, but they are easy to calculate with either. Financial calculators designed with present value and future value formulas already programmed can greatly simplify calculations of the sort we just encountered in Example 10.2. The basic financial calculator uses five keys that correspond to the inputs for time-value-of-money problems such as bond pricing: n is the number of time periods. For a bond, n equals the number of periods until maturity. If the bond makes semiannual payments, n is the number of half-year periods or, equivalently, the number of semiannual coupon payments. For example, if the bond has 10 years until maturity, you would enter 20 for n because each payment period is one-half year. i is the interest rate per period, expressed as a percentage (not as a decimal, which is required by spreadsheet programs). For example, if the interest rate is 6%, you would enter 6, not .06. PV is the present value. Many calculators require that PV be entered as a negative number because the purchase of the bond entails a cash outflow, while the receipt of coupon payments and face value are cash inflows. FV is the future value or face value of the bond. In general, FV is interpreted as a one-time future payment of a cash flow, which, for bonds, is the face (i.e., par) value. page 293 PMT is the amount of any recurring payment. For coupon bonds, PMT is the coupon payment; for zero-coupon bonds, PMT is zero. Given any four of these inputs, the calculator will solve for the fifth. EXAMPLE 10.3 Bond Pricing on a Financial Calculator We can illustrate how to use a financial calculator with the bond in Example 10.2. To find its price when the annual market interest rate is 8%, you would enter these inputs (in any order): n 60 The bond has a maturity of 30 years, so it makes 60 semiannual payments. i 4 The semiannual interest rate is 4%. FV 1,000 The bond will provide a one-time cash flow of $1,000 when it matures. PMT 40 Each semiannual coupon payment is $40. On most calculators, you now punch the “compute” key (labeled “COMP” or “CPT”) and then enter PV to obtain the bond price, that is, the present value today of the bond’s cash flows. If you do this, you should find a value of −1,000. The negative sign signifies that while the investor receives cash flows from the bond, the price paid to buy the bond is a cash outflow, or a negative cash flow. If you want to find the value of the bond when the interest rate is 10% (the second part of Example 10.2), just enter 5% for the semiannual interest rate (type “5” and then “i”), and then when you compute PV, you will find that it is −810.71. Figure 10.3 shows the price of the 30-year, 8% coupon bond for a range of interest rates including 8%, at which the bond sells at par, and 10%, at which it sells for $810.71. The negative slope illustrates the inverse relationship between prices and yields. The shape of the curve in Figure 10.3 implies that an increase in the interest rate results in a smaller price decline than the price gain resulting from a rate decrease of equal magnitude. This property of bond prices is called convexity because of the convex shape of the bond price curve. This curvature reflects the fact that progressive increases in the interest rate result in progressively smaller reductions in the bond price.6 Therefore, the price curve becomes flatter at higher interest rates. We will return to convexity in the next chapter. FIGURE 10.3 The inverse relationship between bond prices and yields: Price of an 8% coupon bond with 30-year maturity making semiannual coupon payments page 294 CONCEPT c h e c k 10.2 Calculate the price of the bond if the market interest rate falls from 4% to 3% per half-year. Compare the capital gain when the interest rate falls from 4% to 3% to the loss incurred when the rate increases from 4% to 5%. Corporate bonds typically are issued at par value. This means the underwriters of the bond issue (the firms that market the bonds to the public for the issuing corporation) must choose a coupon rate that very closely approximates market yields. If the coupon rate is inadequate, investors will not be willing to pay par value. After the bonds are issued, bondholders may buy or sell bonds in secondary markets. In these markets, bond prices fluctuate inversely with the market interest rate. The inverse relationship between price and yield is a central feature of fixed-income securities. Interest rate fluctuations represent the main source of risk in the bond market, and we devote considerable attention in the next chapter to assessing the sensitivity of bond prices to market yields. For now, however, we simply highlight one key factor that determines that sensitivity, namely, the maturity of the bond. As a general rule, keeping all other factors the same, the longer the maturity of the bond, the greater the sensitivity of its price to fluctuations in the interest rate. For example, consider Table 10.2, which presents the price of an 8% coupon bond at different market yields and times to maturity. For any departure of the interest rate from 8% (the rate at which the bond sells at par value), the change in the bond price is greater for longer times to maturity. Table 10.2 Bond prices at different interest rates (8% coupon bond, coupons paid semiannually) This makes sense. If you buy the bond at par with an 8% coupon rate, and market rates subsequently rise, then you suffer a loss: You have tied up your money earning 8% when alternative investments offer higher returns. This is reflected in a capital loss on the bond—a fall in its market price. The longer the period for which your money is tied up, the greater the loss and, correspondingly, the greater the drop in the bond price. In Table 10.2, the row for one-year maturity bonds shows little price sensitivity—that is, with only one year’s earnings at stake, changes in interest rates are not too threatening. But for 30-year maturity bonds, interest rate swings have a large impact on bond prices. The force of discounting is greatest for the longest-term bonds. This is why short-term Treasury securities such as T-bills are considered the safest. They are free not only of default risk but also largely of price risk attributable to interest rate volatility. Bond Pricing between Coupon Dates Equation 10.2 for bond prices assumes that the next coupon payment is in precisely one payment period, either a year for an annual payment bond or six months for a semiannual payment bond. But you probably want to be able to price bonds all 365 days of the year, not just on the one or two dates each year that it makes a coupon payment! In principle, the fact that the bond is between coupon dates does not affect the pricing problem. The procedure is always the same: Compute the present value of each remaining page 295payment and sum up. But if you are between coupon dates, there will be fractional periods remaining until each payment, and this does complicate the arithmetic computations. Fortunately, bond pricing functions are included in many financial calculators and spreadsheet programs such as Excel. The spreadsheets allow you to enter today’s date as well as the maturity date of the bond and so can provide prices for bonds at any date. As we pointed out earlier, bond prices are typically quoted net of accrued interest. These prices, which appear in the financial press, are called flat prices. The invoice price a buyer actually pays for the bond includes accrued interest. Thus, When a bond pays its coupon, flat price equals invoice price because at that moment accrued interest reverts to zero. However, this will be the exception, not the rule. Excel pricing functions provide the flat price of the bond. To find the invoice price, we need to add accrued interest. Excel also provides functions that count the days since the last coupon payment and thus can be used to compute accrued interest. Spreadsheet 10.1 illustrates how to use these functions. The spreadsheet provides examples using bonds that have just paid a coupon and so have zero accrued interest, as well as a bond that is between coupon dates. Bond Pricing in Excel Excel asks you to input both the date you buy the bond (called the settlement date) and the maturity date of the bond. The Excel function for bond price is =PRICE(settlement date, maturity date, annual coupon rate, yield to maturity, redemption value as percent of par value, number of coupon payments per year) For the 2.25% coupon November 2025 maturity bond highlighted in Figure 10.1, we would enter the values in Spreadsheet 10.1, Column B. Alternatively, we could simply enter the following function in Excel: =PRICE(DATE(2019,11,15), DATE(2025,11,15), .0225, .01715, 100, 2) The DATE function in Excel, which we use for both the settlement and maturity dates, uses the format DATE(year,month,day). The first date is November 15, 2019, when the bond is purchased, and the second is November 15, 2025, when it matures. Notice that the coupon rate and yield to maturity in Excel must be expressed as decimals, not percentages (as they would be in financial calculators). In most cases, redemption value is 100 (i.e., 100% of par value), and the resulting price similarly is expressed as a percent of page 296par value. Occasionally, however, you may encounter bonds that pay off at a premium or discount to par value. One example would be callable bonds, discussed shortly. SPREADSHEET 10.1 Valuing Bonds Using a Spreadsheet The value of the bond returned by the pricing function is 103.038 (cell B12), which is within 1 cent of the price reported in The Wall Street Journal. (The yield to maturity is reported to only three decimal places, which induces some rounding error.) This bond has just paid a coupon. In other words, the settlement date is precisely at the beginning of the coupon period, so no adjustment for accrued interest is necessary. To illustrate the procedure for bonds between coupon payments, let’s apply the spreadsheet to the 5.375% coupon February 2031 maturity bond which also appears in Figure 10.1. Using the entries in column D of the spreadsheet, we find in cell D12 that the (flat) price of the bond is 135.1906, which, except for minor rounding error, matches the price given in The Wall Street Journal. What about the bond’s invoice price? Rows 12 through 16 make the necessary adjustments. The function described in cell C13 counts the days since the last coupon. This day count is based on the bond’s settlement date, maturity date, coupon period (1 = annual; 2 = semiannual), and day count convention (choice 1 uses actual days). The function described in cell C14 counts the total days in each coupon payment period. Therefore, the entries for accrued interest in row 15 are the semiannual coupon multiplied by the fraction of a coupon period that has elapsed since the last payment. Finally, the invoice prices in row 16 are the sum of flat price plus accrued interest. As a final example, suppose you wish to find the price of the bond in Example 10.2. It is a 30-year maturity bond with a coupon rate of 8% (paid semiannually). The market interest rate given in the latter part of the example is 10%. However, you are not given a specific settlement or maturity date. You can still use the PRICE function to value the bond. Simply choose an arbitrary settlement date (January 1, 2000, is convenient) and let the maturity date be 30 years hence. The appropriate inputs appear in column F of the spreadsheet, with the resulting price, 81.071% of face value, appearing in cell F16. 10.3 BOND YIELDS Most bonds do not sell for par value. But ultimately, barring default, they will mature to par value. Therefore, we would like a measure of rate of return that accounts for coupon income as well as the price increase or decrease over the bond’s life. The yield to maturity is the standard measure of the total rate of return. However, it is far from perfect, and we will explore several variations of this measure. Yield to Maturity In practice, investors considering the purchase of a bond are not quoted a promised rate of return. Instead, they must use the bond price, maturity date, and coupon payments to infer the return offered over the life of the bond. The yield to maturity (YTM) is defined as the discount rate that makes the present value of a bond’s payments equal to its price. It is often viewed as a measure of the average rate of return that will be earned on a bond if it is bought now and held until maturity. To calculate the yield to maturity, we solve the bond price equation for the interest rate given the bond’s price. yield to maturity (YTM) The discount rate that makes the present value of a bond’s payments equal to its price. EXAMPLE 10.4 Yield to Maturity Suppose an 8% coupon, 30-year bond is selling at $1,276.76. What average rate of return would be earned by an investor purchasing the bond at this price? We find the interest rate at which the present value of the remaining 60 semiannual payments equals the bond price. Therefore, we solve for r in the following equation: page 297 or, equivalently, These equations have only one unknown variable, the interest rate, r. You can use a financial calculator or spreadsheet to confirm that the solution is r = .03, or 3% per half-year.7 This is the bond’s yield to maturity. The financial press annualizes the semiannual yield using simple interest techniques, resulting in an annual percentage rate or APR. Yields annualized using simple interest are also called bond equivalent yields. Therefore, this bond’s semiannual yield would be doubled and reported as a bond equivalent yield of 6%. The effective annual yield of the bond, in contrast, accounts for compound interest. If one earns 3% interest every six months, then after one year, each dollar invested grows with interest to $1 × (1.03)2 = 1.0609, and the effective annual interest rate on the bond is 6.09%. In Example 10.4, we noted that a financial calculator or spreadsheet can be used to find the yield to maturity on the coupon bond. Here are two examples demonstrating how you can use these tools. Example 10.5 illustrates the use of financial calculators while Example 10.6 uses Excel. EXAMPLE 10.5 Finding Yield to Maturity Using a Financial Calculator Consider the yield to maturity problem in Example 10.4. On a financial calculator, we would enter the following inputs (in any order): n 60 The bond has a maturity of 30 years, so it makes 60 semiannual payments. PMT 40 Each semiannual coupon payment is $40. PV (−)1,276.76 The bond can be purchased for $1,276.76, which on some calculators must be entered as a negative number as it is a cash outflow. FV 1,000 The bond will provide a one-time cash flow of $1,000 when it matures. Given these inputs, you now use the calculator to find the interest rate at which $1,276.76 actually equals the present value of the 60 payments of $40 plus the one-time payment of $1,000 at maturity. On some calculators, you first punch the “compute” key (labeled “COMP” or “CPT”) and then enter i. You should find that i = 3, or 3% semiannually, as we claimed. (Notice that just as the cash flows are paid semiannually, the computed interest rate is a rate per semiannual time period.) The bond equivalent yield will be reported in the financial press as 6%. Excel also contains built-in functions to compute yield to maturity. The following example, along with Spreadsheet 10.2, illustrates. EXAMPLE 10.6 Finding Yield to Maturity Using Excel Excel’s function for yield to maturity is =YIELD(settlement date, maturity date, annual coupon rate, bond price, redemption value as percent of par value, number of coupon payments per year) The bond price used in the function should be the reported, or “flat,” price, without accrued interest. For example, to find the yield to maturity of the bond in Example 10.5, we would use column B of Spreadsheet 10.2. If the coupons were paid only annually, we would change the entry for payments per year to 1 (see cell D8), and the yield would fall slightly to 5.99%. The yield to maturity is the internal rate of return on an investment in the bond. It can be interpreted as the compound rate of return over the life of the bond under the assumption that all bond coupons can be reinvested at that yield,8 and is therefore widely accepted as a proxy for average return. page 298 SPREADSHEET 10.2 Finding Yield to Maturity Using a Spreadsheet (30-year maturity bond, coupon rate = 8%, price = 100% of par) Yield to maturity differs from the current yield of a bond, which is the bond’s annual coupon payment divided by its price. The current yield of the 8%, 30-year bond selling at $1,276.76 is $80/$1,276.76 = .0627, or 6.27% per year. In contrast, we just saw that the effective annual yield to maturity is 6.09%. For this bond, which is selling at a premium over par value ($1,276 rather than $1,000), the coupon rate (8%) exceeds the current yield (6.27%), which exceeds the yield to maturity (6.09%). The coupon rate exceeds current yield because the coupon rate divides the coupon payments by par value ($1,000), which is less than the bond price ($1,276). In turn, the current yield exceeds yield to maturity because the yield to maturity accounts for the built-in capital loss on the bond; the bond bought today for $1,276 will eventually fall in value to $1,000 at maturity. current yield Annual coupon divided by bond price. This example illustrates a general rule: For premium bonds (bonds selling above par value), coupon rate is greater than current yield, which in turn is greater than yield to maturity. For discount bonds (bonds selling below par value), these relationships are reversed (see Concept Check 10.3). premium bonds Bonds selling above par value. discount bonds Bonds selling below par value. It is common to hear people talking loosely about the yield on a bond. In these cases, they almost always are referring to the yield to maturity. CONCEPT c h e c k 10.3 What will be the relationship among coupon rate, current yield, and yield to maturity for bonds selling at discounts from par? Illustrate using the 30-year 8% (semiannual payment) coupon bond assuming it is selling at a yield to maturity of 10%. Yield to Call Yield to maturity is calculated on the assumption that the bond will be held until maturity. What if the bond is callable, however, and may be retired prior to the maturity date? Figure 10.4 illustrates the risk of call to the bondholder. The colored line is the value of a “straight” (that is, noncallable) bond with par value of $1,000, an 8% coupon rate, and a 30-year time to maturity as a function of the market interest rate. If interest rates fall, the bond price, which equals the present value of the promised payments, can rise substantially. Now consider a bond that has the same coupon rate and maturity date but is callable at 110% of par value, or $1,100. When interest rates fall, the present value of the bond’s scheduled payments rises, but the call provision allows the issuer to repurchase the bond at the call price. If the call price is less than the present value of the scheduled payments, the issuer can call the bond at the expense of the bondholder. FIGURE 10.4 Bond prices: Callable and straight debt. Coupon = 8%; maturity = 30 years; semiannual payments The dark line in Figure 10.4 is the value of the callable bond. At high market interest rates, the risk of call is negligible because the present value of scheduled payments is substantially less than the call price; therefore, the values of the straight and callable bonds converge. At lower rates, however, the values of the bonds begin to diverge, with the difference reflecting page 299the value of the firm’s option to reclaim the callable bond at the call price. At very low market rates the present value of scheduled payments significantly exceeds the call price, so the bond is called. Its value at this point is simply the call price, $1,100. This analysis suggests that investors might be more interested in a bond’s yield to call than its yield to maturity, especially if the bond is likely to be called. The yield to call is calculated just like the yield to maturity, except that the time until call replaces time until maturity and the call price replaces the par value. This computation is sometimes called “yield to first call,” as it assumes the issuer will call the bond as soon as it may do so. EXAMPLE 10.7 Yield to Call Suppose the 8% coupon, 30-year-maturity bond sells for $1,150 and is callable in 10 years at a call price of $1,100. Its yield to maturity and yield to call would be calculated using the following inputs: Yield to Call Yield to Maturity Coupon payment $40 $40 Number of semiannual periods 20 periods 60 periods Final payment $1,100 $1,000 Price $1,150 $1,150 Yield to call is then 6.64%. To confirm this on your calculator, input n = 20; PV = (−)1150; FV = 1100; PMT = 40; compute i as 3.32%, or 6.64% bond equivalent yield. In contrast, yield to maturity is 6.82%. To confirm, input n = 60; PV = (−)1150; FV = 1000; PMT = 40; compute i as 3.41%, or 6.82% bond equivalent yield. In Excel, you can calculate yield to call as =YIELD(DATE(2000,1,1), DATE(2010,1,1), .08, 115, 110, 2). Notice that redemption value is 110, that is, 110% of par value. While most callable bonds are issued with an initial period of explicit call protection, an additional implicit form of call protection operates for bonds selling at deep discounts from their call prices. Even if interest rates fall a bit, deep-discount bonds still will sell below the call price and thus will not be vulnerable to a call. Premium bonds selling near their call prices are especially apt to be called if rates fall further. If interest rates fall, a callable premium bond is likely to provide a lower return than could be earned on a discount bond whose potential price appreciation is not limited by the likelihood of a call. This is why investors in premium bonds may be most interested in its yield to call. page 300 CONCEPT c h e c k 10.4 A 20-year maturity 9% coupon bond paying coupons semiannually is callable in five years at a call price of $1,050. The bond currently sells at a yield to maturity of 8% (bond equivalent yield). What is the yield to call? Realized Compound Return versus Yield to Maturity Yield to maturity will equal the rate of return realized over the life of the bond if all coupons are reinvested to earn the bond’s yield to maturity. Consider, for example, a two-year bond selling at par value paying a 10% coupon once a year. The yield to maturity is 10%. If the $100 coupon payment is reinvested at an interest rate of 10%, the $1,000 investment in the bond will grow after two years to $1,210, as illustrated in Figure 10.5, Panel A. The coupon paid in the first year is reinvested and grows with interest to a second-year value of $110, which, together with the second coupon payment and payment of par value in the second year, results in a total value of $1,210. To summarize, the initial value of the investment is V0 = $1,000. The final value in two years is V2 = $1,210. The compound rate of return, therefore, is calculated as follows. With a reinvestment rate equal to the 10% yield to maturity, the realized compound return also equals yield to maturity. realized compound return Compound rate of return on a bond with all coupons reinvested until maturity. But what if the reinvestment rate is not 10%? If the coupon can be invested at more than 10%, funds will grow to more than $1,210, and the realized compound return will exceed 10%. If the reinvestment rate is less than 10%, so will be the realized compound return. Consider the following example. EXAMPLE 10.8 Realized Compound Return Suppose the interest rate at which the coupon can be invested is only 8%. The following calculations are illustrated in Panel B of Figure 10.5. Future value of first coupon payment with interest earnings $100 × 1.08 = $  108 Cash payment in second year (final coupon plus par value) 1,100 Total value of investment with reinvested coupons $1,208 The realized compound return is the compound rate of growth of invested funds, assuming that all coupon payments are reinvested. The investor purchased the bond for par at $1,000, and the investment grew to $1,208. So the realized compound yield is less than 10%: FIGURE 10.5 Growth of invested funds. In Panel A, interest payments are reinvested at 10%, the bond’s yield to maturity. In Panel B, the reinvestment rate is only 8%. Example 10.8 highlights the problem with conventional yield to maturity when reinvestment rates can change over time. However, with future interest rate uncertainty, the rates at which interim coupons will be reinvested are not yet known. Therefore, while realized compound return can be computed after the investment period ends, it cannot be computed in advance without a forecast of future reinvestment rates. This reduces much of the attraction of the realized return measure. We also can calculate realized compound yield over holding periods greater than one period. This is called horizon analysis and is similar to the procedure in Example 10.8. The forecast of total return depends on your forecasts of both the bond’s yield to maturity when page 301you sell it and the rate at which you are able to reinvest coupon income. With a longer investment horizon, however, reinvested coupons will be a larger component of final proceeds. horizon analysis Analysis of bond returns over a multiyear horizon, based on forecasts of the bond’s yield to maturity and the reinvestment rate of coupons. EXAMPLE 10.9 Horizon Analysis Suppose you buy a 30-year, 7.5% (annual payment) coupon bond for $980 (when its yield to maturity is 7.67%) and you plan to hold it for 20 years. You forecast that the bond’s yield to maturity will be 8% when it is sold and that the reinvestment rate on the coupons will be 6%. At the end of your investment horizon, the bond will have 10 years remaining until maturity, so the forecast sales price (using a yield to maturity of 8%) is $966.45. The 20 coupon payments will grow with compound interest to $2,758.92. (This is the future value of a 20-year $75 annuity with an interest rate of 6%.) Based on these forecasts, your $980 investment will grow in 20 years to $966.45 + $2,758.92 = $3,725.37. This corresponds to an annualized compound return of 6.90%: Examples 10.8 and 10.9 demonstrate that as interest rates change, bond investors are subject to two offsetting sources of risk. On the one hand, when rates rise, bond prices fall, which reduces the value of the portfolio. On the other hand, reinvested coupon income will compound more rapidly at those higher rates. This reinvestment rate risk offsets price risk. In the next chapter, we will explore this trade-off in more detail and will discover that by carefully tailoring their bond portfolios, investors can precisely balance these two effects for any given investment horizon. reinvestment rate risk Uncertainty surrounding the cumulative future value of reinvested bond coupon payments. 13.3 DIVIDEND DISCOUNT MODELS Consider an investor who buys a share of Steady State Electronics stock, planning to hold it for one year. The intrinsic value of the share is the present value of the dividend to be received at the end of the first year, D1, and the expected sales price, P1. We will henceforth use the simpler notation P1 instead of E(P1) to avoid clutter. Keep in mind, though, that future prices and dividends are unknown, and we are dealing with expected values, not certain values. We’ve already established that (13.1) While this year’s dividend is fairly predictable, you might ask how we can estimate P1, the year-end price. According to Equation 13.1, V1 (the year-end value) will be If we assume the stock will be selling for its intrinsic value next year, then V1 = P1, and we can substitute this value for P1 into Equation 13.1 to find This equation may be interpreted as the present value of dividends plus sales price for a two-year holding period. Of course, now we need to come up with a forecast of P2. Continuing in the same way, we can replace P2 by (D3 + P3)/(1 + k), which relates P0 to the value of dividends plus the expected sales price for a three-year holding period. More generally, for a holding period of H years, we can write the stock value as the present value of dividends over the H years plus the ultimate sales price, PH. (13.2) Notice the similarity between this formula and the bond valuation formula developed in Chapter 10. Each relates price to the present value of a stream of payments (coupons in the case of bonds, dividends in the case of stocks) and a final payment (the face value of the bond or the sales price of the stock). The key differences for stocks are the uncertainty of dividends, the lack of a fixed maturity date, and the unknown sales price at the horizon date. Indeed, one can continue to substitute for price indefinitely to conclude (13.3) Equation 13.3 states the stock price should equal the present value of all expected future dividends into perpetuity. This formula is called the dividend discount model (DDM) of stock prices. dividend discount model (DDM) A formula stating that the intrinsic value of a firm equals the present value of all expected future dividends. page 400 It is tempting, but incorrect, to conclude from Equation 13.3 that the DDM focuses exclusively on dividends and ignores capital gains as a motive for investing in stock. Indeed, we assume explicitly in Equation 13.1 that capital gains (as reflected in the expected sales price, P1) are part of the stock’s value. The reason only dividends appear in Equation 13.3 is not that investors ignore capital gains. It is instead that those capital gains will reflect dividend forecasts at the time the stock is sold. That is why in Equation 13.2 we can write the stock price as the present value of dividends plus sales price for any horizon date. PH is the present value at time H of all dividends expected to be paid after the horizon date. That value is then discounted back to today, time 0. The DDM asserts that stock prices are determined ultimately by the cash flows accruing to stockholders, and those are dividends. The Constant-Growth DDM Equation 13.3 as it stands is still not very useful because it requires dividend forecasts for every year into the indefinite future. To make the DDM practical, we need some simplifying assumptions. A useful and common first pass is to assume that dividends are trending upward at a stable growth rate that we will call g. For example, if g = .05 and the most recently paid dividend was D0 = 3.81, expected future dividends are Assuming constant growth in Equation 13.3, we can write intrinsic value as This equation can be simplified to (13.4) Notice in Equation 13.4 that we calculate intrinsic value by dividing D1 (not D0) by k − g. If the market capitalization rate for Steady State is 12%, this equation implies that the intrinsic value of a share of Steady State stock is Equation 13.4 is called the constant-growth DDM or the Gordon model, after Myron Gordon, who popularized it. It should remind you of the formula for the present value of a perpetuity. If dividends were expected not to grow, then the dividend stream would be a simple perpetuity, and the valuation formula for such a nongrowth stock would be P0 = D1/k.1 Equation 13.4 generalizes this formula for the case of a growing perpetuity. For any given value of D1, as g increases, the stock price also rises. constant-growth DDM A form of the dividend discount model that assumes dividends will grow at a constant rate. EXAMPLE 13.1 Preferred Stock and the DDM Preferred stock that pays a fixed dividend can be valued using the constant-growth dividend discount model. The growth rate of dividends is simply zero. For example, to value a preferred stock paying a fixed dividend of $2 per share when the discount rate is 8%, we compute page 401 EXAMPLE 13.2 The Constant-Growth DDM High Flyer Industries has just paid its annual dividend of $3 per share. The dividend is expected to grow at a constant rate of 8% indefinitely. The beta of High Flyer stock is 1, the risk-free rate is 6%, and the market risk pre