14 CHAPTER IN THE PREVIOUS chapters on risk and return relationships, we treated securities at a high level of abstraction. We assumed implicitly that a prior, detailed analysis of each security already had been performed and that its risk and return features had been assessed. We now turn 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. A debt security is a claim on a specified periodic stream of income. Debt securities are often called fixedincome securities because they promise either a fixed stream of income or one that is 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 their cash flows is minimal as long as the issuer of the security is sufficiently creditworthy. That makes these securities a convenient 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 the universe of bond markets, including Treasury, corporate, and international bonds. We turn next to bond pricing, showing how bond prices are set in accordance with market interest rates and why bond prices change with those rates. Given this background, we can compare the 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 certain tax rules that apply to debt securities, and show how to calculate aftertax returns. Finally, 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. Credit risk is central to fixed-income derivatives such as collateralized debt obligations and credit default swaps, so we examine these instruments as well. 425 PART IV Bond Prices and Yields 426 PART IV Fixed-Income Securities 14.1 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 the “IOU” of the borrower. The arrangement obligates the issuer to make specified payments to the bondholder on specified dates. A typical coupon bond obligates the issuer to make semiannual payments of interest to the bondholder 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 repays the debt by paying the bond’s par value (equivalently, its face value). The coupon rate of the bond determines the interest payment: The annual payment is 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. To illustrate, a bond with par value of $1,000 and coupon rate of 8% might be sold to a buyer for $1,000. The bondholder is then entitled to a payment of 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 30-year life of the bond, 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 the issue price and the payment of par value at maturity. We will return to these bonds later. Treasury Bonds and Notes U.S. Treasury Quotes MATURITY COUPON BID ASKED ASKED CHANGE YIELD (%) May 15 18 1.000 100.3984 100.4141 -0.0859 0.791 May 15 19 0.875 99.8125 99.8281 -0.0859 0.933 Feb 15 21 7.875 130.5781 130.5938 -0.2656 1.225 Aug 15 25 6.875 144.4141 144.4297 -0.5391 1.670 Aug 15 25 2.000 102.2813 102.2969 -0.3438 1.730 May 15 30 6.250 152.3984 152.4609 -0.7969 1.950 Nov 15 41 3.125 111.7891 111.8203 -0.8750 2.496 May 15 46 2.500 2.595 97.9922 98.0234 -0.9063 Figure 14.1 Prices and yields of U.S. Treasury bonds Source: The Wall Street Journal Online, May 16, 2016. 1 Figure 14.1 is an excerpt from the listing of Treasury issues. Treasury notes are issued with original maturities ranging 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. The highlighted bond in Figure 14.1 matures on May 15, 2046. Its coupon rate is 2.5%. Par value typically is $1,000; thus the bond pays interest of $25 per year in two semiannual payments of $12.50. Payments are made in May and November of each year. Although bonds usually are sold in denominations of $1,000, the bid and ask prices are quoted as a percentage of par value.1 Therefore, the ask price is Recall that the bid price is the price at which you can sell the bond to a dealer. The ask price, which is slightly higher, is the price at which you can buy the bond from a dealer. CHAPTER 14 Bond Prices and Yields 98.0234% of par, or $980.234. The minimum price increment, or tick size, in The Wall Street Journal listing is 1/128, so this bond may also be viewed as selling for 98 3/128 percent of par value.2 The last column, labeled “Asked Yield to Maturity,” is the yield to maturity on the bond based on the ask price. The yield to maturity is a measure of the average rate of return to an investor who purchases the bond for the ask 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 bond prices that you see quoted in the financial pages are not actually the prices that investors 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 30/182 of the semiannual coupon. The sale, or invoice, price of the bond would equal the stated price (sometimes called the flat price) plus the accrued interest. In general, the formula for the amount of accrued interest between two dates is Annual coupon payment Days since last coupon payment Accrued interest = ____________________ × ____________________________ 2 Days separating coupon payments Example 14.1 Accrued Interest Suppose that the coupon rate is 8%. Then the annual coupon is $80 and the semiannual coupon payment is $40. Because 30 days have passed since the last coupon payment, the accrued interest on the bond is $40 × (30/182) = $6.59. 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 (par value plus semiannual interest) on the following day and so should be willing to pay a total price of $1,040 for the bond. The bond price is quoted net of accrued interest in the financial pages and thus appears as $1,000.3 Corporate Bonds Like the government, corporations borrow money by issuing bonds. Figure 14.2 is a sample of listings for a few actively traded corporate bonds. Although some bonds trade electronically on the NYSE Bonds platform, most bonds are traded over-the-counter in 2 Bonds traded on formal exchanges are subject to minimum tick sizes set by the exchange. For example, the minimum price increment on the 2-year Treasury bond futures contract (traded on the Chicago Board of Trade) is 1/128, although longer-term T-bonds have larger tick sizes. Private traders can negotiate their own tick size. For example, one can find price quotes on Bloomberg screens with tick sizes as low as 1/256. 3 In contrast to bonds, stocks do not trade at flat prices with adjustments for “accrued dividends.” Whoever owns the stock when it goes “ex-dividend” receives the entire dividend payment, and the stock price reflects the value of the upcoming dividend. The price therefore typically falls by about the amount of the dividend on the “ex-day.” There is no need to differentiate between reported and invoice prices for stocks. 427 428 PART IV Fixed-Income Securities ISSUER NAME SYMBOL COMMONWEALTH BK AUSTRALIA MEDIUM TERM NT CBAU3828562 2.250% COUPON MATURITY MOODY'S/S&P/ HIGH FITCH LOW LAST Mar 16 17 Aaa//AAA 100.892 100.892 100.892 CHANGE YIELD % 0.0020 1.1102 WALGREENS BOOTS WAG4182650 ALLIANCE INC 4.800% Nov 18 44 Baa2 /BBB /BBB 103.367 100.560 100.560 -2.0100 4.7634 ANHEUSER BUSCH INBEV FIN INC BUD4327481 3.650% Feb 01 26 A3 // 104.593 104.096 104.249 -0.0130 3.1254 HSBC HLDGS PLC HBC3699239 6.100% Jan 14 42 A1 //AA- 129.300 128.850 128.850 SOUTHERN CO SO4365686 1.850% Jul 01 19 Baa2 //A- 100.438 100.324 100.324 -0.0310 1.7411 WESTPAC BKG CORP GOLDMAN SACHS GROUP INC WBK4248362 1.550% May 25 18 Aa2 //AA- 100.246 100.148 100.148 -0.1900 1.4738 GS4302031 4.750% Oct 21 45 A3 /BBB+ /A 107.139 106.419 106.727 0.0500 4.3389 HSBC HLDGS PLC HBC4365146 3.900% May 25 26 A1 // 101.564 100.889 101.564 0.1580 3.7109 NEWELL BRANDS INC NWL4346211 2.600% Mar 29 19 Baa3 //BBB- 103.118 101.774 101.774 0.2360 1.9510 LLOYDS TSB BK PLC LYG3833921 4.200% Mar 28 17 A1 //A+ 102.462 102.389 102.389 -0.0770 1.2682 1.4860 4.2419 Figure 14.2 Listing of corporate bonds Source: FINRA (Financial Industry Regulatory Authority), May 31, 2016. a network of bond dealers linked by a computer quotation system. In practice, the bond market can be quite “thin,” with few investors interested in trading a particular issue at any particular time. The bond listings in Figure 14.2 include the coupon, maturity, price, and yield to maturity of each bond. The “rating” column is the estimation of bond safety given by the three major bond-rating agencies—Moody’s, Standard & Poor’s, and Fitch. Bonds with gradations of A ratings are safer than those with B ratings or below. As a general rule, safer bonds with higher ratings promise lower yields to maturity than other bonds with similar maturities. 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. 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. 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. Holders of called bonds must forfeit their bonds for the call price, thereby giving up the attractive coupon rate on their original investment. To compensate investors for this risk, callable bonds are issued with higher coupons and promised yields to maturity than noncallable bonds. CHAPTER 14 Bond Prices and Yields Concept Check 14.1 Suppose that Verizon issues two bonds with identical coupon rates and maturity dates. One bond is callable, however, whereas the other is not. Which bond will sell at a lower price? 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 is 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’s value over its conversion value. If the bond were selling currently for $950, its premium would be $150. Convertible bondholders benefit from price appreciation of the company’s stock. Again, this benefit comes at a price: Convertible bonds offer lower coupon rates and stated or promised yields to maturity than do nonconvertible bonds. However, 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 20. Puttable Bonds While the callable bond gives the issuer the option to extend or retire the bond at the call date, the 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; in this case, the bondholder will instead reclaim principal, which can be invested at current yields. Floating-Rate Bonds Floating-rate bonds make interest payments that are 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 1-year T-bill rate at the adjustment date is 4%, the bond’s coupon rate over the next year would then be 6%. This arrangement means that the bond always pays approximately current market rates. The major risk involved in floaters has to do with changes in the firm’s financial strength. 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. In this case, the price of the bond will fall. Although the coupon rate on floaters adjusts to changes in the general level of market interest rates, it does not adjust to changes in the financial condition of the firm. 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 stream of dividends. However, unlike bonds, the failure to pay the promised 429 430 PART IV Fixed-Income Securities 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, preferred stockholders’ claims to the firm’s assets have lower priority than those of bondholders but higher priority than those of common stockholders. Preferred stock commonly 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 to the firm. 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 corporation, it pays taxes on only 30% of the dividends received. For example, if the firm’s tax bracket is 35%, and it receives $10,000 in preferred-dividend payments, it will pay taxes on only $3,000 of that income: Total taxes owed on the income will be .35 × $3,000 = $1,050. The firm’s effective tax rate on preferred dividends is therefore only .30 × 35% = 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 may 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. The outstanding feature of these is that 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, if a German firm sells a dollar-denominated bond in the United States, the bond is considered a foreign bond. These bonds are given colorful names based on the countries in which they are marketed. For example, foreign bonds sold in the United States are called Yankee bonds. Like other bonds sold in the United States, they are registered with the Securities and Exchange Commission. Yen-denominated bonds sold in Japan by non-Japanese issuers are called Samurai bonds. British pound-denominated foreign bonds sold in the United Kingdom 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 United States (not just in Europe), although London is the largest market for Eurodollar bonds. 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 United Kingdom, and so on. CHAPTER 14 Bond Prices and Yields Innovation in the Bond Market Issuers constantly develop innovative bonds with unusual features; these issues illustrate that bond design can be extremely flexible. The novel bonds discussed next will give you a sense of the potential variety in security design. 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 as the discount rate rises, 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 Pulp Fiction and other films. Domino’s Pizza has issued bonds with payments backed by revenues from its pizza franchises. 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. Catastrophe Bonds Oriental Land Company, which manages Tokyo Disneyland, issued a bond in 1999 with a final payment that depended on the occurrence of an earthquake near the park. More recently, FIFA (the Fédération Internationale de Football Association) issued catastrophe bonds with payments that would have been halted if terrorism had forced the cancellation of the 2006 World Cup. These bonds are a way to transfer “catastrophe risk” from the firm to the capital markets. Investors in these bonds receive compensation for taking on the risk in the form of higher coupon rates. But in the event of a catastrophe, the bondholders will give up all or part of their investments. “Disaster” can be defined by total insured losses or by criteria such as wind speed in a hurricane or Richter level in an earthquake. 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. 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 United States 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 3-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 14.1 shows how the bond’s 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; because the coupon rate is 4%, the coupon payment is 4% of this amount, or $40.80. Notice that par value increases by the inflation rate, and because the coupon payments are 4% of par, they too increase in proportion to the general price level. Therefore, the cash flows paid by the bond are fixed in real 431 432 PART IV Fixed-Income Securities Table 14.1 Principal and interest payments for a Treasury Inflation Protected Security Time Inflation in Year Just Ended 0 1 2 3 2% 3 1 Coupon + Par Value Payment $1,000.00 1,020.00 1,050.60 1,061.11 $40.80 42.02 42.44 Principal Repayment $ 0 0 1,061.11 = Total Payment $ 40.80 42.02 1,103.55 terms. When the bond matures, the investor receives a final coupon payment of $42.44 plus the (price-level-indexed) repayment of principal, $1,061.11.4 The nominal rate of return on the bond in the first year is Interest + Price appreciation 40.80 + 20 Nominal return = ________________________ = _________ = 6.80% Initial price 1,000 The real rate of return is precisely the 4% real yield on the bond: 1 + Nominal return 1.0608 Real return = _______________ − 1 = ______ − 1 = .04, or 4% 1 + Inflation 1.02 One can show in a similar manner (see Problem 18 in the end-of-chapter problems) 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 mid-2016, the real yield on long-term TIPS bonds was about 0.9%. 14.2 Bond Pricing Because a bond’s coupon and principal repayments all occur months or years in the future, the price an investor would be willing to pay for a claim to those payments 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. The cash flows from a bond consist of coupon payments until the maturity date plus the final payment of par value. Therefore, Bond value = Present value of coupons + Present value of par value 4 By the way, total nominal income (i.e., coupon plus that year’s increase in principal) is treated as taxable income in each year. CHAPTER 14 Bond Prices and Yields If we call the maturity date T and call the interest rate r, the bond value can be written as T Coupon Par value Bond value = ∑ _______t + ________ (1 + r)T t = 1 (1 + r) (14.1) The summation sign in Equation 14.1 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 14.1 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 1 1 that lasts for T periods when the interest rate equals r is __r 1 − _T . We call this [ (1 + r) ] 1 expression the T-period annuity factor for an interest rate of r.5 Similarly, we call _______T (1 + r) the 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 1 1 1 Price = Coupon × __ 1 − _T + Par value × ______T r[ (1 + r) ] (1 + r) (14.2) = Coupon × Annuity factor(r, T ) + Par value × PV factor(r, T ) Example 14.2 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 60 $40 $1,000 Price = ∑ +______t + _______ (1.04)60 t+=+1 (1.04) = $40 × Annuity factor(4%, 60) + $1,000 × PV factor(4%, 60) (14.3) 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 14.2, perform these calculations on any financial calculator (see Example 14.3 below), use a spreadsheet program (see column F of Spreadsheet 14.1), or use 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 5 Here is a quick derivation of the formula for the present value of an annuity. An annuity lasting T periods can be viewed as equivalent to a perpetuity whose first payment comes at the end of the current period less another perpetuity whose first payment comes at the end of the (T + 1)st period. The immediate perpetuity net of the delayed perpetuity provides exactly T payments. We know that the value of a $1 per period perpetuity is $1/r. 1 1 Therefore, the present value of the delayed perpetuity is $1/r discounted for T additional periods, or __r × ______T . (1 + r) The present value of the annuity is the present value of the first perpetuity minus the present value of the delayed 1 1 perpetuity, or __r 1 − _T . [ (1 + r) ] 433 434 PART IV Fixed-Income Securities would not sell at par value. For example, if the interest rate were to rise to 10% (5% per six months), the bond’s price would fall by $189.29 to $810.71, as follows: $40 × Annuity factor(5%, 60) + $1,000 × PV factor(5%, 60) = $757.17 + $53.54 = $810.71 At a higher interest rate, the present value of the payments to be received by the bondholder is lower. Therefore, bond prices fall as market interest rates rise. This illustrates a crucial general rule in bond valuation.6 Bond prices are tedious to calculate without a spreadsheet or a financial calculator, but they are easy to calculate with either. Financial calculators designed with present and future value formulas already programmed can greatly simplify calculations of the sort we just encountered in Example 14.2. The basic financial calculator uses five keys that correspond to the inputs for time-value-of-money problems such as bond pricing: 1. n is the number of time periods. In the case of a bond, n equals the number of periods until the bond matures. 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, since each payment period is one-half year. 2. i is the interest rate per period, expressed as a percentage (not as a decimal). For example, if the interest rate is 6%, you would enter 6, not .06. 3. PV is the present value. Many calculators require that PV be entered as a negative number, in recognition of the fact that purchase of the bond is a cash outflow, while the receipt of coupon payments and face value are cash inflows. 4. 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. 5. PMT is the amount of any recurring payment. For coupon bonds, PMT is the coupon payment; for zero-coupon bonds, PMT will be zero. Given any four of these inputs, the calculator will solve for the fifth. We can illustrate with the bond presented in Example 14.2. Example 14.3 Bond Pricing on a Financial Calculator To find the bond’s price when the annual market interest rate is 8%, you would enter these inputs (in any order): n i FV PMT 6 60 4 1,000 40 The bond has a maturity of 30 years, so it makes 60 semiannual payments. The semiannual market interest rate is 4%. The bond will provide a one-time cash flow of $1,000 when it matures. Each semiannual coupon payment is $40. Here is a trap to avoid. You should not confuse the bond’s coupon rate, which determines the interest paid to the bondholder, with the market interest rate. Once a bond is issued, its coupon rate is fixed. When the market interest rate increases, investors discount any fixed payments at a higher discount rate, which implies that present values and bond prices fall. CHAPTER 14 435 Bond Prices and Yields 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 14.2), just enter 5% for the semiannual interest rate (type “5” and then “i”), and when you compute PV, you will find that it is −810.71. Figure 14.3 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 $810.71. The negative slope illustrates the inverse relationship between prices and yields. The shape of the curve in Figure 14.3 implies that an increase in the interest rate results in a price decline that is smaller than the price gain resulting from a decrease of equal magnitude in the interest rate. This property of bond prices is called convexity because of the convex shape of the bond price curve. This curvature reflects the fact that progressive increases in the interest rate result in progressively smaller reductions in the bond price.7 Therefore, the price curve becomes flatter at higher interest rates. We return to convexity in Chapter 16. Concept Check 14.2 Calculate the price of the 30-year, 8% coupon bond for a market interest rate of 3% per half-year. Compare the capital gain for the interest rate decline to the loss incurred when the rate increases from 4% to 5%. 7 4,000 3,500 3,000 Bond Price ($) Corporate bonds typically are issued at par value. This means that 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. In a primary issue, the underwriters attempt to sell the newly issued bonds directly to their customers. If the coupon rate is inadequate, investors will not pay par value for the bonds. 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 fixedincome securities. Interest rate fluctuations represent the main source of risk in the fixed-income market, and we devote 2,500 2,000 1,500 1,000 810.71 500 0 0 5 8 10 Interest Rate (%) 15 20 Figure 14.3 The inverse relationship between bond prices and yields. Price of an 8% coupon bond with 30-year maturity making semiannual payments The progressively smaller impact of interest rate increases results largely from the fact that at higher rates the bond is worth less. Therefore, an additional increase in rates operates on a smaller initial base, resulting in a smaller price decline. 436 PART IV Fixed-Income Securities Table 14.2 Bond prices at different interest rates (8% coupon bond, coupons paid semiannually) Bond Price at Given Market Interest Rate Time to Maturity 2% 4% 6% 8% 10% 1 year 10 years 20 years 30 years 1,059.11 1,541.37 1,985.04 2,348.65 1,038.83 1,327.03 1,547.11 1,695.22 1,019.13 1,148.77 1,231.15 1,276.76 1,000.00 1,000.00 1,000.00 1,000.00 981.41 875.35 828.41 810.71 considerable attention in Chapter 16 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 price to fluctuations in the interest rate. For example, consider Table 14.2, 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. 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 14.2, the row for 1-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 to be the safest. In addition to being free of default risk, they are also largely free of price risk attributable to interest rate volatility. Bond Pricing between Coupon Dates Equation 14.2 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 payment 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 most spreadsheet programs such as Excel. The spreadsheet allows 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. The nearby box shows you how. 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 actual invoice price that a buyer pays for the bond includes accrued interest. Thus, Invoice price = Flat price + Accrued interest eXcel APPLICATIONS: Bond Pricing E xcel and most other spreadsheet programs provide built-in functions to compute bond prices and yields. They typically ask 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 2.5% coupon May 2046 maturity bond highlighted in Figure 14.1, we would enter the values in Spreadsheet 14.1. (Notice that in spreadsheets, we must enter interest rates as decimals, not percentages). Alternatively, we could simply enter the following function in Excel: = PRICE(DATE(2016,5,15), DATE(2046,5,15), .025, .02595, 100, 2) The DATE function in Excel, which we use for both the settlement and maturity date, uses the format DATE(year,month,day). The first date is May 15, 2016, when the bond is purchased, and the second is May 15, 2046, when it matures. Most bonds pay coupons either on the 15th or the last business day of the month. Notice that the coupon rate and yield to maturity are expressed as decimals, not percentages. In most cases, redemption value is 100 (i.e., 100% of par value), and the resulting price similarly is expressed as a percent of par 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. The value of the bond returned by the pricing function is 98.0282 (cell B12), which nearly matches the price reported in 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 A B 2.5% coupon bond, maturing May 15, 2046 Settlement date Maturity date Annual coupon rate Yield to maturity Redemption value (% of face value) Coupon payments per year Flat price (% of par) Days since last coupon Days in coupon period Accrued interest Invoice price Spreadsheet 14.1 Bond Pricing in Excel 5/15/2016 5/15/2046 0.025 0.02595 100 2 98.0282 0 184 0 98.0282 Table 14.1. (The yield to maturity is reported to only three decimal places, which results in a little 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, consider the 2% coupon August 2025 bond, also appearing in Figure 14.1. Using the entries in column D of the spreadsheet, we find in cell D12 that the (flat) price of the bond is 102.2977, which matches the price given in the figure except for a few cents’ rounding error. What about the bond’s invoice price? Rows 13 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 price in row 16 is the sum of the flat price plus accrued interest. As a final example, suppose you wish to find the price of the bond in Example 14.2. 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, 81.0707% of face value, appearing in cell F16. C Formula in column B = DATE (2016, 5, 15) = DATE (2046, 5, 15) =PRICE(B4,B5,B6,B7,B8,B9) =COUPDAYBS(B4,B5,2,1) =COUPDAYS(B4,B5,2,1) =(B13/B14)*B6*100/2 =B12+ B15 D E 2% coupon bond, maturing August 2025 F G 8% coupon bond, 30-year maturity 5/15/2016 8/15/2025 0.02 0.0173 100 2 1/1/2000 1/1/2030 0.08 0.1 100 2 102.2977 90 182 0.495 102.7922 81.0707 0 182 0 81.0707 eXcel Please visit us at www.mhhe.com/Bodie11e 437 438 PART IV Fixed-Income Securities 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 exceptional case, not the rule. Excel pricing functions provide the flat price of the bond. To find the invoice price, we need to add accrued interest. Fortunately, Excel also provides functions that count the days since the last coupon payment date and thus can be used to compute accrued interest. The nearby box also illustrates how to use these functions. The box provides examples using a bond that has just paid a coupon, and so has zero accrued interest, as well as a bond that is between coupon dates. 14.3 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 both current income and 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, an investor considering the purchase of a bond is not quoted a promised rate of return. Instead, the investor must use the bond price, maturity date, and coupon payments to infer the return offered by the bond over its life. The yield to maturity (YTM) is defined as the interest rate that makes the present value of a bond’s payments equal to its price. This interest rate is often interpreted 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. Example 14.4 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. This is the rate consistent with the observed price of the bond. Therefore, we solve for r in the following equation: or, equivalently, 60 $40 $1,000 $1,276.76 = ∑ +______t + _______ (1 + r+)60 t!=+1 (1 + r+) 1,276.76 = 40 × Annuity factor(r, 60) + 1,000 × PV factor(r, 60) These equations have only one unknown variable, the interest rate, r. As we will see in a moment, you can use a financial calculator or spreadsheet to confirm that the solution is r = .03, or 3%, per half-year. This is the bond’s yield to maturity. The financial press reports yields on an annualized basis, and annualizes the bond’s 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, the semiannual yield would be doubled and reported in the newspaper as a bond equivalent yield of 6%. The effective annual yield of the bond, however, 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%. CHAPTER 14 Bond Prices and Yields In Example 14.4, we asserted 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 14.5 illustrates the use of financial calculators while Example 14.6 uses Excel. Example 14.5 n PMT PV FV 60 40 (−)1,276.76 1,000 Finding the Yield to Maturity Using a Financial Calculator The bond has a maturity of 30 years, so it makes 60 semiannual payments. Each semiannual coupon payment is $40. 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. 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 each 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 to have the interest rate computed. If you do so, you will 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 that you can use to find yield to maturity. Example 14.6, along with Spreadsheet 14.2, illustrates these functions. Example 14.6 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 semiannual payment bond in Example 14.4, we would use column B of Spreadsheet 14.2. 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%. A 1 2 3 4 5 6 7 8 9 10 11 12 B C Semiannual coupons Settlement date Maturity date Annual coupon rate Bond price (flat) Redemption value (% of face value) Coupon payments per year Yield to maturity (decimal) D E Annual coupons 1/1/2000 1/1/2030 0.08 127.676 100 2 1/1/2000 1/1/2030 0.08 127.676 100 1 0.0600 0.0599 The formula entered here is: =YIELD(B3,B4,B5,B6,B7,B8) Spreadsheet 14.2 Finding yield to maturity in Excel 439 440 PART IV Fixed-Income Securities The bond’s yield to maturity is the internal rate of return on an investment in the bond. The yield to maturity 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 Yield to maturity is widely accepted as a proxy for average return. Yield to maturity differs from the current yield of a bond, which is the bond’s annual coupon payment divided by the bond price. For example, for the 8%, 30-year bond currently selling at $1,276.76, the current yield would be $80/$1,276.76 = .0627, or 6.27%, per year. In contrast, recall 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. Examples 14.4, 14.5, and 14.6 illustrate 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 14.3). 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 Check 14.3 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 maturity, 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? How should we measure average rate of return for bonds subject to a call provision? Figure 14.4 illustrates the risk of call to the bondholder. The top curve is the value of a “straight” (i.e., noncallable) bond with par value $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 $1,100. When interest rates fall, the present value of the bond’s scheduled payments rises, but the call provision allows the issuer to repurchase the bond at the call price. If the call price is less than the present value of the scheduled payments, the issuer may call the bond back from the bondholder. The lower curve in Figure 14.4 is the value of the callable bond. At high interest rates, the risk of call is negligible because the present value of scheduled payments is less than 8 If the reinvestment rate does not equal the bond’s yield to maturity, the compound rate of return will differ from YTM. This is demonstrated below in Examples 14.8 and 14.9. CHAPTER 14 Example 14.7 2,000 1,800 1,600 Prices ($) 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 the value of the firm’s option to reclaim the callable bond at the call price. At very low rates, the present value of scheduled payments 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 bond market analysts might be more interested in a bond’s yield to call rather than 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. Straight Bond 1,400 1,200 1,100 1,000 Callable Bond 800 600 400 200 0 3 4 5 6 7 8 9 10 11 12 13 Interest Rate (%) Figure 14.4 Bond prices: Callable and straight debt (coupon = 8%; maturity = 30 years; semiannual payments) 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: Coupon payment Number of semiannual periods Final payment Price 441 Bond Prices and Yields Yield to Call Yield to Maturity $40 20 periods $1,100 $1,150 $40 60 periods $1,000 $1,150 Yield to call is then 6.64%. [To confirm this on a calculator, input n = 20; PV = (−)1150; FV = 1100; PMT = 40; compute i as 3.32%, or 6.64% bond equivalent yield.] 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 input as 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 subject to a call. Premium bonds that might be selling near their call prices, however, 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 442 PART IV Fixed-Income Securities appreciation is not limited by the likelihood of a call. Investors in premium bonds therefore may be more interested in the bond’s yield to call than its yield to maturity because it may appear to them that the bond will be retired at the call date. Concept Check 14.4 a. The yield to maturity on two 10-year maturity bonds currently is 7%. Each bond has a call price of $1,100. One bond has a coupon rate of 6%, the other 8%. Assume for simplicity that bonds are called as soon as the present value of their remaining payments exceeds their call price. What will be the capital gain on each bond if the market interest rate suddenly falls to 6%? b. 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%. What is the yield to call? Realized Compound Return versus Yield to Maturity A: Reinvestment Rate = 10% $1,100 Cash Flow: Time: 0 $100 1 2 $1,100 Future Value: = $1,100 100 × 1.10 = $ 110 $1,210 B: Reinvestment Rate = 8% $1,100 Cash Flow: Time: 0 Future Value: $100 1 2 $1,100 = $1,100 100 × 1.08 = $ 108 $1,208 Figure 14.5 Growth of invested funds Yield to maturity will equal the rate of return realized over the life of the bond if all coupons are reinvested at an interest rate equal to the bond’s yield to maturity. Consider, for example, a 2-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 14.5, 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: V 0 (1 + r)2 = V 2 $1,000 (1 + r)2 = $1,210 r = .10 = 10% With a reinvestment rate equal to the 10% yield to maturity, the realized compound return equals yield to 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. CHAPTER 14 Example 14.8 Bond Prices and Yields Realized Compound Return If the interest rate earned on the first coupon is less than 10%, the final value of the investment will be less than $1,210, and the realized compound return will be less than 10%. To illustrate, suppose the interest rate at which the coupon can be invested is only 8%. The following calculations are illustrated in Figure 14.5, Panel B. 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 this investment grew to $1,208. V 0+(1 + r+)2 = V 2 $1,000+(1 + r+)2 = $1,208 r = .0991 = 9.91% Example 14.8 highlights the problem with conventional yield to maturity when reinvestment rates can change over time. Conventional yield to maturity will not equal realized compound return. However, in an economy with future interest rate uncertainty, the rates at which interim coupons will be reinvested are not yet known. Therefore, although 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. Forecasting the realized compound yield over various holding periods or investment horizons is called horizon analysis. The forecast of total return depends on your forecasts of both the price of the bond when you sell it at the end of your horizon and the rate at which you are able to reinvest coupon income. The sales price depends in turn on the yield to maturity at the horizon date. With a longer investment horizon, however, reinvested coupons will be a larger component of your final proceeds. Example 14.9 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 plan to hold it for 20 years. Your forecast is 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 expiration, so the forecast sales price (using a yield to maturity of 8%) will be $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%.) On the basis of 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%: V 0+(1 + r+)20 = V 20 $980+(1 + r+)20 = $3,725.37 r = .0690 = 6.90% 443 444 PART IV Fixed-Income Securities Examples 14.8 and 14.9 demonstrate that as interest rates change, bond investors are actually subject to two sources of offsetting 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 will offset the impact of price risk. In Chapter 16, 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. 14.4 Bond Prices over Time A bond will sell at par value when its coupon rate equals the market interest rate. In these circumstances, the investor receives fair compensation for the time value of money in the form of the recurring coupon payments. No further capital gain is necessary to provide fair compensation. When the coupon rate is lower than the market interest rate, the coupon payments alone will not provide investors as high a return as they could earn elsewhere in the market. To receive a competitive return on such an investment, investors also need some price appreciation on their bonds. The bonds, therefore, must sell below par value to provide a “built-in” capital gain on the investment. Example 14.10 Fair Holding-Period Return To illustrate built-in capital gains or losses, suppose a bond was issued several years ago when the interest rate was 7%. The bond’s annual coupon rate was thus set at 7%. (We will suppose for simplicity that the bond pays its coupon annually.) Now, with three years left in the bond’s life, the interest rate is 8% per year. The bond’s market price is the present value of the remaining annual coupons plus payment of par value. That present value is9 $70 × Annuity factor(8%, 3) + $1,000 × PV factor(8%, 3) = $974.23 which is less than par value. In another year, after the next coupon is paid and remaining maturity falls to two years, the bond would sell at $70 × Annuity factor(8%, 2) + $1,000 × PV factor(8%, 2) = $982.17 thereby yielding a capital gain over the year of $7.94. If an investor had purchased the bond at $974.23, the total return over the year would equal the coupon payment plus capital gain, or $70 + $7.94 = $77.94. This represents a rate of return of $77.94/$974.23, or 8%, exactly the rate of return currently available elsewhere in the market. Concept Check 14.5 At what price will the bond in Example 14.10 sell in yet another year, when only one year remains until maturity? What is the rate of return to an investor who purchases the bond when its price is $982.17 and sells it one year hence? 9 Using a calculator, enter n = 3, i = 8, PMT = 70, FV = 1000, and compute PV. CHAPTER 14 Bond Prices and Yields 445 Price (% of Par Value) When bond prices are set according to the present value formula, any dis160 count from par value provides an anticipated capital gain that will augment 140 a below-market coupon rate by just 120 enough to provide a fair total rate of return. Conversely, if the coupon rate 100 exceeds the market interest rate, the 80 interest income by itself is greater than that available elsewhere in the market. 60 Investors will bid up the price of these 40 bonds above their par values. As the Coupon = 12% Coupon = 4% bonds approach maturity, they will fall 20 in value because fewer of these above0 market coupon payments remain. The 0 5 10 15 20 25 30 resulting capital losses offset the large Time (years) coupon payments so that the bondholder again receives only a competitive rate of return. Figure 14.6 Price path of two 30-year maturity bonds, each sellProblem 14 at the end of the chapter ing at a yield to maturity of 8%. Bond price approaches par value as asks you to work through the case of maturity date approaches. the high-coupon bond. Figure 14.6 traces out the price paths (net of accrued interest) of two bonds, each selling at a yield to maturity of 8%. One bond has a coupon rate above 8%, while the other has a coupon rate below 8%. The low-coupon bond enjoys capital gains as price steadily approaches par value, whereas the high-coupon bond suffers capital losses.10 We use these examples to show that each bond offers investors the same total rate of return. Although the capital gains versus income components differ, the price of each bond is set to provide competitive rates, as we should expect in well-functioning capital markets. Security returns all should be comparable on an after-tax risk-adjusted basis. If they are not, investors will try to sell low-return securities, thereby driving down their prices until the total return at the now-lower price is competitive with other securities. Prices should continue to adjust until all securities are fairly priced in that expected returns are comparable, given appropriate risk and tax adjustments. We see evidence of this price adjustment in Figure 14.1. Compare the two bonds maturing in August 2025. One has a coupon rate of 6.875%, while the other’s coupon rate is only 2%. But the higher coupon rate on the first bond does not mean that it offers a higher return; instead, it sells at a higher price. The yields to maturity on the two bonds are nearly equal, both just about 1.7%. This makes sense, since investors should care about their total return, including both coupon income as well as price change. In the end, prices of similarmaturity bonds adjust until yields are pretty much equalized. Of course, the yields across bonds in Figure 14.1 are not all precisely equal. Clearly, longer term bonds at this time offered higher promised yields, a common pattern and one that reflects the relative risks of the bonds. We will explore the relationship between yield and time to maturity in the next chapter. 10 If the market interest rate is volatile, the price path will be “jumpy,” vibrating around the price path in Figure 14.6 and reflecting capital gains or losses as interest rates fluctuate. Ultimately, however, the price must reach par value at the maturity date, so the price of the premium bond will fall over time while that of the discount bond will rise. 446 PART IV Fixed-Income Securities Yield to Maturity versus Holding-Period Return In Example 14.10, the holding-period return and the yield to maturity were equal. The bond yield started and ended the year at 8%, and the bond’s holding-period return also equaled 8%. This turns out to be a general result. When the yield to maturity is unchanged over the period, the rate of return on the bond will equal that yield. As we noted, this should not be surprising: The bond must offer a rate of return competitive with those available on other securities. However, when yields fluctuate, so will a bond’s rate of return. Unanticipated changes in market rates will result in unanticipated changes in bond returns and, after the fact, a bond’s holding-period return can be better or worse than the yield at which it initially sells. An increase in the bond’s yield acts to reduce its price, which reduces the holding-period return. In this event, the holding-period return is likely to be less than the initial yield to maturity.11 Conversely, a decline in yield will result in a holding-period return greater than the initial yield. Example 14.11 Yield to Maturity versus Holding-Period Return Consider a 30-year bond paying an annual coupon of $80 and selling at par value of $1,000. The bond’s initial yield to maturity is 8%. If the yield remains at 8% over the year, the bond price will remain at par, so the holding-period return also will be 8%. But if the yield falls below 8%, the bond price will increase. Suppose the yield falls and the price increases to $1,050. Then the holding-period return is greater than 8%: $80 + ($1,050 − $1,000+) Holding-period return = ___________________ = .13, or 13% $1,000 Concept Check 14.6 Show that if yield to maturity increases, then holding-period return is less than initial yield. For example, suppose in Example 14.11 that by the end of the first year, the bond’s yield to maturity is 8.5%. Find the one-year holdingperiod return and compare it to the bond’s initial 8% yield to maturity. Here is another way to think about the difference between yield to maturity and holding-period return. Yield to maturity depends only on the bond’s coupon, current price, and par value at maturity. All of these values are observable today, so yield to maturity can be easily calculated. Yield to maturity is commonly interpreted as a measure of the average rate of return if the investment in the bond is held until the bond matures. In contrast, holding-period return is the rate of return over a particular investment period and depends on the market price of the bond at the end of that holding period; of course this price is not known today. Because bond prices over the holding period will respond to unanticipated changes in interest rates, holding-period return can at most be forecast. 11 We have to be a bit careful here. When yields increase, coupon income can be reinvested at higher rates, which offsets the impact of the initial price decline. If your holding period is sufficiently long, the positive impact of the higher reinvestment rate can more than offset the initial price decline. But common performance evaluation periods for portfolio managers are no more than one year, and over these shorter horizons the price impact will almost always dominate the impact of the reinvestment rate. We discuss the trade-off between price risk and reinvestment rate risk more fully in Chapter 16. CHAPTER 14 447 Bond Prices and Yields Zero-Coupon Bonds and Treasury Strips 30 27 24 21 18 15 12 9 6 3 0 Price ($) Original-issue discount bonds are less common than coupon bonds issued at par. These are bonds that are issued intentionally with low coupon rates that cause the bond to sell at a discount from par value. The most common example of this type of bond is the zero-coupon bond, which carries no coupons and provides all of its return in the form of price appreciation. Zeros provide only one cash flow to their owners, on the maturity date of the bond. U.S. Treasury bills are examples of short-term zero-coupon instruments. If the bill has face value of $10,000, the Treasury issues or sells it for some amount less than $10,000, agreeing to repay $10,000 at maturity. All of the investor’s return comes in the form of price appreciation. Longer-term zero-coupon bonds are commonly created from coupon-bearing notes and bonds. A bond dealer who purchases a Treasury coupon bond may ask the Treasury to break down the cash flows to be paid by the bond into a series of independent securities, where each security is a claim to one of the payments of the original bond. For example, a 10-year coupon bond would be “stripped” of its 20 semiannual coupons, and each coupon payment would be treated as a stand-alone zero-coupon bond. The maturities of these bonds would thus range from six months to 10 years. The final payment of principal would be treated as another stand-alone zero-coupon security. Each of the payments is now treated as an independent security and is assigned its own CUSIP number (by the Committee on Uniform Securities Identification Procedures). The CUSIP number is the security identifier that allows for electronic trading over the Fedwire system, a network that connects all Federal Reserve banks and their branches. The payments are still considered obligations of the U.S. Treasury. The Treasury program under which coupon stripping is performed is called STRIPS (Separate Trading of Registered Interest and Principal of Securities), and these zero-coupon securities are called Treasury strips. What should happen to prices of zeros as time passes? On their maturity dates, zeros must sell for par value. Before maturity, however, they should sell at discounts from par, because of the time value of money. As time passes, price should approach par value. In fact, if the interest rate is constant, a zero’s price will increase at exactly the rate of interest. To illustrate, consider a zero with 30 years until maturity, and suppose the market interest rate is 1,000 900 10% per year. The price of the bond today is $1,000/ 800 (1.10)30 = $57.31. Next year, with only 29 years 700 until maturity, if the yield is still 10%, the price will 600 29 be $1,000/(1.10) = $63.04, a 10% increase over 500 400 its previous-year value. Because the par value of 300 the bond is now discounted for one less year, its 200 price has increased by the 1-year discount factor. 100 Figure 14.7 presents the price path of a 30-year 0 zero-coupon bond for an annual market interest rate of 10%. The bond’s price rises exponentially, Time (years) not linearly, until its maturity. Today Maturity Date After-Tax Returns The tax authorities recognize that the “built-in” price appreciation on original-issue discount (OID) bonds such as zero-coupon bonds represents an implicit interest payment to the holder of Figure 14.7 The price of a 30-year zero-coupon bond over time at a yield to maturity of 10%. Price equals $1,000/(1.10)T, where T is time until maturity. 448 PART IV Fixed-Income Securities the security. The IRS, therefore, calculates a price appreciation schedule to impute taxable interest income for the built-in appreciation during a tax year, even if the asset is not sold or does not mature until a future year. Any additional gains or losses that arise from changes in market interest rates are treated as capital gains or losses if the OID bond is sold during the tax year. Example 14.12 Taxation of Original-Issue Discount Bonds Continuing with the example in the text, if the interest rate originally is 10%, the 30-year zero will be issued at a price of $1,000/1.1030 = $57.31. The following year, the IRS will calculate what the bond price would be if the yield were still 10%. This is $1,000/1.1029 = $63.04. Therefore, the IRS imputes interest income of $63.04 − $57.31 = $5.73. This amount is subject to tax. Notice that the imputed interest income is based on a “constant yield method” that ignores any changes in market interest rates. If interest rates actually fall, let’s say to 9.9%, the bond price will be $1,000/1.09929 = $64.72. If the bond is sold, then the difference between $64.72 and $63.04 will be treated as capital gains income and taxed at the capital gains tax rate. If the bond is not sold, then the price difference is an unrealized capital gain and will not result in taxes in that year. In either case, the investor must pay taxes on the $5.73 of imputed interest at the rate on ordinary income. The procedure illustrated in Example 14.12 applies as well to the taxation of other original-issue discount bonds, even if they are not zero-coupon bonds. Consider, as an example, a 30-year maturity bond that is issued with a coupon rate of 4% and a yield to maturity of 8%. For simplicity, we will assume that the bond pays coupons once annually. Because of the low coupon rate, the bond will be issued at a price far below par value, specifically at $549.69. If the bond’s yield to maturity is still 8%, then its price in one year will rise to $553.66. (Confirm this for yourself.) This would provide a pretax holding-period return (HPR) of exactly 8%: $40 + ($553.66 − $549.69) HPR = _____________________ = .08 $549.69 The increase in the bond price based on a constant yield, however, is treated as interest income, so the investor is required to pay taxes on the explicit coupon income, $40, as well as the imputed interest income of $553.66 − $549.69 = $3.97. If the bond’s yield actually changes during the year, the difference between the bond’s price and the constant-yield value of $553.66 will be treated as capital gains income if the bond is sold. Concept Check 14.7 Suppose that the yield to maturity of the 4% coupon, 30-year maturity bond falls to 7% by the end of the first year and that the investor sells the bond after the first year. If the investor’s federal plus state tax rate on interest income is 38% and the combined tax rate on capital gains is 20%, what is the investor’s after-tax rate of return? CHAPTER 14 Bond Prices and Yields 14.5 Default Risk and Bond Pricing Although bonds generally promise a fixed flow of income, that income stream is not riskless unless the investor can be sure the issuer will not default on the obligation. While U.S. government bonds may be treated as free of default risk, this is not true of corporate bonds. Therefore, the actual payments on these bonds are uncertain, for they depend to some degree on the ultimate financial status of the firm. Bond default risk, usually called credit risk, is measured by Moody’s Investor Services, Standard & Poor’s Corporation, and Fitch Investors Service, all of which provide financial information on firms as well as quality ratings of large corporate and municipal bond issues. International sovereign bonds, which also entail default risk, especially in emerging markets, also are commonly rated for default risk. Each rating firm assigns letter grades to the bonds of corporations and municipalities to reflect their assessment of the safety of the bond issue. The top rating is AAA or Aaa, a designation awarded to only about a dozen firms. Moody’s modifies each rating class with a 1, 2, or 3 suffix (e.g., Aaa1, Aaa2, Aaa3) to provide a finer gradation of ratings. The other agencies use a + or − modification. Those rated BBB or above (S&P, Fitch) or Baa and above (Moody’s) are considered investment-grade bonds, whereas lower-rated bonds are classified as speculative-grade or junk bonds. Defaults on low-grade issues are not uncommon. For example, almost half of the bonds rated CCC by Standard & Poor’s at issue have defaulted within 10 years. Highly rated bonds rarely default, but even these bonds are not free of credit risk. For example, in 2001 WorldCom sold $11.8 billion of bonds with an investment-grade rating. Only a year later, the firm filed for bankruptcy and its bondholders lost more than 80% of their investment. Certain regulated institutional investors such as insurance companies have not always been allowed to invest in speculative-grade bonds. Figure 14.8 provides the definitions of each bond rating classification. Junk Bonds Junk bonds, also known as high-yield bonds, are nothing more than speculative-grade (low-rated or unrated) bonds. Before 1977, almost all junk bonds were “fallen angels,” that is, bonds issued by firms that originally had investment-grade ratings but that had since been downgraded. In 1977, however, firms began to issue “original-issue junk.” Much of the credit for this innovation is given to Drexel Burnham Lambert, and especially its trader Michael Milken. Drexel had long enjoyed a niche as a junk bond trader and had established a network of potential investors in junk bonds. Firms not able to muster an investment-grade rating were happy to have Drexel (and other investment bankers) market their bonds directly to the public, as this opened up a new source of financing. Junk issues were a lower-cost financing alternative than borrowing from banks. High-yield bonds gained considerable notoriety in the 1980s when they were used as financing vehicles in leveraged buyouts and hostile takeover attempts. Shortly thereafter, however, the junk bond market suffered. The legal difficulties of Drexel and Michael Milken in connection with Wall Street’s insider trading scandals of the late 1980s tainted the junk bond market. At the height of Drexel’s difficulties, the high-yield bond market nearly dried up. Since then, the market has rebounded dramatically. However, the average credit quality of newly issued high-yield debt issued today is higher than the average quality in the boom years of the 1980s. Of course, junk bonds are still more vulnerable to economic distress than 449 450 PART IV Fixed-Income Securities Bond Ratings Very High Quality Standard & Poor’s AAA High Quality AA A BBB Speculative BB B Very Poor CCC D Moody’s Aaa Aa A Baa Ba B Caa C At times both Moody’s and Standard & Poor’s have used adjustments to these ratings: S&P uses plus and minus signs: A+ is the strongest A rating and A− the weakest. Moody’s uses a 1, 2, or 3 designation, with 1 indicating the strongest. Moody’s S&P Aaa AAA Debt rated Aaa and AAA has the highest rating. Capacity to pay interest and principal is extremely strong. Aa AA Debt rated Aa and AA has a very strong capacity to pay interest and repay principal. Together with the highest rating, this group comprises the highgrade bond class. A A Baa BBB Ba B Caa Ca BB B CCC CC C D C D Debt rated A has a strong capacity to pay interest and repay principal, although it is somewhat more susceptible to the adverse effects of changes in circumstances and economic conditions than debt in higher-rated categories. Debt rated Baa and BBB is regarded as having an adequate capacity to pay interest and repay principal. Whereas it normally exhibits adequate protection parameters, adverse economic conditions or changing circumstances are more likely to lead to a weakened capacity to pay interest and repay principal for debt in this category than in higher-rated categories. These bonds are medium-grade obligations. Debt rated in these categories is regarded, on balance, as predominantly speculative with respect to capacity to pay interest and repay principal in accordance with the terms of the obligation. BB and Ba indicate the lowest degree of speculation, and CC and Ca the highest degree of speculation. Although such debt will likely have some quality and protective characteristics, these are outweighed by large uncertainties or major risk exposures to adverse conditions. Some issues may be in default. This rating is reserved for income bonds on which no interest is being paid. Debt rated D is in default, and payment of interest and/or repayment of principal is in arrears. Figure 14.8 Definitions of each bond rating class Source: Stephen A. Ross and Randolph W. Westerfield, Corporate Finance, Copyright 1988 (St. Louis: Times Mirror/Mosby College Publishing, reproduced with permission from the McGraw-Hill Companies, Inc.). Data from various editions of Standard & Poor’s Bond Guide and Moody’s Bond Guide. investment-grade bonds. During the financial crisis of 2008–2009, prices on these bonds fell dramatically, and their yields to maturity rose equally dramatically. The spread between yields on B-rated bonds and Treasuries widened from around 3% in early 2007 to an astonishing 19% by the beginning of 2009. CHAPTER 14 451 Bond Prices and Yields Determinants of Bond Safety Bond rating agencies base their quality ratings largely on an analysis of the level and trend of some of the issuer’s financial ratios. The key ratios used to evaluate safety are: 1. Coverage ratios—Ratios of company earnings to fixed costs. For example, the times-interest-earned ratio is the ratio of earnings before interest payments and taxes to interest obligations. The fixed-charge coverage ratio includes lease payments and sinking fund payments with interest obligations to arrive at the ratio of earnings to all fixed cash obligations (sinking funds are described below). Low or falling coverage ratios signal possible cash flow difficulties. 2. Leverage (e.g., debt-to-equity) ratios—A too-high leverage ratio indicates excessive indebtedness, signaling the possibility the firm will be unable to earn enough to satisfy the obligations on its bonds. 3. Liquidity ratios—The two most common liquidity ratios are the current ratio (current assets/current liabilities) and the quick ratio (current assets excluding inventories/current liabilities). These ratios measure the firm’s ability to pay bills coming due with its most liquid assets. 4. Profitability ratios—Measures of rates of return on assets or equity. Profitability ratios are indicators of a firm’s overall financial health. The return on assets (earnings before interest and taxes divided by total assets) or return on equity (net income/ equity) are the most popular of these measures. Firms with higher returns on assets or equity should be better able to raise money in security markets because they offer prospects for better returns on the firm’s investments. 5. Cash flow-to-debt ratio—This is the ratio of total cash flow to outstanding debt. Moody’s periodically computes median values of selected ratios for firms in several rating classes, which we present in Table 14.3. Of course, ratios must be evaluated in the context of industry standards, and analysts differ in the weights they place on particular ratios. Nevertheless, Table 14.3 demonstrates the tendency of ratios to improve along with the firm’s rating class. Many studies have tested whether financial ratios can in fact be used to predict default risk. One of the best-known series of tests was conducted by Edward Altman, who used EBITA/Assets (%) Operating profit margin (%) EBITA to interest coverage (multiple) Debt/EBITDA (multiple) Debt/(Debt + Equity) Funds from operations/Total debt (multiple) Retained cash flow/Net debt (multiple) Aaa Aa A Baa Ba B C 20.9% 22.0% 28.9 0.58 19.3% 1.335 1.3 15.6% 17.1% 15.1 2.03 50.2% 0.385 0.3 13.8% 17.6% 9.7 1.83 38.6% 0.425 0.4 10.9% 14.1% 5.9 2.58 46.2% 0.296 0.3 9.1% 11.2% 3.5 3.41 51.7% 0.206 0.2 7.1% 8.9% 1.7 5.26 72.0% 0.120 0.1 4.0% 4.1% 0.6 8.35 98.0% 0.031 0.0 Table 14.3 Financial ratios by rating class Note: EBITA is earnings before interest, taxes, and amortization. EBITDA is earnings before interest, taxes, depreciation, and amortization. Source: Moody’s Financial Metrics, Key Ratios by Rating and Industry for Global Non-Financial Corporations, December 2013. 452 PART IV Fixed-Income Securities ROE discriminant analysis to predict bankruptcy. With this technique a firm is assigned a score based on its financial characteristics. If its score exceeds a cut-off value, the firm is deemed creditworthy. A score below the cut-off value indicates significant bankruptcy risk in the near future. To illustrate the technique, suppose that we were to collect data on the return on equity (ROE) and coverage ratios of a sample of firms, and then keep records of any corporate bankruptcies. In Figure 14.9 we plot the ROE and coverage ratios Coverage Ratio for each firm, using X for firms that eventually went bankrupt and O for those that remained solvent. Clearly, the X and O firms show different Figure 14.9 Discriminant analysis patterns of data, with the solvent firms typically showing higher values for the two ratios. The discriminant analysis determines the equation of the line that best separates the X and O observations. Suppose that the equation of the line is .75 = .9 × ROE + .4 × Coverage. Then, based on its own financial ratios, each firm is assigned a “Z-score” equal to .9 × ROE + .4 × Coverage. If its Z-score exceeds .75, the firm plots above the line and is considered a safe bet; Z-scores below .75 foretell financial difficulty. Altman found the following equation to best separate failing and nonfailing firms: EBIT Sales Shareholders’ equity Z = 3.1 __________ + 1.0 ______ + .42 _________________ Total assets Assets Total liabilities Retained earnings Working capital +.85 _______________ + .72 ______________ Total assets Total assets where EBIT = earnings before interest and taxes.12 Z-scores below 1.23 indicate vulnerability to bankruptcy, scores between 1.23 and 2.90 are a gray area, and scores above 2.90 are considered safe. Concept Check 14.8 Suppose we add a new variable equal to current liabilities/current assets to Altman’s equation. Would you expect this variable to receive a positive or negative coefficient? Bond Indentures A bond is issued with an indenture, which is the contract between the issuer and the bondholder. Part of the indenture is a set of restrictions that protect the rights of the bondholders. Such restrictions include provisions relating to collateral, sinking funds, dividend 12 Altman’s original work was published in Edward I. Altman, “Financial Ratios, Discriminant Analysis, and the Prediction of Corporate Bankruptcy,” Journal of Finance 23 (September 1968). This equation is from his updated study, Corporate Financial Distress and Bankruptcy, 2nd ed. (New York: Wiley, 1993), p. 29. Altman’s analysis is updated and extended in W. H. Beaver, M. F. McNichols, and J-W. Rhie, “Have Financial Statements Become Less Informative? Evidence from the Ability of Financial Ratios to Predict Bankruptcy,” Review of Accounting Studies 10 (2005), pp. 93–122. CHAPTER 14 Bond Prices and Yields policy, and further borrowing. The issuing firm agrees to these protective covenants in order to market its bonds to investors concerned about the safety of the bond issue. Sinking Funds Bonds call for the payment of par value at the end of the bond’s life. This payment constitutes a large cash commitment for the issuer. To help ensure the commitment does not create a cash flow crisis, the firm agrees to establish a sinking fund to spread the payment burden over several years. The fund may operate in one of two ways: 1. The firm may repurchase a fraction of the outstanding bonds in the open market each year. 2. The firm may purchase a fraction of the outstanding bonds at a special call price associated with the sinking fund provision. The firm has an option to purchase the bonds at either the market price or the sinking fund price, whichever is lower. To allocate the burden of the sinking fund call fairly among bondholders, the bonds chosen for the call are selected at random based on serial number.13 The sinking fund call differs from a conventional bond call in two important ways. First, the firm can repurchase only a limited fraction of the bond issue at the sinking fund call price. At most, some indentures allow firms to use a doubling option, which allows repurchase of double the required number of bonds at the sinking fund call price. Second, while callable bonds generally have call prices above par value, the sinking fund call price usually is set at the bond’s par value. Although sinking funds ostensibly protect bondholders by making principal repayment more likely, they can hurt the investor. The firm will choose to buy back discount bonds (selling below par) at market price, while exercising its option to buy back premium bonds (selling above par) at par. Therefore, if interest rates fall and bond prices rise, firms will benefit from the sinking fund provision that enables them to repurchase their bonds at below-market prices. In these circumstances, the firm’s gain is the bondholder’s loss. One bond issue that does not require a sinking fund is a serial bond issue, in which the firm sells bonds with staggered maturity dates. As bonds mature sequentially, the principal repayment burden for the firm is spread over time, just as it is with a sinking fund. One advantage of serial bonds over sinking fund issues is that there is no uncertainty introduced by the possibility that a particular bond will be called for the sinking fund. The disadvantage, however, is that bonds of different maturity dates are not interchangeable, which reduces the liquidity of the issue. Subordination of Further Debt One of the factors determining bond safety is total outstanding debt of the issuer. If you bought a bond today, you would be understandably distressed to see the firm tripling its outstanding debt tomorrow. Your bond would be riskier than it appeared when you bought it. To prevent firms from harming bondholders in this manner, subordination clauses restrict the amount of additional borrowing. Additional debt might be required to be subordinated in priority to existing debt; that is, in the event of bankruptcy, subordinated or junior debtholders will not be paid unless and until the prior senior debt is fully paid off. 13 Although it is less common, the sinking fund provision also may call for periodic payments to a trustee, with the payments invested so that the accumulated sum can be used for retirement of the entire issue at maturity. 453 454 PART IV Fixed-Income Securities Dividend Restrictions Covenants also limit the dividends firms may pay. These limitations protect the bondholders because they force the firm to retain assets rather than paying them out to stockholders. A typical restriction disallows payments of dividends if cumulative dividends paid since the firm’s inception exceed cumulative retained earnings plus proceeds from sales of stock. Collateral Some bonds are issued with specific collateral behind them. Collateral is a particular asset that the bondholders receive if the firm defaults on the bond. If the collateral is property, the bond is called a mortgage bond. If the collateral takes the form of other securities held by the firm, the bond is a collateral trust bond. In the case of equipment, the bond is known as an equipment obligation bond. This last form of collateral is used most commonly by firms such as railroads, where the equipment is fairly standard and can be easily sold to another firm should the firm default. Collateralized bonds generally are considered safer than general debenture bonds, which are unsecured, meaning they do not provide for specific collateral. Credit risk of unsecured bonds depends on the general earning power of the firm. If the firm defaults, debenture owners become general creditors of the firm. Because they are safer, collateralized bonds generally offer lower yields than general debentures. Figure 14.10 shows the terms of a huge $6.5 billion bond issue by Apple in 2015. We have added some explanatory notes alongside the terms of the issue. Yield to Maturity and Default Risk Comment Description of Bond 1. Interest of 3.45% will be payable on February 9 and August 9 of each year. Thus every 6 months each note will pay interest of (.0345/2) × $1,000 = $17.25. ISSUE: Apple Inc. 3.45% Notes 2. Investors will be repaid the $1,000 face value in 2045. DUE: February 9, 2045 3. Moody’s bond rating is Aa, the second-highest-quality rating. RATING: Aa 4. A trustee is appointed to look after investors’ interest. TRUSTEE: Issued under an indenture between Apple and The Bank of New York Mellon Trust Company 5. The bonds are registered. The registrar keeps a record of who owns the bonds. REGISTERED: Issued in registered, book entry form 6. The company is not obliged to repay any of the bonds on a regular basis before maturity. SINKING FUND: None 7. The company has the option to buy back the notes. The redemption price is the greater of $1,000 or a price that is determined by the value of an equivalent Treasury bond. CALLABLE: In whole or in part at any time 8. The notes are senior debt, ranking equally with all Apple’s other unsecured senior debt. SENIORITY 9. The notes are not secured; that is, no assets have been set aside to protect the noteholders in the event of default. However, if Apple sets aside assets to protect any other bondholders, the notes will also be secured by these assets. This is termed a negative pledge clause. SECURITY: The notes are unsecured. However, “if Apple shall incur, assume or guarantee any Debt, … it will secure … the debt securities then outstanding equally and ratably with … such Debt.” 10. The principal amount of the issue was $2 billion. The notes were sold at 99.11% of their principal value. OFFERED: $2,000,000,000 at 99.11% 11. The book runners are the managing underwriters to the issue and maintain the book of securities sold. JOINT BOOK - RUNNING MANAGERS: Goldman, Sachs; Deutsche Bank Securities Figure 14.10 Apple’s 2015 bond issue. Because corporate bonds are subject to default risk, we must distinguish between the bond’s promised yield to maturity and its expected yield. The promised or stated yield will be realized only if the firm meets the obligations of the bond issue. Therefore, the stated yield is the maximum possible yield to maturity of the bond. The expected yield to maturity must take into account the possibility of a default. For example, at the height of the financial crisis in October 2008, as Ford Motor Company struggled, its bonds due in 2028 were rated CCC and were selling at about 33% of par value, resulting in a yield to maturity of about 20%. Investors did not really believe the expected rate of return on these bonds was 20%. They recognized that there was a decent chance that bondholders would not receive all the payments promised in the bond contract and that the yield based on expected cash flows was far less than the yield based on promised cash flows. As it turned out, of course, Ford weathered the storm, and investors who purchased its bonds made a very nice profit: The bonds were selling in mid2016 for about 117% of par value, about 3.5 times their value in 2008. CHAPTER 14 Example 14.13 Bond Prices and Yields Expected versus Promised Yield to Maturity Suppose a firm issued a 9% coupon bond 20 years ago. The bond now has 10 years left until its maturity date, but the firm is having financial difficulties. Investors believe that the firm will be able to make good on the remaining interest payments, but at the maturity date, the firm will be forced into bankruptcy, and bondholders will receive only 70% of par value. The bond is selling at $750. Yield to maturity (YTM) would then be calculated using the following inputs: Coupon payment Number of semiannual periods Final payment Price Expected YTM Stated YTM $45 20 periods $700 $750 $45 20 periods $1,000 $750 The stated yield to maturity, which is based on promised payments, is 13.7%. Based on the expected payment of $700 at maturity, however, the yield to maturity is only 11.6%. The stated yield to maturity is greater than the yield investors actually expect to earn. Example 14.13 suggests that when a bond becomes more subject to default risk, its price will fall, and therefore its promised yield to maturity will rise. Similarly, the default premium, the spread between the stated yield to maturity and that on otherwisecomparable Treasury bonds, will rise. However, its expected yield to maturity, which ultimately is tied to the systematic risk of the bond, will be far less affected. Let’s continue Example 14.13. Example 14.14 Default Risk and the Default Premium Suppose that the condition of the firm in Example 14.13 deteriorates further, and investors now believe that the bond will pay off only 55% of face value at maturity. Because of the higher risk, investors now demand an expected yield to maturity of 12% (i.e., 6% semiannually), which is .4% higher than in Example 14.13. But the price of the bond will fall from $750 to $688 [n = 20; i = 6; FV = 550; PMT = $45]. At this price, the stated yield to maturity based on promised cash flows is 15.2%. While the expected yield to maturity has increased by .4%, the drop in price has caused the promised yield to maturity to rise by 1.5%. Concept Check 14.9 What is the expected yield to maturity in Example 14.14 if the firm is in even worse condition? Investors expect a final payment of only $500, and the bond price has fallen to $650. To compensate for the possibility of def...
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