The Valuation of Long-Term Securities

Transcription

1 4 The Vauation of Long-Term Securities Contents Distinctions Among Vauation Concepts Liquidation Vaue versus Going-Concern Vaue Book Vaue versus Market Vaue Market Vaue versus Intrinsic Vaue Bond Vauation Perpetua Bonds Bonds with a Finite Maturity Preferred Stock Vauation Common Stock Vauation Are Dividends the Foundation? Dividend Discount Modes Rates of Return (or Yieds) Yied to Maturity (YTM) on Bonds Yied on Preferred Stock Yied on Common Stock Summary Tabe of Key Present Vaue Formuas for Vauing Long-Term Securities Key Learning Points Questions Sef-Correction Probems Probems Soutions to Sef-Correction Probems Seected References Objectives After studying Chapter 4, you shoud be abe to: Distinguish among the various terms used to express vaue, incuding iquidation vaue, going-concern vaue, book vaue, market vaue, and intrinsic vaue. Vaue bonds, preferred stocks, and common stocks. Cacuate the rates of return (or yieds) of different types of ong-term securities. List and expain a number of observations regarding the behavior of bond prices. 73

2 Part 2 Vauation What is a cynic? A man who knows the price of everything and the vaue of nothing. OSCAR WILDE In the ast chapter we discussed the time vaue of money and expored the wonders of compound interest. We are now abe to appy these concepts to determining the vaue of different securities. In particuar, we are concerned with the vauation of the firm s ong-term securities bonds, preferred stock, and common stock (though the principes discussed appy to other securities as we). Vauation wi, in fact, underie much of the ater deveopment of the book. Because the major decisions of a company are a interreated in their effect on vauation, we must understand how investors vaue the financia instruments of a company. Distinctions Among Vauation Concepts The term vaue can mean different things to different peope. Therefore we need to be precise in how we both use and interpret this term. Let s ook briefy at the differences that exist among some of the major concepts of vaue. Liquidation vaue The amount of money that coud be reaized if an asset or a group of assets (e.g., a firm) is sod separatey from its operating organization. Going-concern vaue The amount a firm coud be sod for as a continuing operating business. Book vaue (1) An asset: the accounting vaue of an asset the asset s cost minus its accumuated depreciation; (2) a firm: tota assets minus iabiities and preferred stock as isted on the baance sheet. Market vaue The market price at which an asset trades. Intrinsic vaue The price a security ought to have based on a factors bearing on vauation. Liquidation Vaue versus Going-Concern Vaue Liquidation vaue is the amount of money that coud be reaized if an asset or a group of assets (e.g., a firm) is sod separatey from its operating organization. This vaue is in marked contrast to the going-concern vaue of a firm, which is the amount the firm coud be sod for as a continuing operating business. These two vaues are rarey equa, and sometimes a company is actuay worth more dead than aive. The security vauation modes that we wi discuss in this chapter wi generay assume that we are deaing with going concerns operating firms abe to generate positive cash fows to security investors. In instances where this assumption is not appropriate (e.g., impending bankruptcy), the firm s iquidation vaue wi have a major roe in determining the vaue of the firm s financia securities. Book Vaue versus Market Vaue The book vaue of an asset is the accounting vaue of the asset the asset s cost minus its accumuated depreciation. The book vaue of a firm, on the other hand, is equa to the doar difference between the firm s tota assets and its iabiities and preferred stock as isted on its baance sheet. Because book vaue is based on historica vaues, it may bear itte reationship to an asset s or firm s market vaue. In genera, the market vaue of an asset is simpy the market price at which the asset (or a simiar asset) trades in an open marketpace. For a firm, market vaue is often viewed as being the higher of the firm s iquidation or going-concern vaue. Market Vaue versus Intrinsic Vaue Based on our genera definition for market vaue, the market vaue of a security is the market price of the security. For an activey traded security, it woud be the ast reported price at which the security was sod. For an inactivey traded security, an estimated market price woud be needed. The intrinsic vaue of a security, on the other hand, is what the price of a security shoud be if propery priced based on a factors bearing on vauation assets, earnings, future 74

3 4 The Vauation of Long-Term Securities prospects, management, and so on. In short, the intrinsic vaue of a security is its economic vaue. If markets are reasonaby efficient and informed, the current market price of a security shoud fuctuate cosey around its intrinsic vaue. The vauation approach taken in this chapter is one of determining a security s intrinsic vaue what the security ought to be worth based on hard facts. This vaue is the present vaue of the cash-fow stream provided to the investor, discounted at a required rate of return appropriate for the risk invoved. With this genera vauation concept in mind, we are now abe to expore in more detai the vauation of specific types of securities. Bond Vauation Bond A ong-term debt instrument issued by a corporation or government. Face vaue The stated vaue of an asset. In the case of a bond, the face vaue is usuay $1,000. Coupon rate The stated rate of interest on a bond; the annua interest payment divided by the bond s face vaue. A bond is a security that pays a stated amount of interest to the investor, period after period, unti it is finay retired by the issuing company. Before we can fuy understand the vauation of such a security, certain terms must be discussed. For one thing, a bond has a face vaue. 1 This vaue is usuay $1,000 per bond in the United States. The bond amost aways has a stated maturity, which is the time when the company is obigated to pay the bondhoder the face vaue of the instrument. Finay, the coupon rate, or nomina annua rate of interest, is stated on the bond s face. 2 If, for exampe, the coupon rate is 12 percent on a $1,000-face-vaue bond, the company pays the hoder $120 each year unti the bond matures. In vauing a bond, or any security for that matter, we are primariy concerned with discounting, or capitaizing, the cash-fow stream that the security hoder woud receive over the ife of the instrument. The terms of a bond estabish a egay binding payment pattern at the time the bond is originay issued. This pattern consists of the payment of a stated amount of interest over a given number of years couped with a fina payment, when the bond matures, equa to the bond s face vaue. The discount, or capitaization, rate appied to the cash-fow stream wi differ among bonds depending on the risk structure of the bond issue. In genera, however, this rate can be thought of as being composed of the risk-free rate pus a premium for risk. (You may remember that we introduced the idea of a market-imposed trade-off between risk and return in Chapter 2. We wi have more to say about risk and required rates of return in the next chapter.) Perpetua Bonds Conso A bond that never matures; a perpetuity in the form of a bond. The first (and easiest) pace to start determining the vaue of bonds is with a unique cass of bonds that never matures. These are indeed rare, but they hep iustrate the vauation technique in its simpest form. Originay issued by Great Britain after the Napoeonic Wars to consoidate debt issues, the British conso (short for consoidated annuities) is one such exampe. This bond carries the obigation of the British government to pay a fixed interest payment in perpetuity. The present vaue of a perpetua bond woud simpy be equa to the capitaized vaue of an infinite stream of interest payments. If a bond promises a fixed annua payment of I forever, its present (intrinsic) vaue, V, at the investor s required rate of return for this debt issue, k d, is 1 Much ike criminas, many of the terms used in finance are aso known under a number of different aiases. Thus a bond s face vaue is aso known as its par vaue, or principa. Like a good detective, you need to become famiiar with the basic terms used in finance as we as their aiases. 2 The term coupon rate comes from the detachabe coupons that are affixed to bearer bond certificates, which, when presented to a paying agent or the issuer, entite the hoder to receive the interest due on that date. Nowadays, registered bonds, whose ownership is registered with the issuer, aow the registered owner to receive interest by check through the mai. 75

4 Part 2 Vauation I I I V = ( 1+ kd) ( 1+ kd) ( 1+ kd) I = ( 1 + k ) t t = 1 d (4.1) = I(PVIFA kd, ) (4.2) which, from Chapter 3 s discussion of perpetuities, we know shoud reduce to V = I /k d (4.3) Thus the present vaue of a perpetua bond is simpy the periodic interest payment divided by the appropriate discount rate per period. Suppose you coud buy a bond that paid $50 a year forever. Assuming that your required rate of return for this type of bond is 12 percent, the present vaue of this security woud be V = $50/0.12 = $ This is the maximum amount that you woud be wiing to pay for this bond. If the market price is greater than this amount, however, you woud not want to buy it. Bonds with a Finite Maturity Nonzero Coupon Bonds. If a bond has a finite maturity, then we must consider not ony the interest stream but aso the termina or maturity vaue (face vaue) in vauing the bond. The vauation equation for such a bond that pays interest at the end of each year is V = = I I I MV n ( 1+ k ) ( 1+ k ) ( 1+ k ) ( 1+ k ) n d I MV + t ( 1+ k ) ( 1+ k ) n t = 1 d d d (4.4) = I(PVIFA kd,n) + MV(PVIF kd,n) (4.5) where n is the number of years unti fina maturity and MV is the maturity vaue of the bond. We might wish to determine the vaue of a $1,000-par-vaue bond with a 10 percent coupon and nine years to maturity. The coupon rate corresponds to interest payments of $100 a year. If our required rate of return on the bond is 12 percent, then $100 $100 $100 $1,000 V = (1.12) 1 (1.12) 2 (1.12) 9 (1.12) 9 = $100(PVIFA 12%,9 ) + $1,000(PVIF 12%,9 ) Referring to Tabe IV in the Appendix at the back of the book, we find that the present vaue interest factor of an annuity at 12 percent for nine periods is Tabe II in the Appendix reveas under the 12 percent coumn that the present vaue interest factor for a singe payment nine periods in the future is Therefore the vaue, V, of the bond is V = $100(5.328) + $1,000(0.361) = $ $ = $ The interest payments have a present vaue of $532.80, whereas the principa payment at maturity has a present vaue of $ (Note: A of these figures are approximate because the present vaue tabes used are rounded to the third decima pace; the true present vaue of the bond is $ ) d n d 76

5 4 The Vauation of Long-Term Securities If the appropriate discount rate is 8 percent instead of 12 percent, the vauation equation becomes $100 $100 $100 $1,000 V = (1.08) 1 (1.08) 2 (1.08) 9 (1.08) 9 = $100(PVIFA 8%,9 ) + $1,000(PVIF 8%,9 ) Looking up the appropriate interest factors in Tabes II and IV in the Appendix, we determine that V = $100(6.247) + $1,000(0.500) = $ $ = $1, In this case, the present vaue of the bond is in excess of its $1,000 par vaue because the required rate of return is ess than the coupon rate. Investors woud be wiing to pay a premium to buy the bond. In the previous case, the required rate of return was greater than the coupon rate. As a resut, the bond has a present vaue ess than its par vaue. Investors woud be wiing to buy the bond ony if it sod at a discount from par vaue. Now if the required rate of return equas the coupon rate, the bond has a present vaue equa to its par vaue, $1,000. More wi be said about these concepts shorty when we discuss the behavior of bond prices. Zero-coupon bond A bond that pays no interest but ses at a deep discount from its face vaue; it provides compensation to investors in the form of price appreciation. Zero-Coupon Bonds. A zero-coupon bond makes no periodic interest payments but instead is sod at a deep discount from its face vaue. Why buy a bond that pays no interest? The answer ies in the fact that the buyer of such a bond does receive a return. This return consists of the gradua increase (or appreciation) in the vaue of the security from its origina, beow-face-vaue purchase price unti it is redeemed at face vaue on its maturity date. The vauation equation for a zero-coupon bond is a truncated version of that used for a norma interest-paying bond. The present vaue of interest payments component is opped off, and we are eft with vaue being determined soey by the present vaue of principa payment at maturity, or MV V = ( + ) 1 k n d (4.6) = MV(PVIF kd,n) (4.7) Suppose that Espinosa Enterprises issues a zero-coupon bond having a 10-year maturity and a $1,000 face vaue. If your required return is 12 percent, then V = $1,000 (1.12) 10 = $1,000(PVIF 12%,10 ) Using Tabe II in the Appendix, we find that the present vaue interest factor for a singe payment 10 periods in the future at 12 percent is Therefore: V = $1,000(0.322) = $322 If you coud purchase this bond for $322 and redeem it 10 years ater for $1,000, your initia investment woud thus provide you with a 12 percent compound annua rate of return. Semiannua Compounding of Interest. Athough some bonds (typicay those issued in European markets) make interest payments once a year, most bonds issued in the United States pay interest twice a year. As a resut, it is necessary to modify our bond vauation 77

6 Part 2 Vauation equations to account for compounding twice a year. 3 For exampe, Eqs. (4.4) and (4.5) woud be changed as foows (4.8) = (I/2)(PVIFA kd /2,2n) + MV(PVIF kd /2,2n) (4.9) where k d is the nomina annua required rate of interest, I/2 is the semiannua coupon payment, and 2n is the number of semiannua periods unti maturity. Take Note V = 2n t = 1 I / 2 MV + t (1 + k / 2 ) (1 + k / 2) d Notice that semiannua discounting is appied to both the semiannua interest payments and the ump-sum maturity vaue payment. Though it may seem inappropriate to use semiannua discounting on the maturity vaue, it isn t. The assumption of semiannua discounting, once taken, appies to a infows. d 2n To iustrate, if the 10 percent coupon bonds of US Bivet Corporation have 12 years to maturity and our nomina annua required rate of return is 14 percent, the vaue of one $1,000-par-vaue bond is V = ($50)(PVIFA 7%,24 ) + $1,000(PVIF 7%,24 ) = ($50)(11.469) + $1,000(0.197) = $ Rather than having to sove for vaue by hand, professiona bond traders often turn to bond vaue tabes. Given the maturity, coupon rate, and required return, one can ook up the present vaue. Simiary, given any three of the four factors, one can ook up the fourth. Aso, some speciaized cacuators are programmed to compute bond vaues and yieds, given the inputs mentioned. In your professiona ife you may very we end up using these toos when working with bonds. TIP TIP Remember, when you use bond Eqs. (4.4), (4.5), (4.6), (4.7), (4.8), and (4.9), the variabe MV is equa to the bond s maturity vaue, not its current market vaue. Preferred Stock Vauation Preferred stock A type of stock that promises a (usuay) fixed dividend, but at the discretion of the board of directors. It has preference over common stock in the payment of dividends and caims on assets. Most preferred stock pays a fixed dividend at reguar intervas. The features of this financia instrument are discussed in Chapter 20. Preferred stock has no stated maturity date and, given the fixed nature of its payments, is simiar to a perpetua bond. It is not surprising, then, that we use the same genera approach as appied to vauing a perpetua bond to the vauation of preferred stock. 4 Thus the present vaue of preferred stock is V = D p /k p (4.10) 3 Even with a zero-coupon bond, the pricing convention among bond professionas is to use semiannua rather than annua compounding. This provides consistent comparisons with interest-bearing bonds. 4 Virtuay a preferred stock issues have a ca feature (a provision that aows the company to force retirement), and many are eventuay retired. When vauing a preferred stock that is expected to be caed, we can appy a modified version of the formua used for vauing a bond with a finite maturity; the periodic preferred dividends repace the periodic interest payments and the ca price repaces the bond maturity vaue in Eqs. (4.4) and (4.5), and a the payments are discounted at a rate appropriate to the preferred stock in question. 78

7 4 The Vauation of Long-Term Securities Ask the Foo QWhat s preferred stock? AWe generay avoid investing in preferred stocks, but we re happy to expain them. Like common stock, a share of preferred stock confers partia ownership of a company to its hoder. But unike common stock, hoders of preferred stock usuay have no voting privieges. Shares of preferred stock often pay a guaranteed fixed dividend that is higher than the common stock dividend. Preferred stock isn t reay for individua investors, though. The shares are usuay purchased by other corporations, which are attracted by the dividends that give them income taxed at a ower rate. Corporations aso ike the fact that preferred stockhoders caims on company earnings and assets have a higher priority than that of common stockhoders. Imagine that the One-Legged Chair Co. (ticker: WOOPS) goes out of business. Many peope or firms with caims on the company wi want their due. Creditors wi be paid before preferred stockhoders, but preferred stockhoders have a higher priority than common stockhoders. Source: The Motey Foo ( Reproduced with the permission of The Motey Foo. where D p is the stated annua dividend per share of preferred stock and k p is the appropriate discount rate. If Margana Cipher Corporation had a 9 percent, $100-par-vaue preferred stock issue outstanding and your required return was 14 percent on this investment, its vaue per share to you woud be V = $9/0.14 = $64.29 Common Stock Vauation Common stock Securities that represent the utimate ownership (and risk) position in a corporation. The theory surrounding the vauation of common stock has undergone profound change during the ast few decades. It is a subject of considerabe controversy, and no one method for vauation is universay accepted. Sti, in recent years there has emerged growing acceptance of the idea that individua common stocks shoud be anayzed as part of a tota portfoio of common stocks that the investor might hod. In other words, investors are not as concerned with whether a particuar stock goes up or down as they are with what happens to the overa vaue of their portfoios. This concept has important impications for determining the required rate of return on a security. We sha expore this issue in the next chapter. First, however, we need to focus on the size and pattern of the returns to the common stock investor. Unike bond and preferred stock cash fows, which are contractuay stated, much more uncertainty surrounds the future stream of returns connected with common stock. Are Dividends the Foundation? When vauing bonds and preferred stock, we determined the discounted vaue of a the cash distributions made by the firm to the investor. In a simiar fashion, the vaue of a share of common stock can be viewed as the discounted vaue of a expected cash dividends provided by the issuing firm unti the end of time. 5 In other words, 5 This mode was first deveoped by John B. Wiiams, The Theory of Investment Vaue (Cambridge, MA: Harvard University Press, 1938). And, as Wiiams so apty put it in poem form, A cow for her mik/a hen for her eggs/and a stock, by heck/for her dividends. 79

8 Part 2 Vauation = Dt + k ) t (4.11) (4.12) where D t is the cash dividend at the end of time period t and k e is the investor s required return, or capitaization rate, for this equity investment. This seems consistent with what we have been doing so far. But what if we pan to own the stock for ony two years? In this case, our mode becomes V = t = 1 (1 e where P 2 is the expected saes price of our stock at the end of two years. This assumes that investors wi be wiing to buy our stock two years from now. In turn, these future investors wi base their judgments of what the stock is worth on expectations of future dividends and a future seing price (or termina vaue). And so the process goes through successive investors. Note that it is the expectation of future dividends and a future seing price, which itsef is based on expected future dividends, that gives vaue to the stock. Cash dividends are a that stockhoders, as a whoe, receive from the issuing company. Consequenty, the foundation for the vauation of common stock must be dividends. These are construed broady to mean any cash distribution to sharehoders, incuding share repurchases. (See Chapter 18 for a discussion of share repurchase as part of the overa dividend decision.) The ogica question to raise at this time is: Why do the stocks of companies that pay no dividends have positive, often quite high, vaues? The answer is that investors expect to se the stock in the future at a price higher than they paid for it. Instead of dividend income pus a termina vaue, they rey ony on the termina vaue. In turn, termina vaue depends on the expectations of the marketpace viewed from this termina point. The utimate expectation is that the firm wi eventuay pay dividends, either reguar or iquidating, and that future investors wi receive a company-provided cash return on their investment. In the interim, investors are content with the expectation that they wi be abe to se their stock at a subsequent time, because there wi be a market for it. In the meantime, the company is reinvesting earnings and, everyone hopes, enhancing its future earning power and utimate dividends. Dividend Discount Modes D1 D2 D V = ( 1 + k ) ( 1 + k ) ( 1 + k ) D1 D2 P ( 1 + k ) ( 1 + k ) ( 1 + k ) Dividend discount modes are designed to compute the intrinsic vaue of a share of common stock under specific assumptions as to the expected growth pattern of future dividends and the appropriate discount rate to empoy. Merri Lynch, CS First Boston, and a number of other investment banks routiney make such cacuations based on their own particuar modes and estimates. What foows is an examination of such modes, beginning with the simpest one. Constant Growth. Future dividends of a company coud jump a over the pace; but, if dividends are expected to grow at a constant rate, what impications does this hod for our basic stock vauation approach? If this constant rate is g, then Eq. (4.11) becomes V D g 0( 1 + ) D0( 1 + g) = ( 1 + k ) ( 1 + k ) e e e D0( 1 + g) ( 1 + k ) (4.13) where D 0 is the present dividend per share. Thus the dividend expected at the end of period n is equa to the most recent dividend times the compound growth factor, (1 + g) n. This may not ook ike much of an improvement over Eq. (4.11). However, assuming that k e is greater e e 2 e 2 e e e 80

9 4 The Vauation of Long-Term Securities than g (a reasonabe assumption because a dividend growth rate that is aways greater than the capitaization rate woud impy an infinite stock vaue), Eq. (4.13) can be reduced to 6 Rearranging, the investor s required return can be expressed as V = D 1 /(k e g) (4.14) k e = (D 1 /V ) + g (4.15) The critica assumption in this vauation mode is that dividends per share are expected to grow perpetuay at a compound rate of g. For many companies this assumption may be a fair approximation of reaity. To iustrate the use of Eq. (4.14), suppose that LKN, Inc. s dividend per share at t = 1 is expected to be $4, that it is expected to grow at a 6 percent rate forever, and that the appropriate discount rate is 14 percent. The vaue of one share of LKN stock woud be V = $4/( ) = $50 For companies in the mature stage of their ife cyce, the perpetua growth mode is often reasonabe. TIP TIP A common mistake made in using Eqs. (4.14) and (4.15) is to use, incorrecty, the firm s most recent annua dividend for the variabe D 1 instead of the annua dividend expected by the end of the coming year. Conversion to an Earnings Mutipier Approach With the constant growth mode, we can easiy convert from dividend vauation, Eq. (4.14), to vauation based on an earnings mutipier approach. The idea is that investors often think in terms of how many doars they are wiing to pay for a doar of future expected earnings. Assume that a company retains a constant proportion of its earnings each year; ca it b. The dividend-payout ratio (dividends per share divided by earnings per share) woud aso be constant. Therefore, and (1 b) = D 1 /E 1 (4.16) (1 b)e 1 = D 1 where E 1 is expected earnings per share in period 1. Equation (4.14) can then be expressed as V = [(1 b)e 1 ]/(k e g) (4.17) 6 If we mutipy both sides of Eq. (4.13) by (1 + k e )/(1 + g) and subtract Eq. (4.13) from the product, we get Because we assume that k e is greater than g, the second term on the right-hand side approaches zero. Consequenty, V(k e g) = D 0 (1 + g) = D 1 V = D 1 /(k e g) This mode is sometimes caed the Gordon Dividend Vauation Mode after Myron J. Gordon, who deveoped it from the pioneering work done by John Wiiams. See Myron J. Gordon, The Investment, Financing, and Vauation of the Corporation (Homewood, IL: Richard D. Irwin, 1962). 81

10 Part 2 Vauation where vaue is now based on expected earnings in period 1. In our earier exampe, suppose that LKN, Inc., has a retention rate of 40 percent and earnings per share for period 1 are expected to be $6.67. Therefore, Rearranging Eq. (4.17), we get V = [(0.60)$6.67]/( ) = $50 Earnings mutipier = V/E 1 = (1 b)/(k e g) (4.18) Equation (4.18) thus gives us the highest mutipe of expected earnings that the investor woud be wiing to pay for the security. In our exampe, Earnings mutipier = (1 0.40)/( ) = 7.5 times Thus expected earnings of $6.67 couped with an earnings mutipier of 7.5 vaues our common stock at $50 a share ($ = $50). But remember, the foundation for this aternative approach to common stock vauation was nevertheess our constant growth dividend discount mode. No Growth. A specia case of the constant growth dividend mode cas for an expected dividend growth rate, g, of zero. Here the assumption is that dividends wi be maintained at their current eve forever. In this case, Eq. (4.14) reduces to V = D 1 /k e (4.19) Not many stocks can be expected simpy to maintain a constant dividend forever. However, when a stabe dividend is expected to be maintained for a ong period of time, Eq. (4.19) can provide a good approximation of vaue. 7 Growth Phases. When the pattern of expected dividend growth is such that a constant growth mode is not appropriate, modifications of Eq. (4.13) can be used. A number of vauation modes are based on the premise that firms may exhibit above-norma growth for a number of years (g may even be arger than k e during this phase), but eventuay the growth rate wi taper off. Thus the transition might we be from a currenty above-norma growth rate to one that is considered norma. If dividends per share are expected to grow at a 10 percent compound rate for five years and thereafter at a 6 percent rate, Eq. (4.13) becomes V = (4.20) Note that the growth in dividends in the second phase uses the expected dividend in period 5 as its foundation. Therefore the growth-term exponent is t 5, which means that the exponent in period 6 equas 1, in period 7 it equas 2, and so forth. This second phase is nothing more than a constant-growth mode foowing a period of above-norma growth. We can make use of that fact to rewrite Eq. (4.20) as foows: 5 t D D V = ( ) (4.21) t 5 t = 1 ( 1 + ke) ( 1 + ke) ( ke 0. 06) If the current dividend, D 0, is $2 per share and the required rate of return, k e, is 14 percent, we coud sove for V. (See Tabe 4.1 for specifics.) 5 5 t = 1 t D0( 1.10) + t ( 1 + k ) t = 6 t V = + t t $ 2 ( 1.10 ) 1 $ = 1 ( 1.14) ( 1.14) ( ) = $ $22.13 = $31.12 e t D5( 1.06) t ( 1 + k ) e 5 7 AT&T is one exampe of a firm that maintained a stabe dividend for an extended period of time. For 36 years, from 1922 unti December 1958, AT&T paid $9 a year in dividends. 82

11 4 The Vauation of Long-Term Securities Tabe 4.1 Two-phase growth and common stock vauation cacuations PHASE 1: PRESENT VALUE OF DIVIDENDS TO BE RECEIVED OVER FIRST 5 YEARS END OF PRESENT VALUE CALCULATION PRESENT VALUE YEAR (DIVIDEND PVIF 14%,t ) OF DIVIDEND 1 $2(1.10) 1 = $ = $ (1.10) 2 = = (1.10) 3 = = (1.10) 4 = = (1.10) 5 = = t $(. 2110) or t t = 1 (. 114) = $8.99 PHASE 2: PRESENT VALUE OF CONSTANT GROWTH COMPONENT Dividend at the end of year 6 = $3.22(1.06) = $3.41 Vaue of stock at the end of year 5 = D 6 /(k e g) = $3.41/( ) = $42.63 Present vaue of $42.63 at end of year 5 = ($42.63)(PVIF 14%,5 ) = ($42.63)(0.519) = $22.13 V = $ $22.13 = $31.12 PRESENT VALUE OF STOCK The transition from an above-norma rate of dividend growth coud be specified as more gradua than the two-phase approach just iustrated. We might expect dividends to grow at a 10 percent rate for five years, foowed by an 8 percent rate for the next five years and a 6 percent growth rate thereafter. The more growth segments that are added, the more cosey the growth in dividends wi approach a curviinear function. But no firm can grow at an abovenorma rate forever. Typicay, companies tend to grow at a very high rate initiay, after which their growth opportunities sow down to a rate that is norma for companies in genera. If maturity is reached, the growth rate may stop atogether. Rates of Return (or Yieds) So far, this chapter has iustrated how the vauation of any ong-term financia instrument invoves a capitaization of that security s income stream by a discount rate (or required rate of return) appropriate for that security s risk. If we repace intrinsic vaue (V) in our vauation equations with the market price (P 0 ) of the security, we can then sove for the market required rate of return. This rate, which sets the discounted vaue of the expected cash infows equa to the security s current market price, is aso referred to as the security s (market) yied. Depending on the security being anayzed, the expected cash infows may be interest payments, repayment of principa, or dividend payments. It is important to recognize that ony when the intrinsic vaue of a security to an investor equas the security s market vaue (price) woud the investor s required rate of return equa the security s (market) yied. Market yieds serve an essentia function by aowing us to compare, on a uniform basis, securities that differ in cash fows provided, maturities, and current prices. In future chapters we wi see how security yieds are reated to the firm s future financing costs and overa cost of capita. Yied to maturity (YTM) The expected rate of return on a bond if bought at its current market price and hed to maturity. Yied to Maturity (YTM) on Bonds The market required rate of return on a bond (k d ) is more commony referred to as the bond s yied to maturity. Yied to maturity (YTM) is the expected rate of return on a bond if bought at its current market price and hed to maturity; it is aso known as the bond s interna rate 83

12 Part 2 Vauation of return (IRR). Mathematicay, it is the discount rate that equates the present vaue of a expected interest payments and the payment of principa (face vaue) at maturity with the bond s current market price. For an exampe, et s return to Eq. (4.4), the vauation equation for an interest-bearing bond with a finite maturity. Repacing intrinsic vaue (V) with current market price (P 0 ) gives us P 0 = n I MV + t (1 + k ) (1 + k ) n t = 1 d d (4.22) If we now substitute actua vaues for I, MV, and P 0, we can sove for k d, which in this case woud be the bond s yied to maturity. However, the precise cacuation for yied to maturity is rather compex and requires bond vaue tabes, or a sophisticated handhed cacuator, or a computer. Interpoation. If a we have to work with are present vaue tabes, we can sti determine an approximation of the yied to maturity by making use of a tria-and-error procedure. To iustrate, consider a $1,000-par-vaue bond with the foowing characteristics: a current market price of $761, 12 years unti maturity, and an 8 percent coupon rate (with interest paid annuay). We want to determine the discount rate that sets the present vaue of the bond s expected future cash-fow stream equa to the bond s current market price. Suppose that we start with a 10 percent discount rate and cacuate the present vaue of the bond s expected future cash fows. For the appropriate present vaue interest factors, we make use of Tabes II and IV in the Appendix at the end of the book. V = $80(PVIFA 10%,12 ) + $1,000(PVIF 10%,12 ) = $80(6.814) + $1,000(0.319) = $ A 10 percent discount rate produces a resuting present vaue for the bond that is greater than the current market price of $761. Therefore we need to try a higher discount rate to handicap the future cash fows further and drive their present vaue down to $761. Let s try a 15 percent discount rate: V = $80(PVIFA 15%,12 ) + $1,000(PVIF 15%,12 ) = $80(5.421) + $1,000(0.187) = $ This time the chosen discount rate was too arge. The resuting present vaue is ess than the current market price of $761. The rate necessary to discount the bond s expected cash fows to $761 must fa somewhere between 10 and 15 percent. To approximate this discount rate, we interpoate between 10 and 15 percent as foows: 8 Interpoate Estimate an unknown number that ies somewhere between two known numbers. X $ X 0.05 $ $ YTM $ $ $ (0.05) ($103.12) = Therefore, X = $ $ = Mathematicay, we can generaize our discount-rate interpoation as foows: where i L = discount rate that is somewhat ower than the investment s YTM (or IRR), i H = discount rate that is somewhat higher than the investment s YTM, PV L = present vaue of the investment at a discount rate equa to i L, PV H = present vaue of the investment at a discount rate equa to i H, PV YTM = present vaue of the investment at a discount rate equa to the investment s YTM, which (by definition) must equa the investment s current price. 84

13 4 The Vauation of Long-Term Securities In this exampe X = YTM Therefore, YTM = X = = , or percent. The use of a computer provides a precise yied to maturity of percent. It is important to keep in mind that interpoation gives ony an approximation of the exact percentage; the reationship between the two discount rates is not inear with respect to present vaue. However, the tighter the range of discount rates that we use in interpoation, the coser the resuting answer wi be to the mathematicay correct one. For exampe, had we used 11 and 12 percent, we woud have come even coser to the true yied to maturity. Behavior of Bond Prices. On the basis of an understanding of Eq. (4.22), a number of observations can be made concerning bond prices: Bond discount The amount by which the face vaue of a bond exceeds its current price. Bond premium The amount by which the current price of a bond exceeds its face vaue. 1. When the market required rate of return is more than the stated coupon rate, the price of the bond wi be ess than its face vaue. Such a bond is said to be seing at a discount from face vaue. The amount by which the face vaue exceeds the current price is the bond discount. 2. When the market required rate of return is ess than the stated coupon rate, the price of the bond wi be more than its face vaue. Such a bond is said to be seing at a premium over face vaue. The amount by which the current price exceeds the face vaue is the bond premium. 3. When the market required rate of return equas the stated coupon rate, the price of the bond wi equa its face vaue. Such a bond is said to be seing at par. TIP TIP If a bond ses at a discount, then P 0 < par and YTM > coupon rate. If a bond ses at par, then P 0 = par and YTM = coupon rate. If a bond ses at a premium, then P 0 > par and YTM < coupon rate. 4. If interest rates rise so that the market required rate of return increases, the bond s price wi fa. If interest rates fa, the bond s price wi increase. In short, interest rates and bond prices move in opposite directions just ike two ends of a chid s seesaw. Interest-rate (or yied) risk The variation in the market price of a security caused by changes in interest rates. From the ast observation, it is cear that variabiity in interest rates shoud ead to variabiity in bond prices. This variation in the market price of a security caused by changes in interest rates is referred to as interest-rate (or yied) risk. It is important to note that an investor incurs a oss due to interest-rate (or yied) risk ony if a security is sod prior to maturity and the eve of interest rates has increased since time of purchase. A further reationship, not as apparent as the previous four observations, needs to be iustrated separatey. 5. For a given change in market required return, the price of a bond wi change by a greater amount, the onger its maturity. In genera, the onger the maturity, the greater the price fuctuation associated with a given change in market required return. The coser in time that you are to this reativey arge maturity vaue being reaized, the ess important are interest payments in determining the market price, and the ess important is a change in market required return on the market price of the security. In genera, then, the onger the maturity of a bond, the greater the risk of price change to the investor when changes occur in the overa eve of interest rates. Figure 4.1 iustrates our discussion by comparing two bonds that differ ony in maturity. The price sensitivities of a 5-year bond and a 15-year bond are shown reative to changes in market required rate of return. As expected, the bond with the onger term to maturity shows a greater change in price for any given change in market yied. [A points on the two curves are based on the use of pricing Eq. (4.22).] 85

14 Part 2 Vauation Figure 4.1 Price yied reationship for two bonds where each price yied curve represents a set of prices for that bond for different assumed market required rates of return (market yieds) One ast reationship aso needs to be addressed separatey, and it is known as the coupon effect. 6. For a given change in market required rate of return, the price of a bond wi change by proportionay more, the ower the coupon rate. In other words, bond price voatiity is inversey reated to coupon rate. The reason for this effect is that the ower the coupon rate, the more return to the investor is refected in the principa payment at maturity as opposed to interim interest payments. Put another way, investors reaize their returns ater with a ow-coupon-rate bond than with a high-coupon-rate bond. In genera, the further in the future the buk of the payment stream, the greater the present vaue effect caused by a change in required return. 9 Even if high- and ow-coupon-rate bonds have the same maturity, the price of the ow-coupon-rate bond tends to be more voatie. YTM and Semiannua Compounding. As previousy mentioned, most domestic bonds pay interest twice a year, not once. This rea-word compication is often ignored in an attempt to simpify discussion. We can take semiannua interest payments into account, however, when determining yied to maturity by repacing intrinsic vaue (V) with current market price (P 0 ) in bond vauation Eq. (4.8). The resut is P 0 = 2n t = 1 I / 2 MV + t (1 + k / 2 ) (1 + k / 2) d (4.23) Soving for k d /2 in this equation woud give us the semiannua yied to maturity. The practice of doubing the semiannua YTM has been adopted by convention in bond circes to provide the annuaized (nomina annua) YTM or what bond traders woud ca the bond-equivaent yied. The appropriate procedure, however, woud be to square 1 pus the semiannua YTM and then subtract 1: that is, (1 + semiannua YTM) 2 1 = (effective annua) YTM 9 The interested reader is referred to James C. Van Horne, Financia Market Rates and Fows, 6th ed. (Upper Sadde River, NJ: Prentice Ha, 2001), Chap. 7. d 2n 86

15 4 The Vauation of Long-Term Securities As you may remember from Chapter 3, the (effective annua) YTM just cacuated is the effective annua interest rate. Yied on Preferred Stock Substituting current market price (P 0 ) for intrinsic vaue (V) in preferred stock vauation Eq. (4.10), we have P 0 = D p /k p (4.24) where D p is sti the stated annua dividend per share of preferred stock, but k p is now the market required return for this stock, or simpy the yied on preferred stock. Rearranging terms aows us to sove directy for the yied on preferred stock: k p = D p /P 0 (4.25) To iustrate, assume that the current market price per share of Acme Zarf Company s 10 percent, $100-par-vaue preferred stock is $ Acme s preferred stock is therefore priced to provide a yied of k p = $10/$91.25 = 10.96% Yied on Common Stock The rate of return that sets the discounted vaue of the expected cash dividends from a share of common stock equa to the share s current market price is the yied on that common stock. If, for exampe, the constant dividend growth mode was appropriate to appy to the common stock of a particuar company, the current market price (P 0 ) coud be said to be P 0 = D 1 /(k e g) (4.26) Soving for k e, which in this case is the market-determined yied on a company s common stock, we get k e = D 1 /P 0 + g (4.27) From this ast expression, it becomes cear that the yied on common stock comes from two sources. The first source is the expected dividend yied, D 1 /P 0 ; whereas the second source, g, is the expected capita gains yied. Yes, g wears a number of hats. It is the expected compound annua growth rate in dividends. But, given this mode, it is aso the expected annua percent change in stock price (that is, P 1 /P 0 1 = g) and, as such, is referred to as the capita gains yied. Question What market yied is impied by a share of common stock currenty seing for $40 whose dividends are expected to grow at a rate of 9 percent per year and whose dividend next year is expected to be $2.40? Answer The market yied, k e, is equa to the dividend yied, D 1 /P 0, pus the capita gains yied, g, as foows: k e = $2.40/$ = = 15% 87

16 Part 2 Vauation Summary Tabe of Key Present Vaue Formuas for Vauing Long-Term Securities (Annua Cash Fows Assumed) SECURITIES BONDS 1. Perpetua I V = (1 + k 2. Finite maturity, nonzero coupon I MV V = + (1 t + k ) (1 + k ) EQUATION (4.1), (4.3) = I(PVIFA kd,n) + MV(PVIF kd,n) (4.5) 3. Zero coupon MV V = (4.6) (1 + k n d) = MV(PVIF k d,n) (4.7) PREFERRED STOCK 1. No ca expected D D V = p p = t t = 1 (1 + kp) kp 2. Ca expected at n V = = D p (PVIFA k p,n) + (ca price)(pvif k p,n) COMMON STOCK Constant growth V = n n t = 1 d d D ca price + t n (1 + k ) (1 + k ) n p t = 1 p p t D ( 1 + g) D1 = t (1 + k ) ( k g) 0 t = 1 e I = k t t = 1 d) d e (4.4) (4.10) (see footnote 4) (4.14) Key Learning Points The concept of vaue incudes iquidation vaue, goingconcern vaue, book vaue, market vaue, and intrinsic vaue. The vauation approach taken in this chapter is one of determining a security s intrinsic vaue what the security ought to be worth based on hard facts. This vaue is the present vaue of the cash-fow stream provided to the investor, discounted at a required rate of return appropriate for the risk invoved. The intrinsic vaue of a perpetua bond is simpy the capitaized vaue of an infinite stream of interest payments. This present vaue is the periodic interest payment divided by the investor s required rate of return. The intrinsic vaue of an interest-bearing bond with a finite maturity is equa to the present vaue of the interest payments pus the present vaue of principa payment at maturity, a discounted at the investor s required rate of return. The intrinsic vaue of a zero-coupon bond (a bond that makes no periodic interest payments) is the present vaue of the principa payment at maturity, discounted at the investor s required rate of return. The intrinsic vaue of preferred stock is equa to the stated annua dividend per share divided by the investor s required rate of return. Unike bonds and preferred stock, for which the future cash fows are contractuay stated, much more 88

17 4 The Vauation of Long-Term Securities uncertainty surrounds the future stream of returns connected with common stock. The intrinsic vaue of a share of common stock can be viewed as the discounted vaue of a the cash dividends provided by the issuing firm. Dividend discount modes are designed to compute the intrinsic vaue of a share of stock under specific assumptions as to the expected growth pattern of future dividends and the appropriate discount rate to empoy. If dividends are expected to grow at a constant rate, the formua used to cacuate the intrinsic vaue of a share of common stock is V = D 1 /(k e g) (4.14) In the case of no expected dividend growth, the equation above reduces to V = D 1 /k e (4.19) Finay, when dividend growth is expected to differ during various phases of a firm s deveopment, the present vaue of dividends for various growth phases can be determined and summed to produce the stock s intrinsic vaue. If intrinsic vaue (V) in our vauation equations is repaced by the security s market price (P 0 ), we can then sove for the market required rate of return. This rate, which sets the discounted vaue of the expected cash infows equa to the security s market price, is aso referred to as the security s (market) yied. Yied to maturity (YTM) is the expected rate of return on a bond if bought at its current market price and hed to maturity. It is aso known as the bond s interna rate of return. Interest rates and bond prices move in opposite directions. In genera, the onger the maturity for a bond, the greater the bond s price fuctuation associated with a given change in market required return. The ower the coupon rate, the more sensitive that reative bond price changes are to changes in market yieds. The yied on common stock comes from two sources. The first source is the expected dividend yied, and the second source is the expected capita gains yied. Questions 1. What connection, if any, does a firm s market vaue have with its iquidation and/or goingconcern vaue? 2. Coud a security s intrinsic vaue to an investor ever differ from the security s market vaue? If so, under what circumstances? 3. In what sense is the treatment of bonds and preferred stock the same when it comes to vauation? 4. Why do bonds with ong maturities fuctuate more in price than do bonds with short maturities, given the same change in yied to maturity? 5. A 20-year bond has a coupon rate of 8 percent, and another bond of the same maturity has a coupon rate of 15 percent. If the bonds are aike in a other respects, which wi have the greater reative market price decine if interest rates increase sharpy? Why? 6. Why are dividends the basis for the vauation of common stock? 7. Suppose that the controing stock of IBM Corporation was paced in a perpetua trust with an irrevocabe cause that cash or iquidating dividends woud never be paid out of this trust. Earnings per share continued to grow. What woud be the vaue of the company to the stockhoders? Why? 8. Why is the growth rate in earnings and dividends of a company ikey to taper off in the future? Coud the growth rate increase as we? If it did, what woud be the effect on stock price? 9. Using the constant perpetua growth dividend vauation mode, coud you have a situation in which a company grows at 30 percent per year (after subtracting out infation) forever? Expain. 10. Tammy Whynot, a cassmate of yours, suggests that when the constant growth dividend vauation mode is used to expain a stock s current price, the quantity (k e g) represents the expected dividend yied. Is she right or wrong? Expain. 89

18 Part 2 Vauation 11. $1,000 US Government Treasury Bond FREE! with any $999 Purchase shouted the headine of an ad from a oca furniture company. Wow! Like, for sure, this is ike getting the furniture ike free, said your friend Heather Dawn Tiffany. What Heather overooked was the fine print in the ad where you earned that the free bond was a zerocoupon issue with a 30-year maturity. Expain to Heather why the free $1,000 bond is more in the nature of an advertising comeon rather than something of arge vaue. $ 1000 U.S. GOVERNMENT TREASURY BOND FREE! WITH ANY $ 999 PURCHASE ANY COMBINATION OF ANY ITEMS TOTALING $ 999 OR MORE AND YOU GET A $ 1000 U.S. GOVERNMENT BOND FREE. Sef-Correction Probems 1. Fast and Loose Company has outstanding an 8 percent, four-year, $1,000-par-vaue bond on which interest is paid annuay. a. If the market required rate of return is 15 percent, what is the market vaue of the bond? b. What woud be its market vaue if the market required return dropped to 12 percent? To 8 percent? c. If the coupon rate were 15 percent instead of 8 percent, what woud be the market vaue [under Part (a)]? If the required rate of return dropped to 8 percent, what woud happen to the market price of the bond? 2. James Conso Company currenty pays a dividend of $1.60 per share on its common stock. The company expects to increase the dividend at a 20 percent annua rate for the first four years and at a 13 percent rate for the next four years, and then grow the dividend at a 7 percent rate thereafter. This phased-growth pattern is in keeping with the expected ife cyce of earnings. You require a 16 percent return to invest in this stock. What vaue shoud you pace on a share of this stock? 3. A $1,000-face-vaue bond has a current market price of $935, an 8 percent coupon rate, and 10 years remaining unti maturity. Interest payments are made semiannuay. Before you do any cacuations, decide whether the yied to maturity is above or beow the coupon rate. Why? a. What is the impied market-determined semiannua discount rate (i.e., semiannua yied to maturity) on this bond? b. Using your answer to Part (a), what is the bond s (i) (nomina annua) yied to maturity? (ii) (effective annua) yied to maturity? 4. A zero-coupon, $1,000-par-vaue bond is currenty seing for $312 and matures in exacty 10 years. a. What is the impied market-determined semiannua discount rate (i.e., semiannua yied to maturity) on this bond? (Remember, the bond pricing convention in the United States is to use semiannua compounding even with a zero-coupon bond.) b. Using your answer to Part (a), what is the bond s (i) (nomina annua) yied to maturity? (ii) (effective annua) yied to maturity? 5. Just today, Acme Rocket, Inc. s common stock paid a $1 annua dividend per share and had a cosing price of $20. Assume that the market expects this company s annua dividend to grow at a constant 6 percent rate forever. a. Determine the impied yied on this common stock. b. What is the expected dividend yied? c. What is the expected capita gains yied? 90