CHAPTER 5: EQUITY MARKETS

Transcription

1 CHAPTER 5: EQUITY MARKETS Overview Whe we buy stock i a corporatio, we exchage cash for a share of hoped-for future profits of the compay. These profits ca come via capital appreciatio ad through receipt of divideds. Ulike bodholders or other creditors, shareholders are ot guarateed ay retur. Should the compay go bakrupt, commo shareholders will be at the very ed of a log lie of ivestors. There is substatial risk i the stock market, but there is also the potetial for sigificat returs. Moder portfolio theory holds that there is a relatioship betwee the riskiess of a asset ad the retur demaded by ivestors: the more risky the asset, the higher the retur it should ear. There are may ecoomic theories aimed at evaluatio of a ivestor s risk profile. The risk profile of a ivestor is characterized by a quadratic utility fuctio. The geeral assumptio is that ivestors are risk-averse. Valuatio Overview What is a fair price for a asset? What factors ifluece the price? Is the value we see i the pages of the fiacial press correct? How do we accout for the riskiess of a asset? Although prices are discovered i the marketplace, there are some tools that we ca employ to price a asset. Effectively, prices are drive by expectatios of future earigs, so today s price is calculated as the discouted cash flow of expected future earigs. Accordig to Bejami Graham, earigs power of a firm is the ultimate source of value. So i order to price ay asset, we must have a good uderstadig of its curret ad future growth prospects. We also must accout for the iheret risk of a asset. This risk is embedded i the price of the asset through the correct choice of discout rate. How to choose this discout rate is, itself, the topic of discussio that cotiues eve today. While there is a etire chapter o risk later i this book, for ow we will decompose equity risk ito two sources: systematic risk, which all stocks are subject to ad usystematic risk, which is the compay-specific portio of risk. First, we must summarize the Efficiet Market Hypothesis, which is a theory of how prices are set i the market. The Efficiet Market Hypothesis The efficiet market hypothesis (EMH) or Radom Walk Theory is the foudatio of moder portfolio theory. This hypothesis asserts that future prices are radom. They may follow some mea path related to expected growth of corporate earigs, but prices fluctuate accordig to radom evets. are three levels of the efficiet markets hypothesis: the weak form, the semi-strog form, ad the strog form. The Weak Form of the EMH asserts that security prices iclude all publicly available tradig data such as price/share, tradig volume, short iterest ad so forth. Thus, studies of price history cotai o iformatio that ca help predict future prices, so chartig ad other forms of techical aalysis do ot work. (The idea here is that sice this iformatio is publicly available, all ivestors have access to it ad so it would already be factored ito the security price.) The Semi-Strog Form of the EMH asserts that security prices embed all kow tradig data as well as compay fudametals. Thus, o meaigful price behavior ca be forecasted by formig relatioships betwee accoutig variables such as earigs/share, divideds/price ad other such fiacial ratios. (The idea here is similar to the weak form: such data ca easily be obtaied from sources such as Bloomberg.com so would already be reflected i the price of the security.) The Strog Form of the EMH asserts that security prices reflect all of the iformatio it is possible to kow about a compay, both public ad private, icludig tradig variables, all accoutig data ad all iformatio kow oly to isiders. The oly variable that ca impact the price of a security is ew iformatio, which by defiitio eters the market radomly ad is ukowable. Uder this form, stock prices follow a radom walk. The Strog Form icludes the weak form as well as the semistrog form as show below.

2 Weak Form All public tradig iformatio Semi-Strog Form All public compay fudametal data Strog Form All relevat iformatio, both public ad private The Divided Discout Model The value of ay compay is the preset value of expected future cash flows that will accrue to ivestors. The ivestors iclude debt holders, preferred divided holders ad commo equity shareholders. I order to value of the price of the commo stock, we have to work with cash flows expected to accrue to the commo shareholders. So, the simplest possible model is the divided discout model. Here, we just forecast of the future flow of divideds, discout by the appropriate cost of equity capital to calculate the preset value of the divided stream. This gives the price of the stock. The first simple case is where the compay pays a divided that is expected to remai stable over time. This might be the case for a mature compay. The the price of the stock is calculated usig the perpetuity model already developed: P CS I the above formula, DIV is the stable divided ad r CE is the cost of equity capital, which is ot observable. Oe way to estimate it is to use the CAPM. Aother way is to ask maagemet what their hurdle rate for ew projects is. DIV r Example Suppose that the Caterpillar Corporatio (CAT) pays a aual divided of $.40 ad their cost of equity capital is 5%. What is the price of the stock? Aswer If ivestors expect a perpetual stream of $.40 divideds ad the cost of equity capital is expected to remai costat, the DIV $.40 P CS $9.33. r 0.5 CE Gordo Growth Model If the divided is expected to grow forever at a costat rate g, the stock price will be higher tha that calculated above. I this case, the price of the stock is calculated usig CE

3 ( + g) DIV0 PCS where DIV 0 is the curret divided. I this case it ca be see that the spread rce g betwee the cost of equity capital ad the growth rate of the divided is the primary driver of stock price. Example Suppose that the Caterpillar Corporatio (CAT) pays a aual divided of $.40 ad their cost of equity capital is 6.5% but the divided is expected to grow at a costat rate of 2.5%/year. What is the price of the stock? Aswer ( + g) $.40( ) DIV0 PCS $.48 r g CE So how well did these models do i pricig Caterpillar? Sice the actual price is curretly $54.7, there must be somethig that these models are ot capturig. Caveats The Gordo Growth model must be used with care. If the growth rate is higher tha the cost of equity capital, a egative price/share will result. A compay ca t grow at a high rate for a ifiite period of time, otherwise it would overtake the etire ecoomy. (Oh, wait, that did happe Microsoft.) We ca costruct more complex divided discout models to accout for compaies that will be experiecig chagig growth rates over time, as, for example, whe a high-tech compay expads rapidly for a period, the begis a trasitio period where sales might begi to stabilize, ad evetually the compay eters a mature period of stable growth. Such models iclude but are ot limited to: the two-stage divided discout model, the H Model, the three-stage divided discout model ad so forth. Also, ote that these models oly apply to compaies that pay a divided, or are expected to pay a divided i the future, so ca t be used to value start-ups ad other compaies that may ot pay divideds. For such compaies, we ca use multiples aalysis for valuatio. Accordig to a recet paper by Keeth Frech, of Fama-Frech fame, The percet of firms payig cash divideds falls from 66.5 i 978 to 20.7 i 998. Multiples Aalysis Commo multiples used i valuatio iclude Price/Earigs, Price/Sales, Divided/Price, Book Value/Market Value ad so o. While multiples are very easy to use, we have to be very careful sice we do t kow what the right multiple should be. To value a compay usig a multiple requires a set of comparable compaies for compariso. Also, sice o uderstadig or kowledge of the compay fudametals or outlook is required to use a multiple, it ca give misleadig prices if misused. Nevertheless, let s try out a few of these ratios o some real compaies. Price/Earigs The earigs here mea et icome per share. Net icome is used because it is et of egotiated debt capital that must be paid ad taxes ad operatig expeses such as capital expeditures, so it is theoretically the icome available to pay to ivestors. Sice et icome is oly reported quarterly (at least for US compaies), we have to use historical, or trailig icome. Assume that the average umber of shares held over the period Mar 3, 200 Mar 3, 2002 was millio ad the et icome for this period was $75 millio. The earigs/share $75 millio/343.8 millio shares $2.08/share. If price/share was $54.7, the price/earigs (price/share)/(earigs/share) $54.7/$ We use price/earigs multiples to compare to other stocks or idices, to either assess relative valuatio or attractiveess of the stock, or to estimate price whe other metrics are t available. If we wated to value a ew compay similar to Caterpillar, we might use the P/E ratio. Suppose the similar compay has earigs/share of $.50. The it s price would be calculated as 26.30($.50) $ This could be doe eve if the compay is ot expected to pay ay divideds.

4 Aother way to calculate Price/Earigs ratio is to recogize that compaies ca either pay their et icome out to shareholders via divideds, or reivest it i the compay to fud ew projects. The ratio of et icome paid i the form of divideds is kow as the payout ratio K. Whe compaies are experiecig rapid growth, they geerally plow their earigs back ito the compay but as they mature, they geerally pay out a higher percetage of their earigs as divideds. Thus DIV K E. The Gordo Growth Model ca be modified to allow us to solve for Price/Earigs as: P CS DIV r 0 CE ( + g) KE ( + g) P K( + g) g r 0 CE g so E CS 0 r CE g If K 66.7% for CAT ad the cosesus estimate for log-term growth rate is %, the ( + 0.) P CS E This would imply a curret price of 8.5(E ) $38.5. Warig: It is really ot a good idea to use P/E ratios to value a stock, sice there are so may variables ivolved. The P/E ratio is a quick estimate that assumes that the oly factors ifluecig a stock s price are earigs/share. Eve if the two compaies we are comparig have totally differet risk characteristics, it is possible to forecast the same P/E ratio for them. The P/E ratio is a easy metric to calculate, but should ever be used i place of more sophisticated aalysis. We might look at the tred of P/E ratios over time for a stock or idex to try to get some idea of whether the market is uder or over-priced, but eve this will work oly if the curret eviromet is very similar to that which prevailed historically. We ca see that price/earigs ratios are ot stable over time as compaies frequetly maipulate their earigs through accoutig games. As a example of how we might actually use P/E ratio i a useful way, cosider the stock of TASER, a compay that makes laser stu gus. The stock was laguishig aroud $6/share pre-9/ but shot up to over $8/share after Uited Airlies placed a order for 300 gus. At $575/gu this is equivalet to about $750,000 i reveue. Accordig to UAL s most recet fiacial statemets, they have a fleet size of 543 aircraft, which meas they are buyig 2.4 gus/plae. Lookig at TASER s fiacial statemets, we see twelve-moth trailig earigs of about $608,000. With 2.8 millio shares outstadig, this yields earigs/share of $0.2. Thus the P/E ratio is $8/$ The Dow is curretly tradig at about 20 times earigs. Is a $750,000 sale to UAL sufficiet to justify the icrease i P/E of 85 from 28? To fid out, let s figure out how much reveue would have to be geerated to justify the TASER multiple. With et profit margis of 7.59% this meas that approximately $56,925 of et icome spread over 2.8 millio shares will add about two cets/share. If it is assumed that the compay should trade at about the same multiple as the Dow, a P/E of 20 meas that the EPS should be $0.8875/share. How may gus would have to be sold to justify this? Reveue $0.8875/share times 2.8 millio shares/ $32.7 millio. This implies that about 57,000 gus would have to be sold per year to justify eve a P/E ratio of 20. How large is the market for TASER gus o airlies assumig that all airlies buy i the same ratio as UAL? There are about 5,000 commercial aircraft. Eve if each buy 3 gus/plae, this implies sales of 45,000 gus, oce. So the price must embed expectatios of sellig to police forces ad so o. I ay case, how to justify a P/E ratio of 85? (Based o New York Post article by Christopher Byro, May 28, 2002.) Divided/Price Otherwise kow as Divided Yield, for Caterpillar this ratio would be $.40/$ %. If the divided yield ad divided are kow, this allows us to estimate price. Studies have bee performed o the divided yield of the S&P500 ad other idices i a attempt to forecast market tops ad bottoms. A relatively high divided yield implies that the market is udervalued while a low

5 divided yield implies that the market is overpriced. If mea-reversio holds, the idea is that periods of low divided yields should be followed by fallig prices. The problem is that o oe ca predict exactly whe this will happe. S&P Levels of S&P 500 ad Divided Yield, to Year S&P 500 Divided Yield 8.0% 7.0% 6.0% 5.0% 4.0% 3.0% 2.0%.0% 0.0% Divided Yield % Price/Sales Sice Sales are the ultimate source from which et icome ad cosequet divideds may be paid, this ratio is ofte used. Additioally, sales are cosidered to be less subject to accoutigs maipulatio tha earigs are so that this ratio might be stable over time. The price/sales ratio ca be used for compaies that do t yet have ay reported earigs such as start-ups. Returig to Caterpillar, if we kew that the P/S ratio was 0.95 ad we foud their last reported sales o a site such as SEC filigs or fiace.yahoo.com to be $20.0 billio the the market cap would be calculated as $9 billio. If there are shares outstadig, this leads to a price of $55.25/share. The Price/Sales ratio might be a good valuatio measure for TASER, which has o divideds so ca t be priced usig the divided discout model. If price/sales for comparable compaies (sector: techology, idustry: Electroic Istrumetatio & Cotrols) are obtaied ad averaged, we might fid P/S 2.29 for the idustry, 5.46 for the sector ad 3.4 for the S&P500. If the twelve moth trailig sales are $8 millio/2.8 millio shares $2.85/share, this would give a P/S ratio for the compay of $7.75/ Price computed usig comparables are tabulated below. P/S Calculated Share Price Idustry 2.29 $6.54 Sector 5.46 $5.56 S&P $9.7 TASER 6.23 $7.75 Price/Book Value This ratio compares the curret price of the firm to it s book value. The uderlyig idea is that the compay is able to use its assets to geerates earig. For CAT, the Price/Book Value ratio for the quarter eded 3/3/2002 was The book value is obtaied from the balace sheet ad is assets less liabilities. This residual is what would remai to be distributed to shareholders. If the book value is $6.50/share, this would imply a share price of 3.32(6.50) $ the problem with usig this ratio is that some assets, such as cash ad marketable securities, are recorded at curret market prices but loger term assets are recorded at historical acquisitio cost. This will depress book value ad overstate the P/BV ratio i may cases. For may service based compaies such as

6 cosultig or software firms, the assets are the talets of their employees ad this is ot recorded o ay balace sheet, so the book value will appear very low. I fact, for the first quarter of 2002, Microsoft had a price/book value ratio of 5.4. For a reported book value/share of $0.03, this would imply a price of $54.86 at a time whe Microsoft s price had experieced a high price of $76/share. Service compaies may have higher P/BV ratios sice they have fewer hard assets. Growig compaies makig large capital expeditures may have lower P/BV ratios but as these assets start to depreciate, the ratio should rise. Price/Cash Flow Usig accoutig statemets, the free cash flow to uleveraged equity ca be computed. This cash flow is the flow that would accrue to commo shareholders after ay capital expeditures have bee made; iterest paymets have bee made ad ay preferred divideds paid. It is felt that is more difficult for compaies to play accoutig games with their actual cash flow, so that perhaps this ratio is a better measure of earigs power. For example, for TASER the cash flow/share was $0.28/share which results i a price/cash flow measure of 63. By compariso, the idustry, sector ad S&P500 were 9.92, 30.7 ad respectively. Other Multiples There are a host of other metrics by which performace may be measured. These iclude Reveue/Employee, Net Icome/Employee ad so forth. Retur o a Asset The retur o ay asset is calculated as the sum of price appreciatio over the holdig period plus et cash flows accruig to the ivestor. Thus, if the asset is expected to grow at a rate g over the holdig period, a ivestmet of P today will grow to (+g)p by the ed of the holdig period. The retur will the be: ( + g) P + D P D r + g P P If the cosesus log-term growth expectatios for CAT are %, the curret price is $54.8 ad a divided paymet of $.40 is expected over the year, eglectig compoudig of the quarterly divided, the expected retur over the year would be calculated as $.40 r %. This would give a expected price i oe year of.355($54.8) $62.2 $54.8 for the stock. Note that it is the expected growth rate that drives this calculatio, ot the divided paymet, so if the growth rate is wrog, a ivestor could make the wrog decisio. CAPM About forty years ago, a relatioship betwee a security s retur ad it s risk was developed. This relatioship is kow as the Capital Asset Pricig Model, or CAPM for short. The CAPM posits that all security risk is captured i a parameter kow as beta, ad that there is a liear relatioship betwee the excess retur of the security to the excess retur of the market. The ivestor should be compesated for risk. The greater the risk, the higher the retur the ivestor should require. The CAPM is: R s R f + β(r m R f ) I terms of excess returs, R s - R f β(r m R f ) where R s retur o security, % R f risk-free rate,% R m retur o market portfolio, % β measure of correlatio betwee security ad market portfolio

7 The CAPM ca be used to estimate the required retur o equity capital eeded i security valuatio (it would be the correct discout rate to use), or to determie relative attractiveess of a security. The validity of the CAPM is a fertile topic of academic debate, as are questios over the proper risk free rate to use, what market portfolio should be used, whether a market portfolio eve exists, whether beta is stable over time, whether beta is alive or dead ad so o. Risk-free Rate The risk-free rate chose should correlate with the expected holdig period of the ivestmet. Some use the 3-moth T-bill rate, but for corporate fiace matters or ivestmets which are meat to be held for loger periods, use loger-teor maturities. For example if oe wats the required retur o equity capital for a ew capital expediture expected to last for te years, use the 0 year Treasury Bod ad so o. Market Portfolio The market portfolio is geerally assumed to be uobservable. Theoretically the market portfolio would ecompass all possible ivestmets, but i practice is usually take to be the retur o the S&P500 If we are valuig large-cap US stocks, this is probably acceptable. If you are usig the CAPM to value risky ivestmets i emergig markets, you should choose aother idex which is more appropriate to your ivestmets. Excess Retur The retur over the risk-free rate as i R s R f, R m R f. Historically, the excess returs over the S&P500 idex have averaged about 7%. Beta Measures the co-movemet of the security with the market portfolio. By defiitio, b of the market itself is equal to oe. Stocks that are perfectly correlated with the market, the, will have b. Stocks that are more risky tha the market will have betas exceedig, ad stocks less-risky will have betas lower tha oe. The miimum value of beta is zero. A beta of 3 would mea that if the market had a excess retur of 0%, the stock would be expected to have a excess retur of 30%. Beta ca be calculated via regressio aalysis or looked up from fiacial sources. Kowledge of beta ca help i devisig a portfolio with the desired risk ad retur characteristics. Example Suppose that the risk-free rate is 5%, the market retur is expected to be 2%, ad the compay s beta.74. The required retur o equity is calculated usig the CAPM as or 2.8% over the risk-free rate. R s 5% +.74(2% 5%) 7.8% Example Caterpillar s beta is reported as It is expected to pay divideds of $.40 over the ext year ad pays out 66.7% of et icome as divideds. Is the stock a attractive buy at the curret price of $55? Assume that the risk-free rate is 3% ad the market will retur 0%. Aswer Oe way to evaluate the curret price is to figure out what the required retur o the stock is usig the CAPM ad compare to the curret price. Or, we ca use the curret price to figure out the implied required retur ad compare to that calculated usig the CAPM. Required retur o equity capital usig CAPM is R s 3% (0% 3%) 8.%. The the DIV0 ( + g) price/share should be PCS where g is the iteral growth rate of the compay. This ca rce g be calculated by recogizig the fact that there are oly a few ways for a compay to grow iterally: it ca retai earigs, plowig them back ito ew capital projects to ear the retur o equity, or buy back shares priced below book value, or sellig stock above book value ad ivestig the capital received. Assumig that the compay will plow back earigs, the iteral growth rate g retur o

8 Net Icome equity times earigs retetio ratio, or g ROE(-K). Sice ROE this ca be Commo Equity decomposed ito two sources of retur. Sice Book Value Commo Equity, the Share Share EPS/BV. Earigs Share Net Icome ad Share Net Icome Net Icome ROE Commo Equity Share Share Book Value Side Note: the above calculatio is kow as Dupot Aalysis, i which the retur is decomposed ito sources of retur. Aother useful form is to write ROE Sales Assets NetIcome Assets Sales Equity A quick trick to remember (S/A) (NI/S) (A/E) is to write out the first letters of each term, SANISAE ad remember Southwest Airlies Aouces: Nuts Icrease Sales! (Film) At Eleve The g EPS/BV(-K) ROE(-K). Sice Caterpillar s EPS ad BV/share are $2.08/share ad $6.5/share respectively, the ROE is calculated as 2.6% ad hece g 2.6%(-.667) 4.9%. ( ) $.40 Fially, the price is calculated as P is higher tha this, it is ot a attractive buy at this time. ( AssetTurover)( Pr ofitm arg i)( LeverageRatio) $37.2/share. Sice the curret market price NOTE: There are may perils i usig this formula. For oe thig, it assumes that the compay has reached a mature growth stage ad so is assumed to grow at a costat rate from ow o, which may ot be true if the compay is plaig acquisitios, diversificatios ad so o. A sesitivity aalysis should always be performed. Sice the price is so sesitive to the spread betwee r CE ad g, we ca calculate a rage of prices. I fact, if r CE is decreased by just oe percet, the price would be $49.95 whereas icreasig g by just oe percet results i a share price would be $ The Markowitz Portfolio If we plot all possible (risk, retur) pairs of assets where risk is measured by stadard deviatio, we might get somethig like the followig.

9 Markowitz Efficiet Frotier C Retur, % B A Risk, as Measured bystdev, % The solid lie i the above graph is the efficiet frotier: ivestors holdig portfolios that lie alog the efficiet frotier have the optimal retur for the risk they are willig to bear. If a ivestor was willig to tolerate a stadard deviatio of 4%, he could hold portfolio A. But this is suboptimal as there are portfolios, such as B, that offer a higher retur for the same amout of risk. The highest retur that could be eared would be if a portfolio C which lies alog the efficiet frotier could be foud. Accordig to the theory, there are o portfolios that could be formed lyig outside of the evelope described by the frotier. This goes back to the idea that ivestors are risk-averse. The assumptio is that ivestors are oly cocered with mea retur ad risk of their portfolios. A 2 Suppose that the utility fuctio looks somethig like U E[ RP ] σ [ R P ] where A is some 2 positive costat, which must be determied for each ivestor. The utility fuctio describes a tradeoff betwee risk ad retur with higher utility derived from higher returs, but moderated by risk. Overlayig sample utility fuctios for two differet ivestors o the efficiet frotier gives:

10 Utility Fuctios U A U B UC U U2 U3 Retur, % A B C Risk, as Measured bystdev, % The idifferece curves for a highly risk-adverse ivestor are show above as U A, U B ad U C with U A > U B > U C. This ivestor would be idifferet betwee ay (risk, retur) combiatio alog ay utility curve. It is assumed that the ivestor seeks to maximize his utility so, if he desires to hold the portfolio C o idifferece curve U C, he could ear higher utility for the same risk by holdig portfolio A o idifferece curve U A. It is ot possible to form portfolios havig higher retur for the same risk sice these would lie outside of the efficiet frotier. The optimal portfolio for this ivestor is where the highest utility curve is taget to the efficiet frotier. O the other had, for the utility curves U, U 2 ad U 3 of a less risk-averse ivestor, the optimal portfolio to hold is where the highest utility curve U is taget to the efficiet frotier, which occurs at poit B. The secod ivestor will ear a higher retur tha the first ivestor, but must take o more risk to do so. Derivatio of the CAPM We regress the returs of a stock agaist the returs of the market. The slope of the regressio lie is β. For example, regressig Microsoft s returs agaist the returs of the S&P 500 over the past seve years results i the relatio R MSFT -R F (R S&P500 -R F ). This implies that β.47 for this period. (We should also perform statistical aalysis of the regressio coefficiets. We fid that the t-statistic of the itercept is 0.5, so the itercept is ot sigificat. The slope has a t-statistic of 9. so is sigificat. Sice the stadard deviatio of the slope coefficiet is 0.6, a 95% cofidece iterval is.44 β.79.)

11 MSFT Excess Returs vs S&P 500 Excess Returs Mar May MSFT Excess Returs (%) S&P 500 Excess Returs (%) Recall that if we form a portfolio of securities, the expected retur ad variace of the portfolio ca be calculated as where E [ R ] R + w E[ R ] P f i i ( R ) i f i w i P i i j, σ R wi σ i + 2 wi w jcovi, j, i j Returig to our risk-averse ivestor, suppose he decides to hold oly two assets: some fractio w S of a sigle stock ad w m (- w s ) of the market portfolio. Ca we derive his optimal portfolio? The expected retur of the portfolio will be R P R f + w s (R s -R f ) + w m (R m -R f ) ad the variace will be σ 2 p w 2 s σ 2 s + w 2 m σ 2 m +2w s w m COV(R s,r m ). Differetiatig with respect to w s gives σ w 2 p s 2w σ s 2 s + 2w m COV ( R s, R m ) ad E [ R ] w s p R s R p Assumig that at the optimal poit w s 0 ad w m, 2COV ( R, R ) Also, at the optimal poit, the margial chage i expected retur for chage i variace should be costat regardless of the security held, eve for the market portfolio, so σ w 2 s f s m

12 E σ [ R ] p 2 p Rs R f Rm R f Rm R 2COV ( R, ) 2 (, ) 2 s Rm COV Rm R m 2σ m This leads to the CAPM f R p R f COV, σ 2 ( R ) s Rm ( R R ) β ( R R ) m s f sm s f which implies that the oly source of risk that the market will compesate the ivestor for is market risk, measured by the covariace of the asset retur with the market. This is so because with proper choice of weights, it is possible to diversify away compay-specific (usystematic) risk: the market does ot reward oe for takig diversifiable risk. We will show this i the followig sectio. Please ote that the CAPM is essetially a oe-factor model, as it employs a sigle source of risk the covariace of the retur with the market retur as drivig retur. There are more complex models such as the APT factor model which attempt to quatify retur as the sum of sesitivities to various ecoomic factors. The CAPM assumes that ivestors are mea-variace optimizers ad that the market portfolio, whatever that may be, is the portfolio that gives the optimal retur for a give variace. Aother difficulty with the CAPM is that it provides a relatioship betwee expected excess returs, or ex-ate excess returs, but ivestors observe actual, ex-post returs. Equity Idexes Most busiesses are exposed to the same macroecoomic effects such as iflatio, busiess cycles, iterest rate chages, eergy prices, costs of maufacturig iputs ad so o. If all of these commo factors are lumped together as F, we ca measure the sesitivity of the idividual firm to F by β i. Firms are also exposed to firm-specific risk (such as approval of patets, successful itegratio of ew techology, iovatio, favorable settlemet of litigatio ad so o) ad if this risk is represeted by the term ε i the we ca represet the expected excess retur of the security over the risk-free rate as R s -R f α i + β i (R m -R f )+ε i I this equatio, the itercept α i represets the retur that would be expected i a eutral market where R m -R f 0. β i (R m -R f ) is the excess retur resultig from market exposure ad ε i is the firm-specific risk. The the excess retur is composed of two parts: the systematic excess retur α i + β i (R m -R f ) due to sesitivity to market factors ad the excess retur attributable to usystematic, firm specific risk ε i. This is kow as the idex model ad is very similar to the CAPM, except here we have the term α i. This form is also idetical to the regressio model obtaied earlier by regressig MSFT returs o the S&P500. If we form a equally-weighted portfolio of assets, the excess retur of the portfolio is just the weighted sum of each idividual excess retur. R P R f i w i ( ) ( ) Ri R f α i + β i Rm R f + ε i α i + β i ( Ri R f ) + i i i i ε i if α P α i ad β P β i i i the

13 R P R f α + β P P ( Rm R f ) + i ε i The first two terms are the excess returs due to systematic market risk ad the last are due to omarket (firm-specific) factors. The variace of R P -R f is: σ ( R R ) β σ + σ ( ε ) β σ σ ( ε ) 2 P P f P P p P P + Plottig this: σ 2 (ε)/ Diversifiable Risk Systematic Risk β p 2 σ m 2 Studies have bee performed that show holdig 30 stocks should be sufficiet to diversify the portfolio. Covertible Bods ad Warrats I additio to the stadard debt ad equity istrumets that a corporatio ca issue, there are hybrid istrumets: debt istrumets with exposure to equity, foreig exchage, commodities or currecy markets. These hybrid istrumets iclude dual currecy bods, covertible bods, PERCS ad so o. Corporatios ca issue two types of stock: commo ad preferred. Preferred stock etitles the holder to preferred divideds. Preferred divideds are paid before ay commo stock divideds. The coupo rate quoted is the maximum the compay will pay, as it may pay less if it does ot have the moey. Some preferred stock has the feature of beig covertible ito commo stock at the holder s optio. The corporatio may also issue covertible bods, which are bods etitlig the holder to the ormal coupo paymets ad par value at maturity, but, like covertible preferred stock, have the feature of beig covertible ito commo stock. Sice the process by which we value covertible bods ad covertible preferred stock is the same, we wo t distiguish betwee them i the followig. Warrats are rights to purchase stock at a later date at a specified price, called the subscriptio price. Warrats are ofte attached to debt offerigs but are detachable. If the subscriptio price is lower tha the stock price, the warrat will have itrisic value. Warrats are similar to call optios, but the major

14 differeces are that () call optios are traded o regular exchages through clearighouses ad warrats are ot; (2) exercise of call optios does ot result i the issue of ew stock, but exercise of warrats causes issuace of ew stock, so causes dilutio of shares ad (3) warrats typically have log expiratio dates, spaig several years while commo stock optios (ot withstadig LEAPS) may have expiries of less tha oe year. The famed Black-Scholes equatio was actually origially developed as a meas to value warrats. Covertible Bods Like a regular bod, a covertible bod has a stated coupo, paymet schedule, par value ad maturity. Corporatios issue covertible bods because they provide a low-cost fudig alterative (perhaps 400 to 700 basis poits below the rates they would pay o straight bods.) Ivestors are willig to accept lower rates for the opportuity to participate i stock appreciatio ad the corporatio also beefits because most covertible bods are callable, allowig them to force coversio if eed be. Some covertible bods are eve putable. Covertible bods are becomig icreasigly popular, particularly amog the dot.com compaies, with $6 billio issued i Start up compaies have traditioally bee able to obtai ew capital by issuig equity, which would dilute existig shares, or by issuig debt, which would probably carry a high coupo rate. Covertible bods provide a low-cost alterative. The compay is effectively sellig stock at a premium ad there are favorable accoutig treatmets. Compaies that issue covertible bods are hopig that their stock prices go up so they ca call the bods, forcig coversio ad edig their iterest paymets. Whe a corporatio issues a covertible bod, the term sheet will specify a coversio price ad a coversio ratio. If you kow the coversio price, you ca figure out the coversio ratio ad vice versa. For example, if a compay issues a corporate bod covertible ito 50 shares of commo stock, the coversio ratio is oe bod per 50 shares of stock ad the coversio price is $,000/50 shares $20/share. The oly difficulty i valuatio of covertible bods is decidig upo a optimal coversio strategy. I order to value a covertible bod, we must compare the value of the bod as a straight debt istrumet to the value of the equity it could be coverted ito. Depedig o the price of the uderlyig equity, covertible bods may behave more like equity tha debt. Some compaies are turig to zero-coupo covertible bods as a alterative to raisig capital i the equity market. The depedece of the covertible bod value to the uderlyig equity value ca be depicted i a graph composed of three regios as show below. Value of Covertible Bod Bod-Equivalet Regio Hybrid Regio Straight Bod Value Equity-Equivalet Regio Covertible Value Coversio Value Value of Commo Stock

15 I the bod-equivalet regio, the coversio price is well above the market price of commo stock ad so the bod trades like a straight debt istrumet. I the hybrid regio, the bod has characteristics of debt ad equity: as the share price icreases, the bod icreases i value, but sice it is tradig at a premium, it does ot icrease at the same rate as the stock. I the equity-equivalet regio, the commo shares have appreciated above the coversio price ad the bod the behaves similarly to equity. To determie the optimal coversio strategy we will use the squeezig techique oce agai: the ivestor would make a risk-free profit if he purchased the bod whe the price of commo stock were higher tha the coversio price, sice he could just buy the bod, immediately covert it ad sell the stock. O the other had, if the coversio price were lower tha the curret share price, he could short the bod ad buy the shares. Thus, the optimal coversio strategy would be to covert the bod whe the coversio price is equal to the share price. ParValue of Bod Coversio Pr ice Coversio Ratio Stock splits ad stock divideds will affect the coversio ratio ad coversio price of a covertible bod, sice more shares will exist. A 2 for stock split will make the precedig bod covertible ito 00 shares ($50 x 2) so the ew coversio price will be $0 ($20 x ½) A 20% stock divided will result i 0 more shares for each 50 held, so the coversio ratio will icrease by 20% while the coversio price will declie to $6.67. Parity If the market value of the covertible bod is equal to the market value of the shares it ca be coverted ito, the bod is said to be at parity. Most covertible bods sell at a premium i order to iduce ivestors to hold them (ad that call optio they sold the compay is worth somethig.) Parity Pr ice of Bod Market Pr ice of Commo x Number of Shares Example If commo stock is sellig at $70/share ad the coversio ratio is 20 shares per bod, what price should the bod be to be at parity with the stock? It should sell at $70/share x 20 shares or $,400. Valuatio of Covertible Bods Step Determie the Miimum Value of the Covertible Bod This is the lowest value at which the bod should trade ad is foud by comparig the value of the bod as a straight bod without the coversio feature to the parity value. Step 2 Calculate the ivestmet premium of the bod Step 3 Calculate the premium payback period Step 4 Calculate value of the embedded covertible bod optio Example A iteret start-up issued $500 millio of eight-year, 7% covertible bods whe their curret stock price was $55/share. The bods were covertible ito shares of stock oce the stock hit $90. Assume that the bods are callable ad that the stock pays o divideds. If bods of similar credit quality were yieldig 8%, what is the market coversio price, the market premium ratio, the premium payback period ad the value of the embedded call optio give that the market price of the bod is $200? Assume that the par value of the bod is $,000 ad all coupos are paid aually. Step First, we must calculate the value of a straight bod (oe without the covertible feature.) The price of a eight year, 7% coupo bods priced to yield 8% is 94.7% of par or $94.7. Next, we calculate the coversio value of the bod. We have the coversio ratio k already (but if we did t, we would just calculate it as Par/Coversio price $,000/ ). The the coversio value of the bod is k x curret share price, x $55 $ We have two prices, which oe is right? The miimum price of the bod should be the larger of these two umbers max(straight bod price, coverted share value) $94.7. This is because if we priced the bod at the lower umber

16 $85.76, which assumes immediate coversio, a arb could buy this ad have a straight bod worth $94.7. Step 2 Next, we perform a premium aalysis by comparig this bod to a straight bod. We eed to figure out what we are really payig for the right to acquire shares. Market Pr ice Covertible Bod $,200 Pr emium Over Straight Bod 27%. Straight Value of Bod $94.7 As a side ote, we could also figure out the equivalet yield of this covertible bod compared to a straight bod yieldig 8% as a way of idicatig the premium paid. So what are we payig per share by acquirig them via the covertible bod? If we coverted ow, Market Pr ice of Covertible Bod Market Coversio Pr ice $,200/5.263 $228/share. k The coversio premium that we are payig is the the differece betwee the market coversio price ad the curret market price, $228 - $55 $73. The market coversio premium ratio is the premium over the market price of the shares, $73/$55 47%. Step 3 Is this a good deal? We are payig a premium to hold the covertible bod, but we will receive coupo iterest o it. The breakeve poit will occur where the differece betwee the coupo iterest ad the divideds we would receive had we just purchased the stock outright offsets the premium we pay to hold the bod. (I this case, there ARE o divideds, but the idea is preseted ayway.) As a rule of thumb the payback period should be betwee 3 ad 5 years. Market Pr ice of Bod kp Pr emium Payback Period Coupo o Bod kdiv CS CS $, ($55) 5.48 years. $70 0 Step 4 Fially, the value of the embedded call. If a covertible bod is either callable or putable, it is equivalet to a portfolio cosistig of a straight bod plus a call optio o the stock at the coversio price. So, P covbod P straight + Call Optio. The call optio ca be valued by the Black- Scholes equatio ad for this bod, with T 8 years, S 55, K 90, r f 5% ad σ 35%, the value of the call is $ Sice oe bod is covertible ito shares, so the price of the covertible bod should be *5.263 $,30. If the bod is callable by the corporatio, say at ay time after three years, the ivestor has essetially writte a covered call at the coversio price with maturity three years or greater. The value of the bod is the P covbod P straight + Call Optio Value of Embedded Call. Note that the Black-Scholes equatio ca be used to value the optio o the uderlyig stock, but it caot be used to value the embedded call. This is because the Black-Scholes equatio relies o may assumptios that are ot appropriate i valuig bods. First, it assumes that volatilities are costat which, for bods is icorrect. As a bod approaches maturity the duratio decreases ad the volatility should decrease sice the holder receives a kow quatity par at maturity. Secod, it assumes that short term iterest rates are costat, which will cause a sigificat error i valuatio of iterest-rate sesitive securities. Fially it is assumed that the distributio of termial prices is logormally distributed, with a price of zero as well as ifiitely high price possible (albeit with very low probability.) A bod would ever be worth more tha its par value plus ay accrued coupos so Black-Scholes teds to overprice bod optios. Istead, biomial trees should be used to value this optio. A fial ote: otice how the compay i this example was able to raise $500 millio by sellig these covertible bods. If all are evetually coverted, they will dilute equity by issuig 5.263*$500,000,000/$,000 par 2.6 millio ew shares. Had they simply sold the same umber of

17 shares at a public offerig at the curret market price of $55, they would have raised oly $403 millio. Equity-Specific Derivatives Equity derivatives iclude the familiar calls, puts ad other combiatios. These are valued i the usual way usig Black-Scholes or iterest rate trees. The foudatio for these derivatives is foud i chapter 3. Stock Idex Futures Futures ad forwards o stock idexes work just like the other forwards ad futures we have see, except that istead of physical delivery of a bod, stock or commodity, the cotract is settled i cash. The value of the cotract is calculated as (Idex Value)(Multiplier) where the multiplier used depeds o the idex. For the S&P500 ad S&P400, the multiplier is $250; for the DJIA, $0; for the Russell 2000, $500 ad for the NIKKEI the multiplier is $5. Thus the value of the log positio at time t is V t (Idex Multiplier)(F t -F t- ) where F t is the idex level at time t. The value of the short positio is V t. The multiplier makes it easy to calculate the chage i value of positio for a chage i uit of the uderlyig, it is just Idex Multiplier. Example A ivestor, Willie, believes that the DJIA will rise over the ext three moths ad decides to purchase a future o the idex. She deposits $5,000 i her margi accout ad istructs her futures broker to perform the trasactio. A secod ivestor, Tara, believes that the DJIA will fall over the ext three moths. She also deposits the margi fuds ito her accout ad requests executio of her trade. Whe the trades hit the futures pit i the CBOT, they are settled at There is o cash flow util the ed of the tradig day whe the accouts are marked to market. Each day, Willie s accout is settled accordig to V t (Multiplier)(F t -F t- ) ad Tara s accordig to V t - (Multiplier)(F t -F t- ). Suppose that at the ed of the first day, the value of the futures cotract is The profit to Willie is $0( ) $00 (as promised, $0 for every uit chage i the idex). Tara s accout is debited $00 ad her balace is ow $4,900. The $00 is passed through the clearighouse ad credited to Willie s accout, which ow has a balace of $5,00. This process cotiues o a daily basis util the ivestors reverse out of their positios or the expiratio date of the futures cotract. Suppose that oe moth passes ad the futures cotract is ow tradig at 000. Willie decides to close out her positio. She calls her broker ad asks him to sell her positio. Her total profit is the sum of the profit ad loss over each day, but sice (Multiplier)(F T F T- )+ (Multiplier)(F T- F T-2 )+ + (Multiplier)(F F 0 ) is the same as (Multiplier)(F T F 0 ) it ca be calculated as ($0)(F t -F t- ) $0( ) $,750. Willie walks away with her $6,750. If Tara also closes out her accout o the same day, she would walk away with $5,000 + ($,750 ) $3,250. Alteratively, oe or both of them ca just wait util expiratio. At this date, the futures price must coverge to the spot price of the idex. If the idex is 9920 o this date, Willie will lose a total of $50 for the three moth period ad Tara will realize a profit of $50. The daily ups ad dows could be much more volatile, however. Forwards o stock idexes work the same way, but cash is settled oly at oe time (expiry or closig out of the cotract.) The et chage i portfolio value should be the same whether settled i daily icremets as it is i futures cotracts or at oe poit as i a forward. Valuatio Stock idex futures ca be priced usig spot-futures parity, F ( r ) S e T f δ The divided yield δ is estimated usig historical data or take from the fiacial press. For example, the DJIA the divided yield as of May 3, 2002 was reported to be.86% compared to.57% oe year prior. If the curret level S ad the three-moth risk free rate is.82%, what should be the futures price? ( F e ). 25 0,08. Hedgig with Stock Idex Futures Cotracts Stock idex futures were developed as a hedgig vehicle for portfolios of equity securities. The most commo sceario is that a ivestor is log a portfolio of equities ad wishes to hedge agaist adverse price movemets. He ca hedge by shortig a umber of equivalet forward or futures cotracts. So his total portfolio cosists of a log positio i 0

18 the equity portfolio, Q P ad a short positio i forwards or futures Q F. If the value of the portfolio chages by a amout the the, if the hedge is perfect, the value of the total portfolio should be uchaged: V CS + V F 0. The value of the equity positio is the price times the umber of shares ad the value of the futures cotract is the cotract multiplier times the umber of cotracts N F times the cotract size, times the futures price so a chage i price would result i a correspodig chage i value of the hedged portfolio of: P CS Q CS + (Idex Multiplier)(Cotract Size)N F F 0 where β F is the futures hedge ratio, obtaied by a regressio of historical futures price chage to uderlyig price chage of equities. The R 2 obtaied i the regressio idicates the effectiveess of the hedge, while -R 2 is the basis risk of the hedge. Ideally, the portfolio of securities should be hedged with a istrumet havig R 2. This is ot always possible ad so there will be basis risk. Example A portfolio maager has $00,000,000 worth of equity securities. Cocered about his upcomig performace-based bous, he fears a loss i portfolio value over the ext five moths. He wats to hedge his portfolio to protect agaist a aticipated market fall. Sice he has mostly largecap, US domestic stocks i the portfolio, he chooses the 6 moth futures cotract o the S&P500 as his hedgig vehicle. The curret value of the idex is 074 ad six moth futures are 090. The umber of cotracts required for a hedge are: N F The Quatity of Equities to be Hedged $00,000,000 β β -367β F. F N Defiig β F F P CS ( Idex Multiplier) ( CotractSize) F P CS we have N F Q CS F β F ( Idex Multiplier )( CotractSize) F ( IdexMultiplier)( CotractSize) $250( 090) He rus a regressio ad determies β F to be 0.85 so he will eed to sell short 367(0.85) 32 futures cotracts. (You must always roud up or dow to a iteger umber). Suppose at the ed of five moths, the S&P500 idex is at 065 ad the futures cotract is at 080. What is his gai/(loss) o his portfolio, both hedged ad uhedged? Uhedged Iitial value of equity portfolio $00,000,000 Gai/(Loss) o stocks $00,000, ($72,29) 074 Net Value of Portfolio $99,287,70 Net Gai/(Loss) (7 bp) Hedged (eglectig trasactios costs) Iitial value of equity portfolio $00,000,000 Gai/(Loss) o stocks $00,000, ($72,29) 074 Gai/(Loss) o futures positio 32x$250x( ) $780,000 Net Value of Portfolio $00,067,70 Net Gai/(Loss) 7 bp The hedgig istrumet was ot perfect so there is some basis risk: risk that at termiatio the spread betwee the spot price of the uderlyig ad the futures price of the hedgig istrumet will ot be zero. Also, this hedge should be rebalaced o a daily basis to track fluctuatios i β F. This is called tailig the hedge. But the maager did better tha he would have, had he ot hedged. Q CS

19 Equity Swaps Suppose a ivestor desires exposure to Japaese stocks for diversificatio of a $00,000,000 portfolio but does ot wish to, or is uable to, make direct ivestmets i Japaese equities. A corporate treasurer holds a positio i the NIKKEI idex, but is ucomfortable with this exposure ad wishes istead to receive floatig-rate LIBOR. (Perhaps this treasurer is required to make floatig rate paymets to yet aother ivestor ad wats to hedge these paymets.) These two ivestors ca each receive the exposure they desire by eterig ito a equity swap. The ivestor agrees to pay the treasurer LIBOR i exchage for receivig the retur o the NIKKEI idex o a otioal amout, here agreed to be $00,000,000. No otioal is exchaged i this trasactio ad the paymets are etted. This is a OTC trasactio ad the cash flows would appear as follows: Ivestor NIKKEI Treasurer LIBOR NIKKEI Suppose that the three-moth LIBOR o the date of the agreemet is 5%. The retur o the NIKKEI over the three-moth period is 0%. The the treasurer would pay 0.($00,000,000) $0,000,000 to the ivestor. The ivestor must pay LIBOR/4*$00,000,000 $,250,000 to the treasurer. Sice paymets are etted, the treasurer pays $8,750,000 to the ivestor. Now, the treasurer origially eared $0,000,000 o his log NIKKEI positio. By payig $8,750,000 to the ivestor, he has a et retur of 5% aualized, the LIBOR retur he wated. The ivestor, i tur, receives the full retur from the NIKKEI, the exposure he wated. What if the NIKKEI had lost 5% istead? The treasurer is boud to pay r NIKKEI so would receive $5,000,000 from the ivestor. The ivestor also has to pay LIBOR so the et positio of the treasurer is that his NIKKEI loss is covered, he gets LIBOR ad the ivestor is fully exposed to the 5% loss o the NIKKEI, which is what he bargaied for.

20 Sample Questios ad Aswers. What should be the retur o equity capital of TASER s stock if the risk-free rate is 5%, the expected retur o the market is 0% ad TASER s beta is.78? 2. What should be ext-year s price of TASER if the curret ROE is 7.25%, the compay pays out o divideds ad the curret price is $8? 3. If the et icome for the past twelve moths was $650,000 ad there are 2.8 millio shares outstadig, what is the P/E ratio? If the compay should be tradig at the curret P/E of the Dow, 20, what would the price be? 4. Based o a sector price/sales ratio of 6.54, total twelve-moth trailig sales of 8 millio what should the price of TASER be? 5. Kevi ows 00,000 shares of Ford Motor Compay stock for his portfolio. Ford is curretly tradig at $7.72 with a beta of.05. He is cocered that the share price will fall over the ext three moths but is restricted by his employer from tradig optios o Ford stock, so decides to hedge by usig futures o the S&P500. The curret level of the S&P500 is If the futures price is 085, how may cotracts would he have to buy or sell? If he closes out his positio i two moths whe the idex is at 060 ad the futures price is 075, what is his gai/(loss) compared to the gai/(loss) he would have experieced without hedgig? Assume zero trasactio cost. 6. Kevi quits ad is ow allowed to trade optios o Ford. If the curret risk free rate is 5%, the aual volatility of Ford stock is 5% ad he buys a three-moth put with a strike price of $7, how much does the put cost? How much should a call cost? Assume o divideds will be paid. 7. Kevi wats to buy more Ford shares but woders if they are overvalued. He has checked GM ad Daimler-Chysler s P/E ratios ad they are ad 2.4. He ca t calculate Ford s P/E ratio but uses the divided-discout model to estimate the P/E usig the curret divided of $0.40/year, divided yield 2.25% ad expected growth rate of 5%. What is the P/E ratio o this basis? Assume Ford pays out 30% of its earigs i the form of divideds. 8. Kevi s broker calls him ad tells him about a ew te-year 6% covertible bod issued by Ford Motor Credit which will allow him to covert ito Ford stock whe Ford reaches $25/share. If he buys the bod, what effective price is he payig for the shares ad how log will it take him to break eve? Assume that Ford maitais the divided yield of 2.25%, that the bod is sellig at 0% of par (i.e., $,00) ad makes aual paymets. The te-year treasury bod is curretly yieldig 6.5%. Is this bod a good deal? Aswers. Use the CAPM to fid R s R f + β(r m R s ) 5% +.78(0%-5%) 3.9%. This is the required retur o equity capital that should be used i discoutig future free cash flows to equity. 2. Sice the compay pays o divideds, we ca t use the model DIV P r CE ( + g) KE ( + g) 0 0 g r CE g However, we ca use the ROE. If we assume that TASER will grow oly because it plows back earigs, the growth rate should equal ROE, i fact g ROE(-K) but sice K, the divided payout P + DIV P0 P0 ( + g) + DIV P0 DIV ratio, 0 we have g ROE 7.25%. Sice r + g P0 P0 P0 Thus g r 7.25%. The P P 0 (+g) $8(.725) $2. 3. Earigs/share et icome/umber of shares outstadig $650,000/2.8 millio shares $0.23/share, P/E $8/ If Price is 20 times earigs, the price should be $4.6/share. 4. Price/Sales 6.54, sales/share $8 millio/2.8 millio so price (P/S) (Sales) 6.54(8/2.8) $ The quatity of equities to be hedged umber of shares x share price 00,000 x $7.72. The umber of futures cotracts for the hedge is calculated from