APPENDIX 1 TO CHAPTER 4 Models of Asset Pricig I this appedix, we first examie why diversificatio, the holdig of may risky assets i a portfolio, reduces the overall risk a ivestor faces. The we will see how diversificatio affects the price of assets by developig models of asset pricig. Beefits of Diversificatio Our discussio of the theory of asset demad idicates that most people like to avoid risk; that is, they are risk-averse. Why, the, do most ivestors hold may risky assets rather tha just oe? Does t holdig may risky assets expose the ivestor to more risk? The old warig about ot puttig all your eggs i oe basket holds the key to the aswer: Because holdig may risky assets (called diversificatio) reduces the overall risk a ivestor faces, diversificatio is beeficial. To see why this is so, let s look at some specific examples of how a ivestor fares o his ivestmets whe he is holdig two risky securities. Cosider two assets: commo stock of Frivolous Luxuries, Ic., ad commo stock of Bad Times Products, Ulimited. Whe the ecoomy is strog, which we ll assume is oe-half of the time, Frivolous Luxuries has high sales ad the retur o the stock is 15%; whe the ecoomy is weak, the other half of the time, sales are low ad the retur o the stock is 5%. O the other had, suppose that Bad Times Products thrives whe the ecoomy is weak, so that its stock has a retur of 15%, but it ears less whe the ecoomy is strog ad has a retur o the stock of 5%. Sice both these stocks have a expected retur of 15% half the time ad 5% the other half of the time, both have a expected retur of 10%. However, both stocks carry a fair amout of risk, because there is ucertaity about their actual returs. Suppose, however, that istead of buyig oe stock or the other, Irvig the Ivestor puts half his savigs i Frivolous Luxuries stock ad the other half i Bad Times Products stock. Whe the ecoomy is strog, Frivolous Luxuries stock has a retur of 15%, while Bad Times Products has a retur of 5%. The result is that Irvig ears a retur of 10% (the average of 5% ad 15%) o his holdigs of the two stocks. Whe the ecoomy is weak, Frivolous Luxuries has a retur of oly 5% ad Bad Times Products has a retur of 15%, so Irvig still ears a retur of 10% regardless of whether the ecoomy is strog or weak. Irvig is better off from this W-1
W- Appedix 1 to Chapter 4 strategy of diversificatio because his expected retur is 10%, the same as from holdig either Frivolous Luxuries or Bad Times Products aloe, ad yet he is ot exposed to ay risk. Although the case we have described demostrates the beefits of diversificatio, it is somewhat urealistic. It is quite hard to fid two securities with the characteristic that whe the retur of oe is high, the retur of the other is always low. 1 I the real world, we are more likely to fid at best returs o securities that are idepedet of each other; that is, whe oe is high, the other is just as likely to be high as to be low. Suppose that both securities have a expected retur of 10%, with a retur of 5% half the time ad 15% the other half of the time. Sometimes both securities will ear the higher retur ad sometimes both will ear the lower retur. I this case if Irvig holds equal amouts of each security, he will o average ear the same retur as if he had just put all his savigs ito oe of these securities. However, because the returs o these two securities are idepedet, it is just as likely that whe oe ears the high 15% retur, the other ears the low 5% retur ad vice versa, givig Irvig a retur of 10% (equal to the expected retur). Because Irvig is more likely to ear what he expected to ear whe he holds both securities istead of just oe, we ca see that Irvig has agai reduced his risk through diversificatio. The oe case i which Irvig will ot beefit from diversificatio occurs whe the returs o the two securities move perfectly together. I this case, whe the first security has a retur of 15%, the other also has a retur of 15% ad holdig both securities results i a retur of 15%. Whe the first security has a retur of 5%, the other has a retur of 5% ad holdig both results i a retur of 5%. The result of diversifyig by holdig both securities is a retur of 15% half of the time ad 5% the other half of the time, which is exactly the same set of returs that is eared by holdig oly oe of the securities. Cosequetly, diversificatio i this case does ot lead to ay reductio of risk. The examples we have just examied illustrate the followig importat poits about diversificatio: 1. Diversificatio is almost always beeficial to the risk-averse ivestor sice it reduces risk uless returs o securities move perfectly together (which is a extremely rare occurrece).. The less the returs o two securities move together, the more beefit (risk reductio) there is from diversificatio. 1 Such a case is described by sayig that the returs o the two securities are perfectly egatively correlated. We ca also see that diversificatio i the example above leads to lower risk by examiig the stadard deviatio of returs whe Irvig diversifies ad whe he does t. The stadard deviatio of returs if Irvig holds oly oe of the two securities is 0.5 115% 10% 0.5 15% 10% 5%. 1 Whe Irvig holds equal amouts of each security, there is a probability of 4 that he will ear 5% o both 1 (for a total retur of 5%), a probability of 4 that he will ear 15% o both (for a total retur of 15%), ad a probability of 1/ that he will ear 15% o oe ad 5% o the other (for a total retur of 10%). The stadard deviatio of returs whe Irvig diversifies is thus 0.5 115% 10% 0.5 15% 10% 0.5 110% 10% 3.5%. Sice the stadard deviatio of returs whe Irvig diversifies is lower tha whe he holds oly oe security, we ca see that diversificatio has reduced risk.
Models of Asset Pricig W-3 Diversificatio ad Beta I the previous sectio, we demostrated the beefits of diversificatio. Here, we examie diversificatio ad the relatioship betwee risk ad returs i more detail. As a result, we obtai a uderstadig of two basic theories of asset pricig: the capital asset pricig model (CAPM) ad arbitrage pricig theory (APT). We start our aalysis by cosiderig a portfolio of assets whose retur is R p x 1 R 1 1 x R p x R (1) where R p = the retur o the portfolio of assets R i = the retur o asset i x i = the proportio of the portfolio held i asset i The expected retur o this portfolio, E(R p ), equals E1R p E1x 1 R 1 E1x R p E1x R x 1 E1R 1 x E1R p x E1R () A appropriate measure of the risk for this portfolio is the stadard deviatio of the portfolio s retur ( p ) or its squared value, the variace of the portfolio s retur ( ), which ca be writte as p E[R p E(R p )] E[{x 1 R 1 p x R } {x 1 E(R 1 ) p x E(R )}] E[x 1 {R 1 E(R 1 )} p x {R E(R)}] This expressio ca be rewritte as p p E[{x 1 [R 1 E(R 1 )] p x [R E(R)]} {R p E(R p )}] x 1 E[{R 1 E(R 1 )} {R p E(Rp)}] p x E[{R E(R )} {R p E(R p )}] This gives us the followig expressio for the variace for the portfolio s retur: p x 1 1p x p x p (3) p where the covariace of the retur o asset i with the portfolio s retur E[{R i E(R i )} {R p E(Rp)}] Equatio 3 tells us that the cotributio to risk of asset i to the portfolio is x i p. By dividig this cotributio to risk by the total portfolio risk ( p ), we have the proportioate cotributio of asset i to the portfolio risk: x i p > p The ratio p > p tells us about the sesitivity of asset i s retur to the portfolio s retur. The higher the ratio is, the more the value of the asset moves with chages i the value of the portfolio, ad the more asset i cotributes to portfolio risk. Our algebraic maipulatios have thus led to the followig importat coclusio: The margial cotributio of a asset to the risk of a portfolio depeds ot
W-4 Appedix 1 to Chapter 4 o the risk of the asset i isolatio, but rather o the sesitivity of that asset s retur to chages i the value of the portfolio. If the total of all risky assets i the market is icluded i the portfolio, the it is called the market portfolio. If we suppose that the portfolio, p, is the market portfolio, m, the the ratio m > m is called the asset i s beta; that is: m > m (4) where the beta of asset i A asset s beta, the, is a measure of the asset s margial cotributio to the risk of the market portfolio. A higher beta meas that a asset s retur is more sesitive to chages i the value of the market portfolio ad that the asset cotributes more to the risk of the portfolio. Aother way to uderstad beta is to recogize that the retur o asset i ca be cosidered as beig made up of two compoets oe that moves with the market s retur (R m ) ad the other a radom factor with a expected value of zero that is uique to the asset ( ) ad so is ucorrelated with the market retur: R i 1 R m (5) The expected retur of asset i ca the be writte as It is easy to show that i the above expressio is the beta of asset i we defied before by calculatig the covariace of asset i s retur with the market retur usig the two equatios above: m E[{R i E(R i )} {R m E(R m )}] E[{ [R m E(R m ) } {R m E(R m )}] However, sice is ucorrelated with R m,e[{ } {R m E(R m )}] 0. Therefore, Dividig through by gives us the followig expressio for : m E1R i E1R m m m m > m which is the same defiitio for beta we foud i Equatio 4. The reaso for demostratig that the i Equatio 5 is the same as the oe we defied before is that Equatio 5 provides better ituitio about how a asset s beta measures its sesitivity to chages i the market retur. Equatio 5 tells us that whe the beta of a asset is 1.0, its retur o average icreases by 1 percetage poit whe the market retur icreases by 1 percetage poit; whe the beta is.0, the asset s retur icreases by percetage poits whe the market retur
Models of Asset Pricig W-5 icreases by 1 percetage poit; ad whe the beta is 0.5, the asset s retur oly icreases by 0.5 percetage poit o average whe the market retur icreases by 1 percetage poit. Equatio 5 also tells us that we ca get estimates of beta by comparig the average retur o a asset with the average market retur. For those of you who kow a little ecoometrics, this estimate of beta is just a ordiary least squares regressio of the asset s retur o the market retur. Ideed, the formula for the ordiary least squares estimate of m > m is exactly the same as the defiitio of earlier. Systematic ad Nosystematic Risk We ca derive aother importat idea about the riskiess of a asset usig Equatio 5. The variace of asset i s retur ca be calculated from Equatio 5 as ad sice E[R i E(R i )] E{ [R m E(R m )} ] is ucorrelated with market retur: The total variace of the asset s retur ca thus be broke up ito a compoet that is related to market risk, i m, ad a compoet that is uique to the asset,. The i m compoet related to market risk is referred to as systematic risk, ad the compoet uique to the asset is called osystematic risk. We ca thus write the total risk of a asset as beig made up of systematic risk ad osystematic risk: Total asset risk systematic risk osystematic risk (6) Systematic ad osystematic risk each have aother feature that makes the distictio betwee these two types of risk importat. Systematic risk is the part of a asset s risk that caot be elimiated by holdig the asset as part of a diversified portfolio, whereas osystematic risk is the part of a asset s risk that ca be elimiated i a diversified portfolio. Uderstadig these features of systematic ad osystematic risk leads to the followig importat coclusio: The risk of a well-diversified portfolio depeds oly o the systematic risk of the assets i the portfolio. We ca see that this coclusio is true by cosiderig a portfolio of assets, each of which has the same weight o the portfolio of (1/). Usig Equatio 5, the retur o this portfolio is R p 11> a which ca be rewritte as m 11> a R p R m 11> a R m 11> a
W-6 Appedix 1 to Chapter 4 where the average of the s 11> a the average of the s 11> a If the portfolio is well diversified so that the s are ucorrelated with each other, the usig this fact ad the fact that all the s are ucorrelated with the market retur, the variace of the portfolio s retur is calculated as p m 11>1average variece of As gets large the secod term, (1/)(average variace of ), becomes very small, so that a well-diversified portfolio has a risk of m, which is oly related to systematic risk. As the previous coclusio idicated, osystematic risk ca be elimiated i a well-diversified portfolio. This reasoig also tells us that the risk of a welldiversified portfolio is greater tha the risk of the market portfolio if the average beta of the assets i the portfolio is greater tha 1; however, the portfolio s risk is less tha the market portfolio if the average beta of the assets is less tha 1. The Capital Asset Pricig Model (CAPM) We ca ow use the ideas we developed about systematic ad osystematic risk ad betas to derive oe of the most widely used models of asset pricig the capital asset pricig model (CAPM) developed by William Sharpe, Joh Liter, ad Jack Treyor. Each cross i Figure 1 shows the stadard deviatio ad expected retur for each risky asset. By puttig differet proportios of these assets ito portfolios, we ca geerate a stadard deviatio ad expected retur for each of the portfolios usig Equatios ad 3. The shaded area i the figure shows these combiatios of stadard deviatio ad expected retur for these portfolios. Sice risk-averse ivestors always prefer to have higher expected retur ad lower stadard deviatio of the retur, the most attractive stadard deviatio expected retur combiatios are the oes that lie alog the heavy lie, which is called the efficiet portfolio frotier. These are the stadard deviatio expected retur combiatios risk-averse ivestors would always prefer. The capital asset pricig model assumes that ivestors ca borrow ad led as much as they wat at a risk-free rate of iterest, R f. By ledig at the risk-free rate, the ivestor ears a expected retur of R f ad his ivestmet has a zero stadard deviatio because it is risk-free. The stadard deviatio expected retur combiatio for this risk-free ivestmet is marked as poit A i Figure 1. Suppose a ivestor decides to put half of his total wealth i the risk-free loa ad the other half i the portfolio o the efficiet portfolio frotier with a stadard deviatio expected retur combiatio marked as poit M i the figure. Usig Equatio, you should be able to verify that the expected retur o this ew portfolio is halfway betwee R f ad E(R m ); that is, [R f E(R m )]/. Similarly, because the covariace betwee the risk-free retur ad the retur o portfolio M must ecessarily be zero, sice there is o ucertaity about the retur o the risk-free loa, you should also be able to verify, usig Equatio 3, that the stadard deviatio of the retur o the ew portfolio is halfway betwee zero ad, that is, (1/). m m
Models of Asset Pricig W-7 Expected Retur E(R) E(R m ) R f Opportuity Locus C E(R m ) R f E(R m ) R f A Efficiet Portfolio Frotier B M FIGURE 1 1/ m m m Stadard Deviatio of Retus Risk Expected Retur Trade-off The crosses show the combiatio of stadard deviatio ad expected retur for each risky asset. The efficiet portfolio frotier idicates the most preferable stadard deviatio expected retur combiatios that ca be achieved by puttig risky assets ito portfolios. By borrowig ad ledig at the risk-free rate ad ivestig i portfolio M, the ivestor ca obtai stadard deviatio expected retur combiatios that lie alog the lie coectig A, B, M, ad C. This lie, the opportuity locus, cotais the best combiatios of stadard deviatios ad expected returs available to the ivestor; hece the opportuity locus shows the trade-off betwee expected returs ad risk for the ivestor. The stadard deviatio expected retur combiatio for this ew portfolio is marked as poit B i the figure, ad as you ca see it lies o the lie betwee poit A ad poit M. Similarly, if a ivestor borrows the total amout of her wealth at the riskfree rate R f ad ivests the proceeds plus her wealth (that is, twice her wealth) i portfolio M, the the stadard deviatio of this ew portfolio will be twice the stadard deviatio of retur o portfolio M, m. O the other had, usig Equatio, the expected retur o this ew portfolio is E(R m ) E(R m ) R f, which equals E(R m ) R f. This stadard deviatio expected retur combiatio is plotted as poit C i the figure. You should ow be able to see that both poit B ad poit C are o the lie coectig poit A ad poit M. Ideed, by choosig differet amouts of borrowig ad ledig, a ivestor ca form a portfolio with a stadard deviatio expected retur combiatio that lies aywhere o the lie coectig poits A ad M. You may have oticed that poit M has bee chose so that the lie coectig poits A ad M is taget to the efficiet portfolio frotier. The reaso for choosig poit M i this way is that it leads to stadard deviatio expected retur combiatios alog the lie that are the most desirable for a risk-averse ivestor. This lie ca be thought of as the opportuity locus, which shows the best combiatios of stadard deviatios ad expected returs available to the ivestor.
W-8 Appedix 1 to Chapter 4 The capital asset pricig model makes aother assumptio: All ivestors have the same assessmet of the expected returs ad stadard deviatios of all assets. I this case, portfolio M is the same for all ivestors. Thus whe all ivestors holdigs of portfolio M are added together, they must equal all of the risky assets i the market, which is just the market portfolio. The assumptio that all ivestors have the same assessmet of risk ad retur for all assets thus meas that portfolio M is the market portfolio.therefore, the R m ad m i Figure 1 are idetical to the market retur, R m, ad the stadard deviatio of this retur, m, referred to earlier i this appedix. The coclusio that the market portfolio ad portfolio M are oe ad the same meas that the opportuity locus i Figure 1 ca be thought of as showig the tradeoff betwee expected returs ad icreased risk for the ivestor. This trade-off is give by the slope of the opportuity locus, E(R m ) R f, ad it tells us that whe a ivestor is willig to icrease the risk of his portfolio by m, the he ca ear a additioal expected retur of E(R m ) R f. The market price of a uit of market risk, m, is E(R m ) R f. E(R m ) R f is therefore referred to as the market price of risk. We ow kow that market price of risk is E(R m ) R f ad we also have leared that a asset s beta tells us about systematic risk, because it is the margial cotributio of that asset to a portfolio s risk. Therefore the amout a asset s expected retur exceeds the risk-free rate, E(R i ) R f, should equal the market price of risk times the margial cotributio of that asset to portfolio risk, [E(R m ) R f ]. This reasoig yields the CAPM asset pricig relatioship: E(R i ) R f [E(R m ) R f ] (7) This CAPM asset pricig equatio is represeted by the upward slopig lie i Figure, which is called the security market lie. It tells us the expected retur that the market sets for a security give its beta. For example, it tells us that if a security has a beta of 1.0 so that its margial cotributio to a portfolio s risk is the same as the market portfolio, the it should be priced to have the same expected retur as the market portfolio, E(R m ). To see that securities should be priced so that their expected retur beta combiatio should lie o the security market lie, cosider a security like S i Figure, which is below the security market lie. If a ivestor makes a ivestmet i which half is put ito the market portfolio ad half ito a risk-free loa, the the beta of this ivestmet will be 0.5, the same as security S. However, this ivestmet will have a expected retur o the security market lie that is greater tha that for security S. Hece ivestors will ot wat to hold security S ad its curret price will fall, thus raisig its expected retur util it equals the amout idicated o the security market lie. O the other had, suppose there is a security like T which has a beta of 0.5 but whose expected retur is above the security market lie. By icludig this security i a well-diversified portfolio with other assets with a beta of 0.5, oe of which ca have a expected retur less tha that idicated by the security lie (as we have show), ivestors ca obtai a portfolio with a higher expected retur tha that obtaied by puttig half ito a risk-free loa ad half ito the market portfolio. This would mea that all ivestors would wat to hold more of security T, ad so its price would rise, thus lowerig its expected retur util it equaled the amout idicated o the security market lie. The capital asset pricig model formalizes the followig importat idea: A asset should be priced so that is has a higher expected retur ot whe it has a greater risk i isolatio, but rather whe its systematic risk is greater.
Models of Asset Pricig W-9 Expected Retur E(R) E(R m ) R f T S Security Market Lie FIGURE Security Market Lie 0.5 1.0 Beta The security market lie derived from the capital asset pricig model describes the relatioship betwee a asset s beta ad its expected retur. Arbitrage Pricig Theory Although the capital asset pricig model has proved to be very useful i practice, derivig it does require the adoptio of some urealistic assumptios; for example, the assumptio that ivestors ca borrow ad led freely at the risk-free rate, or the assumptio that all ivestors have the same assessmet of expected returs ad stadard deviatios of returs for all assets. A importat alterative to the capital asset pricig model is the arbitrage pricig theory (APT) developed by Stephe Ross of M.I.T. I cotrast to CAPM, which has oly oe source of systematic risk, the market retur, APT takes the view that there ca be several sources of systematic risk i the ecoomy that caot be elimiated through diversificatio. These sources of risk ca be thought of as factors that may be related to such items as iflatio, aggregate output, default risk premiums, ad/or the term structure of iterest rates. The retur o a asset i ca thus be writte as beig made up of compoets that move with these factors ad a radom compoet that is uique to the asset ( ): R i 1 1factor 1 1factor p k 1factor k (8) Sice there are k factors, this model is called a k-factor model. The 1 i, p k, describe the sesitivity of the asset i s retur to each of these factors. Just as i the capital asset pricig model, these systematic sources of risk should be priced. The market price for each factor j ca be thought of as E(R factor j ) R f, ad hece the expected retur o a security ca be writte as: E1R i R f 1 3E1R factor 1 R f 4 p k 3E1R factor k R f 4 (9)
W-10 Appedix 1 to Chapter 4 This asset pricig equatio idicates that all the securities should have the same market price for the risk cotributed by each factor. If the expected retur for a security were above the amout idicated by the APT pricig equatio, the it would provide a higher expected retur tha a portfolio of other securities with the same average sesitivity to each factor. Hece ivestors would wat to hold more of this security, ad its price would rise util the expected retur fell to the value idicated by the APT pricig equatio. O the other had, if the security s expected retur were less tha the amout idicated by the APT pricig equatio, the o oe would wat to hold this security, because a higher expected retur could be obtaied with a portfolio of securities with the same average sesitivity to each factor. As a result, the price of the security would fall util its expected retur rose to the value idicated by the APT equatio. As this brief outlie of arbitrage pricig theory idicates, the theory supports a basic coclusio from the capital asset pricig model: A asset should be priced so that it has a higher expected retur ot whe it has a greater risk i isolatio, but rather whe its systematic risk is greater. There is still substatial cotroversy about whether a variat of the capital asset pricig model or the arbitrage pricig theory is a better descriptio of reality. At the preset time, both frameworks are cosidered valuable tools for uderstadig how risk affects the prices of assets.