Chapter 5 Portfolios, Efficiency and the Capital Asset Pricing Model

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1 Chater 5 Portfolios, fficiency and the Caital sset Pricing Model The obectives of this chater are to enable you to: Understand the rocess of cobining of securities into ortfolios Understand easureent of ortfolio return and risk reciate the iortance of diversification Understand ho diversification is related to security returns covariance and ortfolio size Consider the iact of internationalization on ortfolio risk and efficiency Cobine riskless and risky securities to for an efficient ortfolio Understand distinctions beteen arket and fir-secific risk Understand the relationshi beteen return and risk Coute beta and aly it to risk-adusted discount rates for resent value analysis 5.. INTODUCTION In Chater 4, e learned about assessing the return and risk of a single security or investent. In this chater, e ill learn ho to do the sae for a ortfolio. ortfolio is sily a collection of investents. The entire set of an investor s holdings is considered to be his ortfolio. It ay be reasonable for an investor to be concerned ith the erforance of individual securities only to the extent that their erforance affects the erforance of his overall ortfolio of investents. Thus, the erforance of the ortfolio is of riary iortance to the investor. The return of an investor's ortfolio is sily a eighted average of the returns of the individual securities that corise his ortfolio. The exected return of a ortfolio ay be calculated either as a function of otential ortfolio returns and their associated robabilities (a eighted average of otential returns as in Chater 4) or as a sile eighted average of the exected individual security returns. Hoever, the risk of the ortfolio is soehat ore colicated to deterine. Generally, the ortfolio variance or standard deviation of returns ill be less than a eighted average of the individual security variances or standard deviations. This reduction on ortfolio risk ill be intensified as the ortfolio becoes ore diversified; that is, ortfolio risk is reduced hen the selected securities are ore dissiilar and hen the nuber of securities in the ortfolio increases. 5.B. POTFOLIO TUN The exected return of a ortfolio can be calculated using quation (.0) here the subscrit () designates the ortfolio and the subscrit () designates one articular outcoe out of () otential outcoes:

2 (4.0) P. Chater 5 Thus, the exected return of a ortfolio is sily a eighted average of the otential ortfolio returns here the outcoe robabilities serve as the eights. This is ust ho e couted the exected return of a security in Chater 4. For any ortfolio anageent alications, it is useful to exress ortfolio return as a function of the returns of the individual securities that corise the ortfolio. This is often because e ant to kno ho a articular security ill affect the return and risk of our overall holdings or ortfolio. For exale, consider a ortfolio ade u of to securities, one and to. The exected return of security one is 0% and the exected return of security to is 0%. If forty ercent of the dollar value of the ortfolio is invested in security one (that is, [ ] =.40), and the reainder is invested in security to ([ ] =.60), the exected return of the ortfolio ay be deterined by quation (5.): (5.) i i i (. 4.0) (.6.0). 6 n. The subscrit (i) designates a articular security, and eights [ i ] are the ortfolio roortions. That is, a security eight ( i ) secifies ho uch oney is invested in Security (i) relative to the total aount invested in the entire ortfolio. For exale, [ ] is: = $ invested in security _ Total $ invested in the ortfolio Thus, ortfolio return is sily a eighted average of individual security returns. 5.C. POTFOLIO VINC Because risky securities often behave quite differently, the variance of ortfolio returns is not sily a eighted average of individual security variances. In fact, in this section, e ill deonstrate that cobining securities into ortfolios ay actually result in risk levels loer than those of any of the securities corising the ortfolio. That is, in soe instances, e can cobine a series of highly risky assets into a relatively safe ortfolio. The risk of this ortfolio in ters of variance of returns can be deterined by solving the folloing double suation: n n (5.) i i i i n n i. Consider the ortfolio constructed in section 5.B. If the standard deviation of returns on securities one and to ere.0 and.30, resectively, and the correlation coefficient ( i ) beteen returns on the to securities ere.5, the resultant standard deviation of the ortfolio ould be.3, i i

3 Portfolios, fficiency and the Caital sset Pricing Model the square root of its.0504 variance level: (5.3) ( ) ( ) ( ) ( ). 3 ; More generally,.0504 (5.4) ( ( ) ( ) ) ( ) Notice that both counters (i) and () are set equal to one to begin the double suation rocess. Thus, in the first set of arentheses of quations (5.3) and (5.4), since both (i) and () equal one, both ortfolio eights ( i ) and ( ) equal.4; ( i ) and ( ) equal.. The coefficient of correlation beteen any variable and itself ust be one; therefore, ( ) equals one. fter variables are substituted into quation (5.) for (i) equals and () equals, the counter of the inside suation is increased to to. Thus, in the second set of arentheses, (i) equals and () equals. Hence, ( i ) equals.4, ( ) equals.6, ( i ) equals., and ( ) equals.3. Since the nuber of securities corising the ortfolio (n) is to, the inside suation is coleted. We no increase the counter of the outside suation (i) to to and begin the inside suation over again (by setting equal to ). Thus, in the third set of arentheses, (i) equals to and () equals one. The correlation coefficient ( ) ust equal.5 because it ust be identical to ( ). We no increase the counter of the inside suation to to; in the fourth set of arentheses both (i) and () equal to. Since both counters no equal (n), (5.3) can be silified and solved. It is iortant to realize that (i) and () are erely counters; they do not necessarily refer to any secific security consistently throughout the suation rocess. By silifying the exressions in the first and fourth sets of arentheses, and cobining the ters in the second and third sets, one can silify quation (5.3): (5.5) (.4. ) (.6.3 ) ( ) Therefore, hen a ortfolio is corised of to securities, its variance can be deterined by quation (5.6): (5.6) ) ( ) ( ) ( quation (5.6) allos us to deterine ortfolio variance ithout having to ork through the double suation only hen (n) equals to. Larger ortfolios require the use of soe for of quation (5.). Hoever, the nuber of sets of arentheses to ork through and then add is equal to the nuber of securities in the ortfolio squared (n ). quation (5.) can be silified to an equation ith a for siilar to that of (5.6). For exale, if the ortfolio ere to be corised of three securities, quation (5.6) ould change to: (5.6.a) ( ( ) ( 3 )( ) ( ) ( 3 3 ) 3 ) 3 3

4 4 Chater 5 You ay find it useful to derive quation (5.6.a) fro quation (5.). In any case, notice the siilarity in the atterns of variables beteen equations (5.6) and (5.6.a). If fifty securities ere to be included in the investor's ortfolio, 500 exressions ust be solved and then added for quation (5.). This ortfolio ould require solutions to 75 exressions for solving the ore sile exression (5.6.a). Obviously, as the nuber of securities in the ortfolio becoes large, couters becoe quite useful in orking through the reetitive calculations. The equations are not difficult to solve, they are erely reetitive and tie-consuing. In our first exale, the eighted average of the standard deviation of returns of the to securities one and to is 6%, yet the standard deviation of returns of the ortfolio they cobine to ake is only 3%. Clearly, soe risk has been diversified aay by cobining the to securities into the ortfolio. In fact, the risk of a ortfolio ill alost alays be loer than the eighted average of the standard deviations of the securities that corise that ortfolio. For a ore extree exale of the benefits of diversification, consider to securities, three and four, hose otential return outcoes are erfectly inversely related. Data relevant to these securities is listed in Table (5.). If outcoe one occurs, security three ill realize a return of 30%, and security four ill realize a 0% return level. If outcoe to is realized, both securities ill attain returns of 0%. If outcoe three is realized, securities three and four ill attain return levels of 0% and 30%, resectively. If each outcoe is equally likely to occur ([P i ] is.333 for all outcoes), the exected return level of each security is 0%; the standard deviation of returns for each security is The exected return of a ortfolio cobining the to securities is 0% if each security has equal ortfolio eight ([ 3 ] = [ 4 ] =.5), yet the standard deviation of ortfolio returns is zero. Thus, to relatively risky securities have been cobined into a ortfolio that is virtually risk-free. TBL 5.: Portfolio return ith erfectly inversely correlated securities. W 3 =W 4 = 0.5 i 3i 4i i P i Notice in the revious aragrah that e first cobined securities three and four into a ortfolio and then found that ortfolio's return given each outcoe. The ortfolio's return is 0% regardless of the outcoe; thus, it is risk free. The sae result could have been obtained by finding the variances of securities three and four fro quation (4.), the correlation coefficient beteen their returns fro quations (4.3) and (4.4), then solving for ortfolio variance ith quation (5.5) as in Table (5.). 4

5 Portfolios, fficiency and the Caital sset Pricing Model TBL 5.: Portfolio return ith erfectly inversely correlated securities. Given: = = = = Then: 3, ( ) ( ) ( ) ( ) The ilication of the to exales rovided in this chater is that security risk can be diversified aay by cobining the individual securities into ortfolios. Thus, the old stock arket adage "Don't ut all your eggs in one basket" really can be validated atheatically. Sreading investents across a variety of securities does result in ortfolio risk that is loer than the eighted average risks of the individual securities. This diversification is ost effective hen the returns of the individual securities are at least soehat unrelated; or better still, inversely related as ere securities three and four in the revious exale. For exale, returns on a retail food coany stock and on a furniture coany stock are not likely to be erfectly ositively correlated; therefore, including both of the in a ortfolio ay result in a reduction of ortfolio risk. Fro a atheatical ersective, the reduction of ortfolio risk is deendent on the correlation coefficient of returns ( i ) beteen securities included in the ortfolio. Thus, the loer the correlation coefficients beteen these securities, the loer ill be the resultant ortfolio risk. In fact, as long as ( i ) is less than one, hich, realistically is alays the case, soe reduction in risk can be realized fro diversification. Consider Figure (5.). The correlation coefficient beteen returns of securities C and D is one. The standard deviation of returns of any ortfolio cobining these to securities is a eighted average of the returns of the to securities' standard deviations. Diversification here yields no benefits. In Figure (5.), the correlation coefficient beteen returns on Securities and B is.5. Portfolios cobining these to securities ill have standard deviations less than the eighted average of the standard deviations of the to securities. Given this loer correlation coefficient, hich is ore reresentative of "real orld" correlations, there are clear benefits to diversification. In fact, e can see in Figures (5.3) and (5.4) that decreases in correlation coefficients result in increased diversification benefits. Loer correlation coefficients result in loer risk levels at all levels of exected return. Thus, an investor ay benefit by constructing his ortfolio of securities ith lo correlation coefficients. 5

6 6 Chater 5 W D= W C= W D=0 W C= W D=.5 W C=.5 C D C D CD Figure 5.: elationshi beteen ortfolio return and risk hen CD = B W B=.5 W =.5 0 W B=.6 W =.4 B 0 B B B Figure 5.: elationshi beteen ortfolio return and risk hen B =.5 6

7 Portfolios, fficiency and the Caital sset Pricing Model B W B=.5 W =.5 0 W B=.6 W =.4 B 0 B B B Figure 5.3: The relationshi beteen ortfolio return and risk hen F = W H=0 W G= W H= W G=0 G H H GH.0.0 G Figure 5.4: The relationshi beteen ortfolio return and risk hen GH =- 7

8 Chater Derivation Box 5. Deriving Portfolio Variance In Chater Four, e discussed finding variance of returns based on either otential or actual historical returns. Portfolio variance ay also be found as a function of otential or historical ortfolio returns. Hoever, it is often useful to exress ortfolio variance as a function of individual security characteristics. For exale e ay have estiates of security variance and covariance levels (based on historical estiates) but have no inforation regarding robabilities to associate ith outcoes. Furtherore, it is useful to kno exactly ho changing ortfolio eights ill affect ortfolio variances. To derive the variance of ortfolio () as a function of security variances, covariances and eights, e begin ith our standard variance exression as a function of () otential ortfolio return outcoes () and associated robabilities. (5.) P ) ( For the sake of silicity, let the nuber of securities (n) in our ortfolio equal to. Fro our ortfolio return exression, e ay coute ortfolio variance as follos: () P ) ( Next, e colete the square for quation () and cobine ters ultilied by the to eights to obtain: (B) P ) )( ( ) ( ) ( Next, e bring the suation ter inside the brackets: (C) P P P ) )( ( ) ( ) (

9 Portfolios, fficiency and the Caital sset Pricing Model We colete our derivation by noting our definitions fro Chater Four for variances and covariances as follos: (5.6) The rocess for deriving variances for larger ortfolios ould be siilar. 5.D. GLOBL POTFOLIO DIVSIFICTION In their text on ortfolio analysis, lton and Gruber reort that the average correlation coefficient beteen returns on to randoly selected stocks of U.S. cororations is aroxiately.40. This correlation is substantially higher than the dollar return correlation beteen randoly selected U.S. stocks and randoly selected stocks fro other countries, hich is likely to range fro about. to.35. Since the U.S. stock arket corises soehere beteen thirty and forty ercent of orld stock arkets, ale oortunity exists for erican investors to diversify their ortfolio risk ithout sacrificing ortfolio return. The doestic investor faces several additional risks investing outside of doestic arkets: Country isk: Countries face varying levels of olitical and econoic stability. Foreign return variances ill vary in foreign countries. Hoever, country risk beteen any countries ill often be quite lo. xchange isk: Currency exchange is sily the trading or saing of currencies. The currency exchange rate is sily the nuber of units of one currency that ust be exchanged for another; the exchange rate reresents the costs of currencies. The exchange rate beteen dollars and a foreign currency ill certainly affect the dollar denoinated return on an investent ade in that country. For exale, if an erican fir invests in the United Kingdo, all of the British rofits ill be generated in ounds. These ounds ust be exchanged for dollars before they can be sent in the U.S. Since the dollar exchange rate (the value of the dollar) varies over tie, one cannot be certain exactly ho any dollars can be urchased ith rofits denoinated in ounds. This is clearly a source of risk to erican investors. On the other hand, the lo covariance beteen values of currencies fro different countries ill serve to irove overall ortfolio diversification. lthough foreign investents are likely to have higher risk (variance) levels for the erican investor than the tyical doestic investent, they still reresent an oortunity for ericans to reduce ortfolio risk ithout sacrificing return. This is due to the articularly lo correlation coefficients beteen erican and foreign securities. In fact the folloing is offered in suort of globalizing investent ortfolios: lton, din J. and Martin J. Gruber. Modern Portfolio Theory and Investent nalysis, fourth edition. Ne York: John Wiley & Sons, Inc.:.5. 9

10 Chater 5. Portfolio risk at any return level ill be loer for a globally diversified ortfolio than for a doestic ortfolio.. Portfolio return at any risk level ill be higher for a globally diversified ortfolio than for a doestic ortfolio. 3. Feer securities fro global arkets ill be required to attain a given ortfolio diversification and risk level than ould be required fro only a doestic arket. This is significant because larger ortfolios tyically require larger brokerage fees to acquire and are ore costly and tie consuing to anage. lton and Gruber [99] reort that the average correlation coefficient beteen returns on U.S. securities is aroxiately.40. The correlation coefficient beteen to randoly selected 00 security ortfolios, one dran fro NYS stocks and the other selected fro MX stocks exceeds.90. Hoever, correlations beteen stock indices of different international arkets is significantly saller than these values, as indicated by the Table 5.3: TBL 5.3: Correlation Coefficients Beteen Market Indices us'lu'a Bel. Can. Fra. Ita. Ja. Net. Si. U.K. W.Ger ustralia ustria.03 Belgiu Canada France Italy Jaan Netherlands Sitzerland U.K W.Gerany U.S Source: Joy, Panton, eilly and Martin: Financial evie, 976) These correlation coefficients are all based on aounts converted into U.S. dollars. 5.. FFICINCY ND DOMINNC Because investors refer as uch return and as little risk as ossible, the ost efficient ortfolios are those ith the folloing characteristics:. Saller risk than all ortfolios ith identical or larger returns and. Greater return than all ortfolios ith identical or less risk. One ortfolio doinates a second hen one of the folloing three conditions is et:. the first ortfolio has both higher return and saller risk levels than does the second, 0

11 Portfolios, fficiency and the Caital sset Pricing Model. both ortfolios have identical variance but the first ortfolio has a higher return level than does the second, or 3. both ortfolios have identical returns but the first ortfolio has a saller variance than does the second. ortfolio is considered doinant if it is not doinated by any other ortfolio. Thus, the ost efficient ortfolios are all doinant. 5.F. CONSTUCTION OF TH FFICINT FONTI Consider a arket here the average coefficient of correlation beteen returns on securities is.8. (This is not really an unrealistic assution.) For sake of silicity, assue that there exist in this arket five securities, () through (). Cobine securities () and (B) into a ortfolio. eturn and risk cobinations of the resultant ortfolio ill fall soehere on the curve extending beteen the to securities, deending on their relative eights (See Figure 5.5). Siilarly, securities (B) and (C) can be cobined into ortfolios as can securities (C) and (D), and (D) and () (See Figure 5.6). We have constructed a series of curves reresenting risk-return cobinations of an infinite nuber of to security ortfolios. These resultant ortfolios theselves can be cobined into additional ortfolios. For exale, consider ortfolios (B) and (BC) in Figure 5.6. These ortfolios can be cobined into further ortfolios as can ortfolios (BC) and (CD) as ell as (CD) and (D). The resultant ortfolios can all be cobined into additional ortfolios. Notice that as the ortfolios becoe ore diversified, they becoe ore efficient. Thus, the curves reresenting the risk-return cobinations of these ortfolios fall further to the northeast on the risk-return sace. Hoever, the benefits of this diversification ust reach a liit. This is because the ortfolios that are being cobined are ore correlated than the individual securities that they contain. On the curve indicating this liit, further diversification cannot result in ore efficient ortfolios. The uard sloing ortion of this curve is called the fficient Frontier. The ost efficient ortfolios of risky assets ill have risk-return cobinations falling on the efficient frontier. The fficient Frontier reresents the risk-return cobinations of the ost efficient ortfolios in the feasible region. (The feasible region is sily the risk-return cobinations of all ortfolios available to investors.) Thus, the fficient Frontier is the left-ost, uerost boundary of the feasible region. B Figure 5.5: Portfolio risk-return levels hen B =.8

12 Chater 5 BC CD D C D B B Figure 5.6: Portfolio risk-return levels hen i =.8 for all i and ; ortfolios are each corised of to securities through. fficient Frontier D CDD BCC D CD D Feasible egion BC C BB C B B Figure 5.7: fficient frontier and feasible region

13 Portfolios, fficiency and the Caital sset Pricing Model.0 f f Lending ortfolios f.5 f 0 Borroing ortfolios Figure 5.8: Cobinations of risky asset ortfolio and the risk-free asset 5.G. TH ISK-F SST In reality, there exists no risk-free asset. Hoever, for coutational uroses, it is useful to assue the existence of such an asset. Historical evidence suggests that short-ter United States Treasury bills have been aong the ost reliable in actually realizing the returns exected by investors. By urchasing treasury bills, an investor is loaning the governent oney. The United States governent has roven to be an extreely reliable debtor (at least it akes good on all of its treasury bills). Treasury bills are fully backed by the full faith and credit of the U.S. governent, hich has substantial resources due to its ability to tax citizens and create oney. Thus, these securities are safer than the safest of cororate bonds or short-ter notes. They ay even be safer than U.S. F.D.I.C. backed savings accounts and certificates of deosit, hich are backed only by the liited resources available to banks and to the F.D.I.C. The resources of the F.D.I.C. are liited to assets it receives fro articiating coercial and savings banks. (Hoever, Congress has assed a resolution roising to back the F.D.I.C. if it runs out of funds, although this rocess can be slo and ight be uncertain.) Thus, the Treasury bill is ractically risk-free and robably the safest of all investents. Because the United States Treasury bill sees to be the safest of all investents, its characteristics are often used as surrogates for the characteristics of the risk-free asset. By definition, the variance (or, standard deviation) of exected returns on the risk-free asset is zero. Thus, an investor urchasing such an asset ill certainly receive the return he originally exected. Though this asset is riskless, the investor ill require a return, coensating 3

14 Chater 5 hi for inflation and his tie value of oney. This risk-free rate of return (r f ) can be aroxiated ith the short-ter Treasury bill rate. The risk-free asset can be cobined ith any ortfolio of risky assets. Such a ortfolio ill have a risk-return cobination hich is sily a eighted average of the risky ortfolio's and the risk-free asset's risk-return cobinations. For exale, consider a ortfolio of risky assets ith exected return and standard deviation levels of 0% and 0% and a risk-free asset ith an exected return of 5%. If the ortfolio and the risk-free asset ere cobined into a ne ortfolio ith equal eights ( f = =.5), the resultant ortfolio ould have exected return and standard deviation levels of 7.5% and 0%: (. 5.0) (.5.05). 75 (.5.0 ) (.5 0 ) ( ) (. 5.04) (0) (0).0. Notice that the correlation coefficient beteen returns on any risky asset and the risk-free asset ust be zero. Thus, both ortfolio exected returns and ortfolio standard deviations ill be a linear cobination of the individual security returns and standard deviations only hen ( i ) = or, as in this case, hen a risk-free asset is cobined ith a risky investent. If an investor has the oortunity to borro oney at the risk-free rate of return (r f ), he has the oortunity to create a negative eight ( f ) for the risk-free asset. For exale, if an investor had an initial ealth level of $000, but ished to invest $3000 in a risky asset ith an exected return of 0%, he could borro $000 at the risk-free rate of 5% if the lender ere certain the investor ould fulfill his debt obligation. Since the investor is borroing oney rather than lending (buying Treasury bills is, in effect, lending the governent oney), the eight associated ith the risk-free asset is negative. Because the total su invested in the risky asset is three ties as great as the investor's initial ealth level, ( ) is equal to 3. The investor's exected ortfolio return level is 0%, higher than the return of either of the assets corising the ortfolio: (.05) (3.0). 0 Notice that the su borroed is tice as great as the investor's initial ealth level, thus ( f ) is equal to -. The standard deviation of returns on the ortfolio is.6: (.0 ) (3 0 ) ( ) (3.0 ) Notice that the ortfolio standard deviation is higher than the standard deviations of either of the assets corising the ortfolio. Therefore, borroing oney (creating leverage) erits the investor to increase his exected returns; hoever, he ust also face additional risk. Whether an 4

15 Portfolios, fficiency and the Caital sset Pricing Model investor ill borro, and exactly ho uch he ill borro ill be deterined later in this chater. Consider an investor ho has the oortunity to invest in a cobination of a risk-free asset and one of several risky ortfolios () through () deicted in Figure 5.9. Which of these five ortfolios is the best to cobine ith the risk-free asset? Notice that the ortfolios ith risk-return cobinations on the line connecting the risk-free asset and ortfolio (C) doinate all other ortfolios available to the investor. Thus, any ortfolio hose risk-return cobination falls on lines extending through ortfolios (), (B), (D), and () ill be doinated by soe ortfolio hose risk-return cobination is deicted on the line extending through ortfolio(c). This line has a steeer sloe than all other lines beteen the risk-free asset and risky ortfolios. The investor's obective is to choose that ortfolio of risky assets enabling hi to axiize the sloe of this line; that is, the investor should ick that ortfolio ith the largest ossible ( ), here ( ) is defined by quation (5.7): (5.7) r f Therefore, the investor should invest in soe cobination of ortfolio (C) and the risk-free asset. If the curve connecting ortfolios () through () ere the fficient Frontier, then ortfolio (C) ould be referred to as the arket ortfolio. This is because every risk averse investor in the arket should select this ortfolio of risky assets to cobine ith the riskless asset. Notice that the line extending through ortfolio (C) is tangent to the curve at oint (C).. D C B Sloe= Figure 5.9: Cobination of risk-free asset ith one of five ortfolios of risky assets 5

16 Chater 5 5.H. TH CPITL MKT LIN The best ortfolio of risky assets to cobine ith the risk-free security lies on the fficient Frontier, tangent to the line extending fro the risk-free security. This line is referred to as the Caital Market Line (CML). Notice that ortfolios on the Caital Market Line doinate all ortfolios on the fficient Frontier. If a risk-free security exists, the Caital Market Line reresents risk-return cobinations of the best ortfolios of securities available to investors. Thus, an investor's risk-return cobinations are constrained by the Caital Market Line. M Caital Market Line M Market Portfolio fficient Frontier r f M Figure 5.0: The Caital Market Line The ost efficient ortfolio on the fficient Frontier to cobine ith the riskless asset is referred to as the Market Portfolio (deicted by [M] in Figure 5.0). Thus, the Market Portfolio lies at a oint of tangency beteen the fficient Frontier and the Caital Market Line. ll investors should hold ortfolios of risky assets hose eights are identical to those of the Market Portfolio. The Caital Market Line cobines the Market Portfolio ith the riskless asset. This line can be divided into to arts: the lending ortion and the borroing ortion. If an investor invests at oint (M) on the Caital Market Line, all of his oney is invested in the Market Portfolio. If he invests to the left of (M), his ortfolio is a lending ortfolio. That is, he has urchased treasury bills, in effect, lending the governent oney, and invested the reainder of his funds in the Market Portfolio. If he invests to the right of oint (M), he has a borroing ortfolio. In this case, he has invested all of his funds in the Market Portfolio and borroed additional oney at the risk-free rate to invest in the Market Portfolio. ll investors ill invest at soe risk-return cobination on the Caital Market Line. xactly hich risk-return cobination an investor ill choose ill deend on the investor s level of risk aversion. 6

17 Portfolios, fficiency and the Caital sset Pricing Model 5.I. INTODUCTION TO TH CPITL SST PICING MODL The Caital sset Pricing Model (CPM) rovides us ith a theory of equilibriu in caital arkets. This eans that the CPM ill exlain ho investors rice securities in the arketlace, as long as the assutions underlying the theory are fulfilled. The assutions that underlie the CPM are as follos:. Caital arkets are erfectly efficient. This eans that security rices fully reflect all available inforation at all ties. Characteristics of erfectly efficient caital arkets include: zero transactions costs; that is, investors, cororations and institutions buy and sell securities ithout incurring brokerage or other transactions fees. no taxes on investent incoe. investors have equal access to inforation on a costless basis. no single investor s transactions can influence the arket rice of any security.. Security returns are norally distributed ilying that investors ho axiize exected utility focus only on exected return and risk levels of their ortfolios. 3. ll assets are arketable and infinitely divisible. 4. No restrictions are laced on short-sales (borroing securities, selling the and reurchasing later). 5. Investors all have identical exectations regarding security exected return and risk levels. 6. There exists a risk free security ith no restrictions on borroing and lending. 7. Investor lanning horizons are for a single tie eriod. If these assutions hold, e can use the Caital sset Pricing Model to estiate the risk of an investent. This risk easure can then be used to calculate a discount rate for couting the resent value of that investent. The urchase/sale decision is ade on the investent s resent value relative to the investent s arket rice. 5.J. SYSTMTIC ND UNSYSTMTIC ISK The risk associated ith a security ight be classified as either systeatic or unsysteatic risk. Systeatic risk is that ortion of a security's risk that is related to variance of the arket ortfolio. Systeatic risk is sensitivity to arket ortfolio fluctuations. Thus, systeatic risk is often referred to as arket-related risk. Unsysteatic risk is that ortion of a security's variance that is unrelated to risk of the arket ortfolio; that is, unsysteatic risk (or fir-secific or unique risk) is unique to the security under analysis. For exale, a labor strike or death of the coany's chief executive officer ay affect fir erforance and ay be regarded as a fir-secific or unsysteatic risk. Figure 5. deicts the iact of ortfolio size (nuber of securities) on the risk of a randoly selected ortfolio of stocks. The average standard deviation of a randoly selected stock fro the Ne York Stock xchange is aroxiately.40. bout half of this risk ight be regarded as being arket-related and the reainder is fir-secific. s additional randoly selected stocks are added to this equally eighted ortfolio, ortfolio risk declines, quickly at first 7

18 Portfolio Standrd Deviation Chater 5 but then at a sloer rate as the ortfolio becoes better diversified. The diversification eliinates aroxiately 99% of fir-secific risk ith investent in as fe as 30 securities, reducing overall risk to arket-related only. The standard deviation of the arket ortfolio is aroxiately 0% Nuber of Securities Figure 5.: Portfolio Size and isk eduction The required return of any security ill be related to a risk-free coonent coensating investors for inflation and their tie values of oney and reius coensating investors for both arket related and unique risk (though, as e shall see later, unique risk coensation ill be zero for a erfectly diversified ortfolio). Market related risk of a security can be easured as the standard deviation of returns associated ith that security relative to the standard deviation of returns associated ith the arket ortfolio ultilied by the coefficient of correlation beteen returns on the security and the arket ortfolio: (5.8) i i,. Thus, Beta ( i ) easures the risk of security (i) relative to the risk of the arket ortfolio. The coefficient of correlation acts as a sort of "fudge factor" relating to the reduction in ortfolio risk realized by including security (i) in a ell- diversified ortfolio. If the standard deviation of exected returns on security () and the arket ortfolio ere.4 and., resectively, and the correlation coefficient beteen returns on the to ere.75, the Beta ( ) of security () ould be.5:.4 i, Notice that quation (5.8) can be reritten as follos: 8

19 Portfolios, fficiency and the Caital sset Pricing Model i M im COV ( i, M ) (5.9) i Betas are norally couted on the basis of historical standard deviations and correlation coefficients. Many investors believe that the relative stability over tie of these statistical easures ustifies the use of "historical" betas. Historical betas are couted on the basis of historical standard deviations and correlation coefficients. Frequently, analysts ill coute historical stock betas, standard deviations and correlation coefficients based on five years of historical onthly returns data. Several investent advisory services such as Value Line ill rovide historical betas for a large nuber of idely traded stocks. By definition, the arket ortfolio (or "average" security) requires a risk reiu of (r - r f ). Therefore, an investor requires this reiu to coensate for the risk associated ith the "average" security. In fact, the systeatic risk reiu required by any investor for any security (i) is: (5.0) (rr i r f ) i ( r r f ). Notice that, since ( ) ( ), and (, ), the beta of the arket ortfolio equals one., If an investor assues additional unsysteatic (unique) risk by the urchase of a security, he ill require an unsysteatic risk reiu. This reiu ill be unrelated to the arket ortfolio; thus, it ill be unique to each individual security. The total risk reiu required by an investor for the urchase of security () ill be: (5.) (rr rf ) ( r rf ) Covariances beteen the non-arket related return coonents beteen securities in a ell-diversified ortfolio ill, on average, equal zero. We find, using tie-series data, that on average, securities earn returns equal to the risk-free return lus their required systeatic risk reiu coonents. Therefore, the unsysteatic risk reiu required by an investor holding a ell-diversified ortfolio ill be zero. We can ignore the (µ ) coonent of quation (5.) for shareholders ith ell-diversified ortfolios. Therefore, the return required by any investor to urchase any security (i) ill be: (5.) rr i r f i ( r r f ) This is the Caital sset Pricing Model. The CPM enables us to deterine the required return for any investent, given its risk characteristics and the current risk-free rate of return. Thus, an investor ust exect to receive the required return on an investent in order to urchase it. If caital arkets function according to the assutions outlined in Section 5.I, security rices ill be such that the exected returns on all securities ill equal their required returns. lthough any of these assutions do not hold in reality, the CPM still rovides us ith good aroxiations of required security returns. Nonetheless, there exist a nuber of variations of the CPM that have been adusted to adat to ore realistic arket conditions. 9

20 Chater 5 If the required return on the arket ortfolio ere.4 and the current risk-free rate ere.06, the required return for security () described in Section (5.J) ould be.8: rr = ( ) =.8. 5.K. ISK-DJUSTD DISCOUNT TS The required rate of return generated by the Caital sset Pricing Model rovides an excellent risk-adusted discount rate useful for evaluating a variety of investents. This risk-adusted discount rate reflects inflation and investors' tie value of oney through its risk-free coonent. The CPM reflects investors' risk-return references through the arket systeatic risk reiu (r - r f ). Furtherore, the odel reflects the risk of the security under evaluation through the Beta ( i ) coonent. Therefore, any security can be evaluated by deterining its Beta ( i ), lugging it into the CPM, and then discounting roected cash flos using the required rate of return as a discount rate: (5.8) i i, (5.) rr i r f i ( r r f ) n CFt (5.3) PVi ( rr t t i ). Thus, if security () ere exected to ay a $0 dividend in each of the next three years and then be sold for $00, its current value ould be $8.6: PV $0 (.8) $0 (.8) $0 $00 3 (.8) 8.6 0

21 Portfolios, fficiency and the Caital sset Pricing Model CHPT FIV QUSTIONS ND POBLMS 5.. n investor is considering cobining Douglas Coany and Tilden Coany coon stock into a ortfolio. Fifty ercent of the dollar value of the ortfolio ill be invested in Douglas Coany stock; the reainder ill be invested in Tilden Coany stock. Douglas Coany stock has an exected return of six ercent and an exected standard deviation of returns of nine ercent. Tilden Coany stock has an exected return of tenty ercent and an exected standard deviation of thirty ercent. The coefficient of correlation beteen returns of the to securities has been shon to be.4. Coute the folloing for the investor's ortfolio: a. exected return b. exected variance c. exected standard deviation 5.. Work through each of your calculations in Proble 5. again assuing the folloing eights rather than those given originally: a. 00% Douglas Coany stock; 0% Tilden Coany stock b. 75% Douglas Coany stock; 5% Tilden Coany stock c. 5% Douglas Coany stock; 75% Tilden Coany stock d. 0% Douglas Coany stock;00% Tilden Coany stock 5.3. Ho do exected return and risk levels change as the ortfolio roortions invested in Tilden Coany stock increase? Why? Preare a grah ith exected ortfolio return on the vertical axis and ortfolio standard deviation on the horizontal axis. Plot the exected returns and standard deviations for each of the ortfolios hose eights are defined in Probles 5. and 5.. Describe the sloe of the curve connecting the oints on your grah The coon stocks of the Landon Coany and the Burr Coany are to be cobined into a ortfolio. The exected return and standard deviation levels associated ith the Landon Coany stock are five and telve ercent, resectively. The exected return and standard deviation levels for Burr Coany stock are ten and tenty ercent. The ortfolio eights ill each be 50%. Find the exected return and standard deviation levels of this ortfolio if the coefficient of correlation beteen returns of the to stocks is: a. b..5 c. 0 d. -.5 e Describe ho the coefficient of correlation beteen returns of securities in a ortfolio affect the return and risk levels of that ortfolio n investor is considering cobining Securities and B into an equally eighted ortfolio. This investor has deterined that there is a tenty ercent chance that the econoy ill erfor very ell, resulting in a thirty ercent return for security and a tenty ercent for security B. The investor estiates that there is a fifty ercent chance that the econoy ill erfor only

22 Chater 5 adequately, resulting in telve ercent and ten ercent returns for Securities and B. The investor estiates a thirty ercent robability that the econoy ill erfor oorly, resulting in a negative nine ercent return for Security and a zero ercent return for Security B. These estiates are suarized as follos: outcoe robability ai bi a. What is the ortfolio return for each of the otential outcoes? b. Based on each of the outcoe robabilities and otential ortfolio returns, hat is the exected ortfolio return? c. Based on each of the outcoe robabilities and otential ortfolio returns, hat is the standard deviation associated ith ortfolio returns? d. What are the exected returns of each of the to securities? e. What are the standard deviation levels associated ith returns on each of the to securities? f. What is the coefficient of correlation beteen returns of the to securities? g. Based on your ansers to art d in this roble, find the exected ortfolio return. Ho does this anser coare to your anser in art b? h. Based on your ansers to arts e and f, hat is the exected deviation of ortfolio returns? Ho does this anser coare to your anser in art c? 5.7. n investor has cobined securities X, Y and Z into a ortfolio. He has invested $000 in Security X, $000 into Security Y and $3000 into Security Z. Security X has an exected return of 0%; Security Y has an exected return of 5% and security Z has an exected return of 0%. The standard deviations associated ith Securities X, Y and Z are %, 8% and 4%, resectively. The coefficient of correlation beteen returns on Securities X and Y is.8; the correlation coefficient beteen X and Z returns is.7; the correlation coefficient beteen Y and Z returns is.6. Find the exected return and standard deviation of the resultant ortfolio n investor ishes to cobine Stevenson Coany stock and Sith Coany stock into a riskless ortfolio. The standard deviations associated ith returns on these stocks are 0% and 8% resectively. The coefficient of correlation beteen returns on these to stocks is -. What ust be each of the ortfolio eights for the ortfolio to be riskless? 5.9. ssue that the coefficient of correlation beteen returns on all securities equals zero in a given arket. There are an infinite nuber of securities in this arket, all of hich have the sae standard deviation of returns (assue that it is.5). What ould be the ortfolio return standard deviation if it included all of these infinite nuber of securities in equal investent aounts? Why? (Deonstrate your solution atheatically.) 5.0. Which of the folloing ortfolios are doinant? Portfolio xected eturn Standard Deviation a.05 0

23 Portfolios, fficiency and the Caital sset Pricing Model b c..5 d.5.04 e.8.0 f Portfolios X and Y are doinant ortfolios fro hich the fficient Frontier can be constructed. Portfolio X has an exected return of 6% and a standard deviation of 5%. Portfolio Y has an exected return of % and a standard deviation of 5%. The coefficient of correlation beteen returns on these ortfolios is (-.5). Construct a grah ith exected return and standard deviation axes and lot the coordinates for ortfolios ith the folloing eights: x = y = 0 x =.75 y =.5 x =.5 y =.5 x =.5 y =.75 x = 0 y = a. Construct the fficient Frontier for this security arket. (Note: eeber that only these to ortfolios are required for the fficient Frontier construction since they are the only doinant ortfolios.) b. Based on your grah of the fficient Frontier and a five ercent risk-free rate of return, estiate the exected return and standard deviation levels of the arket ortfolio. c. Based on your solutions to arts a and b, construct the Caital Market Line. d. What is the sloe of the Caital Market Line in Part c? 5.. re extreely risk averse investors likely to be borroers or lenders? Ho does risk aversion affect borroing levels? Why? 5.3. Given that correlation coefficients beteen doestic securities exceed correlation coefficients beteen doestic and foreign securities, ho ould exanding the feasible region to include foreign securities affect the fficient Frontier? Ho ould this exansion affect the Caital Market Line? 5.4. stock currently selling for $60 has a historical standard deviation of.5 and a coefficient of correlation ith the arket ortfolio of.4. Over the sae eriod, the historical standard deviation of the arket ortfolio as.6. Investors anticiate dividends of $ er share in one year at hich tie the stock can be sold for $65. Deterine the folloing for the stock if the current Treasury Bill rate is.05 and the required return on the arket ortfolio is.: a. The stock Beta. b. The required return of the stock. c. The discount rate to be associated ith cash flos fro the stock. d. The resent value of cash flos associated ith the stock. e. Whether the stock constitutes a good investent Historical returns for Holes Coany stock, Warren Coany stock and the arket 3

24 ortfolio along ith Treasury Bill (T-Bill) rates are suarized in the folloing chart: Chater 5 Year Holes Co. Warren Co. Market T-Bill 996 % 4% 0% 6% 997 8% 0% 4% 6% 998 7% % 6% 6% 999 3% -3% % 6% 000 0% 9% 8% 6% a. Calculate return standard deviations for each of the stocks and the arket ortfolio. b. Calculate correlation coefficients beteen returns on each of the stocks and returns on the arket ortfolio. c. Preare grahs for each of the stocks ith axes ( it - ft ) and ( t - ft ) here it is the historical return in year (t) for stock (i) ; t and ft are historical arket and risk-free returns in tie (t). Plot regression lines for each of the to stocks; that is, try to fit a single line for each stock that is as close as ossible to all of the data oints. d. Calculate Betas for each of the stocks. Ho do your Betas coare to the sloes of the regression lines that you dre? 5.6. Consider the Holes and Warren stocks hose historical returns are given in Proble 5.5. ssue an investor had cobined each of the stocks into a ortfolio such that half of his ealth as invested in each of the stocks at the beginning of each year. Calculate the folloing for the investor's ortfolio: a. Historical returns for each of the five years. b. Historical ortfolio standard deviation for the five year eriod. c. Historical correlation coefficient beteen the arket ortfolio and the investor's ortfolio. d. The ortfolio Beta. e. Ho does this ortfolio Beta coare to the Betas of the individual stocks? 5.7. Ho ight a cororation calculate a Beta for one of its assets? Why ight this calculation be soehat ore difficult than calculating a stock Beta? 5.8. Calculate the Beta of a risk-free asset If investors can diversify aay unsysteatic risk by constructing ortfolios, hy are cororate anagers so concerned ith the riskiness of their individual fir's oerations? 4