Capital Asset Pricing

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Jared Underwood
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1 Caital Asset Pricig The otio of a erfect hedge which was itroduced i the last lecture is colicated but iterestig. A siler but siilar cocet is the ossibility of a risk free retur. Cosider a asset F that has a retur of r f ad has o risk. That is, the retur o F occurs without variace; r f is ot a rado variable; it has o stadard deviatio. I this case we ca cosider a ortfolio that is coosd of F ad soe risky asset X. The risk-retur frotier i this case will be sily a straight lie joiig the risk-retur cobiatio of X with the retur of F o the vertical axis. The sloe of this lie is just the rise over the ru, or Er ( x) rf x Cobiig the risk-free asset with two risky assets gives us the bullet-shaed risky-asset ivestet frotier with a lak laid dow o it steig fro the risk-free retur to the highest oit o the bullet. Thus, with two risky assets ad oe risk free, the cosuer is able to ejoy the beefits of ortfolio diversificatio alog with the risk-reducig choices offered by the risk free asset. Next, cosider the ivestet risk-retur frotier whe there are uerous assets i additio to the risk free asset. With uerous risky assets available to the ivestor, ortfolio wealth ca be sread across ay or all. I this case the exected ortfolio retur ca be writte as: Er ( ) = ωier ( i) i= where ω i are the weights assiged to each of the risky assets that the ivestor chooses. If the ivestor chooses oly risky assets the ω s su to oe. Siilarly, the variace of the ortfolio retur ca be defied as: V( r )= i= i= j ωω where s ii is the sae thig as the variace of the retur o the ith asset, s i. The ivestor s riskretur frotier is still bullet shaed eve though it is ore difficult to deict the frotier i this case because of the ultidiesioal variatio i the ortfolio. The ortfolio ca vary by chagig the weight assiged to each of assets, ad ortfolio variace is ore colicated because of ultile covariaces. Cosider the sile exale of three assets, X, Y, ad Z. With three assets, oe ca iagie a bullet betwee two of the assets. Each oit o the two asset bullet defies a relative ortfolio weight for the first two assets. Let α be the set of relative ortfolio weights that geerate a ivestet frotier betwee assets X ad Y. The two asset bullet ca the be erged with the third asset. For each of the relative ortfolio weights allocatig α to X ad (-α ) to Y, there are a set of weights betwee the asset Z ad the ortfolio of X ad Y. Call these weights α. For istace, if α is.5 ad α is.5, the 5% of the ortfolio is i X, 5% i Y, ad 50% i Z. The bullet associated with the three asset ortfolio is the eveloe of all the bullets that ca be geerated as all ossible cobiatios of α ad α are cosidered. By eveloe, we ea that the bullet is the ost efficiet cobiatios of the three assets, that is, the highest retur relative to each ossible level of risk. See. 73+ i Coelad ad Westo for a good discussio of the algebra ortfolio variace. i j ij 4.doc; Revised: Seteber 9, 005; M.T. Maloey

2 I like fashio, a ivestet frotier for all assets exists. As the ortfolio is exaded i ters of the uber of assets, fro a subset to all risky ivestet otios N, a globally efficiet eveloe ivestet frotier results. It is soewhat istructive to cosider what haes to the variace of this global ortfolio as the share i ay asset chages. This ca be foud by defiig ortfolio variace as give i eqt. () over all assets N ad the differetiatig with resect to the share held i the ith asset: N V( r ) = ω + ω ω j= i i i j ij j i 3 Now cosider what haes to ortfolio variace as N goes to ifiity. I this liit, the idividual orfolio weights go to zero. This eas that the first ter o the right-had side of eqt. (3) becoes ifiitely sall. The secod ter, however, is sued over the ifiitely large N so it does ot vaish. The ilicatio is that the ow variace of a asset becoes a trivial cooet of ortfolio variace, while its covariace with other assets cotiues to be iortat. Give the efficiet frotier of ivestet otios idetified by the global evelo of all bullets, the ivestor ca lik this to the risk free rate. The risk free rate is a oit o the vertical axis of the risk-retur sace. This oit ca be rojected to a tagecy with the efficiet riskretur frotier. Where this tagecy occurs defies the ost efficiet of the efficiet ortfolios. We call this the arket ortfolio. The retur associated with this ortfolio is the arket retur. The lie fro the risk-free retur through this oit is called the caital arket lie. The caital arket lie is iortat because it idetifies the ost efficiet ivestet otios facig the cosuer. By holdig a cobiatio of the arket ortfolio ad the risk-free retur the cosuer ca beat ay other ortfolio of risky assets. If the cosuer wats less risk tha the arket, the cosuer holds soe risk-free asset ad soe arket. If the cosuer wats ore risk tha the arket, the cosuer ca borrow at the risk-free rate ad ivest i the arket. The cosuer essetially searates the ortfolio of risky assets fro the riskless asset. The cosuer axiizes utility by choosig the oit o the caital arket lie where the highest ossible idifferece curve is taget. Asset Pricig ad the Market Portfolio The stage is ow set to ull the rabbit out of the hat. We have idetified cosuer behavior i ters of utility axiizatio i the face of risk aversio. We have show how ortfolio diversificatio aog risky assets offers suerior risk-retur tradeoffs coared to holdig sigle risky assets. We have show that give a risk-free asset, there is a uique ortfolio of choice for risky assets. Now we show how assets are riced. Sily eough, give that there is a uique ortfolio of choice for risky assets, that is, give that there is oe ortfolio that everyoe holds, each asset ust be riced ito that ortfolio. The silest way to thik about this is to cosider oly two risky assets, X with the higher risk ad retur ad Y with the lower. What if whe the risk-free lak was laid dow o the bullet of the two assets it hit oly the higher retur oit? This would ea, of course, that ivestors would oly be iterested i holdig the oe asset. This would cause eole to bid dow the rice of Y ad bid u the rice of X. However, these rice chages would chage the returs to the two 4.doc; Revised: Seteber 9, 005; M.T. Maloey

3 assets. As the rice of Y falls, its retur icreases; as the rice of X icreases, its retur is drive dow. This chages the bullet ad the arket ortfolio, but i the two asset case, we ca iagie how the equilibriu develos. The sae thig haes whe there are ay risky assets. 3 Now that the rabbit is out of the hat, all we eed do is show that it is a real buy by lettig it ru aroud a bit. To do this it is elighteig to cosider how arket ortfolios that are iefficiet look whe coared to the efficiet oe. We defie a ortfolio of two assets: Oe is the efficiet, arket ortfolio treated as oe asset. That is, let there be a utual fud that is coosed of all risky assets i their efficiet weights. This utual fud is available to ivestors to hold i ay aout. The secod asset we wish to cosider is a idividual risky asset, X i, which has a retur r i. The exected ortfolio retur ad ortfolio risk are defied just as they were i our last lecture whe we cosidered ortfolios of X ad Y. Here we cosider what haes to the diversificatio frotier whe the ivestor chooses to hold a ortio of his wealth, ω, i asset i ad the rest, (-ω), i the arket. 4 Exected Portfolio Retur: E(r ) = ωe(r i ) + (-ω)e(r ) 4 Portfolio Risk s = [ω s i + (-ω) s + ω (-ω) s i ] / 5 We ow do what we refraied fro doig i the last lecture i order to heighte the draatic effect. Let s differetiate both eqt s (4) ad (5) with resect to ω. The derivative of eqt. (4) is sile. Er ( ) = Er ( i) Er ( ) 6 ω The derivative of ortfolio risk is a little essy but it will silify icely i a oet: ω = [ ω i ( ω) + i 4ω i] 7 We wish to evaluate eqt. (7) at the oit where ω is equal to zero. This requires a little exlaiig. The arket ortfolio cotais asset i i the efficiet weight ω i. The questio osed by eqt. (7) is, what haes to ortfolio risk whe this asset is icreased beyod its otial aout? By evaluatig eqt. (7) at ω = 0 we ca deterie the effect o ortfolio risk whe this occurs. Reeber that the arket ortfolio as a oit o the efficiet ivestet frotier. The arket ortfolio is arbitrarily chose fro the efficiet frotier by its tagecy with the lie fro the risk-free retur. Every asset has a efficiet weight i the arket ortfolio. This weight is differet all alog the efficiet frotier. What the derivative i eqt. (7) asks is how does ortfolio risk r=(p +D)/P 0. 3 I Share's origial aer, he shows the global ivestet frotier becoig flat alog the caital arket lie so that coetitive ricig of assets actually creates ay "arket" ortfolios with various risk ad returs that are liear cobiatios. 4 Where all assets icludig i are held i value weighted roortios. These are the otial ω j s. 4.doc; Revised: Seteber 9, 005; M.T. Maloey 3

4 chage as the efficiet ortfolio weight of asset i chages ovig away fro the efficiet weight or fro a iefficiet cobiatio toward the efficiet oe. Matheatically, eqt. (7) evaluated at ω = 0 reduces to oly a few ters: ω ω= 0 i = 8 The result is that the chage i ortfolio risk i the eighborhood of the efficiet frotier is equal to the differece betwee the covariace of asset i with the arket ius the variace of the arket divided by the stadard deviatio of the arket. The sloe of the bullet that akes u the efficiet frotier at the arket ortfolio ca be writte as the ratio of eqt s (6) ad (8): Er ( ) ω= 0 Er ( ) = ω = ω ω= 0 Er ( ) Er ( ) i i 9 Equatio (9) tells us the sloe of the bullet as the ortfolio share of asset i aroaches efficiecy. Fro the caital arket lie, we kow the sloe of the efficiet frotier. Fro () we ca write: Er ( ) rf 0 As the sloe of the bullet show by eqt. (9) aroaches efficiecy, it aroaches the sloe of the caital arket lie. At the efficiet share for asset i, equatios (9) ad (0) are equal. By likig eqt s (9) ad (0), we ca deduce what haes as the ortfolio share of asset i coverges to the otial. Er ( Er Er r i) ( ) ( ) f = i Rewritig gives: i [ Er ( i) rf] = [ Er ( ) rf] = βi[ Er ( ) rf] Equatio () is it the Caital Asset Pricig Model, or CAPM for short. It tells us that the exected retur o asset i et of the risk free retur is liearly related to the exected arket retur et of the risk free. The ilicatio is that if the arket retur icreases, the the retur to asset i ust icrease as well, ad the aout that it will icrease is equal to the icrease i the arket retur ties the liear correlatio of the arket retur ad the ith asset's retur. Equatio () is rereseted by the failiar arket odel, which is usually see i the coo for r = α + βr + ε, i i 4.doc; Revised: Seteber 9, 005; M.T. Maloey 4

5 where α = ( β) r f, ad β is foud by the least-squares estiator, β Liear Pricig ad Covergece to a Equilibriu = i. It is elighteig to reflect o the uaces of the CAPM that are revealed whe we cosider what the odel says if there is o risk free asset. With o risk free, ivestors face the global ivestet frotier. Based o their ow referece fuctios they attet to for ortfolios that are idiosycratic i their coositio. That is, a relative risk taker will choose a ortfolio coosed fro ore high retur ad high risk assets, while a relative risk averter will choose ore low retur low risk assets. Whe there is a risk free asset, all ivestors choose the sae ortfolio of risky assets. With o risk-free asset, whe ivestors are choosig aog alterative ortfolios of risky assets, the aout that they are willig to ay for assets differs. Relatively risk averse cosuers will be willig to ay slightly ore for low risk assets, while relatively risk lovig cosuer will be willig to ay slightly ore for high risk assets. As relative risk lovers bid dow the retur o high risk stocks, it chages the ivestet frotier that faces relative risk averters. Siilarly, as relative risk averters readjust their ortfolios, it chages the frotier yet agai. The roble is that there is o assurace that with differet ivestors searchig for differet ortfolios alog the global ivestet frotier, assets will be riced cosistetly. Assets are still valued based o their relative covariaces. However, the relative rakigs ay ot be the sae across all ivestors. The liear ricig betwee ay oe asset ad all the rest disaears. Soe Notes o Cooud Returs Wealth accuulatio over the holdig eriod ca be described i ters of the geoetric cooud forula: P t= P 0 = ( + r) = ( + r ) t H where r t is the stochastic retur o a asset durig each seget of tie, to, ad r H is a costat rate of iterest that reflects the average of the stochastic retur over the etire holdig eriod,. Rewritig gives us the holdig eriod retur exlicitly: r H / = ( + r ) t= t which ca also be exressed i ters of logs: l( + rt ) / t= µ rh e e = = Rewritig, takig logs, ad exected values, we have: E[l( r + )] = E[ l( + r) / ] =µ H 4.doc; Revised: Seteber 9, 005; M.T. Maloey 5 t

6 which says that the exected value of the log of the holdig eriod retur lus oe is equal to the exected value of the average of the log of the stochastic retur ad this is equal to a costat, u. Let (+r t ) = x, which is a rado variable such that its atural log, l(x), is distributed orally with ea, µ, ad stadard deviatio,. The desity of x is the called the log-oral. It looks siilar to the oral with soe adjustets: 5 f( x) = e x π [l( x ) µ ] The edia of (+r t ) is exected value, we ca write: e µ ad the exected value 6 of (+r t ) is Ex ( ) = e µ+. Fro the µ = l( Ex ( )) which says: Ε l( + r t ) = l(+e(r t )) - / Fro this we ca calculate the holdig eriod retur fro the average of the stochastic returs: / rh [l( + E( r)) / ] = e This akes ituitive sese because the ati-log fuctio coresses the egative tail of the desity ad skews the distributio. The average of the stochastic returs is higher tha the holdig eriod returs because of the skewess ad this is accouted for by subtractig oe-half the variace. Sice is the stadard deviatio of log(+r) we eed a trasforatio there also. This gets a little essy, but for sall value of, the stadard deviatio of x ad of l(x) are very close. By sall values, I ea. or less. If the stadard deviatio of r is., the is.09; if the stadard deviatio of r is.05, the is This eas that you ca just use the stadard deviatio of r i the calculatio. The exact forula is: µ+ V() r = e ( e ) Figure : Noral ad Log-Noral Desity 5 If x~f(x) the g(x) ~g'(x) f(g(x)), where g() is a ootoic trasforatio. 6 The exected value ca be derived fro a oet geeratig fuctio. 4.doc; Revised: Seteber 9, 005; M.T. Maloey 6

7 4.doc; Revised: Seteber 9, 005; M.T. Maloey 7

8 Soe ubers o S&P 500 Idex icludig divideds, // reset. Aualized Average daily returs:.0358% 8.767% Std. dev.: Cooud retur:.0307% 8.040% Estiated cooud retur:.0308% Nuber of days [/3/'50-9/7/'05]: 4009 Average othly retur o S&P 500:.78% 8.733% Stadard deviatio of othly returs:.0399 Cooud retur:.648% 8.059% Estiated cooud retur:.648% Nuber of oths [/3/'50-8/3/'05]: 667 Average aual retur o S&P 500: 9.387% Stadard deviatio of aual returs:.676 Cooud retur: 8.08% Estiated cooud retur: 7.86% Nuber of years ['50-'04] 54 The aualized returs are derived usig the cooud forula fro the cooud retur fro the shorter eriod, i.e., the aualized value for othly returs is (+othly cooud retur) to the ower of. See below the grah of the uit oral variate of daily returs coared to the true uit oral distributio. The icture trucates the largest values of the returs ad still the ictures differ arkedly. 4.doc; Revised: Seteber 9, 005; M.T. Maloey 8

9 4.doc; Revised: Seteber 9, 005; M.T. Maloey 9

2.6 Rational Functions and Their Graphs

2.6 Rational Functions and Their Graphs .6 Ratioal Fuctios ad Their Graphs Sectio.6 Notes Page Ratioal Fuctio: a fuctio with a variable i the deoiator. To fid the y-itercept for a ratioal fuctio, put i a zero for. To fid the -itercept for a