Prncples of Fnance Grzegorz Trojanowsk Lecture 6: Captal Asset Prcng Model Prncples of Fnance - Lecture 6 1 Lecture 6 materal Requred readng: Elton et al., Chapters 13, 14, and 15 Supplementary readng: Luenberger, Chapter 7 Alexander et al., Chapter 10 Prncples of Fnance - Lecture 6 1
Lecture 6: Checklst By the end of ths lecture you should: Understand the nature of the relatonshp between the beta of an asset, ts rsk and ts expected return Be able to derve the standard captal asst prcng model (CAPM) Be able to show how devatons from the CAPM lead to explotable proft opportuntes Understand the relatonshp between the equlbrum expected return of an asset and ts equlbrum prce Be able to derve the CAPM when there s no rsk-free lendng or borrowng Prncples of Fnance - Lecture 6 3 Last lectures recap (1) In mean-varance analyss, rsky assets are charactersed by ther expected return and varance (or standard devaton) The feasble set defnes the expected return and varance of all possble combnatons, or portfolos, of rsky assets Most of these portfolos are neffcent n the sense that there exst other portfolos whch offer the same expected return wth lower standard devaton, or the same standard devaton wth hgher expected return The set of portfolos that are not domnated n ths way s known as the effcent set Prncples of Fnance - Lecture 6 4
Last lectures recap () When there s no rsk-free asset, the effcent set s the segment of the envelope (mnmum varance set), above the GMV portfolo When there s a rsk-free asset, the effcent set s the tangent from the rsk-free asset to the feasble set of rsky assets, and s known as the captal market lne, or CML Investors choose to nvest n a specfc effcent portfolo dependng on ther rsk preferences Prncples of Fnance - Lecture 6 5 Last lectures recap (3) Captal market lne (CML) Portfolo expected return Market portfolo (M) Rsk-free rate (r f ) Portfolo standard devaton Prncples of Fnance - Lecture 6 6 3
Last lectures recap (4) The nputs to mean-varance analyss are the expected return vector and the varancecovarance matrx As the number of stocks ncreases, calculaton of the nputs to mean-varance analyss rapdly becomes nfeasble, owng to the large number of parameters that must be estmated However, the calculaton of the nputs to mean-varance analyss s greatly smplfed by the use of factor models (or ndex models) Prncples of Fnance - Lecture 6 7 Last lectures recap (5) The smplest form of factor model s the sngle factor model, whch assumes that ndvdual returns are drven by a sngle common factor r t = α + β r + ε where E(r M ε ) = 0, E(ε ) = 0, and E(ε ε j ) = 0 Usng the sngle factor model, we can wrte the total rsk of an ndvdual stock as M Mt σ = β σ + σ where σ = var( ), σ = var( ) and σ ε = var( ) r t ε t M r M ε Prncples of Fnance - Lecture 6 8 4
Last lectures recap (6) σ ε The term dsappears when the asset s combned n a well dversfed portfolo: Ths s known as dversfable or non-systematc or dosyncratc rsk Only the term β σ M contrbutes to the portfolo s varance, and so t s only for ths term that nvestors wll be rewarded by the market n terms of expected return: Ths s known as non-dversfable, or systematc or market rsk Prncples of Fnance - Lecture 6 9 Market portfolo (1) So far we have consdered the nvestment opportuntes for an ndvdual nvestor; let us now consder the aggregate nvestments of all nvestors When there s a rsk-free asset, the effcent set s the CML All portfolos on ths effcent set nvolve two nvestments: one n the tangency portfolo M and one n the rsk free asset Prncples of Fnance - Lecture 6 10 5
Market portfolo () Suppose that all nvestors had the same nformaton about expected returns and varances all nvestors wll then face exactly the same mean-varance dagram Consequently all nvestors wll nvest n a combnaton of the rsk free asset and the same tangency portfolo, M If M s the portfolo that all nvestors nvest n, and no nvestor nvests n any other portfolo, then M s rather a specal portfolo; n partcular, t must contan all stocks n the stock market n proporton to ther market value Prncples of Fnance - Lecture 6 11 Market portfolo (3) Ths can be explaned as follows: f the portfolo M contans 3% of stock X (by value), and every nvestor nvests n M, and consequently nvests 3% of ther wealth n stock X then stock X must comprse 3% of the total market value of all stocks Ths specal portfolo, M, s known as the market portfolo Prncples of Fnance - Lecture 6 1 6
Market portfolo: Example Suppose we have a market that comprses sx shares wth share prces and numbers of shares outstandng as follows We can easly compute the consttuton of the market portfolo A B C D E 1 Market 3 Stock Prce Shares Outstandng Market Value Portfolo Weght =B4*C4 4 5 A B 10.34.6 00 550,068.00 1,43.00 18.50% 11.1% =D4/D$11 6 C 4.55 600,730.00 4.43% 7 D 6.00 350,100.00 18.79% 8 E 9.3 50,330.00 0.85% 9 F 14.10 50 705.00 6.31% 10 11 Total market value 11,176.00 =SUM(D4:D9) Prncples of Fnance - Lecture 6 13 Market portfolo (4) The feasble set s the set of all rsky assets (stocks, bonds, property, human captal) and so n prncple, the market portfolo s the portfolo of all of these assets n proporton to ther market value In practce (.e. n academc or professonal emprcal work) the market portfolo s proxed by a broad based equty portfolo (such as the FTSE100, S&P500, Nkke 5), sometmes a portfolo of equtes and bonds, and sometmes ncludng nternatonal assets Prncples of Fnance - Lecture 6 14 7
Equlbrum asset prcng models (1) Consder the problem of tryng to ascertan the far expected return (or, equvalently, the equlbrum prce) for an asset Ideally, we would lke a model that gave us the far (or equlbrum) expected return for any rsky asset, gven the characterstcs of that asset Such a model s known as an asset prcng model In the mean-varance framework, the CML gves the equlbrum expected return of effcent assets and portfolos Prncples of Fnance - Lecture 6 15 Equlbrum asset prcng models () But what about neffcent assets and portfolos? Although nvestors wll not hold neffcent portfolos ndvdually, they must stll have an equlbrum expected return snce they are held as part of the market portfolo Equlbrum asset prcng models allow nvestors to prce neffcent portfolos as well as effcent ones The most mportant equlbrum prcng models are the captal asset prcng model and the arbtrage prcng theory Prncples of Fnance - Lecture 6 16 8
The Captal Asset Prcng Model The essental dfference between an effcent asset and an neffcent asset s that the rsk of an effcent asset comprses solely undversfable (or systematc, or market) rsk In contrast, the rsk of an neffcent asset comprses both undversfable and dversfable rsk In general, expected return s the reward for bearng rsk However, an nvestor can expect no extra expected return for holdng an asset that has dversfable rsk, snce ths, by defnton, can just be dversfed away Expected return should therefore the reward for bearng undversfable rsk Ths s the bass of the most mportant model of equlbrum asset returns, namely the captal asset prcng model, or CAPM Prncples of Fnance - Lecture 6 17 Assumptons of the CAPM (1) Investors make decsons based solely on the expected return and the varance of returns Investors have homogenous nformaton There are no transacton costs There s no personal taxaton Unlmted short sales are allowed Prncples of Fnance - Lecture 6 18 9
Assumptons of the CAPM () All assets are nfntely dvsble Indvdual nvestors cannot nfluence the prce of an asset There are unlmted opportuntes for lendng and borrowng at the rsk-free rate All assets are marketable Prncples of Fnance - Lecture 6 19 CAPM: A heurstc dervaton (1) We have seen that when asset returns are generated by a sngle factor model, the approprate measure of an asset s rsk s ts beta We can ratonalse ths by notng that under the sngle ndex model, the varance (or total rsk) of an asset or portfolo can be decomposed nto non-dversfable rsk, β σ M, and dversfable rsk, σ ε The dversfable rsk of an asset can be dversfed away, and so the only rsk that s mportant to nvestors s ts non-dversfable rsk, whch s determned by ts beta Prncples of Fnance - Lecture 6 0 10
CAPM: A heurstc dervaton () Another way of ratonalsng the use of beta as a measure of rsk s to note that whle nvestors care about the varance of returns (snce they are rsk averse), t s the varance of ther overall portfolo that s mportant, not the varance of ndvdual assets When consderng an ndvdual asset, therefore, an nvestor wll consder the contrbuton that t makes to hs or her portfolo, whch s determned by the covarance between the ndvdual asset and the portfolo Prncples of Fnance - Lecture 6 1 CAPM: A heurstc dervaton (3) We have seen that f all nvestors hold the same portfolo then ths must be the market portfolo, and so nvestors evaluate a stock by ts covarance wth the market portfolo, or equvalently, ts beta Thus, there should be a postve relatonshp between the beta of and asset and ts expected return Prncples of Fnance - Lecture 6 11
CAPM: A heurstc dervaton (4) It s straghtforward to show that f we form a portfolo of any two stocks then both the expected return and the beta of the portfolo wll be a weghted average of the expected return and the beta of the two stocks t comprses Thus, n the absence of arbtrage opportuntes, the equlbrum relatonshp between expected return and rsk should be not only postve, but also lnear Prncples of Fnance - Lecture 6 3 CAPM: A heurstc dervaton (5) We can therefore wrte the equlbrum relatonshp between expected return and beta as the formula for a straght lne E(r ) = a + bβ We already know two ponts on ths straght lne: the rsk-free asset, whch has E(r ) = r f and β = 0, and the market portfolo, whch has E(r ) = E(r M ) and β = 1 The formula of the straght lne that goes through these two ponts s therefore gven by E(r ) = r f + [E(r M ) - r f ]β Ths s the captal asset prcng model, or CAPM Prncples of Fnance - Lecture 6 4 1
CAPM: A more rgorous dervaton (1) The effcent fronter of rsky assets was found by solvng Ε( r p ) δ maxθ = σ The weghts of the tangency portfolo assocated wth δ are gven by λωw = E(R) - δ Note that prevously, we defned z = λw and solved for z; we then rescaled the elements of z so that they summed to unty n order to fnd w p Prncples of Fnance - Lecture 6 5 CAPM: A more rgorous dervaton () When there s a rsk free asset, we can fnd the weghts of the market portfolo by settng δ = r f, yeldng λωw = E(R) - r f Ths s a set of N equatons (one for each asset ) of the form N λw σ = Ε( r ) r for = 1, K N j= 1 j j f, The term on the left hand sde of ths equaton s smply λ cov(r, r M ) and so we can wrte ths equaton as E(r ) = r f + λ cov(r, r M ) Prncples of Fnance - Lecture 6 6 13
CAPM: A more rgorous dervaton (3) Ths equaton must hold for any asset, and hence also for the market portfolo tself: E(r M ) = r f + λ cov(r M, r M ) = r f + λ var(r M ) Ε( rm ) rf Ths can be solved to yeld λ = var( rm ) Substtutng nto the orgnal prcng formula gves Ε E( r ) r M f ( r ) = rf + cov( r, rm ) = rf + [ E( rm ) var( rm ) Ths s the same equaton as derved earler, namely the CAPM r f ] β Prncples of Fnance - Lecture 6 7 CAPM: A more rgorous dervaton (4) The value [E(r ) r f ] s the expected excess return of asset over the rsk free rate The CAPM states that the expected excess return of an asset s proportonal to the expected excess return on the market portfolo, whch s called the equty rsk premum Prncples of Fnance - Lecture 6 8 14
Securty Market Lne (1) The lnear relatonshp between rsk and expected return s called the securty market lne or SML, and can be represented graphcally as follows Securty Market Lne Prncples of Fnance - Lecture 6 9 Securty Market Lne () The SML gves the equlbrum expected return for any asset, gven ts rsk, as measured by beta If an asset devates from the SML then there s an arbtrage opportunty that can be exploted by ether buyng the asset (f t les above the SML) or short sellng the asset (f t les below the SML) Prncples of Fnance - Lecture 6 30 15
Estmatng the SML (1) The CAPM says that all shares should le on the SML In practce, f we estmate the expected return and beta for a set of shares, we wll fnd that most shares wll not le exactly on the SML, even f the CAPM holds Ths s because of measurement error We therefore have the emprcal CAPM relatonshp r = r + ( E( r ) r ) ˆ β + ε r f M where s the sample mean return for stock (our estmate of E(r )), βˆ s our estmate of the beta of stock (whch we computed n the prevous lecture) and ε s the error term f Prncples of Fnance - Lecture 6 31 Estmatng the SML () In order to estmate the SML, we can use regresson analyss, and estmate the followng regresson r = a + b ˆ β + ε The estmated parameter a wll be our estmate of the rsk free rate, r f, and the estmated parameter b wll be our estmate of the equty rsk premum, [E(r M ) r f ] Prncples of Fnance - Lecture 6 3 16
Estmatng the SML: Example (1) Consder the sx shares from the last lecture, together wth estmates of ther expected return and beta A B C D E F G H 1 AMR BS GE HR MO UK SP500 1974-0.3505-0.1154-0.446-0.107-0.0758 0.331-0.647 3 1975 0.7083 0.47 0.3719 0.7 0.013 0.3569 0.370 : : : : : : : : : 9 1981-0.064-0.04-0.075-0.747 0.0913 0.0479-0.0491 10 198 1.064-0.1493 0.6968-0.615 0.43 0.0456 0.141 11 1983 0.194 0.3680 0.3110 1.868 0.066 0.640 0.51 1 Mean 0.03 0.0531 0.1501 0.159 0.105 0.110 0.138 13 14 Alpha 0.0197-0.0811-0.01-0.0080 0.0701 0.0598 0.0000 15 Beta 1.480 1.0839 1.3107 1.991 0.6 0.4939 1.0000 =AVERAGE(G:G11) =SLOPE(B:B11,$H$:$H$11) Prncples of Fnance - Lecture 6 33 Estmatng the SML: Example () We can regress the mean return on the estmated beta for these sx stocks ether usng the Tools Data Analyss Regresson functon (see prevous lecture) or usng the SLOPE and INTERCEPT functons A B =INTERCEPT(B1:G1,B15:G15) 3 33 Intercept Slope 0.0766 0.0545 =SLOPE(B1:G1,B15:G15) 34 R-Squared 0.793 =RSQ(B1:G1,B15:G15) The R-squared measures the goodness-of-ft of the regresson: the estmated beta explans about 8% of the varaton n mean returns Our estmated SML s gven by the equaton Ε( r ) = 0.0766 + 0.0546β Prncples of Fnance - Lecture 6 34 17
Equlbrum expected return vs. equlbrum prce (1) The CAPM gves the equlbrum expected return on a securty or portfolo of securtes gven ts beta However, from ths, we can obtan an expresson for the equlbrum prce of the securty or portfolo of securtes The expected return on a securty s Ε( p ( ) 1) p Ε r = p where p 0 s the current prce of the asset and E(p 1 ) s the expected future prce of the asset 0 0 Prncples of Fnance - Lecture 6 35 Equlbrum expected return vs. equlbrum prce () Re-arrangng, we have the followng expresson for the current prce of the asset: Ε( p ) 1 p0 = 1+ r + β ( Ε( r ) r ) f Ths can be nterpreted as the present value of E(p 1 ), calculated usng a dscount rate that s equal to the rsk free rate plus a term that represents the rsk of the securty M f Prncples of Fnance - Lecture 6 36 18
CAPM wth no rskless borrowng or lendng (1) Suppose that we have no rsk-free asset and hence no rskless lendng or borrowng Although there s no rsk-free asset, we can dentfy the rsk-free rate would have to exst n order to yeld the current market portfolo and denote ths by r f The market portfolo correspondng to ths rsk free rate s gven by the soluton to the equatons N ' λw j σ j = Ε( r ) rf for = 1, K, N j= 1 Prncples of Fnance - Lecture 6 37 CAPM wth no rskless borrowng or lendng () The term on the left hand sde of ths equaton s smply λ cov(r, r M ) and so we can wrte ths as E(r ) = r f + λ cov(r, r M ) Ths equaton must hold for any asset, and hence for the market portfolo E(r M ) = r f + λ cov(r M, r M ) Ε( rm ) rf ' Ths can be solved to yeld λ = var( rm ) Substtutng nto the orgnal prcng formula gves ' ' Ε( rm ) rf ' ' Ε( r ) = rf + cov( r, rm ) = rf + ( Ε( rm ) rf ) β var( r ) M Prncples of Fnance - Lecture 6 38 19
CAPM wth no rskless borrowng or lendng (3) Although there s no rsk free asset, there are an nfnte number of assets that have β = 0, and that offer an expected return of r f Defnng Z as the mnmum varance zero-beta asset, we have E(r ) = E(r Z ) + [E(r M ) E(r Z )] β Ths s the zero-beta CAPM, and can be used to fnd the equlbrum prce of rsky assets n the absence of a rsk free asset Prncples of Fnance - Lecture 6 39 CAPM wth no rskless borrowng or lendng (4) Prncples of Fnance - Lecture 6 40 0
CAPM wth no rskless borrowng or lendng (5) Investors now satsfy ther rsk-return preferences by holdng dfferent combnatons of Z and M Z must le below the mnmum varance portfolo, S, and so no nvestor wll hold just Z, snce t s an neffcent portfolo The effcent fronter s now the feasble set that les above S Note also that snce n aggregate nvestors hold the market portfolo, the net holdng of Z must be zero Prncples of Fnance - Lecture 6 41 CAPM wth no rskless borrowng or lendng (6) The SML s unchanged n the absence of a rsk free asset, except that ts ntercept s now the expected return on the zero beta asset, rather than the rsk free rate Prncples of Fnance - Lecture 6 4 1
Past exam questons: CAPM (1) Queston 3 (00), Part b: Consder a world where there are only the followng two rsky assets: # of shares Prce per share E(r) Std. dev. Stock A 50.00 1% 0% Stock B 150 1.50 16% 30% The correlaton coeffcent between the two shares' returns s 0.. What s the composton of the market portfolo? What s ts expected return and standard devaton? What are the CAPM betas of the two shares? Assumng that the CAPM holds, what s the rsk free rate? Prncples of Fnance - Lecture 6 43 Past exam questons: CAPM () Other CAPM-related questons (001-00): Defne and explan the term 'market portfolo'. Why are the followng assumptons requred for the dervaton of the standard CAPM: () that nvestors have homogenous nformaton, () assets are nfntely dvsble? Explan brefly how the absence of a rsk-free asset affects the standard CAPM. Prncples of Fnance - Lecture 6 44