Capital asset pricing model, arbitrage pricing theory and portfolio management

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1 Captal asset prcng model, arbtrage prcng theory and portfolo management Vnod Kothar The captal asset prcng model (CAPM) s great n terms of ts understandng of rsk decomposton of rsk nto securty-specfc rsk and market rsk. Before we dscuss the CAPM, t would be mportant to understand rsk of portfolos. Rsk of portfolos: Markowtz made a semnal contrbuton to theory of nvestments way back n 1959 when he propounded hs effcent portfolo theory. Markowtz contended, and establshed wth basc statstcs, that when a rsky securty s combned wth another rsky securtes, unless the two are closely correlated, the rsk of the portfolo (that s, the two securtes put together) does not go up t comes down. To understand ths pont, let us frst understand the rsk of securtes, and then we move on to rsk of portfolos. Rsk of securtes: Let us take an example. Example 1 Let us suppose there are two securtes, wth the followng returns profle: Scenaro Securty A Probablty Securty B Probablty Expected returns The expected returns n the last row have been computed by multplyng the returns n each scenaro, by the probablty of the scenaro, and addng up the products. That s to say: n R 1 R P where: R : the expected returns from the portfolo R : the returns n scenaro (1)

2 P : the probablty of scenaro It s qute clear that Securty B has hgher expected returns than Securty A. However, t s also apparent, even to naked eye, that the varablty of returns of Securty B s substantally hgher. The varablty of returns s captured by computng the standard devaton of the two securtes, whch we do below: Securty (R- ((R- Securty (R- ((R- Scenaro A Probablty R )^2 R )^2)*P B Probablty R )^2 R )^2)*P Expected returns Standard devaton The standard devaton has been computed by addng up Col 5/ Col 9 respectvely, and takng the root of the same. 2 ( R R) P (2) Makng a portfolo: We have so far seen the ndvdual returns of the two securtes. An nvestor has a partcular sum of money to nvest, whch he may nvest ether entrely n Securty A, or entrely n Securty B, or he may hold varous combnatons of A and B. Let us suppose these optons are summed n the followng table: Composton of Portfolo Securty A Securty B Portfolo returns There mght be nfnte ways of combnng Securty A and B, but we have taken above 11 scenaros, changng the weght of the two securtes from 100 of Securty A, sldng t

3 down to 0. The expected returns of the portfolo, n Col 4 above, s smply the returns from Securty A and Securty B, weghted n ther respectve proportons as gven by Col 2 and 3. That s to say, R p = R A.X A + R B.X B Or generalzng: n R p = R.X (3) 1 Where R p : Expected return from the Portfolo : Proporton of securty n the portfolo X Needless to say, as we add more of Securty B to the portfolo, the returns from the portfolo contnue to go up. Ths s qute obvous, as B gves hgher returns. But then B has hgher rsk too. Does that mean, as B s added to the portfolo, the rsk of the portfolo also goes up? That s exactly where Martowtz made a sgnfcant pont, holdng that as doses of Securty B are added to Securty A, whle the expected return goes up, the rsk does not go up, at least upto a partcular level. Portfolo rsk: The rsk or the standard devaton of the portfolo s gven by: n n 2 p X X j j 1 j1 and (4) j = j j where X, X j etc are the proportons of the respectve assets n the portfolo. j s the correlaton between -th asset and j-th asset. s also referred to as the co-varance of - th and j-th securty. Needless to say, to get the p from Eq 4, all we have to do t to take ts square root. The above formula s ntutvely understandable. As n case of the mean, the standard devaton s also the weghted average of the standard devatons of the two securtes; however, t s the correlaton that s makng a dfference here. If the correlaton s 1, the covarance s the same as the weghted average of the standard devatons. However, where correlaton s less than 1, t causes the covarance of the portfolo to come down. Example 2 Let us assume we have two securtes whch have the followng rsk return profle j

4 Securty A Securty B Mean returns Standard devaton 3 6 Let us assume that the correlaton between the returns of the two securtes s 0.25 or 25. We the above realgnment, the correlaton between Securty A and Securty B s approxmately 25. Now, f we put a correlaton assumpton of 25, and compute portfolo rsk as per Eq. 4, let us say, wth 90 of Stock A and 10 of Stock B, we get a portfolo standard devaton of , whch s less than 3, the standard devaton of securty A only. Ths may, at frst sght seem a lttle strange we added a rsker securty (B), and yet, the combned result s less rsky than the sngle securty. It s lke mxng chlly wth sugar, and the result beng sweeter! However, on further reflecton, t s not dffcult to understand ths the correlaton between the two securtes s low. Wth the lower degree of correlaton acts as a rsk absorpton devce the rsk comes down even though, addng Securty B to the portfolo, the returns go up. Wth dfferent combnatons of Securty A and Securty B, keepng correlaton of 0.25, the rsks/returns look as follows: Proporton of A Proporton of B Portfolo SD Portfolo returns If we were to plot these results on a graph (rsk on X axs and returns on Y axs), the graph looks lke the one below:

5 16.00 Portfolo rsk returns Note the bulge of the graph towards the left ths ndcates the reducton n rsk wth ncreasng returns upto a pont, beyond whch the rsk starts ncreasng wth ncreasng returns. The least rsky poston for an nvestor s the left-most pont on the bulgng curve. However, ths pont need not necessarly be the deal choce for the nvestor, as the nvestor may, ndeed, be comfortable wth a hgher dose of rsk, but wth ncreased returns. It would not be dffcult to understand that as the correlaton between the two securtes s ncreased, the bulgng curve starts gettng flatter. The Table below shows the portfolo standard devaton for dfferent correlaton levels: Correlaton between A and B Proporton of A

6 CAPM model The rsk of the market dctates the returns from the market as the rsk goes up, the returns of the market also go up. Ths relatonshp s gven by the captal market lne. As for an ndvdual securty, the relatonshp between the market returns (Rm) and the returns from the ndvdual securty j (Rj) depends on the senstvty of Rj wth Rm. The slope of the lne that relates Rj wth Rm s called the beta of securty j. If the beta s more than 1, the securty s more senstve Generalzed formula for Rj: R j = R f + beta (R m R f ) Where R j - expected return on securty j R f - rsk free rate of return R m market returns Generalzed formula for beta: covarance of securty j wth market dvded by varance of the market Beta = ( m Im )/ 2 m Placng equaton 2 n equaton 1, we have the followng result: o The spread provded by the market s a functon of the devaton of the market. Ths spread, dvded by the market devaton, multpled by devaton of securty j, multpled by ts correlaton, provdes the spread gven by securty j The rsk ntroduced by the beta s rsk derved from the market ths rsk s, therefore, the market rsk or systematc rsk. Ths rsk s non-dversfable. The actual return on a partcular securty wll nclude the error term, that s, the devaton between the realzed return and the expected return. Ths error term s subject to reducton by dversfcaton. In other words, ths rsk s dversfable rsk, also called dosyncratc rsk or unsystematc rsk. Arbtrage Prcng Theory (APT) The essence of the CAPM was that the prcng of the ndvdual securty n the market s done based on ts beta, that s, senstvty of the stock to the market returns. In the actual realzed returns, there wll be a dfference to the dosyncratc error term e, but the expected value of e equals zero. At the same tme, e s dversfable. Hence, the CAPM

7 beleves the only factor that affects returns from the partcular securty s the market return. APT has been propounded by Ross. APT s also an equlbrum model explanng that the market process brngs securty prces ultmately at an equlbrum. The arbtrage prcng theory seeks to explan the process of prcng of securtes n the market as the process whereby nvestors try and explot arbtrage opportuntes untl arbtrage opportuntes are completely klled. That s, the market reaches an equlbrum when there s no opportunty to make arbtrage or rskless proft. Where do we say there s an arbtrage opportunty? When do we a say a stock s too cheap or too overprced? Analysts try and dentfy factors that explan market prces, and then correlate the movement of those factors wth the movement of the prces. There mght be several such factors explanng market prces: the factors mght be macroeconomc, or ndustryspecfc. In other words, the factor model sees a lnear relatonshp between varables that explan the market prces, and senstvty of changes n each of these varables wth the changes n the securty prces. These varables or factors may also be perceved as the rsks that affect the prce of the securty. In the CAPM, the only model that explaned changes n the securty prces/returns was the market return. In the APT, the factors may be several. Hence, CAPM was a snglefactor model; the APT s a mult-factor model. In the APT model, the prce of the securty s explaned as follows: Two factor equaton: R j = a + b 1j F 1 + b 2j F 2 + e j Mult factor model: Rj = a + b 1j F 1 + b 2j F 2 + b mj F m + e j (APT1) (APT2) Let us try understandng APT1. The frst term a s the return when the causatve factors or rsks have zero value. Ths may be perceved as the rsk-free rate. F 1 and F 2 are factors that affect the prces of securty j, and B 1 and B 2 are degree to whch the factors affect the returns from the securty, that s, the senstvty of the returns from securty j to the respectve factors. These are the betas. There are separate betas for each of the factors hence, we have b 1j, b 2j and so on. The last term n the equaton s the error term or the varablty of the realzed return from the return explaned by the betas and the factors. As n case of the CAPM, the expected value of e wll be zero that s, t wll have gans and losses that wll neutralze. The extenson to mult-factor model (Eq APT-2) s not very dffcult. We have smply extended the equaton to nclude multple factors wth ther respectve betas. In APT, there are as many betas as there are factors that affect the prce of the securty.

8 Yet another way to understand the APT would be to look at the prcng of a securty as composed of rsk free rate, and rsk premums representng dfferent rsks. The factors may also be perceved as dfferent rsk premums, wth the betas beng the multplers for these dfferent rsk premums. Why s t called arbtrage prcng? If the betas of two dfferent securtes wth a gven factor F1 are known, then the expected Sharpe Index model Wllam Sharpe s Sharpe Index model or sngle ndex model s actually a precursor to the CAPM, and s a smplfcaton of Markowtz. Under the Markowtz model, the rsk of the portfolo s affected by covarance of pars of securtes f we were to extend the formula for rsk under Markowtz below, there wll n * (n-1)/2 co-varances. n n 2 p X X j j 1 j1 Instead, Sharpe suggested that the requred nputs under the Martowtz model may be smplfed by lookng at the correlaton of the securty wth a broader market ndex, nstead of pars of securtes. Hence, the return of the ndvdual securty can be seen as: R j = a + b j R m + e j Where R m s return from the market, and the beta s the senstvty of the stock to the market.

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