CAPM and Black-Litterman

Transcription

1 CAPM and Black-Ltterman Ing-Haw Cheng y Prnceton Unversty November, 28 Abstract Ths teachng note descrbes CAPM and the Black-Ltterman portfolo optmzaton process. Keywords: CAPM, Black-Ltterman Ths s a teachng note accompanyng the ECO467/FIN567 course, "Insttutonal Fnance," taught by Prof. Markus Brunnermeer at Prnceton Unversty. y Department of Economcs, Prnceton Unversty, Prnceton, NJ 8544, e-mal: cheng@prnceton.edu,

2 CAPM To truly understand Black-Ltterman, t s essental to understand the CAPM as an equlbrum model, and not just a prcng relaton. uppose there are two dates, t = ;, and N rsky assets whose gross returns are dstrbuted normally, R~N (; ). Note that R s N, s N, and s N N. Asset returns R f at t = (normalze the prce of the rsk-free asset s ) wth probablty. Let Q = R R f denote the N vector of excess returns, where s an N vector of s.. The Model There are a contnuum of 2 [; ] agents n the economy who take prces as gven. Let denote the N vector of dollar demand (not quantty) of each agent (.e., let j = j p j f j s the quantty of shares of asset j demanded by agent ). Each agent chooses an N vector of dollar demand to maxmze tme wealth, whch follows the budget dynamcs W = R f W + R = R f W + (R R f ) = R f W + Q The symbol denotes "transpose." Agents have mean-varance utlty, so they solve the problem max U = max E W 2 V ar W s:t: W = R f W + Q where s the rsk-averson for agent. Lettng = E [Q] = E [R] R f denote the N vector of expected excess returns, note that E W = R f W + V ar W = so that the objectve functon s equvalent to max 2 ()

3 The rst order condtons for each agent are = so that = (2) where = = s the rsk tolerance of agent _. 2 Let denote the N vector of outstandng value of shares for the rsky assets. In equlbrum, the market clearng condton says that Z = whch mples Z = Note that () s actually an N-equaton maxmzaton problem. The rst order condton comes from where, usng some matrx calculus, d d n 2 @x = a = A = x A + A = 2x A f A s symmetrc for any conformable matrx A and vectors a; x. 2 Note that n the sngle asset case, ths delvers the classc lnear demand equaton = E [v] R f P V ar (v) where v s the payo and s the quantty of shares demanded. case s equvalent to = E [R] V ar (R) To get the equaton for, note that R f (3) To see ths, observe that (2) n the sngle-asset = E [R] R f V ar (R) = P E [v] R f P V ar (v) = E [v] =P R f V ar (v=p ) = P 2 (E [v] =P R f ) V ar (v) whch delvers (3) snce = P: 2

4 Thus equlbrum excess returns must satsfy = (4) where = R s the aggregate (average) rsk averson of market partcpants..2 Equlbrum Beta De ne the market rsk premum as and note that Q M = Q M = E [Q M ] = = usng (4). Note also that Cov (Q; Q M ) = V ar (Q M ) = 2 Usng (4), we can then wrte = = ( )2 = M (5) for = Cov (Q; Q M) V ar (Q M ) (6) Note, however, that ths secton s really just notaton; equaton (4) s the central result that gves us the classc relatonshp between expected returns and demand. 3

5 .3 What dd we learn? Equatons (5) and (6) are the famlar expressons of found n typcal undergraduate texts. The usual nterpretaton of s that t represents the "senstvty of returns to market returns." Whle ths statement s correct, the dervatons above hghlght the deeper equlbrum nterpretaton of : equatons (5) and (6) say that s the contrbuton of an asset to the total varance of the aggregate portfolo, and that an asset earns a hgh rsk premum f and only f t contrbutes a sgn cant amount of varance to the aggregate portfolo. Ths s a powerful nsght..3. Predctons Three spec c predctons that fall out of ths nsght, n order of ncreasng strength, are that. Asset has explanatory power for expected excess returns. 2. The premum for unt of s the market premum, M. 3. An asset provdes postve expected excess returns only f t bears market rsk. Predcton seems to hold. Roughly, t says that " matters." That s, f you run a tme-seres regresson of an asset s return on the tme-seres of market returns, you should obtan a statstcally sgn cant, non-zero coe cent. Roughly, Predcton 2 says the followng. uppose you estmate for each stock by runnng a tme-seres regresson of excess returns on market returns. Then, n a cross-sectonal regresson, you regress (tme-seres) average returns on. If you plot the resultng lne, the slope of your lne should be M. Typcally, Predcton 2 s rejected - the slope s smaller. Predcton 3 says that M s the only factor that explans expected excess returns. However, we know that a number of other factors, such as the value factor, HML, and the sze factor, MB, also tend to explan returns, so ths predcton s rejected. Fama and French (Journal of Economc Perspectves, 24) has an n-depth dscusson of these predctons..3.2 Relaton to Black-Ltterman Black-Ltterman uses ths nsght of CAPM as the startng pont for formng portfolo weghts. pec cally, f you are a manager that s benchmarked aganst an underlyng portfolo, the Black- Ltterman procedure suggests usng mpled returns from (5) and (6) as the mean for (normally 4

6 dstrbuted) pror belefs. These mples returns by rst estmatng and usng as nputs and from the modeler. Whle the actual Black-Ltterman procedure can be understood just by takng these formulas as gven, understandng why t works requres really understandng the equlbrum ntuton of descrbed above. 2 Black-Ltterman In the typcal portfolo optmzaton problem, the nancal modeler estmates two tems. Frst, the modeler estmates the vector of expected excess returns,. econd, the modeler estmates the matrx. The modeler then uses these two as nputs n computng optmal rsky portfolo weghts w where w =, where s the N vector of ones. (Note that w are value-weghts, not quantty-weghts.) That s, the nancal modeler computes the w that maxmzes the harpe rato, and then allocates a proporton ( ) to the rsk-free asset, and to the optmal rsky portfolo. In contrast, Black-Ltterman replaces the estmaton of based on hstorcal data wth the mpled returns from CAPM, usng (5) or (4). The thought experment s as follows: suppose we take the observed market portfolo as the startng pont for an optmal portfolo. If we beleve that the world s close to equlbrum, then ths should be a reasonable approxmaton. However, we mght thnk that we have some nformaton or belefs that the market portfolo weghts have not ncorporated yet. The nsght of Black-Ltterman s that, n a world where returns are dstrbuted normally, Bayes rule gves us a smple way to update the market portfolo wth our own nformaton. Ths can be accomplshed n ve steps. 2. Compute uppose we have T observatons of returns on our N assets. Denote the observaton of excess returns n perod s as q s, and let Q be the T N stacked matrx of qs. (Recall that q s s N, so Q s lterally taken by stackng q on top of q 2, and so forth, all the way untl q T.) There are two possbltes - N s small, or N s large. If N s small, a reasonable estmator of s the sample covarance matrx. To revew, recall that, by de nton, the covarance matrx s = E (Q E [Q]) (Q E [Q]) = E QQ 5

7 so the analagous sample estmator s ^ = T TX q s qs qq s= where Ths can be re-wrtten as q = T TX s= q s ^ = T Q Q qq (7) If N s large, the number of parameters to be estmated (N (N + ) =2) s large, and so the number of observatons T requred so that (7) converges wll be extraordnarly large. There s a small scence behnd how to compute covarance matrces for large N - for more nformaton, see Chapter 8 of Ltterman (23). 2.2 Compute CAPM Impled Returns. We wsh to calculate = M = ( )2 Ths step requres some nput from the nancal modeler. One procedure s to conjecture a ^ M based on a benchmark portfolo and compute the ^ of each asset wth ths benchmark portfolo based on (6) wth a conjectured vector. For example, f the benchmark s the &P5 (a P5 value-weghted ndex of 5 stocks), ; = p q, and = j= p jq j, and ^ M s the hstorcal rsk-premum of the &P5 over a sutable rsk-free rate. Alternatvely, f the benchmark s an equal-weght portfolo, = =K, =, and ^ M s the hstorcal rsk-premum of ths equal-weght portfolo. From these conjectures, the modeler can easly compute ^ = ^ ^ 6

8 and ^ = ^ ^ M for the modeler s choce of M. Note that ths procedure s equvalent, from an economcs standpont, to conjecturng aggregate rsk-averson, and computng (4) drectly usng ^ and a spec ed, but often t s easer from a practtoner s perspectve to conjecture a ^ M nstead of. 2.3 Express Con dence n CAPM. The rst two steps have gven us an "ntal estmate," or a set of pror belefs, about the dstrbuton of returns. That s, from what we have computed above, our current belefs about the dstrbuton of returns s The BL approach allows the modeler, at ths step, to express hs or her con dence n the CAPM model as a whole, through a parameter. Black-Ltterman s Q~N ^; ^ That s, the set of pror belefs used by A typcal value of s the neutral value of. Hgher values of express lower con dence n CAPM. 2.4 Express Vews. Now that we have a set of "pror belefs," the fact that the underlyng model mples these belefs are normally dstrbuted gves us an easy way to ncorporate any news or prvate nformaton the nancal modeler may have about returns. (N N) and M (N ); such that To be spec c, a set of vews are two matrces, P P = M and a dagonal N N matrx that express con dence n these vews. For example, suppose N = 3. The matrces % 225 P = ; M = 6 5% ; = express the vew that asset wll earn an excess return of 7% wth con dence 5%, and that asset 2 wll outperform asset 3 by 5%, wth con dence 25%. Note that s n unts of (% 2 ). 7

9 It s mportant to remember that these vews represent Gaussan belefs. That s, the vew that asset wll earn an excess return of 7% wth con dence 5% essentally means that the modeler wshes to ncorporate the belef that s dstrbuted N :7; :5 2. The second belef corresponds to the belef that 2 3 are dstrbuted N :5; : Compute Posteror Belefs and Portfolo Weghts. Once we have all these parameters, the rest s mechancal applcaton of Bayes rule and some extra mathematcal manpulaton. Gven our vews, the posteror belef dstrbuton of returns - that s, the combnaton of the belefs mpled by CAPM and those of the modeler, combned usng Bayes rule - s dstrbuted normally. ^ P OT = ^ P OT = Denote these belefs as Q P OT, t follows that Q P OT ~N ^P OT ; ^ P OT ^ + P P ^ ^ + P M ^ + P P From these belefs, t s easy to compute back the mpled portfolo weghts. Recall that, f our vews are correct, then from equaton (4), = as an equlbrum relaton. vews are correct, t must be that Then, f our BL ^ P OT = ^ P OT BL where BL s the mpled value of shares that the BL portfolo mples holdng. Then the portfolo weghts w BL are gven by w BL BL BL = ^ ^ P OT P OT ^ ^ P OT P OT The quantty of shares demanded s clearly BL = BL :=P where ":=" s element-by-element dvson. 8

10 3 Fnal Thoughts Why s Black-Ltterman mportant? From a theoretcal perspectve, t s an elegant way to move from observed returns to portfolo weghts by assumng that observed returns are generated from CAPM and then applyng Bayes rule to ncorporate our prvate belefs. The nancal modeler then has a way to center portfolo weghts around a gven benchmark portfolo and then use these vews to derve new portfolo weghts. From a practcal perspectve, BL s very mportant because the Markowtz optmzaton process often gves you nfeasble portfolo weghts. For example, weghts of 3% are not uncommon. Note that the nal step does not requre a "re-optmzaton" process - the equlbrum relaton (4) gves us everythng we need! 9