3.3 Heterogeneous Beliefs and the Alpha
3.3.2 CAPM with Heterogeneous Beliefs
While the previous section showed the relation between the Alpha, the SML and the CML for any given investor, this section extends the analysis towards heterogeneous investors. In the standard CAPM investors differ with respect to their initial endowments and their degree of risk aversion, but they share the same beliefs about the expected returns and covariance of returns. Now we allow the investors to also differ with respect to their beliefs on the assets’ expected returns, i.e., in principle we could have thati ¤ j for any two investorsiandj. However, we keep the assumption that investors agree on the covariances of the assets. This can be justified by two descriptive and one pragmatic argument.23First, errors in means are much more detrimental to the agents’ utility than errors in covariances. To see this, let
optWD 1
iCOV1.iRf1/
be the optimal portfolio of agenti, allowing for short sales. Then the optimal level of utility is
Vi;opt WD 1
2i.iRf1/0COV1.iRf1/:
Hence errors in covariances are of linear order to the utility while errors in means change the utility in a quadratic way.24 Second, covariances tend to be better predictable, since they are less time-dependent. Take as an example bonds and
23See Gerber and Hens [GH06] for a generalization towards heterogeneous beliefs on covariances.
24Note that we have defined the utility on gross returns, i.e., expected returns are larger than one.
stocks: In the medium run (2–3 years) bond and stock returns are negatively correlated. In a boom, stocks shoot up but bonds do poorly, in an economic recession, bonds do fine but stocks do depreciate. However, whether on medium-run horizons stock returns are higher than bond returns is much more difficult to predict since this would include a prediction on the stage of the business cycle. Finally, there is a pragmatic reason to keep the assumption of homogeneous covariance expectations which is perhaps most compelling – at least from a didactical point of view: the assumption of heterogeneous expectations on means is already sufficient to explain all the phenomena we mentioned in the introduction of this section. – So why should we make things more complicated than necessary?
In the following we derive the SML for the case of heterogeneous beliefs. We state the result in a proposition and then give the proof of it.
Proposition 3.5 In the CAPM with heterogeneous beliefs the Security Market Line holds for the average beliefs, i.e., for all assets kD1; : : : ;K,
NkRf Dˇk;M.NMRf/;
where as usualˇk;M WD cov
Rk;RMı var
RM
andNM WDPI
iD1aii, with aiWD
ri i=PI
jD1 rj
j and ri D wif=PI
jD1wif, wif D .1i0/W0i, where W0i denotes the first period income.
Proof In the CAPM with heterogeneous beliefs an investor maximizes .iRf1/0ii
2i0COVi:
The first-order condition isCOViD 1i.iRf1/. Multiplying this equation with the relative financial wealth of investori, which is given by ri WD wif=PI
jD1wif, where wif denotes the financial wealth of investor i, and summing up over all investors on the market, we getCOVM D PI
iD1ri
i.iRf1/,M WDPI iD1rii, which by the definition ofRMis equivalent to
cov Rk;RM
D XI
iD1
ri
i.ikRf/; kD1; : : : ;K:
In these expressionsMk denotes the relative market capitalization of assetk. We haveMk D PI
iD1riik, i.e., the relative market capitalization of assetkis equal to the average percentage of wealth the investors put into assetk.
Multiplying the last expression withMk and summing up, we get
var RM
D XI iD1
ri
i.i;MRf/ where i;MWD XK kD1
ikMk:
3.3 Heterogeneous Beliefs and the Alpha 117
Dividing cov Rk;RM
and var RM
byPI iD1 ri
i we obtain:
cov Rk;RM PI
iD1 ri i
D.NkRf/; where NkWD XI
iD1 ri i
PI jD1 rj
j
„ ƒ‚ …
Dai
ik;
and
var RM PI
iD1 ri i
D.NMRf/; where NMWD XI
iD1
aii;M:
EliminatingPI iD1 ri
i from the last equation and inserting into the previous one yields cov
Rk;RM
var.RM/ .NMRf/Dˇk;M.NMRf/D.NkRf/;
which reads for assetkas.NkRf/Dˇk;M.NMRf/. ut This is theSecurity Market Line (SML)with average expectations, as shown in Fig.3.14. Note that the averaging is done taking into account both the relative wealth and also the risk aversion of the agents. The wealthier and the less risk averse agents determine the average more than the poor and more risk averse agents. Since agents have the same covariance expectations, the Beta factors are as in the model with homogeneous beliefs.
When we have heterogeneous beliefs, just like with heterogeneous investor sets, we get individual security lines along which all assets are lined up if they form an optimal portfolio. The derivation is done as above. We consider the first order condition for maximizing the mean-variance utility function, multiply each equation with the portfolio share and add these equations up to eliminate the risk aversion Fig. 3.14 The Security
Market Line for average expectations
μ
Rf
β SML
μ¯M
1
μ¯k−Rf =βk(¯μM−Rf) whereβk=COVkM/σ2M
μ
Rf
β SML
μiλi,opt
1
μik−Rf =βkλi,opt(μλi,opt−Rf)
whereβkλi,opt= cov(Rk, Rλi,opt)/σ2(Rλi,opt)
Fig. 3.15 Security Line of investori
parameteri. As a result we obtain the individual security line: for allkwe have ikRf Dˇk;i.iRf.1i0//;
whereˇk;i D cov
Rk;Riı var
Ri
(Fig.3.15).
If an investor happens to have beliefs equal to the average belief, i.e., ifi D N then he will hold a portfolio of risky assets that coincides with the market portfolio. In general this may not be the case and agents can have under-diversified portfolios,25 two-fund separation fails and a new asset with positive Alpha vis-à- vis the market portfolio can have negative Alpha for some agents. The last point is exemplified in the exercise book. We come back to this point later, after we have analyzed the zero-sum game property.