4.4 Special Cases: CAPM, APT and Behavioral CAPM
4.4.1 Deriving the CAPM by ‘Brutal Force of Computations’
Note that we have up to now always made the first of these assumptions. For the sake of completeness we state it explicitly since the risk-free asset plays a special role in the CAPM. To make use of this special role we need to separate the risk-free asset from the risky assets. To this end we introduce the following notation. For vectors and matrices we defineA D .1;AO/whereAO is the SK matrix ofrisky assets.
By.A/O D ..AO0/; : : : ; .AOK// we denote the vector of mean payoffs of assets in a matrixA. Similarly,O COV.A/O D .cov.Ak;Aj//k;jD1;:::;K denotes (as before) the variance-covariance matrix associated with a matrixA. Note that the variance of a portfolio of assets can be written as
2.AO/O DO0AO0.prob/AOO.AOO/.AO/O 0DO0cov.AO/:O
Equipped with this notation, we analyze the decision problem of a mean-variance agent, in a setting where there is no first period consumption and endowments are spanned:
Omax
i2RKC1Vi..ci/; 2.ci// such that XK kD0
qkOi;kD XK kD0
qkAi;kDwi;
wherecisWDPK
kD0AksOi;k,sD1; : : : ;S.
Recall that we defined the risk-free rate byq0 WD1=Rf. From the budget equation we can then express the units of the risk-free asset held byO0 DRf.wiOq0/O . Hence, we can eliminate the budget restriction and re-write the maximization problem as
Omax
i2RKVi
RfwiC..A/O Rfq/O 0Oi; 2.AOOi/ :
178 4 Two-Period Model: State-Preference Approach
The first order condition is21:.A/O RfqO Dicov.A/O Oi, whereiWD @@Vi
Vi.; 2/ is the agent’s degree of risk aversion.22Solving for the portfolio we obtain
OiD 1
iCOV.A/O 1..A/O Rfq/:O
From the first order condition we see that any two different agents,iandi0, will form portfolios whose ratio of risky assets, Oi;k=Oi;k0 D Oi0;k=Oi0;k0, are identical.
This is because the first order condition is a linear system of equations differing across agents only by a scalar,i. This is again thetwo-fund separation property, since every agent’s portfolio is composed out of two funds, the risk-free asset and a composition of risky assets that is the same for all agents, i.e.,Oi D.Oi;0;Oi;1O/, iD1; : : : ;I.
Dividing the first order condition byiand summing up over all agents, we obtain X
i
1
i .A/O RfqO
Dcov.A/O X
i
Oi: From the equality of demand and supply of assets we know thatP
iOiDP
iiA DW OM, where the sum of all assets available is denoted by assetM, the market portfolio.
Accordingly, denote the market portfolio’s payoff byAOM D AOOM and let the price of the market portfolio beqOMDqO0OM. Then we get:
.A/O RfqO
DX
i
1 i
1
cov.A/O OM:
Multiplying both sides with the market portfolio yields an expression from which we can derive the harmonic mean of the agents’ risk aversions:
X
i
1 i
1
D
.AOM/RfqOM 2.AOM/ :
21We assume that the mean-variance utility functionVi.; /is quasi-concave so that the first order condition is necessary and sufficient to describe the solution of the maximization problem.
This is, for example, the case for the standard mean-variance functionVi.; /WD2i2, since it is even concave.
22Note that @@VVii is the slope of the indifference curve in a diagram with the mean as a function of the standard deviation.
Substituting this back into the former equation, we finally get the asset pricing rule:
RfqO D.A/O
.AOM/RfqOM
2.AOM/ cov.A;O AOM/:
Hence, the price of any assetkis equal to its discounted expected payoff, adjusted by the covariance of its payoffs to the market portfolio. Writing this more explicitly we have derived:
qkD .Ak/ Rf
cov.Ak;AM/ var.AM/
.AM/ Rf
qM
! :
We see that the present price of an asset is given by its expected payoff discounted to the present minus a risk premium that increases the higher the covariance to the market portfolio. This is a nice asset pricing rule in economic terms and it is quite easy to derive the analog in finance terms. To this end multiply the resulting expression byRf and divide it byqk and qM. We then obtain the by now well- known expression relating the asset excess returns to the excess return of the market portfolio:
.Rk/Rf Dˇk..RM/Rf/ where ˇkD cov.Rk;RM/ 2.RM/ ; which we have already seen in Sect.3.2.
Being equipped with the economic and the finance version of the SML we can revisit the claim based on the finance SML that increasing the systematic risk of an asset is a good thing for the asset according to the SML since it increases its returns.
This suggests that a hedge fund could do better than a mutual fund by simple taking more risk. The logic of the CAPM is quite the opposite: increasing the risk, the investors dorequirea higher return on the asset. The economic SML reveals that this is obviously not a good thing for the shares since the investors’ demand for a higher return will be satisfied by adecreased price. Hence, the value of the hedge fund decreases!
What does the SML tell us about the likelihood ratio process? Recall from the general risk-return decomposition that
.Rk/Rf D cov.`;Rk/; kD1; ;K:
Similarly the SML yields
.Rk/Rf Dcov.RM;Rk/.RM/Rf
2.RM/ :
180 4 Two-Period Model: State-Preference Approach
Thus we get
cov.`;Rk/Dcov.RM;Rk/.RM/Rf
2.RM/ :
Hence, the likelihood ratio process is a linear functional of the market return` D abRM for some parametersa,b, where b D ..RM/Rf/=2.RM/anda is obtained from.`/Dab.RM/D1. ThusaD1Cb.RM/.23