The CAPM discussed in this chapter is based on the assumption that all the investors choose a combination of the risk-free asset and a portfolio of risky assets. According to efficient portfolio theory, this portfolio of risky assets is the tangency portfolio for all investors. Thus, the market portfolio is equivalent to the tangency portfolio so that the market portfolio has the properties of the tangency portfolio. It is important to note that the optimality of the tangency portfolio in this context is based on the fact that investors combine their portfolio of risky assets with the risk-free asset.
In this section, we consider a version of the CAPM that holds without relying on the existence of a risk-free asset. There are two important implica- tions of this change for the CAPM. The more obvious one is that we cannot include the risk-free rate,μf, in the model. The second, less obvious, change is that the tangency portfolio is no longer the unambiguous optimal portfolio
and, hence, we may no longer assume that the market portfolio is equivalent to the tangency portfolio.
Instead, we assume that each investor holds a portfolio of risky assets that lies on the efficient frontier, but these portfolios may vary by investor.
According to Propositions 5.2, portfolios constructed from assets lying on the efficient frontier are also on the efficient frontier provided that the mean return of the portfolio is greater than the mean return on the minimum vari- ance portfolio. Hence, we may still assume that the market portfolio lies on the efficient frontier. However, it is not necessarily equal to the tangency portfolio.
Let Rm denote the return on the market portfolio. We assume that if there is another portfolio, with return Rp, such that E(Rp) = E(Rm), then Var(Rp)≥Var(Rm); alternatively, if Var(Rp) = Var(Rm), then E(Rp)≤ E(Rm). Note that these assumptions state simply that the market portfolio lies on the efficient frontier.
The proof of the CAPM given in Proposition 7.1 is based on the fact that the market portfolio has the maximum Sharpe ratio among all assets and, hence, modifying the weight given to any asset cannot increase the Sharpe ratio. For the version of the CAPM considered in this section, we use a similar argument based on the efficiency of the market portfolio.
Let Ri denote the return on asset i. Suppose that we can construct a portfolio consisting of asset i together with the market portfolio that has the same expected return as the market portfolio; then the variance of that portfolio must be at least as large as that of the market portfolio. We may try to use this fact to establish a relationship similar to that in the SML.
However, it is clear that such an approach cannot work—unless asset i has the same expected return as the market portfolio, a portfolio including both assetiand the market portfolio cannot have the same expected return as the market portfolio. Hence, we need to include a third asset in the portfolio.
Because we would like the eventual result to focus on the relationship between the return on assetiand the return on the market portfolio, we might consider an asset with a return that is uncorrelated with the market return.
Let Rz denote the return on an asset satisfying Cov(Rz, Rm) = 0 and E(Rz)= E(Rm). At the end of this section it will be shown that such a portfolio exists. Note that Cov(Rz, Rm) = 0 implies that the value of beta corresponding toRz is zero; therefore, the asset with returnRz is known as thezero-beta portfolio.
Proposition 7.3. Let Rm denote the return on the market portfolio and let Rzdenote the return on the corresponding zero-beta portfolio; letμm=E(Rm) and μz=E(Rz). Consider an asset with return Ri; let μi=E(Ri) and let βi =Cov(Ri, Rm)/Var(Rm). Then
μi−μz=βi(μm−μz). (7.19)
Proof. For a real numberθ, consider the zero-investment portfolio with return Ri+ (θ−1)Rm−θRz=θ(Ri−Rz) + (1−θ)(Ri−Rm); (7.20) hence, this portfolio places weight 1 on asset i, weight θ−1 on the market portfolio, and weight −θon the zero-beta portfolio. Note that the expected value of (7.20) is
θ(μi−μz) + (1−θ)(μi−μm).
Let
θ0= μm−μi
μm−μz
and let
R0=θ0(Ri−Rz) + (1−θ0)(Ri−Rm).
Then
E(R0) = μm−μi
μm−μz
(μi−μz) +
1−μm−μi
μm−μz
(μi−μm)
= (μm−μi)(μi−μz) + (μi−μz)(μi−μm) μm−μz
= 0.
Thus, R0 is the return on a zero-investment portfolio that has zero expected return. Because the market portfolio is on the efficient frontier, it now follows from Corollary 5.2 that Cov(R0, Rm) = 0. Note that
Cov(R0, Rm) = Cov(Ri+ (θ0−1)Rm−θ0Rz, Rm)
= Cov(Ri, Rm) + (θ0−1)Var(Rm)
= (βi−(1−θ0)) Var(Rm). (7.21) Therefore,
βi= 1−θ0
and, using the expression forθ0,
βi= 1−μm−μi
μm−μz
= μi−μm
μm−μz
,
proving the result.
That is, a form of the SML holds withμzreplacingμf. The pricing model given by (7.19) is known as thezero-beta CAPM.
Note that Proposition 7.3 requires only that the market portfolio is on the efficient frontier, which is weaker than the condition that the market portfolio is the tangency portfolio required in Proposition 7.1. Hence, one might consider the possibility of weakening the conditions of Proposition 7.1 to require only that the market portfolio is on the efficient frontier, using the method of proof used in Proposition 7.3 with the risk-free asset playing the role of the zero-beta portfolio. However, in Proposition 7.3, it is important to keep
in mind that the efficiency of the market portfolio is with respect to all assets under consideration; if a risk-free asset is available, then the market portfolio must be efficient with respect to portfolios that include the risk-free asset.
Thus, such efficiency requires that the market portfolio is equivalent to the tangency portfolio; that is, it is not possible to use the approach of Proposition 7.3 to weaken the conditions used to establish the SML in Proposition 7.1.
Existence of the Zero-Beta Portfolio
Proposition 7.3 is based on the existence of the zero-beta portfolio; thus, we now show that such a zero-beta portfolio always exists, provided that the market portfolio is not the same as the minimum-variance portfolio. It may be shown that the market portfolio is equivalent to the minimum-variance portfolio if and only if all investors hold the minimum-variance portfolio.
Lemma 7.2. LetRmdenote the return on the market portfolio, with variance σ2m, and let Rmv denote the return on the minimum-variance portfolio, with variance σ2mv. If σ2m>σ2mv, then there exists a portfolio with return Rz such that Cov(Rz, Rm) = 0 and E(Rz)=E(Rm).
The returnRz may be written
Rz= 1
σ2mv−σ2m(σ2mvRm−σ2mRmv).
Proof. Consider the portfolio with return φRm+ (1−φ)Rmv for some real number φ. Recall that, according to Proposition 5.2, the covariance of Rmv with the return on any other portfolio is equal to Var(Rmv). Therefore,
Cov(Rmv, Rm) = Var(Rmv) =σ2mv. It follows that, for any real numberφ,
Cov(φRm+ (1−φ)Rmv, Rm) =φVar(Rm) + (1−φ)Var(Rmv)
=φσ2m+ (1−φ)σ2mv. Let
φz= σ2mv σ2mv−σ2m and define
Rz =φzRm+ (1−φz)Rmv. Then
Cov(Rz, Rm) =φzσ2m+ (1−φz)σ2mv= 0.
Note that because σ2mv<σ2m, φz<0 and the efficiency of the market portfolio implies that E(Rmv)<E(Rm). It follows that
E(Rz) = E(Rm) + (1−φz)[E(Rmv)−E(Rm)]<E(Rm);
that is, E(Rz)= E(Rm), as required.