6. Discount Factors, Betas, and Mean-Variance Frontiers

The figure below summarizes every path between the three views. A discount factor, a reference variable for betas, and a frontier return all carry the same information; given any one, you can find the others.

Figure 6.1　Relation between three views of asset pricing. Law of one price \(\Rightarrow m\) exists; \(E(RR')\) nonsingular \(\Rightarrow R^{mv}\) exists; each arrow is a construction from one representation to another.

Historically, Roll (1977) pointed out the link between the mean-variance frontier and beta pricing; Ross (1978) and Dybvig and Ingersoll (1982) the link between linear discount factors and beta pricing; Hansen and Richard (1987) the link between a discount factor and the frontier.

6.1 From Discount Factors to Beta Representations

\(m,x^*,R^*\) can each be the single factor in a single-beta representation. Using \(m\). From \(1=E(mR^i)=E(m)E(R^i)+\operatorname{cov}(m,R^i)\), multiply and divide by \(\operatorname{var}(m)\) and set \(\gamma\equiv1/E(m)\):

This gives a single-beta representation. Key point: the consumption model can equivalently be stated as "mean returns are linear in the regression betas on \((c_{t+1}/c_t)^{-\gamma}\)," and the slope \(\lambda_m\) is not a free parameter (it should equal that factor's variance-to-mean ratio). The factor risk premium for marginal utility growth is negative, \(\lambda_m<0\): positive expected returns go with positive correlation with consumption growth, hence negative correlation with marginal utility \(m\).

Using \(x^*\) and \(R^*\). It is often desirable to use a payoff as the factor rather than a real variable like consumption growth (avoiding measurement problems). Since \(x^*=\operatorname{proj}(m|X)\) is still a discount factor, \(1=E(mR^i)=E(x^*R^i)=E(x^*)E(R^i)+\operatorname{cov}(x^*,R^i)\), which solves to

Because \(R^*\) is the minimum-second-moment return on the lower frontier, it has a negative expected excess return (factor risk premium \(\lambda_{R^*}=-\operatorname{var}(R^*)/E(R^*)<0\)); \(\gamma=1/E(x^*)\) is its zero-beta rate. Special cases to rule out: \(E(m),E(x^*),E(R^*)\) cannot be zero (else a zero or infinite risk-free rate); \(\operatorname{var}(m)\) etc. cannot be zero (pure risk neutrality, where all expected returns equal the risk-free rate).

6.2 From Mean-Variance Frontier to a Discount Factor and Beta Representation

Geometric intuition. \(m=x^*\) points the same way as \(R^*\), and the frontier is \(R^*+wR^{e*}\). Take a frontier \(R^{mv}\), stretch it (\(bR^{mv}\)), then subtract some of the 1 vector (\(a\)) — since \(R^{e*}\) is generated by the unit vector, this kills the \(R^{e*}\) component and returns to \(x^*\). If \(R^{mv}\) were off the frontier, \(a+bR^{mv}\) (with \(b\ne0\)) would point partly in the \(n\) direction, which \(x^*\) does not — hence "must be on the frontier." Algebraically, set \(m=a+b(R^*+wR^{e*}+n)\), fix \(a,b\) by pricing \(R^*\) and \(R^{e*}\), then verify that pricing an arbitrary \(x^i=y^iR^*+w^iR^{e*}+n^i\) requires \(E(nn^i)=0\) for all \(x^i\Rightarrow n=0\). Chaining this section with §6.1 takes us from the frontier to a beta model.

6.3 Factor Models and Discount Factors

The cleanest case is when all test assets are excess returns. Normalize \(E(m)=1\), i.e. \(m=1+[f-E(f)]'b\). Given \(0=E(mR^e)\), expanding gives \(E(R^e)=-\operatorname{cov}(R^e,f')b\), and "from covariance to beta" (multiply by \(\operatorname{var}(f)^{-1}\operatorname{var}(f)\)):

where \(\beta\) are the multiple-regression coefficients of excess returns on the factors. When the test assets are returns (not excess), one must additionally track the constant in \(m\) and the zero-beta rate in the beta model, giving \(\gamma\equiv1/E(m)=1/a\) and \(\lambda\equiv-\gamma E[mf]\). Given either form one can construct the other (not uniquely: one may add to \(m\) anything orthogonal to returns, or add spurious factors with zero \(\beta\) and \(\lambda\)).

Equation (6.12) shows the factor risk premium \(\lambda\) is the price of the factor: testing \(\lambda\ne0\) is testing whether "the factor is priced." More precisely \(\lambda\) is the factor's price less its risk-neutral valuation; if the factor is a return with price one, \(\lambda=E(f)-\gamma\). Note the "factors" need not be returns, need not be orthogonal, and need not be serially uncorrelated or mean-zero — these are special cases, not requirements for a factor pricing representation.

Factor-mimicking portfolios. When factors are not returns, one often uses their mimicking portfolios \(f^*=\operatorname{proj}(f|X)\) (or the corresponding returns/excess returns). \(m=b'f^*\) carries the same pricing information on \(X\) as \(m=b'f\), since \(E(b'fx)=E[b'(\operatorname{proj}f|X)x]=E[b'f^*x]\); it gives a beta representation too, now with betas being regression coefficients on \(f^*\) rather than \(f\).

6.4 Discount Factors and Beta Models to the Mean-Variance Frontier

The reverse also holds: given \(m\), construct \(x^*=\operatorname{proj}(m|X)\) and \(R^*=x^*/E(x^{*2})\) — the minimum-second-moment return, on the frontier. If a beta model holds (\(m=b'f\)), then \(x^*\) is linear in \(f^*=\operatorname{proj}(f|X)\), so \(R^*\) is linear in the factor-mimicking payoffs/returns — i.e. a linear combination of factor-mimicking portfolios is on the frontier. With one risk-free-rate proxy added, any frontier return is a linear function of these factor returns.

6.5 Three Risk-Free Rate Analogues

With no risk-free rate (the unit payoff not in the payoff space), three returns each retain one property of the risk-free rate; all are on the frontier in \(R^*+wR^{e*}\) form:

• Zero-beta return \(R^{\gamma}\): the frontier return uncorrelated with \(R^*\); its mean is the zero-beta rate \(\gamma\) (found in Figure 6.3 by extending the tangency at \(R^*\) to the vertical axis). Note generally \(\gamma\ne1/E(m)\): projecting \(m\) onto \(X\) preserves pricing on \(X\) but not predictions outside \(X\) (like the risk-free rate).
• Minimum-variance return: \(R^{\min}=R^*+\dfrac{E(R^*)}{1-E(R^{e*})}R^{e*}\), retaining the property that its weight on \(R^{e*}\) equals its own mean (\(R^{\min}=R^*+E(R^{\min})R^{e*}\)).
• Constant-mimicking return: \(R=\operatorname{proj}(1|X)/p[\operatorname{proj}(1|X)]\), the return closest to the unit payoff, with weight \(\gamma\): \(R=R^*+\dfrac{E(R^{*2})}{E(R^*)}R^{e*}\).

Figure 6.3　Zero-beta rate \(\gamma\) and zero-beta return \(R^{\gamma}\) for \(R^*\). \(\gamma=E(R^{*2})/E(R^*)=1/E(x^*)\), found by extending the tangency at \(R^*\) to the vertical axis; it lies above the minimum-variance return.

With a risk-free rate, all three reduce to \(R^f=R^*+R^fR^{e*}\), and \(R^f=\dfrac{E(R^{*2})}{E(R^*)}=\dfrac{E(R^*)}{1-E(R^{e*})}=\dfrac{\operatorname{var}(R^*)}{E(R^*)E(R^{e*})}\).

6.6 Mean-Variance Special Cases with No Risk-Free Rate

The special cases all stem from "\(E(m)\), the price of a unit payoff, or the risk-free rate must not be zero or infinite." In a complete arbitrage-free market \(m>0\) so \(E(m)>0\); with a risk-free rate \(E(m)=1/R^f\) is finite. In an incomplete market you might happen to pick an \(m\) with \(E(m)=0\) — just pick another. Specifically:

• Frontier to discount factor: any frontier return works except the constant-mimicking return (it gives infinite \(m\)).
• Frontier to beta model: any frontier return works as reference except the minimum-variance return (it gives an infinite zero-beta rate). This is the famous Roll (1977), Hansen-Richard (1987) theorem.

The two exclude different returns, reflecting the distinction between the minimum-variance return (projecting returns on 1; closest to extensions of 1) and the constant-mimicking return (projecting 1 on the payoff space; the payoff closest to 1).

Summary

The three languages — discount factor, beta, mean-variance frontier — carry the same information and are freely interchangeable (Figure 6.1). The most important bridge is §6.3: a factor model \(\iff m=b'f\), folding CAPM/APT/ICAPM into the \(p=\mathbb E(mx)\) framework. The empirical implications of these equivalence theorems come in Chapter 7.

Problems

1. In the argument that \(R^{mv}=R^*+wR^{e*}\) implies \(m=a+bR^{mv}\), must we rule out risk neutrality? (Hint: what is \(R^{e*}\) when the economy is risk neutral?)
2. With factor-mimicking portfolios (6.13), the predicted expected returns match those from the factors themselves. But are the \(\gamma^*,\lambda^*,\beta^*\) the same as the original \(\gamma,\lambda,\beta\)?
3. Suppose the CAPM holds, \(m=a-bR^m\) prices a set of assets, and there is a risk-free rate \(R^f\). Express \(R^*\) in terms of the moments of \(R^m,R^f\).
4. If you express the frontier as a linear combination of factor-mimicking portfolios, do their relative weights change as you sweep the frontier, or stay the same? (Start with the risk-free-rate case.)
5. For an arbitrary frontier return \(R^*+wR^{e*}\), find its zero-beta return and zero-beta rate, and show your rate reduces to \(R^f\) when there is one.
6. When the economy is risk neutral with no risk-free rate, show the zero-beta, minimum-variance, and constant-mimicking returns are again equivalent (but not equal to the risk-free rate). What is \(R^*\) in this economy?