5. Stochastic Discount Factor

Suppose a model tells us a particular SDF \(m\). For example, the consumption-based SDF from the CRRA utility in (1.7):

with two unknown parameters \(\beta,\gamma\). Let the payoff space be spanned by gross returns \(\mathcal X_{t+1}=\{1+r_t^{1,t+1},\,1+r_t^{2,t+1},\dots,1+r_t^{N,t+1}\}\). Then the fundamental pricing equation is

Define the pricing error that calibrates the consumption-based SDF:

The correct \((\beta,\gamma)\) should make its unconditional expectation zero. Using instrument variables \(\mathbf y_t\) in the period-\(t\) information set, construct the unconditional moment restrictions

and estimate \(\beta,\gamma\) by GMM (Generalized Method of Moments). See the later chapter for GMM details.

Proposition 5.1. Let the basis of the payoff space \(\mathcal X\) include \(N\) risky payoff vectors \(\mathbf x\) and one risk-free asset with payoff \(1+r_f\) and price \(1\). Then there is a unique SDF \(x^*\in\mathcal X\), given in closed form.

Proposition 5.2. Normalizing payoffs by their prices, consider the gross-return payoff space

with \(R_i=x_i/p(x_i)\). Let the risk-free gross return be \(R_f\) and \(\mathbf R=[R_1,\dots,R_N]'\). Then the SDF \(x^*\in\mathcal X_R\) is pinned down by Proposition 5.1:

Rewriting (5.7) shows \(x^*\) is an affine function of the gross return of the mean-variance tangency portfolio:

so that \(x^*=\alpha+\beta R_{ee}\), where \(R_{ee}\) is the gross return of the mean-variance frontier tangency portfolio.

From (5.4), compute the variance of \(x^*\):

Minimum-variance property. Any admissible SDF \(m\) can be written \(m=x^*+\varepsilon\), with \(\mathbb E[\varepsilon x]=0\ \forall x\in\mathcal X\) and \(\mathbb E[\varepsilon]=0\) (since the constant \(1\in\mathcal X\)):

So \(x^*\) is the minimum-variance SDF (equivalently the minimum second-moment portfolio).

Theorem 5.1 (Hansen–Jagannathan 1991). The ratio of the standard deviation of the SDF to its mean is no smaller than the absolute Sharpe ratio attainable by any portfolio.

Figure 5.1: Mean-variance frontier for gross returns (left) and for SDFs (right). On the left, the tangent line through \(1/\mathbb E[m]\) has slope equal to the maximum absolute Sharpe ratio; on the right the same bound is drawn in the \((\mathbb E[m],\sigma_m)\) plane — any given \(\mathbb E[m]\) implies a risk-free rate \(R_f=1/\mathbb E[m]\), and feasible SDFs must lie above the hyperbola.

Since consumption-growth volatility \(\sigma_c\) is tiny, meeting the bound forces an implausibly large risk aversion \(\gamma\) — this is the equity premium puzzle.

(5.13) and (5.17) look like two different bounds but are equivalent. First, from (5.11) and (5.13),

For comparison, normalize each basis payoff by its price into gross returns \(\mathbf R\) (so \(p(R_j)=1\)) (5.22), and substitute \(\mathbb E[x^*]=1/R_f\) into (5.21):

Then plug the maximum Sharpe ratio \(\sqrt{(\mathbb E[\mathbf R]-R_f\mathbf 1)'\Sigma_{\mathbf R}^{-1}(\mathbb E[\mathbf R]-R_f\mathbf 1)}\) into (5.17) and simplify:

(5.23) and (5.24) are identical — the bound from the SDF projection structure and the bound from the return Sharpe ratio are equivalent, and yield the same calculation in estimation.

Each period has a unique (conditional) SDF \(x^*_{t+1}\in\mathcal X_t\). A general SDF is the marginal-utility ratio

To raise the SDF's volatility toward the H–J bound, one sometimes multiplies (5.25) by an extra random term \(g_{t+1}=1+\varepsilon_{t+1}\) (with \(\varepsilon_{t+1}\) orthogonal to \(\mathcal X_t\) and mean zero):

\(g_{t+1}\) can be estimated, but lacks a clear economic foundation — the main drawback of such preference-shock specifications.

With several candidate SDFs (different utility forms), how do we pick the best? Hansen and Jagannathan (1997) propose the distance from a candidate SDF \(y\) to the closest true SDF \(m\) as the criterion:

subject to:

\(\boldsymbol\alpha_{\mathbf x}\) are the pricing errors from using \(y\) as the SDF; the larger \(\delta_g\), the worse the candidate model.

Theorem 4.2 gives the unique SDF \(x^*\in\mathcal X\) inside the payoff space. Its gross return is

Since \(x^*\in\mathcal X\), \(R^*\) is the gross return of an asset on the frontier (shown below to be the minimum second-moment frontier portfolio).

Figure 5.2: Mean-variance frontier. (a) without a risk-free asset; (b) with a risk-free asset, the frontier becomes the capital allocation line (CAL) through \(R_f\). The \(N\) individual assets scatter inside the frontier; \(R^*\) corresponds to the minimum second-moment portfolio.

Define the excess return space as the set of zero-price payoffs:

i.e. projecting the constant \(1\) onto \(\mathcal Z\) gives \(R^{e*}\).

Treat the original payoff space \(\mathcal X\) as the equivalent gross-return payoff space \(\mathcal X_R\) (5.37); we can then do mean-variance analysis on all its gross returns.

Theorem 5.2 (three-way decomposition). For \(\forall R^i\in\mathcal X_R\),

where \(R^*\) is given by (5.32), \(\omega^i\) is a constant specific to return \(i\), and \(\eta^i\in\mathcal X\) is orthogonal to both \(R^*\) and \(R^{e*}\).

Theorem 5.3. \(R^*\) is the minimum second-moment frontier portfolio. Lemma 5.1. The minimum second-moment portfolio lies on the mean-variance frontier.

Any frontier portfolio has \(\eta^i=0\): if \(\eta^{mv}\neq0\), then \(\hat R^{mv}=R^*+\omega^{mv}R^{e*}\in\mathcal X_R\) has the same mean as \(R^{mv}\) but strictly smaller variance, contradicting that \(R^{mv}\) is on the frontier. Hence

Theorem 5.4. Any portfolio \(R^{mv}\) on the mean-variance frontier is a linear combination of \(R^*\) and \(R^{e*}\): \(R^{mv}=R^*+\omega^{mv}R^{e*}\).

Proposition 5.3. For \(\forall R^i\in\mathcal X_R\), \(\mathbb E[R^i]\) admits a Beta pricing representation with respect to \(R^*\).

Proposition 5.4. For \(\forall R^{mv}\in\mathcal X_R\), \(R^{mv}=R^*+\omega^{mv}R^{e*}\) if and only if returns admit a Beta pricing representation with respect to \(R^{mv}\).

Introduce a filtration \(\mathcal F_0\subset\mathcal F_1\subset\cdots\subset\mathcal F_t\). In an incomplete market, estimating a conditional expectation from asset information carries error, but it suffices that the error be orthogonal to the \(\mathcal F_t\) information: \(\mathbb E[(z_{t+1}-y_t)z_t]=0\), where \(y_t\) is an \(\mathcal F_t\)-measurable random variable.

Conditionalize pricing period by period. The (conditional) price of security \(y_{t+1}\) given \(\mathcal F_t\) is

By the law of iterated expectations, \(\mathbb E[p_t(y_{t+1})]=\mathbb E[x^*_{t+1}y_{t+1}]\) — conditional pricing agrees with unconditional pricing in expectation. The gross return of the conditional SDF is

Similarly define the conditional excess-return space \(\mathcal Z_t=\{z_{t+1}\in\mathcal X_t:\ p_t(z_{t+1})=0\}\) and obtain \(R^{e*}_{t+1}\) by projection: \(\mathbb E[(1-R^{e*}_{t+1})z_{t+1}\mid\mathcal F_t]=0\) (5.47). Analogous to Theorem 5.4, any portfolio on the conditional mean-variance frontier is a linear combination of \(R^*_{t+1}\) and \(R^{e*}_{t+1}\):

where \(\omega^{mv}_t\) is a time-varying constant.

The unconditional CAPM is, from (2.25),

Conditionalizing it period by period gives the conditional CAPM (w.r.t. the conditional mean-variance frontier):

Taking unconditional expectations of both sides of (5.50) leaves an extra covariance term between \(\beta^i_t\) and the conditional premium:

• When the covariance of \(\beta^i_t\) with the conditional market premium \(\pi_t\) is non-zero, the conditional CAPM holding does not imply the unconditional CAPM — explaining why conditional models can fit data while unconditional ones are empirically rejected.
• For the unconditional CAPM to also hold, \(\beta^i_t\) must be uncorrelated with \(\pi_t\) (e.g. both constant).
• Time-varying parameters (time-varying \(\beta\), time-varying premium) thus enter asset pricing, motivating the conditional factor models and time-varying loadings of Section 6.3.