# 4 The stochastic discount factor approach.

Modern asset pricing theory posits the existence of a stochastic discount factor, [m.sub.t+1], which is a scalar random variable, such that the following equation holds:

E([m.sub.t+1]pR.sub.t+1] - [1.bar]| [Z.sub.t]) = 0, (4.1)

where [R.sub.t+1] is the vector of primitive asset gross returns (payoff divided by price), [1.bar] is an N-vector of ones and [Z.sub.t] denotes the public information set available at time t. Virtually all asset pricing models may be viewed as specifying a particular stochastic discount factor, [m.subt+1]. The elements of the vector [m.sub.t+1][R.sub.t+1] may be viewed as "risk adjusted" gross returns. The returns are risk adjusted by "discounting" them, or multiplying by the discount factor, [m.sub.t+1], so that the expected "present value" per dollar invested is equal to one dollar. Thus, [m.sub.t+1] is called a stochastic discount factor (SDF). We say that a SDF "prices" the assets if Equation (4.1) is satisfied.

Re-arranging Equation (4.1) reveals that the expected return is determined by the SDF model as:

[E.sub.t]([R.sub.t+1]) = [[[E.sb.t]([m.sub.t+1])].sup.-1] + [Cov.sub.t]{-[m.sub.t+1]/[E.sub.t]([m.sub.t+1]);[R.sub.t+1]}, (4.2) where [Cov.sub.t] {.,.] is the conditional covariance given the information at time t and [[[E.sub.t]([m.sub.t+1])].sup.-1] is the risk-free or expected "zero-beta" return, known at time t. Thus, predicted excess returns differ across funds in proportion to the conditional covariances of their returns with the SDF.

Chen and Knez (1996) were the first to develop SDF alphas for fund performance. For a given SDF we may define a fund's conditional SDF alpha following Chen and Knez (1996) and Farnsworth et al. (2002) as:

[[alpha].sub.pt] = E([m.sub.t+1][R.sub.p,t+1] | [Z.sub.t]) - 1. (4.3)

Consider an example where [m.sub.t+1] is the intertemporal marginal rate of substitution for a representative investor: [m.sub.t+1] = u'([C.sub.t+1])/u'([C.sub.t]), where u_(C) is the marginal utility of consumption. In this case, Equation (4.1) is the Euler equation which must be satisfied in equilibrium. If the consumer has access to a fund for which the conditional alpha is not zero he or she will wish to adjust the portfolio, purchasing more of the fund if alpha is positive and less if alpha is negative.

The SDF alpha is the risk adjusted excess return on the fund, relative to that of a benchmark portfolio that is assumed to be correctly priced by the SDF. If [R.sub.Bt+1] is the benchmark, then Equation (4.3) implies [[alpha].sub.p]t = E([m.sub.t+1][[R.sub.p,t+1] - [R.sub.Bt+1]] | [Z.sub.t]).

The SDF alpha depends on the model for the SDF, and theoretically the SDF is not unique unless markets are complete. Thus, different SDFs can produce different measured performance. This mirrors the classical approaches to performance evaluation, where performance is sensitive to the chosen benchmark. For example, analogous to results from Roll (1978), for any SDF that prices the primitive assets there exists another SDF that also prices the primitive assets and re-orders the ranking of the nonzero performance measures across funds.

4.1 Relation to the Beta Pricing Approach

The SDF and traditional beta pricing methods are simply related, as the existence of an SDF that is linear in a set of factors is equivalent to the existence of a beta pricing model based on the same factors. In this case the difference between the two alphas is simply a matter of scale. To illustrate, assume that [m.sub.t+1] = a + [br.sub.mt+1]. Substituting into Equation (4.3), and expanding the expectation of the product into the product of the expectations and rearranging, we find that the SDF alpha in (4.3) is equal to E(m) times the alpha in the beta pricing model (see Ferson (1995, 2003) for more discussion).

The unconditional SDF alpha that is formed ignoring [Z.sub.t] is the unconditional mean of the conditional alphas, where the expectation is taken across the states. In this respect SDF alphas differ from beta pricing model alphas. The conditional SDF alpha given Z is [alpha](Z) = E(mR - 1|Z) and the unconditional alpha is [[alpha].sub.u] = E(mR - 1), so E([alpha](Z)) = [[alpha].sub.a]. The conditional alpha of a beta pricing model, in contrast, is the SDF alpha divided by the risk-free rate. When the risk-free rate is time varying, Jensen's inequality implies that the expected value of the conditional alpha in the beta pricing model is not the unconditional alpha. Ferson and Schadt (1996) find that average conditional alphas and unconditional alphas from beta pricing models can differ empirically for equity style funds.

4.2 Estimation of SDF Alphas

While the conditional SDF alpha, [[alpha].sub.pt], is in general a function of [Z.sub.t], it is simpler to discuss the estimation of [[alpha].sub.p] = E([[alpha].sub.pt]). A useful approach for estimating unconditional SDF alphas is to form a system of equations as follows:

[u1.sub.t] = [[m.sub.t+1][R.sub.t+1] - [1.bar]] [cross product] [Z.sub.t]

[u2.sub.t] = [[alpha].sb.p] - [[m.sub.t+1][R.sub.t+1] +1. (4.4)

The sample moment condition is g = [T.sup.-1][[SIGMA].sub.t]([u1'.sub.t],[u2'.sub.t]). We can use the Generalized Method of Moments (Hansen, 1982) to simultaneously estimate the parameters of the SDF model and the fund's SDF alpha.

The system (4.4) may also be estimated using a two-step approach, where the parameters of the model for [m.sub.t+1] are estimated in the first step and the fitted SDF is used to estimate alphas in the second step. Farnsworth et al. (2002) find that simultaneous estimation is dramatically more efficient. However, a potential problem with the simultaneous approach is that the number of moment conditions grows substantially if many funds are to be evaluated, and there are more funds than months in most studies.

Fortunately, Farnsworth et al. (2002) show that we can estimate the joint system (4.4) separately for each fund without loss of generality. Estimating a version of system (4.4) for one fund at a time is equivalent to estimating a system with many funds simultaneously. The estimates of ap and the standard errors for any subset of funds are invariant to the presence of another subset of funds in the system.