Conditional performance evaluation refers to the measurement of performance of a managed portfolio taking into account the information that was available to investors at the time the returns were generated. An example of an unconditional measure is Jensen’s alpha based on the capital asset pricing model (CAPM). Unconditional measures may assign superior performance to managers who form dynamic strategies using publicly available information. Since any investor could have done the same (because the information is public) it is undesirable to label this as superior performance. In addition the distribution of returns on assets which managers invest in is known to change as the public information changes.
Recent empirical work has found that incorp orating public information variables such as dividend yields and interest rates is important in explaining expected returns. Conditional performance evaluation brings these insights to the portfolio performance problem. For instance, Ferson and Schadt (1996) assume that the beta conditional on a vector Zw of information variables has a linear functional form:
where is a vector of deviations of ZM from its mean vector. The coefficient 00,? is an average beta, and the vector Bp measures the response of the conditional beta to the information variables.
Applying this model of conditional beta to Jensen’s alpha regression equation yields the following model for conditional performance evaluation:
where the a? can now be interpreted as a conditional alpha. Ferson and Schadt find that the inclusion of conditioning information changes inferences slightly in that the distribution of alphas seems to shift to the right, the region of superior performance. This can be easily extended to the case of a model with multipl e factors (perhaps motivated by the APT) by including the cross products of each benchmark with the information variables.
Christopherson et al. (1996) make the additional extension of allowing the conditional alpha to vary with the information variables. They model alpha as a linear function of zM
which generates the modified model
They find that conditional models seem to have more power to detect persistence of performance relative to unconditional models.
In a recent paper Chen and Knez (1996) extend the theory of performance evaluation to the case of general asset pricing models. Modern asset pricing theory identifies models on the basis of the stochastic discount factors (SDFs) which they imply. For any asset pricing model the SDF is a scalar random variable mM such that for any claim which provides a (random) time t + 1 payoff of V+1 the price of the claim at time t is given by
where Q, is the public information set at time t. Suppose that there are N assets available to investors and that prices are non-zero. Since mt+ 1 is the same for all assets we have that
where Rt+1 is the vector of primitive asset gross returns (payoffs divided by price) and 1 is an N-vector of ones.
Let Rp denote the gross return on a portfolio formed of the primitive assets. Rp may be expressed as x’R. where x is a vector of portfolio weights. These weights may change over time according to the information available to the person who manages the portfolio. Suppose that this person has only public information. Then we can write x(Q() to indicate this dependence on the public information set. Such a portfolio must satisfy
since x depends only on Q( and the elements of x sum to one.
Since performance evaluation is involved with identifying managers who form portfolios using superior information (which is not in Q( at time t) it is natural to speak of abnormal performance as a situation in which the above does not hold. In particular define the alpha of a fund as
If we choose predetermined information variables Zt _ 1 as above and assume that these variables are in fiM, we can apply the law of iterated expectations to both sides of the above equation to obtain a conditional alpha measur e of performance. Unconditional performance evaluation amounts to taking the unconditional expectation.
Farnsworth et al. (1996) empirically investigate several conditional and unconditional formulations of mt+l including a SDF version of the CAPM, various versions of multifactor models where the factors are specified to be economic variables, the numeraire portfolio of Long (1990), and a primitive efficient SDF which is the payoff on a portfolio which is constructed to be mean-variance efficient (th is case is also examined in Chen and Knez, 1996). Their results showed that inference base d on the SDF formulation of the CAPM differ from those obtained using Jensen’s alpha approach even though the same market index was used.
Whether these results show that the SDF framework is superior is still an open question. Future research should try to determine if SDF models are better at pricing portfolios which are known to use only public information. If they do not then another reason must be found for the difference. It does appear that inclusion of conditioning information sharpens inferences on performance. Future work may help determine what information specifically should be included in order to perform conditional performance evaluation.
