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Financial Markets and Portfolio Management

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01-11-2014

Why not use SDF rather than beta models in performance measurement?

Authors: Jonas Gusset, Heinz Zimmermann

Published in: Financial Markets and Portfolio Management | Issue 4/2014

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Abstract

This paper analyzes performance measurement based on stochastic discount factors, compared to beta models traditionally used in computing funds’ (Jensen) alphas. From a theoretical point of view, standard alphas suffer from several limitations. Our paper addresses this issue from an empirical point of view using a sample of Swiss mutual funds from 2000 to 2011. Our results suggest that the key for a “fair” comparison between stochastic discount function (SDF) and beta models is the specification of the set of primitive assets used to calibrate the SDF function. Once this is established, the size of (absolute) performance differences considerably decreases between the two model families. However, there are sizeable performance deviations in the cross-section of funds if conditioning information is incorporated in the tests, up to some 20 basis points per month, or about 2.3 % per year. In almost all cases, the SDF-alphas are lower than the standard (Jensen) alphas. In absolute terms, the average SDF-based underperformance of the funds is way larger than the average total expense ratio (TER) of the funds, both in a conditional and unconditional setting.

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Stochastic discount factor models have been used in performance measurement by Chen and Knez (1996), Dahlquist and Söderlind (1999), and Ferson et al. (2006).

In order to avoid confusion, excess performance based on beta models is called “alpha” in this paper, while excess performance based on SDF-models is called “SDF-alpha” or SDF pricing error.

The relationship is unambiguous only at the margin, i.e., for infinitesimal portfolio changes (Dybvig and Ross 1985b). However, as shown by Gibbons et al. (1989), in Table 6, an asset with a positive (Jensen) alpha can exhibit a negative weight in the optimal orthogonal portfolio (OOP). [The OOP is a portfolio allocated to a set of mispriced assets, which portfolio weights proportional to the (Jensen) alphas computed with respect to a (possibly inefficient) benchmark portfolio. The OOP can be regarded as the “optimal” orthogonal complement of the benchmark portfolio which, with appropriate weights, can be combined to an efficient portfolio; see Jobson and Korkie (1982) or MacKinlay (1995) for applications to active management and asset pricing.] Ferson (2010) concludes from this observation: “So, even if a positive alpha is attractive at the margin to a mean variance investor, it might not imply buying a positive alpha fund given a realistic discrete response.” In contrast, the sign of the SDF-alpha is unambiguous with respect to the marginal expected utility from discrete changes in assets’ weights.

Respectively, with minimum pricing error.

See: Jagannathan and Wang (2002), and Kan and Zhou (2002).

The logarithmic yield is computed because of superior time series properties (Lewellen 2004).

The results are not different if the MSCI Switzerland TR market returns are used.

As a robustness test, we substitute the SPI by the MSCI Switzerland TR Index as benchmark portfolio in the CAPM and FF model. The results are very similar. The Spearman rank correlation coefficients between the alphas are 0.9842 and 0.9981 for the unconditional and conditional models, respectively.

To simplify notation, we skip time indices in this section.

Remember that the conditional SDF refers to the scaling of risk factors, while the un/conditional HJD refers to the scaling of primitive returns.

However, these performance discrepancies strongly depend on the applied model, i.e., the performance differences are much less pronounced when we use, for example, the Fama and French three-factor model.

The subsequent figures are based on the SDF_CAPM(9SPI) specification.

Notice that \({E}( {{m}_{{t}+1} })=1/{E}( {{R}_{{f},{t}+1} })\), where \({R}_{{f},{t}+1} \) is the gross return on the risk-free security. Including the risk-free asset in the set of primitive assets is suggested by Dahlquist and Söderlind (1999) and Farnsworth et al. (2002). They show that the incorporation of the risk-free security in the set of primitive assets helps to correctly identify the conditional mean of the SDF.

In general, the alphas from the beta model and the alphas from the SDF model in (6) differ in the unconditional as well as in the conditional setting. The issue is discussed in detail in Söderlind (1999). We denote both alphas by \(\alpha _{P} \) to keep notation simple.

A potential problem of the simultaneous GMM procedure is that the number of moment conditions grows substantially with the number of funds to be evaluated. Thus, we estimate the system (7) separately for each fund. Nevertheless, Farnsworth et al. (2002) prove that separate estimation is not restrictive as the estimated fund’s alpha and its standard error are invariant to the number of funds in the system. Estimating the system for one fund at a time is therefore equivalent to estimating the system with all funds in the sample simultaneously.

We only specify models where factors are returns or excess returns on traded assets (spread portfolios, mimicking portfolios).

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Title: Why not use SDF rather than beta models in performance measurement?
Authors: Jonas Gusset
Heinz Zimmermann
Publication date: 01-11-2014
Publisher: Springer US
Published in: Financial Markets and Portfolio Management / Issue 4/2014
Print ISSN: 1934-4554
Electronic ISSN: 2373-8529
DOI: https://doi.org/10.1007/s11408-014-0235-z

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