Ebook Evaluating Asset Pricing Models Using the Second Hansen Jagannathan Distance
The fundamental theorem of asset pricing, one of the cornerstones of neoclassical finance, establishes the equivalence between the absence of arbitrage and the existence of a positive stochastic discount factor (hereafter SDF) that correctly prices all assets. The main purpose of this paper is to develop asset pricing tests that fully reflect the implications of the fundamental theorem of as set pricing. Specifically, we develop a systematic approach for estimating, testing, and comparing asset pricing models based on the second Hansen Jagannathan distance (hereafter HJD).
The first and second HJDs developed by Hansen and Jagannathan (1997) measure specification errors of SDF models by least squares distances between an SDF model and the set of admissible SDFs that can correctly price a set of test assets. The first HJD considers the set of all admissible SDFs, which we denote as. The second HJD considers only the smaller set of strictly positive admissible SDFs, which we denote as. The positivity constraint of the second HJD guarantees the admissible SDFs to be arbitrage free and is important for pricing derivatives associated with the test assets. Hansen and Jagannathan (1997) show that while the first HJD represents the maximum pricing error of a portfolio of the test assets with a unit norm, the second HJD represents the minmax bound of the pricing errors of a portfolio of both the test assets and their related derivatives with a unit norm. Obviously, the second HJD represents a more stringent criterion for evaluating asset pricing models and is generally larger than the first HJD.
There are important advantages of using the second HJD in evaluating asset pricing models, although the existing empirical literature has mainly focused on the first HJD. Conceptually, the second HJD reflects more fully the implications of the fundamental theorem of asset pricing than the first HJD. While the first HJD only tests whether an SDF model can correctly price the test assets, the second HJD further tests whether the SDF model is strictly positive. As a result, the second HJD is more powerful than the first HJD in detecting misspecified SDF models that can price the test assets but are not strictly positive, a situation that is especially likely to happen to linear factor models.
Dybvig and Ingersoll (1982) show that linear asset pricing models are not arbitrage free and are not appropriate for pricing derivatives because their SDFs take negative values in certain states of the world. Although linear factor models are seldom used directly to price derivatives, they have been widely used in performance evaluation of mutual funds and hedge funds. Mutual funds and hedge funds often employ dynamic trading strategies which generate option2like payoffs.( Most hedge funds directly trade derivatives, and their returns exhibit option2like features.) Grinblatt and Titman (1987), Glosten and Jagannathan (1994), Ferson and Khang (2002), and Ferson, Henry, and Kisgen (2006), among others, have emphasized the importance of the positivity constraint for mutual fund performance evaluation. The fast growing hedge fund industry and the need to evaluate hedge fund performances further highlights the significance of the second HJD for empirical asset pricing studies.
Even for applications that do not involve derivatives, there are still important advantages of using the second HJD in estimating and comparing asset pricing models. For example, one is likely to obtain more robust and reliable parameter estimates using the second HJD than the first HJD, especially for linear factor models. The first HJD chooses the parameters of a linear model to minimize the pricing errors of the test assets. However, such estimated models could be far from, since in many cases the estimated SDF models have to take negative values with high probabilities to price the test assets. Therefore, models estimated using the first HJD are likely to overfit the test assets and to perform poorly out of sample. The second HJD helps to overcome the overfitting problem because it chooses parameters to minimize the distance between an SDF model and. As a result, the second HJD provides more realistic assessments of model performance and leads to estimated SDF models that are closer to.
Moreover, the second HJD is more powerful than the first HJD in distinguishing the relative performances of models that have small pricing errors of the same set of test assets. According to Lewellen, Nagel, and Shanken (2006) (hereafter LNS), it is difficult to differentiate models that have been developed to explain the cross sectional returns of the 25 size and book to market (hereafter BM) portfolios of Fama and French (1993) using traditional methods, because these models tend to have small pricing errors for the test assets by construction. While LNS (2006) suggest several interesting ways to improve the traditional methods, the second HJD represents a powerful measure of relative model performances. SDF models with similar pricing errors for the Fama French portfolios may have very different probabilities in taking negative values and thus can be differentiated based on the second HJD. In addition, since the second HJD measures the distance between an SDF model and, a model with a smaller second HJD is likely to be a better model. Given that most asset pricing models are approximations of reality and likely to be misspecified, it is important to have a powerful measure like the second HJD to compare the relative performances of misspecified models.
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