A Brilliantly Formulated Asset Pricing Model can be Intrinsically Flawed

What is it about?

Suppose there exist two competing formulaic approaches to the calibration of the performance of the same entity. Absent the existence of an independently and objectively formulated measure of the appropriateness and robustness of the two competing approaches, which of the two competing approaches ought to be preferred to the other is nigh impossible to discern. Suppose the entity in question is the stock market and that the objective of the competing formulaic approaches is an assessment of the robustness with which the assets in the market have been priced by investors. Since stock markets are meaningfully populated only whenever each listed stock possesses some unique risk-return properties that distinguishes it from any other competing stock, one formulaic approach outperforms others only whenever it's formulation is most robust to all of the different ways in which one stock can uniquely differ from others. By natural extension, a measure that is robust to the detection of the formulaic approach that outperforms all others also must be robust to all of the different ways in which one stock can uniquely differ from others. This study arrives at the very first measure - a distance measure - which robustly juxtaposes the relative performance of different heterogeneous formulations of asset pricing models. The importance of the availability of such a measure is, perhaps, first clearly articulated in MacKinlay (1995), then more recently reiterated in Kozak et al. (2018). The evidence that the distance measure - which is formally and theoretically developed in this study - is the canonical (best) approach to the juxtaposition of the performance of competing asset pricing models, but yet a measure that had yet to be developed in any prior study, is articulated in studies, such as, Vuong (1989), Pastor (2000), or Barillas and Shanken (2018). For evidence that, feasibly, heterogeneous specifications of asset pricing models yield very different outcomes, see Pástor and Stambaugh (2000).

Featured Image

Photo by Jackson Jost on Unsplash

Photo by Jackson Jost on Unsplash

Why is it important?

Ideally any measure that is robust to a juxtaposition of the robustness of any two formulaic approaches to the parsing of the performance of an entity juxtaposes some specific output from each competing model that either is already known or can be shown to be a metric for the robustness of any and all competing models. Prior to this study, no such measure had yet to be developed. For concreteness, none of the prior studies which juxtapose the robustness of heterogeneous specifications of asset pricing models juxtapose some output from the competing models. Rather, the studies generate a post-implementation statistic - e.g. the Likelihood Ratio (LR) or ANOVA - that is specific to each model, then compare the specific realizations of the statistic that are generated by the competing models - a reasoned approximation to the development of a measure, but yet one that falls short of the ideal. The use of the LR or ANOVA falls short of the ideal, because either is affected by the specific factors that are incorporated into each competing model. But if the competing models are, rather necessarily, required to be heterogeneously formulated, there is arrival at an endogeneity, namely that the performance of each model is not independent of the number of parameters, appropriate or otherwise, that are incorporated into each competing model. In presence of the enumerated caveat, typically such empirical structures only juxtapose models that are nested in each other, a restriction which prevents the most interesting comparisons of asset pricing models, namely models whose structures are totally different. In addition to the enumerated weakness, none of the prior studies are able to apply just one stock or one portfolio of stocks to a juxtaposition of the robustness of the heterogeneously formulated asset pricing models. Rather all stocks of interest are pooled into one grand empirical structure; the extent, as such, to which the competing models are robust to the pricing of the different stocks is unknown; ideally, however, as is argued in Pastor (2000), it is inferences in respect of the variability of the robustness with which the competing models price either different stocks or different portfolios that is the pragmatic outcome of a juxtaposition of the different competing models. For additional discussions of the weaknesses of the enumerated hitherto approaches, see studies, such as, Zhou (1991), Shanken (1985), or Raponi, Robotti, and Zaffaroni (2020). This study applies the error terms that are spanned by each competing asset pricing model towards the generation, using a regression model, of a distance measure. A positive distance measure, which is indicated by a positive partial effect, indicates that the reference model outperforms the alternate competing model (for details, see the manuscript). The formal theory that spans the distance measure, equivalently the empirical structure, establishes that a positive partial effect indicates the reference model prices agents' rationality (the first regression model that is motivated) or agents' systemic priors, e.g. trading schedules (the second regression model that is motivated), or both, better than the competing model. In presence of the newly formulated empirical structure, so long as each model in a menu of competing asset pricing models incorporates a market portfolio factor, the relative robustness of all are directly comparable. In stated respect, in presence of 50 competing models, the robustness of the models with reference to either a specific asset or specific portfolio can be ranked from 1.00 to 50.00. For concreteness, regardless whether an empirically formulated asset pricing model is an affine model, a stochastic volatility model, or a stochastic volatility plus jumps model, so long as each competing model incorporates a market portfolio factor as an asset pricing factor, the relative robustness of each model is directly and robustly decipherable. With models that are juxtaposed not then restricted to be nested in each other, there is arrival, not only at an ideological (formal theoretical) improvement, but also at a pragmatic improvement. The pragmatic value of this study's distance measure has articulation as follows. Suppose an agent seeks to hold the S&P500. Feasibly, there exists some asset pricing model that is most robust to the pricing of the S&P500. It is solely this study's distance measure that can robustly sensitize such an agent to the asset pricing model that is most robust to the pricing of the S&P500. Suppose, contrarily, that the same agent has interest in some new stock, such as, Palantir. Ab initio, the probability, to wit, the model that best prices Palantir does not coincide with the model that best prices a diversified portfolio, such as the S&P500 is high, is greater than 50 percent. Yet again, it is solely this study's distance measure that can sensitize the agent to the need to switch from some Model A to some alternate Model B. Concurring with the claims in the immediate preceding, whereas Pástor and Stambaugh (2000) are able to show that heterogeneous specifications of asset pricing models yield heterogeneous outcomes, the study is unable to rank the relative robustness of the competing heterogeneous models. A revisit of that study, namely the imposition of this study's distance measure on that study's menu of competing asset pricing models will, however, in a new complementing study (were some agent to engage in such an effort), facilitate a ranking of the competing models.

Perspectives