The Gibbs Posterior and Parametric Portfolio Choice

Abstract

Parametric portfolio policies may experience estimation risk. I develop a generalized Bayesian framework that updates priors, delivering a posterior distribution over characteristic tilts and out-of-sample returns that is the unique belief-updating rule consistent with the investor's utility function, requiring no model for the return generating process. The Gibbs posterior is the closest distribution to the prior in Kullback-Leibler divergence subject to utility maximization. The posterior's scaling parameter $λ$ controls the weight placed on data relative to the prior. I develop a KNEEDLE algorithm to select optimal $λ^*$ in-sample by trading off posterior precision against numerical fragility, eliminating the need for out-of-sample validation. I apply this to U.S. equities (1955-2024), and confirm characteristic-based gains concentrate pre-2000. I find that $λ^*$ varies meaningfully with risk aversion and depends on higher-order moments.

Figures (18)

• Figure 1: Figure 1. Identification Frontier$\Sigma$ is the posterior covariance matrix. I consider the -log det $\Sigma$ a function of $\Sigma$'s condition number, $\kappa$. I project -log det $\Sigma$ onto log $\kappa$ to obtain the slope $m$. I construct the second derivative with respect to $\kappa$ as $\tfrac{-m}{\kappa^2}$. This information deceleration is the vertical axis in this figure. I use a KNEEDLE method to identify $\lambda^*$. This is for the case of the 240 months ending in December 1996, and log utility.
• Figure 2: Figure 2. Identification Frontier$\Sigma$ is the posterior covariance matrix. I consider the -log det $\Sigma$ a function of $\Sigma$'s condition number, $\kappa$. I project -log det $\Sigma$ onto log $\kappa$ to obtain the slope $m$. I construct the second derivative with respect to $\kappa$ as $\tfrac{-m}{\kappa^2}$. This information deceleration is the vertical axis in this figure. I use a KNEEDLE method to identify $\lambda^*$. This is for the case of the 240 months ending in December 1996, and a power utility function with coefficient of relative risk aversion, $\gamma$, = 6.
• Figure 3: Figure 3a. Gibbs posterior on $\theta$, by year. Gibbs posterior of the $\theta$ coefficients for the log utility function. Posteriors are constructed conditional on the preceding 240 months of data, and $\lambda^*$. The "whiskers" show the 2.5%ile - 97.5%ile Gibbs posterior bands. The "box" shows the Gibbs posterior interquartile range, and the bar inside the box is the Gibbs posterior median.
• Figure 4: Figure 3b. Gibbs posterior on $\theta$, by year. Gibbs posterior of the $\theta$ coefficients for the log utility function. Posteriors are constructed conditional on the preceding 240 months of data, and $\lambda^*$. The "whiskers" show the 2.5%ile - 97.5%ile Gibbs posterior bands. The "box" shows the Gibbs posterior interquartile range, and the bar inside the box is the Gibbs posterior median.
• Figure 5: Figure 4a. Gibbs posterior on $\theta$, by year. Gibbs posterior of the $\theta$ coefficients for the power utility function with $\gamma = 2$. Posteriors are constructed conditional on the preceding 240 months of data, and $\lambda^*$. The "whiskers" show the 2.5%ile - 97.5%ile Gibbs posterior bands. The "box" shows the Gibbs posterior interquartile range, and the bar inside the box is the Gibbs posterior median.