# Bayesian Nonparametric Learning of How Skill Is Distributed across the Mutual Fund Industry

By Fisher, Mark; Jensen, Mark J. et al. | Federal Reserve Bank of Atlanta, Working Paper Series, March 1, 2019 | Go to article overview

# Bayesian Nonparametric Learning of How Skill Is Distributed across the Mutual Fund Industry

Fisher, Mark, Jensen, Mark J., Tkac, Paula, Federal Reserve Bank of Atlanta, Working Paper Series

1 Introduction

Since the seminal article by Jensen (1968) estimating the skill level of managed mutual funds has been widely researched and debated (see Elton & Gruber 2013). In addition to measuring the skill of a fund, others have investigated how skill is the distributed across the industry. For instance, Kosowski et al. (2006), Barras et al. (2010), Fama & French (2010) and Ferson & Chen (2015) take a frequentist approach and estimate the population distribution by bootstrapping the estimated skill of the funds. Both Chen et al. (2017) and Harvey & Liu (2018) model the population distribution with a finite mixture of normals and estimate the mixture parameters with an EM algorithm. Barras et al. (2018) use a nonparametric method to estimate the population distribution but do not use the information from the population in the estimation of a fund's skill.

Jones & Shanken (2005) estimate the population distribution from a parametric Bayesian perspective using a hierarchical normal prior for skill. Others like Pastor & Stambaugh (2002 b) assume a distribution for the population. Baks et al. (2001), Pastor & Stambaugh (2002 a), and Avramov & Wermers (2006) also assume they know the cross-sectional distribution. Each finds their estimate of skill to be sensitive to the choice of the population distribution.

To our knowledge, no one has estimated mutual fund skill by letting the population distribution be entirely unknown, estimating it, and using it to infer the skill level of the funds. We do this here by modeling the unknown population distribution with a Bayesian, nonparametric, hierarchical prior. This nonparametric prior is an infinite mixture of normals with unknown mixture weights, locations, scales, and mixture order. We infer these mixture unknowns with an unsupervised learning (1) approach where we partition the panel of mutual funds into a finite number of groups (mixture clusters) where the members of a group all have the same average stock-picking ability and variability (mixture location and scale). (2)

We leverage these random partitions to resolve the uncertainty in the skill level of a member fund by pooling the information of the group's other funds. Sharing the groups information is especially important in resolving the uncertainty around the skill level of newer funds with short performance histories. Partitioning the funds also eliminates the global shrinkage issues that plague parametric hierarchical priors. Extraordinarily skilled funds are allowed to have their own group and not have their estimate of skill shrunk towards the global industry average as in Jones & Shanken (2005).

Using return data from the entire actively managed, US domestic equity fund, industry, we find the population distribution of skill to be fat-tailed, slightly skewed towards better stock-picking ability, and having three modes. These three modes are i) a minor mode where skill is extraordinarily high, ii) a secondary mode where funds lose money for its investors, and iii) a primary mode at a skill level where funds cover the average fees charged investors. As a result of our nonparametric population distribution, there is a greater chance a fund will be extraordinarily skilled relative to a normally distributed population. We also see that the exceptionally skilled and unskilled funds that we uncover with our nonparametric population distribution look rather ordinary under a normal population distribution.

We organize the paper in the following manner. In Section 2 we present a mutual fund investor's investment decision and how he applies Bayes rule to update both his understanding of the population distribution of skill and the potential skill of a fund. Section 3 describes our nonparametric, hierarchical, Dirichlet process mixture, prior for the skill level of the funds and the initial population distribution for this nonparametric prior. We then describe in Section 4 the Bayesian nonparametric learning that comes with the Dirichlet process, followed by the model's Markov Chain Monte Carlo sampler in Section 5. …

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