Are Mutual Fund Investors Bayesian Learners?
University of California at Irvine
University of California, Irvine – Paul Merage School of Business
January 22, 2016
Abstract:
Many financial models assume that financial market participants learn following Bayes’ rule. In this paper, we use a new measure of investor disagreement to study the learning speed of mutual fund investors about manager skill. Although the precision of estimates of manager skill increases quickly over time, investors’ disagreement about manager skill does not decline for many years. Investors’ non-Bayesian learning is not due to the inconsistency of potential performance measures used by investors, or investing frictions, but is likely due to limited attention. Overall, our results suggest market participants learn much more slowly than currently assumed.
Are Mutual Fund Investors Bayesian Learners? – Introduction
One difficulty all financial market participants face when making decisions is uncertainty. How investors resolve uncertainty (i.e. learn) has important implications for financial theory and thus is an important building block for several theoretical models in a variety of settings. For example, Pastor and Veronesi (2003, 2006) find that learning can explain why higher growth rate uncertainty leads to higher stock prices. Timmerman (1983) and Brennan and Xia (2001) show that learning by investors can explain why stocks are more volatile than their underlying dividends. Berk and Green (2004) document that investors learning about mutual fund managers’ skill levels can explain why mutual fund performance is not persistent, even if managers have skill. In almost all of these models, the cornerstone of how market participants learn is Bayes’ rule.
However, to date, there is little empirical documentation of how fast investors learn. In this paper, we fill this gap by examining the learning speed of mutual fund investors about manager skill. When a new mutual fund is introduced, limited information about the manager’s ability is available to investors. Thus, the ‘skill’ estimate for that new fund is highly uncertain and investors are likely to disagree about the quality of the manager. As time passes, however, information is revealed, allowing market participants to form a more precise skill estimate. Investors’ assessment of the manager’s skill will converge, leading to a reduction in disagreement. Thus, to measure learning speed, we observe how quickly investor disagreement declines. If investors are learning as quickly as financial models imply, the decline in disagreement should be related to the decline in uncertainty about manager skill.
Most financial models that incorporate learning assume that investors learn following Bayes’ rule. Given some original parameter estimate and signal process, Bayes’ rule provides estimates of the posterior variance of the particular parameter of interest. Specifically, under Bayes’ rule, investors have an original belief of some unknown parameter that is normally distributed 
where are the mean and standard deviation of investors’ prior belief. Assuming a signal about arrives at each period t:
where The posterior variance of after T periods is:
Under this framework, the posterior variance is a convex function of the passage of time. In other words, the posterior variance declines most quickly in the early time periods and then gradually slows down over time. For example, in Figure 1, we plot the posterior variances of a parameter using a signal variance of one and 0.2.
Thus, if investors learn following Bayes’ rule, we should see that the rate of convergence in disagreement has a similar relation with time.
Of course, learning does not necessarily lead to reduction in disagreement if a subset of investors can learn from private signals unobserved by other investors (e.g., Jiang and Sun (2014)). However, one of the advantages to studying learning in the mutual fund industry is that it is highly unlikely that some investors have asymmetric information about manager skill that could lead to persistent disagreement. Mutual funds are subject to a large litany of public disclosure. For example, daily returns are available to investors and funds must disclose their portfolio holdings quarterly. They must also disclose proxy voting while other disclosures provide information on the manager, the flows of the fund, expenses, balance sheet, and so forth. Given that all of these data are publicly available and are typically aggregated by various service provides, it is reasonable to believe all significant signals about manager skills are public.
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