Model uncertainty, performance persistence and flows

Abstract

Model uncertainty makes it difficult to draw clear inference about mutual fund performance persistence. I propose a new performance measure, Bayesian model averaged (BMA) alpha, which explicitly accounts for model uncertainty. Using BMA alphas, I find evidence of performance persistence in a large sample of US funds. There is a positive and asymmetric relation between flows and past BMA alphas, suggesting that fund investors respond to the information in BMA alphas. My findings are robust to various sensitivity analyses, including alternative measures of post-ranking performance, flows and total net assets, and alternative econometric model specifications.

Appendices

Appendix A

See Table 15.

Table 15 Variables in mutual fund return models

Appendix B

2.1 Prior distributions and specification of prior hyperparameters

To facilitate discussion in this and subsequent sections, I rewrite Eq. 1 as:

$$ r_{i} = Z_{i} \phi_{i} + u_{i} $$

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where r _i is the S × 1 vector containing the S observations of r _i,t (I assume the fund has monthly returns for S months); Z _i  = (l _S , X _i ) is the S × (K + 1) matrix containing l _S , the S × 1 unit vector in the leftmost column and X _i , the S × K matrix containing the explanatory variables specific to the jth model; $\phi_{i} = (\alpha_{i} ,\beta_{i,1} , \ldots ,\beta_{i,K} )^{\prime}$, the (K + 1) × 1 coefficient vector and u _i is the S × 1 vector containing the disturbance terms. To facilitate subsequent exposition, define the (K × 1) sub-vector, $b_{i} = (\beta_{i,1} , \ldots ,\beta_{i,K} )$. Following Pastor and Stambaugh (2002a), I employ the natural conjugate normal-inverted gamma prior for σ ² _u and ϕ _i . Specifically, the prior for σ ² _u follows an inverted gamma distribution (Zellner 1971),

$$ \sigma_{u}^{2} \sim IG(\underline{\nu } ,\underline{s}^{2} ) $$

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where “IG” stands for inverted gamma and $\underline{\nu }$ and $\underline{s}^{2}$ are parameters of the inverted gamma distribution. Conditional on σ ² _u , α _i and b _i are normally distributed

$$ \alpha_{i} |\sigma_{u}^{2} \sim N\left( {\underline{{\alpha_{i} }} ,{\frac{{\sigma_{u}^{2} }}{{E\left( {\sigma_{u}^{2} } \right)}}}\underline{{\sigma_{\alpha }^{2} }} } \right) $$

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$$ b_{i} |\sigma_{u}^{2} \sim N\left( {\underline{{b_{i} }} ,{\frac{{\sigma_{u}^{2} }}{{E\left( {\sigma_{u}^{2} } \right)}}}\underline{{V_{b} }} } \right) $$

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where $\underline{{\alpha_{i} }}$ is the prior mean of α _i , $\underline{{b_{i} }}$ is the prior mean vector of b _i , $\underline{{\sigma_{\alpha }^{2} }}$ is the marginal prior variance of α _i (“prior variance of alpha”) and $\underline{{V_{b} }}$ is the marginal prior covariance matrix of b _i . I assume α _i and b _i are independent of each other. Given this assumption, ϕ _i is multivariate normal

$$ \phi_{i} |\sigma_{u}^{2} \sim N\left( {\underline{{\phi_{i} }} ,\sigma_{u}^{2} \underline{{V_{\phi } }} } \right) $$

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where $\phi_{i} = \left( {\underline{{\alpha_{i} }} ,\underline{{b^{\prime}_{i} }} } \right)^{\prime }$ and $\underline{{V_{\phi } }}$ is defined as

$$ \underline{{V_{\phi } }} = {\frac{1}{{E\left( {\sigma_{u}^{2} } \right)}}}\left[ {\begin{array}{*{20}c} {\underline{{\sigma_{\alpha }^{2} }} } & 0 \\ 0 & {\underline{{V_{b} }} } \\ \end{array} } \right] $$

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The diagonal structure of $\underline{{V_{b} }}$ stems from the assumed independence of α _i and b _i . To implement Bayesian estimation, I specify values for the prior hyperparameters $\underline{{\alpha_{i} }}$, $\underline{{\sigma_{\alpha }^{2} }}$, $\underline{s}^{2}$, $\underline{\nu }$, $\underline{{b_{i} }}$, and $\underline{{V_{b} }}$. I already discussed the specification of $\underline{{\alpha_{i} }}$ and $\underline{{\sigma_{\alpha }^{2} }}$ in Sect. 3.1. Thus, it only remains to describe specification of the remaining hyperparameters. Following Pastor and Stambaugh (2002a) and Busse and Irvine (2006), I employ an empirical Bayes approach in specifying values for $\underline{s}^{2}$, $\underline{\nu }$, $\underline{{b_{i} }}$, and $\underline{{V_{b} }}$.⁴³ Each fund is viewed as a draw from the cross section of funds with the same investment objective. Thus, prior uncertainty about a fund’s parameter is driven by the cross sectional variation in that parameter. For each investment objective, I select all funds having at least 60 months of data and compute the OLS estimate of b _i for each fund.⁴⁴ Then I set $\underline{{b_{i} }}$ equal to the sample mean of the OLS estimates and $\underline{{V_{b} }}$ equal to the covariance matrix of the OLS estimates. Each OLS regression also yields $\hat{\sigma }_{u}^{2}$, the estimate of σ ² _u . To explain how I specify $\underline{s}^{2}$ and $\underline{\nu }$, I introduce the first and second moments of σ ² _u . Based on Zellner (1971, pp. 371–372),

$$ E\left( {\sigma_{u}^{2} } \right) = {\frac{{\underline{\nu } \underline{s}^{2} }}{{\underline{\nu } - 2}}} $$

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$$ {\text{Var}}\left( {\sigma_{u}^{2} } \right) = {\frac{{2\underline{\nu } \underline{s}^{2} }}{{(\underline{\nu } - 2)^{2} (\underline{\nu } - 4)}}} $$

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By substituting Eq. 17 into Eq. 18, I can express $\underline{\nu }$ as

$$ \underline{\nu } = 4 + {\frac{{2\left( {E\left( {\sigma_{u}^{2} } \right)} \right)^{2} }}{{{\text{Var}}\left( {\sigma_{u}^{2} } \right)}}} $$

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I insert the cross sectional mean and variance of $\hat{\sigma }_{u}^{2}$ into the right-hand side of Eq. 19 and evaluate that expression. $\underline{\nu }$ is set equal to the next largest integer of the resulting value on the right-hand side of Eq. 19. Once I have solved for $\underline{\nu }$, I use that value, the cross sectional mean of $\hat{\sigma }_{u}^{2}$ and Eq. 18 to solve for $\underline{s}^{2}$.

2.2 Posterior distribution

I derive the posterior distributions of ϕ _i and σ ² _u defined with respect to the jth model.⁴⁵ Given the distributional assumptions of u _i,t , the likelihood function of r _i is normal

$$ p(r_{i} |Z_{i} ,\phi_{i} ,\sigma_{u} ) = {\frac{1}{{(2\pi )^{S/2} \sigma_{u}^{2} }}}\exp \left\{ { - {\frac{1}{{2\sigma_{u}^{2} }}}(r_{i} - Z_{i} \phi_{i} )^{\prime}(r_{i} - Z_{i} \phi_{i} )} \right\} $$

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The prior pdf of σ ² _u is

$$ p(\sigma_{u} ) = {\frac{2}{{\Upgamma \left( {{{\underline{\nu } } \mathord{\left/ {\vphantom {{\underline{\nu } } 2}} \right. \kern-\nulldelimiterspace} 2}} \right)}}}\left( {{\frac{{\underline{\nu } \underline{s}^{2} }}{2}}} \right){\frac{1}{{\sigma_{u}^{{\underline{\nu } + 1}} }}}\exp \left\{ { - {\frac{{\underline{\nu } \underline{s}^{2} }}{{2\sigma_{u}^{2} }}}} \right\} $$

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where $\Upgamma (.)$ denotes the gamma function. Conditional on σ _u , the prior pdf of ϕ _i is

$$ p(\phi_{i} |\sigma_{u} ) = {\frac{1}{{(2\pi )^{(k + 1)/2} \left| {\underline{{V_{\phi } }} } \right|^{1/2} \sigma_{u}^{k + 1} }}}\exp \left\{ { - {\frac{1}{{2\sigma_{u}^{2} }}}\left( {\phi_{i} - \underline{{\phi_{i} }} } \right)^{\prime } \left( {\phi_{i} - \underline{{\phi_{i} }} } \right)} \right\} $$

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The product of Eqs. 20, 21 and 22 yields the joint posterior pdf of ϕ _i and σ _u

$$ \begin{aligned} p(\phi_{i} ,\sigma_{u} |Z_{i} ,r_{i} ) \propto & p(r_{i} |Z_{i} ,\phi_{i} ,\sigma_{u} )p(\phi_{i} |\sigma_{u} )p(\sigma_{u} ) \\ & \propto {\frac{1}{{\sigma_{u}^{{\underline{\nu } + 1}} }}} \times {\frac{1}{{\sigma_{u}^{k + 1} }}} \times {\frac{1}{{\sigma_{u}^{S} }}} \times \exp \left\{ { - {\frac{1}{{2\sigma_{u}^{2} }}}\left( {r_{i} - Z_{i} \phi_{i} } \right)^{\prime } \left( {r_{i} - Z_{i} \phi_{i} } \right)} \right\} \\ & \times \exp \left\{ { - \frac{1}{2}\left( {\phi_{i} - \underline{{\phi_{i} }} } \right)^{\prime } \underline{{V_{\phi } }}^{ - 1} \left( {\phi_{i} - \underline{{\phi_{i} }} } \right)} \right\} \times \exp \left\{ { - {\frac{{\underline{\nu } \underline{s}^{2} }}{{2\sigma_{u}^{2} }}}} \right\} \\ \end{aligned} $$

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Rewriting $(r_{i} - Z_{i} \phi_{i} )^{\prime}(r_{i} - Z_{i} \phi_{i} ) + \left( {\phi_{i} - \underline{{\phi_{i} }} } \right)^{\prime } \underline{{V_{\phi } }}^{ - 1} \left( {\phi_{i} - \underline{{\phi_{i} }} } \right)$ as

$$ r_{i}^{\prime } r_{i} + \underline{{\phi_{i} }}^{\prime } \underline{{V_{\phi } }}^{ - 1} \underline{{\phi_{i} }} + \left( {\phi_{i} - \overline{{\phi_{i} }} } \right)^{\prime } \left( {\underline{{V_{\phi } }}^{ - 1} + Z_{i}^{\prime } Z_{i} } \right)\left( {\phi_{i} - \overline{{\phi_{i} }} } \right) - \overline{{\phi_{i} }}^{\prime } \left( {\underline{{V_{\phi } }}^{ - 1} + Z_{i}^{\prime } Z_{i} } \right)\overline{{\phi_{i} }} $$

and rearranging the terms in the exponents gives us

$$ p(\phi_{i} ,\sigma_{u} |Z_{i} ,r_{i} ) \propto {\frac{1}{{\sigma_{u}^{k + 1} }}}\exp \left\{ { - {\frac{1}{{2\sigma_{u}^{2} }}}\left( {\phi_{i} - \overline{{\phi_{i} }} } \right)^{\prime } \overline{{V_{\phi } }}^{ - 1} \left( {\phi_{i} - \overline{{\phi_{i} }} } \right)} \right\} \times {\frac{1}{{\sigma^{{\bar{\nu } + 1}} }}}\exp \left\{ { - {\frac{{\bar{\nu }\bar{s}^{2} }}{{2\sigma_{u}^{2} }}}} \right\} $$

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where $\overline{{V_{\phi } }} = \left( {\underline{{V_{\phi } }}^{ - 1} + Z^{\prime}_{i} Z_{i} } \right)^{ - 1} ,\overline{{\phi_{i} }} = \overline{{V_{\phi } }} \left( {\underline{{V_{\phi } }}^{ - 1} \underline{{\phi_{i} }} + Z^{\prime}_{i} r_{i} } \right),\bar{\nu } = S + \underline{\nu } ,$ and $\bar{\nu }\bar{s}^{2} = \underline{\nu } \underline{s}^{2} + r_{i}^{\prime } r_{i} + \underline{{\phi_{i} }}^{\prime } \underline{{V_{\phi } }}^{ - 1} \underline{{\phi_{i} }} - \overline{{\phi_{i} }}^{\prime } \left( {\underline{{V_{\phi } }}^{ - 1} + Z_{i}^{\prime } Z_{i} } \right)\overline{{\phi_{i} }}$. Conditional on σ _u , the posterior pdf of ϕ _i is multivariate normal and the posterior pdf of σ _u is inverted gamma

$$ \phi_{i} |Z_{i} ,r_{i} ,\sigma_{u} \sim N\left( {\overline{{\phi_{i} }} ,\sigma_{u}^{2} \overline{{V_{\phi } }} } \right) $$

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$$ \sigma_{u}^{2} \sim IG(\bar{\nu },\bar{s}^{2} ) $$

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$$ E\left( {\sigma_{u}^{2} |Z_{i} ,r_{i} } \right) = {\frac{{\bar{\nu }\bar{s}^{2} }}{{\bar{\nu } - 2}}} = \tilde{\sigma }_{u}^{2} $$

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In addition, the marginal posterior ϕ _i follows a multivariate t distribution with mean and variance given by

$$ E\left( {\phi_{i} |D,M_{j} } \right) = \bar{\phi }_{i} ,{\text{Var}}\left( {\phi_{i} |D,M_{j} } \right) = {\frac{{\bar{\nu }\bar{s}^{2} }}{{\bar{\nu } - 2}}}\overline{{V_{\phi } }} $$

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For the jth model, the Bayesian estimate of alpha, $E(\alpha_{i} |D,M_{j} )$, is the (1,1) element of $\bar{\phi }_{i}$.

2.3 Marginal likelihood and posterior model probability

The derivation of the posterior distribution leads us to the discussion of the marginal likelihood, $p(D|M_{j} )$, since it requires certain quantities from the prior and posterior distributions. Given the normal-inverted gamma natural conjugate prior, the marginal likelihood under the jth model has an analytical form⁴⁶

$$ p(D|M_{j} ) = c_{j} \left[ {{{\left| {\overline{{V_{{\phi_{j} }} }} } \right|} \mathord{\left/ {\vphantom {{\left| {\overline{{V_{{\phi_{j} }} }} } \right|} {\left| {\underline{{V_{{\phi_{j} }} }} } \right|}}} \right. \kern-\nulldelimiterspace} {\left| {\underline{{V_{{\phi_{j} }} }} } \right|}}} \right]^{1/2} \left( {\bar{\nu }_{j} \bar{s}_{j}^{2} } \right)^{{ - \overline{{\nu_{j} }} /2}} $$

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$$ c_{j} = {\frac{{\Upgamma \left( {{{\bar{\nu }_{j} } \mathord{\left/ {\vphantom {{\bar{\nu }_{j} } 2}} \right. \kern-\nulldelimiterspace} 2}} \right)\left( {\underline{{\nu_{j} }} \underline{{s_{j}^{2} }} } \right)^{{{{\underline{{\nu_{j} }} } \mathord{\left/ {\vphantom {{\underline{{\nu_{j} }} } 2}} \right. \kern-\nulldelimiterspace} 2}}} }}{{\Upgamma \left( {{{\underline{{\nu_{j} }} } \mathord{\left/ {\vphantom {{\underline{{\nu_{j} }} } 2}} \right. \kern-\nulldelimiterspace} 2}} \right)\pi^{S/2} }}} $$

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where $\left| {\underline{{V_{{\phi_{j} }} }} } \right|$ and $\left| {\overline{{V_{{\phi_{j} }} }} } \right|$ denote the determinants of $\underline{{V_{{\phi_{j} }} }}$ and $\overline{{V_{{\phi_{j} }} }}$ respectively. The subscript j reminds us that the various quantities in Eqs. 29 and 30 are computed under the jth model.

The jth model’s posterior model probability, $p(M_{j} |D)$, is computed as (see, e.g., Hoeting et al. 1999)

$$ p\left( {M_{j} |D} \right) = {\frac{{p(D|M_{j} )p(M_{j} )}}{{\sum\nolimits_{j = 1}^{26} {p(D|M_{j} )p(M_{j} )} }}} $$

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where $p(M_{j} )$ is the prior model probability of the jth model. For this study, each model under consideration receives equal prior model probability, i.e., $p(M_{j} ) = p(M_{k} )\,\,\forall \,j,k$. Thus, the jth model’s posterior probability simplifies to

$$ p(M_{j} |D) = {{p(D|M_{j} )} \mathord{\left/ {\vphantom {{p(D|M_{j} )} {\sum\nolimits_{j = 1}^{26} {p(D|M_{j} )} }}} \right. \kern-\nulldelimiterspace} {\sum\nolimits_{j = 1}^{26} {p(D|M_{j} )} }} $$

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Note that the denominator in Eq. 32 need not sum to 1. To the extent that models used by researchers do not capture all the models that can explain the data, the sum of the marginal likelihoods will not be 1. Posterior model probabilities are still well-defined in this instance. As long as a model’s marginal likelihood is the highest amongst all other models under consideration, it will receive the highest posterior model probability. Conversely, a model with the lowest marginal likelihood will receive the lowest posterior model probability.