**Luck Versus Skill In The Cross Section Of Mutual Fund Returns**

University of Chicago – Finance

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Tuck School of Business at Dartmouth; National Bureau of Economic Research (NBER)

December 14, 2009

*Tuck School of Business Working Paper No. 2009-56 *

*Chicago Booth School of Business Research Paper*

*Journal of Finance, Forthcoming*

**Abstract: **

The aggregate portfolio of U.S. equity mutual funds is close to the market portfolio, but the high costs of active management show up intact as lower returns to investors. Bootstrap simulations suggest that few funds produce benchmark adjusted expected returns sufficient to cover their costs. If we add back the costs in expense ratios, there is evidence of inferior and superior performance (non-zero true alpha) in the extreme tails of the cross section of mutual fund alpha estimates.

### Luck Versus Skill In The Cross Section Of Mutual Fund Returns – Introduction

There is a constraint on the returns to active investing that we call equilibrium accounting. In short (details later), suppose that when returns are measured before costs (fees and other expenses), passive investors get passive returns; that is, they have zero Alpha (abnormal expected return) relative to passive benchmarks. This means active investment must also be a zero sum game – aggregate Alpha is zero before costs. If some active investors have positive Alpha before costs, it is dollar for dollar at the expense of other active investors. After costs, that is, in terms of net returns to investors, active investment must be a negative sum game. (Sharpe (1991) calls this the arithmetic of active management.)

We examine mutual fund performance from the perspective of equilibrium accounting. For example, at the aggregate level, if the value-weight (VW) portfolio of active funds has a positive Alpha before costs, we can infer that the VW portfolio of active investments outside mutual funds has a negative Alpha. In other words, active mutual funds win at the expense of active investments outside mutual funds. In fact, we find that the VW portfolio of active funds that invest primarily in U.S. equities is close to the market portfolio, and estimated before expenses, its Alpha relative to common benchmarks is close to zero. Since the VW portfolio of active funds produces Alpha close to zero in gross (pre-expense) returns, Alpha estimated on the net (post-expense) returns realized by investors is negative by about the amount of fund expenses.

The aggregate results imply that if there are active mutual funds with positive true Alpha, they are balanced by active funds with negative Alpha. We test for the existence of such funds. The challenge is to distinguish skill from luck. Given the multitude of funds, many have extreme returns by chance. A common approach to this problem is to test for persistence in fund returns, that is, whether past winners continue to produce high returns and losers continue to underperform (for example, Grinblatt and Titman (1992), Carhart (1997)). Persistence tests have an important weakness. They rank funds on short-term past performance, so there may be little evidence of persistence because the allocation of funds to winner and loser portfolios is largely based on noise.

We take a different tack. We use long histories of individual fund returns and bootstrap simulations of return histories to infer the existence of superior and inferior funds. We compare the actual cross-section of fund Alpha estimates to the results from 10,000 bootstrap simulations of the cross-section. The returns of the funds in a simulation run have the properties of actual fund returns, except we set true Alpha to zero in the return population from which simulation samples are drawn. The simulations thus describe the distribution of Alpha estimates when there is no abnormal performance in fund returns. Comparing the distribution of Alpha estimates from the simulations to the cross-section of Alpha estimates for actual fund returns allows us to draw inferences about the existence of skilled managers.

For fund investors the simulation results are disheartening. When Alpha is estimated on net returns to investors, the cross-section of precision-adjusted Alpha estimates, t(Alpha), suggests that few active funds produce benchmark adjusted expected returns that cover their costs. Thus, if many managers have sufficient skill to cover costs, they are hidden by the mass of managers with insufficient skill. On a practical level, our results on long-term performance say that true Alpha in net returns to investors is negative for most if not all active funds, including funds with strongly positive Alpha estimates for their entire histories.

Mutual funds look better when returns are measured gross, that is, before the costs included in expense ratios. Comparing the cross-section of t(Alpha) estimates from gross fund returns to the average cross-section from the simulations suggests that there are inferior managers whose actions reduce expected returns and there are superior managers who enhance expected returns. If we assume that the cross section of true Alpha has a normal distribution with mean zero and standard deviation ?, then ? around 1.25% per year seems to capture the tails of the cross section of Alpha estimates for our full sample of actively managed funds.

The estimate of the standard deviation of true Alpha, 1.25% per year, does not imply much skill. It suggests, for example, that fewer than 16% of funds have Alpha greater than 1.25% per year (about 0.10% per month), and only about 2.3% have Alpha greater than 2.50% per year (about 0.21% per month) – before expenses. The simulation tests have power. If the cross section of true Alpha for gross fund returns is normal with mean zero, the simulations strongly suggest that the standard deviation of true Alpha is between 0.75% and 1.75% per year. Thus, the simulations rule out values of ? rather close to our estimate, 1.25%. The power traces to the fact that a large cross section of funds produces precise estimates of the percentiles of t(Alpha) under different assumptions about ?, the standard deviation of true Alpha. This precision allows us to put ? in a rather narrow range.

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