#### Skill Versus Luck And Investment Performance by Caltech

California Institute Of Technology

Pasadena, CA 91125

626 564-2001

**Abstract**

This article presents a simple procedure for assessing the relative impact of luck and skill in determining investment performance. The procedure is then applied to the large cap value managers. The results are consistent with earlier work that suggests that the great majority of the cross-sectional variation in fund performance is due to random noise.

### Introduction: The basic problem of Skill Versus Luck

Successful investing, like most activities in life, is based on a combination of skill and serendipity. Distinguishing between the two is critical for forward looking decision making because skill is relatively permanent while serendipity, or luck, by definition is not. An investment manger who is skillful this year presumably will be skillful next year. An investment manager who was lucky this year is no more likely to be lucky next year than any other manager.

The problem is that skill and luck are not independently observable. Instead all that can be observed is their combined impact which is here called performance. The central question, therefore, is to determine how much can be learned about skill by observing performance. It turns out that there is a straightforward way to investigate that question based on application of the bivariate normal distribution. Though the results presented here are well known in statistics, they are not commonly applied in the context of assessing portfolio managers. As shown, they can serve as the basis of a simple and useful model for assessing the skill of competing fund managers. To illustrate how the model works, I use the procedure to analyze the performance of large cap equity managers tracked by Morningstar. It turns out, as one might expect given the volatility of asset prices, that the relative performance of managers in any given year provides little information about management skill.

### A simple model for assessing luck and skill

To develop the model, assume that there exists a measure of performance, p, that reflects the sum of skill, s, and luck, L. This formulation has a straightforward interpretation in terms of portfolio management. In that context, p represents the observed return on a specific portfolio, s represents the added expected return due to the skill of the investment manager, and L represents the impact of idiosyncratic risk on the portfolio’s return over the observed holding period.

More specifically, assume that both luck and skill are normally distributed in the cross section and that

where s ~ n(E(s), sd(s)) and L ~ n(0, sd(L)). By definition, the mean of the luck distribution is zero. Because p = L + s, p and s are distributed as bivariate normal with mean vector [E(s), 0] and covariance matrix

Because luck cannot be correlated with skill, otherwise there would be a predictable component of luck, it follows that

For the bivariate normal distribution, it is well known from the statistical literature1 that

Substituting the relations from (2) into (3) gives,

Because E(L) = 0, it follows that E(p) = E(s) so equation (4) can be written,

Equation (5) is the basic model. It states that when performance is observed in excess of the mean, the assessment of skill is adjusted upward, but not all the way to the observed level of performance, p. Instead, the assessment of s is adjusted upward by the observed superior performance, p-E(s), times the ratio of the variance of s to the variance of p. Therefore, the assessment of skill based on the observation of performance depends critically on var(s)/var(p).

Notice that in both limiting cases, equation (5) makes intuitive sense. If var(L) is much larger than var(s), then var(p) >> var(s) in which case E(s|p) goes to E(s). That is reasonable because if performance is dominated by luck, then observation of performance should play little role in the assessment of skill. On the over hand, if var(s) >> var(L) then var(s) is approximately equal to var(p), which implies E(s|p) goes to p. That makes sense because if luck has a relatively minor impact on performance, then observed performance is a precise measure of skill.

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