### Why Indexing Works

Bartlit Beck Herman Palenchar & Scott LLP

University of Chicago – Booth School of Business

University of Oxford – Mathematical Institute

October 12, 2015

**Abstract: **

We develop a simple stock selection model to explain why active equity managers tend to underperform a benchmark index. We motivate our model with the empirical observation that the best performing stocks in a broad market index perform much better than the other stocks in the index. While randomly selecting a subset of securities from the index increases the chance of outperforming the index, it also increases the chance of underperforming the index, with the frequency of underperformance being larger than the frequency of overperformance. The relative likelihood of underperformance by investors choosing active management likely is much more important than the loss to those same investors of the higher fees for active management relative to passive index investing. Thus, the stakes for finding the best active managers may be larger than previously assumed.

### Why Indexing Works – Introduction

The tendency of active equity managers to underperform their benchmark index (e.g., Lakon-ishok, Shleifer, and Vishny (1992), Gruber (1996)) is something of a mystery. It is one thing for active equity managers to fail to beat the benchmark index, since that may imply only a lack of skill to do better than random selection. It is quite another to find that most active equity managers fail to keep up with the benchmark index, since that implies that active equity managers are doing something that systematically leads to underperformance.

We develop a simple stock selection model that builds on the underemphasized empirical fact that the best performing stocks in a broad index perform much better than the other stocks in the index, so that average index returns depend heavily on the relatively small set of winners (e.g., J.P. Morgan (2014)). In our model, randomly selecting a small subset of securities from the index maximizes the chance of outperforming the index – the allure of active equity management – but it also maximizes the chance of underperforming the index, with the chance of underperformance being larger than the chance of overperformance. To illustrate the idea, consider an index of five securities, four of which (though it is unknown which) will return 10% over the relevant period and one of which will return 50%. Suppose that active managers choose portfolios of one or two securities and that they equally-weight each investment. There are 15 possible one or two security portfolios.” Of these 15, 10 will earn returns of 10%, because they will include only the 10% securities. Just 5 of the 15 portfolios will include the 50% winner, earning 30% if part of a two security portfolio and 50% if it is the single security in a one security portfolio. The mean average return for all possible actively-managed portfolios will be 18%, while the median actively-managed portfolio will earn 10%. The equally-weighted index of all 5 securities will earn 18%. Thus, in this example, the average active-management return will be the same as the index (see Sharpe (1991)), but two-thirds of the actively-managed portfolios will underperform the index because they will omit the 50% winner.

Our paper continues as follows. In Section 2, we develop our simple stock selection model. In Section 3, we present simulation results. Section 4 concludes.

### Simple Model of Stock Selection from an Index

We consider a benchmark index that contains N stocks . Let the dynamics of stock over time be given by a geometric Brownian motion

where for simplicity we consider the volatility to be constant for all stocks. We assume that stock drifts are distributed , which generates a small number of extreme winners, a small number of extreme losers, and a large number of stocks with drifts centered around with standard deviation . While our model implies unpriced covariance among securities and a lack of learning, much theory and evidence suggests that the learning problem is too difficult over the lifetimes of most investors to pay much attention to that modeling limitation (e.g., Merton (1980), Jobert, Platania, and Rogers (2006)).1

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