EMPIRICAL EVIDENCE: YOU MUST USE PROBABILITY-WEIGHTED RETURN TO SIZE POSITIONS

Portfolio managers often ask me for empirical proof for why they need to calculate probability-weighted return to size positions. Because I do not have empirical evidence, my argument is that probability-weighted return is a precise measurement of an asset's impact on the portfolio and, to maximize overall returns, you must make your best ideas your biggest positions. Their concern is that they do not have confidence in their calculated probability-weighted return or their ability to differentiate between the quality of investments in their portfolio.
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I always make a few points in return:

1) If you can differentiate between an asset that deserves to be in the portfolio and an asset that should not, then there is clearly some differentiation between quality of assets in the portfolio.

2) Whether you calculate probability-weighted return or not, you are estimating risk-reward when making investment decisions so you might as well calculate it to cut down on mistakes.

3) Monte Carlo simulations show probability-weighted return as a superior method of portfolio construction when compared to other position sizing methodologies.

4) Probabilistic portfolio management is the competitive advantage of some of the smartest and most successful funds that I have spoken with.

All of the reasons mentioned above are compelling but a little academic support is nice. Below are four academic papers that show why it is critical to make your best ideas your largest positions and how not calculating your best idea can put you in a 6% return hole compared to your peers.

Excerpts (Once you click on the link, push download, then select the SSRN site):
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"Best Ideas" – Randy Cohen, Christopher Polk, and Bernhard Silli (2009) – 1 to 4% improvement

We examine the performance of stocks that represent managers' "Best Ideas." We find that the stock that active managers display the most conviction towards ex-ante, outperforms the market, as well as the other stocks in those managers' portfolios, by approximately one to four percent per quarter depending on the benchmark employed. The results for managers' other high-conviction investments (e.g. top five stocks) are also strong. The other stocks managers hold do not exhibit significant outperformance. This leads us to two conclusions. First, the U.S. stock market does not appear to be efficiently priced, since even the typical active mutual fund manager is able to identify stocks that outperform by economically and statistically large amounts. Second, consistent with the view of Berk and Green (2004), the organization of the money management industry appears to make it optimal for managers to introduce stocks into their portfolio that are not outperformers. We argue that investors would benefit if managers held more concentrated portfolios.

When asked to talk about his portfolio, the typical investment manager will identify a position therein and proceed to describe the opportunity and the investment thesis with tremendous conviction and enthusiasm. Frequently the listener is overwhelmed by the persuasiveness of the passionate presentation. This leads to a natural follow-up question: how many investments make up the portfolio. Informed that the answer is, e.g., 150, the questioner will often wonder how anyone could possess such depth of knowledge and passion for so many disparate companies. Pressed to answer, investment managers have been known to sheepishly confess that their portfolio contains a few core high-conviction positions, the "best ideas", and then a large number of additional positions which may have less expected excess return but which serve to "round out" the portfolio.

This paper asks a related simple question. What if each mutual fund manager had only to pick a few stocks, their best ideas? Could they outperform under those circumstances? We document strong evidence that they could, as the best ideas of active managers generate up to an order of magnitude more alpha than their portfolio as whole, depending on the performance benchmark.

We argue that this presents powerful evidence that the typical mutual fund managers can, indeed, pick stocks. The poor overall performance of mutual fund managers in the past is not due to a lack of stock-picking ability, but rather to institutional factors that encourage them to overdiversify, i.e. pick more stocks than their best alpha-generating ideas.

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"The Value of Active Mutual Fund Management" – Hsiu-Lang Chen, Narasimhan Jegadeesh, Russ Wermers (1999) – 2% improvement

When we examine mutual fund trades, we find that stocks that the funds actively buy have significantly higher returns than stocks that they actively sell. This return difference is roughly two percent during the one-year holding period following the trades, adjusted for the characteristics of the stocks that are traded.

We find that stocks that funds newly buy have significantly higher returns than stocks they newly sell. This is true for large stocks as well as small stocks, and for value stocks as well as growth stocks. The evidence that stocks actively traded by the funds outperform stocks that are passively held from prior periods suggests that mutual funds hold stocks longer than the horizon over which they can predict returns, possibly because of a preference to avoid high transaction costs or capital gains taxes.

Overall, our evidence is suggestive of the funds possessing superior stock-selection skills.

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"Fund Managers Who Take Big Bets: Skilled or Overconfident" – Klass Baks, Jeffrey Busse, and Clifton Green (2006) – 4% improvement

We document a positive relation between mutual fund performance and managers' willingness to take big bets in a relatively small number of stocks. Focused managers outperform their more broadly diversified counterparts by approximately 30 basis points per month, or roughly 4% annualized. The results hold for mimicking portfolios based on fund holdings as well as when returns are measured net of expenses. Concentrated managers outperform precisely because their big bets outperform the top holdings of more diversified funds.

The findings lend support to the notion that the managers who tilt their portfolios toward their favorite stocks assess correctly the relative merits of stocks overall as well as within their portfolios. By contrast, funds whose portfolio weights more closely approximate a uniform distribution display less ability to correctly sort stocks within their portfolio according to future performance. Overall, our results suggest that concentrated fund managers do have some ability to correctly pick stocks.

Using a variety of performance measures, we find that concentrated fund managers outperform their diversified counterparts. This result lends support to the notion that the managers who are confident in their ability assess correctly the relative merits of stocks overall as well as within their portfolios. By contrast, funds whose portfolio weights more closely approximate a uniform distribution display less ability to correctly sort stocks within their portfolio according to future performance. Overall, our results suggest that focused fund managers do have some ability to correctly pick stocks.

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"The Information Content of Revealed Beliefs in Portfolio Holdings"- Tyler Shumway, Maciej Szefler, and Kathy Yuan (2009) – 2 to 6% improvement

In this paper, we elicit heterogeneous fund manager beliefs on expected stock returns from funds' portfolio holdings at each quarter-end. Revealed beliefs are extracted by assuming that each fund manager aims to outperform a certain benchmark portfolio by choosing an optimal risk-return tradeoff. We then construct a measure of fund managers' forecasting ability—the belief accuracy index (BAI)—by correlating a manager's revealed beliefs on stock returns with the subsequently realized returns. We measure the differences in beliefs between funds with high BAI and all other funds, the belief difference index (BDI). Sorting stocks based on BDI, we find that the annualized return difference between the top and bottom decile is about two to six percent.