#### Seeking Alpha? It’s a Bad Guideline For Portfolio Optimization via SSRN

Hebrew University of Jerusalem – Jerusalem School of Business Administration

California Institute of Technology

April 28, 2015

**Abstract:**

Alpha is the most popular measure for evaluating the performance of both individual assets and funds. The alpha of an asset with respect to a given benchmark portfolio measures the change in the portfolio’s Sharpe ratio driven by a marginal increase in the asset’s portfolio weight. Thus, alpha indicates which assets should be marginally over/underweighted relative to the benchmark weights, and by how much. This study shows that alpha is actually a bad guideline for portfolio optimization. The reason is that alpha only measures the effects of infinitesimal changes in the portfolio weights. For small but finite changes, which are those relevant to investors, the optimal weight adjustments are almost unrelated to the alphas. In fact, in many cases the optimal adjustment is in the opposite direction of alpha – it may be optimal to reduce the weight of an asset with a positive alpha, and vice versa. Rather than employing alphas as a guideline, one can do much better by direct optimization with the desired constraint on the distance from the benchmark.

### Seeking Alpha? It’s A Bad Guideline For Portfolio Optimization – Introduction

Alpha is the primary measure for the performance of individual assets as well as mutual funds.1 It measures excess average return over and above the return expected given the asset’s risk exposure. There are two ways to view alpha, and to employ it. The first is as a measure of a fund manager’s stock selection and market timing abilities.2 The second is as a guideline for the investor wishing to optimize her portfolio. The alpha of an asset, calculated with respect to a given benchmark portfolio, measures the change in the portfolio’s Sharpe ratio driven by a marginal increase of the asset’s weight in the portfolio. The vector of alphas is thus the direction of marginal adjustment in portfolio-weight space that yields the maximal increase in the portfolio’s Sharpe ratio. Alphas tell the investor how to best marginally adjust her portfolio relative to the benchmark: increase the weight of assets with positive alphas, and decrease the weight of assets with negative alphas (and do so proportionately to the absolute size of alpha). Indeed, many investors seek alpha and tilt their portfolios toward assets with positive alphas. This paper is about this second view of alpha, as a tool for portfolio optimization.

We examine the usefulness of alpha as a guideline for portfolio optimization by examining the increase in the Sharpe ratio obtained by shifting the portfolio weights “in the direction” of the alpha vector. We find that while alpha indeed indicates the best way to make an infinitesimal adjustment to the portfolio weights (by its mathematical definition), it is not useful as a practical guideline, where finite adjustments are considered. Rather than adjusting the weights according to alphas, the investor can do much better by directly optimizing the portfolio with the desired constraint on the degree of deviation from the benchmark portfolio weights. When finite adjustments are considered, adjustments according to alpha are not only sub-optimal, but may in fact be in the “wrong direction” – in many cases the weight of an asset with a positive alpha should optimally be reduced, and vice versa.

The reason that alpha is a bad guideline for portfolio optimization is that it is an indication only about the best infinitesimal shift in portfolio weights. Once the portfolio is shifted, alphas may change considerably. Thus, making large adjustments in the direction of the original alphas could be very sub-optimal. It is making a large step in the direction of the local gradient in order to search for a global maximum. This is closely related to the well-known fact that a small change in the benchmark portfolio can lead to a big change in the beta-expected-return relationship, and thus to a big change in the assets’ alphas (Roll and Ross 1994). This implies that while making a small shift of the portfolio weights in the direction of the alpha vector increases the portfolio’s Sharpe ratio, as one changes the portfolio, the alphas may also change quite dramatically. Therefore, continuing to move “in the direction” of the original alpha vector beyond the initial infinitesimal shift may be far from optimal, as this is no longer “the right direction”.