There is a great deal of evidence supporting the existence of alternative sources of excess returns, such as value, momentum and low risk. Factor investing is real; In fact, these factors are observed in virtually every market and asset class around the world. And they have persisted for many decades. Yet no amount of evidence can prove conclusively that these factors will persist. In the end, investors need to do their own homework, and assign a level of confidence.

Surprisingly, it may make sense to add exposure to factor premia even if investors are skeptical of factor persistence.

As a case study, let’s assume the market factor (MKT) has an expected Sharpe ratio of 1 (the units don’t matter for this exercise, but assuming a Sharpe of 1 makes the math simpler). Let’s also assume three theoretical alternative factor premia, Factor 1 (F1), Factor 2 (F2) and Factor 3 (F3) exhibit an empirical Sharpe ratio of 1, on par with the market. Further assume that they are uncorrelated to the market and to each other. This is broadly consistent with the historical evidence on factor premia, and Arbitrage Pricing Theory.

Now consider a skeptical investor who is only 25% certain that the alternative risk factors will produce competitive risk-adjusted returns in the future. The probability-weighted expected Sharpe ratio of the factors is therefore 25% times the expected Sharpe of 1, or 0.25. Recall that a Sharpe of 1 reflects 1 unit of excess return for 1 unit of risk. A probability-weighted Sharpe of 0.25 reflects 0.25 units of excess return per 1 unit of risk.

Will these factors improve the performance of the portfolio under these assumptions?

Let’s examine a portfolio with a 50% exposure to the MKT factor; the other 50% is evenly distributed among the three alternative factor premia, with exposures of 16.7% each. The expected *excess return* of the portfolio is then (0.5 * 1) + (.167 * 0.25) + (.167 * 0.25) + (.167 * 0.25) = 0.625.

Now, since all factors have a volatility of 1 and are uncorrelated, the covariance matrix is:

MKT | F1 | F2 | F3 | |

MKT | 1 | 0 | 0 | 0 |

F1 | 0 | 1 | 0 | 0 |

F2 | 0 | 0 | 1 | 0 |

F3 | 0 | 0 | 0 | 1 |

Simple matrix math yields a portfolio volatility of 0.58, which produces an expected portfolio Sharpe ratio of 0.625/0.58=1.08. This is higher than the Sharpe ratio of the market portfolio on its own. In fact, under similar assumptions about relative Sharpe ratios and correlations, a rational investor would allocate at least 50% of the portfolio to alternative premia, unless he was less than 16% confident that the factors will persist*.

While this is a stylized example, it reflects the reality of both factor investing and investor wariness. And it makes plain that investors do not need to have complete faith in factor persistence to warrant some exposure.

*Recall that our case study assumed that the market portfolio has a Sharpe of 1. If the probability weighted Sharpe ratios for the non-market factors are 0.16 (16% * 1), then the factors are expected to produce 0.16 units of excess return per 1 unit of risk. As such, the expected excess return of the portfolio would be (0.5 * 1) + (0.167 * 0.16) + (0.167 * 0.16) + (0.167 * 0.16) = 0.58. Since we found that the expected volatility of the portfolio is 0.58, the portfolio would have an expected Sharpe ratio of 1, consistent with the market.