A uniformly distributed random portfolio

Abstract

In this study, we propose a uniformly distributed random portfolio as an alternative benchmark for portfolio performance evaluation. The uniformly distributed random portfolio is analogous to an enumeration of all feasible portfolios without any prior on the market. Therefore, the relative ranking of a portfolio can be evaluated without peer group information. We derive a closed-form expression for the probability distribution of the Sharpe ratio of a uniformly distributed random portfolio, and conduct comparative analysis with US equity mutual funds. We find that the uniformly distributed random portfolio properly captures the historical performance distribution of equity mutual funds. In addition, we evaluate performance of cap-weighted equity portfolios via uniformly distributed random portfolios.

Keywords:

• Portfolio performance evaluation
• Random portfolio
• Active investment

Acknowledgements

The authors would like to thank the editor and two anonymous reviewers for their valuable comments.

Notes

¹ w =  means that the portfolio consists of only risk-free asset, and thus, the Sharpe ratio cannot be defined. As we are interested in the probability distribution of the Sharpe ratio, excluding a single point in does not affect our analysis.

² A unit hypersphere in the original space does not become a unit hypersphere in a -transformed space. However, due to the scale-invariance property of the Sharpe ratio, considering a unit hypersphere in a -transformed space does not affect our analysis.

³ Note that the Sharpe ratio of the optimal tangent portfolio calculated in the proof of Proposition 2 is . Thus, if we short the optimal tangent portfolio, the minimum Sharpe ratio would be .

⁴ The regularized incomplete beta function is defined as , where is the incomplete beta function and is the beta function.

⁵ Concise formulas for the surface area of a hyperspherical cap and the intersection of two hyperspherical caps are given in Li (2008) and Lee and Kim (2014), respectively.

⁶ Source: http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html.

⁷ Note that both UDRP and Monte Carlo have no prior on the future market behaviour, and thus, they have almost zero skewness.

Additional information

Funding

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2015R1D1A1A01059339) and the KAIST High Risk High Return Project (HRHRP).