A mixture of Gaussians approach to mathematical portfolio oversight: the EF3M algorithm

Abstract

An analogue can be made between: (a) the slow pace at which species adapt to an environment, which often results in the emergence of a new distinct species out of a once homogeneous genetic pool and (b) the slow changes that take place over time within a fund, mutating its investment style. A fund’s track record provides a sort of genetic marker, which we can use to identify mutations. This has motivated our use of a biometric procedure to detect the emergence of a new investment style within a fund’s track record. In doing so, we answer the question: What is the probability that a particular PM’s performance is departing from the reference distribution used to allocate her capital? The EF3M algorithm, inspired by evolutionary biology, may help detect early stages of an evolutionary divergence in an investment style and trigger a decision to review a fund’s capital allocation.

Keywords:

• Skewness
• Kurtosis
• Mixture of Gaussians
• Moment matching
• Maximum likelihood
• EM algorithm

JEL Classification:

• C13
• C15
• C16
• C44

Acknowledgments

We wish to thank Robert Almgren (Quantitative Brokers, NYU), Tony Anagnostakis (Moore Capital), Marco Avellaneda (NYU), David H. Bailey (Lawrence Berkeley National Laboratory), Sid Browne (Guggenheim Partners), John Campbell (Harvard University), Peter Carr (Morgan Stanley, NYU), David Easley (Cornell University), Ross Garon (SAC Capital), Robert Jarrow (Cornell University), Andrew Karolyi (Cornell University), David Leinweber (Lawrence Berkeley National Laboratory), Attilio Meucci (Kepos Capital, NYU), Maureen O’Hara (Cornell University), Riccardo Rebonato (PIMCO, University of Oxford), Luis Viceira (HBS), and participants at Morgan Stanley’s Monthly Quant Seminar for their helpful comments.

Notes

¹ This tiebreak step is not essential to the algorithm. Its purpose is to deliver one and only one solution for each run, based on the researcher’s confidence on the fourth and fifth moments. In absence of a view on this regard, the researcher may ignore the tiebreak and use every solution to which the algorithm converges (one or more per run).

² In each repetition we use the same , however the values of p differ between runs, as they are drawn from a uniform distribution.

³ This (non-Normal) is on the cumulative returns, not the simple returns.

⁴ We are thankful to the referee for suggesting this Section.