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# How Bayesians Solve the Markowitz Problem

January 13, 2019

Understanding of the “Markowitz problem” has changed in the 60+ years since Harry Markowitz’ publication of an article in the Journal of Finance that outlined the basics of modern portfolio theory.

The problem is that portfolio theory requires an investor to estimate risk, return, and correlation from market data, meaning from past data. This leaves a lot of room for those “unknown unknowns” that you bump into when driving forward while looking in the rearview mirror.

Living with Partial Information

An entire library of literature has developed that applies filtering and learning techniques given circumstances of partial information. That is the situation facing investors and their portfolio managers. This line of research owes much to Peter Lakner of the Statistics and Operations Research Department, New York University, who wrote seminal articles on the subject in 1995 and 1998. Lakner assumes that there is a “Brownian motion” or “drift” in asset prices, but that the drift is not directly observable. Nonetheless, rational consumers will seek to maximize total expected utility just as rational investors will seek to maximize risk-adjusted return. The question is how one goes about doing that in the face of and without denying one’s ignorance.

The new face of the old Markowitz problem, then, is the optimal portfolio strategy for an investor in the face of drift.

A new contribution to this question comes from Carmine De Franco, OSSIAM’s quantitative analyst, Johann Nicolle, also of OSSIAM, and Huyen Pham, of the Universite Paris Diderot. Their purpose is to show that a “Bayesian learning strategy” outperforms other strategies for solving the problem.

Path Specificity

The idea of a “Bayesian learning strategy” is derived from Bayesian probability, which in turn is derived from the work of the mid-18th century mathematician Thomas Bayes. The idea is that when we speak of the probability of any event, we speak of the latest adjustment of our own expectation. We are constantly updating our expectations, so that one’s hypothesis of the likelihood of H on Tuesday (the prior) is modified by the perceived events or new datum of Wednesday into a posterior probability. The Bayesian notion of probability then is path-specific—where I am is dependent on where I have been.

A Bayesian learning strategy embraces this path specificity. It is a strategy that combines past experiences with new learning in a mathematically coherent way to form the next step in the growth of the learner’s body of knowledge.

Applying this to portfolios, De Franco et al. wrote that this assumes “that the investor has an a priori view of the risky assets and their expected returns, but she is uncertain about how good her forecast is.”

BL versus NL Strategies

In the scenario they develop for back testing, through the period starting from January 2000 and continuing to June 2018, Bayesian learning delivers an annualized performance of 5.96% while the non-Bayesian approach would have achieved only 3.96%. The next bit of arithmetic is simple enough: “Incorporating uncertainty and learning from the data yields an annualized 2% excess return.”

What about the risk metrics? The annualized vol is slightly higher for BL, but the maximum drawdown figure is better for BL than for NL. The Sharpe ratio is higher for BL (1.08) than for NL (0.80). Interestingly: both the BL and the NL increase as leverage increases. Specifically, in each case increase in leverage brings “higher performance but also higher volatilities, and the volatility grows faster than the performance.”

The better performance of BL comes as one would expect it to come, from the ability to process information, as indicated by an information ratio of 0.55 in its favor.

Rebalancing the NL?

Would the gaps have been closed, or would the comparison have gone the other way, had the NL method employed for this test been re-balanced more frequently? The authors don’t test for a range of rebalancing frequencies, but they do observe that more rebalancing carries turnover and transaction costs and takes on board a “significant amount of noise coming from daily, short term, price movements.”

Testing for the consequences of rebalancing would be tricky, too, because practitioners generally employ higher rebalancing frequencies for some parts of the portfolio than the other, creating innumerable possibilities, “beyond the scope of this paper.”