Monday, November 20, 2017

Portfolio Optimization

Deciding what percentage of your portfolio to allocate to bonds, domestic stocks, foreign stocks, etc. can be challenging. Any attempt to optimize this allocation is necessarily based on assumptions. It’s dangerous to blindly follow optimized allocation percentages without examining the assumptions built into the optimization process.

In his book Finance for Normal People, Meir Statman tells the story of investment consultants choosing asset allocation percentages for a large U.S. public pension fund. The consultants used Harry Markowitz’s mean-variance portfolio theory to calculate an optimal portfolio for the pension fund. But then they modified all the percentages.

Statman’s explanation for why the consultants changed the percentages is that the managers of the pension fund wanted more than the “utilitarian benefits of higher expected return”; they wanted “expressive and emotional benefits, including the benefits of conformity to the portfolio conventions of this pension fund and similar pension funds.”

Statman may be correct in this assessment, but there is another reason for rejecting the recommended percentages from mean-variance portfolio theory. This reason is based on strictly utilitarian benefits and not expressive or emotional benefits.

The basis for mean-variance portfolio theory is that investment returns follow what is called a lognormal distribution. This model does a decent job for most of the range of possible investment returns, but it vastly underestimates the chances of extreme events. Unfortunately, making sure you can survive extreme events is a very important part of portfolio allocation.

If returns really conformed to mean-variance portfolio theory, then rational investors would be using a lot of leverage (investing with borrowed money). To compensate for the tendency of mean-variance portfolio theory to recommend risky portfolios, we usually choose a low value for the standard deviation of portfolio returns we can tolerate. This helps but isn’t a perfect solution.

There are other stable distributions that do a better job of modeling extreme investment returns. Unfortunately, they are harder to work with. In fact, their standard deviations are infinite.

Statman may be right that the primary reason why people deviate from portfolios optimized by mean-variance portfolio theory is that they seek expressive and emotional benefits. However, trying to protect portfolios against extreme events is another good reason.


  1. Hi MJ. I haven't read the book you refer to, but in my work, I've helped a number of institutional investors make such portfolio choices. The "accepted standard of care" is mean-variance portfolio theory as the optimization approach, with scenario stress-testing to set boundary conditions for extreme conditions (which, as you say, are not adequately covered by MVPT).
    In my experience, the dirty little secret isn't these days that extremes would be neglected, but that both in the mean-variance calibration as well as in the stress scenarios, there is great sensitivity to the historical timeframes used to choose the assumptions. And so your answer is analytically quite unstable, provoking -- not absurdly -- decisionmakers to deviate from the analytics much more cheerfully.
    In addition, while most pension funds have admirably long time horizons, driven by their long-term asset-liability matching, their management teams, Boards, and other stakeholders do not. The fund may well have 20+ years to recover from a misstep, or a market downturn may be a genuine buying opportunity, but it's hard to live that with zen-like calm when your mark-to-market valuation is down 25%, journalists are asking whether retirees should be worried -- and when, as usual, if you look hard enough, there is something about your decisions that in retrospect makes you have egg on your face.

    1. @Martin: The fundamental problem is an inability to know the probability distribution of future returns. The fact that returns are not lognormal is usually dealt with by choosing a low target for variance (otherwise the model calls for absurd levels of leverage). Having "solved" that problem, the next is to choose expected return, variance, and correlation parameters for the different asset classes. Unfortunately, as you say, the computed "optimal" allocation is very sensitive to these parameters. The best we can do is to use historical returns from some period to estimate these parameters, but all that matters is the future. In the end, mean-variance portfolio theory makes the whole process look scientific, but it's questionable whether the optimization has much influence over the final allocation chosen. And that's probably a good thing.

  2. I experimented with MPT when setting up a strategic allocation years ago. Seeing the allocation was extremely sensitive to the correlation coefficients, and realizing those are not well known and actually should be modeled as time varying, I ignored its results.

    Even within a particular asset class, annual returns are non-stationary process. Also, multi-year returns are not independent (e.g. the business cycle), and MPT does not account for that. From time to time, I use a non-Markov process that builds in mean reversion in an ad hoc way, inserting that into a Monte-Carlo simulation I run. But any particular choice is arbitrary and does not necessarily model market returns well enough to be of value in the Monte Carlo results.

    1. @Anonymous: My experience was similar. When I assigned ranges to expected returns, variances, and correlations, the range of portfolio percentages was wild. I found no useful way to use MPT.