## Why You Can’t Optimize Your Investments Using Modern Portfolio Theory

I recently finished reading Anti-Fragile, the follow-up to The Black Swan and Fooled By Randomness. It’s the kind of book I like where I can nod along with half of what it’s saying and violently disagree with the other half. It is a bit strange at times – for example the author says at the start that “this time I don’t care and I’m going to call out everyone who is wrong”, and then he proceeds to use pseudonyms for many people. There are also many points in the book where he says “before I claimed I was going to be technical but I was lying, and now I’m really going to get technical” – and it turns out to be superficial again. Amusingly there is a section at the back that claims to explain things in graphs for people who don’t like words; my estimate is that 20% of the space there is graphs and 80% is writing.

I’ve explained before some of the author’s ideas that I disagree with. However in the back of the book there is a great analysis of why Modern Portfolio Theory is wrong, and the reasoning applies to a lot of other methods people use to make investment decisions. The way MPT is usually applied is to optimize a portfolio by using the past correlations of the asset classes to find the ideal mix that will give you the maximum return for a given amount of risk. At one time there were claims that you could use this to create a mix of stocks that were as safe as bonds. I stopped hearing that around 2008 for some reason.

I’ve been aware of some shortcomings for a while. One is the measure of risk, using volatility to express how risky a portfolio is. This doesn’t cut it for many people. I like volatility because it creates opportunities for profit. The only real risk for me, and many others, is not having enough money whether that’s from actual losses or from subpar investment returns.

Another concern with MPT is that it’s too precise. You can calculate probabilities to 10 decimal places but the result isn’t really meaningful since the uncertainty overwhelms the precision. But the biggest problem goes beyond this. If you calculate the correlation between assets you can get a precise number, but you will find that calculating it over different time periods gives you different results. This means that there is no one right answer and your results could change based on how you choose to do the calculation, which sort of defeats the whole purpose.

Anti-Fragile chimes in with another criticism of MPT that is often overlooked (and Taleb claims that a proper analysis of this error would have prevented the Nobel prize awarded for the idea). The mathematical way to explain it is that using an average value as an input to a function gives you different results than if you compute the function for several possible values and then take the average of those results. To put it more simply, the problem comes from using averages as an input to the formulas.

For a simple example let’s say we expect the S&P 500 index to return 7% in an average year, and bonds to return 3% in an average year. Based on that and known correlations you might say that you want 80% of your portfolio in the S&P 500 and 20% in bonds. Now let’s break down the averages. Let’s say we expect stocks to have returns anywhere from -20% to 34% in one year, and we expect bonds to have returns anywhere from -4% to 10% in one year. If we look at the extremes, in years when stocks return 35% and bonds return -4% we want 100% in stocks. And in years where stocks return -20% and bonds return 10% we would want 100% in bonds. The average of those two allocation is 50% in stocks and 50% in bonds.

The result is is that the average of the allocations you would want in different years is not the same as the allocation you would want in an average year (and average years are rare since approximately 2/3 of years have stock returns that are either negative or above-average). If you apply this reasoning it could lead to all sorts of different conclusions, from a shifted allocation to deciding that you don’t want to own stocks at all because the results in some years are unacceptable even if the average is good. And that all comes from understanding that the real world is messier than the neat calculations used in MPT.

For a while I’ve used rules like this in many areas. It might seem too conservative (or sometimes even too risky). But it pays off when something unexpected happens. And something unexpected always happens; if you look back at the biggest drop in the stock market, it was unexpectedly large since there had never been a drop that large before. This tells us that the biggest fall in the past is not the biggest one possible.

Instead of needing endless calculations with precise results that only make sense if everything happens as predicted, I prefer knowing that I’m prepared for just about anything that could happen. That doesn’t necessarily mean a portfolio with lots of bonds and gold. Right now I have very little in bonds because I would rather own stocks even if they are falling. I’m comfortable with the full range of possible outcomes from owning stocks but not all the potential outcomes from owning bonds. This principle is the reason I can stick to my decision even if everyone else is saying it’s not optimal.

I agree that optimizing real world portfolios along the efficient frontier is just madness. However, MPT does have important insights for investors. The key one in my opinion is the insight that one should not look at the risk return characteristics of an asset class in isolation but also at how an asset class interacts with the rest of the portfolio. And to be fair, it seems to me that Markowitz was well aware that expected returns and correlations might be quite different from past values.

http://www.math.ust.hk/~maykwok/courses/ma362/07F/markowitz_JF.pdf

Thanks for the comment CC! It sounds like the theory helped spread some other good ideas such as asset allocation as a fundamental part of the portfolio. Unfortunately some people take that to mean that everything associated with it is universally applicable. As long as we avoid that misconception we can do well.

I think your point that MPT is not strictly “universally applicable” is an important point. MPT is a statistical model built on general principles, not a deterministic equation with a single best answer. MPT results (plural) depend on assets under consideration, time horizon (preferably long to capture business cycles), and the investor’s risk tolerance. That said, the precision/confidence/significance/stability/uncertainty of MPT results are solely dependent on the algorithm implementation and statistical metrics of interest, not the underlying MPT model. In short, MPT is a tool, and the user bears some responsibility for using the tool properly. It is a fair criticism that historical prices reflect past market conditions, but the virtue is that historical prices are 1) objective data, not speculation, and 2) real price dynamics in response to real market conditions (albiet past conditions). Also, it is not surprising that actual performance departs from MPT results in any given year because MPT is a statistical model. The anomalies are best approached with rebalancing to cash in winners and position losers for a rebound. Finally, it is regrettable that someone claimed MPT can reduce stock volatility to that of bonds. MPT can not reduce risk to arbitrary low levels. Properly used, MPT is a tool for reducing non-systematic risk among assets, but it can not protect from a broad economic downturn that hits all markets (equity, credit, real estate, commodity), as happened in 2008. Anyway, the strongest endorsement for MPT is that numerous mutual funds, hedge funds, trusts, and endowments use the principles of MPT to balance and reduce risk. I hope this helps explain the value of MPT, and the potential for confusion by over-interpreting isolated results.