The infallible portfolio?

In this short essay, Ricardo Fenholz of Columbia University makes what seems (to me at least) to be a rather incredible claim: that it's relatively easy to construct a portfolio that is guaranteed to outperform the S&P 500 over one year (or any other interval you like) and also has a limited downside during that year. The idea is the take the S&P 500 Index and tweak it a little, creating a portfolio with less weight in stocks with higher capitalization, and more weight in those with less capitalization, and presto -- you have something guaranteed to outperform the S&P 500, he claims. Is that possible? That easy? Here's a little more detail:

To understand how this works, let’s consider the S&P 500 U.S. stock index. Suppose that we wish to invest some money in S&P 500 stocks for one year. Currently, Apple has a total market capitalization of roughly $500 billion, making it the largest stock in the S&P 500 and equal to approximately 4% of the total capitalization of the entire index. Suppose that we believe it is very unlikely or impossible that either Apple or any other corporation’s capitalization will be equal to more than 99% of the total S&P 500 capitalization for this entire year during which we plan to invest. As long as this turns out to be true, then it is actually pretty simple to construct a portfolio containing S&P 500 stocks that is guaranteed to outperform the S&P 500 index over the course of the year and that has a limited downside relative to this index. In essence, we can construct a portfolio that will never fall below the value of the S&P 500 index by more than, say, 5% and that is guaranteed to achieve a higher value than the S&P 500 index by the end of the year.[1]

This is not a trivial proposition. If we combine a long position in this outperforming portfolio together with a short position in the S&P 500 index, then we have a trading strategy that requires no initial investment, has a limited downside, and is guaranteed to produce positive wealth by the end of the year. According to standard financial theory, this should not be possible.[2] Furthermore, the assumptions that guarantee that our portfolio will outperform the S&P 500 index appear entirely reasonable. After all, not for one day in the more than 50-year history of the S&P 500 has one corporation’s market capitalization come anywhere close to equaling even 50% of the total capitalization of the market. A 99% share of total market capitalization would essentially amount to there being only one corporation in the entire U. S. for an entire year. This seems like neither a likely outcome nor one that investors should take seriously when constructing their portfolios.

What does a portfolio made up of S&P 500 stocks that is guaranteed to outperform the S&P 500 index look like? There are many different ways in which such a portfolio can be constructed, but one feature common to all such portfolios is that relative to the S&P 500 index itself, they place more weight on those stocks with small total market capitalizations and less weight on those stocks with large total market capitalizations. The weight that an index such as the S&P 500 places on each individual stock is equal to the ratio of that stock’s total market capitalization relative to all stocks’ total market capitalizations taken together. In the case of Apple, then, the S&P 500 index would place a weight of roughly 4% in this individual stock while those portfolios that use HFT to outperform this index would instead place a weight of less than 4% in Apple stock.

The only condition for this to work, he suggests, is that the assumption that no stock in the market comes to dominate the market in the sense that its market capitalization comes to be a high fraction of that of the entire market. This is, as he notes, a fairly weak assumption, although the weaker you make the assumption, the longer the time interval over which this idea apparently works.

Now, I'm not doubting the veracity of this claim. I'm just stunned that such a simple recipe could work, and can't see the intuition behind it. What if the high cap stocks happen to perform brilliantly next year, relative to the lower cap stocks? Wouldn't this portfolio with underweighted high cap stocks then underperform the S&P Index? I've had a quick look at the paper Fernholz references as a detailed support of his claim, and there he explains the conditions for the theorem to hold in slightly different terms:

The conditions mandate, roughly, that the largest stock have "strongly negative" rate of growth, resulting in a sufficiently strong repelling drift away from an appropriate boundary; and that all other stocks have "sufficiently high" rates of growth.

That sounds very different from the quite plausible assumption about no market dominance of a single stock. Indeed, this seems like saying that, if one assumes that large cap stocks will perform poorly, and small caps stock better, then we can build a portfolio guaranteed to outperform the S&P 500 Index by weighting small cap stocks more heavily. Isn't that like assuming we know the future?

But maybe I'm wrong. I'd be interested in the thoughts of others. The paper is quite dense and light on intuitive discussion of the logic. Fernholz suggests that perhaps the existence of these superior portfolios -- which require continuous rebalancing by high-frequency buying and selling of many stocks -- explains some of the very high profits consistently earned by quantitative high-frequency hedge funds such as Renaissance Technology's Medallion Fund. I think I find more convincing the analysis of Lo and Khandani which seemed to suggest that much of the performance of quant hedge funds over the past decade or so can be accounted for by fairly vanilla long short equity strategies, with increasing use of leverage in the mid 2000s (used to maintain high reported earnings even as raw earnings fell off due to competition).

Why diversification doesn't work

You're standing in your canoe, on a beautiful Canadian lake, taking photos of the wildlife, occasionally fishing. Why standing, not sitting? Well, you've read about those disturbing studies that show how sitting is really bad for your long term health; how every hour of television viewing, for example, takes about 20 minutes off your life expectancy, and why the same is probably true for sitting at the computer, sitting reading a book, whatever. So you're standing and that's OK because you're balanced and stable, with your weight distributed uniformly.

Of course, anyone with even a few minutes of experience in a canoe knows this isn't as safe as it seems. What really matters isn't how well-balanced you are when the canoe rests peacefully, but what happens when a few waves come along, kicked up by rednecks passing in a souped-up bass trawler (I lived in rural Virginia for several years, so I know the experience). As you shift your stance to stay upright, and the boat shifts, that balanced distribution vanishes and you can easily tip. Stability demands balance in the midst of the boat's dynamics, not only in the static peace beforehand.

As it turns out, this same lesson applies to investment portfolios -- a new paper in Nature Scientific Reports shows just how important this insight may be.

Famously, of course, Harry Markowitz introduced the idea of diversification into investing back in the 1950s (at least he formalized the idea, which was probably around long before). Using information on the mathematical correlations between the returns of the different stocks in a portfolio, you can choose a weighted portfolio to minimize the overall portfolio of volatility for any expected return. This is maybe the most basic of all results in mathematical finance.

But it doesn't work; it suffers from the same problem as the balanced man in the canoe. This is clear from any number of studies over the past decade which show that the correlations between stocks change when markets move up or down. If the market suddenly plunges downward, you would hope that your well-diversified portfolio, invested as it is in stocks that tend to move unlike one another, would be OK. But when markets move significantly down (or up), it turns out, the correlations are no longer what they were. Trending markets induce strong correlations among stocks that aren't there beforehand, and aren't obvious from long-term averages. So the risks to a portfolio are actually much larger than the simple diversification analysis suggests -- just as the risk of a canoe tipping is much more than it seems to a man standing balanced on a peaceful lake.

The new paper by physicist Tobias Preis and colleagues makes this point with probably the largest data set used so far, looking at the stocks in the DJIA over about 70 years. It's a fairly simple analysis (modulo some nitty gritty details). Roughly, they look at the correlations between different stocks in the DJIA and see how these correlations depend on the recent average return of the DJIA. Are the correlations stable? Or do they go up as the market begins to move? The figure below showing the average correlation coefficient versus the return indicates that the result is clearly the latter: a trending market, in either direction, induces significant correlations among the DJIA stocks.

One of the interesting things here is that this link holds on many different timescales, from 10 days up through two months. The worrying thing for an investor, of course, is that these correlations make the risks of large losses significantly larger than they would appear to be on the basis of long-term correlations alone. As the authors conclude:

... a “diversification breakdown” tends to occur when stable correlations are most needed for portfolio protection. Our findings, which are qualitatively consistent with earlier findings^{42, 44} but quantitatively different, could be used to anticipate changes in mean correlation of portfolios when financial markets are suffering significant losses. This would enable a more accurate assessment of the risk of losses.

 As any canoeist knows, dynamics really matter.