Examining the World through the Lense of Analytics

In part 2 of this series, we examined the return profiles of several simulations of the Martingale strategy and comparing it to a simpler system of flat wagers. Under one set of assumptions, we saw this profile:

Here, the Martingale player will generally outperform the minimum bet player most of the time, but will suffer catastrophic losses a little more than 10% of the time. Let’s continue to use our simulation, but depart from the casino. Assume now that the odds are in the favor of the gambler by flipping the probability of a win to 52%. Here is the full set of assumptions:

In this scenario, the Martingale player is expected to outperform the minimum bet player by nearly four to one:

Now that the odds are in the simulated players’ favor, they are each expected to win. However, the Martingale player has a higher expectation of profit. The system is amplifying the 2% edge.

On favorability of outcomes, the two systems look roughly equivalent, with profits in a little more than three quarters of simulations:

Once again, however, the Martingale player has a higher risk of catastrophic loss. The table minimum player simply cannot lose his entire $10k by betting $20 for 300 rounds. His total possible loss would be $6k, but that is so unlikely that we see no losses in the simulation over $1k. On the other hand, the Martingale player sees a 4% probability of losing everything:

Note that a naïve interpretation of this could be that the probability of complete loss is not statistically significant (technically speaking, if you did a one-tailed test with a 5% p-value). The new histogram shows the return profiles:

The Martingale player may make as much as $3500, a 35% return, while the table minimum player maxes out short of $1200. In any given simulation, the Martingale player is likely to come out with a higher return. However, when he suffers a loss, it will average 18 times as much.

The Martingale system is inherently more risky than the flat bet, but it will likely lead to a higher return. When would this sort of return profile be attractive?

A mutual fund manager is in charge of investing his clients’ money in securities. His job is to "grow" money by assembling a portfolio, and he earns his living by taking a percentage of both assets and profits. The idea is that a dedicated, experienced professional will be able to do the research needed to make good decisions. Giving your money to a fund manager is a well-informed decision similar to having a surgeon take out your spleen or having an architect design your home.

It’s a nice idea, flawed only by its incorrectness. In fact, no fund manager can consistently beat the
market (assuming you ignore those doing insider trading). Nassim Taleb argued convincingly in *Fooled by
Randomness* that even those who appear to consistently beat the market are simply fortunate – if you start
out with tens of millions of investors, you are likely to get one Warren Buffett. You just don’t know who it
will be when you start.

So what is a mutual fund manager to do? No amount of fundamental research will teach you how to predict the future. No matter how hard you study a firm’s filings, you will not know how their market share will change over the next year. Interviewing the board of directors won’t inform you whether the dollar will strengthen and what effect it will have on the firm.

Consider, then, our graph, but re-imagine the options as debt securities. The red investment is a US Savings bond, which used to be quaintly known as a "risk-free" investment. It will give you a modest return with little variability. You don’t know exactly how much it will be worth because you don’t know what will happen with inflation, but you can be fairly sure that you will make some money.

The blue option is a riskier debt security. It may be a Greek government bond or a corporate bond. It may be a subprime mortgage-backed security. Whatever it is, it offers a different return profile than the red option. The expected profit, remember, is over four times as high as the less risky option.

Assume then, that these are the only two options available to a mutual fund manager. In the long run, pursuing the blue option will lead to catastrophe. His fund will lose all of its money. In the short run, however, it may appear profitable. In a given year, he is likely to earn a 30% or 40% return, while a fund manager pursuing the red option would be lucky to crack 10%.

In fact, a money manager’s incentives will push him towards the riskier option. If his gamble pays off, he will be rewarded handsomely in performance fees. He will literally make millions. If it fails, however, it is not his money at risk; only his clients will lose money. In fact, he may even be able to blame the loss on “unforeseeable” events: a housing market crash, tsunami, or any other macroeconomic event. The only thing a fund manager has at risk is his reputation and his job.

In short, a mutual fund manager is likely to pursue the riskier scenario because it is likely to pay off in the short run, which will be long enough to enrich him personally. He also shares little to no risk with his investors.

Furthermore, fund managers are measured against each other. Specifically, they are considered successful if they outperform the average fund manager. Those who fail and close up shop are removed from the average, which means the negative returns from a Martingale-like strategy will be minimally felt. The vast majority of funds pursuing such a strategy will have a positive result in any given year, and those going bust will quickly exit.

It is also important to note that the term “riskier” described in this section is not the same as the models that professional money managers use. Classic portfolio theory states that the way to measure risk is to look at the dispersion of possible outcomes. A low-risk asset will produce returns in a narrow band, while a high risk asset will have a wider range of potential returns.

Put simply, a “safe” stock will either rise or fall a small amount, while a “risky” stock will rise or fall much more. The degree of volatility of a stock is often referred to as its “beta”. This is an elegant theory backed up by well-formed mathematical proofs. Unfortunately, it quite simply does not apply to assets with non-normal return functions.

Fundamentally, this theory assumes that extreme events on either tail of the probability curve are equally likely to occur. The chance of an absolute loss is countered by the chance of an enormous gain. As we have seen, this is simply not true for some types of security. If you loan someone money, they have a non-zero chance of defaulting, but a zero chance of paying you back at a hundred times the agreed-upon interest.

Of course, not all fund managers rely on portfolio management techniques that assume normal returns. Enough of them do, however, to set the standard of performance, as A simple google search will show.

Continue on to Part 5 of our Series, *The General Theory of Martingale*.