Models have the capacity for behaving badly, warns Emanuel Derman. Yet modeling market behavior is still essential, even if it comes with risk. The solution is to be aware of modeling’s limitations and act accordingly. There’s danger here if we let ourselves become blinded by the numbers, but modeling can tell us a lot about the risks we face.

As a simple example, consider the profile of the stock market in terms of its return distributions. The standard practice is to assume a normal distribution. If that was an accurate description of how markets behave, extreme losses would relatively rare and the distribution of negative and positive returns would be symmetric. That’s a good first approximation, but no one should stop there. Indeed, history isn’t always destiny, but it can’t be ignored either.
Average monthly returns for U.S. stocks since 1881 are tightly clustered between the range of -5% and +7.5%. This range accounts for 95% of returns. Of those returns, more than 80% are flat to positive. In other words, history suggests that monthly returns are likely to be moderately in the black for any given month in the long run.

If we shift the investment horizon to longer-term holding periods, the return distribution leans further into positive territory. For example, the second chart below shows how 10-year periods stack up for the U.S. stock market. Performances in the range of zero to 12.5% represent nearly 89% of the distribution.

Profiling the market’s long run performance distribution provides us with some useful context, but it’s not terribly practical if we want to understand the limits to this sunny profile. Risk modeling for most investors is focused on one question: How bad can it get?
It’s well known that financial markets suffer from tail risk and so it’s only natural that we spend some time reviewing these events as a rough guide for thinking about the future. An obvious starting point is profiling the worst-case scenarios. For instance, on a monthly basis, the deepest 5% of losses for the equity market since 1881 range from a 5.9% retreat down to a crushing 26.5% implosion. Yes, it can get quite ugly, even for average monthly results. If you’re expecting a normal distribution, you’re asking for trouble… eventually.

Applying a conditional Value-at-Risk, or CVaR, analysis of the historical data for the 5% tail tells us that the weighted average of the expected loss is 9.7%. In other words, we should be prepared for a monthly loss of around 10% on average if our investment horizon for U.S. stocks is one month.
How do the numbers compare if the holding period is 10 years? The next chart summarizes the results.

If we have the discipline to hold pat for 10 years, the expected loss in the 5% tail drops to 4.9%, based on a weighted average of the historical data. The message is that by increasing the holding period to 10 years from one month cuts the average expected loss in the worst 5% of cases by around one-half.
The question is whether the 50% drop in the expected loss in the tail by committing to a dramatically longer holding period is a worthwhile tradeoff? The answer isn’t obvious, or at least not if we’re looking for a general rule for every investor. Risk tolerance varies from investor to investor, as do financial objectives, net worth, and other factors. Even if we’re using the average investor as a guide, there are other ways beyond increasing the holding period to manage risk. Still, if we’re intent on developing additional perspective on this topic the next step might be estimating CVaR in the 5% tail for a range of different holding periods and comparing the results, perhaps with an eye on choosing the optimal investment horizon for a given set of investor assumptions.
History can only reveal so much about the future, of course. All models behave badly at times in part because all models are slaves to historical data in some degree. Models can help us minimize the element of surprise, but they can’t tell us what the future holds. Understanding the distinction is the first step for intelligently modeling market behavior.
Ultimately, we, the humans, are in charge–not the models. Investors too often act as if the relationship works in reverse. As Derman observes in Models.Behaving.Badly: Why Confusing Illusion with Reality Can Lead to Disaster, on Wall Street and in Life, “One has to treat people as responsible for their actions, and yet also recognize that they can’t help what they do.”
Some hazards in finance (and life) simply aren’t subject to quantification.