PART TWO

RISK FORECASTING

To measure risk as the standard deviation of returns is fraught with issues. But if we understand its limitations, this measure has the advantage of simplicity.

—JPP

Risk is easier to forecast than return. For example, based on a very large sample of daily US stock returns data from 1927 to 2018, the correlation in volatility from one month to the next is +69%.¹ This number is much higher than any of the one-month forward correlations we’ve discussed so far on the return side. That’s without even trying: I just used the equal-weighted 21-day volatility as a predictor of volatility over the next 21 days. A caveat is that volatility is not always a good proxy for risk, especially if we define risk as exposure to loss. But ultimately, risk models are more deterministic than return models, which often require judgment calls, as I’ve argued in Part One of this book.

Therefore, many economists and financial analysts focus on risk models to satisfy their “physics envy.” The ARCH model and its many extensions are an example of a class of models where advanced mathematics seems to add predictive value. Mark Kritzman, in a 1991 Financial Analysts Journal column titled “What Practitioners Need to Know . . . About Estimating Volatility, Part 2,” demystifies the approach. His introduction is worth quoting directly:

Circle the correct answer:

Autoregressive Conditional Heteroscedasticity (ARCH) is:

1.   a computer programming malfunction in which a variable is assigned more than one definition, causing the program to lock into an infinite loop;

2.   a psychological disorder characterized by a reversion to earlier behavior patterns when confronted with unpleasant childhood memories;

3.   in evolution, a reversion to a more primitive life form caused by inadequate diversity within a species;

4.   a statistical procedure in which the dependent variable in a regression equation is modeled as a function of the time-varying properties of the error term.

To the uninitiated, all the above definitions might seem equally plausible. Moreover, the correct definition, (4.), may still not yield an intuitively satisfying description of ARCH.

This column is intended as a child’s guide to ARCH, which is to say that it contains no equations. Our goal is to penetrate the cryptic jargon of ARCH so that at the very least you will feel comfortable attending social events hosted by members of the American Association of Statisticians. Of course, you should not expect that familiarity with ARCH will actually cause you to have fun at such events.

I suppose the Financial Analysts Journal was more relaxed back then. Before I interviewed with him for the first time, I read all Mark’s papers. The last sentence made me spit out my coffee: “Of course, you should not expect that familiarity with ARCH will actually cause you to have fun at such events.” For years I’ve tried to emulate his clever sense of humor, during conference presentations for example, with inconsistent success.

In a 2005 review of the literature on how to forecast risk, Ser-Huang Poon and Clive Granger summarize 93 published papers on the topic. Think of Poon and Granger’s article as the summary of a giant, multiyear, multiauthor horse race to find the best model. They compare the effectiveness of historical, ARCH, stochastic volatility, and option-implied models.

Historical models include the random walk model, which I used in my example on US stocks when I simply assumed that next month’s volatility would be the same as this month’s (plus or minus some unpredictable noise term). This model is perhaps the easiest to implement, and Poon and Granger conclude that it’s very tough to beat. To clarify, this model doesn’t assume volatility is random. It simply assumes that the most recent observation is the most relevant to forecast volatility over the following period—it assumes persistence. Other versions of the historical models use averages of past volatilities. These averages can be equal-weighted, they can be exponentially weighted (where more weight is put on recent observations), or they can be based on some form of fitted weights that maximize predictability. Historical models perform better than ARCH-type models in at least half the studies reviewed by Poon and Granger.

However, the ARCH models more explicitly fit the short-term persistence (predictability) in volatility, sometimes referred to as volatility clustering. [In 2003, the model’s inventor, Robert F. Engle, won the Nobel Prize for “methods of analyzing economic time series with time-varying volatility (ARCH).”] Stephen Marra (2015) explains that conditional heteroscedasticity “refers to the notion that next period’s volatility is conditional on the volatility in the current period as well as the time varying nature of volatility.” Then he explains that the generalized autoregressive conditional heteroscedasticity, or GARCH version, “incorporates changes in the error term—or fluctuations in volatility—and tracks the persistence of volatility as it fluctuates around its long-term average.” Several other versions of the ARCH model have been proposed to incorporate fat tails, asymmetries in volatility (the fact that volatility spikes up more than down), exponential weights, dynamic correlations, etc.

However, Marra shows that for US stocks, most sophisticated models, whether of the historical or ARCH classes, barely outperform the random walk model. The differences in model effectiveness don’t look statistically significant. Other issues with sophisticated models include publication bias (only the good results get published), as well as a related, important issue: the possibility that these models may overfit the in-sample data.

It’s hard to argue that one specific model should perform consistently better than to simply extrapolate recent volatility. Aside from a slight advantage for volatility estimates derived from options prices, Poon and Granger find that across 93 academic studies, there’s no clear winner of the great risk forecasting horse race. And option-implied volatilities aren’t a silver bullet either. Options don’t trade for a wide range of assets anyway. In contrast with Poon and Granger’s conclusion, Marra’s results reveal that the option- implied approach performed the worst out of the eight models he backtested.

Of course, while no single model has surfaced as the most effective, some models perform better than others for specific asset classes and during specific time periods or market regimes. But in the end, the important takeaway remains that short-term volatility, from one month to one year ahead, is quite predictable, even with simple models. We’ll discuss longer-term forecasts shortly, but for now, let us discuss how investors can take advantage of this short-term predictability.

Note

1.   Fama-French data library (http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html). Time period: January 7, 1927 to June 29, 2018. Stock returns are net of the risk-free rate.