Recent episodes in the financial markets have preceded a flourish of commentary and analysis about market behavior during periods of high volatility. The majority of these analyses conclude the correlations between markets rise when any major market experiences a large negative return. The best example of this was October 1987. This leads to the question "Why does portfolio diversification disappear just when I need it?"
Our research was not intended to answer that question, but rather to test the validity of its underlying assumption. Do the markets behave differently during extreme periods as opposed to normal times?
While it is easy to use anecdotal descriptions, objective analysis requires gathering information from all extreme events. To do this, we need a definition of extreme and normal periods. One way is to define any returns that lie within two standard deviations of the mean return as normal, or inside observations, and the remaining returns as extreme, or outliers. We show this in EXHIBIT 1 for monthly returns of the U.S. stock market from 1973 to 1998.
Defining extremes
The outliers are shown in red and include the -20% return of October 1987 as well as the 16% return of October 1974. This approach can be used to define extreme months based on the returns of the U.S. stock market. However, we are interested in extreme observations across multiple investments. What about months that were inside observations for U.S. stocks but included large movements in U.S. bond returns? EXHIBIT 2 shows the most extreme positive and negative returns of the Lehman Long Government Bond index from 1973 to 1998. Not one of these months registers as extreme for U.S. stocks.
To account for the possibility of extreme events in other markets, we need to perform a more sophisticated sampling technique. What about looking at the months when the bond market experienced returns outside of 2 standard deviations around its mean? One might be tempted to add those periods to the sample of extreme stock market returns. This temptation must be avoided. A simple way to see the flaw of this approach is to imagine looking for extreme months across 1,000 investments. Assuming they are not perfectly correlated to each other, probably every month would be extreme for at least one of these series.
Besides this flaw, the statistics generated from such a method introduce errors in the correlation calculations. Imagine you create random returns for stocks and bonds that are normally distributed. By design we know there is no difference between extreme and normal events. When you choose extreme months based on the returns of any single investment, you create inside and outlier samples that have different correlations.
There also another scenario we want to address. There are events that are not exceptional for any specific investment but are unusual in combination with the correlation across investments. For example, imagine an event where stocks return 1.5 standard deviations above the mean and bonds return 1.5 standard deviations below it. If the correlation between them is 0.4, then this observation is an outlier.
In "Optimal portfolios in Good Times and Bad," published in the Financial Analysts Journal, May/June 1999, we define a sampling approach that fits a multivariate normal distribution to the investment series of interest and determines whether a period is extreme by taking into account the returns across all series. Using this method, some of the months that appear to be outliers based the U.S. stock return are inside observations.
For this method, the number of series, the time period and threshold probability (i.e. how rare an event has to be in order to be called an outlier) affect the results. To test whether markets behave differently during extreme times, we want to compare correlations across markets between outlier and inside months.
We have created an example that uses monthly returns for U.S. stocks, U.S. bonds and Japanese stocks from 1980 to 1998. We chose these investments as proxies for the two most significant choices a global investor has for creating a diversified portfolio - asset class diversification and country diversification. The bar chart above shows the correlation of bonds and Japanese stocks to U.S. stocks for the entire period, inside months, and outlier months.
For the entire period, the correlation of U.S. stocks to Japan (0.27) was lower than for U.S. stocks to bonds (0.32). Country and asset class diversification are about equally significant. Exclude observations that lie in the 5% tails of the fitted distribution and the correlation to U.S. bonds rises to 0.41 while it falls to 0.23 for Japan. Under "normal" conditions, country diversification dominates asset class diversification. For the outlier sample, the correlation to bonds falls to 0.05 while the correlation to Japan rises to 0.38. The benefits of country diversification are reduced during extreme periods, while asset class diversification improves. Therefore, while correlations are different during extreme periods, they do not all move in the same direction. Obviously, this is a broad statement based on one example, but it addresses the validity of the initial question: "Why does diversification disappear just when I need it?" While this might be valid for a global equity investor, it does not hold for an asset allocator. An asset allocator realized greater diversification than expected during extreme periods.
So what do we do now?
Markets behave differently between extreme and normal periods. One can take three simple actions to forecast future correlations:
Exclude outliers. When we use historic returns, we assume that each past observation represents a possible future outcome. Suppose you estimate future risk using the last 20 years of monthly returns. Now go back to 1988 and calculate the risk and correlations of U.S. stocks and bonds. The historical risk and correlations have changed completely because of one month, October 1987. When you use all 240 months to estimate risk and correlation, you implicitly assume an October 1987 event might occur once every 20 years. If you believe October 1987 actually is an event that occurs once every 100 years, then you want to de-emphasize it when calculating your risk and correlation estimates. One simple way to reduce the impact of October 1987 in the forecast is to exclude it. Had you done that, you would have ended up with much better predictions of what actually transpired since 1988.
Focus on outliers. We use equal samples of time in order to define an observation. Our analysis uses monthly intervals. Why not daily? Well, we happen to know that prices are not set every day - on some days the markets are closed. In some months, we experience many significant events, while in others, not much happens. Both of these samples are given equal weight. Many of the "normal" observations merely record the passage of time. If we focus on the outlier sample, we mitigate this problem.
Another rationale for the focus on outliers is that these are the instances when you really care about portfolio diversification. If a plan sponsor receives an unscheduled call from a trustee, is it more likely that the markets are in a normal period or an extreme period?
Blend observations. History is a blend of inside and outlier observations. Maybe you don't think that history will repeat itself. Perhaps you have a view that the future will be riskier than the past. Then you can overweight the outlier sample in the estimate of future risk and correlation. We believe that allowing the investor to define the threshold probability of what makes an extreme event and to set the probability that extreme events will occur in the future is the most effective way to deal with this issue.
