In 1952 Harry Markowitz introduced to the investment community a new approach for constructing portfolios. His innovation, which today is known as mean-variance optimization, requires portfolio managers to estimate expected returns, standard deviations, and correlations. With this information, Markowitz showed how to combine assets optimally so that for a particular level of expected return the optimally combined assets would offer the lowest possible level of expected risk. A continuum of these portfolios plotted in dimensions of expected return and standard deviation is called the efficient frontier.

In the years following Markowitz' innovation, the investment community gradually embraced his prescription for constructing portfolios, but with an important caveat. To the extent that the recommended solution departed from industry norms or prior expectations, investors typically imposed constraints to force a more palatable solution. For example, an investor might instruct the optimizer to find the combination of assets with the lowest standard deviation for a particular expected return, subject to the constraint that no more than 10% of the portfolio be allocated to foreign assets or non-traditional investments and that no less than 40% be allocated to domestic equities. The reason for such constraints is that investors are reticent to depart from the crowd when there is a significant chance they will be wrong. The matrix below illustrates this point.

If we accept that investors care not only about how they perform in an absolute sense, but also about how their performance stacks up against other investors, there are four possible outcomes. An investor achieves favourable absolute returns and at the same time outperforms his or her peers, which would be great, as represented by Quadrant I. Alternatively, an investor might beat the competition but fall short of an absolute target (Quadrant II). Or, an investor might generate a high absolute return but underperform the competition (Quadrant III). These results would probably be tolerable because the investor produces superior performance along at least one dimension. However, what might be very unpleasant is a situation in which an investor generates an unfavourable absolute result and at the same time performs poorly relative to other investors (Quadrant IV). It is the fear of this outcome that induces investors to conform to the norm. A superior approach to constructing a portfolio is to encompass both absolute and relative measures of risk in an unconstrained optimzation process.

In 1995 George Chow introduced such a technique.1 In order to describe Chow's innovation, it first might be helpful to review how it is that an optimizer identifies the best combinations of assets.

Optimization is a process by which we determine the most favourable tradeoff between the competing interests of risk reduction and return enhancement. The fulcrum that balances these interests is called risk aversion. An optimizer identifies the optimal portfolio by maximizing the following quantity:

Expected Return - Risk Aversion X Risk
The optimizer accepts our assumptions about the expected returns, standard deviations, and correlations of the various assets and then figures out the allocations that yield the highest value for the above expression. The quantity called risk aversion measures how many units of expected return we are willing to give up in order to reduce risk by one unit. If we are infinitely risk averse, the optimizer will identify the portfolio with the lowest volatility, and if we are infinitely risk tolerant, the optimizer will select the asset with the highest expected return. By gradually reducing the value for risk aversion from an arbitrarily high value to zero, the optimizer will trace out a continuum of portfolios in dimensions of expected return and standard deviation that comprise Markowitz' efficient frontier.

Let's now return to the issue of wrong and alone. When investors employ an optimizer to identify portfolios along the efficient frontier, they instruct it to abide by constraints that reflect their notion of normalcy. This constrained optimization is an ad hoc procedure for dealing with aversion to tracking error. Tracking error is a measure of relative risk. Just as standard deviation measures dispersion around an average value, tracking error also measures dispersion, but instead around a benchmark's returns. It is literally the standard deviation of relative returns.

If we care only about relative performance, we could define our returns net of a benchmark and optimize in dimensions of expected relative return and tracking error. This approach would address our concern about deviating from the norm, assuming our benchmark represents normal investment choices. However, we would fail to address any concern we might have about absolute results. Chow proposed an elegant solution. He showed how to augment the quantity to be maximized by the optimizer to include both measures of risk, to wit:

Expected Return - Risk Aversion X Risk - Tracking Aversion X Tracking Error2
This measure of investor satisfaction simultaneously addresses concerns about absolute performance and relative performance. Instead of producing an efficient frontier in two dimensions, however, this optimization process, which Chow calls mean-variance-tracking error optimization, produces an efficient surface in three dimensions: expected return, standard deviation, and tracking error.

The efficient surface is bounded on the upper left by the traditional mean-variance efficient frontier, which is comprised of efficient portfolios in dimensions of expected return and standard deviation. The leftmost portfolio on the mean-variance efficient frontier is the minimum risk portfolio. The right boundary of the efficient surface is the mean-tracking error efficient frontier. It comprises portfolios that offer the highest expected return for varying levels of tracking error. The leftmost portfolio on the mean-tracking error efficient frontier is the benchmark portfolio, because it has no tracking error. The efficient surface is bounded on the bottom by combinations of the minimum risk portfolio and the benchmark portfolio. All of the portfolios that lie on this surface are efficient in three dimensions.

This approach will almost certainly yield a result that is superior to constrained mean-variance optimization. For a given combination of expected return and standard deviation, it will produce a portfolio with less tracking error. Or for a given combination of expected return and tracking error, it will identify a portfolio with a lower standard deviation. Or finally, for a given combination of standard deviation and tracking error, it will find a portfolio with a higher expected return than a constrained mean-variance optimization. In fact, the only way in which Chow's procedure would fail to improve upon a constrained mean-variance optimization is if the investor knew in advance what constraints were optimal. But, of course, this knowledge could only come from a mean-variance-tracking error optimization.

Quiet vs. Turbulent Regimes
We measure returns as a function of units of time such as days or months. This approach is somewhat arbitrary in that in some return intervals there were no significant events to cause asset prices to change. The returns in these periods merely reflect noise. In other intervals, however, several important events may have occurred. The typical estimation of standard deviations and correlations assigns as much weight to the intervals with no significant events as it does to the event filled intervals.

It may be informative to distinguish event-related returns from returns arising merely from noise and to estimate risk parameters from these event-related returns. One approach is to associate "multivariate outliers" with event-related returns. An outlier, given a return series for a single asset, is straightforward to identify. It is simply a return that falls outside a chosen confidence interval around the expected return. For example, if we wished to select outliers based on a confidence interval of 25%, we would simply select the returns that fall within the tails comprising 25% of the return distribution, 12.5% on either side.

A multivariate outlier, by contrast, represents a set of contemporaneous returns across several assets that is unusual for one or more reasons. Perhaps just one of the returns is significantly above or below its mean. Alternatively, a pair of returns that on average are highly correlated may be sufficiently different to render the period unusual. Thus a multivariate outlier could reflect the unusual performance of one or more assets in isolation, or an interaction that is out of character for a particular combination of assets.

Once we have identified a sample of outliers3, we can use these to estimate standard deviations and correlations that are more likely to coincide with turbulent periods. We can also use the remaining "inside" observations to estimate standard deviations that are likely to be associated with quiet times.

Endnotes
1. G. Chow, "Portfolio Selection Based on Return, Risk, and Relative Performance," Financial Analysts Journal, March-April 1995.

2. Technically, risk is equal to variance or standard deviation squared and tracking error should be squared in this expression.

3. The methodology for identifying multivariate outliers is described in, "Chow, G., E. Jacquier, M. Kritzman, and K. Lowry, "Optimal Portfolios in Good Times and Bad," Financial Analysts Journal, May/June 1999.

Mark Kritzman is a founding partner of Windham Capital Management and a managing partner of State Street Associates, LLC.