Morningstar Advisor - April/May 2010 - 24

Gray Matters

Third, we substitute conditional value at risk, which only looks at tail risk, for standard deviation, which looks at average variation. Fourth, the Markowitz model used a covariance matrix to model the distribution of returns on asset classes; we replace this with a scenariobased model that can be generated with Monte Carlo simulation and can incorporate any number of distributions. Finally, we exploit new statistical technologies pioneered by Savage in the field of probability management. Savage invented the Distribution String, or DIST, which encapsulates thousands of trials as a single data element or cell. It eliminates the main disadvantage of the scenario-based approach—the need to store and process large amounts of data.
The Scenario Approach

The scenario-based approach has two main advantages over a distribution-curve approach. One, it is highly flexible. Nonlinear instruments such as options, for example, can be modeled in a straightforward manner. Second, it is mathematically manageable. For example, portfolio returns are simply weighted averages of asset-class returns within the scenarios. In this way, the distribution of a portfolio can be derived from the distributions of the asset classes without working complicated equations that might lack analytical solutions; only straightforward portfolio arithmetic is needed. In standard scenario analysis, there is no precise graphical representation of return distributions. Histograms serve as approximations, such as those shown in Exhibit 1. We augment the scenario approach by employing a smoothing technique so that smooth curves represent return distributions. Exhibit 2 shows the distribution curve of annual returns for large-company stocks under our approach. Comparing Exhibit 2 with the large-company-stock histogram in Exhibit 1, we can see that the smooth distribution curve retains the properties of the historical distribution while showing the distribution in a more aesthetically pleasing and precise form. Furthermore, our model makes it possible to bring all of the power of continuous mathematics (previously enjoyed only by models based on continuous distributions) to the scenario approach. In Exhibit 2, the green line is what we get when we use mean-variance analysis and assume that returns follow a lognormal distribution. The blue line is what we get when we use our smoothed scenario-based approach. The area under the blue solid line to the left of the red vertical segment shows that the 5th-percentile return under our model is negative 25.8%, meaning there is 5% probability of a return of less than negative 25.8%. Under the lognormal model, however, the probability of the return being less than negative 25.8% is only 1.6%.

This illustrates how a mean-variance model can woefully underestimate the probability of tail events. As Kaplan et al. (2009) discuss, tail events have occurred often throughout the history of capital markets all over the world. Therefore, it is important for asset-allocation models to assign nontrivial probabilities to them.
Reward Over the Long Term

The second enhancement we make to MVO is to use geometric mean. In MVO, reward is measured by expected return, which is a forecast of arithmetic mean. Over long periods of time, however, investors are not concerned with simple averages of return; rather, they are concerned with the accumulation of wealth. We use forecast long-term geometric mean as the measure of reward, because investors who plan on repeatedly reinvesting in the same strategy over an indefinite period would seek the highest rate of growth for the portfolios as measured by geometric mean.
Downside of Standard Deviation

One of the limitations of the traditional mean-variance optimization framework is that it assumes that the distribution of returns for the assets in the optimization can be described simply by mean and variance alone. The most common depiction of this assumption is to draw the distribution of each asset class as a symmetrical bell-shaped curve. As illustrated in Exhibit 1, however, the return distributions of different asset classes don’t always follow a symmetrical bell-shaped curve. Some assets have distributions that are skewed to the left or right, while others have distributions that are skinnier or fatter in the tails than others. Over the years, various alternatives have been put forth to replace mean-variance optimization with an optimization framework that takes into account the non-normal features of return distributions. Some researchers have proposed using distribution curves that exhibit skewness and kurtosis (that is, ones that have fat tails), while others have proposed using large numbers of scenarios based on historical data or Monte Carlo simulation.

Our third enhancement deals with risk. Much has been written about how investors are not concerned merely with the degree of dispersion of returns (as measured by standard deviation), but with how much wealth they could lose. Many “downside” risk measures have been proposed to replace standard deviation as the measure of risk in strategic asset allocation. While any one of these could be used, our preference is to use conditional value at risk (CVaR). CVaR is related to value at risk (VaR). VaR describes the left tail in terms of how much capital can be lost over a given period of time. For example, a 5% VaR answers a question of the form: Having invested $10,000, there is a 5% chance of losing $X or more in 12 months. (The “or more” implications of VaR are sometimes overlooked by investors, with serious consequences.) Applying this idea to returns, the 5% VaR is the negative of

24 Morningstar Advisor April/May 2010
Morningstar Advisor - April/May 2010

Table of Contents for the Digital Edition of Morningstar Advisor - April/May 2010
