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It's Hard to Distinguish
Skill from Luck

Seven Quantitative
Insights into Active
Management -
A Summary



Forecast Return
Distribution in Aegis
Risk Manager 2.0

The Market
Impact Model™ - Part 3

Automating the
Investment Process



Ron Kahn Looks
Back on 11 Years
at BARRA


The BARRA Brainteaser
for Summer 1998
Solution to the Spring
1998 Brainteaser



A Beautiful Mind

Forecast Return Distribution in Aegis Risk Manager 2.0

by Claes Lekander and Paul Green

With the release of the BARRA Aegis System™ Risk Manager 2.0, BARRA has introduced the new capability of creating forecast return distributions. This feature takes the art of equity risk forecasting several steps beyond traditional risk measures and provides a more complete picture of portfolio risk. By applying the multiple-factor risk model in new ways, we have implemented a methodology that estimates all potential portfolio return outcomes and their associated probabilities. From the forecast return distribution we derive the following key summary measures of portfolio risk previously not available in the Aegis System: Value at Risk (VAR)/Return at Risk (RAR), skewness, and kurtosis.

The VAR and RAR measures indicate the minimum value or return that, at a user-defined level of probability, is at risk of being lost. RAR is simply the return from our cumulative return density function at the specified probability level. Multiplying RAR by the portfolio value gives us VAR. Skewness captures the extent to which returns are asymmetrically distributed, and kurtosis represents the degree of “peakedness” of the distribution.

Background

BARRA risk forecasts have traditionally focused on a single measure of risk: the standard deviation of return. This number, when computed versus a benchmark, is often called active risk or tracking error. It has been a common practice to infer return probabilities from this standard deviation by assuming that returns are normally distributed. For example, an index fund with a tracking error of 2% would be expected to return +/- 2% of the index return roughly two-thirds of the time.

Our research has shown that this connection of tracking error and probability, focusing on the center of the distribution, is fairly accurate for the majority of portfolios. The analysis is less accurate at the tails of the distribution. Observed equity portfolio return distributions tend to exhibit “fat tails”—meaning that extreme returns are more likely to occur than is predicted by the normal distribution. When relying on RAR or VAR as a measure of risk, the type of distribution used can significantly impact the decisions that are made. By not accounting for the presence of “fat tails,” the assumption of normality will generally cause the RAR/VAR parameters to be underestimated. Our newly implemented methodology more accurately represents the true distribution of portfolio returns and, as a result, leads to more reliable VAR/RAR analysis.

We use an empirical approach to estimate the forecast return distribution. The actual shape of the distribution depends on the underlying risk exposures and will therefore vary from one portfolio to another. Instead of historical portfolio returns, our methodology combines current portfolio holdings and BARRA multiple-factor modeling to compute better estimates of future return distributions. The details of our methodology are described in the next section.

Modeling approach

Estimating the forecast return distribution is a three-step process. We begin by simulating a distribution based only on common factors. We then merge this distribution with a normal distribution representing specific returns. In the third and final step, we adjust the distribution to match the standard deviation predicted by the risk model.

Step 1

Using the portfolio’s current factor exposures and the full history of monthly factor returns for the model in use, we compute monthly factor-related returns for the portfolio:

where:

= portfolio s factor-related return in month t,

= portfolio's current exposure to factor i, and

= return for factor i in month t.

The monthly factor-related portfolio returns are equally weighted across the model history to form a return distribution. We then shift the distribution so that the mean return is zero.

Step 2

We assume that specific asset returns are normally distributed. Even though the forecast specific returns for single assets tend to be non-normally distributed, this assumption works well in diversified portfolios because specific returns are uncorrelated across assets.

We combine the portfolio’s factor-related and specific return distribution. The resulting distribution for a portfolio with a high proportion of specific risk will tend to closely resemble a normal distribution. Conversely, a portfolio whose risk is dominated by common factor sources is likely to deviate from a normal distribution because of the non-normal distribution of the underlying factor returns. This effect is particularly pronounced when we compare total return distributions, where common factors dominate, with active return distributions, where specific risk typically represents a high proportion of the active risk. (See the figures in the Appendix to this article for more details.)

We should note that the specific risk contribution to the forecast return distribution has been modeled for diversified portfolios. When used for single assets or heavily concentrated portfolios (fewer than 20 assets), the results will not be as accurate.

Step 3

In the first two steps, we built the distribution by equally weighting the underlying monthly observations. BARRA risk models use sophisticated techniques, including exponential smoothing and GARCH, to forecast the standard deviation of return with greater accuracy. As a final adjustment, we take advantage of this superior forecast by scaling the return distribution to match the portfolio’s predicted standard deviation.

Case study

In Aegis Risk Manager, the end result of this three-step process is shown in either total return units or active return units relative to a user-specified benchmark portfolio, such as the S&P 500 or MSCI EAFE®. In the following example we will use the U.S. E3 model, but this approach is available in all BARRA equity models, including the Global Equity Model.

For an active manager judged relative to a benchmark, the first step to viewing the forecast active return distribution is specifying a benchmark and choosing whether to view the distribution in percentage terms (“Return-at-Risk”) or value terms (“Value-at-Risk”), as shown in the Risk Settings screen (see FIGURE 1). In our case we will select the S&P 500 benchmark and the Value-at-Risk view. Also, while Aegis users have the option of entering predicted returns, in this example we will not specify any; the system therefore defaults to a zero mean return.

Figure 1 - Aegis Risk Manager Risk Settings screen: Selecting the benchmark and return distribution parameters

The resulting distribution is shown in FIGURE 2. Note the near-normal appearance of the curve and the low skewness (-.08) and kurtosis (0.14). These observations are typical of a traditional active management strategy with moderate tracking error—in this case, 3.65%. Note also that users can select any probability percentage to calculate the VAR. At the default value of 5%, the sample portfolio exhibits $5.81 million of VAR.

Figure 2 - Forecast Active Return Distribution, in VAR terms

To shift from viewing the return distribution in active units to total return units, we deselect the benchmark. The resulting distribution (FIGURE 3), as expected, is less smooth than the active distribution shown in FIGURE 2 and has higher skewness (0.53) and kurtosis (1.88).

Figure 3 - Forecast Total Return Distribution in VAR terms

Summary

By examining the forecast return distribution based on current security holdings, we can develop a more complete understanding of potential return outcomes than is possible by relying solely on the standard deviation of return. RAR/VAR analysis benefits greatly from our combined empirical and model-based approach, which estimates probabilities in the tails with far greater accuracy than would a normal distribution. Possible uses of this analysis include enhanced dialogues with clients and consultants, quantification of potential losses at different probability levels and versus different benchmarks, assessment of relative value among competing requests for capital allocation, and verification of risk levels for fiduciary and regulatory purposes.

Appendix: Effects of specific risk on a distribution

To see the effects of specific risk on a return distribution, let’s look at two forecast return distributions—one in total risk units and one in active risk units.

Our first example is the S&P 500 as of July 31, 1998, analyzed in total risk units. In variance terms, the specific risk of the S&P 500 at this date was 3.82 and the total risk was 217.95, indicating that specific risk was responsible for only approximately 1.75% of the predicted total variance of the S&P 500 (see FIGURE 4). The forecast return distribution is somewhat “lumpy” as a result of this near-total dominance of common factor risk and also exhibits significant skewness and kurtosis (see FIGURE 5).

Figure 4 - Total Risk Decomposition of S&P 500 Index

Figure 5 - Forecast Total Return Distribution, in RAR terms

Contrast this with an active return distribution—in this case, the S&P 100 Index compared with a benchmark of the S&P 500. At the end of July 1998, in variance terms this portfolio exhibited 5.00 units of specific risk, compared with 8.17 units of active risk, for a share of specific risk in the overall active risk of over 61% (see FIGURE 6). This much larger specific risk share results in a “smoother,” nearly normal forecast active return distribution (see FIGURE 7).

Figure 6 - Active risk decomposition of S&P 100 Index versus S&P 500 Index

Figure 7 - Forecast Active Return Distribution, in RAR terms

Suggested readings

For further reading about the concepts of equity return distributions, forecasting techniques, and Value at Risk, we suggest the following articles:

Balzer, Leslie, “Measuring Investment Risk: A Review,” Journal of Investing, Fall 1995.

Kahn, Ronald N., “Distribution of Equity Returns,” BARRA Newsletter, November/ December 1990.

-----, “Mutual Fund Risk,” BARRA Research Insights, BARRA 1997.

Rosenberg, Barr, “Prediction of Common Stock Investment Risk,” Journal of Portfolio Management, Fall 1984.

Simons, Katerina, “Value at Risk—New Approaches to Risk Management,” New England Economic Review, September/ October 1996.