BARRA Releases New
Equity Trading Risk Model

by Michel Bishopp and Rakesh Manani

In January BARRA released the U.K. Equity Trading Model for traders, broker/dealers, and other investment professionals who work with very short investment horizons. Delivered through the Windows-based BARRA Aegis System™, the U.K. Trading Model utilizes BARRA's state-of-the-art risk modeling techniques to provide the equity trader with an integrated package of risk management and basket construction tools to help make complex buy-and-sell decisions. BARRA intends to adapt additional equity models to this shorter horizon as market demand dictates.

The U.K. Equity Trading Model provides superior risk forecasts for equity books over a one-day to two-week horizon, compared with BARRA's traditional monthly horizon for market players concerned with longer-term risk forecasts. Incorporating a new daily covariance matrix, coupled with daily prices delivered through BARRALINK, the U.K. Trading Model provides up-to-date, accurate risk forecasts in a trading context.

Model estimation

The mechanics of building the model are as follows: Each month a cross-sectional regression is performed across all assets in the estimation universe, yielding a set of factor returns. The historical time series of these factor returns are then used to construct the factor covariance matrix, which is the heart of the model. A separate specific variance model is then built, which captures the idiosyncratic component of asset variance.

Exponential weighting of factor returns and a GARCH¹ model of market volatility are incorporated into the estimation of the covariance matrix to provide superior forecasts of shorter-term portfolio volatility. This is accomplished by relaxing the assumption that markets are stable over very long periods of time, thus making the model more responsive to current market conditions. These extra modeling elements serve to improve the short-term forecasts of the common factor components of asset variance.

Testing

This testing method uses a Black-Scholes options pricing framework to price a one-day-to-expiry at-the-money put and a one-day-to-expiry at-the-money call on the same portfolio. The package price of the put plus call is the sum of the two. Each agent will price this straddle using his own volatility forecasting model. Since the price of an option is a positive function of the volatility of the underlying asset, the agent with the lower volatility forecast will believe that the agent with the higher forecast has overvalued the straddle, and vice versa. The straddle is then traded between the two agents at the average of their "valuation" prices. After one day has passed, the options mature and the profits to each agent are calculated using the observed price of the underlying portfolio.

This test evaluates the respective profits each agent would have realized if he had bought or sold the straddle based on the price implied by his volatility forecast. That is, the agent will "buy volatility" if offered the straddle at a lower price than that implied by his own volatility forecast, and he will "sell volatility" if the reverse is the case. On average, the agent with the superior volatility forecasting model will benefit from "profits" at the expense of the agent with the less superior volatility forecasting model. For this article we have applied this testing method to the FTSE 100³ daily index. The two competing models for which we present results are the volatility forecasts of the FTSE 100 generated from the U.K. Trading Model and an "Average Historical Volatility" forecast (see Table 1). We ran the test over the period January 1990 to November 1995 on a daily basis. The Average Historical Volatility forecast was calculated as the standard deviation of the daily FTSE 100 returns (ex-dividend) across the testing period.

Table 1 shows the daily "profit" to both the Average Historical Volatility (AHV) model and the U.K. Trading Model, along with the corresponding statistics. The t-statistic shown is for the Null Hypothesis that the respective forecasting model earns no profit. Here we see that the 10.67% annual profit earned using the U.K. Trading Model volatility forecast for the FTSE 100 is statistically significant at the 99% confidence level. These results highlight the superiority of BARRA's U.K. Trading Model volatility forecasts over conventional historical methods as indicators of future volatility.

Table 1: Daily profit to Average Historical Volatility (AHV) Model and U.K. Trading Model

The U.K. Equity Trading Model—An example

The forecast of short-term total risk obtained from the BARRA Aegis System allows the trader to quantify the Value-at-Risk (VaR) contained within his position. As an example, Figure 1 illustrates the VaR forecast by the Trading Model of a £10 million U.K. equity book. It is interesting to note how this VaR increases with the number of trade days in the typical parabolic fashion.

Figure 1: The BARRA Trading Model's Value-at-Risk forecast of a £10 million U.K. equity book

The risk statistics available within Aegis do not end with this basic VaR number. In Figures 2 and 3 we see the total risk decomposed into its common factor and specific components. This multifactor modeling approach is unique in allowing a breakdown of this risk along intuitive and understandable dimensions.

Figure 2: Risk decomposition of a £10 million U.K. equity book as shown by the BARRA Aegis System

Figure 3: Schematic of risk decomposition of a £10 million U.K. equity book

As Figures 2 and 3 show, the majority of the total risk (11.24%) is due to common factor risk—that is, risk due to assets that share similar characteristics. Breaking this down further, we see the major contributor to this common factor risk is the industry breakdown (11.07%). The Trading Style factors show much less impact, at 72 basis points of risk.

Having looked at the broad-based total risk decomposition, the trader can now drill down further and identify exactly which industry or trading style plays are adding to or detracting from this risk. Further inspection reveals the individual names that contribute most and least to position risk.

In Figure 4 we see that Highland Distilleries contributes least to portfolio risk, and Barratt Development contributes most to portfolio risk. Furthermore, the Marginal Contribution to Risk (MCTR) number displayed quantifies the change in total risk that would result from increasing our position in any name. For example, if we were to increase our position in Highland Distilleries by 1% of book value, the forecast volatility of the book would increase by just under 7 basis points. The identification of the assets that contribute most and least to portfolio risk also allows the trader to sequence trades so that the risk of the aggregate position is minimized.

Figure 4: Marginal Contribution to Risk ranking, £10 million U.K. equity book

The powerful Aegis Trade Scenario feature allows the trader to see the potential impact on position risk of asset swaps. Let's assume that in order to minimize the position risk of our example book we wish to trade Highland Distilleries with Barratt Development. Using the Trade Scenario tool, the trader can experiment with different holdings in the respective stocks and see the resulting impact on the volatility of the book (see Figure 5).

Figure 5: Using the Aegis Trade Scenario tool to assess the impact of asset swaps on position risk

Increasing our position in Highland Distilleries to 5.08% of our book value and financing this by an equal sterling decrease in our Barratt Development position will result in an overall book volatility decrease of 30 basis points (11.41% minus 11.11%). Taken to the limit—that is, when all positions have identical marginal contributions to risk—the trader can rest assured that no more risk-reducing trades exist within the current book.

The Aegis Best Hedge tool allows market makers, traders running proprietary books, and risk managers to offset the market component of the total risk they are bearing. The U.K. Trading Model calculates the optimal number of futures contracts that should be shorted in order to offset this market risk. In the example shown in Figure 6, we have chosen the FTSE 100 index as a prospective hedge instrument. (Note that any portfolio, both real and synthetic, can be used.) After the hedge has been applied, the resulting total risk is 2.44%. This risk is classified as residual to the market and must be borne by the trader, as it cannot be offset by the futures hedge.

Figure 6: The result of using the Aegis Best Hedge tool—FTSE 100 index futures lower total risk

The Best Hedge tool is flexible and can calculate the optimal hedge using multiple contracts, even when holding assets which are constituents of multiple indexes.

Conclusion

Notes