MEASURES OF EXTREME LOSS

In portfolio management, a common measure of risk is the standard deviation of return, also known as volatility. Volatility reflects the typical range of returns that a manager might expect to see. Researchers, including Bertsimas¹ and Goldberg and Hayes,² have encouraged investors to supplement volatility with measures of extreme risk that more clearly portray the depth of potential losses. The magnitude of extreme losses may not be apparent even from the best volatility forecasts.

One measure of extreme risk is expected shortfall, or simply, shortfall. Shortfall is a measure of the expected loss over a given horizon, which in our study we define as one day. It represents how much a portfolio is expected to lose on a bad day. Loss can be measured either in terms of the value of the portfolio or the return of the portfolio. Focusing on return, we define the loss of a portfolio P to be: L_P = –r_P; thus, L_P measures the magnitude of the loss.

A more precise definition of the expected shortfall ES_P, of portfolio P, is

Here, VaR_P denotes the portfolio's Value at Risk. The portfolio suffers a return worse than VaR_P or less, no more than 5% of the time.³ Thus, shortfall is the expected size of the loss, given that the loss is among the 5% worst losses the portfolio experiences.

It is useful to understand how an asset's return is likely to respond when a portfolio sustains extreme losses. Does it tend to fall more or less than the portfolio, or not at all? Just as beta describes this behavior on average, shortfall beta captures this behavior during periods of extreme loss.

The shortfall beta, β, of asset i with respect to portfolio P can be written as

where L_i is the loss to asset i.

To estimate the beta of an asset, we first compute the expected shortfall of the portfolio and the expected shortfall of the asset. To do this, we simulate returns to the asset (or portfolio) using its current exposures and a history of daily factor returns over the last four years, ignoring the specific return. We then compute the sample shortfall. The methodology is similar to that developed by Goldberg and Hayes⁴ and outlined in Barra Extreme Risk Analytics Guide⁵; however there a longer period to estimate shortfall is used. Further details are provided in the appendix to this chapter.