Equity Valuation and Portfolio Management

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 ...