Do Risk Limits Add Value?
According to the Wall Street adage, one of the best ways to make money is not to lose it. The importance of binding and enforceable internal limits has long been recognized in financial firms, yet not always observed in practice. Five years after the collapse of the Lehman Brothers, it appears that lax exposure limits could have been one of the major causes of the firm’s failure (U. S. Bankruptcy Court 2010).
Let us consider an argument about two investment funds presented by Lo (2001). Fund A has a portfolio with an expected return of 10 % p. a. and an annual volatility of 75 %. Fund B replicates the portfolio of Fund A, but enforces a stop-loss limit every time its annual returns fall down to —20 %. Assuming that the portfolio returns of Fund A follow a log-normal distribution, it can be shown that Fund B would
exhibit not only a lower volatility of 67 %, but also a higher expected return of 21 %, more than double that of Fund A. Thus, truncating the loss tail of the returns distribution reduces its variance and skews it towards the positive side. For a normal distribution with mean д and variance a2, a one-sided truncation at level z (i. e. enforcing a stop-loss limit) ensures a higher expected value and a lower variance:
д C a x X(a) > m,
X > z) = a2 x (1 -1 (a)) < a2(X),
where a = (z-д)/a, X(a) = ¥(a)/(1 – Ф(а)), 1(a) = X(a) (X(a) – a),
ф(-)—the probability density function of the standard normal distribution,
Ф(-)—the cumulative probability function of the standard normal distribution.
Besides stop-loss limits, truncated distributions appear in other risk management applications including margin trading (loss tail truncation at a margin call level), private equity funds (profit tail truncation at an exit level), and hedging (e. g. twosided truncation at strike prices in bull/bear spreads).
Focusing on stop-loss limits in our further analysis, let us consider how these ex-post limits can be embedded into ex-ante exposure limits.