The Cost of Negative Returns

We study the impact of negative returns on the health of a given financial portfolio. It is often the case that a series of significant negative returns trigger a credit event such as a downgrade in rating, or even a default of the portfolio owner. We focus our attention on a Weighted Average of Ordered Returns, which is a statistic that allows us to weight returns according to their relative adverse impact. We use an option pricing approach to derive the theoretical price and properties of a forward, a swap contract, a call and a put option written on the Weighted Average of Ordered Returns under different assumptions about the distribution of returns. The models of returns considered in this paper are defined by the following underlying price processes: geometric Brownian motion, Merton model with Poisson jumps, and GARCH model. We present a convergence result which states that the price of a forward on the Weighted Average of Ordered Returns converges to the theoretical law invariant coherent risk measure. Finally, we show that the forward price process itself satisfies the axioms of a dynamic coherent risk measures.