Monotone tail functions: definitions, properties, and application to risk-reducing strategies

This paper studies properties of functions having monotone tails. We extend Theorem 1 of Dhaene et al. (2002a) and show how the tail quantiles of a random variable transformed with a monotone tail function can be expressed as the transformed tail quantiles of the original random variable. The main result is intuitive, in that Dhaene et al. (2002a)’s properties still hold, but only for certain quantile values. However, the proof presents some complications that arise especially when the function involved has discontinuities. We consider different situations where monotone tail functions occur and can be useful, such as the evaluation of the payoff of option trading strategies and the present value of insurance contracts providing both death and survival benefits. The paper also applies monotone tail functions to study quadrant perfect dependence, and shows how this dependence structure integrates within the framework of monotone tail functions. Moreover, we apply the theory to the problem of risk reduction and investigate conditions on a hedger ensuring efficient reductions of required economic capital.