Lambda-Quantiles as Fixed Points

Abstract

Lambda-quantiles are risk measures presenting properties that may justify their use in practical applications of actuarial mathematics, financial mathematics and beyond. With respect to coherent or convex risk measures, lambda-quantiles remain finite when facing heavy-tailed potential losses with infinite expectations. With respect to the value at risk, lambda-quantiles may reflect an interesting advantage, since the value at risk does not distinguish high-risk situations beyond the chosen confidence level. This paper extends the notion of lambda-quantile, and the lambda-fixed point risk measures will be introduced. Six concrete examples will be given, and the lambda-quantile will be one of them. Several properties will be established, a representation theorem will be given and risk optimization problems will be addressed. An illustrative actuarial application involving reinsurance contracts and premiums calculation will be presented.

Keywords: Lambda-quantile, lambda-fixed point risk measure, risk representation, risk optimization.

JEL Classification: G10, G22, C02.