Fast return-level estimates for flood insurance via an improved Bennett inequality for random variables with differing upper bounds

The k-year return levels of insurance losses due to flooding can be estimated by simulating and then summing a large number of independent losses for each of a large number of hypothetical years of flood events, and replicating this a large number of times. This leads to repeated realisations of the total losses over each year in a long sequence of years, from which return levels and their uncertainty can be estimated; the procedure, however, is highly computationally intensive. We develop and use a new, Bennett-like concentration inequality in a procedure that provides conservative but relatively accurate estimates of return levels at a fraction of the computational cost. Bennett's inequality provides concentration bounds on deviations of a sum of independent random variables from its expectation; it accounts for the different variances of each of the variables but uses only a single, uniform upper bound on their support. Motivated by the variability in the total insured value of insurance risks within a portfolio, we consider the case where the bounds on the support can vary by an order of magnitude or more, and obtain tractable concentration bounds. Simulation studies and application to a representative portfolio demonstrate the substantial improvement of our bounds over those obtained through Bennett's inequality. We then develop an importance sampling procedure that repeatedly samples the loss for each year from the distribution implied by the concentration inequality, leading to conservative estimates of the return levels and their uncertainty.