Numerical Ultimate Ruin Probabilities under Interest Force

Abstract

This work addresses the issue of ruin of an insurer whose portfolio is exposed to insurance risk arising from the classical surplus process. Availability of a positive interest rate in the financial world forces the insurer to invest into a risk free asset. We derive a linear Volterra integral equation of the second kind and apply an order four Block-by-block method in conjuction with the Simpson rule to solve the Volterra equation for ultimate ruin. This probability is arrived at by taking a linear combination of some two solutions to the Volterra integral equation. The several numerical examples given show that our results are excellent and reliable.

Risk, Ruin and Survival

“…Sundt and Teugels (1995) gave an extensive treatment of the ruin probability with constant interest force and obtained approximations, as well as upper and lower bounds. Kasozi and Paulsen (2005) used numerical methods such as the block-by-block method and the Simpson rule to approximate the ultimate ruin probabilities under a constant rate of interest. Cai et al (2009) considered the well-known Gerber-Shiu function under the risk model of liquid reserves and constant interest on the surplus, and more recently, Schmidli (2015) studied a variant of the discounted penalty function where a penalty applies when the surplus process leaves a finite interval.…”

Minimizing an Insurer’s Ultimate Ruin Probability by Reinsurance and Investments

“…They derived a linear VIE-2 and applied a fourth-order block by-block method of Paulsen et al [39] in conjunction with Simpson's rule to solve the Volterra integral equation for the optimal dividend barrier. In another study, Kasozi and Paulsen [40] applied a fourth-order block-by-block method to the numerical solution of the Volterra integral equation (VIE) for ultimate ruin in the Cramér-Lundberg model compounded by a constant force of interest. More pertinent literature on the block-by-block method is available, for example, in [41,42].…”

Numerical Ruin Probability in the Dual Risk Model with Risk-Free Investments

“…Sundt and Teugels (1995) gave an extensive treatment of the ruin probability with constant interest force and obtained approximations, as well as upper and lower bounds. Kasozi and Paulsen (2005) used numerical methods such as the block-by-block method and the Simpson rule to approximate the ultimate ruin probabilities under a constant rate of interest. Cai et al (2009) considered the well-known Gerber-Shiu function under the risk model of liquid reserves and constant interest on the surplus, and more recently, Schmidli (2015) studied a variant of the discounted penalty function where a penalty applies when the surplus process leaves a finite interval.…”