Two-dimensional forward and backward transition rates

TL;DR

This paper extends the concept of forward transition rates to two dimensions to better evaluate life insurance liabilities, especially for path-dependent cash flows and second-order moments, enhancing both prospective and retrospective calculations.

Contribution

It introduces two-dimensional forward and backward transition rates, enabling more accurate modeling of life insurance reserves and path-dependent cash flows without relying solely on Markov assumptions.

Findings

Two-dimensional forward transition rates facilitate calculation of second-order moments.

Backward transition rates enable retrospective analysis of life insurance liabilities.

Extension improves modeling of path-dependent cash flows.

Abstract

Forward transition rates were originally introduced with the aim to evaluate life insurance liabilities market-consistently. While this idea turned out to have its limitations, recent literature repurposes forward transition rates as a tool for avoiding Markov assumptions in the calculation of life insurance reserves. While life insurance reserves are some form of conditional first-order moments, the calculation of conditional second-order moments needs an extension of the forward transition rate concept from one dimension to two dimensions. Two-dimensional forward transition rates are also needed for the calculation of path-dependent life insurance cash-flows as they occur upon contract modifications. Forward transition rates are designed for doing prospective calculations, and by a time-symmetric definition of so-called backward transition rates one can do retrospective calculations.