Pricing variable annuities with multi-layer expense strategy

TL;DR

This paper develops a mathematical framework for pricing variable annuities with a multi-layer fee structure, modeling the underlying asset with a hyper-exponential jump diffusion process, and derives explicit formulas for fair fees and related quantities.

Contribution

It introduces a novel approach to price variable annuities with layered fees using hyper-exponential jump diffusion models and provides explicit formulas for key quantities.

Findings

Derived closed-form formulas for fair fee rates.

Computed total fees and deduction times for insurers.

Numerical examples illustrating the theoretical results.

Abstract

We study the problem of pricing variable annuities with a multi-layer expense strategy, under which the insurer charges fees from the policyholder's account only when the account value lies in some pre-specified disjoint intervals, where on each pre-specified interval, the fee rate is fixed and can be different from that on other interval. We model the asset that is the underlying fund of the variable annuity by a hyper-exponential jump diffusion process. Theoretically, for a jump diffusion process with hyper-exponential jumps and three-valued drift, we obtain expressions for the Laplace transforms of its distribution and its occupation times, i.e., the time that it spends below or above a pre-specified level. With these results, we derive closed-form formulas to determine the fair fee rate. Moreover, the total fees that will be collected by the insurer and the total time of deducting fees are also computed. In addition, some numerical examples are presented to illustrate our results.