On Reinsurance by Capital Injections in a Brownian perturbed Risk Model
Zied Ben Salah*, American University in Cairo; Jose Garrido, Concordia University
In this paper we consider a risk model where the deficits after ruin can be covered by reinsurance contracts with different levels of retention. To allow the insurance company surviving after ruin, the reinsurer has to inject additional capitals. Assuming that these are obtained through a reinsurance agreement, the problem is to determine the reinsurance premium using the discounted expected value of capital injections. Inspired by results of Z. Ben Salah (2014) , we show that an explicit formula for the reinsurance premium exists in a setting involving accumulated claims modeled by a subordinator, and Brownian perturbation. We illustrate this result by specific examples when there is no Brownian perturbation.
The retrospective loss random variable and its relevance in actuarial science
Emiliano Valdez*, University of Connecticut
In this paper, we define a retrospective loss random variable and mathematically demonstrate that its expectation is the retrospective reserve which in turn equals the prospective reserve. By defining an associated random variable for the retrospective reserve, similar to the prospective loss random variable for the prospective reserve, we can explore various properties of the retrospective loss random variable. We demonstrate that the retrospective loss random variable provides us with valuable historical information on how actual experience varies from reserving assumptions and whether it is significant enough to adjust the prospective reserves for the business. The paper concludes with a model of a block of inforce policies with actual experience different from reserving assumptions, and a rigorous and consistent methodology on how prospective reserves could be adjusted based on the realized retrospective loss random variable. This is joint work with J. Vadiveloo, G. Niu and G.Gan.
Optimal decumulation into annuity after retirement: a stochastic control approach.
Nicolas Langrené*, CSIRO; Thomas Sneddon, CSIRO; Geoffrey Lee, CSIRO; Zili Zhu, CSIRO
Around the world, public and private pension systems continue to shift gradually from traditional defined benefit plans to defined contribution plans. Under this new system, individuals contribute to their own individual retirement account, and must manage it throughout retirement. In particular, contrary to defined benefit plans, individuals face longevity risk, i.e. the risk that their retirement savings are not enough to cover their consumption if they were to live longer than expected.
A natural product to hedge this longevity risk would be an inflation-protected life annuity, which, in exchange of an immediate lump sum payment, delivers a steady stream of payments to the purchaser as long as he lives. However, most retirees do not voluntarily annuitize any of their savings. Prospect theory provides some insights on this puzzle: framed as an investment product, an annuity is seen as a massive, unappealing bet on one’s life expectancy. In particular, the perceived risk of losing all one’s entire principal if dying shortly after the annuity purchase outweighs the more distant risk of running out of money if living longer than expected.
Between these two opposites that are no annuitization and immediate full annuitization, we propose an intermediate strategy: progressive decumulation into annuity. Entering gradually into annuity keeps longevity risk at bay, while mitigating shortcomings of immediate full annuitization: inflexibility, risk of illiquidity, and risk of quick loss of the whole principal. Moreover, if the dynamic shift into annuity takes heed of the market conditions, the retiree can keep benefiting from the equity premium.
Mathematically, finding the best gradual shift into annuity can be expressed as a simple stochastic control problem. We solve it numerically using an extension of the least-squares Monte Carlo algorithm, as it allows for great flexibility on the dynamics of the risk factors and on the mortality model. More precisely, in order to obtain realistic and consistent simulations of the risk factors over the whole retirement period, we extend the classical Wilkie investment model into a more general cascade structure containing all the relevant variables in the economy.
Several objective functions were tested: expected utility, probability of ruin, and more sophisticated combinations. In each case, allowing for dynamic purchase of annuity improves substantially the financial situation of the retiree. This is illustrated on the dynamic evolution of the distribution of the net wealth of the retiree. Moreover, the algorithm provides the explicit policy to follow over time to obtain these observed improvements in practice.
Our results can either be used by retirees to make informed decisions on annuity purchase over time, or by insurance companies to package this dynamic strategy into a new kind of annuity package featuring delayed, gradual initial payment and equity-linked, inflation-protected gradual payout streams.