Retrospective Reserves and Bonus with Policyholder Behavior

Abstract

Legislation imposes insurance companies to project their assets and liabilities in various financial scenarios. Within the setup of with-profit life insurance, we consider retrospective reserves and bonus, and we study projection of balances with and without policyholder behavior. The projection resides in a system of differential equations of the savings account and the surplus, and the policyholder behavior options surrender and conversion to free-policy are included. The inclusion results in a structure where the system of differential equations of the savings account and the surplus is non-trivial. We consider a case where we are able to find accurate differential equations and suggest an approximation method to project the savings account and the surplus, including policyholder behavior, in general. To highlight the practical applications of the results in this paper, we study a numerical example.

An Intrinsic Value Approach to Valuation with Forward–Backward Loops in Dividend Paying Stocks

“…The redistribution of the bonus contains certain degrees of freedom for the insurance company and depends on their dividend strategy. References [4,5] describe a projection model of the balance sheet of a with-profit life insurance company where a bonus is used to buy more insurance (spoken of as additional benefits). The model from [4,5] contains simplifying assumptions about the future dividend strategy, and an obvious extension of the model is to allow for a broader range of dividend strategies.…”

“…References [4,5] describe a projection model of the balance sheet of a with-profit life insurance company where a bonus is used to buy more insurance (spoken of as additional benefits). The model from [4,5] contains simplifying assumptions about the future dividend strategy, and an obvious extension of the model is to allow for a broader range of dividend strategies. A relevant extension of the model from [4,5] is to allow the future dividend strategy of the company to depend on the market reserve of future payments and future additional benefits and, in that case, a dependence structure similar to our model setup arises.…”

“…In [4,5], the authors derive the dynamics of the so-called savings account and the surplus of a with-profit insurance contract (see Equations ( 8) and ( 9) in [4]), where the dividend strategy depends on the savings account and the surplus only. If we allow the dividend strategy to also depend on the market reserve, V, the dynamics of the savings account X and the surplus Y are in the form…”

“…The structure and the idea of our approach are much easier to comprehend in a version of the problem where the jump processes and the domain specific notions of profit-sharing are peeled away. What is important is that the forward-backward dependence structure that we study in this paper arises as a natural and practically relevant extension of the model from [4,5], when allowing for dividends to depend on the market reserve.…”

“…The underlying assets of the company are reduced by dividends to policy holders and, thus, the assets, the dividends, and the liability value of policy holder dividends are completely entangled. The simpler case, where dividends depend on the assets (and other quantities that are easily calculated at every time point in a simulation), has only recently been formalized, for example, in [4,5]. The Future Management Actions in with-profit insurance naturally motivate our study but we stick to the corporate finance story when we present our method and numerical example below.…”