For insurance companies in Europe, the introduction of Solvency II leads to a tightening of rules for solvency capital provision. In life insurance, this especially affects retirement products that contain a significant portion of longevity risk (e.g., conventional annuities). Insurance companies might react by price increases for those products, and, at the same time, might think of alternatives that shift longevity risk (at least partially) to policyholders. In the extreme case, this leads to so-called tontine products where the insurance company’s role is merely administrative and longevity risk is shared within a pool of policyholders. From the policyholder’s viewpoint, such products are, however, not desirable as they lead to a high uncertainty of retirement income at old ages. In this article, we alternatively suggest a so-called tonuity that combines the appealing features of tontine and conventional annuity. Until some fixed age (the switching time), a tonuity’s payoff is tontine-like, afterwards the policyholder receives a secure payment of a (deferred) annuity. A tonuity is attractive for both the retiree (who benefits from a secure income at old ages) and the insurance company (whose capital requirements are reduced compared to conventional annuities). The tonuity is a possibility to offer tailor-made retirement products: using risk capital charges linked to Solvency II, we show that retirees with very low or very high risk aversion prefer a tontine or conventional annuity, respectively. Retirees with medium risk aversion, however, prefer a tonuity. In a utility-based framework, we therefore determine the optimal tonuity characterized by the critical switching time that maximizes the policyholder’s lifetime utility.

Financial products are priced using risk-neutral expectations justified by hedging portfolios that (as accurate as possible) match the product’s payoff. In insurance, premium calculations are based on a real-world best-estimate value plus a risk premium. The insurance risk premium is typically reduced by pooling of (in the best case) independent contracts. As hybrid life insurance contracts depend on both financial and insurance risks, their valuation requires a hybrid valuation principle that combines the two concepts of financial and actuarial valuation. The aim of this paper is to present a novel three-step projection algorithm to valuate hybrid contracts by decomposing their payoff in three parts: a financial, hedgeable part, a diversifiable actuarial part, and a residual part that is neither hedgeable nor diversifiable. The first two parts of the resulting premium are directly linked to their corresponding hedging and diversification strategies, respectively. The method allows for a separate treatment of unsystematic, diversifiable mortality risk and systematic, aggregate mortality risk related to, for example, epidemics or population-wide improvements in life expectancy. We illustrate our method in the case of CAT bonds and a pure endowment insurance contract with profit and compare the three-step method to alternative valuation operators suggested in the literature.

In a typical equity-linked life insurance contract, the insurance company is entitled to a share of return surpluses as compensation for the return guarantee granted to the policyholders. The set of possible contract terms might, however, be restricted by a regulatory default constraint — a fact that can force the two parties to initiate sub-optimal insurance contracts. We show that this effect can be mitigated if regulatory policy is more flexible. We suggest that the regulator implement a traffic light system where companies are forced to reduce the riskiness of their asset allocation in distress. In a utility-based framework, we show that the introduction of such a system can increase the benefits of the policyholder without deteriorating the benefits of the insurance company. At the same time, default probabilities (and thus solvency capital requirements) can be reduced.

The tendency of insurance providers to refrain from offering long-term guarantees on investment or mortality risk has shifted attention to mutual risk pooling schemes like (modern) tontines, pooled annuities or group self annuitization schemes. While the literature has focused on mortality risk pooling schemes, this paper builds on the advantage of pooling mortality and morbidity risks, and their inherent natural hedge. We introduce a modern “life-care tontine”, which in addition to retirement income targets the needs of long-term care (LTC) coverage for an ageing population. In contrast to a classical life-care annuity, both mortality and LTC risks are shared within the policyholder pool by mortality and morbidity credits, respectively. Technically, we rely on a backward iteration to deduce the smoothed cashflows pattern and the separation of cash-flows in a fixed withdrawal and a surplus from the two types of risks. We illustrate our results using real life data, demonstrating the adequacy of the proposed tontine scheme.