Lee-Carter goes risk-neutral: an application to the Italian annuity market

We consider a class of stochastic intensities of mortality that generalizes the model proposed by Lee and Carter (1992), allowing general diffusions to drive the mortality time-trend. We analyze the stability of such class of intensities under measure changes and show how a risk-neutral version of the generalized Lee-Carter model can be employed for fair valuation purposes. We provide an example of model calibration based on the Italian annuity market

Bayesian Poisson Log-Bilinear Mortality Projections

Mortality projections are major concerns for public policy, social security and private insurance. This paper implements a Bayesian log-bilinear Poisson regression model to forecast mortality. Computations are carried out using Markov Chain Monte Carlo methods in which the degree of smoothing is learnt from the data. Comparisons are made with the approach proposed by Brouhns, Denuit & Vermunt (2002a,b), as well as with the original model of Lee & Carter (1992)

Longevity-indexed annuities

This paper addresses the problem of the sharing of longevity risk between an annuity provider and a group of annuitants. An appropriate longevity index is designed in order to adapt the amount of the periodic payments in life annuity contracts. This accounts for unexpected longevity improvements experienced by a given reference population. The approach described in the present paper is in contrast with Group Self-Annuitization where annuitants bear their own risk. Here, the annuitants only bear the non-diversifiable risk that the future mortality trend departs from that of the reference forecast. In that respect, the life annuities discussed in this paper are substitutes for reinsurance and securitization of longevity risk

Some positive dependence notions, with applications in actuarial sciences.

The paper is devoted to the study of several notions of positive dependence among risks, namely association, linear positive quadrant dependence, positive orthant dependence and conditional increasingness in sequence. Various examples illustrate the usefulness of these notions in an actuarial context.Dependence; Applications; Actuarial; Science;

Does positive dependence between individual risks increase stop-loss premiums?.

Actuaries intuitively feel that positive correlations between individual risks reveal a more dangerous situation compared to independence. The purpose of this short note is to formalize this natural idea. Specifically, it is shown that the sum of risks exhibiting a weak form of dependence known as positive cumulative dependence is larger in convex order than the corresponding sum under the theoretical independence assumption.Dependence; Risk;

The economics of insurance: a review and some recent developments.

The present paper is devoted to different methods of choice under risk in an actuarial setting. The classical expected utility theory is first presented, and its drawbacks are underlined. A second approach based on the so-called distorted expectation hypothesis is then described. It will be seen that the well-known stochastic dominance as well as the stop-loss order have common interpretations in both theories, while defining higher degree stochastic orders leads to different concepts. The aim of this paper is to emphasize the similarities of the two approaches of choice under risk as well as to point out their major differences.Economics; Insurance;

Correlated risks, bivariate utility and optimal choices

In this paper, we consider a décision-maker facing a financial risk flanked by a background risk, possibly non-financial, such as health or environmental risk. A decision has to be made about the amount of an investment (in the financial dimension) resulting in a future benefit either in the same dimension (savings) or in the order dimension (environmental quality or health improvement). In the first case, we show that the optimal amount of savings decreases as the pair of risks increases in the bivariate increasing concave dominance rules of higher degrees which express the common preferences of all the decision-makers whose two-argument utility function possesses direct and cross derivatives fulfilling some specific requirements. Roughly speaking, the optimal amount of savings decreases as the two risks become "less positively correlated" or marginally improve in univariate stochastic dominance. In the second case, a similar conclusion on optimal investment is reached under alternative conditions on the derivatives of the utility function.bivariate higher order increasing concave stochastic dominance, precautionary savings, background risk, dependence

Stochastic approximations of present value functions.

The aim of the paper is to apply the method proposed by Denuit, Genest and Marceau (1999) for deriving stochastic upper and lower bounds on the present value of a sequence of cash flows, where the discounting is performed under a given stochastic return process. The convex approximation provided by Goovaerts, Dhaene and De Schepper (1999) and Goovaerts and Dhaene (1999) is then compared to these stochastic bounds. On the basis of several numerical examples, it will be seen that the convex approximation seems reasonable.Value; Functions;

A simple proof that comonotonic risks have the convex-largest sum.

In the recent actuarial literature, several proofs have been given for the fact that if a random vector X(1), X(2), …, X(n) with given marginals has a comonotonic joint distribution, the sum X(1) + X(2) + … + X(n) is the largest possible in convex order. In this note we give a lucid proof of this fact, based on a geometric interpretation of the support of the comonotonic distribution.Risk; Actuarial; Distribution;