The purpose of this paper is to show that, for the classical risk model, explicit expressions for survival probabilities in a finite time horizon can be obtained through the inversion of the double Laplace transform of the distribution of time to ruin. To do this, we consider Gerber and Shiu (1998) and a particular value for their penalty function. Although other methods to address the problem exist, we find this approach, perhaps, more direct and simple. For the analytic inversion, we have applied twice, after some algebra, the Laplace complex inversion formula.

This article summarizes some main results in modern portfolio theory. First, the Markowitz approach is presented. Then the capital asset pricing model is derived and its empirical testability is discussed. Afterwards Neumann–Morgenstern utility theory is applied to the portfolio problem. Finally, it is shown how optimal risk allocation in an economy may lead to portfolio insurance.

This (mostly) expository paper describes the importance of hedging to the pricing of modern financial products and how hedging may be achieved even when the traditional Black-Scholes assumptions are absent.

We consider estimating tail events using exponential families of importance sampling distributions. When the cannonical sufficient statistic for the exponential family mimics the tail behaviour of the underlying cumulative distribution function, we can achieve bounded relative error for estimating tail probabilities. Examples of rare event simulation from various distributions including Tukey's g&h distribution are provided.

In this paper, we consider the risk–return trade-off for variable annuities in a Black–Scholes setting. Our analysis is based on a novel explicit allocation of initial wealth over the payments at various horizons. We investigate the relationship between the optimal consumption problem and the design of variable annuities by deriving the optimal so-called assumed interest rate for an investor with constant relative risk aversion preferences. We investigate the utility loss due to deviations from this. Finally, we show analytically how habit-formation-type smoothing of financial market shocks over the remaining lifetime leads to smaller year-to-year volatility in pension payouts, but to increases in the longer-term volatility.

One of the basic functions of risk theory is the so-called generalised Poisson function F(x), which gives the probability that the total amount of claims ξ does not exceed some given limit x during a year (or during some other fixed time period). For F(x) is obtained the well known expansion

where n is the expected number of claims during this time period and Sk*(x) is the k:th convolution of the distribution function S(z) of the size of one claim. The formula (1) is, however, much too inconvenient for numerical computations and for most other applications. One of the main problems of risk theory, which is still partly open, is to find suitable methods to compute, or at least to approximate, the generalised Poisson function.

A frequently used approximation is to replace F(x) by the normal distribution function having the same mean and standard deviation as F as follows:

where α1 and α2 are the first zero-moments of S(z):

SM(Z) is here again the distribution function of the size of one claim. To obtain more general results a reinsurance arrangement is assumed under which the maximum net retention is M. Hence the portfolio on the company's own retention is considered. If the reinsurance is of Excess of Loss type, then

where S(z) is the distribution function of the size of one total claim.

We investigate the implications of a dual approach to the graduation of the force of mortality based on the modelling of the exposures as gamma random variables, as opposed to the modelling of the numbers of deaths as Poisson random variables.

Parameter estimation in case of loglinear modelled claim cost distribution characteristics is mathematically tractable, especially with the Inverse Gaussian and Lognormal distribution.

We determine the optimal asset allocation to bonds and stocks using an annually recalculated virtual annuity (ARVA) spending rule for DC pension plan decumulation. Our objective function minimizes downside withdrawal variability for a given fixed value of total expected withdrawals. The optimal asset allocation is found using optimal stochastic control methods. We formulate the strategy as a solution to a Hamilton–Jacobi–Bellman (HJB) Partial Integro Differential Equation (PIDE). We impose realistic constraints on the controls (no-shorting, no-leverage, discrete rebalancing) and solve the HJB PIDEs numerically. Compared to a fixed-weight strategy which has the same expected total withdrawals, the optimal strategy has a much smaller average allocation to stocks and tends to de-risk rapidly over time. This conclusion holds in the case of a parametric model based on historical data and also in a bootstrapped market based on the historical data.

In most textbooks and papers that deal with the stochastic theory of life contingencies, the stochastic approach is restricted to the computation of expectations and higher order moments. For a wide class of insurances on a single life, we derive the distribution and the probability density function of the benefit and the loss functions. Both the continuous and the discrete case are considered.

We analyze optimal consumption in the life cycle model by introducing life and pension insurance contracts. The model contains a credit market with biometric risk, and market risk via risky securities. This idealized framework enables us to clarify important aspects of life insurance and pension contracts. We find optimal pension plans and life insurance contracts where the benefits are state dependent. We compare these solutions both to the ones of standard actuarial theory, and to policies offered in practice. Implications of this include what role the insurance industry may play to improve welfare. The relationship between substitution of consumption and risk aversion is highlighted in the presence of a consumption puzzle. One problem related portfolio choice is discussed the horizon problem. Finally, we present some comments on longevity risk and cohort risk.

Whittaker graduation is applied to the spatial smoothing of insurance data. Such data (e.g. claim frequency) form a surface over the 2-dimensional geographic domain to which they relate. Observations on this surface are subject to sampling error. They need to be smoothed spatially if a reliable estimate of the underlying surface is to be obtained.

A measure of smoothness of a surface has been defined. This has been incorporated in 2-dimensional Whittaker graduation to effect the necessary smoothing. The details of this are worked out in Section 4. The procedure is illustrated by numerical example in Section 5. The Bayesian interpretation of this form of spatial smoothing is discussed, and used to assist in the selection of the Whittaker relativity constant.

Recently, there has been an increasing interest from life insurers to assess their portfolios' mortality risks. The new European prudential regulation, namely Solvency II, emphasized the need to use mortality and life tables that best capture and reflect the experienced mortality, and thus policyholders' actual risk profiles, in order to adequately quantify the underlying risk. Therefore, building a mortality table based on the experience of the portfolio is highly recommended and, for this purpose, various approaches have been introduced into actuarial literature. Although such approaches succeed in capturing the main features, it remains difficult to assess the mortality when the underlying portfolio lacks sufficient exposure. In this paper, we propose graduating the mortality curve using an adaptive procedure based on the local likelihood. The latter has the ability to model the mortality patterns even in presence of complex structures and avoids relying on expert opinions. However, such a technique fails to offer a consistent yet regular structure for portfolios with limited deaths. Although the technique borrows the information from the adjacent ages, it is sometimes not sufficient to produce a robust life table. In the presence of such a bias, we propose adjusting the corresponding curve, at the age level, based on a credibility approach. This consists in reviewing the assumption of the mortality curve as new observations arrive. We derive the updating procedure and investigate its benefits of using the latter instead of a sole graduation based on real datasets. Moreover, we look at the divergences in the mortality forecasts generated by the classic credibility approaches including Hardy–Panjer, the Poisson–Gamma model and the Makeham framework on portfolios originating from various French insurance companies.

A common procedure for experience rating is to use Whitney's credibility formula with the manual premium per risk unit estimated by the observed average claim amount per risk unit. As pointed out by Whitney himself, this observed average also needs to be subjected to credibility adjustment. This suggestion is pursued in the paper and an experience rating procedure developed in which there are two stages:

(i) revision of the prior expected manual premium in accordance with the observed average claim amount per risk unit;

(ii) experience rating of the individual risk classes by the usual methods but using the manual premium obtained in stage (i).

Both stages of this procedure involve credibility indices, for which formulae are developed.

Conditions under which the “common procedure” referred to above is fairly reliable are found and it is seen, somewhat surprisingly, that the procedure is often more valid than one expects on intuitive grounds. Care is necessary, however, when there are wide differences in size between different risk classes.

This paper deals with optimal investment and redistribution of the free reserves connected to life and pension insurance contracts in form of dividends and bonus. Formulated appropriately this problem can be viewed as a modification of Merton’s problem of optimal consumption and investment with a very particular form of consumption and utility hereof. Both are linked to a finite state Markov chain. We distinguish between utility optimization of dividends, where a semi-explicit result is obtained, and utility optimization of bonus payments. The latter connects to the financial notion of durable goods and allows for an explicit solution only in very special cases.

Spatial data are a rich source of information for actuarial applications: knowledge of a risk’s location could improve an insurance company’s ratemaking, reserving or risk management processes. Relying on historical geolocated loss data is problematic for areas where it is limited or unavailable. In this paper, we construct spatial embeddings within a complex convolutional neural network representation model using external census data and use them as inputs to a simple predictive model. Compared to spatial interpolation models, our approach leads to smaller predictive bias and reduced variance in most situations. This method also enables us to generate rates in territories with no historical experience.

Assume that an insurance company pays dividends to its shareholders whenever the surplus process is above a given threshold. In this paper we study the expected amount of dividends paid, and the expected time to ruin in the compound Poisson risk process perturbed by a Brownian motion. Two models are considered: In the first one the insurance company pays whatever amount exceeds a given level b as dividends to its shareholders. In the second model, the company starts to pay dividends at a given rate, smaller than the premium rate, whenever the surplus up-crosses the level b. The dividends are paid until the surplus down-crosses the level a, a < b . We assume that the claim sizes are phase-type distributed. In the analysis we apply the multidimensional Wald martingale, and the multidimensional Asmussesn and Kella martingale.

This paper is devoted to the study of the initial reserve, as a function of the retention limit, needed to assure that the probability of ruin, at the end of a certain period of time, is not higher than an agreed value, for an excess of loss treaty. To assess the probability of ruin, the normal and the normal power approximation are used. It is shown that the initial reserve is not in general an increasing function of the retention, having a minimum under fair assumptions.

In this paper a method for determining benchmark rates for the excess of loss reinsurance of a Motor Third Party Liability insurance portfolio will be developed based on observed market rates. The benchmark rates are expressed as a percentage of the expected premium income that is available to cover the whole risk of the portfolio. The rates are assumed to be based on a compound process with a heavy tailed severity, such as Burr or Pareto distributions. In the absence of claim data these assumptions propagate the theoretical benchmark rate component of the regression model.

Given the whole set of excess of loss reinsurance rates in a given market, the unknown parameters are estimated within the framework of quasi-likelihood estimation. This framework makes it possible to select a theoretical benchmark rate model and to choose a parsimonious submodel for describing the observed market rates over a 4-years observation period. This method is applied to the Belgian Motor Third Party Liability excess of loss rates observed during the years 2001 till 2004.

In this paper, we find explicit expressions for the moments of the fund level and the value of the total contribution when arithmetic or geometric rates of return are modeled by a moving average process of order q and when a proportional control is applied to the contributions. Our approach is based on the bilinear Markovian representation.