In this paper we extend the work of Reinhard and Snoussi (2000) by developing a recursive system for finding the distribution of the surplus prior to ruin in a discrete semi-Markov risk model.

In this paper we analyse, in a contingent-claims framework, one of the most common life insurance policies sold in Italy during the last two decades. The policy, of the endowment type, is initially priced as a standard one, given a mortality table and a technical interest rate. Subsequently, at the end of each policy year, the insurance company grants a bonus, which is credited to the mathematical reserve and depends on the performance of a special investment portfolio. More precisely, this bonus is determined in such a way that the total interest rate credited to the insured equals a given percentage (participation level) of the annual return on the reference portfolio and anyway does not fall below the technical rate (minimum interest rate guaranteed, henceforth). Moreover, if the contract is paid by periodical premiums, it is usually stated that the annual premium is adjusted at the same rate of the bonus, and thus the benefit is also adjusted in the same measure. In such policy the variables controlled by the insurance company (control-variables, henceforth) are the technical rate, the participation level and, in some sense, the riskiness of the reference portfolio measured by its volatility. However, as it is intuitive, not all sets of values for these variables give rise to a fair contract, i.e. to a contract priced consistently with the usual assumptions on financial markets and, in particular, with no-arbitrage. We derive then necessary and sufficient conditions under which each control-variable is determined by a fair pricing of the contract, given the remaining two ones.

For the purpose of Value-at-Risk (VaR) analysis, a model for the return distribution is important because it describes the potential behavior of a financial security in the future. What is primarily, is the behavior in the tail of the distribution since VaR analysis deals with extreme market situations. We analyze the extension of the normal distribution function to allow for fatter tails and for time-varying volatility. Equally important to the distribution function are the associated parameter values. We argue that parameter uncertainty leads to uncertainty in the reported VaR estimates. There is a tradeoff between more complex tail-behavior and this uncertainty. The “best estimate”-VaR should be adjusted to take account of the uncertainty in the VaR. Finally, we consider the VaR forecast for a portfolio of securities. We propose a method to treat the modeling in a univariate, rather than a multivariate, framework. Such a choice allows us to reduce parameter uncertainty and to model directly the relevant variable.

In this paper we will show how to set up a practical bonus-malus system with a finite number of classes. We will use the actual claim amount and claims frequency distributions in order to predict the future observed claims frequency when the new bonus-malus system will be in use. The future observed claims frequency is used to set up an optimal bonus-malus system as well as the transient and stationary distributions of the drivers in the new bonus-malus system. When the number of classes as well as the transition rules of the new bonus-malus system have been adopted, the premium levels are obtained by minimizing a certain distance between the levels of the practical bonus-malus system and the corresponding optimal bonus-malus system. Some iterations are necessary in order to reach stabilization of the future observed claims frequency and the levels of the practical bonus-malus system.

The purpose of the paper is to use the age of claims in the prediction of risks. A dynamic random effects model on longitudinal count data is presented, and estimated on the portfolio of a major Spanish insurance company. The estimated autocorrelation coefficients of stationary random effects are decreasing. A consequence is that the predictive ability of a claim decreases with the lag between the period of risk prediction and the period of occurrence. There is a wide gap between the long term properties of actuarial and real-world experience rating schemes. This gap can be partly filled if the age of claims is taken into account in the actuarial model.

We are dealing with the ruin probability and the expected ruin time in a two state Markov model where the premium is the reciprocal of an integer and the initial surplus is a multiple of the premium.