This study investigates a combined optimal financing, reinsurance and dividend distribution problem for a big insurance portfolio. A manager can control the surplus by buying proportional reinsurance, paying dividends and raising money dynamically. The transaction costs and liquidation values at bankruptcy are included in the risk model. Under the objective of maximising the insurance company's value, we identify the insurer's joint optimal strategies using stochastic control methods. The results reveal that managers should consider financing if and only if the terminal value and the transaction costs are not too high, less reinsurance is bought when the surplus increases or dividends are always distributed using the barrier strategy.

The paper considers the problem of finding an upper bound for the Stop loss premium.

We will start with a brief sketch of the practical context in which this problem is relevant.

If it is reasonable to assume, that the accumulated claims variable of the underlying risk can be represented by a Compound Poisson Process, the following data are needed for fixing the Stop loss premium:

— the claims intensity,

— the distribution of the claimsizes (jump-size variable).

In practical situations it is usually possible to find a reasonable estimate for the claims intensity (expected number of claims in a given period).

Generally speaking, however, it is not so easy to get sufficient data on the claimsize distribution. Ordinarily only its mean is known. This deficiency in information can of course be offset by assuming the unknown distribution to be one of the familiar types, such as Exponential, Gamma or Pareto.

Stop loss premiums are however very sensitive to variations in the type of claimsize distribution and consequently it can make a lot of difference in the result what particular choice is made.

To gain some insight into the consequences of a specific assumption, it is useful to know within what range the premium can move for varying distributional suppositions. This means establishing an upper bound and a lower bound. The lower bound is trivially obtained if the mass of the claimsize distribution is solely concentrated at its mean. The upper bound on the other hand should correspond to the “worst” possible claimsize distribution. This means, that we have to look for a distribution which maximizes the Stop loss premium.

The bonus system used by Finnish insurers is as follows: Policies are divided into 4 bonus classes, C1, C2, C3, C4, each of them being initially placed in class C1. After one claimless year, a policy placed in class Ch is transferred to class Ch+1 (h ≤ 3) from the beginning of the following year. After a claim year, each policy is entered back to class C1. (Whether the number of claims is one or more does not play any role in this transition system.) From the initial premium, i.e. the premium in class C1, a reduction of pi, per cent is granted in class Ci. In numbers these percentages are as follows:

In present investigations there is a trend to replace this scale by another in which mere contingency would not cause too great a change in the premium and in which the „test period” would be longer.

Let us suppose that we want to find a suitable percentage scale P = (P1, P2, …, Pm) for a bonus system (Cl, C2, …, Cm) with a given transition mechanism according to which for each bonus class Ch two bonus classes Ci and Cj are given to which the policy is transferred after one year from Ch according as there have been claims or not.

The theory of extreme values is a special branch of mathematical statistics and was mainly treated by E. J. Gumbel [4]). This theory has only been applied in a few cases to problems in the insurance business. The first practical application to insurance known to the author of the present paper is due to A. Thépaut who has invented a new reinsurance system called ECOMOR [5]. According to this system the reinsurer covers the excess risk for the m largest claims and the ceding company retains an amount equal to the (m + I) largest claim. The credit for having pointed out the importance of the theory of extreme values belongs to R. E. Beard [1]. Recently E. Franckx [3] has found a most remarkable result by disclosing the general form of the distribution for the largest claim occurring in a certain accounting period.

The present paper starts from the consideration that not only is the distribution of major claims, which might be eliminated by means of reinsurance, of interest to an insurer but also the distribution of the remaining total loss after excluding the largest claims. The nature of this distribution is important not only in connection with stability and security, but also for statistical investigations of the observed claim ratio. The credibility of such an investigation might be greatly improved if a suitable number of major claims were excluded. To simplify matters, the present paper considers the case where only the largest claim is excluded.

We consider the optimal financing (capital injections) and dividend payment problem for a Brownian motion model in the case of restricted dividend rates. The company has no obligation to inject capitals and therefore, the bankruptcy risk is present. Capital injections, if any, will incur both fixed and proportional transaction costs and dividend payments incur proportional transaction costs. The aim is to find the optimal strategy to maximize the expected present value of dividend payments minus the total cost of capital injections up to the time of bankruptcy. The problem is formulated as a mixed impulse-regular control problem. We address the problem via studying three cases of two auxiliary functions. We derive important analytical properties of the auxiliary functions and use them to study the value function and then identify the optimal control strategy. We show that the optimal dividend control is of threshold type and the optimal financing strategy prescribes to either never inject capitals or inject capitals only when the surplus reaches 0 with a fixed lump sum amount.

We determine the optimal robust strategy of an individual who seeks to maximize the (penalized) probability of reaching a bequest goal when she is uncertain about the drift of the risky asset and her hazard rate of mortality. We assume the individual can invest in a Black–Scholes market. We solve two optimization problems with ambiguity. The first is to maximize the penalized probability of reaching a bequest goal without life insurance in the market. In the second problem, in addition to investing in the financial market, the individual is allowed to purchase term life insurance to help her reach her bequest goal. As the individual becomes more ambiguity averse concerning the drift of the risky asset, she becomes more conservative with her investment strategy. Also, as she becomes more ambiguity averse about her hazard rate of mortality, numerical work indicates she is more likely to buy life insurance when the ambiguity towards the return of the risky asset is not too large.

The paper introduces an alternative approach to the traditional experience rating theory in automobile insurance. The approach is based on a simple theory of how high deductibles financed by loans maintain the risk differentiation in an automobile insurance arrangement. Thus the approach differs totally from the usual bonus-malus classes as well as from the credibility based experience rating ideas. The paper is of a theoretical nature and leads up to a mathematical description of how the approach may be optimalized within the framework of a risk model.

Upper and lower bounds are derived for the stop-loss premium of compound distributions with fixed claim number distribution and known mean, variance and range for the claim severity distribution.

We consider the optimal dividend control problem to find an optimal strategy under the constraint that dividend rates is restricted such that the expected total discounted dividends are maximized for an insurance company. The evolution of the reserve is modeled by a diffusion process with drift and volatility coefficients modulated by an observable Markov chain. We consider the regime-switching threshold strategy which pays out dividends at the maximal possible rate when the current reserve is above some critical level dependent on the regime of the Markov chain at the time, and pays nothing when the reserve is below that level. We give sufficient conditions under which such type of strategy is optimal for the regime-switching model.

A simple risky situation is studied in the framework of consumption theory. Saving is shown to be a substitute to insurance. Two new concepts, riskbearing budget and effective risk coverage, are introduced in order to give a more accurate insight into the optimal risk-bearing decision. The effect of a variation in current consumption and in wealth upon the optimal insurance coverage is analysed.

The α-level Conditional Tail Expectation (CTE) of a continuous random variable X is defined as its conditional expectation given the event {X > qα} where qα represents its α-level quantile. It is well known that the empirical CTE (the average of the n (1 – α) largest order statistics in a sample of size n) is a negatively biased estimator of the CTE. This bias vanishes as the sample size increases, but in small samples can be significant. In this article it is shown that an unbiased nonparametric estimator of the CTE does not exist. In addition, the asymptotic behavior of the bias of the empirical CTE is studied, and a closed form expression for its first order term is derived. This expression facilitates the study of the behavior of the empirical CTE with respect to the underlying distribution, and suggests an alternative (to the bootstrap) approach to bias correction. The performance of the resulting estimator is assessed via simulation.

The existence of long memory in mortality data improves the understandings of features of mortality data and provides a new approach for establishing mortality models. The findings of long-memory phenomena in mortality data motivate us to develop new mortality models by extending the Lee–Carter (LC) model to death counts and incorporating long-memory model structure. Furthermore, there are no identification issues arising in the proposed model class. Hence, the constraints which cause many computational issues in LC models are removed. The models are applied to analyse mortality death count data sets from three different countries divided according to genders. Bayesian inference with various selection criteria is applied to perform the model parameter estimation and mortality rate forecasting. Results show that multivariate long-memory mortality model with long-memory cohort effect model outperforms multivariate extended LC cohort model in both in-sample fitting and out-sample forecast. Increasing the accuracy of forecasting of mortality rates and improving the projection of life expectancy is an important consideration for insurance companies and governments since misleading predictions may result in insufficient funds for retirement and pension plans.

A one-parameter version of the generalised Poisson distribution provided by Consul and Jain (1973) is considered in this paper. The distribution is unimodal with a zero vertex and over-dispersed. A generalised linear model related to this distribution is also presented. Its parameters can be estimated by using a Fisher-Scoring algorithm which is equivalent to iteratively reweighted least squares. Due to its flexibility and capacity to describe highly skewed data with an excessive number of zeros, the model is suitable to be applied in insurance settings as an alternative to the negative binomial and zero-inflated model.

Credibility weighting is helpful in many insurance applications where sparse data crave information from other sources of data. In this paper we aim at estimating a hazard curve using the nonparametric kernel method, where a credibility weighting principle is used locally, so that areas of sparse data for one subgroup can be alleviated by available information from other subgroups. The credibility estimator is found through a Hilbert space projection formulation of Buhlmann-Straub's credibility approach.

In the present paper we extend a recursive algorithm developed by Vernic (1999) for compound distributions with bivariate counting distribution and univariate severity distributions to more general multivariate counting distributions.

Several “short cut” methods exist to approximate the total amount of claims ( = χ) of an insurance collective. The classical one is the normal approximation

where and σx are the mean value and standard deviation of x. Φ is the normal distribution function.

It is well-known that the normal approximation gives acceptable accuracy only when the volume of risk business is fairly large and the distribution of the amounts of the individual claims is not “too dangerous”, i.e. not too heterogeneous (cf. fig. 2).

One way to improve the normal approximation is the so called NP-method, which provides for the standardized variable a correction Δz

where

is the skewness of the distribution F(χ). Another variant (NP3) of the NP-method also makes use of the moment μ4, but, in the following, we limit our discussion mainly to the variant (2) (= NP2).

If Δz is small, a simpler formula

is available (cf. fig. 2).

Another approximation was introduced by Bohman and Esscher (1963). It is based on the incomplete gamma function

where

Experiments have been made with both formulae (2) and (4); they have been applied to various F functions, from which the exact (or at least controlled) values are otherwise known. It has been proved that the accuracy is satisfactory provided that the distribution F is not very “dangerous”.

Goovaerts and de Vylder (1983) provided a stable recursive algorithm for calculating the probability of ultimate ruin. Their algorithm yielded bounds for this probability. It is shown that in practice their method may be inherently unstable because it is based on the subtraction of nearly equal numbers. An alternative to this type of subtraction is provided. It is proved that their algorithm converges only at a linear rate to the true value. It is suggested that this slow rate of convergence be improved via an application of the Richardson extrapolation technique.

According to the provisions of the Swedish tables of rates the premium payable on a certain policy both for Third Party Liability and for Car Damage Insurance (without clause for particular self retention) is equal to the initial premium reduced with regard to points earned as related to years with no indemnities. One such point is earned for each year during which no indemnity has been paid on the policy; upon an indemnity one or two points are lost. The detailed rules for the reduction have been altered from time to time, the existing rules have been valid from the year 1952, these rules will here be called system II, while system I denotes the rules valid during the years immediately previous to 1952. The systems are described in the following table.

The probabilities for the occurrence of a claim and for the occurrence of a lapse for a policy can be considered functions of two variables, the points obtained in the bonus scale, b, and the “duration” i.e. the time elapsed since the policyholder signed his first policy, τ. For simplicity two kinds of models have been constructed, one assuming the probabilities to depend only on τ, the other assuming the probabilities to depend only on b.

We consider a dependent portfolio of insurance contracts. Asymptotic tail probabilities of the ECOMOR and LCR reinsurance amounts are obtained under certain assumptions about the dependence structure.

The new Solvency II Directive and the upcoming IFRS 17 regime bring significant changes to current reporting of insurance entities, and particularly in relation to valuation of insurance liabilities. Insurers will be required to valuate their insurance liabilities on a risk-adjusted basis to allow for uncertainty inherent in cash flows that arise from the liability of insurance contracts. Whilst most European-based insurers are expected to adopt the Cost of Capital approach to calculate reserve risk margin — the risk adjustment method commonly agreed under Solvency II and IFRS 17, there is one additional requirement of IFRS 17 to also disclose confidence level of the risk margin.

Given there is no specific guidance on the calculation of confidence level, the purpose of this paper is to explore and examine practical ways of estimating the risk margin confidence level measured by Probability of Sufficiency (PoS). The paper provides some practical approximation formulae that would allow one to quickly estimate the implied PoS of Solvency II risk margin for a given non-life insurance liability, the risk profile of which is specified by the type and characteristics of the liability (e.g. type/nature of business, liability duration and convexity, etc.), which, in turn, are associated with

• • the level of variability measured by Coefficient of Variation (CoV);

• • the degree of Skewness per unit of CoV; and

• • the degree of Kurtosis per unit of CoV2.