On the probability function in the collective theory of risk

Abstract

Upon reading Dr. LUNDBERG'S paper ?Über die Wahrscheinlichkeitsfunktion einer Risikenmaase?¹ and trying to penetrate it along my own lines of thought, I found another way of deducing some of his formulas, giving the results in a form that directly invites a fairly simple approximation of the probability function. Though time has not permitted my going deeper into the problem, I propose here to give a brief account of the method.     

Some properties of the ruin function in the collective theory of risk

Abstract

It is well known that the chief aim of all theory of risk is to attain a sort of objective and somehow confirmed opinion of how and to which extent an insurance company ought to reinsure its risks in order that the probability of ruin by random fluctuations of the risk process shall become so small that it can be overlooked in practice.     

On a nonparametric estimator for ruin probability in the classical risk model

In this paper, we present a nonparametric estimator for ruin probability in the classical risk model with unknown claim size distribution. We construct the estimator by Fourier inversion and kernel density estimation method. Under some conditions imposed on the kernel, bandwidth and claim size density, we present some large sample properties of the estimator. Some simulation studies are also given to show the finite sample performance of the estimator.     

On a risk measure inspired from the ruin probability and the expected deficit at ruin

In this paper, we study a risk measure derived from ruin theory defined as the amount of capital needed to cope in expectation with the first occurrence of a ruin event. Specifically, within the compound Poisson model, we investigate some properties of this risk measure with respect to the stochastic ordering of claim severities. Particular situations where combining risks yield diversification benefits are identified. Closed form expressions and upper bounds are also provided for certain claim severities.     

From ruin theory to solvency in non-life insurance

We start from ruin theory considerations in the classical Cramér–Lundberg model. We modify these considerations step by step so that finally we arrive at today’s solvency assessments for non-life insurance companies. These modifications include discussions about time horizons, risk measures, financial returns, and valuation of insurance liabilities.     

In the insurance business risky investments are dangerous: the case of negative risk sums

We investigate models with negative risk sums when the company invests its reserve into a risky asset whose price follows a geometric Brownian motion. Our main result is an exact asymptotic of the ruin probabilities for the case of exponentially distributed benefits. As in the case of non-life insurance with exponential claims, the ruin probabilities are either decreasing with a rate given by a power function (the case of small volatility) or equal to one identically (the case of large volatility). The result allows us to quantify the share of reserve to invest into such a risky asset to avoid a catastrophic outcome, namely the ruin with probability one. We address also the question of smoothness of the ruin probabilities as a function of the initial reserve for generally distributed jumps.     

Research in collective risk theory

Abstract

An insurance company can be considered as an adjustment institution for the policyholders. The individual risks of the policyholders are taken over by the company at the price of a comparatively small stake, the premium. This is so calculated that the premiums from all the policyholders will, according to statistical experience, on the average cover the company's payments for claims. With respect to unfavourable random deviations from the average, the premiums contain security loadings. For the same purpose the company also makes other precautions. The most important of these are reinsurance and the building up of adjustment funds. On the other hand, extensive precautions increase the price of the insurance. Therefore the objective fixing of the precautions in order to get a satisfactory solidity as well as a reasonable price constitutes a weighing problem, demanding a measure of the effect of diverse precautions.     

Asymptotic ruin probabilities for a discrete-time risk model with dependent insurance and financial risks

This contribution focuses on a discrete-time risk model in which both insurance risk and financial risk are taken into account and they are equipped with a wide type of dependence structure. We derive precise asymptotic formulas for the ruin probabilities when the insurance risk has a dominatedly varying tail. In the special case of regular variation, the corresponding formula is proved to be uniform for the time horizon.     

Note on the collective theory of risk

Abstract

What follows has grown out of a discussion with Carl Philipson following a lecture [1] on the collective theory of risk. Although I give here nothing else but a refined interpretation of Paul Lévy's form (see, e.g., [2], p. 322) of identically distributed random variables the result still seems of interest for all those working in the field of collective risk theory. I thank Carl Philipson for stimulating my interest in this matter.     

On finite-time ruin probabilities for classical risk models

Abstract

Generally, the return of premiums without interest or with simple interest is provided for. It might, however, be worth while to notice that less complicated formulre are needed for the return of premiums with compound interest.     

On a conjecture related to the ruin probability for nonhomogeneous exponentially distributed claims

Recently, some recursive formulas have been obtained for the ruin probability evaluated at or before claim instants for a surplus process under the assumptions that the claim sizes are independent, nonhomogeneous Erlang distributed, and independent of the inter-claim revenues, which are assumed to be independent, identically distributed, following an arbitrary distribution. Based on numerical examples, a conjecture has also been stated relating the order in which the claims arrive to the magnitude of the corresponding ruin probability. In this paper, we prove this conjecture in the particular case when the claims are all exponentially distributed with different parameters.     

A ruin problem for the insurance of expanding portfolios

Abstract

The purpose of this short note is to demonstrate the power of very straightforward branching process methods outside their traditional realm of application. We shall consider an insurance scheme where claims do not necessarily arise as a stationary process. Indeed, the number of policy-holders is changing so that each of them generates a random number of new insurants. Each one of these make claims of random size at random instants, independently but with the same distribution for different individuals. Premiums are supposed equal for all policy-holders. It is proved that there is, for an expanding portfolio, only one premium size which is fair in the sense that if the premium is larger than that, then the profit of the insurer grows infinite with time, whereas a smaller premium leads to his inevitable ruin. (Branching process models for the development of the portfolio may seem unrealistic. However, they do include the classical theory, where independent and identically distributed claims arise at the points of a renewal process.)     

A review of the collective theory of risk

Abstract

1. Ackermann, W. G. 1939. Eine Erweiterung des Poissonschen Grenzwertsatzes und ihre Anwendung auf Risikoprobleme der Sachversicherung. Berlin.     

保费收入随机化条件下的保险公司破产概率模型研究

保费收入是保险公司破产概率的重要影响因素。传统的保险公司破产概率模型常将保费收入过程看作连续的确定性过程,然而在现实中,保费收入过程却是一个离散的随机过程。本文用复合泊松过程描述保费收入,从而将确定性保费收入条件下的破产概率模型拓展到随机化保费收入条件下的破产概率模型,在此基础上模拟计算了保险公司破产概率,并比较分析了不同的保险资金投资模式对破产概率的影响。     

On the severity of ruin in a Markov-modulated risk model

We consider a Markov-modulated risk model in which the claim inter-arrivals, amounts and premiums are influenced by an external Markovian environment process. A system of Laplace transforms of the probabilities of the severity of ruin, given the initial environment state, is established from a system of integro-differential equations derived by Snoussi [The severity of ruin in Markov-modulated risk models Schweiz Aktuarver. Mitt., 2002, 1, 31–43]. In the two-state model, explicit formulas for probabilities of the severity of ruin are derived, when the initial reserve is zero or when both claim amount distributions are from the rational family. Numerical illustrations are also given.     

On semiparametric estimation of ruin probabilities in the classical risk model

The ruin probability of an insurance company is a central topic in risk theory. We consider the classical Poisson risk model when the claim size distribution and the Poisson arrival rate are unknown. Given a sample of inter-arrival times and corresponding claims, we propose a semiparametric estimator of the ruin probability. We establish properties of strong consistency and asymptotic normality of the estimator and study bootstrap confidence bands. Further, we present a simulation example in order to investigate the finite sample properties of the proposed estimator.     

Exact solutions for ruin probability in the presence of an absorbing upper barrier

Abstract

An explicit solution for the probability of ruin in the presence of an absorbing upper barrier was developed by Segerdahl (1970) for the particular case in which both the interoccurrence times between successive claims and the single claim amounts follow an exponential distribution with unit mean. In this paper we show that his method of solution may be extended to produce explicit solutions for two more general types of single claim amount distribution. These are the gamma distribution, denoted γ(a), where a is an integer, and the mixed exponential distribution. Comparisons are drawn between this approach when the upper barrier tends to infinity, and the classical solution for ruin probability in these particular cases given in Cramér (1955).