Notes on collective risk theory

Abstract

A complete proof of existence of a probability measure m the space Ω of all sample functions was given by Cramér [4]. For a finitc period, a simplified proof was given in my paper [2]. The latter proof could be restricted to the space of sample functions having only a finite number of jumps, as the probability of an infinite number of jumps is zero in this case. In fact, dividing the space Ω into disjunct subspaces Ω_n containing exactly n jumps we have: The measure of Ω_n m the case of a finite period of length x is: Thus and consequently P (Ω_∞) = 0. Therefore the set Ω_∞ and all its subsets can be neglected.     

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.     

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.     

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.     

A note on a classical result in the collective risk theory

Abstract

In his paper “Über einige risikotheoretische Fragestellungen” (SAT 1942: 1–2, p. 43) C.-O. Segerdahl generalizes the theory of ruin probability ψ(u) to the case where interest is continuously added to the risk reserve u at the rate δ′.     

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.     

A generalization of the collective theory of risk in regard to fluctuating basic-probabilities

Abstract

Generally in the theory of risk one starts from the two fundamental assumptions that the basic-probabilities are constant and that the deviations occurring may be interpreted as random fluctuations. Thus, the theory of risk appears as an application of the ordinary probability-theory, which starts from the binomial distribution and leads to the distributions of Poisson and Gauss.     

A note on different models of stochastic processes dealt with in the collective theory of risk

Abstract

1. For the definition of general processes with special regard to those concerned in Collective Risk Theory reference is made to Cramér (Collective Risk Theory, Skandia Jubilee Volume, Stockholm, 1955). Let the independent parameter of such a process be denoted by τ, with the origin at the point of departure of the process and on a scale independent of the number of expected changes of the random function. Denote with p(τ, n)dt the asymptotic expression for the conditional probability of one change in the random function while the parameter passes from τ to τ + dτ: relative to the hypothesis that n changes have occurred, while the parameter passes from 0 to τ. Assume further—unless the contrary is stated—that the probability of more than one change, while the parameter passes from τ to τ + dτ, is of smaller order than dτ.     

On the probability of ruin in the collective risk theory for insurance enterprises with oly negative risk sums

Abstract

1. The determination of the probability that an insurance company once in the future will be brought to ruin is a problem of great interest in insurance mathematics. If we know this probability, it does not only give us a possibility to estimate the stability of the insurance company, but we may also decide which precautions, in the form of f. ex. reinsurance and loading of the premiums, should be taken in order to make the probability of ruin so small that in practice no ruin is to be feared.     

Remarks on risk theory

§ 1. The Scheme U(n).

Following remarks are the result of some deliberations which I have made in order to find a simple starting point for rough estimates of the risk reserves necessary under given conditions. They do not appear with any pretensions. The idea, upon which my thoughts were based, is far from original: I have hoped to be able to simplify the theory by simplifying in a high degree the hypotheses and by confining myself to small numbers and arithmetical methods.     

A theory of risk disclosure

In this paper, we consider the price effects of risk disclosure. We develop a model in which investors are uncertain about the variance of a firm’s cash flows and the firm releases an imperfect signal regarding this variance. In our model, uncertainty over the riskiness of a firm’s cash flows leads to a variance uncertainty premium in its price. We demonstrate that risk disclosure decreases the firm’s cost of capital by reducing this premium and that the market response to risk disclosure is small when the expected level of risk is high. Moreover, we find that firms acquire and disclose more risk information when their cash flow risk is greater than expected. Finally, we demonstrate that in a multi-asset setting, only risk disclosure concerning systematic risks will impact the cost of capital.     

On Hattendorff's theorem in the theory of risk

Abstract

1. Two of the most important measures of dispersion are the {istandard }deviation and the {iaverage deviation}¹ which, if we are concerned with the financial effects of deviations from an assumed mortality, are called the {imean risk} and the {iaverage risk} and are denoted by {iM} and {iR} respectively.     

A fiscal theory of sovereign risk

Under certain monetary-fiscal regimes the risk of default and thus the emergence of sovereign risk premiums are inevitable. This paper argues that in this context even small differences in the specification of monetary policy can have enormous effects on the equilibrium behavior of default rates and risk premiums. Under some monetary policy rules studied, the conditional expectation of default rates and sovereign risk premiums are constant, so movements in these variables always arrive as a surprise. Under other monetary regimes considered, the equilibrium default rate and the sovereign risk premium are serially correlated and therefore forecastable. The paper also studies the consequences of delaying default. It characterizes environments under which procrastinating on default is counterproductive.     

Liquidity risk and arbitrage pricing theory

Classical theories of financial markets assume an infinitely liquid market and that all traders act as price takers. This theory is a good approximation for highly liquid stocks, although even there it does not apply well for large traders or for modelling transaction costs. We extend the classical approach by formulating a new model that takes into account illiquidities. Our approach hypothesizes a stochastic supply curve for a securitys price as a function of trade size. This leads to a new definition of a self-financing trading strategy, additional restrictions on hedging strategies, and some interesting mathematical issues.Received: 1 November 2003, Mathematics Subject Classification: 60G44, 60H05, 90A09JEL Classification: G11, G12, G13Umut Çetin: This work was performed while Dr. Çetin was at the Center for Applied Mathematics, Cornell UniversityPhilip Protter: Supported in part by NSF grant DMS-0202958 and NSA grant MDA-904-03-1-0092 The authors wish to thank M. Warachka and Kiseop Lee for helpful comments, as well as the anonymous referee and Associate Editor for numerous helpful suggestions, which have made this a much improved paper.     

An introduction into Lundberg's theory of risk

Abstract

In this number of the journal a paper of Dr. Filip Lundberg is published, in which he thoroughly deals with certain problems of the theory of risk. As all the former works of Dr. Lundberg about the theory of risk with only one exception (the paper “Über die Theorie der Rückversicherung” in the transactions of the Congress of Actuaries in Wien 1909) are published in the Swedish language and consequently inaccessible to the international insurance world, a simultaneous report of some of the fundamental ideas in this former production perhaps will offer some interest. Though the latest paper follows a special line, the starting points and the manner of consideration are unaltered, and hence an aquaintance with the simpler problems which here will be dealt with will be rather illustrative.     

Systematic risk and the theory of the firm: A reexamination

A recent microeconomic model of the determinants of equity betas (Subrahmanyam and Thomadakis 1980) is generalized by including risky human capital in the market portfolio and allowing a general covariance structure between the model's sources of uncertainty. This provides an explanation of the ambiguous effect of operating leverage on beta by viewing human capital and equity contributors as risk sharers in the firm's output risk. This explanation may help to clarify the apparent conflict with the previous literature. The relationship between systematic risk and monopoly power is rederived and shown to depend upon a plausible condition on the correlation between wage rate and price uncertainty. Finally, the public policy implications of this analysis are presented.     

An optimal stopping problem in risk theory

Abstract

In classical risk theory often stationary premium and claim processes are considered. In some cases it is more convenient to model non-stationary processes which describe a movement from environmental conditions, for which the premiums were calculated, to less favorable circumstances. This is done by a Markov-modulated Poisson claim process. Moreover the insurance company is allowed to stop the process at some random time, if the situation seems unfavorable, in order to calculate new premiums. This leads to an optimal stopping problem which is solved explicitly to some extent.     

运用系统科学理论加强银行风险管理

自商业银行诞生之日起,风险就与之相伴相随。随着商业银行业务发展的多元化和国际化,其风险也呈现出复杂多变的特征。从对象上看,已经由单一的信贷风险发展为包括信用风险、市场风险、操作风险等在内的多类型风险;从性质上看,从最初的局部风险演变为全球风险。风险存在于商业银行业务发展的每一个环节中,商业银行提供金融服务的过程也就是承担和控制风险的过程。