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.     

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.     

也谈竞争力

首先,抢抓机遇需要我们有一双善于发现商机的眼睛;其次,抢抓机遇需要我们有善舞长袖的本领;最后,抢抓机遇需要我们有化危为机的能力。     

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.     

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.     

Remarks on Basel II

In this paper Le Pan focuses on some key issues related to Basel II, such as complexity, impact on capital and home‐host implementation, and offers ten thoughts on the post Basel II world.     

Remarks on iteration.

1. Iteration is an operation with many aspects; we shall here only occupy ourselves with iteration as a means of numerical determination of a real root of an equation. Nor shall we go into the whole of the theory of this subject ² See, for instance, Whittaker & Robinson: The Calculus of Observations, or Runge & König: Vorlesungen über numerisches Rechnen. , our principal object being to show how the process of iteration which is often futile in its primitive form may be improved by a suitable combination of three consecutive values. But before doing this, it will be convenient to recall briefly some of the principles involved in iteration.     

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.     

Corporate risk management and dividend signaling theory

This article investigates the effect of corporate risk management on dividend policy. We extend the signaling framework of Bhattacharya [1979. Bell Journal of Economics 10, 259–270] by including the possibility of hedging the future cash flow. We find that the higher the hedging level, the lower the incremental dividend. This result is intuitive. It is in line with studies suggesting that cash flows’ predictability decreases the marginal gain from costly signaling through dividends and the assertion that corporate hedging decreases cash flow volatility. It is also in line with the purported positive relation between information asymmetry and dividend policy (e.g., Miller and Rock [1985. The Journal of Finance 40, 1031–1051]) and the assertion that risk management alleviates the information asymmetry problem (e.g., DaDalt et al. [2002. The Journal of Future Markets 22, 261–267]). Our theoretical model has testable implications.     

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.     

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′.     

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 bridge from ruin theory to credit risk

The structural model uses the firm-value process and the default threshold to obtain the implied credit spread. Merton’s (J Finance 29:449–470, 1974) credit spread is reported too small compared to the observed market spread. Zhou (J Bank Finance 25:2015–2040, 2001) proposes a jump-diffusion firm-value process and obtains a credit spread that is closer to the observed market spread. Going in a different direction, the reduced-form model uses the observed market credit spread to obtain the probability of default and the mean recovery rate. We use a jump-diffusion firm-value process and the observed credit spread to obtain the implied jump distribution. Therefore, the discrepancy in credit spreads between the structural model and the reduced-form model can be removed. From the market credit spread, we obtain the implied probability of default and the mean recovery rate. When the solvency-ratio process in credit risk and the surplus process in ruin theory both follow jump-diffusion processes, we show a bridge between ruin theory and credit risk so that results developed in ruin theory can be used to develop analogous results in credit risk. Specifically, when the jump is Logexponentially distributed, it results in a Beta distributed recovery rate that is close to market experience. For bonds of multiple seniorities, we obtain closed-form solutions of the mean and variance of the recovery rate. We prove that the defective renewal equation still holds, even if the jumps are possibly negative. Therefore, we can use ruin theory as a methodology for assessing credit ratings.      

A decision theoretic formulation of insurance risk theory

Abstract

In a number of papers Borch has shown how certain insurance problems can be formulated using the concept of utility. (See Borch [3], [4], [5], [6], [7] and [8].) Borch's work is used as a building block in Part I of this report, which presents a Bayesian decision theoretic formulation of some of the main aspects of insurance risk theory. Part I makes use of the concepts of utility and subjective probability. It is admitted that these concepts are more commonly associated with individuals rather than groups of individuals such as insurance companies. However, in this report, we will refer to an insurance company as an individual (albeit a neuter one) and assume that it can quantify its preferences for consequences and its opinions about the occurrence of events. Further, we assume that a company “behaves” according to certain rules of consistent behavior which imply that when presented with several risky courses of action, the company will take the action which has the greatest expected utility. Formal treatments of assumptions that lead to this mode of behavior can be found in Savage [17] and Pratt, Raiffa, and Schlaifer [15].     

新春贺辞

2008年的脚步渐渐远去,2009年的钟声已经敲响.又是一个充满希望与挑战的一年.     

Remarks on semi-invariants and incomplete moments

Let us consider a general discontinuous frequency distribution where the x^pi -S are the values of the variable x, and f(x^pi) is the probability that x will take the value x^pi . We will assume that that is to say: x must take one of the values x^pi(i = 0, 1, 2, 3, ... n).