Ruin theory

Ruin theory, sometimes referred to as collective risk theory, is a branch of actuarial science that studies an insurer's vulnerability to insolvency based on mathematical modeling of the insurer's surplus.

The theory permits the derivation and calculation of many ruin-related measures and quantities, including the probability of ultimate ruin, the distribution of an insurer's surplus immediately prior to ruin, the deficit at the time of ruin, the distribution of the first drop in surplus given that the drop occurs, etc.

It is also considered as an area of applied probability because most of the techniques and methodologies adopted in ruin theory are based on the application of stochastic processes. Though most problems in ruin theory stem from real-life actuarial studies, it is the mathematical aspects of ruin theory that have drawn much of the attention from actuarial scientists and probabilists in the past few decades.

History

The theoretical foundation of ruin theory, known as the classical compound-Poisson risk model in the literature, was introduced in 1903 by the Swedish actuary Filip Lundberg. [Lundberg, F. (1903) Approximerad Framställning av Sannolikehetsfunktionen, Återförsäkering av Kollektivrisker, Almqvist & Wiksell, Uppsala.] The classical model was later extended to relax assumptions about the inter-claim time distribution, the distribution of claim sizes, etc. In most cases, the principal objective of the classical model and its extensions was to calculate the probability of ultimate ruin.

Ruin theory received a substantial boost with the articles of Powers [Powers, M.R. (1995) A theory of risk, return, and solvency, Insurance: Mathematics and Economics 17(2): 101-118.] in 1995 and Gerber and Shiu [Gerber, H.U. and Shiu, E.S.W. (1998) On the time value of ruin, North American Actuarial Journal 2(1): 48-78.] in 1998, which introduced the expected discounted penalty function, a generalization of the probability of ultimate ruin. This fundamental work was followed by a large number of papers in the ruin literature deriving related quantities in a variety of risk models.

Classical model

Traditionally, an insurer's surplus has been modeled as the result of two opposing cash flows: an incoming cash flow of premium income collected continuously at the rate of $c$, and an outgoing cash flow due to a sequence of insurance claims $Y\_1,Y\_2,dots,Y\_i,dots,$ that are mutually independent and identically distributed with common distribution function $P(y)$. The arrival of claims is assumed to follow a Poisson process with intensity rate $lambda$, which means that the number of incurred claims $N(t)$ at time $t$ is governed by a Poisson distribution with mean $lambda\; t$. Hence, the insurer's surplus at any time $t$ is given by

:$X(t)=x+ct-sum^\{N(t)\}\_\{i=0\}Y\_i,$

where the insurer's business commences with an initial surplus level $X(0)=x$ under the probability measure $mathbb\{P\}^x$.

The central object of Lundberg's model was to investigate the probability that the insurer's surplus level eventually would fall below zero (making the firm bankrupt). This quantity, called the probability of ultimate ruin, is defined as

:

where the time of ruin is with the convention that $infvarnothing=infty$.

It is well known that the probability of ultimate ruin is the tail probability of a compound-geometric distribution. The exact solutions and asymptotic approximations to the probability of ruin rely largely on techniques of renewal theory.

Expected discounted penalty function

The articles of Powers (1995) and Gerber and Shiu (1998) analyzed the behavior of the insurer's surplus through the expected discounted penalty function, which is commonly referred to as Gerber-Shiu function in the ruin literature. It is arguable whether the function should have been called Powers-Gerber-Shiu function due to the contribution of Powers (1995).

In Powers’ notation, this is defined as

:$m(x)=mathbb\{E\}^x\; [e^\{-delta\; au\}K\_\{\; au\}]$,

where $delta$ is the discounting force of interest, $K\_\{\; au\}$ is a general penalty function reflecting the economic costs to the insurer at the time of ruin, and the expectation $mathbb\{E\}^x$ corresponds to the probability measure $mathbb\{P\}^x$. The function is called expected discounted cost of insolvency in Powers (1995).

In Gerber and Shiu’s notation, it is given as

:

where $delta$ is the discounting force of interest and $w(X\_\{\; au-\},X\_\{\; au\})$ is a penalty function capturing the economic costs to the insurer at the time of ruin (assumed to depend on the surplus prior to ruin $X\_\{\; au-\}$ and the deficit at ruin $X\_\{\; au\}$), and the expectation $mathbb\{E\}^x$ corresponds to the probability measure $mathbb\{P\}^x$. Here the indicator function

It is quite intuitive to interpret the expected discounted penalty function. Since the function measures the actuarial present value of the penalty that occurs at $au$, the penalty function is multiplied by the discounting factor $e^\{-delta\; au\}$, and then averaged over the probability distribution of the waiting time to $au$. While Gerber and Shiu (1998) applied this function to the classical compound-Poisson model, Powers (1995) argued that an insurer’s surplus is better modeled by a family of diffusion processes. There are a great variety of ruin-related quantities that fall into the category of the expected discounted penalty function.

Other finance-related quantities belonging to the class of the expected discounted penalty function include the perpetual American put option, [ Gerber, H.U. and Shiu, E.S.W. (1997) From ruin theory to option pricing, AFIR Colloquium, Cairns, Australia 1997.] the contingent claim at optimal exercise time, and more.

Recent developments

*Sparre-Andersen risk model
*Compound-Poisson risk model with constant interest
*Compound-Poisson risk model with stochastic interest
*Brownian-motion risk model
*General diffusion-process model
*Markov-modulated risk model

ee also

*Poisson process
*Continuous-time Markov process
*Stochastic processes
*Queueing theory

Bibliography

*cite book
author = Gerber, H.U.
year = 1979
title = An Introduction to Mathematical Risk Theory
publisher = S.S. Heubner Foundation Monograph Series 8
location = Philadelphia

*cite book
author = Asmussen S.
year = 2000
title = Ruin Probabilities
publisher = World Scientific Publishing Co.
location = Singapore

References