15.1 Introduction

In examining the nature of the risk associated with a portfolio of business, it is often of interest to assess how the portfolio may be expected to perform over an extended period of time. One approach concerns the use of ruin theory (Panjer and Willmot; 1992). Ruin theory is concerned with the excess of the income (with respect to a portfolio of business) over the outgo, or claims paid. This quantity, referred to as insurer's surplus, varies in time. Specifically, ruin is said to occur if the insurer's surplus reaches a specified lower bound, e.g. minus the initial capital. One measure of risk is the probability of such an event, clearly reflecting the volatility inherent in the business. In addition, it can serve as a useful tool in long range planning for the use of insurer's funds.

We recall now a definition of the standard mathematical model for the insurance risk, see Grandell (1991) and Chapter 14. The initial capital of the insurance company is denoted by $ u$, the Poisson process $ N_{t}$ with intensity (rate) $ \lambda $ describes the number of claims in $ (0,t]$ interval and claim severities are random, given by i.i.d. non-negative sequence $ \{X_{k}\}_{k=1}^{\infty}$ with mean value $ \mu$ and variance $ \sigma^{2}$, independent of $ N_t$. The insurance company receives a premium at a constant rate $ c$ per unit time, where $ c=(1+\theta)\lambda\mu$ and $ \theta>0$ is called the relative safety loading. The classical risk process $ \{R_{t}\}_{t\geq 0 }$ is given by

$\displaystyle R_{t}=u+ct-\sum_{i=1}^{N_{t}}X_{i}.$    

We define a claim surplus process $ \{S_{t}\}_{t \geq 0 }$ as

$\displaystyle S_{t}=u-R_{t}=\sum_{i=1}^{N_{t}}X_{i}-ct.$    

The time to ruin is defined as $ \tau(u)=\inf\{t\geq 0 : R_{t} < 0 \}=\inf\{t\geq 0 :
S_{t}>u\}.$ Let $ L=\sup_{0\leq t<\infty}\{S_{t}\}$ and $ L_T=\sup_{0\leq t<T}\{S_{t}\}$. The ruin probability in infinite time, i.e. the probability that the capital of an insurance company ever drops below zero can be then written as

$\displaystyle \psi(u)=\textrm{P}(\tau(u) < \infty )=\textrm{P}(L>u).$ (15.1)

We note that the above definition implies that the relative safety loading $ \theta$ has to be positive, otherwise $ c$ would be less than $ \lambda\mu$ and thus with probability $ 1$ the risk business would become negative in infinite time. The ruin probability in finite time $ T$ is given by

$\displaystyle \psi(u,T)=\textrm{P}(\tau(u) \leq T )=\textrm{P}(L_{T}>u).$ (15.2)

We also note that obviously $ \psi(u,T)<\psi(u)$. However, the infinite time ruin probability may be sometimes also relevant for the finite time case.

From a practical point of view, $ \psi(u,T)$, where $ T$ is related to the planning horizon of the company, may perhaps sometimes be regarded as more interesting than $ \psi(u)$. Most insurance managers will closely follow the development of the risk business and increase the premium if the risk business behaves badly. The planning horizon may be thought of as the sum of the following: the time until the risk business is found to behave ``badly'', the time until the management reacts and the time until a decision of a premium increase takes effect. Therefore, in non-life insurance, it may be natural to regard $ T$ equal to four or five years as reasonable (Grandell; 1991).

We also note that the situation in infinite time is markedly different from the finite horizon case as the ruin probability in finite time can always be computed directly using Monte Carlo simulations. We also remark that generalizations of the classical risk process, which are studied in Chapter 14, where the occurrence of the claims is described by point processes other than the Poisson process (i.e., non-homogeneous, mixed Poisson and Cox processes) do not alter the ruin probability in infinite time. This stems from the following fact. Consider a risk process $ \tilde{R_t}$ driven by a Cox process $ \tilde{N}_t$ with the intensity process $ \tilde{\lambda} (t)$, namely $ \tilde{R}_{t}=u+(1+\theta)\mu\int_0^t\tilde{\lambda} (s)ds-\sum_{i=1}^{\tilde{N}_{t}}X_{i}.$ Define now $ \Lambda_t = \int_0^t \tilde\lambda(s)ds$ and $ R_t=\tilde{R}(\Lambda^{-1}_t)$. Then the point process $ N_t = \tilde{N}(\Lambda^{-1}_t)$ is a standard Poisson process with intensity 1, and therefore, $ \tilde{\psi}(u)=\textrm{P} (\inf_{t\geq 0}\{\tilde{R_t}\}<0)= \textrm{P} (\inf_{t\geq 0} \{R_t\}<0)=\psi(u)$. The time scale defined by $ \Lambda^{-1}_t$ is called the operational time scale. It naturally affects the time to ruin, hence the finite time ruin probability, but not the ultimate ruin probability.

The ruin probabilities in infinite and finite time can only be calculated for a few special cases of the claim amount distribution. Thus, finding a reliable approximation, especially in the ultimate case, when the Monte Carlo method can not be utilized, is really important from a practical point of view.

In Section 15.2 we present a general formula, called Pollaczek-Khinchin formula, on the ruin probability in infinite time, which leads to exact ruin probabilities in special cases of the claim size distribution. Section 15.3 is devoted to various approximations of the infinite time ruin probability. In Section 15.4 we compare the 12 different well-known and not so well-known approximations. The finite-time case is studied in Sections 15.5, 15.6, and 15.7. The exact ruin probabilities in finite time are discussed in Section 15.5. The most important approximations of the finite time ruin probability are presented in Section 15.6. They are illustrated in Section 15.7.

To illustrate and compare approximations we use the PCS (Property Claim Services) catastrophe data example introduced in Chapter 13. The data describes losses resulting from natural catastrophic events in USA that occurred between 1990 and 1999. This data set was used to obtain the parameters of the discussed distributions.

We note that ruin theory has been also recently employed as an interesting tool in operational risk. In the view of the data already available on operational risk, ruin type estimates may become useful (Embrechts, Kaufmann, and Samorodnitsky; 2004). We finally note that all presented explicit solutions and approximations are implemented in the Insurance library of XploRe. All figures and tables were created with the help of this library.


15.1.1 Light- and Heavy-tailed Distributions

We distinguish here between light- and heavy-tailed distributions. A distribution $ F_{X}(x)$ is said to be light-tailed, if there exist constants $ a>0$, $ b>0$ such that $ \bar{F}_{X}(x)=1-F_{X}(x) \leq a e^{-bx} $ or, equivalently, if there exist $ z>0$, such that $ M_{X}(z)<
\infty$, where $ M_{X}(z)$ is the moment generating function, see Chapter 13. Distribution $ F_{X}(x)$ is said to be heavy-tailed, if for all $ a>0$, $ b>0$: $ \bar{F}_{X}(x) > a e^{-bx},$ or, equivalently, if $ \forall z>0$ $ M_{X}(z)= \infty.$ We study here claim size distributions as in Table 15.1.

Table 15.1: Typical claim size distributions. In all cases $ x\geq 0$.
Light-tailed distributions
Name Parameters pdf
Exponential $ \beta>0$ $ \; f_{X}(x)=\beta \exp(-\beta x)$
Gamma $ \alpha >0$, $ \beta>0$ $ \; f_{X}(x)=\frac{\beta^{\alpha}}{\Gamma (\alpha )} x^{\alpha-1}\exp(-\beta x)$
Weibull $ \beta>0$, $ \tau \geq 1$ $ \; f_{X}(x)=\beta \tau x^{\tau -1}\exp(-\beta x^{\tau})$
Mixed exp's $ \beta_{i}>0$, $ \sum_{i=1}^{n}\limits a_{i}=1 $ $ \; f_{X}(x)=\sum_{i=1}^{n}\limits
\left\{a_{i}\beta_{i}\exp(-\beta_{i}x)\right\}$
Heavy-tailed distributions
Name Parameters pdf
Weibull $ \beta>0$, $ 0 < \tau < 1$ $ \; f_{X}(x)=\beta \tau x^{\tau -1}\exp(-\beta x^{\tau})$
Log-normal $ \mu\in \mathbb{R}$, $ \sigma>0$ $ \; f_{X}(x)=\frac{1}{\sqrt{2\pi}\sigma
x}\exp\left\{-\frac{\left(\ln x -\mu\right)^{2}}{2\sigma^{2}}\right\}$
Pareto $ \alpha >0$, $ \lambda > 0$ $ \; f_{X}(x)=\frac{\alpha}{\lambda+x}\left( \frac{\lambda}{\lambda +x}
\right)^{\alpha}$
Burr $ \alpha >0$, $ \lambda > 0$, $ \tau>0$ $ \; f_{X}(x)=\frac{\alpha\tau\lambda^{\alpha} x^{\tau-1}}{(\lambda +x^{\tau})^{\alpha+1}}$

In the case of light-tailed claims the adjustment coefficient (called also the Lundberg exponent) plays a key role in calculating the ruin probability. Let $ \gamma=\sup_z\left\{M_{X}(z)\right\}<\infty$ and let $ R$ be a positive solution of the equation:

$\displaystyle 1+(1+\theta)\mu R=M_{X}(R), \qquad R<\gamma.$ (15.3)

If there exists a non-zero solution $ R$ to the above equation, we call it an adjustment coefficient. Clearly, $ R=0$ satisfies the equation (15.3), but there may exist a positive solution as well (this requires that $ X$ has a moment generating function, thus excluding distributions such as Pareto and the log-normal). To see the plausibility of this result, note that $ M_{X}(0)=1$, $ M'_{X}(z)<0$, $ M^{''}_{X}(z)>0$ and $ M^{'}_{X}(0)=-\mu$. Hence, the curves $ y=M_{X}(z)$ and $ y=1+(1+\theta )\mu z$ may intersect, as shown in Figure 15.1.

Figure 15.1: Illustration of the existence of the adjustment coefficient. The solid blue line represents the curve $ y=1+(1+\theta )\mu z$ and the dotted red one $ y=M_{X}(z)$.

\includegraphics[width=1.04\defpicwidth]{STFruin01.ps}

An analytical solution to equation (15.3) exists only for few claim distributions. However, it is quite easy to obtain a numerical solution. The coefficient $ R$ satisfies the inequality:

$\displaystyle R<\frac{2\theta \mu}{\mu^{(2)}},$ (15.4)

where $ \mu^{(2)}=\mathop{\textrm{E}}(X_{i}^{2})$, see Asmussen (2000). Let $ D(z)=1+(1+\theta)\mu z-M_{X}(z)$. Thus, the adjustment coefficient $ R>0$ satisfies the equation $ D(R)=0$. In order to get the solution one may use the Newton-Raphson formula

$\displaystyle R_{j+1}=R_{j}-\frac{D(R_{j})}{D'(R_{j})},$ (15.5)

with the initial condition $ R_{0}=2\theta \mu/\mu^{(2)}$, where $ D'(z)=(1+\theta)\mu-M'_{X}(z)$.

Moreover, if it is possible to calculate the third raw moment $ \mu^{(3)}$, we can obtain a sharper bound than (15.4), Panjer and Willmot (1992):

$\displaystyle R<\frac{12\mu\theta}{3\mu^{(2)}+\sqrt{9(\mu^{(2)})^{2}+24\mu\mu^{(3)}\theta}},
$

and use it as the initial condition in (15.5).