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 , the Poisson process
with intensity (rate)
describes the number of claims in
interval and claim severities are random, given by i.i.d. non-negative sequence
with mean value
and variance
, independent of
. The insurance company receives a premium at a constant rate
per unit time, where
and
is called the relative safety loading. The classical risk process
is given by
We define a claim surplus process
as
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We note that the above definition implies that the relative safety loading has to be positive, otherwise
would be less than
and thus with probability
the risk business would become negative in infinite time. The ruin probability in finite time
is
given by
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(15.2) |
From a practical point of view, , where
is related to the planning horizon of the company, may perhaps sometimes be regarded as more
interesting than
. 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
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
driven by a Cox process
with the intensity process
, namely
Define now
and
. Then the point process
is a standard Poisson process with intensity 1, and therefore,
. The time scale defined by
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.
We distinguish here between light- and heavy-tailed distributions. A distribution is said to be light-tailed, if there exist constants
,
such that
or, equivalently, if there exist
, such that
, where
is the moment generating function, see Chapter 13.
Distribution
is said to be heavy-tailed, if for all
,
:
or, equivalently, if
We study here claim size distributions as in Table 15.1.
Light-tailed distributions | ||
Name | Parameters | |
Exponential | ![]() |
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Gamma | ![]() ![]() |
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Weibull | ![]() ![]() |
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Mixed exp's |
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Heavy-tailed distributions | ||
Name | Parameters | |
Weibull | ![]() ![]() |
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Log-normal |
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Pareto | ![]() ![]() |
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Burr | ![]() ![]() ![]() |
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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
and let
be a positive solution of the equation:
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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 satisfies the inequality:
Moreover, if it is possible to calculate the third raw moment , we can obtain a sharper bound than (15.4), Panjer and Willmot (1992):