15.5 Exact Ruin Probabilities in Finite Time

We are now interested in the probability that the insurer's capital as defined by (15.1) remains non-negative for a finite period $ T$ rather than permanently. We assume that the number of claims process $ N_t$ is a Poisson process with rate $ \lambda $, and consequently, the aggregate loss process is a compound Poisson process. Premiums are payable at rate $ c$ per unit time. We recall that the intensity of the process $ N_t$ is irrelevant in the infinite time case provided that it is compensated by the premium, see discussion at the end of Section 15.1.

In contrast to the infinite time case there is no general formula for the ruin probability like the Pollaczek-Khinchin one given by (15.8). In the literature one can only find a partial integro-differential equation which satisfies the probability of non-ruin, see Panjer and Willmot (1992). An explicit result is merely known for the exponential claims, and even in this case a numerical integration is needed (Asmussen; 2000).


15.5.1 Exponential Claim Amounts

First, in order to simplify the formulae, let us assume that claims have the exponential distribution with $ \beta =1$ and the amount of premium is $ c=1$. Then

$\displaystyle \psi(u,T)=\lambda \exp\left\{-(1-\lambda)u\right\}-\frac{1}{\pi}\int_{0}^{\pi}\frac{f_{1}(x) f_{2}(x)}{f_{3}(x)}dx,$ (15.26)

where $ f_{1}(x)=\lambda \exp\left\{2\sqrt{\lambda}T\cos x-(1+\lambda)T+u\left (\sqrt{\lambda}\cos x-1 \right
)\right\},$ $ f_{2}(x)=\cos\left (u\sqrt{\lambda}\sin x\right )-\cos\left (u\sqrt{\lambda}\sin x+2x\right),$ and $ f_{3}(x)=1+\lambda-2\sqrt{\lambda}\cos x.$

Now, notice that the case $ \beta\neq 1$ is easily reduced to $ \beta =1$, using the formula:

$\displaystyle \psi_{\lambda,\beta}(u,T)=\psi_{\frac{\lambda}{\beta},1}(\beta u,\beta T).$ (15.27)

Moreover, the assumption $ c=1$ is not restrictive since we have

$\displaystyle \psi_{\lambda,c}(u,T)=\psi_{\lambda/c,1}(u,cT).$ (15.28)

Table 15.17 shows the exact values of the ruin probability for exponential claims with $ \beta =6.3789\cdot 10^{-9}$ (see Chapter 13) with respect to the initial capital $ u$ and the time horizon $ T$. The relative safety loading $ \theta$ equals $ 30\%$. We see that the values converge to those calculated in infinite case as $ T$ is getting larger, cf. Table 15.2. The speed of convergence decreases as the initial capital $ u$ grows.


Table 15.17: The ruin probability for exponential claims with $ \beta =6.3789\cdot 10^{-9}$ and $ \theta =0.3$ (u in USD billion).
$ u$ 0 $ 1$ $ 2$ $ 3$ $ 4$ $ 5$
             
$ \psi(u,1)$ 0.757164 0.147954 0.025005 0.003605 0.000443 0.000047
$ \psi(u,2)$ 0.766264 0.168728 0.035478 0.007012 0.001288 0.000218
$ \psi(u,5)$ 0.769098 0.176127 0.040220 0.009138 0.002060 0.000459
$ \psi(u,10)$ 0.769229 0.176497 0.040495 0.009290 0.002131 0.000489
$ \psi(u,20)$ 0.769231 0.176503 0.040499 0.009293 0.002132 0.000489
             

26120 STFruin20.xpl