15.6 Approximations of the Ruin Probability in Finite Time

In this section, we present $ 5$ different approximations. We illustrate them on a common claim size distribution example, namely the mixture of two exponentials claims with $ \beta _1=3.5900\cdot 10^{-10}$, $ \beta _2=7.5088\cdot 10^{-9}$ and $ a=0.0584$ (see Chapter 13). Their numerical comparison is given in Section 15.7.


15.6.1 Monte Carlo Method

The ruin probability in finite time can always be approximated by means of Monte Carlo simulations. Table 15.18 shows the output for mixture of two exponentials claims with $ \beta_1$, $ \beta_2$, $ a$ with respect to the initial capital $ u$ and the time horizon $ T$. The relative safety loading $ \theta$ is set to $ 30\%$. For the Monte Carlo method purposes we generated $ 50$ x $ 10000$ simulations. We see that the values approach those calculated in infinite case as $ T$ increases, cf. Table 15.4. We note that the Monte Carlo method will be used as a reference method when comparing different finite time approximations in Section 15.7.


Table 15.18: Monte Carlo results (50 x 10000 simulations) for mixture of two exponentials claims with $ \beta _1=3.5900\cdot 10^{-10}$, $ \beta _2=7.5088\cdot 10^{-9}$, $ a=0.0584$ and $ \theta =0.3$ (u in USD billion).
$ u$ 0 $ 1$ $ 5$ $ 10$ $ 20$ $ 50$
             
$ \psi(u,1)$ 0.672550 0.428150 0.188930 0.063938 0.006164 0.000002
$ \psi(u,2)$ 0.718254 0.501066 0.256266 0.105022 0.015388 0.000030
$ \psi(u,5)$ 0.753696 0.560426 0.323848 0.159034 0.035828 0.000230
$ \psi(u,10)$ 0.765412 0.580786 0.350084 0.184438 0.049828 0.000726
$ \psi(u,20)$ 0.769364 0.587826 0.359778 0.194262 0.056466 0.001244
             

26244 STFruin21.xpl


15.6.2 Segerdahl Normal Approximation

The following result due to Segerdahl (1955) is said to be a time-dependent version of the Cramér-Lundberg approximation given by (15.13). Under the assumption that $ c=1$, cf. relation (15.28), we have

$\displaystyle \psi_S(u,T)= C\exp(-Ru)\Phi\left (\frac{T-um_{L}}{\omega_{L}\sqrt{u}}\right ),$ (15.29)

where $ C=\theta \mu/\left\{M_{X}'(R)-\mu (1+\theta )\right\}$, $ m_{L}=1\left\{\lambda M_{X}'(R)-1\right\}^{-1}$ and $ \omega_{L}^{2}=\lambda
M''_{X}(R)m_{L}^{3}. $


Table 15.19: The Segerdahl approximation for mixture of two exponentials claims with $ \beta _1=3.5900\cdot 10^{-10}$, $ \beta _2=7.5088\cdot 10^{-9}$, $ a=0.0584$ and $ \theta =0.3$ (u in USD billion).
$ u$ 0 $ 1$ $ 5$ $ 10$ $ 20$ $ 50$
             
$ \psi(u,1)$ 0.663843 0.444333 0.172753 0.070517 0.013833 0.000141
$ \psi(u,2)$ 0.663843 0.554585 0.229282 0.092009 0.017651 0.000175
$ \psi(u,5)$ 0.663843 0.587255 0.338098 0.152503 0.030919 0.000311
$ \psi(u,10)$ 0.663843 0.587260 0.359593 0.192144 0.049495 0.000634
$ \psi(u,20)$ 0.663843 0.587260 0.359660 0.194858 0.057143 0.001254
             

26355 STFruin22.xpl

This method requires existence of the adjustment coefficient. This implies that only light-tailed distributions can be used. Numerical evidence shows that the Segerdahl approximation gives the best results for huge values of the initial capital $ u$, see Asmussen (2000).

In Table 15.19, the results of the Segerdahl approximation for mixture of two exponentials claims with $ \beta_1$, $ \beta_2$, $ a$ with respect to the initial capital $ u$ and the time horizon $ T$ are presented. The relative safety loading $ \theta=30\%$. We see that the approximation in the considered case yields quite accurate results for moderate $ u$, cf. Table 15.18.


15.6.3 Diffusion Approximation

The idea of the diffusion approximation is first to approximate the claim surplus process $ S_t$ by a Brownian motion with drift (arithmetic Brownian motion) by matching the first two moments, and next, to note that such an approximation implies that the first passage probabilities are close. The first passage probability serves as the ruin probability.

The diffusion approximation is given by:

$\displaystyle \psi_{D}(u,T)= IG\left(\frac{T\mu_{c}^{2}}{\sigma_{c}^{2}}; -1;\frac{u\vert\mu_{c}\vert}{\sigma_{c}^{2}}\right),$ (15.30)

where $ \mu_{c}=-\lambda\theta\mu$, $ \sigma_{c}^2=\lambda\mu^{(2)}$, and $ IG(\cdot;\zeta;u)$ denotes the distribution function of the passage time of the Brownian motion with unit variance and drift $ \zeta$ from the level 0 to the level $ u>0$ (often referred to as the inverse Gaussian distribution function), namely $ IG(x;\zeta;u)=1-\Phi\left(u/\sqrt{x}-
\zeta\sqrt{x}\right)+\exp\left (2\zeta u\right)$ $ \cdot\Phi\left(-u/\sqrt{x}-\zeta\sqrt{x}\right)$, see Asmussen (2000).


Table 15.20: The diffusion approximation for mixture of two exponentials claims with $ \beta _1=3.5900\cdot 10^{-10}$, $ \beta _2=7.5088\cdot 10^{-9}$, $ a=0.0584$ and $ \theta =0.3$ (u in USD billion).
$ u$ 0 $ 1$ $ 5$ $ 10$ $ 20$ $ 50$
             
$ \psi(u,1)$ 1.000000 0.770917 0.223423 0.028147 0.000059 0.000000
$ \psi(u,2)$ 1.000000 0.801611 0.304099 0.072061 0.001610 0.000000
$ \psi(u,5)$ 1.000000 0.823343 0.370177 0.128106 0.011629 0.000000
$ \psi(u,10)$ 1.000000 0.829877 0.391556 0.150708 0.020604 0.000017
$ \psi(u,20)$ 1.000000 0.831744 0.397816 0.157924 0.024603 0.000073
             

26477 STFruin23.xpl

We also note that in order to apply this approximation we need the existence of the second moment of the claim size distribution.

Table 15.20 shows the results of the diffusion approximation for mixture of two exponentials claims with $ \beta_1$, $ \beta_2$, $ a$ with respect to the initial capital $ u$ and the time horizon $ T$. The relative safety loading $ \theta$ equals $ 30\%$. The results lead to the conclusion that the approximation does not produce accurate results for such a choice of the claim size distribution. Only when $ u=5$ billion USD the results are acceptable, cf. the reference values in Table 15.18.


15.6.4 Corrected Diffusion Approximation

The idea presented above of the diffusion approximation ignores the presence of jumps in the risk process (the Brownian motion with drift is skip-free) and the value $ S_{\tau(u)}-u$ in the moment of ruin. The corrected diffusion approximation takes this and other deficits into consideration (Asmussen; 2000). Under the assumption that $ c=1$, cf. relation (15.28), we have

$\displaystyle \psi_{CD}(u,t) =IG\left(\frac{T\delta_{1}}{u^{2}}+\frac{\delta_{2}}{u}; -\frac{Ru}{2};1+\frac{\delta_{2}}{u}\right),$ (15.31)

where $ R$ is the adjustment coefficient, $ \delta_{1}=\lambda M_{X}''(\gamma_{0}),$ $ \delta_{2}=M_{X}'''(\gamma_{0})
/\left\{3M_{X}''(\gamma_{0})\right\}$, and $ \gamma_{0}$ satisfies the equation: $ \kappa'(\gamma_{0})=0$, where $ \kappa(s)=\lambda\left\{M_X (s)-1\right\}-s$.


Table 15.21: The corrected diffusion approximation for mixture of two exponentials claims with $ \beta _1=3.5900\cdot 10^{-10}$, $ \beta _2=7.5088\cdot 10^{-9}$, $ a=0.0584$ and $ \theta =0.3$ (u in USD billion).
$ u$ 0 $ 1$ $ 5$ $ 10$ $ 20$ $ 50$
             
$ \psi(u,1)$ 0.521465 0.426840 0.187718 0.065264 0.007525 0.000010
$ \psi(u,2)$ 0.587784 0.499238 0.254253 0.104967 0.016173 0.000039
$ \psi(u,5)$ 0.638306 0.557463 0.321230 0.157827 0.035499 0.000251
$ \psi(u,10)$ 0.655251 0.577547 0.347505 0.182727 0.049056 0.000724
$ \psi(u,20)$ 0.660958 0.584386 0.356922 0.192446 0.055610 0.001243
             

26615 STFruin24.xpl

Similarly as in the Segerdahl approximation, the method requires existence of the moment generating function, so we can use it only for light-tailed distributions.

In Table 15.21 the results of the corrected diffusion approximation for mixture of two exponentials claims with $ \beta_1$, $ \beta_2$, $ a$ with respect to the initial capital $ u$ and the time horizon $ T$ are given. The relative safety loading $ \theta$ is set to $ 30\%$. It turns out that corrected diffusion method gives surprisingly good results and is vastly superior to the ordinary diffusion approximation, cf. the reference values in Table 15.18.


15.6.5 Finite Time De Vylder Approximation

Let us recall the idea of the De Vylder approximation in infinite time: we replace the claim surplus process with the one with $ \theta=\bar{\theta}$, $ \lambda=\bar{\lambda}$ and exponential claims with parameter $ \bar{\beta}$, fitting first three moments, see Section 15.3.6. Here, the idea is the same. First, we compute

$\displaystyle \bar{\beta}=\frac{3\mu^{(2)}}{\mu^{(3)}}, \qquad\bar{\lambda}=\fr...
...qquad {\rm and} \qquad\bar{\theta}=\frac{2\mu\mu^{(3)}}{3\mu^{(2)^{2}}}\theta.
$

Next, we employ relations (15.27) and (15.28) and finally use the exact, exponential case formula presented in Section 15.5.1.


Table 15.22: The finite time De Vylder approximation for mixture of two exponentials claims with $ \beta _1=3.5900\cdot 10^{-10}$, $ \beta _2=7.5088\cdot 10^{-9}$, $ a=0.0584$ and $ \theta =0.3$ ($ u$ in USD billion).
$ u$ 0 $ 1$ $ 5$ $ 10$ $ 20$ $ 50$
             
$ \psi(u,1)$ 0.528431 0.433119 0.189379 0.063412 0.006114 0.000003
$ \psi(u,2)$ 0.594915 0.505300 0.256745 0.104811 0.015180 0.000021
$ \psi(u,5)$ 0.645282 0.563302 0.323909 0.158525 0.035142 0.000215
$ \psi(u,10)$ 0.662159 0.583353 0.350278 0.183669 0.048960 0.000690
$ \psi(u,20)$ 0.667863 0.590214 0.359799 0.193528 0.055637 0.001218
             

26746 STFruin25.xpl

Obviously, the method gives the exact result in the exponential case. For other claim distributions, the first three moments have to exist in order to apply the approximation.

Table 15.22 shows the results of the finite time De Vylder approximation for mixture of two exponentials claims with $ \beta_1$, $ \beta_2$, $ a$ with respect to the initial capital $ u$ and the time horizon $ T$. The relative safety loading $ \theta=30\%$. We see that the approximation gives even better results than the corrected diffusion one, cf. the reference values presented in Table 15.18.


15.6.6 Summary of the Approximations

Table 15.23 shows which approximation can be used for each claim size distribution. Moreover, the necessary assumptions on the distribution parameters are presented.


Table 15.23: Survey of approximations with an indication when they can be applied
$ \qquad$Distrib. Exp. Gamma Wei- Mix. Log- Pareto Burr
Method bull Exp. normal
Monte Carlo + + + + + + +
Segerdahl + + - + - - -
Diffusion + + + + + $ \alpha>2$ $ \alpha\tau>2$
Corr. diff. + + - + - - -
Fin. De Vylder + + + + + $ \alpha>3$ $ \alpha\tau>3$