20.5 Ruin Probability Criterion when the Initial Capital is Given

Presuming long-run horizon for premium calculation we turn back to ruin theory. Our aim is now to obtain such a level of premium for the portfolio yielding each year the aggregate loss $ W$, which results from a presumed level of ruin probability $ \psi$ and initial capital $ u$. This is done by inverting various approximate formulae for the probability of ruin. Information requirements of different methods are emphasized. Special attention is paid also to the problem of decomposition of the whole portfolio premium.


20.5.1 Approximation Based on Lundberg Inequality

This is a simplest (and crude) approximation method, simply assuming replacement of the true function $ \psi(u)$ by:

$\displaystyle \psi _{Li} \left( u \right) = e^{ - Ru}.$    

At first we obtain the approximation $ R_{(Li)} $ of the desired level of the adjustment coefficient $ R$:

$\displaystyle R_{(Li)} = \frac{ - \ln \psi }{u}.$    

In the next step we make use of the definition of the adjustment coefficient for the portfolio:

$\displaystyle \mathop{\textrm{E}}\left( {e^{RW}} \right) = e^{R\Pi \left( W \right)},$    

to obtain directly the premium formula:

$\displaystyle \Pi \left( W \right) = R^{ - 1}\ln \left\{ {\mathop{\textrm{E}}\left( {e^{RW}} \right)} \right\} = R^{ - 1}C_W \left( R \right),$    

where $ C_W $ denotes the cumulant generating function. The result is well known as the exponential premium formula. It possesses several desirable properties - not only that it is derivable from ruin theory. First of all, by the virtue of properties of the cumulant generating function, it is additive for independent risks. So there is no need to distinguish between marginal and basic premiums for individual risks. By the same reason the formula does not reflect the cross-sectional diversification effect when the portfolio is composed of large number of risks, each of them being small. The formula can be practically applied once we replace the adjustment coefficient $ R$ by its approximation $ R_{(Li)} $.

Under certain conditions we could rely on truncating higher order terms in the expansion of the cumulant generating function:

$\displaystyle \Pi \left( W \right) = \frac{1}{R}C_W \left( R \right) = \mu _W +...
...{1}{2!}R\sigma _W^2 + \frac{1}{3!}R^2\mu _{3,W} + \frac{1}{4!}R^3c_{4,W} + ...,$ (20.9)

and use for the purpose of individual risk pricing the formula (where higher order terms are truncated as well):

$\displaystyle \Pi \left( X \right) = \frac{1}{R}C_X \left( R \right) = \mu _X +...
...c{1}{2!}R\sigma _X^2 + \frac{1}{3!}R^2\mu _{3,X} + \frac{1}{4!}R^3c_{4,X} + ...$ (20.10)

Some insight into the nature of the long-run criteria for premium calculation could be gained by re-arrangement of the formula (20.9). At first we could express the initial capital in units of standard deviation of the aggregate loss: $ U = u\sigma _W^{ -
1} $. Now the adjustment coefficient could be expressed as:

$\displaystyle R = \frac{ - \ln \psi }{U\sigma _W },$    

and premium formula (20.9) as:

$\displaystyle \Pi \left( W \right) = \mu _W + \sigma _W \left\{ {\frac{1}{2!}\f...
...ac{1}{4!}\left( {\frac{ - \ln \psi }{U}} \right)^3\gamma _{2,W} + ...} \right\}$ (20.11)

where in the brackets appear only unit-less figures, that form together the pricing formula for the standardized risk $ \left( {W - \mu _W }
\right)\sigma _W^{ - 1} $. Let us notice that the contribution of higher order terms in the expansion is neglectible when initial capital is large enough. The above phenomenon could be interpreted as a result of risk diversification in time (as opposed to cross-sectional risk diversification). Provided the initial capital is large, the ruin (if it happens at all) will rather appear as a result of aggregation of poor results over many periods of time. However, given the skewness and kurtosis of one-year increment of the risk process, the sum of increments over $ n$ periods has skewness of order $ n^{-\frac{1}{2}}$, kurtosis of order $ n^{-1}$ etc. Hence the larger initial capital, the smaller importance of the difference between the distribution of the yearly increment and the normal distribution. In a way this is how the diversification of risk in time works (as opposed to cross-sectional diversification). In the case of a cross-sectional diversification the assumption of mutual independency of risks plays the crucial role. Analogously, diversification of risk in time works effectively when subsequent increments of the risk process are independent.


20.5.2 ``Zero'' Approximation

The ``zero'' approximation is a kind of naive approximation, assuming replacement of the function $ \psi(u)$ by:

$\displaystyle \psi _0 \left( u \right) = \left( {1 + \theta } \right)^{ - 1}\exp \left( { - Ru} \right),$    

where $ \theta$ denotes the relative security loading, which means that $ \left( {1 + \theta } \right) = \frac{\Pi \left( W \right)}{\mathop{\textrm{E}}(W)}$. The ``zero'' approximation is applicable to the case of Poisson claim arrivals (as opposed to Lundberg inequality, which is applicable under more general assumptions). Relying on ``zero'' approximation leads to the system of two equations:

$\displaystyle {\begin{array}{*{20}c} \Pi \left( W \right) = R^{ - 1}C_W \left( ...
... \frac{\mathop{\textrm{E}}(W)}{\psi \Pi \left( W \right)}.\hfill\\ \end{array}}$    

The system could be solved by assuming at first:

$\displaystyle R^{\left( 0 \right)} = \frac{ - \ln \psi }{u},$    

and next by executing iterations:

$\displaystyle {\begin{array}{*{20}c} \Pi ^{\left( n \right)}\left( W \right) = ...
...trm{E}}(W)}{\psi \Pi ^{\left( n \right)}\left( W \right)},\hfill\\ \end{array}}$    

that under reasonable circumstances converge quite quickly to the solution $ R_{(0)} = \mathop {\lim }\limits_{n \to \infty } R^{\left( n \right)}$, which allows applying formula (20.9) for the whole portfolio and formula (20.10) for individual risks, provided the coefficient $ R$ is replaced by its approximation $ R_{(0)} $.


20.5.3 Cramér-Lundberg Approximation

Premium calculation could also be based on the Cramér-Lundberg approximation. In this case the problem can be reduced also to the system of equations (three this time):

$\displaystyle \Pi\left(W\right)$ $\displaystyle =$ $\displaystyle R^{ - 1}C_W \left( R \right)$  
$\displaystyle R$ $\displaystyle =$ $\displaystyle \frac{1}{u}\left\{ { - \ln \psi + \ln \frac{\mu _Y \theta }{{M}'_Y \left(
R \right) - \mu _Y \left( {1 + \theta } \right)}} \right\}$  
$\displaystyle \left({1 + \theta } \right)$ $\displaystyle =$ $\displaystyle \frac{\Pi \left( W \right)}{\mathop{\textrm{E}}(W)}.$  

where $ {M}'_Y \left( \cdot \right)$ and $ \mu _Y $ denote respectively the first order derivative of the moment generating function and the expectation of the severity distribution. Solution of the system in respect of unknowns $ \Pi \left( W
\right)$, $ \theta$ and $ R$ requires now a bit more complex calculations. Obtained result $ R_{(CL)} $ could be used then to replace $ R$ in formulas (20.9) and (20.10). The method is applicable to the case of Poisson claim arrivals. Moreover, severity distribution has to be known in this case. It can be expected that the method will produce accurate results for large $ u$.


20.5.4 Beekman-Bowers Approximation

This method is often recommended as the one which produces relatively accurate approximations, especially for moderate amounts of initial capital. The problem consists in solving the system of three equations:

$\displaystyle \psi$ $\displaystyle =$ $\displaystyle \left( {1 + \theta } \right)^{ - 1}\left\{ 1 - G_{\alpha ,\beta }
\left( u \right) \right\}$  
$\displaystyle \frac{\alpha }{\beta }$ $\displaystyle =$ $\displaystyle \left( {1 + \theta } \right)\frac{m_{2,Y} }{2\theta
m_{1,Y} }$  
$\displaystyle \frac{\alpha \left( {\alpha + 1} \right)}{\beta ^2}$ $\displaystyle =$ $\displaystyle \left( {1 + \theta }
\right)\left\{ {\frac{m_{3,Y} }{3\theta m_{1,Y} } + 2\left( {\frac{m_{2,Y}
}{2\theta m_{1,Y} }} \right)^2} \right\},$  

where $ G_{\alpha,\beta}$ denotes the cdf of the gamma distribution with parameters $ (\alpha,\beta)$, and $ m_{k,Y}$ denotes the raw moment of order $ k$ of the severity distribution. Last two equations arise from equating moments of the gamma distribution to conditional moments of the maximal loss distribution (provided the maximal loss is positive). Solving this system of equation is a bit cumbersome, as it involves multiple numerical evaluations of the cdf of the gamma distribution. The admissible solution exists provided $ m_{3,Y}m_{1,Y}>m^2_{2,Y}$, that is always satisfied for arbitrary severity distribution with support on the positive part of the axis. Denoting the solution for the unknown $ \theta$ by $ \theta_{BB}$, we can write the latter as a function:

$\displaystyle \theta_{BB} = \theta_{BB}\left(u,\psi, m_{1,Y}, m_{2,Y}, m_{2,Y}\right),$    

and obtain the whole portfolio premium from the equation:

$\displaystyle \Pi_{BB} \left( W\right) = \left( {1 + \theta_{BB} } \right)\mathop{\textrm{E}}(W).$    

Formally, application of the method requires only moments of first three orders of the severity distribution to be finite. However, the problem arises when we wish to price individual risks. Then we have to know the moment generating function of the severity distribution, and it should obey conditions for adjustment coefficient to exist. If this is a case, we can replace the coefficient $ \theta$ of the equation:

$\displaystyle M_Y (r) = 1 + \left( {1 + \theta } \right)m_{1,Y}r$    

by its approximation $ \theta_{BB}$, and thus obtain the approximation $ R_{(BB)}$ of the adjustment coefficient $ R$. It allows calculating premiums according to formulas (20.9) and (20.10). It is easy to verify that there is no danger of contradiction, as both formulas for the premium $ \Pi_{BB}\left(W\right)$ produce the same result $ \left( {1 +
\theta_{BB} } \right)\mathop{\textrm{E}}(W)=R^{-1}_{(BB)}C_W(R_{(BB)})$.


20.5.5 Diffusion Approximation

This approximation method requires the scarcest information. It suffices to know the first two moments of the increment of the risk process, to invert the formula:

$\displaystyle \psi_D (u) = \exp\left( -R_{(D)}u \right),$    

where:

$\displaystyle R_{(D)} = 2\left\{ \Pi(W)-\mu_W \right\}\sigma^{-2}_W,$    

in order to obtain the premium:

$\displaystyle \Pi_D (W) = \mu_W + \frac{\sigma^2_W}{2}\frac{-\log\psi}{u},$    

that again is easily decomposable for individual risks. The formula is equivalent to the exponential formula (20.9), where all terms except the first two are omitted.


20.5.6 De Vylder Approximation

The method requires information on moments of the first three orders of the increment of the risk process. According to the method, ruin probability could be expressed as:

$\displaystyle \psi_{dV} (u) = \frac{1}{1+R_{(D)}\rho}\exp\left( - \frac{R_{(D)}u}{1+R_{(D)}\rho} \right),$    

where for simplicity the abbreviated notation $ \rho\stackrel{\mathrm{def}}{=}\frac{1}{3}\sigma_W\gamma_W$ is used. Setting $ \psi_{dV}\left(u\right)$ equal to $ \psi$ and rearranging the equation we obtain another form of it:

$\displaystyle \left\{ -\log\psi-\log\left( 1+R_{(D)}\rho \right)\right\}\left( 1+R_{(D)}\rho \right) = R_{(D)}u$    

that can be solved numerically in respect of $ R_{(D)}$, to yield as a result premium formula:

$\displaystyle \Pi_{dV} (W) = \mu_W + \frac{\sigma^2_W}{2}R_{(D)},$    

which again is directly decomposable.

When the analytic solution is needed, we can make some further simplifications. Namely, the equation entangling the unknown coefficient $ R_{(D)}$ could be transformed to a simplified form on the basis of the following approximation:

$\displaystyle \hspace*{-6cm}\left( 1+R_{(D)}\rho \right) \log\left( 1+R_{(D)}\rho \right) =$    

$\displaystyle = \left( 1+R_{(D)}\rho \right) \left\{ R_{(D)}\rho - \frac{1}{2}\...
...+ \frac{1}{3}\left( R_{(D)}\rho\right )^3 - \dots \right\} \approx R_{(D)}\rho.$    

Provided the error of omission of higher order terms is small, we obtain the approximation:

$\displaystyle R_{(D)} \approx \frac{-\log\psi}{u + \rho (\log\psi+1)}.$    

The error of the above solution is small, provided the initial capital $ u$ is several times greater than the product $ \rho\left\vert\log\psi+1\right\vert$. Under this condition we obtain the explicit (approximated) premium formula:

$\displaystyle \Pi_{dV^{\ast}} (W) = \mu_W + \frac{\sigma^2_W}{2} \left\{ \frac{-\log\psi}{u+\rho(\log\psi+1)} \right\},$    

where the star symbolizes the simplification made. Applying now the method of linear approximation of marginal cost $ \Pi_{dV^{\ast}}\left( W+X \right) - \Pi_{dV^{\ast}}\left(W\right)$ presented in Section 20.3 yields the result:

$\displaystyle \Pi_{dV^{\ast}} (X) = \mu_X + \frac{ -\log\psi\left\{u+2\rho(\log...
...i (\log\psi + 1)}{6 \left\{u + \rho\left(\log\psi+1\right)\right\}^2}\mu_{3,X}.$    

The reader can verify that the formula $ \Pi_{dV^{\ast}}(\cdot)$ is additive for independent risks, and so it can serve for marginal as well as for basic valuation.


20.5.7 Subexponential Approximation

This method applies to the classical model (Poisson claim arrivals) with thick-tailed severity distribution. More precisely, when the severity cdf $ F_Y$ possesses the finite expectation $ \mu _Y $, then the integrated tail distribution cdf $ F_{L_1}$ (interpreted as the cdf of the variable $ L_1$, being the ``ladder height'' of the claim surplus process) is defined as follows:

$\displaystyle 1-F_{L_1}(x) = \frac{1}{\mu_Y}\int_x^{\infty}\{1-F_Y(y)\}dy.$    

Assuming now that the latter distribution is subexponential (see Chapter 15), we could obtain (applying the Pollaczek-Khinchin formula) the approximation, which should work for large values of initial capital:

$\displaystyle \Pi_S(W) = \mu_W \left[1+\frac{1}{\psi}\left\{1-F_{L_1}(u)\right\}\right].$    

The extended study of consequences of thick-tailed severity distributions can be found in Embrechts, Klüppelberg, and Mikosch (1997).


20.5.8 Panjer Approximation

The Pollaczek-Khinchin formula could be also used in combination with the Panjer recursion algorithm, to produce quite accurate (at the cost of time-consuming calculations) answers in the case of the classical model (Poisson claim arrivals). The method consists of two basic steps. In the first step the integrated tail distribution $ F_{L_1}(x)$ is calculated and discretized. Once this step is executed, we have a distribution of a variable $ \tilde{L}_1$ (discretized version of the ``ladder height'' $ L_1$):

$\displaystyle f_j = \textrm{P}\left( \tilde{L}_1 = jh \right),\qquad j=0,1,2,\dots\vspace{-0.2cm}$    

The second step is based on the fact that the maximal loss $ L=L_1+\dots+L_N$ has a compound geometric distribution. Thus the distribution of the discretized version $ \tilde{L}$ of the variable $ L$ is obtained by making use of the Panjer recursion formula:

$\displaystyle \textrm{P}\left( \tilde{L} = 0 \right) = (1-q)\sum_{j=0}^{\infty}(qf_0)^j,\vspace{-0.2cm}$    

and for $ k=1,2,\dots$:

$\displaystyle \textrm{P}\left( \tilde{L} = kh \right) = \frac{q}{1-qf_0}\sum_{j=1}^k f_j\textrm{P}\left\{\tilde{L}=(k-j)h\right\},\vspace{-0.2cm}$    

where $ q\stackrel{\mathrm{def}}{=}(1+\theta)^{-1}$. Iterations should be stopped when for some $ k_{\psi}$ the cumulated probability $ F_{\tilde{L}}(k_{\psi}h)$ exceeds for the first time the predetermined value $ 1-\psi$. The approximated value of the capital $ u$ at which the ruin probability attains the value $ \psi$ could be set then on the basis of interpolation, taking into account that the ruin probability function is approximately exponential:

$\displaystyle u_{\psi} \stackrel{\mathrm{def}}{=} k_{\psi}h-h\frac{ \log\psi-\l...
...ilde{L}}(k_{\psi}h-h)\right\} - \log\left\{1-F_{\tilde{L}}(k_{\psi}h)\right\}}.$    

Calculations should be repeated for different values of $ \theta$ in order to find such value $ \theta_{Panjer}(\psi,u)$, at which the resulting capital $ u_{\psi}$ approaches the predetermined value of capital $ u$. Then the resulting premium is given by the formula:

$\displaystyle \Pi_{Panjer}(W) = (1+\theta_{Panjer})\mu_W.$    

It should be noted, that it is only the second step of calculations which has to be repeated many times under the search procedure, as the distribution of the variable $ \tilde{L}_1$ remains the same for various values of $ \theta$ being tested. The advantage of the method is that the range of the approximation error is under control, as it is a simple consequence of the width of the discretization interval $ h$ and the discretization method used. The disadvantage already mentioned is a time-consuming algorithm. Moreover, the method produces only numerical results, and therefore, provides no rule for decomposition of the whole portfolio premium for individual risk premiums. Nevertheless, the method could be used to obtain quite accurate approximations, and thus, a reference point to estimate approximation errors produced by simpler methods.

All approximation methods presented in this section are more or less standard, and more detailed information on them can be found in any actuarial textbook, as for example in ``Actuarial Mathematics'' by Bowers et al. (1986, 1997). More advanced analysis can be found in a book ``Ruin probabilities'' by Asmussen (2000) and numerical comparison of this and other approximations are given in Chapter 15.