# 20.6 Ruin Probability Criterion and the Rate of Return

This section is devoted to considering the problem of balancing profitability and solvency requirements. In Section 20.2 a similar problem has already been studied. However, we have considered there return on capital on the single-period basis. Therefore neither the allocation of returns (losses) nor the long run consequences of decision rules applied in this respect were considered. The problem was already illustrated in Example 1. Section 20.6.1 is devoted to presenting the same problem under more general assumptions about the risk process, making use of some of approximations presented in Section 20.5. Section 20.6.2 is devoted to another generalization, where more flexible dividend policy allows for sharing risk between the company and shareholders.

## 20.6.1 Fixed Dividends

First we consider a reinterpretation of the model presented in Example 1. Now the discrete-time version of the model is assumed:

where all events are assumed to be observed once a year, and notations are obviously adapted. The question is the same: to choose the optimal level of initial capital that minimizes the premium given the ruin probability and the dividend rate . The solution depends on how much information we have on the distribution of the variable , and how precise result is required. Provided our information is restricted to the expectation and variance of , we can use the diffusion approximation. This produces exactly the same results as in Example 1, although now we interpret them as an approximated solution. Let us remind that the resulting premium formula reads:

with the accompanying result for the optimal level of capital:

Despite the fact that the premium formula is not additive, we can follow arguments presented in Section 20.3.4, to propose the individual basic premium formula:

and obviously the marginal premium containing loading twice as small as the basic one.

The basic idea presented above can be generalized to cases when richer information on the distribution of the variable allows for more sophisticated methods. For illustrative purposes only the method of De Vylder (in a simplified version) is considered.

Example 3

Our information encompasses also skewness (which is positive), so premium is calculated on the basis of the De Vylder approximation. Allowing for simplification proposed in the previous section, we obtain the minimized function:

Almost as simply as in the Example 1 we get the solutions:

where again the safety loading amounts to . However, in this case the safety loading is smaller than a half of the total premium loading. This time the capital (and so the dividend loading) is larger, because of component proportional to . This complicates also individual risks pricing, as (analogously to formulas considered in Section 20.3.3), the basic premium in respect of this component has to be set arbitrarily.

Comparing problems presented above with those considered in Section 20.5 we can conclude that premium calculation based on ruin theory are easily decomposable as far as the capital backing risk is considered as fixed. Once the cost of capital is explicitly taken into account, we obtain premium calculation formulas much more similar to those derived on the basis of one-year considerations, what leads to similar obstacles when the decomposition problem is considered.

## 20.6.2 Flexible Dividends

So far we have assumed that shareholders are paid a fixed dividend irrespective of the current performance of the company. It is not necessarily the case, as shareholders would accept some share in risk provided they will get a suitable risk premium in exchange. The more general model which encompasses the previous examples as well as the case of risk sharing can be formulated as follows:

where is a dividend due to the year , defined as such a function of the variable that . As dividend is a function of the current year result, it preserves independency of increments of the risk process. Of course, only such definitions of are sensible, which reduce in effect the range of fluctuations of the risk process. The example presented below assumes one of possible (and sensible) choices in this respect.

Example 4

Let us assume that has a gamma distribution, and the dividend is defined as:

which means that shareholders' share in profits amounts to , but they do not participate in losses. Problem is to choose a value of the parameter and amount of capital so as to minimize premium , under the restriction , and given parameters . The problem could be reformulated so as to solve it numerically, making use of the De Vylder approximation.

Solution.

Let us write the state of the process after periods in the form:

with the increment equal . The variable could be then defined as:

According to the De Vylder method ruin probability is approximated by:

where , and ; for simplicity, the number of year has been omitted.

In order to minimize the premium under restrictions:

and under predetermined values of it suffices to express the expectation of a variable and cumulants of order 1, 2, and 3 of the variable as functions of these parameters and variables. First we derive raw moments of order 1, 2, and 3 of the variable . From its definition we obtain:

that (after some calculations) leads to the following results:

where denotes the cdf of gamma distribution with parameters .

In respect of the relation , and taking into account that:

we easily obtain raw moments of the variable :

so as cumulants of this variable, too. Provided we are able to evaluate numerically the cdf of the gamma distribution, all elements needed to construct the numerical procedure solving the problem are completed.

In Example 4 some specific rule of sharing risk by shareholders and the company has been applied. On the contrary, the assumption on the distribution of the variable is of some general advantage, as the shifted gamma distribution is often used to approximate the distribution of the aggregate loss. We will make use of it in Example 6 presented in the next section.