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.
First we consider a reinterpretation of the model presented in Example 1. Now the discrete-time version of the model is assumed:
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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:
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Almost as simply as in the Example 1 we get the solutions:
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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.
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:
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Example 4
Let us assume that has a gamma
distribution, and the dividend is defined as:
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Solution.
Let us write the state of the process after periods in the form:
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According to the De Vylder method ruin probability is approximated by:
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In order to minimize the premium under restrictions:
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where
denotes the cdf of gamma distribution
with parameters
.
In respect of the relation
, and taking into account that:
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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.