# 20.7 Ruin Probability, Rate of Return and Reinsurance

In this section premium calculation is considered under predetermined ruin probability and predetermined rate of dividend, with reinsurance included. At first the example involving fixed dividend is presented.

## 20.7.1 Fixed Dividends

Example 5

We assume (as in Example 2), that the aggregate loss has a compound Poisson distribution with expected number of claims , and with severity distribution being truncated-Pareto distribution with parameters . We assume also that the excess of each loss over the limit is ceded to the reinsurer using the same pricing formula:

The problem lies in choosing such a value of the retention limit and initial capital , which minimize the total premium paid by policyholders, under predetermined values of parameters . The problem could be solved with application of the De Vylder and Beekman-Bowers approximation methods. As allowing for reinsurance leads to numerical solutions anyway, there is no more reason to apply the simplified version of the De Vylder method, as in Example 3.

Solution.

Risk process can be written now as:

The problem takes a form of minimization of the premium under restrictions, which in the case of De Vylder metod take a form:

and in the version based on the Beekman-Bowers approximation method take a form:

Moments of the first three orders of the variable as well as cumulants of variables and are calculated the same way as in Example 2. All these characteristics are functions of parameters and the decision variable .

## 20.7.2 Interpretation of Solutions Obtained in Example 5

 Variants of minimization problems Method of approx. of the ruin probability Retention limit Initial capital Loading V.1: (basic) BB 184.2 416.6 4.17% dV 185.2 416.3 4.16% V.2: BB 179.5 408.2 4.25% dV 180.5 407.9 4.25% V.3: BB 150.1 463.3 4.65% dV 156.3 461.7 4.63% V.4: BB 126.1 406.2 4.13% dV 127.1 406.0 4.13% V.5: BB 139.7 409.0 4.13% dV 140.5 408.8 4.13% V.6: (no reinsurance) BB 500.0 442.9 4.25% dV 500.0 442.7 4.25%

Results of numerical optimization are reported in Table 20.2. In the basic variant of the problem, parameters has been set on the level . In variant 6 the value is assumed, so as this variant represents the lack of reinsurance. Variants 2, 3, 4 and 5 differ from the basic wariant by the value of one of parameters . In variant 2 the dividend rate has been increased so as to obtain the same level of premium, than it is obtained in variant 6. Results could be summarized as follows:

(i)
Reinsurance results either in premium reduction under unchanged rate of dividend (compare variant 6 with wariant 1), or in increase of the rate of dividend under the same premium level (compare variant 2 with variant 1). In both cases the need for capital is also reduced. If we wish to obtain reduction of premium as a result of reinsurance introduced, then the reduction of capital is slightly smaller than in the case when reinsurance serves to enlarge the rate of dividend.
(ii)
Comparison of variants 3 and 1 shows that increasing safety (reduction of parameter from 5% to 2.5%) results in significant growth of the premium. This effect is caused as well by increase of capital (which burdens the premium through larger cost of dividends), as by increase of costs of reinsurance, because of reduced retention limit. It is also worthwhile to notice that predetermining results in significant diversification of results obtained by two methods of approximation. In the case when the difference is neglectible.
(iii)
Results obtained in variants 4 and 5 show that the optimal level of reinsurance is quite sensitive to changes of parameters reflecting costs of reinsurance.

## 20.7.3 Flexible Dividends

In the next example assumptions are almost the same as in Example 5, except that the fixed dividend is replaced by the dividend dependent on financial result by the same manner, as in Example 4.

Example 6

Assumptions on the aggregate loss are the same as in Example 5: compound Poisson truncated-Pareto distribution with parameters . Assumptions concerning available reinsurance (excess of loss over , pricing formulas characterized by parameters and ) are also the same. Dividend is defined as in Example 4, with a suitable correction due to reinsurance allowed:

Now the problem lies in choosing capital , risk-sharing parameter and retention limit so as to minimize premium under the restriction , and predetermined values of parameters characterizing the distribution , parameters characterizing reinsurance costs and parameters characterizing profitability and safety .

Solution.

Under the predetermined values of decision variables and remaining parameters the risk process has a form:

with increment , where the variable is defined as:

The problem differs from that presented in Example 4 by two factors: variable is not gamma distributed, and the premium is now replaced by the constant . However, variable could be approximated by the shifted gamma distribution with parameters chosen so as to match moments of order 1, 2, and 3 of the original variable . Suitable calculations lead to the definition of the variable , that approximates the original variable :

where the variable has a gamma distribution, and the constant equals . Thus we could express moments of the variable as functions of parameters exactly this way, as it is done with respect to variable and parameters in Example 4. It suffices in turn to approximate ruin probability with the De Vylder method:

where and , and where the expected value of dividend satisfies the restriction:

Hence it is clear that the problem of minimization of premium under restrictions , , , , and predetermined values of parameters is in essence analogous to the problem presented in Example 4, and differs only in details. The set of decision variables in Example 4 is now extended by additional variable , and the variable is only approximately gamma distributed.

## 20.7.4 Interpretation of Solutions Obtained in Example 6

Results are presented in Table 20.3. In all variants predetermined values of parameters are the same. In variant 1 (basic) the ruin probability is assumed, and reinsurance is allowed. Variant 2 differs from the basic one by higher safety standard ( , whereas variant 3 differs by lack of reinsurance. In each variant three slightly different versions of the problem have been solved. Version A is a simplified one, assuming fixed dividend rate , so that . Consequently the minimization of the premium is conducted with respect to only. In fact, the results from Table 20.2 are quoted for this version. Versions B and C assume minimization with respect to . Version B plays a role of a basic version, where premium is minimized under the expected rate of dividend . In version C such a rate of dividend has been chosen, that leads (through minimization) to the same premium level, as obtained previously in version A. So two alternative effects of the consent of shareholders to participate in risk could be observed. Effect in terms of reduction of premium (expected rate of dividend remaining unchanged) is observed when we compare version B and A. Effect in terms of increase of the expected rate of dividend (premium being fixed) is observed when versions C and A are compared. Results could be summarized as follows.

In each of three variants, the consent of shareholders for risk participation allows for substantial reduction of premium (loading reduced by about 20%).

It is interesting that shareholder's consent to participate in risk allows for much more radical reduction of premium than reinsurance. It results from the fact that reinsurance costs have been explicitly involved in optimization, whereas the costs of the shareholder's consent to participate in risk'' have not been accounted for.

 Variant of assumptions Version of problem M V.1: , reins. A 5% 185.2 416.3 4.16% - 0 B 5% 189.3 406.0 3.35% 41.7% 5.02% C 8.54% 143.6 305.2 4.16% 48.6% 8.14% V.2: , reins. A 5% 156.3 461.7 4.63% - 0 B 5% 157.0 447.5 3.67% 44.4% 4.94% C 8.96% 122.9 329.7 4.63% 52.3% 8.29% V.3: , no reins. A 5% 500.0 442.7 4.25% - 0 B 5% 500.0 429.8 3.45% 42.0% 4.70% C 8.09% 500.0 340.0 4.25% 48.2% 7.15%

Comparison of versions C with versions A in each variant of the problem allows us to see the outcome (increase of expected rate of dividend) of the shareholder's consent to share risk. In the last column of the table the (relative) standard deviation of dividends is reported; it could serve as a measure of cost'' at which the outcome, in terms of the increment of the expected dividend rate, is obtained.

Comparing versions B and C in variants 1 and 2 we could observe effects of the increment in the expected rate of dividend. Apart from the obvious effect on premium increase, also the reduction of capital could be observed (cost of capital is higher), and at the same time retention limits are reduced. Also the sharing parameter increases, as well as the (relative) standard deviation of dividends .

Comparing variants 1 and 2 (in all versions A, B, and C) we notice the substantial increase of the premium as an effect of higher safety standard (smaller . Also the amount of capital needed increases and the retention limit is reduced. At the same time a slight increase of sharing parameter is observed (versions B and C).