20.4 Rate of Return and Reinsurance Under the Short Term Criterion

This section is devoted to extending the decision problem considered in previous sections by allowing for reinsurance. Then the pricing obey the form:

$\displaystyle \Pi \left( W \right) = \Pi _I \left( {W_I } \right) + \Pi _R \left( {W_R } \right),$    

where the whole aggregate loss $ W$ is subdivided into the share of the insurer $ W_I
$ and that of reinsurer $ W_R $. $ \Pi _I \left( \cdot \right)$ denotes the premium formula applied by the insurer to price his share, set in accordance with $ RBC$ concept. $ \Pi _R \left( \cdot \right)$ symbolizes the pricing formula used by the reinsurer. Provided formula $ \Pi _R \left( \cdot \right)$ is accurate enough to reflect the existing offer of the reinsurance market, we could compare various variants of subdivision of the variable $ W$ into components $ W_I
$ and $ W_R $, looking for such subdivision which optimizes some objective function.


20.4.1 General Considerations

No matter which particular objective function is chosen, the space of possible subdivisions of the variable $ W$ has to be reduced somehow. One of the most important cases is when the variable $ W$ has a compound Poisson distribution, and the excess of loss reinsurance is chosen. Denoting by $ N$ the number of claims, we could define for each claim amount $ Y_i $, $ i =
1,2,...N$ its subdivision into the truncated loss $ \underline{Y}_{M,i} \stackrel{\mathrm{def}}{=} \min \left\{ {Y_i ,M} \right\}$ and the excess of loss $ \overline{Y}_{M,1} \stackrel{\mathrm{def}}{=} \max \left\{ {Y_i - M,0} \right\}$ and then define variables representing subdivision of the whole portfolio:

$\displaystyle W_I = \underline{Y}_{M,1} + ... + \underline{Y}_{M,N} ,$    

$\displaystyle W_R = \overline{Y}_{M,1} + ... + \overline{Y}_{M,N} ,$    

both having compound Poisson distributions too, with characteristics being functions of the subdivision parameter $ M$.

Assuming that capital of the insurer is not flexible, and that the current amount $ u$ of capital is smaller than the amount $ RBC\left( W \right)$ necessary to accept solely the whole portfolio, we could simply find such value of $ M$, for which $ RBC\left( {W_I } \right) = u$. In the case when the current amount of capital is in excess, it is still relevant to assess such portion of the capital, which should serve as a protection for insurance operations. The excess of capital over this amount can be treated separately, as being free of prudence requirements when investment decisions are undertaken.

It is more interesting to assume that the amount of capital is flexible, and to choose the retention limit $ M$ to minimize the total premium $ \Pi \left( W
\right)$ given parameters $ r^\ast $, $ s$, and $ \varepsilon$. The objective function reflects the aim of maximizing competitiveness of the company. If the resulting premium (after being charged by respective cost loadings) is lower than that acceptable by the market, we can revise assumptions. Revised problem could consist in maximizing expected rate of return given the premium level and parameters $ \eta$ and $ \varepsilon$. This would mean getting higher risk premium than that offered by the capital market. Reasonable solutions could be expected in the case when reinsurance premium formula $ \Pi _R \left( \cdot \right)$ contains loadings proportional primarily to the expected value, and its sensitivity to the variance (more so as to skewness and kurtosis) is small. This could be expected as a result of transaction costs on the one hand, and larger capital assets of reinsurers on the other. Also the possibility to diversify risk on the world-wide scale work in the same direction, increasing transaction costs and at the same time reducing the reinsurer's exposure to risk.


20.4.2 Illustrative Example

Example 2

Aggregate loss $ W$ has a compound Poisson distribution with truncated-Pareto severity distribution, with cdf given for $ y \geqslant 0$ by the formula:

$\displaystyle F_Y \left( y \right) = \left\{ {{\begin{array}{*{20}c} {1 - \left...
... 1 \hfill & {{\rm when}\quad y \geqslant M_0 } \hfill \\ \end{array} }} \right.$    

Variable $ W$ is subdivided into retained part $ \underline{W}_M $ and ceded part $ \overline{W}_M $, that given the subdivision parameter $ M \in \left( {0,M_0 }
\right]$ have a form:

$\displaystyle \underline{W}_M = \underline{Y}_{M,1} + ... + \underline{Y}_{M,N} ,$    

$\displaystyle \overline{W}_M = \overline{Y}_{M,1} + ... + \overline{Y}_{M,N} .$    

We assume that reinsurance pricing rule can be reflected by the formula:

$\displaystyle \Pi _R \left( {\overline{W}_M } \right) = \left( {1 + re_0 } \rig...
... {\overline{W}_M } \right) + re_1 \textrm{Var}\left( {\overline{W}_M } \right),$    

and that insurer's own pricing formula is:

$\displaystyle \Pi _I \left( {\underline{W}_M } \right) = \mathop{\textrm{E}}\le...
...lon } \right) - \mathop{\textrm{E}}\left( {\underline{W}_M } \right)} \right\},$    

with a respective approximation of the quantile of the variable $ \underline{W}_M $.

For expository purposes we take the following values of parameters:

(i)
Parameters of the Pareto distribution $ \left( {\alpha ,\lambda } \right) =
\left( {\tfrac{5}{2},\;\tfrac{3}{2}} \right)$, with truncation point $ M_0 =
500$;
(ii)
Expected value of the number of claims $ \mathop{\textrm{E}}(N) = \lambda_P =
1000$;
(iii)
Substitution rate $ s = 2$;
(iv)
Remaining parameters (in the basic variant of the problem) $ \varepsilon =
2\% $, $ r^\ast = 10\% $, $ re_0 = 100\% $, $ re_1 = 0.5\% $.

Problem consists in choosing the retention limit $ M \in \left( {0,M_0 }
\right]$ that minimize the total premium $ \Pi \left( W \right) = \Pi _I
\left( {\underline{W}_M } \right) + \Pi _R \left( {\overline{W}_M } \right)$.

Solution.

First step is to express moments of first four orders of variables $ \underline{Y}_M $ and $ \overline{Y}_M $ as functions of parameters $ \left(
{\alpha ,\lambda ,M_0 } \right)$ and the real variable $ M$. Expected value of the truncated-Pareto variable with parameters $ \left( {\alpha ,\lambda ,M} \right)$ equals by definition:

$\displaystyle \int\limits_0^M {y\frac{\alpha \lambda ^\alpha }{\left( {\lambda ...
...{x^{\alpha + 1}}dx + } M \left( {1 + \frac{M}{\lambda }} \right)^{ - \alpha } =$    

$\displaystyle = \alpha \lambda\int\limits_1^{1 + \frac{M}{\lambda }} {\left( {x...
... \alpha - 1}} \right)dx + } M\left({1 + \frac{M}{\lambda }}\right)^{ - \alpha }$    

that, after integration and reordering of components produces the following formula:

$\displaystyle m_1 = \frac{\lambda }{\alpha - 1}\left\{ {1 - \left( {1 + \frac{M}{\lambda }} \right)^{1 - \alpha }} \right\}.$    

Similar calculations made for moments of higher order yield the recursive equation:

$\displaystyle m_{k,\alpha } = \frac{\lambda }{\alpha - 1}\left\{ {\alpha m_{k -...
...a } - M^{k - 1}\left( {1 +
\frac{M}{\lambda }} \right)^{1 - \alpha }} \right\},$      
$\displaystyle \quad
k = 2,3,...$      

where the symbol $ m_{{\rm K},{\rm A}} $ means for $ {\rm A} > 0$ just the moment of order $ {\rm K}$ of the truncated-Pareto variable with parameters $ \left( {{\rm A},\lambda ,M} \right)$. No matter whether A is positive or not, in order to start the recursion we take:

$\displaystyle m_{1,{\rm A}} = \left\{ {{\begin{array}{*{20}c} {\frac{\lambda }{...
...c{M}{\lambda }} \right)} \hfill & {{\rm when\quad A} = 1} \end{array} }}\right.$    

The above formulas could serve to calculate raw moments as well of the variable $ \underline{Y}_M $ as the variable $ Y$, provided we replace $ M$ by $ M_0
$. Having calculated moments for both variables $ \underline{Y}_M $ and $ Y$ already, we make use of the relation:

$\displaystyle \mathop{\textrm{E}}({Y^k}) = \sum\limits_{j = 0}^k {\left( {{\beg...
... \mathop{\textrm{E}}\left( {\underline{Y}_M^{k - j} \overline{Y}_M^j } \right),$ (20.8)

to calculate moments of the variable $ \overline Y _M $. In the above formula we read $ \underline{Y}_M^0 $ and $ \overline Y _M^0 $ as equal one with probability one. Mixed moments appearing on the RHS of formula (20.8) can be calculated easily as positive values of the variable $ \overline Y _M $ happen only when $ \underline{Y}_M = M$. So mixed moments equal simply:

$\displaystyle \mathop{\textrm{E}}\left( {\underline{Y}_M ^m\overline Y _M ^n} \right) = M^mE\left(
{\overline Y _M ^n} \right)$

for arbitrary $ m, n > 0$.

The second step is to express cumulants of both variables $ \underline{W}_M $ and $ \overline{W}_M $ as a product of the parameter $ \lambda_P$ and respective raw moments of variables $ \underline{Y}_M $ and $ \overline Y _M $. Finally, both components $ \Pi _I \left( {\underline{W}_M } \right)$ and $ \Pi _R
\left( {\overline{W}_M } \right)$ of the total premium are expressed as a function of parameters $ \left(
{\lambda_P ,\alpha ,\lambda ,M_0 ,\varepsilon ,r^\ast ,s,re_0 ,re_1 } \right)$ and the decision parameter $ M \in \left( {0,M_0 }
\right]$. Now the search of such a value of $ M$ that minimizes the total premium $ \Pi \left( W
\right)$ is a quite feasible numerical task. Optimal retention level and related minimal premium entail optimal amount of capital $ u_{opt} = \left( {r^\ast } \right)^{
- 1}\left\{ {\Pi \left( {W_I } \right) - \mathop{\textrm{E}}({W_I})}
\right\}$.


20.4.3 Interpretation of Numerical Calculations in Example 2

The problem described in Example 2 has been solved in several different variants of assumptions on parameters. Variants 1-5 consist in minimization of the total premium, in variant 1 the parameters are $ \left( {s, \varepsilon, r^\ast, re_0, re_1}\right)$ $ =$ $ \left({2, 2\%, 10\%, 100\%, 0.5\%}\right)$. In variants 2, 3, 4 and 5 value of one of parameters $ \left( {\varepsilon
,r^\ast ,re_0 ,re_1 } \right)$ is modified and in variant 6 there is no reinsurance, $ \left( {s,\varepsilon ,r^\ast } \right)$ are as in variant 1. Variant 7 consists in maximization of $ r^\ast $ where $ \left( {\varepsilon ,\eta ,re_0 ,re_1 } \right)$ are as in variant 1, and premium loading equals $ 4.47{\%}$. Results are presented in Table 20.1.


Table 20.1: Optimal choice of retention limit $ M$. Basic characteristics of the variable $ W$: $ {\mathop {\textrm {E}}(W)=999.8}$, $ { \sigma \left ( W \right )=74.2}$, $ {\gamma \left ( W \right )=0.779}$, $ {\gamma _2 \left ( W \right )=2.654}$
Optimization variants Quantile approx. method for $ \underline{W}_M $ Retention Limit $ M$ RBC Loading $ \left( {\frac{\Pi \left( W \right)}{\mathop{\textrm{E}}(W)} - 1} \right)$
V.1:

(basic)

FC20.6 114.5 386.6 4.11%
FC20.5 106.5 385.2 4.13%
V.2:

$ r^\ast = 8\% $

FC20.6 114.5 483.3 4.11%
FC20.5 106.5 481.5 4.13%
V.3:

$ \varepsilon = 4\% $

FC20.6 129.7 382.3 4.03%
FC20.5 134.7 382.1 4.01%
V.4:

$ re_0 = 50\% $

FC20.6 79.8 373.3 4.03%
FC20.5 76.3 372.1 4.03%
V.5: $ re_1 = 0.25\% $ FC20.6 95.5 380.0 4.05%
FC20.5 90.9 379.0 4.06%
V.6:

(no reinsurance)

FC20.6 500.0 446.6 4.47%
FC20.5 500.0 475.1 4.75%
V.7: $ r^\ast = 11.27\% $ FC20.6 106.0 372.3 4.47%
$ r^\ast = 11.22\% $ FC20.5 99.6 371.5 4.47%

34145 STFrein01.xpl

Reinsurance reduces the required level of RBC, which coincides either with premium reduction (compare variant 1 and 6) or with increase of the expected rate of return (compare variant 7 and 6). Reinsurance also reduces difference between results obtained on the basis of two different approximation methods (FC20.6 and FC20.5). In variant 6 (no reinsurance) the difference is quite large, which is caused by the fairly long right tail of the distribution of the variable $ Y$.

Comparison of variants 2 and 1 confirms that the choice of a smaller expected rate of return (given substitution rate) automatically raises the need for capital, leaving the premium level unchanged (and therefore also the optimal retention level).

Comparison of variants 3 and 1 shows that admission of greater loss probability  $ \varepsilon$ causes reduction of premium, which coincides with substantial reduction of the need for reinsurance cover, and slight reduction in the need for capital. It is worthwhile to notice that replacement of $ \varepsilon =
2\% $ by $ \varepsilon = 4\% $ entails reversing the relation of results obtained by two approximation methods. Formula FC20.5 leads to smaller retention limits when safety standard is high (small $ \varepsilon )$, and to larger retention limits when safety standard is relaxed (large $ \varepsilon$).

Comparison of variants 4 and 5 with variant 1 illustrates the obvious rule that it does pay off to reduce retention limits when reinsurance is cheap, and to increase them when reinsurance is expensive.

It could happen in practice that pricing rules applied by reinsurers differ by lines of business. When the portfolio $ W = W_1 + ... + W_n $ consists of $ n$ business lines, for which the market offers reinsurance cover priced on the basis of different formulas $ \Pi _{1,R} \left( \cdot \right)$, ..., $ \Pi
_{n,R} \left( \cdot \right)$, the natural generalization of the problem lies in minimization of the premium (or maximization of the rate $ r^\ast )$ made by choosing $ n$ retention limits $ M_1 $, ..., $ M_n $, for each of business lines separately. Separation of business lines makes it feasible to assume different severity distributions, too.