An evolutionary strategy for multiobjective reinsurance optimization

Abstract

In this work we tackle a multiobjective reinsurance optimization problem (MOROP) from the point of view of an insurance company. The MOROP seeks to find a reinsurance program that optimizes two conflicting objectives: the maximization of the expected value of the profit of the company and the minimization of the risk of the insurance losses retained by the company. To calculate these two objectives we built a probabilistic model of the portfolio of risks of the company. This model is embedded within an evolutionary strategy (ES) that approximates the efficient frontier of the MOROP using a combination of four classical reinsurance structures: surplus, quota share, excess-of-loss and stop-loss. Computational experiments with the risks of a specific line of business of a large Colombian general insurance company show that the proposed evolutionary strategy outperforms the classical non-dominated sorting genetic algorithm. Moreover, the analysis of the solutions in the efficient frontier obtained with our ES gave several insights to the company in terms of the structure and properties of the solutions for different risk-return trade-offs.

Appendix A: Evaluation of the objective functions

In this appendix we describe the numerical approach used to evaluate the objective functions $f_1$ and $f_2$ for a given reinsurance program $y=(l,a,p,s)$.

1.1 A.1 Distribution functions involved in the evaluation of the objective functions

In addition to the random variables N, X, K, Z, $\overline{X}^{({\mathrm{SP}})}$, $\overline{X}^{({\mathrm{QS}})}$, $\overline{X}^{({\mathrm{XL}})}$, $\underline{X}^{({\mathrm{LR}})}$, $\overline{S}^{({\mathrm{SP}})}$, $\overline{S}^{({\mathrm{QS}})}$, $\overline{S}^{({\mathrm{XL}})}$, $\underline{S}^{({\mathrm{LR}})}$, $\overline{S}^{({\mathrm{SL}})}$, and $\underline{S}$ introduced previously in Section 4.3, let us define the random variables

$$ \underline{X}^{({\mathrm{SP}})} = \min (l,K)Z, $$

(19)

and

$$ \underline{X}^{({\mathrm{PR}})} = a\min (l,K)Z, $$

(20)

representing, respectively, the portion of claim X retained by the insurer after the coverage of the surplus reinsurance and the portion of claim X retained by the insurer after the coverage of the two proportional reinsurance treaties (i.e., after surplus and quota share).

With these definitions it is easy to see that the following equalities hold:

$$ E\left( \overline{S}^{({\mathrm{SP}})}\right)= E(N)E\left( \overline{X}^{({\mathrm{SP}})}\right) = E(N)\left[ E(X)-E\left( \underline{X}^{({\mathrm{SP}})}\right) \right] $$

(21)

$$ E\left( \overline{S}^{({\mathrm{QS}})}\right)= E(N)E\left( \overline{X}^{({\mathrm{QS}})}\right) = E(N)\left[ E\left( \underline{X}^{({\mathrm{SP}})}\right) -E\left( \underline{X}^{({\mathrm{PR}})}\right) \right] $$

(22)

$$ E\left( \overline{S}^{({\mathrm{XL}})}\right)= E(N)E\left( \overline{X}^{({\mathrm{XL}})}\right) = E(N)\left[ E\left( \underline{X}^{({\mathrm{PR}})}\right) -E\left( \underline{X}^{({\mathrm{LR}})}\right) \right] $$

(23)

$$ E\left( \overline{S}^{({\mathrm{SL}})}\right)= E\left( \underline{S}^{({\mathrm{LR}})}\right) -E\left( \underline{S}\right) . $$

(24)

Using these four equalities the first objective function in problem (16) can be rewritten as

$$ \begin{aligned} f_1(l,a,p,r)&= E(N)\Big [\lambda _{{\mathrm{SR}}}\left( E(X)-E\left( \underline{X}^{({\mathrm{SP}})}\right) \right) + \lambda _{{\mathrm{SR}}}\left( E\left( \underline{X}^{({\mathrm{SP}})}\right) -E\left( \underline{X}^{({\mathrm{PR}})}\right) \right) \nonumber \\ &\quad+ \lambda _{{\mathrm{XL}}}\left( E\left( \underline{X}^{({\mathrm{PR}})}\right) -E\left( \underline{X}^{({\mathrm{LR}})}\right) \right) \Big ] + \lambda _{{\mathrm{SL}}}\left( E\left( \underline{S}^{({\mathrm{LR}})}\right) -E\left( \underline{S}\right) \right) .\end{aligned} $$

(25)

From this last equation and Equation (15) we see that the evaluation of the objective functions $f_1$ and $f_2$ requires the calculation of statistics of random variables X, $\underline{X}^{({\mathrm{SP}})}$, $\underline{X}^{({\mathrm{PR}})}$, $\underline{X}^{({\mathrm{LR}})}$, $\underline{S}^{({\mathrm{LR}})}$ and $\underline{S}$. Therefore, to evaluate the objective functions we need to derive the distribution functions of these random variable in terms of the distribution functions $P_N(n)$, $F_K(k)$ and $F_Z(z)$, which characterize the losses of the insurance portfolio.

The corresponding expressions for the required distribution functions are shown in the second column of Table 4. In this table, we note that $f_Y(y)$ denotes the density of a general continuous random variable Y with distribution function $F_Y(y)$. Similarly, if Y is a discrete random variable with distribution function $P_Y(y)$, then $p_Y(y)$ denotes its corresponding probability mass function. Furthermore, the notation $F^{*j}$ denotes the j-fold convolution of function F with itself.

Table 4 Distribution functions of the random variables involved in the computation of the objective functions

1.2 A.2 Discretization of the distribution functions

Several of the distribution functions in Table 4 involve integrals over K, and hence, the first step in the numerical evaluation of the objective functions entails the discretization of its distribution function, $F_K(k)$. We discretize $F_K(k)$ at points $[k_0,k_1,k_2, \ldots ,k_{n_k-1}] = [m_k, m_k + \delta _k,m_k + 2\delta _k, \ldots , M_k]$, using the so-called rounding method (Embrechts and Frei 2008) so that the equivalent (discrete) probability mass function, $p_K(k)$, is given by:

$$ p_K(k_0)= F_K\left(m_k+\frac{\delta _k}{2}\right) $$

(26)

$$ p_K(k_i)= F_K(m_k+i\delta _k + \frac{\delta _k}{2})-F_K\left(m_k+i\delta _k - \frac{\delta _k}{2}\right), \quad i = 1, \ldots , n_k-2 $$

(27)

$$ p_K(k_{n_k-1})= 1-F_K\left(M_k- \frac{\delta _k}{2}\right), $$

(28)

where $\delta _k$ is the discretization step and $m_k$ and $M_k = m_k + (n_k-1)\delta _k$ are the minimum and maximum discretization values. Although alternative discretization (arithmetization) approaches, such as the local moment matching method (Gerber, 1982) or the nearest second moment method (Vilar 2000), are possible, the rounding method has shown to produce satisfactory results in our application.

The second step involves the computation of discrete equivalents of the distributions functions of X, $\underline{X}^{({\mathrm{SP}})}$, $\underline{X}^{({\mathrm{PR}})}$, $\underline{X}^{({\mathrm{LR}})}$, $\underline{S}^{({\mathrm{LR}})}$ and $\underline{S}$. To facilitate this computation, its convenient to use a common discretization grid $[x_0,x_1,x_2, \ldots ,x_{n_x-1}] = [m_x, m_x + \delta _x,m_x + 2\delta _x, \ldots , M_x]$, where $n_x=2^r$ with integer r; $\delta _x$ is the discretization step; and $m_x$ and $M_x = m_x + (n_x-1)\delta _x$ are the minimum and maximum discretization values.

With this discretization grid, we can compute the equivalent (discrete) probability mass functions for X, $p_X(x)$, as follows:

$$ p_X(x_0)= \hat{F}_X\left(m_x+\frac{\delta _X}{2}\right) $$

(29)

$$ p_X(x_j)= \hat{F}_X\left(m_x+j\delta _x + \frac{\delta _x}{2}\right)-\hat{F}_X\left(m_x+j\delta _x - \frac{\delta _x}{2}\right), \quad j = 1, \ldots , n_x-2 $$

(30)

$$ p_X(x_{n_x-1})= 1-\hat{F}_X\left(M_x- \frac{\delta _x}{2}\right), $$

(31)

where $\hat{F}_X(x) = \sum _{i=0}^{n_k-1} F_{Z}\left( \frac{x}{k_i}\right) p_{K}(k_i)$ is the numerical approximation of $F_X(x)$. We can analogously compute the equivalent (discrete) probability mass functions $p_{\underline{X}^{({\mathrm{SP}})}}(x)$, $p_{\underline{X}^{({\mathrm{PR}})}}(x)$, $p_{\underline{X}^{({\mathrm{LR}})}}(x)$, and $p_{\underline{S}}(x)$, using the numerically approximated distributions $\hat{F}_{\underline{X}^{({\mathrm{SP}})}}(x)$, $\hat{F}_{\underline{X}^{({\mathrm{PR}})}}(x)$, $\hat{F}_{\underline{X}^{({\mathrm{LR}})}}(x)$, and $\hat{F}_{\underline{S}}(x)$, given in Column 3 of Table 4. In these approximations, we note that $u_l = \min \{ i \in \{0,1,\ldots ,n_k-1\}: k_i > l\}$, $P_K(k_{u_l})= \sum _{i=0}^{u_l} p_K(k_{i})$, $x_p = \min \{ x_j \in \{0,1,\ldots ,n_x-1\}: x_j > p\}$ and $x_r = \min \{ x_j \in \{0,1,\ldots ,n_x-1\}: x_j > r\}$.

The distribution function of $\underline{S}^{({\mathrm{LR}})}$, $F_{\underline{S}^{({\mathrm{LR}})}}(x)$, involves the iterated convolution of the distribution function of $\underline{X}^{({\mathrm{LR}})}$, $F_{\underline{X}^{({\mathrm{LR}})}}(x)$. Therefore, we resort to the fast Fourier transform (FFT) (as discussed in Robertson, 1992) to efficiently discretize $\underline{S}^{({\mathrm{LR}})}$. Specifically, we apply the FFT to calculate the (discrete) probability mass functions, $p_{\underline{S}^{({\mathrm{LR}})}}(x)$, and then compute the approximated distribution function, $\hat{F}_{\underline{S}^{({\mathrm{LR}})}}(x)$, as follows:

$$\begin{aligned}&\hat{F}_{\underline{S}^{({\mathrm{LR}})}}(x_j) = \sum \limits _{h=0}^j p_{\underline{S}^{({\mathrm{LR}})}}(x_h), \quad j = 0, \ldots , n_x-2\nonumber \\&\quad \hat{F}_{\underline{S}^{({\mathrm{LR}})}}(M_x) = 1. \end{aligned}$$

(32)

1.3 A.3 Numerical evaluation of the objective functions

Given the discrete probability mass functions $p_{X}(x)$, $p_{\underline{X}^{({\mathrm{SP}})}}(x)$, $p_{\underline{X}^{({\mathrm{PR}})}}(x)$, $p_{\underline{X}^{({\mathrm{LR}})}}(x)$, $p_{\underline{S}^{({\mathrm{LR}})}}(x)$, and $p_{\underline{S}}(x)$, it is now straightforward to evaluate the objective functions $f_1$ and $f_2$.

We evaluate $f_1$ using Equation (25) with the expected values of X, $\underline{X}^{({\mathrm{SP}})}$, $\underline{X}^{({\mathrm{PR}})}$, $\underline{X}^{({\mathrm{LR}})}$, $\underline{S}^{({\mathrm{LR}})}$ and $\underline{S}$ computed using the corresponding probability mass functions. For example, the expected value of X is computed as $E(X)=\sum _{j=0}^{n_x-1}x_jp_{X}(x_j)$. The other expected values are computed in a similar manner.

The numerical evaluation of $f_2$ depends on the measure of risk, $\rho$, used in the optimization problem. In particular, the numerical expression for the VaR at level $\alpha$ is given by (c.f. Equation (15))

$$ {\mathrm{VaR}}_{\alpha }(\underline{S})= x_{u_{\alpha }} , $$

(33)

where the subindex $u_{\alpha } = \min \{ j \in \{0,1,\ldots ,n_x-1\}: \hat{F}_{\underline{S}}(x_{j}) \ge \alpha \}$ is found using a binary search over the table of all the possible values of $\hat{F}_{\underline{S}}(x)$.