Optimal Insurance

In this section the optimal insurance is regarded as a special case of optimal investment. Then the expected utility at the end of the next year

$$U(x,a)=\sum_{k=1}^M u(y^k) p(y^k),$$ (15)

where $p(y^k)$ is a probability to get a wealth $y^k$,
$u(y^k)$ is an utility function of this wealth $y^k$.
Here $a=(a_i,\ i=1,...,m)$ is a vector which components $a_i$ define rates of insurance charges for different objects $i=1,...,m$. Suppose that
$y^k=\sum_{i=1}^m c_i (x_i)$
and

$$c_i(x_i)= \left\{ \begin{array}{ll} z_i- a_i x_i & \mbox{if $\delta_i=1$}\\ (1-a_i) x_i & \mbox{if $\delta_i=0$} \end{array}. \right.$$ (16)

The product $a_i x_i$ denotes insurance charge of the object $i$,
$x_i \le z_i$ is insurance policy of the object $i$,
$z_i$ is market value of the object $i$,
$\delta_i=1$, if the object $i$ survives,
$\delta_i=0$, otherwise,
$p_i=P\{\delta_i=1\}$ is a survival probability of the object $i$.
For example:
$p(y^1)=p_1 \prod_{i \neq 1} (1-p_i)$,
$y^1=c_1( x_1)+\sum_{i=2}^m c_i(x_i)$,
where
$c_1(x_1)= z_1-a_1 x_1,\\ c_i(x_i)=(1-a_i)x_i, \ i=2,...,m$.