20.1 Introduction

In this chapter, setting the appropriate level of insurance premium is considered in a broader context of business decisions, concerning also risk transfer through reinsurance, and the rate of return on capital required to ensure solvability. Furthermore, the long term dividend policy, i.e. the rule of subdividing the financial result between the company and shareholders, is analyzed.

The problem considered throughout this chapter can be illustrated by a simple example.

Example 1

Let us consider the following model of a risk process describing a capital of an insurer:

$\displaystyle R_{t} = u + (c-du)t - S_{t}, \qquad t\geq 0,$    

where $ R_{t}$ denotes the current capital at time $ t$, $ u=R_{0}$ stands for initial capital, $ c$ is the intensity of premium inflow, and $ S_{t}$ is the aggregate loss process - amount of claim's outlays over the period $ (0,t]$. The term $ du$ represents the intensity of outflow of dividends paid to shareholders with $ d$ being the dividend rate. Let us assume that increments of the amount of claims process $ S_{t+h}-S_{t}$ are for any $ t,h>0$ normally distributed $ N(\mu h, \sigma^{2}h)$ and mutually independent. Below we consider premium calculation under two cases.

First case: $ d=0$. In this case the probability of ruin is an exponential function of the initial capital:

$\displaystyle \psi(u) = \exp(-Ru), \qquad u\geq 0,$    

where the adjustment coefficient $ R$ exists for $ c>\mu$, and equals then $ 2(c-\mu)\sigma^{-2}$. The above formula can be easily inverted to render the intensity of premium $ c$ for a given capital $ u$ and predetermined level $ \psi$ of ruin probability:

$\displaystyle c = \mu + \frac{-\log(\psi)}{2u}\sigma^{2}.$    

Given the safety standard $ \psi$, the larger the initial capital $ u$ of the company is, the more competitive it is (since it can offer the insurance cover at a lower price $ c$). However, a more realistic result is considered when we assume positive cost of capital.

Second case: $ d>0$. Now the problem of competitiveness is reduced to the problem of minimizing the premium by choosing the optimal level of capital backing insurance risk:

$\displaystyle c = \mu + \frac{-\log(\psi)}{2u}\sigma^{2} + du.$    

The solution reads:

$\displaystyle u_{opt} = \sigma\sqrt{\frac{-\log(\psi)}{2d}},$

                

$\displaystyle c_{opt} = \mu + \sigma\sqrt{-2d\log{\psi}},$

where exactly one half of the loading $ (c_{opt}-\mu)$ serves to finance dividends and the other half serves as a safety loading (retained in the company).

Having already calculated the total premium, we face the problem of decomposing it into premiums for individual risks. In order to do that we should first identify the random variable $ W=S_{t+1}-S_{t}$ as a sum of independent risks $ X_{1},\dots,X_{n}$, and the intensity of premium $ c$ as a whole-portfolio premium $ \Pi(W)$, which has to be decomposed into individual premiums $ \Pi(X_{i})$. The decomposition is straightforward when the total premium is calculated as in the first case above:

$\displaystyle \Pi(X_{i}) = \mathop{\textrm{E}}(X_{i}) + \frac{-\log(\psi)}{2u}\sigma^{2}(X_{i}),$    

which is due to additivity of variance for independent risks. The premium formula in the second case contains the safety loading proportional to the standard deviation and thus is no more additive. This does not mean that reasonable decomposition rules do not exist - rather that their derivation is not so straightforward.

In this chapter, various generalizations of the basic problem presented in the Example 1 are considered. These generalizations make the basic problem more complex on the one hand, but closer to real-life situations on the other. Additionally, these generalizations do not yield analytical results and, therefore, we demonstrate in several examples how to obtain numerical solutions.

First of all, Example 1 assumes that the safety standard is expressed in terms of an acceptable level of ruin probability. On the contrary, Sections 2, 3, and 4 are devoted to the approach based on the distribution of the single-year loss function. Section 2 presents the basic problem of joint decisions on premium and capital needed to ensure safety in terms of shareholder's choice of the level of expected rate of return and risk. Section 3 presents in more details the problem of decomposition of the whole-portfolio premium into individual risks premiums. Section 4 presents the problem extended by allowing for reinsurance, where competitiveness is a result of simultaneous choice of the amount of capital and retention level. This problem has not been illustrated in Example 1, as in the case of the normal distribution of the aggregate loss and usual market conditions there is no room to improve competitiveness through reinsurance.

Sections 5, 6, and 7 are devoted again to the approach based on ruin probability. However, Section 5 departs from the simplistic assumptions of Example 1 concerning the risk process. It is shown there how to invert various approximate formulas for the ruin probability in order to calculate premium for the whole portfolio as well as to decompose it into individual risks. Section 6 exploits results of Section 5 in the context of positive cost of capital. In that section a kind of flexible dividend policy is also considered, and the possibility to improve competitiveness this way is studied. Finally, Section 7 presents an extension of the decision problem by allowing for reinsurance cession.

Throughout this chapter we assume that we typically have at our disposal incomplete information on the distribution of the aggregate loss, and this incomplete information set consists of cumulants of order 1, 2, 3, and possibly 4. The rationale is that sensible empirical investigation of frequency and severity distributions could be done only separately for sub-portfolios of homogeneous risks. Cumulants for the whole portfolio are then obtained just by summing up figures over the collection of sub-portfolios, provided that sub-portfolios are mutually independent. The existence of cumulants of higher orders is assured by the common practice of issuing policies with limited cover exclusively (which in many countries is even enforced by law). Consequences of the assumption are that both the quantile of the current year loss and the probability of ruin in the long run will be approximated by formulas based on cumulants of the one-year aggregate loss $ W$.

The chapter is based on Otto (2004), a book on non-life insurance mathematics. However, general ideas are heavily borrowed from the seminal paper of Bühlmann (1985).