18.1 Premium Calculation Principles

Let $ X$ denote a non-negative random variable describing the size of claim (risk, loss) with the distribution function $ F_X(t) $. Moreover, we assume that the expected value $ \mathop{\textrm{E}}(X)$, the variance $ \mathop{\textrm{Var}}(X)$ and the moment generating function $ M_X(z)=\mathop{\textrm{E}}(e^{zX})$ exist.

The simplest premium (calculation principle) is called pure risk premium and it is equal to the expectation of claim size variable:

$\displaystyle P=\mathop{\textrm{E}}(X).$     (18.1)

This premium is often applied in life and some mass lines of business in non-life insurance. As it is known from the ruin theory, the pure risk premium without any kind of loading is insufficient since, in the long run, the ruin is inevitable even in the case of substantial (though finite) initial reserves. Nevertheless, the pure risk premium can be - and still is - of practical use because, for one thing, in practice the planning horizon is always limited, and for another, there are indirect ways of loading a premium, e.g. by neglecting interest earnings (Straub; 1988).

The future claims cost $ X$ may be different from its expected value $ \mathop{\textrm{E}}(X)$ and the estimator $ \widehat{\mathop{\textrm{E}}(X)}$ drawn from past may be different from the true $ \mathop{\textrm{E}}(X)$. To reflect this fact, the insurer can impose the risk loading on the pure risk premium.

The pure risk premium with safety (security) loading given by

$\displaystyle P_{SL}(\theta)=(1+\theta)\mathop{\textrm{E}}(X), \quad \theta\geq 0,$     (18.2)

where $ \theta$ and $ \theta \mathop{\textrm{E}}(X)$ are the relative and total safety loadings, respectively, is very popular in practical applications. This premium is an increasing linear function of $ \theta$ and it is equal to the pure risk premium for $ \theta=0$ .

The pure risk premium and the premium with safety loading are sometimes criticised because they do not depend on the degree of fluctuation of $ X$. Thus, two other rules have been proposed. The first one, denoted here by $ P_{V}(a)$ and given by

$\displaystyle P_{V}(a)=\mathop{\textrm{E}}(X)+a \mathop{\textrm{Var}}(X), \quad a\geq 0,$     (18.3)

is called the $ \sigma^2$-loading principle or the variance principle. In this case the premium depends not only on the expectation but also on the variance of the loss. The premium given by (18.3) is an increasing linear function of $ a$ and it is obvious that for $ a=0$ it is equal to the pure risk premium.

The other one, denoted here by $ P_{SD}(b)$ and given by

$\displaystyle P_{SD}(b)=\mathop{\textrm{E}}(X)+b \sqrt{\mathop{\textrm{Var}}(X)}, \quad b\geq 0,$     (18.4)

is called the $ \sigma$-loading principle or the standard deviation principle. In this case the premium depends on the expectation and also on the standard deviation of the loss. The premium given by (18.4) is an increasing linear function of $ b$ and clearly for $ b=0$ it reduces to the pure risk premium. Both the $ \sigma^2$- and $ \sigma$-loading principles are widely used in practice, but there is a discussion which one is better. If we consider two risks $ X_1$ and $ X_2$, the $ \sigma$-loading is additive and the $ \sigma^2$-loading not in case $ X_1$ and $ X_2$ are totally dependent, whereas the contrary is true for independent risks $ X_1$ and $ X_2$. Although in many cases the additivity is required from premium calculation principles, there are also strong arguments against additivity based on the idea that the price of insurance ought to be the lower the larger number of the risk carriers are sharing the risk.

The rules described so far are sometimes called ``empirical'' or ``pragmatic''. Another approach employs the notion of utility (Straub; 1988). The so-called zero utility principle states that the premium $ P_{U}$ for a risk $ X$ should be calculated such that the expected utility is (at least) equal to the zero utility. This principle yields a technical minimum premium in the sense that the risk $ X$ should not be accepted at a premium below $ P_{U}$. In the trivial case zero utility premium equals the pure risk premium. A more interesting case is the exponential utility which leads to a premium, denoted here by $ P_{E}(c)$ and called the exponential premium, given by

$\displaystyle P_{E}(c)=\frac{\ln M_{X}(c)}{c}=\frac{\ln \mathop{\textrm{E}}(e^{cX})}{c}, \quad c>0.$     (18.5)

This premium is an increasing function of the parameter $ c$ that measures the risk aversion and $ \lim_{c \rightarrow 0}P_{E}(c)=\mathop{\textrm{E}}(X)$. It is worth noticing that the zero utility principle yields additive premiums only in the trivial and the exponential utility cases (Gerber; 1980). As the trivial utility is just a special case of exponential utility corresponding to the limit $ c\rightarrow 0$, additivity characterizes the exponential utility.

Another interesting approach to the problem of premium calculations is the quantile premium, denoted here by $ P_{Q}(\varepsilon)$, is given by

$\displaystyle P_{Q}(\varepsilon)=F^{-1}_{X}(1-\varepsilon),$     (18.6)

where $ \varepsilon\in(0,1)$ is small enough. As it can be easily seen, it is just the quantile of order $ (1-\varepsilon)$ of the loss distribution and this means that the insurer wants to get the premium that covers $ (1 - \varepsilon)\cdot 100\%$ of the possible loss. A reasonable range of the parameter $ \varepsilon$ is usually from 1% to 5%.