19.2 General Formulae for Premiums Under Deductibles

Let $ X$ denote a non-negative continuous random variable describing the size of claim (risk, loss), $ F(t)$ and $ f(t)$ its distribution and probability density functions, respectively, and $ h(x)$ the payment function corresponding to a deductible. We consider here the simplest premium which is called the pure risk premium, see Chapter 18. The pure risk premium $ P$ (as we consider only pure risk premium we will henceforth use the term premium meaning pure risk premium) is equal to the expectation, i.e.

$\displaystyle P=\mathop{\textrm{E}}(X),$     (19.1)

and we assume that the expected value $ \mathop{\textrm{E}}(X)$ exists.

In the case of no deductible the payment function is obviously of the form $ h(x)=x $. This means that if the loss is equal to $ x$, the insurer pays the whole claim amount and $ P=\mathop{\textrm{E}}(X)$.

We express formulae for premiums under deductibles in terms of the so-called limited expected value function (levf), namely

$\displaystyle L(x)=\mathop{\textrm{E}}\{\min(X,x)\}=\int_0^xyf(y)dy+x\left\{1-F(x)\right\},$   $\displaystyle x>0.$ (19.2)

The value of this function at a point $ x$ is equal to the expected value of the random variable $ X$ truncated at the point $ x$. The function is a very useful tool for testing the goodness of fit an analytic distribution function to the observed claim size distribution function and was already discussed in Chapter 13.

In the following sections we illustrate premium formulae for the most important types of deductibles. All examples were created with the insurance library of XploRe.


19.2.1 Franchise Deductible

One of the deductibles that can be incorporated in the contract is the so-called franchise deductible. In this case the insurer pays the whole claim, if the agreed deductible amount is exceeded. More precisely, under the franchise deductible of $ a$, if the loss is less than $ a$ the insurer pays nothing, but if the loss equals or exceeds $ a$ claim is paid in full. This means that the payment function can be described as (Figure 19.1)


$\displaystyle h_{FD(a)}(x)= \left\{\begin{array}{ll} 0, & x<a,\\ x, & \textrm{otherwise}. \end{array}\right.$     (19.3)

Figure 19.1: The payment function under the franchise deductible (solid blue line) and no deductible (dashed red line).

It is worth noticing that the franchise deductible satisfies properties (i), (iii) and (iv), but not property (ii). This deductible can even work against property (ii). Since if a loss occurs, the policyholder would prefer it to be greater than or equal to the deductible.

The pure risk premium under the franchise deductible can be expressed in terms of the premium in the case of no deductible and the corresponding limited expected value function:

$\displaystyle P_{FD(a)}=P-L(a)+a\left\{1-F(a)\right\}.$     (19.4)

It can be easily noticed that this premium is a decreasing function of $ a$. When $ a=0$ the premium is equal to the no deductible case and if $ a$ tends to infinity the premium tends to zero.


19.2.2 Fixed Amount Deductible

Figure 19.2: The payment function under the fix amount deductible (solid blue line) and no deductible (dashed red line).

An agreement between the insured and the insurer incorporating a deductible $ b$ means that the insurer pays only the part of the claim which exceeds amount $ b$. If the size of the claim falls below this amount, the claim is not covered by the contract and the insured receives no indemnification. The payment function is thus given by

$\displaystyle h_{FAD(b)}(x)=\max(0,x-b),$     (19.5)

see Figure 19.2. The fixed amount deductible satisfies all the properties (i)-(iv).

The premium in the case of the fixed amount deductible has the following form in terms of the premium under the franchise deductible.

$\displaystyle P_{FAD(b)}=P-L(b)=P_{FD(b)}-b\left\{1-F(b)\right\}.$     (19.6)

As previously, this premium is a decreasing function of $ b$, for $ b=0$ it gives the premium in the case of no deductible and if $ b$ tends to infinity, it tends to zero.


19.2.3 Proportional Deductible

In the case of the proportional deductible with $ c\in (0,1)$, each payment is reduced by $ c\cdot100\% $ (the insurer pays $ 100\%(1-c)$ of the claim). Consequently, the payment function is given by (Figure 19.3)

$\displaystyle h_{PD(c)}(x)=(1-c)x.$     (19.7)

Figure 19.3: The payment function under the proportional deductible (solid blue line) and no deductible (dashed red line).

The proportional deductible satisfies properties (i), (ii), and (iv), but not property (iii), as it implies some compensation for even very small claims.

The relation between the premium under the proportional deductible and the premium in the case of no deductible has the following form.

$\displaystyle P_{PD(c)}=(1-c)\mathop{\textrm{E}}(X)=(1-c)P.$     (19.8)

Clearly, the premium is a decreasing function of $ c$, $ P_{PD(0)}=P $ and $ P_{PD(1)}=0 $.


19.2.4 Limited Proportional Deductible

The proportional deductible is usually combined with a minimum amount deductible so the insurer does not need to handle small claims and with a maximum amount deductible to limit the retention of the insured. For the limited proportional deductible of $ c$ with a minimum amount $ m_1$ and maximum amount $ m_2$ $ (0\leq m_1<m_2) $ the payment function is given by

$\displaystyle h_{LPD(c,m_1,m_2)}(x)= \left\{\begin{array}{ll} 0, & x \leq m_1 ,... ...1-c)x, & m_1/c<x \leq m_2/c,\\ x-m_2, & \textrm{otherwise}, \end{array}\right.$     (19.9)

Figure 19.4: The payment function under the limited proportional deductible (solid blue line) and no deductible (dashed red line).

see Figure 19.4. The limited proportional deductible satisfies all the properties.

The following formula expresses the premium under the limited proportional deductible in terms of the premium in the case of no deductible and the corresponding limited expected value function.

$\displaystyle P_{LPD(c,m_1,m_2)}=P-L(m_1)+c\left\{L\left(\frac{m_1}{c}\right)-L\left(\frac{m_2}{c}\right)\right\}.$     (19.10)

Sometimes only one limitation is incorporated in the contract, i.e. $ m_1=0 $ or $ m_2=\infty $. It is easy to check that the limited proportional deductible with $ m_1=0 $ and $ m_2=\infty $ reduces to the proportional deductible.


19.2.5 Disappearing Deductible

There is another type of deductible that is a compromise between the franchise and fixed amount deductible. In the case of disappearing deductible the payment depends on the loss in the following way: if the loss is less than an amount of $ d_1 >0$, the insurer pays nothing; if the loss exceeds $ d_2$ ($ d_2>d_1$) amount, the insurer pays the loss in full; if the loss is between $ d_1 $ and $ d_2$, then the deductible is reduced linearly between $ d_1 $ and $ d_2$. Therefore, the larger the claim, the less of the deductible becomes the responsibility of the policyholder. The payment function is given by (Figure 19.5)

$\displaystyle h_{DD(d_1,d_2)}(x)= \left\{\begin{array}{ll} 0, & x \leq d_1 ,\\ ... ..._1)}{d_2-d_1}, & d_1<x \leq d_2,\\ x, & \textrm{otherwise}. \end{array}\right.$     (19.11)

Figure 19.5: The payment function under the disappearing deductible (solid blue line) and no deductible (dashed red line).

This kind of deductible satisfies properties (i), (iii), and (iv), but similarly to the franchise deductible it works against (ii).

The following formula shows the premium under the disappearing deductible in terms of the premium in the case of no deductible and the corresponding limited expected value function

$\displaystyle P_{DD(d_1,d_2)}=P+\frac{d_1}{d_2-d_1}L(d_2)-\frac{d_2}{d_2-d_1}L(d_1).$     (19.12)

If $ d_1=0 $, the premium does not depend on $ d_2$ and it becomes the premium in the case of no deductible. If $ d_2$ tends to infinity, then the disappearing deductible reduces to the fix amount deductible of $ d_1 $.