19.3 Premiums Under Deductibles for Given Loss Distributions

In the preceding section we showed a relation between the pure risk premium under several deductibles and a limited expected value function. Now, we use the relation to present formulae for premiums in the case of deductibles for a number of loss distributions often used in non-life actuarial practice, see Burnecki, Nowicka-Zagrajek, and Weron (2004). To this end we apply the formulae for levf for different distributions given in Chapter 13. The log-normal, Pareto, Burr, Weibull, gamma, and mixture of two exponential distributions are typical candidates when looking for a suitable analytic distribution, which fits the observed data well, see Aebi, Embrechts, and Mikosch (1992), Burnecki, Kukla, and Weron (2000), Embrechts, Klüppelberg, and Mikosch (1997), Mikosch (1997), Panjer and Willmot (1992), and Chapter 13. In the log-normal and Burr case the premium formulae will be illustrated on a real-life example, namely on the fire loss data, already analysed in Chapter 13. For illustrative purposes, we assume that the total amount of risk $ X$ simply follows one of the fitted distributions, whereas in practice, in the individual and collective risk model framework (see Chapter 18), in order to obtain an annual premium under a per occurrence deductible we would have to multiply the premium by a number of policies and mean number of losses per year, respectively, since in the individual risk model

$\displaystyle \mathop{\textrm{E}}\left\{\sum_{k=1}^{n}h\left(X_k\right)\right\}=n\mathop{\textrm{E}}\left\{h\left(X_k\right)\right\},$

provided that the claim amount variables are identically distributed, and in the collective risk model

$\displaystyle \mathop{\textrm{E}}\left\{\sum_{k=1}^{N}h\left(X_k\right)\right\}=\mathop{\textrm{E}}(N)\mathop{\textrm{E}}\{h\left(X_k\right)\}.$


19.3.1 Log-normal Loss Distribution

Consider a random variable $ Z$ which has the normal distribution. Let $ X=e^Z $. The distribution of $ X$ is called the log-normal distribution and its distribution function is given by

$\displaystyle F(t)=\Phi\left(\frac{\ln t-\mu}{\sigma}\right)=\int_0^t\frac{1}{\...
...
y}\exp\left\{{-\frac{1}{2}\left(\frac{\ln{y}-\mu}{\sigma}\right)^2}\right\}dy,$      

where $ t,\sigma>0 $, $ \mu \in R $ and $ \Phi(.) $ is the standard normal distribution function, see Chapter 13. For the log-normal distribution the following formulae hold:

(a) franchise deductible premium

$\displaystyle P_{FD(a)}$ $\displaystyle =$ $\displaystyle \exp\left(\mu+\frac{\sigma^2}{2}\right)\left\{1-\Phi\left(\frac{\ln a-\mu-\sigma^2}{\sigma}\right)\right\},$  

(b) fixed amount deductible premium
$\displaystyle P_{FAD(b)}$ $\displaystyle =$ $\displaystyle \exp\left(\mu+\frac{\sigma^2}{2}\right)\cdot$  
       
  $\displaystyle \cdot$ $\displaystyle \left\{1-\Phi\left(\frac{\ln b-\mu-\sigma^2}{\sigma}\right)\right\}
-b\left\{1-\Phi\left(\frac{\ln b-\mu}{\sigma}\right)\right\},$  

(c) proportional deductible premium
$\displaystyle P_{PD(c)}=(1-c)\exp\left(\mu+\frac{\sigma^2}{2}\right),$      

(d) limited proportional deductible premium
$\displaystyle P_{LPD(c,m_1,m_2)}$ $\displaystyle =$ $\displaystyle \exp\left(\mu+\frac{\sigma^2}{2}\right)
\left\{1-\Phi\left(\frac{\ln m_1-\mu-\sigma^2}{\sigma}\right)\right\}$  
       
  $\displaystyle +$ $\displaystyle m_1\left\{\Phi\left(\frac{\ln m_1-\mu}{\sigma}\right)-\Phi\left(\frac{\ln(m_1/c)-\mu}{\sigma}\right)\right\}$  
       
  $\displaystyle +$ $\displaystyle \left\{\Phi\left(\frac{\ln (m_1/c)-\mu-\sigma^2}{\sigma}\right)-\Phi\left(\frac{\ln
(m_2/c)-\mu-\sigma^2}{\sigma}\right)\right\}\cdot$  
       
  $\displaystyle \cdot$ $\displaystyle c\exp\left(\mu+\frac{\sigma^2}{2}\right)+m_2\left\{\Phi\left(\frac{\ln (m_2/c)-\mu}{\sigma}\right)-1\right\},$  

(e) disappearing deductible premium
$\displaystyle P_{DD(d_1,d_2)}$ $\displaystyle =$ $\displaystyle \frac{\exp\left(\mu+\sigma^2/2\right)}
{d_2-d_1}\cdot$  
       
  $\displaystyle \cdot$ $\displaystyle \left\{d_2-d_1 +d_1\Phi\left(\frac{\ln d_2-\mu-\sigma^2}{\sigma}\right)
-d_2\Phi\left(\frac{\ln d_1-\mu-\sigma^2}{\sigma}\right)\right\}$  
       
  $\displaystyle +$ $\displaystyle \frac{d_1d_2}{d_2-d_1}\left\{\Phi\left(\frac{\ln d_1-\mu}{\sigma}\right)-\Phi\left(\frac{\ln d_2-\mu}{\sigma}\right)\right\}.$  

We now illustrate the above formulae using the Danish fire loss data. We study the log-normal loss distribution with parameters $ \mu=12.6645$ and $ \sigma=1.3981$, which best fitted the data. Figure 19.6 depicts the premium under franchise and fixed amount deductibles in the log-normal case.

Figure 19.6: The premium under the franchise deductible (thick blue line) and fixed amount deductible (thin red line). The log-normal case.

Figure 19.7 shows the effect of parameters $ c$, $ m_1$, and $ m_2$ of the limited proportional deductible. Clearly, $ P_{LPD(c,m_1,m_2)}
$ is a decreasing function of these parameters.

Figure 19.7: The premium under the limited proportional deductible with respect to the parameter $ m_2$. The thick blue solid line represents the premium for $ c=0.2 $ and $ m_1=100$ $ 000$ DKK, the thin blue solid line for $ c=0.4 $ and $ m_1=100$ $ 000$ DKK, the dashed red line for $ c=0.2 $ and $ m_1=1$ million DKK, and the dotted red line for $ c=0.4 $ and $ m_1=1$ million DKK. The log-normal case.

Finally, Figure 19.8 depicts the influence of parameters $ d_1 $ and $ d_2$ of the disappearing deductible. Markedly, $ P_{DD(d_1,d_2)} $ is a decreasing function of the parameters and we can observe that the effect of increasing $ d_2$ is rather minor.

Figure 19.8: The premium under the disappearing deductible with respect to the parameter $ d_2$. The thick blue line represents the premium for $ d_1=100$ $ 000$ DKK and the thin red line the premium for $ d_1=500$ $ 000$ DKK. The log-normal case.


19.3.2 Pareto Loss Distribution

The Pareto distribution function is defined by

$\displaystyle F(t)=1-\left(\frac{\lambda}{\lambda+t}\right)^{\alpha},$      

where $ t,\alpha,\lambda>0 $, see Chapter 13. The expectation of the Pareto distribution exists only for $ \alpha >1$. For the Pareto distribution with $ \alpha >1$ the following formulae hold:

(a) franchise deductible premium

$\displaystyle P_{FD(a)}$ $\displaystyle =$ $\displaystyle \frac{1}{\alpha-1}(a\alpha+\lambda)\left(\frac{\lambda}{a+\lambda}\right)^{\alpha},$  

(b) fixed amount deductible premium
$\displaystyle P_{FAD(b)}$ $\displaystyle =$ $\displaystyle \frac{1}{\alpha-1}(b+\lambda)\left(\frac{\lambda}{b+\lambda}\right)^{\alpha},$  

(c) proportional deductible premium
$\displaystyle P_{PD(c)}$ $\displaystyle =$ $\displaystyle (1-c)\frac{\lambda}{\alpha-1},$  

(d) limited proportional deductible premium
$\displaystyle P_{LPD(c,m_1,m_2)}$ $\displaystyle =$ $\displaystyle \frac{1}{\alpha-1}(m_1+\lambda)\left(\frac{\lambda}{m_1+\lambda}\right)^{\alpha}$  
       
  $\displaystyle +$ $\displaystyle \frac{c}{\alpha-1}\Bigg\{\left(\frac{m_2}{c}+\lambda\right)\left(\frac{\lambda}{m_2/c+\lambda}\right)^{\alpha}$  
       
    $\displaystyle -\left(\frac{m_1}{c}
+\lambda\right)\left(\frac{\lambda}{m_1/c+\lambda}\right)^{\alpha}\Bigg\},$  

(e) disappearing deductible premium
$\displaystyle P_{DD(d_1,d_2)}$ $\displaystyle =$ $\displaystyle \frac{1}{(\alpha-1)(d_2-d_1)}\cdot$  
       
  $\displaystyle \cdot$ $\displaystyle \left\{d_2(d_1+\lambda)\left(\frac{\lambda}{d_1+\lambda}\right)^{...
...ha} -d_1(d_2+\lambda)\left(\frac{\lambda}{d_2+\lambda}\right)^{\alpha}\right\}.$  


19.3.3 Burr Loss Distribution

Experience has shown that the Pareto formula is often an appropriate model for the claim size distribution, particularly where exceptionally large claims may occur. However, there is sometimes a need to find heavy tailed distributions which offer greater flexibility than the Pareto law. Such flexibility is provided by the Burr distribution which distribution function is given by

$\displaystyle F(t)=1-\left(\frac{\lambda}{\lambda+t^{\tau}}\right)^{\alpha},$      

where $ t$, $ \alpha $, $ \lambda $, $ \tau>0$, see Chapter 13. Its mean exists only for $ \alpha \tau >1 $. For the Burr distribution with $ \alpha \tau >1 $ the following formulae hold:

(a) franchise deductible premium

$\displaystyle P_{FD(a)}$ $\displaystyle =$ $\displaystyle \frac{\lambda^{{1}/{\tau}}\Gamma\left(\alpha-1/\tau\right)\Gamma\...
...}{\tau},\alpha-\frac{1}{\tau},\frac{a^{\tau}}{\lambda+a^{\tau}}\right)\right\},$  

(b) fixed amount deductible premium
$\displaystyle P_{FAD(b)}$ $\displaystyle =$ $\displaystyle \frac{\lambda^{{1}/{\tau}}\Gamma\left(\alpha-1/\tau\right)\Gamma\left(1+1/\tau\right)}{\Gamma(\alpha)}\cdot$  
       
  $\displaystyle \cdot$ $\displaystyle \left\{1-\textnormal{B}\left(1+\frac{1}{\tau},\alpha-\frac{1}{\ta...
...\tau}}\right)\right\}
-b\left(\frac{\lambda}{\lambda+b^{\tau}}\right)^{\alpha},$  

(c) proportional deductible premium
$\displaystyle P_{PD(c)}$ $\displaystyle =$ $\displaystyle (1-c)\frac{\lambda^{{1}/{\tau}}\Gamma(\alpha-1/\tau)\Gamma(1+1/\tau)}{\Gamma(\alpha)},$  

(d) limited proportional deductible premium
$\displaystyle P_{LPD(c,m_1,m_2)}$ $\displaystyle =$ $\displaystyle \frac{\lambda^{{1}/{\tau}}\Gamma\left(\alpha-1/\tau\right)\Gamma\left(1+1/\tau\right)}{\Gamma(\alpha)}\cdot$  
       
  $\displaystyle \cdot$ $\displaystyle \Bigg\{1-\textnormal{B}\left(1+\frac{1}{\tau},\alpha-\frac{1}{\tau},\frac{m_1^{\tau}}{\lambda+m_1^{\tau}}\right)$  
       
    $\displaystyle +c\textnormal{B}\left(1+\frac{1}{\tau},\alpha-\frac{1}{\tau},\frac{(m_1/c)^{\tau}}{\lambda+(m_1/c)^{\tau}}\right)$  
       
    $\displaystyle -c\textnormal{B}\left(1+\frac{1}{\tau},\alpha-\frac{1}{\tau},\frac{(m_2/c)^{\tau}}{\lambda+(m_2/c)^{\tau}}\right)\Bigg\}$  
       
  $\displaystyle -$ $\displaystyle m_1\left(\frac{\lambda}{\lambda+m_1^{\tau}}\right)^{\alpha}+m_1\left(\frac{\lambda}{\lambda+(m_1/c)^{\tau}}\right)^{\alpha}$  
       
  $\displaystyle -$ $\displaystyle m_2\left(\frac{\lambda}{\lambda+(m_2/c)^{\tau}}\right)^{\alpha},$  

(e) disappearing deductible premium
$\displaystyle P_{DD(d_1,d_2)}$ $\displaystyle =$ $\displaystyle \frac{\lambda^{{1}/{\tau}}\Gamma\left(\alpha-1/\tau\right)\Gamma\left(1+1/\tau\right)}{\Gamma(\alpha)}\cdot$  
       
  $\displaystyle \cdot$ $\displaystyle \Bigg [\frac{d_2-d_1+d_1\textnormal{B}\Big\{1+1/\tau,\alpha-1/\tau,d_2^{\tau}/(\lambda+d_2^{\tau})\Big\}}{d_2-d_1}$  
       
       
    $\displaystyle -\frac{d_2\textnormal{B}\Big\{1+
1/\tau,\alpha-1/\tau,d_1^{\tau}/(\lambda+d_1^{\tau})\Big\}}{d_2-d_1}\Bigg]$  
       
  $\displaystyle +$ $\displaystyle \frac{d_2d_1}{d_2-d_1}
\left\{\left(\frac{\lambda}{\lambda+d_2^{\...
...ght)^{\alpha}-\left(\frac{\lambda}{\lambda+d_1^{\tau}}\right)^{\alpha}\right\},$  

where the functions $ \Gamma(\cdot)$ and $ \textnormal{B}(\cdot,\cdot,\cdot)$ are defined as: $ \Gamma(a)=\int_{0}^{\infty} y^{a-1}e^{-y}dy$ and $ \textnormal{B}(a,b,x)=\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}\int_{0}^{x}y^{a-1}(1-y)^{b-1}dy.$

In order to illustrate the preceding formulae we consider the fire loss data. analysed in Chapter 13. The analysis showed that the losses can be well modelled by the Burr distribution with parameters $ \alpha=0.8804$, $ \lambda=8.4202\cdot 10^6$ and $ \tau=1.2749$. Figure 19.9 depicts the premium under franchise and fixed amount deductibles for the Burr loss distribution.

Figure 19.9: The premium under the franchise deductible (thick blue line) and fixed amount deductible (thin red line). The Burr case.

In Figure 19.10 the influence of the parameters $ c$, $ m_1$, and $ m_2$ of the limited proportional deductible is illustrated. Figure 19.11 shows the effect of the parameters $ d_1 $ and $ d_2$ of the disappearing deductible.

Figure 19.10: The premium under the limited proportional deductible with respect to the parameter $ m_2$. The thick solid blue line represents the premium for $ c=0.2 $ and $ m_1=100$ $ 000$ DKK, the thin solid blue line for $ c=0.4 $ and $ m_1=100$ $ 000$ DKK, the dashed red line for $ c=0.2 $ and $ m_1=1$ million DKK, and the dotted red line for $ c=0.4 $ and $ m_1=1$ million DKK. The Burr case.

Figure 19.11: The premium under the disappearing deductible with respect to the parameter $ d_2$. The thick blue line represents the premium for $ d_1=100$ $ 000$ DKK and the thin red line the premium for $ d_1=500$ $ 000$ DKK. The Burr case.


19.3.4 Weibull Loss Distribution

Another frequently used analytic claim size distribution is the Weibull distribution which is defined by

$\displaystyle F(t)=1-\exp\left(-\beta t^\tau\right),$      

where $ t,$ $ \tau ,$ $ \beta>0$, see Chapter 13.

For the Weibull distribution the following formulae hold:

(a) franchise deductible premium

$\displaystyle P_{FD(a)}$ $\displaystyle =$ $\displaystyle \frac{\Gamma\left(1+1/\tau\right)}{\beta^{1/\tau}} \left\{1-\Gamma\left(1+\frac{1}{\tau},\beta a^{\tau}\right)\right\},$  

(b) fixed amount deductible premium
$\displaystyle P_{FAD(b)}$ $\displaystyle =$ $\displaystyle \frac{\Gamma\left(1+1/\tau\right)}{\beta^{1/\tau}} \left\{1-\Gamm...
...+\frac{1}{\tau},\beta b^{\tau}\right)\right\}-
b\exp\left(-\beta b^\tau\right),$  

(c) proportional deductible premium
$\displaystyle P_{PD(c)}$ $\displaystyle =$ $\displaystyle \frac{(1-c)}{\beta^{1/\tau}} \Gamma\left(1+\frac{1}{\tau}\right),$  

(d) limited proportional deductible premium
$\displaystyle P_{LPD(c,m_1,m_2)}$ $\displaystyle =$ $\displaystyle \frac{ \Gamma\left(1+1/\tau\right)}{\beta^{1/\tau}}\left\{1-\Gamma\left(1+\frac{1}{\tau},\beta m_1^{\tau}\right)\right\}$  
       
  $\displaystyle +$ $\displaystyle \frac{c\Gamma\left(1+1/\tau\right)}{\beta^{1/\tau}}\Gamma\left\{1+\frac{1}{\tau},\beta\left(\frac{m_1}{c}\right)^{\tau}\right\}$  
       
  $\displaystyle -$ $\displaystyle \frac{c\Gamma\left(1+1/\tau\right)}{\beta^{1/\tau}}\Gamma\left\{1+\frac{1}{\tau},\beta \left(\frac{m_2}{c}\right)^{\tau}\right\}$  
       
  $\displaystyle -$ $\displaystyle m_1\exp\left(-\beta m_1^{\tau}\right)+m_1\exp\left\{-\beta \left(\frac{m_1}{c}\right)^\tau\right\}$  
       
  $\displaystyle -$ $\displaystyle m_2\exp\left\{-\beta\left(\frac{m_2}{c}\right)^{\tau} \right\},$  

(e) disappearing deductible premium
$\displaystyle P_{DD(d_1,d_2)}$ $\displaystyle =$ $\displaystyle \frac{ \Gamma\left(1+1/\tau\right)}{\beta^{1/\tau}(d_2-d_1)}\Bigg\{d_2-d_1+d_1\Gamma\left(1+\frac{1}{\tau},\beta
d_2^{\tau}\right)$  
       
    $\displaystyle -d_2\Gamma\left(1+\frac{1}{\tau},\beta{d_1}^{\tau}\right)\Bigg\}$  
       
  $\displaystyle +$ $\displaystyle \frac{d_1d_2}{d_2-d_1}\left\{\exp\left(-\beta d_2^\tau\right)-\exp\left(-\beta d_1^\tau\right)\right\},$  

where the incomplete gamma function $ \Gamma(\cdot,\cdot)$ is defined as

$\displaystyle \Gamma(a,x)=\frac{1}{\Gamma(a)}\int_{0}^{x}y^{a-1}e^{-y}dy.$


19.3.5 Gamma Loss Distribution

All four presented above distributions suffer from some mathematical drawbacks such as lack of a closed form representation for the Laplace transform and nonexistence of the moment generating function. The gamma distribution given by

$\displaystyle F(t)=F(t,\alpha,\beta)=\int_{0}^t\frac{\beta^{\alpha}}{\Gamma(\alpha)}y^{\alpha-1}e^{-\beta y}dy,$      

for $ t,\alpha,\beta>0 $ does not have these drawbacks, see Chapter 13. For the gamma distribution the following formulae hold:

(a) franchise deductible premium

$\displaystyle P_{FD(a)}$ $\displaystyle =$ $\displaystyle \frac{\alpha}{\beta} \left\{1-F\left(a,\alpha+1,\beta\right)\right\},$  

(b) fixed amount deductible premium
$\displaystyle P_{FAD(b)}$ $\displaystyle =$ $\displaystyle \frac{\alpha}{ \beta} \left\{1-F\left(b,\alpha+1,\beta\right)\right\}-b\left\{1-F\left(b,\alpha,\beta\right)\right\},$  

(c) proportional deductible premium
$\displaystyle P_{PD(c)}$ $\displaystyle =$ $\displaystyle \frac{(1-c)\alpha}{\beta},$  

(d) limited proportional deductible premium
$\displaystyle P_{LPD(c,m_1,m_2)}$ $\displaystyle =$ $\displaystyle \frac{\alpha}{ \beta} \left\{1-F\left(m_1,\alpha+1,\beta\right)\right\}$  
       
  $\displaystyle +$ $\displaystyle \frac{c\alpha}{ \beta}\left\{F\left(\frac{m_1}{c},\alpha+1,\beta\right)-F\left(\frac{m_2}{c},\alpha+1,\beta\right)\right\}$  
  $\displaystyle +$ $\displaystyle m_1\left\{F\left(m_1,\alpha,\beta\right)-F\left(\frac{m_1}{c},\alpha,\beta\right)\right\}$  
       
  $\displaystyle -$ $\displaystyle m_2\left\{1-F\left(\frac{m_2}{c},\alpha,\beta\right)\right\},$  

(e) disappearing deductible premium
$\displaystyle P_{DD(d_1,d_2)}$ $\displaystyle =$ $\displaystyle \frac{\alpha}{\beta(d_2-d_1)}\Bigg[d_2\left\{1-F\left(d_1,\alpha+1,\beta\right)\right\}$  
       
    $\displaystyle -d_1\left\{1-F\left(d_2,\alpha+1,\beta\right)\right\}\Bigg]$  
       
  $\displaystyle +$ $\displaystyle \frac{d_1d_2}{d_2-d_1}\left\{F\left(d_1,\alpha,\beta\right)-F\left(d_2,\alpha,\beta\right)\right\}.$  


19.3.6 Mixture of Two Exponentials Loss Distribution

The mixture of two exponentials distribution function is defined by

$\displaystyle F(t)=1-a\exp\left(-\beta_1t\right)-(1-a)\exp\left(-\beta_2t\right),$      

where $ 0\leq a \leq 1$ and $ \beta_1, \beta_2>0$, see Chapter 13. For the mixture of exponentials distribution the following formulae hold:

(a) franchise deductible premium

$\displaystyle P_{FD(c)}$ $\displaystyle =$ $\displaystyle \frac{a}{\beta_1}\exp\left(-\beta_1c\right)+\frac{1-a}{\beta_2}\exp\left(-\beta_2c\right)$  
       
  $\displaystyle +$ $\displaystyle c\left\{a\exp\left(-\beta_1c\right)+(1-a)\exp\left(-\beta_2c\right)\right\},$  

(b) fixed amount deductible premium
$\displaystyle P_{FAD(b)}$ $\displaystyle =$ $\displaystyle \frac{a}{\beta_1}\exp\left(-\beta_1b\right)+\frac{1-a}{\beta_2}\exp\left(-\beta_2b\right),$  

(c) proportional deductible premium
$\displaystyle P_{PD(c)}$ $\displaystyle =$ $\displaystyle (1-c)\left(\frac{a}{\beta_1}+\frac{1-a}{\beta_2}\right),$  

(d) limited proportional deductible premium
$\displaystyle P_{LPD(c,m_1,m_2)}$ $\displaystyle =$ $\displaystyle \frac{a}{\beta_1}\exp\left(-\beta_1m_1\right)+\frac{1-a}{\beta_2}\exp\left(-\beta_2m_1\right)$  
       
  $\displaystyle +$ $\displaystyle \frac{ca}{\beta_1}\left\{\exp\left(-\beta_1\frac{m_2}{c}\right)-\exp\left(-\beta_1\frac{m_1}{c}\right)\right\}$  
       
  $\displaystyle +$ $\displaystyle \frac{c(1-a)}{\beta_2}\left\{\exp\left(-\beta_2\frac{m_2}{c}\right)-\exp\left(-\beta_2\frac{m_1}{c}\right)\right\},$  

(e) disappearing deductible premium
$\displaystyle P_{DD(d_1,d_2)}$ $\displaystyle =$ $\displaystyle \frac{a}{\beta_1}\left\{\frac{d_2}{d_2-d_1}\exp\left(-\beta_1d_1\right)-\frac{d_1}{d_2-d_1}\exp\left(-\beta_1d_2\right)\right\}$  
       
  $\displaystyle +$ $\displaystyle \frac{1-a}{\beta_2}\left\{\frac{d_2}{d_2-d_1}\exp\left(-\beta_2d_1\right)-\frac{d_1}{d_2-d_1}\exp\left(-\beta_2d_2\right)\right\}.$