**4. Application**

#### **4.1 First two moments**

For the numerical illustration, suppose that *<sup>X</sup>* � *pExp <sup>β</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup> <sup>80</sup> � �<sup>þ</sup> ð Þ <sup>1</sup> � *<sup>p</sup> Exp <sup>β</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup> <sup>200</sup> � �, the inter-claim time distribution parameters <sup>λ</sup><sup>1</sup> <sup>¼</sup> <sup>2</sup>*;* 4 and λ<sup>2</sup> ¼ 1, the interest rate δ ¼ 3% (**Tables 1**–**4**). We use three different values for the copula parameter θ ¼ �1*;* 0*;* 1, *p* ¼ <sup>1</sup>*=*<sup>3</sup> and fix the time *t* ¼ 1*;* 10*;* 100. The *m*th moment of *X* is

$$\begin{split} \mu\_m &= p \frac{m!}{\beta\_1^m} + (\mathbf{1} - p) \frac{m!}{\beta\_2^m} \text{ and } \mu\_m' = \underset{0}{\int}{\infty} m \mathbf{x}^{m-1} (\mathbf{1} - F\_X(\mathbf{x}))^2 d\mathbf{x} = \mu\_m = p \frac{m!}{(2\beta\_1)^m} + \\ & (\mathbf{1} - p) \frac{m!}{(2\beta\_2)^m} .\end{split} \tag{72}$$


**Table 1.** *E Z*½ � *<sup>d</sup>*ð Þ*t for* λ*<sup>1</sup>* ¼ *1*, λ*<sup>2</sup>* ¼ *10*, δ ¼ *3*%*.*

## *Probability, Combinatorics and Control*


*G t*ðÞ¼ *E Z*½ �þ *<sup>d</sup>*ð Þ*t θVar Z*½ � *<sup>d</sup>*ð Þ*t ,* (75)

*Var Z*½ � *<sup>d</sup>*ð Þ*<sup>t</sup>* <sup>p</sup> *,* (76)

*Zd*ð Þ*<sup>t</sup>* ð Þ <sup>1</sup> � *<sup>ε</sup> ,* (77)

*Zd*ð Þ*<sup>t</sup>* ð Þ� <sup>1</sup> � *<sup>ε</sup> E Z*½ � *<sup>d</sup>*ð Þ*<sup>t</sup>* (78)

Denote by θ . 0 the safety loading. The standard deviation principle defines the

*G t*ðÞ¼ *E Z*½ �þ *<sup>d</sup>*ð Þ*<sup>t</sup>* <sup>θ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*Moments of the Discounted Aggregate Claims with Delay Inter-Occurrence Distribution…*

The standard deviation principle defines the loaded premium as:

where *ε* is smallest (for example: *ε* ¼ 0*:*5%*,* 1%*,* 2*:*5%*,* 5%Þ.

*M t*ðÞ¼ *<sup>F</sup>*�<sup>1</sup>

knows the form of *FZd*ð Þ*<sup>t</sup>* , then he can apply the quantile principle.

In this case, the safety loading *M t*ð Þ is given by

tion of *Zd*ð Þ*t* using the matching moments technique.

*G t*ðÞ¼ *<sup>F</sup>*�<sup>1</sup>

The principles of standard deviation and variance only require partial information on the distribution of the random variable, *Zd*ð Þ*t* , i.e., its expectation and its

Often, the actuary only has this information for different reasons (time con-

If the actuary has more information about the random variable, *Zd*ð Þ*t* i.e., he

But he does not know much about *FZd*ð Þ*<sup>t</sup>* , then he can approximate the distribu-

We have derived exact expressions for all the moments of the DCDPRV process using renewal arguments, again disproving the popular belief that renewal techniques cannot be applied in the presence of economic factors. Our results, for the DCDPRV process, are consistent: (i) with the results of Léveillé et al. [15] for θ ¼ 0*,* λ<sup>1</sup> 6¼ λ<sup>2</sup> and for θ ¼ 0*,* λ<sup>1</sup> ¼ λ2, (ii) with the results of Bargès et al. [8] for

Within this framework, further research is needed to get exact expressions (or approximations) of certain functional of the f g *Zd*ð Þ*t ; t*≥ 0 process, as stop-loss pre-

Our models have applications in reinsurance, house insurance and car insurance. They can also be used in evaluation of health programs, finance, and other areas. For example, consider the case of a male currently aged 25 who is starting a defined contribution (DC) pension plan and is planning to retire in, say, 40 years at the age of 65. He anticipates that when he reaches that age he will convert his accumulated pension fund into a life annuity in order to hedge his own longevity

where *M t*ðÞ¼ *θVar Z*½ � *<sup>d</sup>*ð Þ*t* .

loaded premium as:

*4.2.3 The standard deviation principle*

*DOI: http://dx.doi.org/10.5772/intechopen.88699*

where *M t*ðÞ¼ <sup>θ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*4.2.4 The quantile principle*

variance.

**5. Conclusion**

θ 6¼ 0*,* λ<sup>1</sup> ¼ λ2.

**289**

miums and ruin probabilities.

straints, information …).

*Var Z*½ � *<sup>d</sup>*ð Þ*<sup>t</sup>* <sup>p</sup> .

**Table 2.**

*E Z*½ � *<sup>d</sup>*ð Þ*t for* λ*<sup>1</sup>* ¼ *5*, λ*<sup>2</sup>* ¼ *10*, δ ¼ *3*%*.*

