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


**Table 4.** *Std Z*½ � *<sup>d</sup>*ð Þ*t for* λ*<sup>1</sup>* ¼ *5*, λ*<sup>2</sup>* ¼ *10*, δ ¼ *3*%*.*

#### **4.2 Premium calculation**

From the results in Section 4.1, we can compute the premium related to the risk of an insurance portfolio represented by *G t*ð Þ, depending on the premium calculation principles adopted by the insurance company. The loaded premium *Zd*ð Þ*t* consists in the sum of the pure premium *E Z*½ � *<sup>d</sup>*ð Þ*t* , the expected value of the costs related to the portfolio, and a loading for the risk *M t*ð Þ as

$$\mathbf{G}(t) = \mathbf{E}[\mathbf{Z}\_d(t)] + \mathbf{M}(t) \tag{73}$$

The loading for the risk differs according to the premium calculation principles.

#### *4.2.1 The expected value principle*

Denote by θ . 0 the safety loading. The expected value principle defines the loaded premium as:

$$\mathbf{G}(\mathbf{t}) = \mathbf{E}[\mathbf{Z}\_d(\mathbf{t})] + \theta \mathbf{E}[\mathbf{Z}\_d(\mathbf{t})],\tag{74}$$

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

#### *4.2.2 The variance principle*

Denote by θ . 0 the safety loading. The variance principle defines the loaded premium as:

*Moments of the Discounted Aggregate Claims with Delay Inter-Occurrence Distribution… DOI: http://dx.doi.org/10.5772/intechopen.88699*

$$\mathbf{G}(\mathbf{t}) = \mathbf{E}[\mathbf{Z}\_d(\mathbf{t})] + \theta \mathbf{Var}[\mathbf{Z}\_d(\mathbf{t})],\tag{75}$$

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

#### *4.2.3 The standard deviation principle*

Denote by θ . 0 the safety loading. The standard deviation principle defines the loaded premium as:

$$\mathbf{G}(\mathbf{t}) = \mathbf{E}[\mathbf{Z}\_d(\mathbf{t})] + \Theta \sqrt{\mathbf{Var}[\mathbf{Z}\_d(\mathbf{t})]},\tag{76}$$

where *M t*ðÞ¼ <sup>θ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *Var Z*½ � *<sup>d</sup>*ð Þ*<sup>t</sup>* <sup>p</sup> .

#### *4.2.4 The quantile principle*

The standard deviation principle defines the loaded premium as:

$$G(t) = F\_{Z\_d(t)}^{-1}(1 - \varepsilon),\tag{77}$$

where *ε* is smallest (for example: *ε* ¼ 0*:*5%*,* 1%*,* 2*:*5%*,* 5%Þ. In this case, the safety loading *M t*ð Þ is given by

$$\mathbf{M}(t) = F\_{Z\_d(t)}^{-1}(1 - \varepsilon) - E[Z\_d(t)] \tag{78}$$

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 variance.

Often, the actuary only has this information for different reasons (time constraints, information …).

If the actuary has more information about the random variable, *Zd*ð Þ*t* i.e., he knows the form of *FZd*ð Þ*<sup>t</sup>* , then he can apply the quantile principle.

But he does not know much about *FZd*ð Þ*<sup>t</sup>* , then he can approximate the distribution of *Zd*ð Þ*t* using the matching moments technique.
