**5. Conclusion**

**4.2 Premium calculation**

*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*%*.*

*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*%*.*

*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*%*.*

*Probability, Combinatorics and Control*

**Table 4.**

**Table 3.**

**Table 2.**

*4.2.1 The expected value principle*

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

*4.2.2 The variance principle*

loaded premium as:

premium as:

**288**

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

**θ** *t* ¼ **1** *t* ¼ **10** *t* ¼ **100** �1 34027.1 29756.44 7954.606 0 24061.85 21053.72 5731.564 1 366.1484 1035.491 1545.671

**θ** *t* ¼ **1** *t* ¼ **10** *t* ¼ **100** �1 33900.85 29646.71 7928.481 0 23972.65 20976.12 5711.548 1 359.8795 1036.223 1542.412

**θ** *t* ¼ **1** *t* ¼ **10** *t* ¼ **100** �1 597.1633 4578.7 16,557 0 554.5237 4535.5 16,513 1 511.8841 4492.3 16,470

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

Denote by θ . 0 the safety loading. The expected value principle defines the

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

*G t*ðÞ¼ *E Z*½ �þ *<sup>d</sup>*ð Þ*t M t*ð Þ (73)

*G t*ðÞ¼ *E Z*½ �þ *<sup>d</sup>*ð Þ*t θE Z*½ � *<sup>d</sup>*ð Þ*t ,* (74)

related to the portfolio, and a loading for the risk *M t*ð Þ as

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 θ 6¼ 0*,* λ<sup>1</sup> ¼ λ2.

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 premiums and ruin probabilities.

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

risk and avoid outliving his own financial resources. The value of his retirement income will depend not only on the value of his pension fund, but also on the price of annuities at the time. Other things being equal, this means that his retirement income prospects will be affected by the distribution on future annuity value: the greater the dispersion of that distribution, the riskier his retirement income will be. For the assessment of the accumulated pension fund and its variability our models can be used. We can suppose that this man makes a deposit to a bank account, and that the time between successive deposits follows a renewal process and the force of interest is stochastic. Our model allows us to calculate the accumulated pension fund and its variability at the age of 65.

**References**

1998;**22**:251-262

2001;**29**:333-344

391-408

49-69

257-273

93-104

**291**

(n) risk process. Insurance:

[1] Dickson DCM, Hipp C. Ruin

probabilities for Erlang(2) risk process. Insurance: Mathematics & Economics.

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

Statistical Association for 2016; Cape Town, South Africa: University of Cape

[10] Adékambi F. Second moment of the discounted aggregate renewal cash flow with dependence. In: Proceedings of the 58th Annual Conference of the South African Statistical Association for 2016; Cape Town, South Africa: University of

[11] Govorun M, Latouche G. Modeling

[12] Adékambi F. Moments of phasetype aging modeling for health

dependent costs. Advances in Decision

[13] Van Noortwijk J, Frangopol D. Two probabilistic life-cycle maintenance models for deteriorating civil infrastructures. Probabilistic

Engineering Mechanics. 2004;**19**(4):

[14] Willmot GE. A note on a class of delayed renewal risk processes. Insurance: Mathematics & Economics.

[15] Léveillé G, Garrido J. Moments of

[16] Baeumer B. On the inversion of the convolution and Laplace transform. Transactions of the American Mathematical Society. 2003;**355**(3):

compound renewal sums with discounted claims. Insurance: Mathematics & Economics. 2001;**28**:

the effect of health: Phase-type approach. European Actuarial Journal.

Town. 2016. pp. 1-8

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

Cape Town. 2017. pp. 1-8

Sciences. 2019;**23**(2):1-28

2014;**4**(1):197-218

345-359

217-231

1201-1212

2004;**34**:251-257

[2] Dickson DCM, Hipp C. On the time to ruin for Erlang(2) risk process. Insurance: Mathematics & Economics.

[3] Li S, Garrido J. On ruin for the Erlang

Mathematics & Economics. 2004;**34**:

[4] Gerber H, Shiu E. The time value of ruin in a sparre Andersen model. North American Actuarial Journal. 2005;**9**:

[5] Li S, Garrido J. On a general class of renewal risk process: Analysis of the Gerber-Shiu function. Advances in Applied Probability. 2005;**37**:836-856

Exponential behaviour in the presence of dependence in risk theory. Journal of Applied Probability. March 2006;**43**(1):

[7] Asimit AV, Badescu AL. Extremes on the discounted aggregate claims in a

Scandinavian Actuarial Journal. 2010;**2**:

[8] Bargès M, Cossette H, Loisel S, Marceau E. On the moments of aggregate discounted claims with dependence introduced by a FGM copula. ASTIN Bulletin: The Journal of

[9] Adékambi F, Dziwa S. Moment of the discounted compound renewal cash flows with dependence: The use of Farlie-Gumbel-Morgenstern copula. In:

[6] Albrecher Hö, Teugels JL.

time dependent risk model.

the IAA. 2011;**41**(1):215-238

Proceedings of the 58th Annual Conference of the South African

Another possible application is in reliability, to model the net present value of aggregate equipment failures costs until its total breakdown. A piece of equipment is deemed to be beyond repair when the repair time exceeds a predetermined gap. Of course, another possible definition of total breakdown is when the cost of repair exceeds a predetermined gap. But, since the cost of repair is defined per unit time, the two definitions are somewhat equivalent.
