Profit testing of profit sharing life insurance policies when asset returns are variance gamma distributed

Abstract

This paper examines the profit testing of life insurance companies that issue participating policies, type B and type A universal life policies, and variable annuities with guaranteed minimum maturity and death benefits, when investment returns are stochastic and modeled by normal or variance gamma distributions. We rely on the stochastic profit testing techniques introduced in Dickson et al. (Actuarial mathematics for life contingent risks, 2nd edn, Cambridge University Press, Cambridge, 2013) to examine the influence of the models’ parameters and of the models themselves on the profit testing indicators. We show that the variance gamma model results in more conservative predictions than the normal model for most cases.

Appendix

In this appendix, we deduct the expression of Eq. (11) by using two representations of the variance gamma process. The first of these representations is Eq. (4), while the second of these representations is a Brownian motion W(t) with drift $\theta t$, both subject to a random time change that follows a gamma process, as in Carr et al. (1998):

$$\begin{aligned} VG(\delta ,\zeta ,\nu ,\theta )=\delta +\theta \Gamma \left( \frac{1}{\nu },\frac{1}{\nu }\right) +\zeta W\left( \Gamma \left( \frac{1}{\nu },\frac{1}{\nu }\right) \right) , \end{aligned}$$

(21)

where $\Gamma \left( \frac{1}{\nu },\frac{1}{\nu }\right)$ is a gamma random variable with shape and size parameters both equal to $\frac{1}{\nu }$.

From Applebaum (2009), Eq. (21) can also be written as

$$\begin{aligned} VG(\delta ,\zeta ,\nu ,\theta )=\delta +\Gamma \left( \frac{1}{\nu },\frac{1}{\mu _p\nu }\right) -\Gamma \left( \frac{1}{\nu },\frac{1}{\mu _q\nu }\right) , \end{aligned}$$

(22)

where $\mu _p=\frac{1}{2}\sqrt{\theta ^2+\frac{2\zeta ^2}{\nu }}+\frac{\theta }{2}$ and $\mu _q=\frac{1}{2}\sqrt{\theta ^2+\frac{2\zeta ^2}{\nu }}-\frac{\theta }{2}$.

Then, Eqs. (4) and (22) can be linked as follows:

$$\begin{aligned} \left\{ \begin{array}{ll} \delta =\mu ^v, &{}\quad \theta =\frac{\eta }{\alpha }-\frac{\eta }{\beta },\\ \nu =\frac{1}{\eta }, &{}\quad \zeta =\sqrt{\frac{2\eta }{\alpha \beta }}. \end{array}\right. \end{aligned}$$

(23)

We now model the correlated variance gamma processes. We use the representation in Eberlein and Madan (2009) that yields:

$$\begin{aligned} VG_i=\delta _i+\theta _i\Gamma \left( \frac{1}{\nu _i},\frac{1}{\nu _i}\right) +\zeta _i \sqrt{\Gamma \left( \frac{1}{\nu _i},\frac{1}{\nu _i}\right) }Z_i, \qquad i=1,2, \end{aligned}$$

(24)

where $Z_i$ are standard normal variables with correlation $\rho '$.

We denote $\Gamma _i=\Gamma \left( \frac{1}{\nu _i},\frac{1}{\nu _i}\right)$ and obtain:

$$\begin{aligned} \hbox {Cov}(VG_1,VG_2)=\zeta _1\zeta _2E(\sqrt{\Gamma _1})E(\sqrt{\Gamma _2})\rho ', \end{aligned}$$

where

$$\begin{aligned} E(\sqrt{\Gamma _i})=\frac{\sqrt{\nu _i}{\widetilde{\Gamma }}(\frac{1}{\nu _i}+\frac{1}{2})}{{\widetilde{\Gamma }}(\frac{1}{\nu _i})}, \end{aligned}$$

and where ${\widetilde{\Gamma }}(\cdot )$ is the gamma function.

Using the parametrization of Eq. (4), we already know that

$$\begin{aligned} \hbox {Cov}(VG_1,VG_2)=\rho \sqrt{V\hbox {ar}(VG_1)}\sqrt{V\hbox {ar}(VG_2)}=\rho \sqrt{\eta _1\eta _2\left( \frac{1}{\alpha _1^2}+\frac{1}{\beta _1^2}\right) \left( \frac{1}{\alpha _2^2}+\frac{1}{\beta _2^2}\right) }, \end{aligned}$$

where $\rho$ is the correlation between the two returns $VG_1$ and $VG_2$. $\rho '$ can be expressed explicitly as a function of the parameters $(\mu ^v_i,\eta _i,\alpha _i,\beta _i)$:

$$\begin{aligned} \rho '=\frac{\rho \sqrt{\alpha _1\alpha _2\beta _1\beta _2\left( \frac{1}{\alpha _1^2}+\frac{1}{\beta _1^2}\right) \left( \frac{1}{\alpha _2^2}+\frac{1}{\beta _2^2}\right) }}{\sqrt{\frac{4}{\eta _1\eta _2}}\frac{{\widetilde{\Gamma }}(\eta _1+\frac{1}{2})}{{\widetilde{\Gamma }}(\eta _1)}\frac{{\widetilde{\Gamma }}(\eta _2+\frac{1}{2})}{{\widetilde{\Gamma }}(\eta _2)}}. \end{aligned}$$

(25)

Finally, we substitute the Eq. (23) into the Eq. (24) and we obtain Eq. (11).