Valuation of mixed life insurance contracts under stochastic correlated mortality and interest rates

Abstract

The need to find a good balance between attractive returns and long-term guarantees has motivated the development of hybrid life insurance products. In these contracts, the premium payments are distributed between a participating and a unit-linked fund, where an additional guarantee fee is applied to the unit-linked return in order to increase the investment guarantee of the participating fund. Moreover, the traditional pricing approach in life insurance is based on models for the financial elements and the mortality elements without any correlations between them. However, recent trends, such as the recent COVID-19 pandemic or the effect of ageing on stock market preferences, motivate us to account for the correlation between changes in demographic trends and the value of financial assets. The purpose of this paper is to price the mixed insurance contract based on longevity and to propose a multi-dimensional approach to explicitly studying the dependence between financial and mortality risks in a joint stochastic continuous time model of interest rates, stock returns, and mortality.

Appendices

Appendix A: Proof of Proposition 1

After defining the T-forward measure, ${\textrm{d}}{\tilde{W}}_t^r = {\textrm{d}}W_t^r + \int _0^t \sigma _r \, B_r(s,T,\theta ) ds$   is a ${{\mathbb {Q}}}^T$-Brownian motion.

We need then to write the dynamic of the intensity of mortality $\mu _x(t)$ under the T-forward measure, that is,

$$\begin{aligned} {\textrm{d}}\mu _x(t) = \,&b \, ({\tilde{\xi }}_t - \mu _x(t)) \, {\textrm{d}}t + \sigma _{\mu } \, \rho _{r,\mu } \, {\textrm{d}}{\tilde{W}}_t^r + \sigma _{\mu } \, {\overline{\rho }}_{G,\mu } \, \sqrt{1 - \rho _{r,\mu }^{2}} \, {\textrm{d}}{\tilde{W}}_t^G \\&+ \, \sigma _{\mu } \, \overline{{\overline{\rho }}}_{H,\mu } \, \sqrt{(1 - \rho _{r,\mu }^{2})(1 - {\overline{\rho }}_{G,\mu }^{2})} \, {\textrm{d}}{\tilde{W}}_t^H \\&+\, \sigma _{\mu } \, \sqrt{(1 - \rho _{r,\mu }^{2})(1 - {\overline{\rho }}_{G,\mu }^{2})(1-\overline{{\overline{\rho }}}_{H,\mu }^{2})} \, {\textrm{d}}{\tilde{W}}_t^{\mu };\\&\quad {\tilde{\xi }}_t = \xi _t - \sigma _{\mu } \, \sigma _r \, \rho _{r,\mu } \frac{B_r(t,T,\theta )}{b}, \end{aligned}$$

where ${\textrm{d}}{\tilde{W}}_t^G = {\textrm{d}}W_t^G,$ ${\textrm{d}}{\tilde{W}}_t^H = {\textrm{d}}W_t^H$ and ${\textrm{d}}{\tilde{W}}_t^{\mu } = {\textrm{d}}W_t^{\mu }$ are ${{\mathbb {Q}}}^T$-Brownian motions.

As a result,

$$\begin{aligned} {{\mathbb {E}}}^{{{\mathbb {Q}}}^T}\left[ e^{- \int _0^T \mu _x(s) ds}\right]&= \, \exp \left( -\mu _x(0) \cdot B_{\mu }(0,T,b) \right. \\&\quad \left. - b \int _0^T {\tilde{\xi }}_s \cdot B_{\mu }(s,T,b) ds + \frac{\sigma _{\mu }^2}{2} \int _0^T B_{\mu }^2(s,T,b) ds \right) \\&= \, \exp \Bigg (-\mu _x(0) \cdot B_{\mu }(0,T,b) - b \int _0^T \xi _s \cdot B_{\mu }(s,T,b) ds \\&\quad +\, \sigma _r \, \sigma _{\mu } \, \rho _{r,\mu } \int _0^T B_r(s,T,\theta ) \cdot B_{\mu }(s,T,b) ds + \frac{\sigma _{\mu }^2}{2} \int _0^T B_{\mu }^2(s,T,b) ds \Bigg )\\&= \, \exp \left( -\mu _x(0) \cdot B_{\mu }(0,T,b) \right. \\&\quad \left. -\, b \int _0^T \xi _s \cdot B_{\mu }(s,T,b) ds + \frac{\sigma _{\mu }^2}{2} \int _0^T B_{\mu }^2(s,T,b) ds \right) \\&\quad \cdot \, \exp \left( \sigma _r \, \sigma _{\mu } \, \rho _{r,\mu } \int _0^T B_r(s,T,\theta ) \cdot B_{\mu }(s,T,b) ds\right) \\&= \, p_x(0,T) \cdot \rho (0,T), \end{aligned}$$

where $\rho (0,T)$ is the term of correlation that arises from the dependence only between the interest rate and the mortality rate, defined in Eq. (16).

Appendix B: Proof of Proposition 2

By computing the Radon–Nikodym derivative, we obtain

$$\begin{aligned} \frac{{\textrm{d}}{{\mathbb {Q}}}^{\mu }}{{\textrm{d}}{{\mathbb {Q}}}^T} \,&= \, \exp \Bigg (- \sigma _{\mu } \, \rho _{r,\mu } \int _0^T B_{\mu }(s,T,b) \, {\textrm{d}}{\tilde{W}}_s^r - \sigma _{\mu } \, {\overline{\rho }}_{G,\mu } \, \sqrt{1-\rho _{r,\mu }^{2}} \int _0^T B_{\mu }(s,T,b) \, {\textrm{d}}{\tilde{W}}_s^G \\&\quad - \sigma _{\mu } \, \overline{{\overline{\rho }}}_{H,\mu } \, \sqrt{(1-\rho _{r,\mu }^{2})(1-{\overline{\rho }}_{G,\mu }^{2})} \int _0^T B_{\mu }(s,T,b) \, {\textrm{d}}{\tilde{W}}_s^H\\&\quad - \sigma _{\mu } \, \sqrt{(1-\rho _{r,\mu }^{2})(1-{\overline{\rho }}_{G,\mu }^{2})(1-\overline{{\overline{\rho }}}_{H,\mu }^{2})} \int _0^T B_{\mu }(s,T,b) \, {\textrm{d}}{\tilde{W}}_s^{\mu }\\&\quad - \frac{1}{2} \sigma _{\mu }^2 \, \rho _{r,\mu }^{2} \int _0^T B_{\mu }^2(s,T,b) \, {\textrm{d}}s - \frac{1}{2} \sigma _{\mu }^2 \, {\overline{\rho }}_{G,\mu }^{2} (1-\rho _{r,\mu }^{2}) \int _0^T B_{\mu }^2(s,T,b) \, {\textrm{d}}s\\&\quad - \frac{1}{2} \sigma _{\mu }^2 \, \overline{{\overline{\rho }}}_{H,\mu }^{2} (1-\rho _{r,\mu }^{2})(1-{\overline{\rho }}_{G,\mu }^{2}) \int _0^T B_{\mu }^2(s,T,b) \, {\textrm{d}}s\\&\quad - \frac{1}{2} \sigma _{\mu }^2 \, (1-\rho _{r,\mu }^{2})(1-{\overline{\rho }}_{G,\mu }^{2})(1-\overline{{\overline{\rho }}}_{H,\mu }^{2}) \int _0^T B_{\mu }^2(s,T,b) \, {\textrm{d}}s\Bigg ). \end{aligned}$$

Then, ${\left\{ \begin{array}{ll} \begin{array}{ll} {\textrm{d}}{\hat{W}}_t^r = {\textrm{d}}{\tilde{W}}_t^r + \sigma _{\mu } \, \rho _{r,\mu } \int _0^t B_{\mu }(s,T,b) {\textrm{d}}s\\ {\textrm{d}}{\hat{W}}_t^G = {\textrm{d}}{\tilde{W}}_t^G + \sigma _{\mu } \, {\overline{\rho }}_{G,\mu } \, \sqrt{1-\rho _{r,\mu }^{2}} \int _0^t B_{\mu }(s,T,b) {\textrm{d}}s\\ {\textrm{d}}{\hat{W}}_t^H = {\textrm{d}}{\tilde{W}}_t^H + \sigma _{\mu } \, \overline{{\overline{\rho }}}_{H,\mu } \, \sqrt{(1-\rho _{r,\mu }^{2})(1-{\overline{\rho }}_{G,\mu }^{2})} \int _0^t B_{\mu }(s,T,b) {\textrm{d}}s\\ {\textrm{d}}{\hat{W}}_t^{\mu } = {\textrm{d}}{\tilde{W}}_t^{\mu } + \sigma _{\mu } \, \sqrt{(1-\rho _{r,\mu }^{2})(1-{\overline{\rho }}_{G,\mu }^{2})(1-\overline{{\overline{\rho }}}_{H,\mu }^{2})} \int _0^t B_{\mu }(s,T,b) {\textrm{d}}s\\ \end{array} \end{array}\right. }$

are ${{\mathbb {Q}}}^{\mu }$-Brownian motions.

The objective is to compute the expectation of $\bigg (\frac{G_T}{G_0} - K\bigg )^{+}$ under the new measure. So, we will express this expression as follows

$$\begin{aligned} \left( \frac{G_T}{G_0} - K\right) ^{+} = \frac{G_T}{G_0} \cdot {\mathbb {1}}_{\frac{G_T}{G_0}>K} - K \cdot {\mathbb {1}}_{\frac{G_T}{G_0}>K}. \end{aligned}$$

Hence,

$$\begin{aligned} {{\mathbb {E}}}^{{{\mathbb {Q}}}^{\mu }}\left[ \left( \frac{G_T}{G_0} - K\right) ^{+}\right] = {{\mathbb {E}}}^{{{\mathbb {Q}}}^{\mu }}\left[ \frac{G_T}{G_0} \cdot {\mathbb {1}}_{\frac{G_T}{G_0}>K}\right] - K \cdot {{\mathbb {E}}}^{{{\mathbb {Q}}}^{\mu }}\Big [{\mathbb {1}}_{\frac{G_T}{G_0}>K}\Big ]. \end{aligned}$$

(37)

Let’s express the dynamic of the fund G under ${{\mathbb {Q}}}^{\mu }$ in order to get the solution of its differential equation. We move first from the risk-neutral measure to the T-forward measure, then to the forward-mortality measure.

Under the forward measure:

$$\begin{aligned}&{\textrm{d}}G_t = (r_t - \sigma _G \, \sigma _r \, \rho _{r,G} \, B_r(t,T,\theta )) \, G_t \, {\textrm{d}}t + \sigma _G \, \rho _{r,G} \, G_t \, {\textrm{d}}{\tilde{W}}_t^r\\ {}&+ \, \sigma _G \, \sqrt{1-\rho _{r,G}^{2}} \, G_t \, {\textrm{d}}{\tilde{W}}_t^G. \end{aligned}$$

Under the forward-mortality measure:

$$\begin{aligned} {\textrm{d}}G_t =&\Big (r_t - \sigma _G \, \sigma _r \, \rho _{r,G} \, B_r(t,T,\theta ) - \sigma _G \, \sigma _{\mu } \, \rho _{G,\mu } \, B_{\mu }(t,T,b)\Big ) \, G_t \, {\textrm{d}}t \\&+ \sigma _G \, \rho _{r,G} \, G_t \, {\textrm{d}}{\hat{W}}_t^r + \sigma _G \, \sqrt{1-\rho _{r,G}^{2}} \, G_t \, {\textrm{d}}{\hat{W}}_t^G. \end{aligned}$$

After integration, we obtain the SDE of the latter expression as follows:

$$\begin{aligned} G_t&= G_0 \cdot \exp \Bigg (\int _0^t r_s {\textrm{d}}s - \sigma _G \, \sigma _r \, \rho _{r,G} \int _0^t B_r(s,T,\theta ) {\textrm{d}}s - \sigma _G \, \sigma _{\mu } \, \rho _{G,\mu } \int _0^t B_{\mu }(s,T,b) {\textrm{d}}s \\&\quad - \, \frac{\sigma _G^2}{2} \cdot t + \sigma _G \, \rho _{r,G} \int _0^t {\textrm{d}}{\hat{W}}_s^r \\&\quad + \, \sigma _G \, \sqrt{1-\rho _{r,G}^2} \int _0^t {\textrm{d}}{\hat{W}}_s^G\Bigg ). \end{aligned}$$

This can also be written as

$$\begin{aligned} \frac{G_T}{G_0}&= \frac{1}{P(0,T)} \cdot \exp \Bigg (- \, \sigma _G \, \sigma _r \, \rho _{r,G} \int _0^T B_r(s,T,\theta ) {\textrm{d}}s - \sigma _G \, \sigma _{\mu } \, \rho _{G,\mu } \int _0^T B_{\mu }(s,T,b) {\textrm{d}}s \\&\quad - \frac{\sigma _r^2}{2} \int _0^T B_r^2(s,T,\theta ) {\textrm{d}}s - \sigma _r \, \sigma _{\mu } \, \rho _{r,\mu } \int _0^T B_r(s,T,\theta ) \cdot B_{\mu }(s,T,b) {\textrm{d}}s \\&\quad - \, \frac{\sigma _G^2}{2} \cdot T + \int _0^T \, (\sigma _G \, \rho _{r,G} + \sigma _r \, B_r(s,T,\theta )){\textrm{d}}{\hat{W}}_s^r + \sigma _G \, \sqrt{1-\rho _{r,G}^{2}} \int _0^T {\textrm{d}}{\hat{W}}_s^G\Bigg ). \end{aligned}$$

Under ${{\mathbb {Q}}}^{\mu },$ $\frac{G_T}{G_0}$ is log-normally distributed which allows us to write Equation (37) as

$$\begin{aligned} {{\mathbb {E}}}^{{{\mathbb {Q}}}^{\mu }}\left[ \left( \frac{G_T}{G_0} - K\right) ^{+}\right] \, =&\, {{\mathbb {E}}}^{{{\mathbb {Q}}}^{\mu }}\left[ \frac{G_T}{G_0} \Big | \frac{G_T}{G_0}>K\right] \cdot {{\mathbb {Q}}}^{\mu }\left[ \frac{G_T}{G_0}>K\right] - K \cdot {{\mathbb {Q}}}^{\mu }\left[ \frac{G_T}{G_0}>K\right] \\ =&\, e^{{\hat{m}}_G+ \frac{{\hat{v}}_G}{2}} \cdot \Phi \left( \frac{{\hat{m}}_G + {\hat{v}}_G - \ln K}{\sqrt{{\hat{v}}_G}}\right) - K \cdot \Phi \left( \frac{{\hat{m}}_G - \ln K}{\sqrt{{\hat{v}}_G}}\right) , \end{aligned}$$

where $\Phi$ is the cumulative distribution function of a standard normal variable and ${\hat{m}}_G$ and ${\hat{v}}_G$ are, respectively, the expectation and the variance of $\ln \left( \frac{G_T}{G_0}\right)$ under the forward-mortality measure, defined in Eqs. (17) and (18).

Appendix C: Proof of Proposition 3

We start by writing the dynamic of the fund F under the forward-mortality measure

$$\begin{aligned} {\textrm{d}}F_t&= \, \big (r_t \, - \varepsilon - \sigma _H \, \sigma _r \, \rho _{r,H} \, B_r(t,T,\theta ) - \sigma _H \, \sigma _{\mu } \, \rho _{H,\mu } \, B_{\mu }(t,T,b)\big ) \, F_t \, {\textrm{d}}t \\&\quad + \, \sigma _H \, \rho _{r,H} \, F_t \, {\textrm{d}}{\hat{W}}_t^r + \sigma _H \, {\overline{\rho }}_{G,H} \, \sqrt{1-\rho _{r,H}^{2}} \, F_t \, {\textrm{d}}{\hat{W}}_t^G \\&\quad +\, \sigma _H \, \sqrt{(1-\rho _{r,H}^{2})(1-{\overline{\rho }}_{G,H}^{2})} \, F_t \, {\textrm{d}}{\hat{W}}_t^H. \end{aligned}$$

After integration,

$$\begin{aligned} F_t&= F_0 \cdot \exp \Bigg (\int _0^t r_s {\textrm{d}}s \, - \varepsilon \cdot t - \sigma _H \, \sigma _r \, \rho _{r,H} \int _0^t B_r(s,T,\theta ) {\textrm{d}}s \\&\quad -\, \sigma _H \, \sigma _{\mu } \, \rho _{H,\mu } \int _0^t B_{\mu }(s,T,b) {\textrm{d}}s \\&\quad - \, \frac{\sigma _H^2}{2} \cdot t + \sigma _H \, \rho _{r,H} \int _0^t {\textrm{d}}{\hat{W}}_s^r + \sigma _H \, {\overline{\rho }}_{G,H} \, \sqrt{1-\rho _{r,H}^2} \int _0^t {\textrm{d}}{\hat{W}}_s^G \\&\quad + \, \sigma _H \, \sqrt{(1-\rho _{r,H}^2)(1-{\overline{\rho }}_{G,H}^{2})} \int _0^t {\textrm{d}}{\hat{W}}_s^H\Bigg ), \end{aligned}$$

hence, the expression of $\frac{F_T}{F_0}$ can be written as

$$\begin{aligned} \frac{F_T}{F_0}&= \frac{1}{P(0,T)} \cdot \exp \bigg ( - \varepsilon \cdot T - \sigma _H \, \sigma _r \, \rho _{r,H} \int _0^T B_r(s,T,\theta ) {\textrm{d}}s \\&\quad -\, \sigma _H \, \sigma _{\mu } \, \rho _{H,\mu } \int _0^T B_{\mu }(s,T,b) {\textrm{d}}s\\&\quad - \, \frac{\sigma _r^2}{2} \int _0^T B_r^2(s,T,\theta ) {\textrm{d}}s \\&\quad -\, \sigma _r \, \sigma _{\mu } \, \rho _{r,\mu } \int _0^T B_r(s,T,\theta ) \cdot B_{\mu }(s,T,b) {\textrm{d}}s - \frac{\sigma _H^2}{2} \cdot T \\&\quad + \, \int _0^T \, (\sigma _H \, \rho _{r,H} + \sigma _r \, B_r(s,T,\theta )){\textrm{d}}{\hat{W}}_s^r \\&\quad +\, \sigma _H \, {\overline{\rho }}_{G,H} \, \sqrt{1-\rho _{r,H}^2} \int _0^T {\textrm{d}}{\hat{W}}_s^G \\&\quad + \, \sigma _H \, \sqrt{(1-\rho _{r,H}^2)(1-{\overline{\rho }}_{G,H}^{2})} \int _0^T {\textrm{d}}{\hat{W}}_s^H\bigg ). \end{aligned}$$

From this end, we can compute ${{\mathbb {E}}}^{{{\mathbb {Q}}}^{\mu }}\left[ \frac{F_T}{F_0}\right] = e^{{\hat{m}}_F+ \frac{{\hat{v}}_F}{2}}$ with

$$\begin{aligned} {\hat{m}}_F&= E\left[ \ln \left( \frac{F_T}{F_0}\right) \right] = \ln \left( \frac{1}{P(0,T)}\right) \, - \varepsilon \cdot T - \frac{\sigma _H^2}{2} \cdot T - \frac{\sigma _H \, \sigma _r \, \rho _{r,H}}{\theta } \, [T - B_r(0,T,\theta )] \\&\quad - \, \frac{\sigma _H \, \sigma _{\mu } \, \rho _{H,\mu }}{b} \, [T - B_{\mu }(0,T,b)]\\&\quad - \, \frac{\sigma _r \, \sigma _{\mu } \, \rho _{r,\mu }}{\theta \cdot b} \, [T + B_{r,\mu }(0,T,\theta +b) - B_r(0,T,\theta ) - B_{\mu }(0,T,b)]\\&\quad - \, \frac{\sigma _r^2}{2 \theta ^2} \, [T + B_r(0,T,2 \theta ) - 2 \, B_r(0,T,\theta )] \end{aligned}$$

$$\begin{aligned} {\hat{v}}_F&= V\left[ \ln \left( \frac{F_T}{F_0}\right) \right] = \sigma _H^2 \cdot T + 2 \cdot \frac{\sigma _H \, \sigma _r \, \rho _{r,H}}{\theta } \, [T - B_r(0, T, \theta )]\\&\quad +\, \frac{\sigma _r^2}{\theta ^2} \, [T + B_r(0,T,2 \theta ) - 2 \, B_r(0,T,\theta )], \end{aligned}$$

where ${\hat{m}}_F$ and ${\hat{v}}_F$ are, respectively, the expectation and the variance of $\ln \left( \frac{F_T}{F_0}\right)$ under the forward-mortality measure.

Appendix D: Proof of Proposition 4

Referring to Sect. 4, the first and the third expectations can be solved explicitly as follows:

$$\begin{aligned} {{\mathbb {E}}}^{{\mathbb {Q}}}\Big [V_T \cdot e^{- \int _0^T r_s ds} \cdot e^{- \int _0^T \mu _x(s)ds}\Big ]&= \pi \cdot P \cdot \frac{p_x(0,T)}{{}_{T}p_{x}} \cdot \rho (0,T) \cdot \Big (P(0,T) \cdot (1+i)^T + \, \beta \cdot c_0^{T}\Big ) \\&\quad + (1-\pi ) \cdot P \cdot p_x(0,T) \cdot \rho ^{\prime }(0,T) \cdot e^{-\varepsilon \cdot T}, \end{aligned}$$

and

$$\begin{aligned} {{\mathbb {E}}}^{{\mathbb {Q}}}\Bigg [\frac{F_{j+1}}{F_0} \cdot e^{- \int _0^{j+1} r_s ds} \cdot e^{- \int _0^{j+1} \mu _x(s)ds}\Bigg ] = p_x(0,j+1) \cdot \rho ^{\prime }(0,j+1) \cdot e^{-\varepsilon \cdot (j+1)}. \end{aligned}$$

Whereas the expectation in the middle can be written as

$$\begin{aligned}{} & {} {{\mathbb {E}}}^{{\mathbb {Q}}}\Bigg [\frac{F_{j+1}}{F_0} \cdot e^{- \int _0^{j+1} r_s ds} \cdot e^{- \int _0^{j} \mu _x(s)ds}\Bigg ]\nonumber \\ {}{} & {} = {{\mathbb {E}}}^{{\mathbb {Q}}}\Bigg [\underbrace{\frac{F_j}{F_0} \cdot e^{- \int _0^j r_s ds} \cdot e^{- \int _0^{j} \mu _x(s)ds}}_{{{\textbf {I}}}} \, \cdot \, \underbrace{ \frac{F_{j+1}}{F_j} \cdot e^{- \int _j^{j+1} r_s ds}}_{{\textbf {II}}}\Bigg ]. \end{aligned}$$

(38)

We then develop the two processes I and II:

$$\begin{aligned} \textrm{I}&= \frac{F_j}{F_0} \cdot e^{- \int _0^j r_s ds} \cdot e^{- \int _0^{j} \mu _x(s)ds} \\&= \exp \Bigg ( - \varepsilon \cdot j - \frac{\sigma _H^2}{2} \cdot j - \mu _x(0) \cdot B_{\mu }(0,j,b) \\&\quad -\, b \int _0^j \xi _s \cdot B_{\mu }(s,j,b) {\textrm{d}}s + \int _0^j (\sigma _H \rho _{r,H} - \sigma _{\mu } \rho _{r,\mu } B_{\mu }(s,j,b)){\textrm{d}}W_s^r \\&\quad +\, \int _0^j (\sigma _H {\overline{\rho }}_{G,H} \sqrt{1-\rho _{r,H}^{2}} - \sigma _{\mu } {\overline{\rho }}_{G,\mu } \sqrt{1-\rho _{r,\mu }^{2}} B_{\mu }(s,j,b)){\textrm{d}}{\overline{W}}_s^G\\&\quad +\, \int _0^j (\sigma _H \sqrt{(1-\rho _{r,H}^{2})(1-{\overline{\rho }}_{G,H}^{2})} - \sigma _{\mu } \overline{{\overline{\rho }}}_{H,\mu } \sqrt{(1-\rho _{r,\mu }^{2})(1-{\overline{\rho }}_{G,\mu }^{2})} B_{\mu }(s,j,b)){\textrm{d}}\overline{{\overline{W}}}_s^H\\&\quad -\, \sigma _{\mu } \sqrt{(1-\rho _{r,\mu }^{2})(1-{\overline{\rho }}_{G,\mu }^{2})(1-\overline{{\overline{\rho }}}_{H,\mu }^{2})} \int _0^j B_{\mu }(s,j,b) {\textrm{d}}\overline{\overline{{\overline{W}}}}_s^{\mu }\Bigg ), \end{aligned}$$

and

$$\begin{aligned} {\textrm{II}}&= \frac{F_{j+1}}{F_j} \cdot e^{- \int _j^{j+1} r_s ds}\\ {}&= \exp \Bigg ( -\varepsilon - \frac{\sigma _H^2}{2} + \sigma _H \rho _{r,H} \int _j^{j+1} {\textrm{d}}W_s^r + \sigma _H {\overline{\rho }}_{G,H} \sqrt{1-\rho _{r,H}^{2}} \int _j^{j+1} {\textrm{d}}{\overline{W}}_s^G\\&\quad +\, \sigma _H \sqrt{(1-\rho _{r,H}^{2})(1-{\overline{\rho }}_{G,H}^{2})} \int _j^{j+1} {\textrm{d}}\overline{{\overline{W}}}_s^H \Bigg ). \end{aligned}$$

The independence between these two processes allows us to split Eq. (38) to

$$\begin{aligned}&{{\mathbb {E}}}^{{\mathbb {Q}}}\Bigg [\frac{F_j}{F_0} \cdot e^{- \int _0^j r_s ds} \cdot e^{- \int _0^{j} \mu _x(s)ds}\Bigg ] \, \cdot \, {{\mathbb {E}}}^{{\mathbb {Q}}}\Bigg [\frac{F_{j+1}}{F_j} \cdot e^{- \int _j^{j+1} r_s ds}\Bigg ]\\&\quad = p_x(0,j) \cdot \rho ^{\prime }(0,j) \cdot e^{-\varepsilon (j+1)}. \end{aligned}$$

Appendix E: Size of the loss in percentage of the premium—annual guarantee contract

See Tables 9 and 10.

Table 9 Size of loss in terms of the premium—annual guarantee without rebalancing

Table 10 Size of loss in terms of the premium—annual guarantee with rebalancing