Evaluation of participating endowment life insurance policies in a stochastic environment

Abstract

Participating life insurance contracts are policies that provide dividends (participation bonuses) based on the insurer’s financial performance. While these products are popular, there exists a gap in the literature for the analysis of these contracts under a stochastic setting. This paper fills this gap by proposing methods to (i) determine performance bonuses, (ii) compute the fair premium of the contract, and (iii) perform risk measurements for participating contracts in a realistic stochastic environment. The specific case of a fixed premium endowment participating contract, where the annual premium remains constant while benefits increase stochastically, is considered. We extend both the variable benefits life insurance approach of Bowers et al. [9] and the compound reversionary bonus mechanism presented in Booth et al. [8] and Bacinello [2] to a stochastic financial market (including stochastic interest rates) and stochastic mortality framework. Monte Carlo simulations provide insight about the sensitivity of premiums to contract specification and the evolution over time of both benefits and risks faced by the insurer.

Appendices

Proof of equation (2.8)

Using the fact that $L_{x+k}$ and $b_k$ ${\mathcal {G}}_{k}$-measurable, the independence between interest rate accrual factors and mortality outcomes and the invariance of mortality dynamics between the physical and risk-neutral measures, (2.7) leads to

$$\begin{aligned} {~}_k V(\Pi ,b_k)= & {} \mathbb {E^Q}\bigg [ \sum _{l=0}^{n-k-1} b_k(1+g)^{l} \frac{D_{x+k+l}}{L_{x+k}}\frac{B_{k}}{ B_{k+l+1}} \\{} & {} + b_k(1+g)^{n-k} \frac{L_{x+n}}{L_{x+k}} \frac{B_k}{B_n} - \Pi \sum _{l=0}^{n-k-1} \frac{L_{x+k+l}}{L_{x+k}}\frac{B_{k}}{ B_{k+l}}\bigg |{\mathcal {G}}_{k} \bigg ]\\= & {} \sum _{l=0}^{n-k-1} \mathbb {E^Q}\bigg [ \frac{D_{x+k+l}}{L_{x+k}} \bigg |{\mathcal {G}}_{k} \bigg ] \mathbb {E^Q}\bigg [ b_k(1+g)^{l} \frac{B_{k}}{ B_{k+l+1}} \bigg |{\mathcal {G}}_{k} \bigg ] \\{} & {} + \mathbb {E^Q}\bigg [\frac{L_{x+n}}{L_{x+k}} \bigg |{\mathcal {G}}_{k} \bigg ] \mathbb {E^Q}\bigg [ b_k(1+g)^{n-k} \frac{B_k}{B_n} \bigg |{\mathcal {G}}_{k} \bigg ]\\{} & {} \quad - \Pi \, \sum _{l=0}^{n-k-1} \mathbb {E^Q}\bigg [\frac{L_{x+k+l}}{L_{x+k}} \bigg |{\mathcal {G}}_{k} \bigg ] \mathbb {E^Q}\bigg [\frac{B_{k}}{ B_{k+l}}\bigg |{\mathcal {G}}_{k} \bigg ]\\= & {} \sum _{l=0}^{n-k-1} ~_{l|}q_{x+k} b_k (1+g)^{l} P(k,k+l+1) \\{} & {} + {}_{n-k}p_{x+k} b_k (1+g)^{n-k} P(k,n)\\{} & {} - \Pi \, \sum _{l=0}^{n-k-1} ~_{l}p_{x+k} P(k,k+l). \end{aligned}$$

Substituting (2.4) and (2.5) in the above yields (2.8). $\square$

Additional details about the financial model

This appendix reports the market model and associated parameters considered in numerical simulations. Such model and parameters are drawn directly from Eghbalzadeh et al. [14]. Parameter estimates were obtained by maximum likelihood estimation on monthly historical time series. Unlike in this manuscript where yearly time periods are considered, the model from Eghbalzadeh et al. [14] was estimated on data with monthly periods. Relations tying yearly returns to monthly values generated by the model are hereby provided.

1.1 Interest rates model

The year-k bank account numéraire is obtained by accruing risk-free interest over 12k months:

$$\begin{aligned} B_{k}=\exp \left( \Delta \sum _{t=0}^{12 k-1} r_t\right) , \quad k=1,\dots ,n, \end{aligned}$$

with $\Delta =1/12$ and $r_t$ being the (annualized) risk-free short rate for the $t+1^{th}$ monthly period. The interest rate term structure dynamics is modeled by the discrete-time arbitrage-free Nelson-Siegel (DTAFNS) model of Eghbalzadeh et al. [14] which is numerically tractable, easily interpretable and which provides an accurate depiction of realized term structures. The DTAFNS model is a three-factor stochastic short-rate model. The short rate is the sum of the first two:

$$\begin{aligned} r_t \equiv X^{(1)}_t + X^{(2)}_t, \end{aligned}$$

(B.1)

where the state variable process $\{X_t\}^{12n}_{t=0}$ defined with time-t factors $X_t= [X^{(1)}_t, X^{(2)}_t, X^{(3)}_t]$ follows auto-regressive dynamics under the physical measure ${\mathbb {P}}$:

$$\begin{aligned} \left[ \begin{array}{c} X^{(1)}_{t+1} -X^{(1)}_t \\ X^{(2)}_{t+1} -X^{(2)}_t \\ X^{(3)}_{t+1} -X^{(3)}_t \end{array} \right] \equiv&\underbrace{\left[ \begin{array}{ccc} -\Sigma _{11}\gamma _1 &{} 0 &{} 0 \\ 0 &{} \lambda - \Sigma _{22}\gamma _2 &{} -\lambda \\ 0 &{} 0 &{} \lambda - \Sigma _{33}\gamma _3 \end{array} \right] }_{ \kappa ^{{\mathbb {P}}} } \underbrace{\left[ \begin{array}{c} 0-X^{(1)}_t \\ \theta ^{{\mathbb {P}}}_2-X^{(2)}_t \\ \theta ^{{\mathbb {P}}}_3-X^{(3)}_t \end{array} \right] }_{ \theta ^{{\mathbb {P}}}-X_t } \nonumber \\&+\underbrace{\left[ \begin{array}{ccc} \Sigma _{11} &{} 0 &{} 0 \\ 0 &{} \Sigma _{22} &{} 0 \\ 0 &{} 0 &{} \Sigma _{33} \end{array} \right] }_{ \Sigma } \left( {\begin{array}{c} Z^{{\mathbb {P}}}_{t+1,1} \\ Z^{{\mathbb {P}}}_{t+1,2} \\ Z^{{\mathbb {P}}}_{t+1,3} \\ \end{array} } \right) , \end{aligned}$$

(B.2)

with $(\theta ^{{\mathbb {P}}},\kappa ^{{\mathbb {P}}},\Sigma )$ being model parameters, $\{ Z^{{\mathbb {P}}}_{t,i} \}^{12n}_{t=1}, i=1,2,3$ being ${\mathcal {F}}$-adapted standard Gaussian white noises with contemporaneous correlation parameters $\rho _{ij} = \text {corr}(Z_{t,i}^{{\mathbb {P}}},Z_{t,j}^{{\mathbb {P}}})$, $t=1,\ldots ,T$ and $i,j=1,2,3$.

Risk-neutral dynamics of the factor process are

$$\begin{aligned} X_{t+1}=X_t+\kappa ^{{\mathbb {Q}}} (\theta ^{{\mathbb {Q}}}-X_t)+\Sigma Z_{t+1}^{{\mathbb {Q}}}, \end{aligned}$$

(B.3)

where

$$\begin{aligned} \kappa ^{\mathbb {Q}} = \left[ \begin{array}{ccc} 0 &{} 0 &{} 0 \\ 0 &{} \lambda &{} -\lambda \\ 0 &{} 0 &{} \lambda \end{array} \right] \text { and } \theta ^{\mathbb {Q}} = \left( {\begin{array}{c} 0 \\ (\theta _2^{{\mathbb {P}}}\kappa ^{{\mathbb {P}}}_{22}+\theta _3^{{\mathbb {P}}}\Sigma _{33}\gamma _3)/\lambda \\ \theta _3^{{\mathbb {P}}} \kappa ^{{\mathbb {P}}}_{33}/\lambda \\ \end{array} } \right) \end{aligned}$$

(B.4)

and processes $Z^{\mathbb {Q}}_{i}=\lbrace Z^{ \mathbb {Q}}_{t,i}\rbrace _{t=1}^{12n}$, $i=1,2,3$ defined through $Z^{\mathbb {Q}}_{t+1,i}=Z^{{\mathbb {P}}}_{t+1,i}-\gamma _i X^{(i)}_{t}$ also are ${\mathcal {F}}$-adapted standard Gaussian white noises under $\mathbb {Q}$ with contemporaneous dependence parameters $\rho _{ij}$, $i,j=1,2,3$.

Proposition 2.1 from Eghbalzadeh et al. [14] shows that under this model the month-t price of a risk-free zero-coupon bond paying one dollar on month T is

$$\begin{aligned} P(t,T) =&A(t,T)\exp \left[ -\Delta {\mathcal {B}}(t,T)^\top X_t\right] , \end{aligned}$$

(B.5)

where ${\mathcal {B}}(t,T)=\left[ {\mathcal {B}}^{(1)}(t,T), \,\, {\mathcal {B}}^{(2)}(t,T), \, \, {\mathcal {B}}^{(3)}(t,T)\right] ^\top$ with

$$\begin{aligned} {\mathcal {B}}^{(1)}(t,T)&= \tau , \quad {\mathcal {B}}^{(2)}(t,T)= \dfrac{1-(1-\lambda )^\tau }{\lambda }, \\ {\mathcal {B}}^{(3)}(t,T)&= \frac{1-(1-\lambda )^{\tau -1}}{\lambda } - (\tau -1) (1-\lambda )^{\tau -1},\\ \log A(t,T)&= -\Delta \theta ^{\mathbb {Q}}_2 \left( {\mathcal {B}}^{(1)}(t,T) - {\mathcal {B}}^{(2)}(t,T)\right) + \Delta \theta ^{\mathbb {Q}}_3 {\mathcal {B}}^{(3)}(t,T) + \frac{1}{2} \Delta ^2\upsilon _\tau ,\\ \upsilon _\tau&= \left( \sum ^3_{i=1} \sum ^3_{j=1}\upsilon ^{(i,j)}_\tau \right) ,\\ \upsilon ^{(1,1)}_\tau&= \Sigma ^2_{11}\dfrac{\tau (\tau -1)(2\tau -1)}{6},\\ \upsilon ^{(2,2)}_\tau&= \frac{\Sigma ^2_{22}}{\lambda ^2} \left( \tau - 2 \left[ \frac{1-(1-\lambda )^\tau }{\lambda }\right] + \frac{1-(1-\lambda )^{2 \tau }}{1-(1-\lambda )^2}\right) ,\\ \upsilon ^{(3,3)}_\tau&= \mathbb {1}_{ \{ \tau> 1\} } \frac{\Sigma ^2_{33}}{\lambda ^2} \Bigg [ \tau -2 + \zeta _0\left( (1-\lambda )^2,\tau -1\right) +\lambda ^2 \zeta _2\left( (1-\lambda )^2,\tau -1\right) \\&\quad - 2\zeta _0\left( (1-\lambda ),\tau -1\right) - 2\lambda \zeta _1\left( (1-\lambda ),\tau -1\right) + 2\lambda \zeta _1\left( (1-\lambda )^2,\tau -1\right) \bigg ],\\ \upsilon ^{(1,2)}_\tau&= \upsilon ^{(2,1)}_\tau = \rho _{12}\Sigma _{11} \Sigma _{22} \frac{1}{\lambda }\left( \frac{\tau (\tau -1)}{2} - \zeta _1 ((1-\lambda ),\tau )\right) ,\\ \upsilon ^{(1,3)}_\tau&= \upsilon ^{(3,1)}_\tau = \mathbb {1}_{ \{\tau> 1\} } \rho _{13}\Sigma _{11} \Sigma _{33} \frac{1}{\lambda } \bigg [\frac{\tau (\tau -1)}{2}-1 -\zeta _0\left( (1-\lambda ),\tau -1\right) \\ {}&-(\lambda +1)\zeta _1\left( (1-\lambda ),\tau -1\right) -\lambda \zeta _2\left( (1-\lambda ),\tau -1\right) \bigg ],\\ \upsilon ^{(2,3)}_\tau&= \upsilon ^{(3,2)}_\tau = \mathbb {1}_{ \{ \tau > 1\} } \rho _{23}\Sigma _{22} \Sigma _{33} \\&\quad \bigg ( \frac{\tau -2- (2-\lambda )\zeta _0\left( (1-\lambda ),\tau -1\right) + (1-\lambda )\zeta _0\left( (1-\lambda )^2,\tau -1\right) }{\lambda ^2}\\&\quad + \frac{- \zeta _1\left( (1-\lambda ),\tau -1\right) + (1-\lambda )\zeta _1\left( (1-\lambda )^2,\tau -1\right) }{\lambda } \bigg ) \end{aligned}$$

and

$$\begin{aligned} \zeta _0(r,\tau )&\equiv \sum _{u=1}^{\tau -1}r^{u} =\dfrac{r-r^{\tau }}{1-r},\\ \zeta _1(r,\tau )&\equiv \sum _{u=1}^{\tau -1}u r^{u} = \dfrac{r- \tau r^{\tau }+(\tau -1) r^{\tau +1}}{(1-r)^2},\\ \zeta _2(r,\tau )&\equiv \sum _{u=1}^{\tau -1}u^2 r^{u} = \dfrac{ -(\tau -1)^2 r^{\tau +2} + (2\tau ^2-2\tau -1)r^{\tau +1} - \tau ^2 r^{\tau } + r^2+r}{(1-r)^3}. \end{aligned}$$

Table 3 provides parameters for the DTAFNS model (B.2)–(B.3), which were obtained through estimation over January 1986 to January 2022 Canadian end-of-month yield curve data.

Table 3 DTAFNS model parameters

1.2 Equity and mixed fund models

The reference account of participating life insurance contracts in which reserves are invested is typically composed of a mix of equity and fixed income securities. The mixed fund model of Eghbalzadeh et al. [14] which represents dynamics of funds invested in both these two classes of asset is therefore used in this work. Such model is an adaptation of the mixed fund model of Augustyniak et al. [1] which uses the DTAFNS term structure model instead of their conventional three-factor Gaussian model.

Again, the considered model parameters are borrowed from Eghbalzadeh et al. [14], who use monthly time periods in their study. The year-k reference account annual log-return $R^{(F)}_{k}$ is thus represented as the sum of the monthly log-returns $\tilde{R}^{(F)}_{k}$:

$$\begin{aligned} R^{(F)}_{k} = \sum ^{12k}_{t=1 +12(k-1)} \tilde{R}^{(F)}_{t}. \end{aligned}$$

The model first considers the dynamics of q equity indices, where index j’s monthly excess log-return dynamics are driven by an EGARCH process:

$$\begin{aligned} \tilde{R}_{t+1,j}^{(S)}-\Delta r_t&= \lambda _j^{(S)}\sqrt{h_{t,j}^{(S)}}-\dfrac{1}{2}h_{t,j}^{(S)} + \sqrt{h_{t,j}^{(S)}}Z_{t+1,j}^{(S)}, \end{aligned}$$

(B.6)

$$\begin{aligned} \log h_{t,j}^{(S)}&=\omega ^{(S)}_j+\alpha _j^{(S)}Z_{t,j}^{(S)}+\gamma ^{(S)}_j\bigg (|Z_{t,j}^{(S)}|-\dfrac{2}{\sqrt{2\pi }}\bigg )+\beta _j^{(S)}\log h_{t-1,j}^{(S)}, \end{aligned}$$

(B.7)

where ($\omega _j^{(S)}, \alpha _j^{(S)}, \gamma _j^{(S)}, \beta _j^{(S)}$) are parameters associated with the conditional volatility, $\lambda _j^{(S)}$ is the equity risk premium parameter, $\{h_{t,j}^{(S)}\}_{t=0}^{12n-1}$ is the GARCH volatility process, and $Z_{j}^{(S)}=\{Z_{t,j}^{(S)}\}_{t=1}^{12n}$, $j=1,\ldots ,q$ are standard Gaussian white noise under ${\mathbb {P}}$ independent of $Z_{t}$ whose contemporaneous correlation is $\Gamma _{ij} = \text {corr} \, \left( Z_{t,i}^{(S)},Z_{t,j}^{(S)}\right)$. The EGARCH process was introduced by Nelson [26], which has the advantageous properties of embedding Black [7]’s leverage effect, of not requiring constraints on parameters during the calibration to ensure positive volatilities and of better capturing the persistence of the volatility shocks than conventional GARCH processes.

The month-t reference account returns $\tilde{R}^{(F)}_{t+1}$ is then represented through an econometric model driven by (i) a linear relationship with respect to term structure shocks and excess returns on the equity indices, and (ii) a basis risk term independent of above terms which also follows an EGARCH process:

$$\begin{aligned} \tilde{R}_{t+1}^{(F)} - \Delta r_t&= \psi _0 + \sum _{i=1}^3 \psi _i\left( X_{t+1}^{(i)}\!-\!\left( 1\!-\!\kappa ^{\mathbb {Q}}_{ii}\right) X_t^{(i)}\right) \!+\! \psi '_3 X_t^{(3)} \nonumber \\&\quad +\! \sum _{j=1}^q \psi ^{(S)}_j \left( \tilde{R}_{t+1,j}^{(S)} - \Delta r_t\right) \!+\!\sqrt{h^{(F)}_{t}}Z^{(F)}_{t+1}, \end{aligned}$$

(B.8)

$$\begin{aligned} \log h_{t}^{(F)}&=\omega ^{(F)}+\alpha ^{(F)}Z_{t}^{(F)}+\gamma ^{(F)}\bigg (|Z_{t}^{(F)}|-\dfrac{2}{\sqrt{2\pi }}\bigg )+\beta ^{(F)} \log h_{t-1}^{(F)}, \end{aligned}$$

(B.9)

where $(\psi _0,\psi _1,\psi _2, \psi _3,\psi '_3,\psi ^{(S)}_1,\ldots ,\psi ^{(S)}_q)$ are parameters associated with the linear mapping between the account returns and risk factors, ($\omega ^{(F)}, \alpha ^{(F)}, \gamma ^{(F)}, \beta ^{(F)}$) are parameters associated with the conditional volatility process $\{h_t^{(F)}\}_{t=1}^{12n-1}$, and $Z_{t}^{(F)}:=\{Z_t^{(F)}\}_{t=1}^{12}$ is a standard Gaussian white noise under ${\mathbb {P}}$ independent of $Z_{t}$ and $Z^{(S)}_{j}$, $j=1,\ldots ,q$.

The considered risk-neutral dynamics of the equity indices and account returns, also provided by Proposition 4.1 of Eghbalzadeh et al. [14], are

$$\begin{aligned} R_{t+1}^{(F)}-\Delta r_t = -\dfrac{1}{2}\bigg (\sigma _t^{(F)}\bigg )^2+\sigma _t^{(F)}\epsilon ^{\mathbb {Q}(F)}_{t+1}, \end{aligned}$$

(B.10)

where

$$\begin{aligned} \bigg (\sigma _t^{(F)}\bigg )^2&\equiv \sum _{i=1}^3\sum _{l=1}^3\psi _i \psi _l \Sigma _{ii} \Sigma _{ll}\rho _{il}+\sum _{j=1}^{q}\sum _{k=1}^q\psi _j^{(S)}\psi _k^{(S)}\Gamma _{j,k} \sqrt{h_{t,j}^{(S)}h_{t,k}^{(S)}}+h_t^{(F)},\\ \phi _t&\equiv \psi _0 + \sum _{i=1}^3 \psi _i \left( \kappa ^{\mathbb {Q}}_{ii} \theta ^{\mathbb {Q}}_i -\lambda \theta ^{\mathbb {Q}}_3 \mathbb {1}_{ \{ i=2\} } \right) + (\psi _2 \lambda +\psi '_3) X_t^{(3)} - \sum _{j=1}^q \psi _j^{(S)} \left( \dfrac{1}{2}h_{t,j}^{(S)}\right) , \\ \epsilon ^{\mathbb {Q}(F)}_{t+1}&\equiv \dfrac{\sum _{i=1}^p\psi _i\Sigma _{ii} Z_{t+1}^{\mathbb {Q}(i)}+\sum _{j=1}^q\psi _j^{(S)}\sqrt{h_{t,j}^{(S)}}Z_{t+1,j}^{\mathbb {Q}(S)} +\sqrt{h_{t}^{(F)}}Z_{t+1}^{\mathbb {Q}(F)}}{\sigma _t^{(F)}},\\ \log h_{t}^{(F)}&=\omega ^{(F)}+\alpha ^{(F)}(Z_{t}^{\mathbb {Q}(F)}-\lambda _{t-1}^{(F)}) +\gamma ^{(F)}\bigg (|Z_{t}^{\mathbb {Q}(F)}-\lambda _{t-1}^{(F)}|-\dfrac{2}{\sqrt{2\pi }}\bigg ) +\beta ^{(F)} \log h_{t-1}^{(F)},\\ \lambda _{t}^{(F)} \equiv&\dfrac{1}{\sqrt{h_t^{(F)}}} \Bigg [\phi _t+\dfrac{1}{2}\bigg (\sigma _t^{(F)}\bigg )^2\bigg ], \end{aligned}$$

and processes $\{ Z_{t,j}^{\mathbb {Q}(S)} \}^{12n}_{t=1} =\{Z_{t,j}^{(S)}+\lambda _j^{(S)} \}^{12n}_{t=1}$, $j=1,\ldots ,q$ and $\{ Z_{t}^{\mathbb {Q}(F)} \}^{12n}_{t=1} =\{Z_{t}^{(F)}+\lambda _{t-1}^{(F)} \}^{12n}_{t=1}$ are also standard Gaussian white noises under $\mathbb {Q}$ retaining the same dependence structure than their physical measure counterpart.

Two equity indices are considered for the equity model (B.6)–(B.7): the S &P/TSX and the S &P 500. Parameters considered are shown in Table 4 and were obtained from a return time series extending from February 1986 to January 2022.

Table 4 Bivariate EGARCH model parameters

Table 5 indicates parameters used for the mixed fund model (B.8)–(B.9), which were obtained based on a time series of net asset value (NAV) returns for the Assumption/Louisbourg Balanced Fund A extending from February 1996 to January 2022.

Table 5 Mixed fund model parameters

Starting values for interest rate factors and volatilities (both for the equity indices and the mixed fund) considered in simulations correspond to inferred values on January 31, 2022 and are given in Table 6.

Table 6 Initial financial risk factors parameters for simulations

These values are used as a starting point for the simulation of the various trajectories of the interest rates, equity indices and mixed fund value.