Policy characteristics and stakeholder returns in participating life insurance: which contracts can lead to a win-win?

Abstract

Participating life insurance contracts and pension plans often include a return guarantee and participation in the surplus of the institution’s result. The final account value in such contracts depends on the investment policy driven by solvency requirements, as well as on the level of market returns, the guarantee and the participation rates. Using a contingent claim model for such contracts, we assume a competitive market with minimum solvency requirements similar to Solvency II. We consider solvency requirements on maturity and 1-year time horizons, as well as contracts with single and periodic premium payments. Through numerical analyses, we link the expected returns for equity holders and policyholders in various situations. Using the return on equity and policyholder internal rate of return along with utility measures, we assess which contract settings optimize the return compromise for both stakeholders in a low-interest-rate environment. Our results extend the academic literature by building on the work by Schmeiser and Wagner (J Risk Insur 82(3):659–686, 2015) and are relevant for practitioners, given the current financial market environment and difficulties in insurance-linked savings plans with guarantees.

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Fig. 1

Data from FINMA (see http://www.finma.ch/en/supervision/insurers/sector-specific-tools/individual-life-insurance) and Swiss National Bank (see data.snb.ch/en)

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Notes

  1. 1.

    In this paper, we focus on the products offered in Switzerland and Germany, where participating life insurance with guarantees is very popular. In Swiss collective life insurance contracts, the interest rate defined by the authorities is to be followed exactly. The lower bound for policyholders’ profit participation holds in German participating life insurance contracts and in Swiss collective life insurance.

  2. 2.

    A short review of insurer defaults linked to interest rate guarantees is provided, for example, in the Introduction in Schmeiser and Wagner [30]. The references cited therein (see their Footnote 2) give more detailed information. Following the introduction of the Solvency II regulation in the European Union, some of the conditions, in particular those linked to the asset allocation have been relaxed and replaced by a more comprehensive risk assessment, see, e.g., Braun et al. [4].

  3. 3.

    In this simple setup, following Schmeiser and Wagner [30], we directly link the investment return to the return credited to the policyholder account. Since in practice smoothing mechanisms for the surplus distribution are in place, our results overestimate the asset volatility. We show that our conceptual findings remain valid by comparing our results with simulations using a much lower asset volatility (compare the results from Tables 4 and 5 in Sect. 4.1 with Tables 8 and 9 reported in the Appendix).

  4. 4.

    We do not consider that a guarantee fund will step in and ensure the interest g, see, e.g. [28, 29].

  5. 5.

    With this consideration, we importantly differentiate our reference setting from Schmeiser and Wagner [30], where a ruin probability of 0.5% is considered for the 10-year contract case.

  6. 6.

    Given that we used continuous compounding for \(r_\mathrm{f}\) and discrete compounding for g (see Eqs. 3, 5), \(r_\mathrm{f}=1.0\%\) does not correspond to the limit point for \(g=1.0\%\). In fact, \(r_\mathrm{f}\) could decrease to \(\log (1+g)\approx 0.995\%\).

  7. 7.

    In their paper, Braun et al. [3] analyze in detail the level of guaranteed interest rate to choose to optimize the policyholders’ utility while keeping the insurance product more valuable than a simple direct investment. The comparison of the utility from insurance contracts and from other investment forms has also been the focus of Schmeiser and Wagner [30, 31]. In this latter study, transaction costs are also taken into account.

  8. 8.

    For the evaluation under the \(\mathbb {Q}\)-measure, \(\mu _\mathrm{B}\) is replaced by \(r_\mathrm{f}\) in Eq. (4), defining \(R_t^\mathbb {Q}\).

  9. 9.

    The inequality holds because the policy is solvent at time \(t-1\), i.e., \(A_{t-1}>P_{t-1}\), and the premium \(\Pi _{t-1}>0\).

  10. 10.

    In fact for \(\frac{g}{\alpha }<\frac{1+g}{\theta _{t-1}}-1\) we have \( \text {Pr}\left[ R_t^\mathbb {P}<\left( \frac{1+g}{\theta _{t-1}}-1\right) ;\,R_{t}^\mathbb {P}<\frac{g}{\alpha }\right] =\text {Pr}\left( R_{t}^\mathbb {P}<\frac{g}{\alpha }\right) \) and we rewrite \(g/\alpha<\frac{1+g}{\theta _{t-1}}-1\iff \theta _{t-1}<\frac{1+g}{1+g/\alpha }=\theta ^*\) where we need \((1+g/\alpha )>0\). In our application with \(\alpha \) close to 100% and g close to zero this condition is fulfilled.

  11. 11.

    In the case where \(\frac{g}{\alpha }>\frac{1+g}{\theta _{t-1}}-1\) we have \(\text {Pr}\left( \frac{g}{\alpha }<R_t^\mathbb {P}<\frac{1-\theta _{t-1}}{\theta _{t-1}-\alpha }\right) =0\) and \(\text {Pr}\left[ R_t^\mathbb {P}<\left( \frac{1+g}{\theta _{t-1}}-1\right) ;R_{t}^\mathbb {P}<\frac{g}{\alpha }\right] =\text {Pr}\left( R_{t}^\mathbb {P}<\frac{1+g}{\theta _{t-1}}-1\right) \).

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Correspondence to Joël Wagner.

Appendix

Appendix

Model implementation

Implementation notes for the model (A): Because no closed-form solutions are available for the policyholder payoff \(L_T\) (Eq. 7) and the equity holder stake \(E_T\) (Eq. 8), we use the Monte Carlo simulation technique. In both models (A) and (B), we generate \(N=100\,000\) different realizations over the period of T years. The iterative calculations at times t of \(A_t\) and \(P_t\) are straightforward. The main challenge is the calculation of the required equity capital \(E_0^*\) and the asset allocation \(\gamma ^*\). For deriving both quantities, we make use of the competitive market constraint (Eq. 13) and the solvency requirement (Eq. 15). The implementation of the algorithm follows the following steps:

  1. 1.

    Generate \(N \times T\) independent identically distributed (iid) random variables (rv) from a standard normal distribution \(W_t^i \sim N(0,1) \), where \(i = 1,2,\ldots ,N\), and \(t = 1,2,\ldots ,T\).

  2. 2.

    For an initial asset allocation \(\gamma \in (0,1)\), calculate \(A_t^{i,\mathbb {P}}(\gamma )\) and \(P_t^{i,\mathbb {P}}(\gamma )\) (cf. Eqs. 3 and 5, respectively) and \(A_t^{i,\mathbb {Q}}(\gamma )\) and \(P_t^{i,\mathbb {Q}}(\gamma )\) at each time t and for each scenario i under the \(\mathbb {P}\)- and \(\mathbb {Q}\)-measures, respectively.Footnote 8

  3. 3.

    Define the capital needed \(E_0^i(\gamma )\) such that \(A_T^{i,{\mathbb {P}}}(\gamma )-P_T^{i,{\mathbb {P}}}(\gamma )=0\) in each scenario i. Order the obtained \(E_0^i(\gamma )\) such that, \(E_0^{i:N}(\gamma ) \ge E_0^{(i+1):N}(\gamma )\) for any \(i\in \{1,2,\cdots ,N\}\). Given the upper bound for the probability of ruin \(\epsilon _T\) define the function \(E_0(\gamma )\) such that \(E_0(\gamma )=E_0^{\lceil \epsilon _T\cdot N\rceil :N}(\gamma )\).

  4. 4.

    Using \(E_0(\gamma )\) for the initial equity capital in \(A_t^{i,\mathbb {Q}}(\gamma )\), introduce the Monte Carlo approximation of the policyholder contract’s net present value under the \(\mathbb {Q}\)-measure

    $$\begin{aligned} \widehat{NPV}(\gamma )=\frac{1}{N}\sum _{i=1}^{N} [P_T^{i,\mathbb {Q}}(\gamma )-\max (P^{i,\mathbb {Q}}_T(\gamma )-A^{i,\mathbb {Q}}_T(\gamma ),0 ) ]\cdot e^{-r_\text {f}\cdot T}-P_0. \end{aligned}$$
  5. 5.

    Numerically find \(\gamma ^*\in (0,1)\) as the solution of \(\widehat{NPV}(\gamma ^*)=0\) and set \(E_0^*=E_0(\gamma ^*)\) using the above. The optimal insurer position is given by the pair \( (E_0^*, \gamma ^* )\).

Implementation notes for the model (B): In the numerical implementation, we will make use of the solvency requirement (18) and competitive market-pricing constraint (12).

  • First, we derive a formula to calculate the asset allocation \(\gamma _t\) from (18), where \(\gamma _t\) at time t is a function of the assets \(A_{t-1}\), the policyholder account \(P_{t-1}\) and the premium \(\Pi _{t-1}\) from the previous period \((t-1)\), as well as the given contract parameters (g and \(\epsilon _1\)) and the given available investments (parameters \(r_\text {f}\), \(\mu _B\), and \(\sigma _\mathrm{B}\)). Consider the event \(\mathbb {A}_t=\left\{ A_{t}-P_{t}<0\right\} \):Footnote 9

    $$\begin{aligned} \mathbb {A}_t&\iff \{\left( A_{t-1}+\Pi _{t-1}\right) \cdot (1+R_t^\mathbb {P} )-\left( P_{t-1}+\Pi _{t-1}\right) \cdot (1+\max (g,\alpha R_t^\mathbb {P} ) )<0 \}\\&\iff \{\theta _{t-1}\cdot (1+R_t^\mathbb {P} )- (1+\max (g,\alpha R_t^\mathbb {P} ) )<0 \},\text { with }\theta _{t-1}=\frac{A_{t-1}+\Pi _{t-1}}{P_{t-1}+\Pi _{t-1}}>1^9&\iff \left\{ \begin{array}{llc} \{\theta _{t-1}\cdot (1+R_t^\mathbb {P} )- (1+\alpha R_t^\mathbb {P} )<0 \} &{} \text { if } g<\alpha R_t^\mathbb {P} &{} \\ \{\theta _{t-1}\cdot (1+R_t^\mathbb {P} )- (1+g )<0 \} &{} \text { otherwise} &{} \\ \end{array}\right. \\&\iff \left\{ \begin{array}{llc} \left\{ R_t^\mathbb {P}<\frac{1-\theta _{t-1}}{\theta _{t-1}-\alpha }\right\} &{}\text { if } g/\alpha< R_t^\mathbb {P} &{} \\ \left\{ R_t^\mathbb {P}<\frac{1+g}{\theta _{t-1}}-1\right\} &{} \text {otherwise} &{}\\ \end{array}\right. \end{aligned}$$

    Thus, from Eq. (18)Footnote 10,Footnote 11,

    $$\begin{aligned}&\text {Pr}\left( \mathbb {A}_t\right) =\epsilon _t \\&\quad \iff \text {Pr}\left( R_t^\mathbb {P}<\frac{1-\theta _{t-1}}{\theta _{t-1}-\alpha }\,\bigg |\,R_t^\mathbb {P}>\frac{g}{\alpha }\right) \cdot \text {Pr}\left( R_t^\mathbb {P}>\frac{g}{\alpha }\right) \\&\qquad +\text {Pr}\left( R_t^\mathbb {P}<\frac{1+g}{\theta _{t-1}}-1\,\bigg |\,R_{t}^\mathbb {P}<\frac{g}{\alpha }\right) \cdot \text {Pr}\left( R_t^\mathbb {P}<\frac{g}{\alpha }\right) =\epsilon _t \\ \end{aligned}$$
    $$\begin{aligned} \iff&\text {Pr}\left( \frac{g}{\alpha }<R_t^\mathbb {P}<\frac{1-\theta _{t-1}}{\theta _{t-1}-\alpha }\right) +\text {Pr}\left[ R_t^\mathbb {P}<\left( \frac{1+g}{\theta _{t-1}}-1\right) ;\,R_{t}^\mathbb {P}<\frac{g}{\alpha }\right] =\epsilon _t \\ \iff&\left\{ \begin{array}{llc} \text {Pr}\left( \frac{g}{\alpha }<R_t^\mathbb {P}<\frac{1-\theta _{t-1}}{\theta _{t-1}-\alpha }\right) +\text {Pr}\left( R_t^\mathbb {P}<\frac{g}{\alpha }\right) =\epsilon _t &{} \text {if}\theta _{t-1}<\theta ^* {^{10}} \\ \text {Pr}\left( R_t^\mathbb {P}<\frac{1+g}{\theta _{t-1}}-1\right) =\epsilon _t &{} \text { otherwise}^{11} \end{array} \right. \\ \iff&\left\{ \begin{array}{llc} \text {Pr}\left( R_t^\mathbb {P}<\frac{1-\theta _{t-1}}{\theta _{t-1}-\alpha }\right) =\epsilon _t &{} \text { if }\theta _{t-1}<\theta ^* &{}\\ \text {Pr}\left( R_t^\mathbb {P}<\frac{1+g}{\theta _{t-1}}-1\right) =\epsilon _t &{} \text { otherwise} &{}\\ \end{array} \right. \\ \iff&\left\{ \begin{array}{llc} \text {Pr}\left[ W^\mathbb {P}_t-W^\mathbb {P}_{t-1}<\frac{1}{\sigma _\mathrm{B}}\ln \left( \frac{\frac{1-\alpha }{\theta _{t-1}-\alpha }-\gamma _t e^{r_\text {f}}}{\left( 1-\gamma _t\right) e^{\mu _B-\sigma _\mathrm{B}^2/2}}\right) \right] =\epsilon _t &{} \text { if }\theta _{t-1}<\theta ^* &{}\\ \text {Pr}\left[ W^\mathbb {P}_t-W^\mathbb {P}_{t-1}\le \frac{1}{\sigma _\mathrm{B}} \ln \left( \frac{\frac{1+g}{\theta _{t-1}}-\gamma _{t} e^{r_\text {f}}}{\left( 1-\gamma _{t}\right) e^{\mu _B-\sigma _\mathrm{B}^2/2}}\right) \right] =\epsilon _t&{} \text { otherwise} &{}\\ \end{array}\right. \\ \iff&\left\{ \begin{array}{llc} \frac{1}{\sigma _\mathrm{B}}\ln \left( \frac{\frac{1-\alpha }{\theta _{t-1}-\alpha }-\gamma _t e^{r_\text {f}}}{\left( 1-\gamma _t\right) e^{\mu _B-\sigma _\mathrm{B}^2/2}}\right) =\Phi ^{-1}\left( \epsilon _t\right) &{} \text { if }\theta _{t-1}<\theta ^* &{}\\ \frac{1}{\sigma _\mathrm{B}} \ln \left( \frac{\frac{1+g}{\theta _{t-1}}-\gamma _{t} e^{r_\text {f}}}{\left( 1-\gamma _{t}\right) e^{\mu _B-\sigma _\mathrm{B}^2/2}}\right) =\Phi ^{-1}\left( \epsilon _t\right) &{} \text { otherwise} &{}\\ \end{array} \right. \end{aligned}$$

    with \(\Phi \) the cumulative normal distribution function. Hence, the asset allocation \(\gamma _t\) at time t is given by:

    $$\begin{aligned} \text {Pr}\left( \mathbb {A}_t\right) =\epsilon _t&\iff \gamma _{t}= \left\{ \begin{array}{llc} \frac{e^{\mu _B-\sigma _\mathrm{B}^2/2+\sigma _\mathrm{B}\Phi ^{-1}\left( \epsilon _t\right) }-\frac{1-\alpha }{\theta _{t-1}-\alpha }}{e^{\mu _B-\sigma _\mathrm{B}^2/2+\sigma _\mathrm{B}\Phi ^{-1}\left( \epsilon _t\right) }-e^{r_\text {f}}} &{}\text { if }\theta _{t-1}<\theta ^*, &{} \\ \frac{e^{\mu _B-\sigma _\mathrm{B}^2/2+\sigma _\mathrm{B}\Phi ^{-1}\left( \epsilon _t\right) }-\frac{1+g}{\theta _{t-1}}}{e^{\mu _B-\sigma _\mathrm{B}^2/2+\sigma _\mathrm{B}\Phi ^{-1}\left( \epsilon _t\right) }-e^{r_\text {f}}}&{} \text { otherwise.} &{} \\ \end{array} \right. \end{aligned}$$
  • Following the implementation of model (A), define the yearly asset allocations \(\gamma _t^i\) in each scenario i (see above). Under these allocations, consider the Monte Carlo estimate of the equity holder’s net present value

    $$\begin{aligned} \widehat{NPV}(E_0)=\frac{1}{N}\sum _{i=1}^{N}\max [A_T^{i^{\mathbb {Q}}}\left( E_0\right) -P_T^{i^{\mathbb {Q}}}\left( E_0\right) ,0 ] e^{-r_\text {f}\cdot T}-E_0, \end{aligned}$$

    where we use \(E_0=E_0(\{\gamma _t^i\})\). The numerical solution of the optimal equity capital \(E_0^*>0\) from (12) comes from \(\widehat{NPV}\left( E_0^*\right) =0\). For calculating \(E_0^*\) define a recursive formula \(E_{0}^{(j)}\) of the expected equity capital at maturity T discounted at \(t=0\),

    $$\begin{aligned} E_{0}^{(j+1)}=\frac{1}{N}\sum _{i=1}^{N}\max [A_T^{i^{\mathbb {Q}}} (E_{0}^{(j)} )-P_T^{i^{\mathbb {Q}}} (E_{0}^{(j)} ),0 ] e^{-r_\text {f}\cdot T}. \end{aligned}$$

    For any initial \( E_0^{(0)}\in \mathbb {R}\) there exists a k such that, \( E_{0}^{(k)}=E_{0}^{(k+1)}=E_0^*\).

In the presentation of our results, we will make use of the following notations. We introduce the yearly average of asset allocation \(\hat{\gamma _t}\),

$$\begin{aligned} \hat{\gamma }_t=\frac{1}{N}\sum _{i=1}^{N}\gamma _{t,i}, \end{aligned}$$
(23)

and the asset allocation average of the portfolio during the whole duration of the contract,

$$\begin{aligned} \bar{\gamma }=\frac{1}{T}\sum _{t=1}^{T}\hat{\gamma }_t. \end{aligned}$$
(24)

Further results for model (A)

See Tables 8 and 9.

Table 8 Model (A)—variation of the interest rate guarantee with volatility \(\sigma _\mathrm{B}/2\)
Table 9 Model (A)—variation of the policyholder participation rate with volatility \(\sigma _\mathrm{B}/2\)

Detailed results for model (B)

See Tables 10, 11 and 12, 12 and 13.

Table 10 Results from the sensitivity analysis in model (B) under variation of the risk-free interest rate, see Fig. 10a, b
Table 11 Results from the sensitivity analysis in model (B) under variation of the interest rate guarantee, see Fig. 10c, d
Table 12 Results from the sensitivity analysis in model (B) under variation of the policyholder participation rate, see Fig. 10e, f
Table 13 Results from the sensitivity analysis in model (B) under variation of the safety level, see Fig. 10g, h

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Mirza, C., Wagner, J. Policy characteristics and stakeholder returns in participating life insurance: which contracts can lead to a win-win?. Eur. Actuar. J. 8, 291–320 (2018). https://doi.org/10.1007/s13385-018-0179-1

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Keywords

  • Life insurance products
  • Interest rate guarantee
  • Policyholder participation
  • Return on investment