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 1year 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 lowinterestrate 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 insurancelinked savings plans with guarantees.
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Notes
 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.
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.
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.
 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 10year contract case.
 6.
 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.
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.
The inequality holds because the policy is solvent at time \(t1\), i.e., \(A_{t1}>P_{t1}\), and the premium \(\Pi _{t1}>0\).
 10.
In fact for \(\frac{g}{\alpha }<\frac{1+g}{\theta _{t1}}1\) we have \( \text {Pr}\left[ R_t^\mathbb {P}<\left( \frac{1+g}{\theta _{t1}}1\right) ;\,R_{t}^\mathbb {P}<\frac{g}{\alpha }\right] =\text {Pr}\left( R_{t}^\mathbb {P}<\frac{g}{\alpha }\right) \) and we rewrite \(g/\alpha<\frac{1+g}{\theta _{t1}}1\iff \theta _{t1}<\frac{1+g}{1+g/\alpha }=\theta ^*\) 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.
In the case where \(\frac{g}{\alpha }>\frac{1+g}{\theta _{t1}}1\) we have \(\text {Pr}\left( \frac{g}{\alpha }<R_t^\mathbb {P}<\frac{1\theta _{t1}}{\theta _{t1}\alpha }\right) =0\) and \(\text {Pr}\left[ R_t^\mathbb {P}<\left( \frac{1+g}{\theta _{t1}}1\right) ;R_{t}^\mathbb {P}<\frac{g}{\alpha }\right] =\text {Pr}\left( R_{t}^\mathbb {P}<\frac{1+g}{\theta _{t1}}1\right) \).
References
 1.
Baloise Group (2015) Baloise sells closed life insurance portfolio in Germany. Media information. https://www.baloise.com/en/home/media/news/2015/baloisesellsclosedlifeinsuranceportfoliogermany.html. Accessed 3 May 2018
 2.
Bernard C, Le Courtois O, QuittardPinon F (2005) Market value of life insurance contracts under stochastic interest rates and default risk. Insur Math Econ 36(3):499–516
 3.
Braun A, Fischer M, Schmeiser H (2015) How to derive optimal guarantee levels in participating life insurance contracts. IVWHSG working paper, University of St. Gallen
 4.
Braun A, Schmeiser H, Schreiber F (2018) Return on riskadjusted capital under solvency ii: implications for the asset management of insurance companies. Geneva Pap Risk Insur Issues Pract 43(3):456–472
 5.
Briys E, de Varenne F (1997) On the risk of insurance liabilities: debunking some common pitfalls. J Risk Insur 64(4):673–694
 6.
Butsic RP (1994) Solvency measurement for propertyliability riskbased capital applications. J Risk Insur 61(4):656–690
 7.
Doherty N, Garven J (1986) Price regulation in propertyliability insurance: a contingentclaims approach. J Fin 41(5):1031–1050
 8.
Eling M, Holder S (2013a) Maximum technical interest rates in life insurance in europe and the united states: an overview and comparison. Geneva Pap Risk Insur Issues Pract 38(2):354–375
 9.
Eling M, Holder S (2013b) The value of interest rate guarantees in participating life insurance contracts: status quo and alternative product design. Insur Math Econ 53(3):491–503
 10.
Eling M, Gatzert N, Schmeiser H (2008) The Swiss Solvency Test and its market implications. Geneva Pap Risk Insur Issues Pract 33(3):418–439
 11.
European Commission (2015) Commission Delegated Decision (EU) 2015/1602. Off J Eur Union L248:95–98
 12.
European Union (1992) Council Directive 92/96/EEC of the European Parliament and of the Council. Off J Eur Communities L360:1–27
 13.
European Union (2002) Directive 2002/83/EC of the European Parliament and of the Council. Off J Eur Communities L345:1–51
 14.
European Union (2009) Directive 2009/138/EC of the European Parliament and of the Council. Off J Eur Union L335:1–155
 15.
European Union (2014) Directive 2014/51/EU of the European Parliament and of the Council. Off J Eur Union L153:1–61
 16.
Federal Assembly of the Swiss Confederation (2004) Insurance Supervision Act (961.01). https://www.admin.ch/opc/de/classifiedcompilation/20022427/index.html. Accessed 3 May 2018
 17.
Gatzert N, Kling A (2007) Analysis of Participating life insurance contracts: a unification approach. J Risk Insur 74(3):547–570
 18.
Gatzert N, Holzmüller I, Schmeiser H (2012) Creating customer value in participating life insurance. J Risk Insur 79(3):645–670
 19.
German Federal Ministry of Justice (2014a) Verordnung über die Mindestbeitragsrückerstattung in der Lebensversicherung (MindZV)
 20.
German Federal Ministry of Justice (2014b) Verordnung über Rechnungsgrundlagen für die Deckungsrückstellungen (DeckRV)
 21.
Grosen A, Jorgensen P (2000) Fair valuation of life insurance liabilities: the impact of interest rate guarantees, surrender options, and bonus policies. Insur Math Econ 26(1):37–57
 22.
Grosen A, Jorgensen P (2002) Life insurance liabilities at market value: an analysis of insolvency risk, bonus policy, and regulatory intervention rules in a barrier option framework. J Risk Insur 69(1):63–91
 23.
International Association of Insurance Supervisors (2016) 2015 Global Insurance Market Report. Technical report
 24.
Killer M (2015) Streit um BVGGelder—Für eine höhere Mindestquote. Neue Zürcher Zeitung 23 January
 25.
Kling A, Richter A, Ruß J (2007a) The impact of surplus distribution on the risk exposure of with profit life insurance policies including interest rate guarantees. J Risk Insur 74(3):571–589
 26.
Kling A, Richter A, Ruß J (2007b) The interaction of guarantees, surplus distribution, and asset allocation in withprofit life insurance policies. Insur Math Econ 40(1):164–178
 27.
Merton RC (1969) Lifetime portfolio selection under uncertainty: the continuoustime case. Rev Econ Stat 51(3):247–257
 28.
Rymaszewski P, Schmeiser H, Wagner J (2012) Under what conditions is an insurance guaranty fund beneficial for policyholders? J Risk Insur 79(3):785–815
 29.
Schmeiser H, Wagner J (2013) The impact of introducing insurance guaranty schemes on pricing and capital structure. J Risk Insur 80(2):273–308
 30.
Schmeiser H, Wagner J (2015) A proposal on how the regulator should set minimum interest rate guarantees in participating life insurance contracts. J Risk Insur 82(3):659–686
 31.
Schmeiser H, Wagner J (2016) What transaction costs are acceptable in life insurance products from the policyholders’ viewpoint? J Risk Fin 17(3):277–294
 32.
Swiss Federal Council (2005) Ordinance on the Supervision of Private Insurance Companies (961.011). https://www.admin.ch/opc/de/classifiedcompilation/20051132/index.html. Accessed 3 May 2018
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Appendix
Appendix
Model implementation
Implementation notes for the model (A): Because no closedform 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.
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.
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.
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.
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.
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 marketpricing 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 \(A_{t1}\), the policyholder account \(P_{t1}\) and the premium \(\Pi _{t1}\) from the previous period \((t1)\), 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}
$$\begin{aligned} \mathbb {A}_t&\iff \{\left( A_{t1}+\Pi _{t1}\right) \cdot (1+R_t^\mathbb {P} )\left( P_{t1}+\Pi _{t1}\right) \cdot (1+\max (g,\alpha R_t^\mathbb {P} ) )<0 \}\\&\iff \{\theta _{t1}\cdot (1+R_t^\mathbb {P} ) (1+\max (g,\alpha R_t^\mathbb {P} ) )<0 \},\text { with }\theta _{t1}=\frac{A_{t1}+\Pi _{t1}}{P_{t1}+\Pi _{t1}}>1^9&\iff \left\{ \begin{array}{llc} \{\theta _{t1}\cdot (1+R_t^\mathbb {P} ) (1+\alpha R_t^\mathbb {P} )<0 \} &{} \text { if } g<\alpha R_t^\mathbb {P} &{} \\ \{\theta _{t1}\cdot (1+R_t^\mathbb {P} ) (1+g )<0 \} &{} \text { otherwise} &{} \\ \end{array}\right. \\&\iff \left\{ \begin{array}{llc} \left\{ R_t^\mathbb {P}<\frac{1\theta _{t1}}{\theta _{t1}\alpha }\right\} &{}\text { if } g/\alpha< R_t^\mathbb {P} &{} \\ \left\{ R_t^\mathbb {P}<\frac{1+g}{\theta _{t1}}1\right\} &{} \text {otherwise} &{}\\ \end{array}\right. \end{aligned}$$Thus, from Eq. (18)^{Footnote 10}^{,}^{Footnote 11},
$$\begin{aligned}&\text {Pr}\left( \mathbb {A}_t\right) =\epsilon _t \\&\quad \iff \text {Pr}\left( R_t^\mathbb {P}<\frac{1\theta _{t1}}{\theta _{t1}\alpha }\,\bigg \,R_t^\mathbb {P}>\frac{g}{\alpha }\right) \cdot \text {Pr}\left( R_t^\mathbb {P}>\frac{g}{\alpha }\right) \\&\qquad +\text {Pr}\left( R_t^\mathbb {P}<\frac{1+g}{\theta _{t1}}1\,\bigg \,R_{t}^\mathbb {P}<\frac{g}{\alpha }\right) \cdot \text {Pr}\left( R_t^\mathbb {P}<\frac{g}{\alpha }\right) =\epsilon _t \\ \end{aligned}$$$$\begin{aligned} \iff&\text {Pr}\left( \frac{g}{\alpha }<R_t^\mathbb {P}<\frac{1\theta _{t1}}{\theta _{t1}\alpha }\right) +\text {Pr}\left[ R_t^\mathbb {P}<\left( \frac{1+g}{\theta _{t1}}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 _{t1}}{\theta _{t1}\alpha }\right) +\text {Pr}\left( R_t^\mathbb {P}<\frac{g}{\alpha }\right) =\epsilon _t &{} \text {if}\theta _{t1}<\theta ^* {^{10}} \\ \text {Pr}\left( R_t^\mathbb {P}<\frac{1+g}{\theta _{t1}}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 _{t1}}{\theta _{t1}\alpha }\right) =\epsilon _t &{} \text { if }\theta _{t1}<\theta ^* &{}\\ \text {Pr}\left( R_t^\mathbb {P}<\frac{1+g}{\theta _{t1}}1\right) =\epsilon _t &{} \text { otherwise} &{}\\ \end{array} \right. \\ \iff&\left\{ \begin{array}{llc} \text {Pr}\left[ W^\mathbb {P}_tW^\mathbb {P}_{t1}<\frac{1}{\sigma _\mathrm{B}}\ln \left( \frac{\frac{1\alpha }{\theta _{t1}\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 _{t1}<\theta ^* &{}\\ \text {Pr}\left[ W^\mathbb {P}_tW^\mathbb {P}_{t1}\le \frac{1}{\sigma _\mathrm{B}} \ln \left( \frac{\frac{1+g}{\theta _{t1}}\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 _{t1}\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 _{t1}<\theta ^* &{}\\ \frac{1}{\sigma _\mathrm{B}} \ln \left( \frac{\frac{1+g}{\theta _{t1}}\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:
$$\begin{aligned} \text {Pr}\left( \mathbb {A}_t\right) =\epsilon _t&\iff \gamma _{t}= \left\{ \begin{array}{llc} \frac{e^{\mu _B\sigma _\mathrm{B}^2/2+\sigma _\mathrm{B}\Phi ^{1}\left( \epsilon _t\right) }\frac{1\alpha }{\theta _{t1}\alpha }}{e^{\mu _B\sigma _\mathrm{B}^2/2+\sigma _\mathrm{B}\Phi ^{1}\left( \epsilon _t\right) }e^{r_\text {f}}} &{}\text { if }\theta _{t1}<\theta ^*, &{} \\ \frac{e^{\mu _B\sigma _\mathrm{B}^2/2+\sigma _\mathrm{B}\Phi ^{1}\left( \epsilon _t\right) }\frac{1+g}{\theta _{t1}}}{e^{\mu _B\sigma _\mathrm{B}^2/2+\sigma _\mathrm{B}\Phi ^{1}\left( \epsilon _t\right) }e^{r_\text {f}}}&{} \text { otherwise.} &{} \\ \end{array} \right. \end{aligned}$$ 
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 \(\hat{\gamma _t}\),
and the asset allocation average of the portfolio during the whole duration of the contract,
Further results for model (A)
Detailed results for model (B)
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Mirza, C., Wagner, J. Policy characteristics and stakeholder returns in participating life insurance: which contracts can lead to a winwin?. Eur. Actuar. J. 8, 291–320 (2018). https://doi.org/10.1007/s1338501801791
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Keywords
 Life insurance products
 Interest rate guarantee
 Policyholder participation
 Return on investment