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The article explores the valuation of multiple premium payment options in participating life insurance contracts, highlighting the importance of interest rate risk and optimal exercise strategies. It introduces a general contract framework and discusses different exercise assumptions. The study employs an extended version of the Least Square Monte Carlo (LSMC) approach to value these options, showing that the price for multiple premium-payment options can be substantial and pose significant risk to life insurance companies. The article also demonstrates that introducing fees for exercising these options does not eliminate their positive value, and that policyholders' exercise behavior can significantly impact the valuation. The numerical results emphasize the impact of interest rate volatility on option values and the necessity for insurers to charge extra for these options to finance adequate risk management measures.
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Abstract
Participating life insurance contracts with investment guarantees typically possess various embedded options. In this paper, we focus on common options with early exercise features, i.e., paid-up options, resumption options, surrender options and the combinations among them. The valuation of these options depends strongly on the exercise strategy applied by the policyholder. We extend the Least Square Monte Carlo (LSMC) method to approximate an optimal exercise strategy and to determine multiple optimal exercise points while incorporating two stochastic sources (asset and interest rate risk). In addition, these findings are compared to results based on two other exercise assumptions. In contrast to earlier studies on this topic, we show in our model setup that premium-payment options can pose a severe risk for life insurers if not priced properly and hence are of great value for policyholders if used strategically. Moreover, we demonstrate that introducing fees for exercising premium-payment options only slightly reduces the option price necessary to finance adequate risk management measures.
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1 Introduction
Participating life insurance contracts with investment guarantees are typically offered with several embedded options. This paper focuses on premium-payment options with early exercise features found in essentially all life insurance contracts with investment guarantees. These options are provided in various forms: A paid-up option allows policyholders to stop premium payments while the main contract continues with adjusted benefits. A resumption option permits policyholders to resume payments after the paid-up option has been exercised (benefits again will be adjusted accordingly). With a surrender option, policyholders can terminate their contracts and receive a surrender amount. With a combined paid-up and surrender option, policyholders may surrender their policy with or without previously exercising the paid-up option. The combination of all three of these options allows policyholders to exercise paid-up, resumption, and surrender options during the contract period.
Popular all over Europe, participating life contracts typically include a cliquet-style option, i.e., a guaranteed yearly interest rate together with a profit sharing scheme (for an overview cf. Mahayni et al. 2021). With this feature, the surrender amount and the adjusted benefit are predetermined. These predetermined values may exceed actual market values. Hence, the premium-payment options may come into money and can be of great value if policyholders use them strategically. In the low interest rate environment, which will continue through the end of 2021, insurers have been particularly challenged with high long-term interest guarantees, which they previously provided to their policyholders. The situation for the insurers can be even more problematic, as policyholders tend to exercise their surrender or paid-up options once the interest rate rebounds (cf. Feodoria and Förstemann 2015). Specifically, the exercise behaviour in respect to premium-payment options and, hence, these option values, depend on future interest rate developments. If the options are not priced adequately and hence no proper risk management has taken place, insurance companies may encounter severe difficulties (cf. the cases of Equitable Life in 2000 or the Hartford in 2009). Current solvency regulation schemes, such as Solvency II in the EU, require insurers to consider lapse risk and offer proper risk management and equity capital for options provided to their customers. Proper models for the premium-payment option valuation and the related risk assessment are thus essential for life insurance companies and should be conducted with care.
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Besides the insurers’ perspective, the aim of this paper is to provide a general framework to policyholders in order to give them the opportunity to use the contracts’ embedded options in an optimal way. For instance, the proposed model could be installed on-line and simply used by inserting individual contract data (e.g. premiums, time to maturity, level of interest rate guarantee, and exercising fees).1 As we show in this paper, the present value of premium payment options in participating life insurance contracts with cliquet-style investment guarantees can easily take up to \(5\%\) of the present value of all policyholders’ premium payments made in the treaty. If insurance companies base their pricing on the option pricing theory and policyholders do not exercise their premium payment options in an optimal manner, a severe wealth transfer takes part to the disadvantages of the insured. This aspect becomes of large general interest taking into account the premium volumes of this kind of products: According to Insurance Europe, more than \(50\%\) of the gross premiums written in 2023 (606.20 billion US Dollar) within the states of the European Union are invested in products of the type focused in this paper2. Hence, even a small percentage of the premium volume caused by a suboptimal exercise strategy sums up to a large amount of money per year lost for old-age provision. The empirical research on policyholders’ option exercising behaviours strongly supports the assumption that premium payment options are often used in a suboptimal manner.3
To the best of our knowledge, this paper is the first to provide a valuation of multiple premium-payment options with two stochastic risk sources (assets and interest rates). To solve the underlying optimal stopping problems, we develop an extended version of the Least Square Monte Carlo (LSMC) approach. In contrast to an earlier paper on this topic (cf. Schmeiser and Wagner 2011) with deterministic term structure, our paper shows that the price for multiple premium-payment options is substantial and can cause a severe risk for life insurance companies. However, additional options possess rather little values, regardless of the underlying market risk: That is, a triple combined paid-up, resumption, and surrender option has almost the same value as a double combined paid-up and surrender option, while this double option values only little more than a single surrender option.4 This surrender option value enlarges strongly with increasing interest rate risk under the optimal-exercise assumption as firstly concluded in Zaglauer and Bauer (2008). We further demonstrate that introducing fees for exercising premium-payment options—as it is commonly done in insurance practice—only slightly reduces the option price necessary to finance adequate risk management measures. This finding stands in contrast to Schmeiser and Wagner (2011) for a deterministic term structure. In the setting of Schmeiser and Wagner (2011), even low fees can eliminate the risk for the insurer and hence no additional payments from the policyholders’ side are necessary for the embedded premium-payment options. Hence, in a realistic setting with stochastic interest rates, a state dependent fee structure as proposed by MacKay et al. (2017) does not eliminate the positive value of premium payment options.
It can be argued that, in real life, policyholders cannot easily apply an optimal stopping strategy because of severe market frictions (such as taxation or informational barriers). In such a case, we show that these options value little and can be even negative if policyholders exercise options without certain optimal strategies. However, a systematic change in the market efficiency and policyholders’ behaviour may take place in the future. Hence, insurers should consider an optimal behaviour from the policyholders’ perspective for risk management purposes.
The remainder of this paper is set out as follows: Sect. 2 discusses the relevant literature. A general contract framework is introduced in Sect. 3. Section 4 puts forth different exercise assumptions in respect to the evaluation of premium-payment options. Section 5 provides numerical results. Finally, Sect. 6 concludes the paper with the central economic implications of our findings.
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2 Related literature
Most of the literature dealing with premium-payment options in participating life insurance contracts with investment guarantees centres on a single surrender option with the assumption that one single premium payment is made up front. Exceptions in the context of the valuation of variable annuities are given by Chi and Lin (2012) and Bernard et al. (2017). Life insurance policies without surrender options are classed as European options, while the insurance policies with surrender options embedded are classed as Bermudan or American options (cf. Grosen and Jørgensen 1997). Surrender options can therefore be valued as the difference between the Bermudan/American option and the European option. To value this option, Bacinello (2005) apply the Recursive Binomial-Tree approach discussed in Cox et al. (1979). First suggested by Longstaff and Schwartz (2001), the Least Square Monte Carlo method (LSMC) is another approach with which to value American options. Bacinello (2008) and Bacinello et al. (2009) have applied this method for the life insurance case. Andreatta and Corradin (2003) compare the Recursive Binomial-Tree approach with the LSMC method and conclude that these two approaches are similarly accurate, while the LSMC can be better applied for a high-dimensional derivative valuation. Bauer et al. (2010) build a general model and compare these numerical valuation approaches. Again, the LSMC was found to be superior because of its efficiency.
Paid-up and resumption options do not exist under the single premium payment assumption. However, life insurance contracts, typically include multiple premium payments during the contract period with multi-period dynamic features. Modelling a multiple premium payment contract, Kling et al. (2006), Gatzert and Schmeiser (2008), and Schmeiser and Wagner (2011) value both single and combined premium-payment options. With geometric Brownian motion for assets and a deterministic interest rate, these studies base their first step on a fair pricing concept: the net present value (NPV) for insurance contracts without premium-payment options must be zero. In the second step, option values are computed as the NPV of the contracts with premium-payment options. In particular, if the options are exercised at their maximum level, the provider may face severe risk (cf. Gatzert and Schmeiser 2008). However, as Kling et al. (2006), Gatzert and Schmeiser (2008), and Reuß et al. (2016) point out, this strategy is not feasible from the policyholders’ viewpoint because a perfect forecast would be necessary. Nevertheless, an assessment on this basis can be interpreted as an upper bound of the option value. Schmeiser and Wagner (2011) value premium-payment options with an optimal stopping strategy first proposed by Andersen (1999). It is showed that the value of premium-payment options is fairly small. The existing literature related to combined premium-payment options assumes a deterministic interest rate. However, this assumption of a constant interest rate is not realistic if the contract’s duration is not short. Life insurance contracts have typically long maturity and the empirical findings confirm the influence of interest rate on policyholders’ option exercising behaviours (cf. Russell et al. 2013 and Kuo et al. 2003 for the US market, Kiesenbauer 2012 for the German market and Kim 2005 for the Korean market).
Comparing LSMC and the optimal stopping strategy, LSMC is slightly biased downwards (cf. Douady 2002). However, the optimal stopping strategy is only feasible when one risk source is considered. Moreover, the LSMC method is fairly efficient, solving the technical challenges which Schmeiser and Wagner (2011) encounter. Thus, the combinations of three or even more options can be handled with this method. Except for the surrender option, which most of the literature focuses on, the payoffs of paid-up and resumption options involve a series of uncertain future cash flows. These payoff schemes and multiple exercise points for combined options make the valuation challenging. Hence, multi-factor models with a closed-form solution, such as those offered by Peterson et al. (2003) cannot be applied in general. The original LSMC method aims to find one optimal exercising point for the option triggering a known cash flow as the case for surrender options in insurance life contracts. In order to value other types of premium-payment options, we develop an adjusted LSMC method, with which policyholders make exercise decisions based on two or more conditional expected values. This adjusted LSMC method allows even more stochastic features (e.g., mortality) and produces multiple optimal exercise decision points.5
3 The model framework
Basic life insurance endowment contracts with a cliquet-style option include two standard features: a guaranteed yearly interest rate (g) and a surplus participation with participation rate (\(\alpha \)) (cf. Grosen and Jørgensen 2000). This basic contract is then extended with different premium-payment options (cf. Schmeiser and Wagner 2011 and Gatzert and Schmeiser 2008). These premium-payment options are assumed to be exercised only at the end of each year, given that the main basic contracts are still in force (i.e., at the end of each contract year, policyholders are alive and the relevant options can still be exercised). Once the option is exercised, the benefit will be adjusted accordingly. We assume that the insurer faces no default risk and hence legitimate payments to the policyholders can always be achieved.6
3.1 Basic contract
A basic life insurance endowment contract lasts T years. Let \(_{t}p_{x}\) be the probability that a policyholder aged x years survives the next t years (with \(t=0...T\) and \(_0p_x=1,\) while \( q_{x}=1-{\,} _{1}p_{x}\) represents the death probability over the next year. Following general actuarial practice, we assume that mortality risk is uncorrelated to financial risk sources and hence is fully diversifiable (cf. Biffis et al. 2010).
Annual premium payments, \(B_t\) for \(t=0...T-1\), are paid by a policyholder at the beginning of year if the policyholder is alive then. Without premium-payment options, premium payments are constant in time, i.e., \(B_t = B^*\). The present value (PV) of premium payments can be written as \(B^*\sum _{t=0}^{T-1}{\,}_{t}p_{x}(1+r^*)^{-t} \), where \(r^*\) is the technical discount rate. Annual premiums are accumulated in the policy account, \(A_t\) at the end of year t for the benefit distribution. The investment return of this accumulated account value includes the guaranteed interest rate g and the surplus participation.
If the policyholder dies during year \(t+1\), a death benefit \(\gamma _{t+1}\) is payable at the end of year \(t+1\) for \(t=0...T-1\). Without premium-payment options exercised, the death benefits are constant, i.e., \(\gamma _{t+1}=\gamma ^*\). If the policyholder survives the whole contract period, the survival benefit is paid. This survival benefit is the policy account \(A_T\), guaranteed with the minimum amount \(\gamma ^*\). Hence, the PV of the benefit payment plus administration costs c (\(c>0\)) can be written as \(\gamma ^*\cdot \left( \sum _{t=0}^{T-1} {\,}_{t}p _{x}q_{x+t} \left( 1+r^* \right) ^{-(t+1)}+A_T{\,}_{T}p_{x}(1+r^*)^{-T}\right) +c\).
According to the actuarial equivalence principle, the PV of the premium payments and that of the death and survival benefits (including the contract’s administration costs) should be identical. As the benefit is guaranteed, the interest guaranteed rate g is used as the technical discount rate \(r^*\) and \(A_T\) equals \(\gamma ^*\) (guaranteed). \(\gamma ^*\), the benefit guaranteed, can then be derived from a fixed premium payment amount \(B^*\). The relationship between \(B^*\) and \(\gamma ^*\) is shown via the following equation:
The guaranteed rate is considered as the lower bound of the contract’s interest rate. With a participating scheme, policyholders receive a surplus return whenever the insurer’s asset return exceeds the guaranteed rate. This surplus is retained in the policy account.7
An annual premium payment can be separated into two parts: \(B_{t}^{R}\) and \(B_{t}^A \). \(B_{t}^{R}\) as \(q_{x+t}\max (\gamma _{t+1}-A_{t},0)\), the annual term life premium, is used to pay the expected difference between the death benefits and the policy account accumulated by the end of year t. The remainder, \(B_{t}^A\), serves as the savings premium:
At the beginning of year \(t+1\) for \(t=0...T-1\), the accumulated policy account contains two parts: the accumulated amount at the end of the previous year, \(A_{t}\), and the annual savings premium, \(B_{t}^A\), on the condition that the policyholder is alive at the end of year t. This policy account earns an annual return at the guaranteed interest rate or surplus participation—whichever is greater. The annual participation rate \(\alpha \) is a fraction of the annual insurer’s investment return in year \(t+1,\) given by \(S_{t+1}/S_t-1, \) with \(S_t\) denoting the value of the insurers’ asset at the end of year t with \(S_t > 0\).
The development of the policy account over time can be formally written as:
The policy account is subject to investment risk, including two risk sources: interest (spot) rate risk and asset risk. We assume that the interest rate can be approximated by \(r_t\), which evolves according to the one-factor Vasicek model (cf. Vasicek 1977). Under the risk-neutral measure \({\mathbb {Q}}\), the interest rate process is given by:
Thereby, \(\lambda \) denotes the (constant) market price of risk, \(\sigma _I\) determines how much randomness \(Z^{\mathbb {Q}}\) is acquired, whereas \(\kappa \) and \(\theta \) are positive constants representing the speed of reversion and the long-term mean. \(Z^{\mathbb {Q}}\) is a Wiener process on a probability space \((\Omega ,\phi ,{\mathbb {Q}})\). The one-period interest rate (with \(dt \simeq \Delta t= 1\)) under the Vasicek model for the risk-neutral measure \({\mathbb {Q}}\), can be approximated by cf. Rabinovitch (1989):
We assume that the return of the assets \(S_{t+1}/S_t\) (with \(t = 0\ldots T-1\)) in the policy account under the risk neutral distribution \({\mathbb {Q}}\) follows a geometric Brownian motion. Under \({\mathbb {Q}}\), the drift for the geometric Brownian motion is given by the stochastic interest rate modeled in Eq. (6). The volatility of the asset process is denoted by \(\sigma _A\) and \(\rho \) indicates the correlation coefficient between the (stochastic) asset development and the (stochastic) interest rate. With W denoting a second Wiener process, we have
The net present value (NPV) of the basic insurance contract is the PV difference between two cash flows: the money paid to the policyholder (or the policy’s beneficiary), and the premiums paid by the policyholder to the insurer. The NPV of the basic contract, denoted by \(\vartheta ^\emptyset \), can be formalised as:
We call a contract fair whenever its NPV is zero (i.e., \(\vartheta ^\emptyset =0\))8. For different parameters, we derive their respective participation rate, \( \alpha \) (with \(0\le \alpha \le 1\)), which leads to a fair condition. The following table summarizes the notations for the basic contracts.
Table 1
Summary of notation in basic contract and the premium-payment options
g
Guaranteed interest rate
\({{\hat{r}}}_t\)
Annual interest rate for year t (\(r_t\simeq {{\hat{r}}}_{t}\))
\(\alpha \)
Participation rate ( \(0\le \alpha \le 1\))
c
Administration costs ( \(c>0\))
\(B_{t}\)
Annual premium payment, paid at the beginning of each year \(t+1\) (\(B_{t}=B^A_{t}+B^R_{t}\))
\(B^R_{t}\)
Annual term life premium
\(B^A_{t}\)
Annual savings premium
\(B^*\)
Constant annual premium payment (for a contract without premium-payment options)
\(\gamma _{t+1}\)
Death benefit paid at the end of year \(t+1\)
\(\gamma ^*\)
Constant death benefit (for a contract without premium-payment options)
\(A_t\)
Policy account value at the end of year t
\(\vartheta ^\emptyset \)
NPV of a basic contract without any premium-payment options
3.2 Modelling premium-payment options
With the investment guaranteed rate and the surplus participation included in the contract, premium-payment options have certain values given two stochastic sources (asset and interest rate risks) in the policy. This option value can be positive or negative depending on the market condition at the exercise points.9
The present value of premium-payment options is computed as the NPV of the contract with the premium-payment options. Since the NPV of the basic contract (without any premium-payment options) is zero, the premium-payment option’s value is regarded as the additional value at the beginning of the contract (\(t=0\)).
The premium-payment options considered in this paper are:
1.
\(\vartheta ^P_\tau \), PV of a single paid-up option with the paid-up option exercised at the end of year \(\tau \), where \(0<\tau \le T\): When a paid-up option is exercised, policyholders stop premium payments while the contract continues with adjusted benefits \(\gamma ^P_\tau \).
2.
\(\vartheta ^{PR}_{\tau ,\nu }\), PV of a combined paid-up and resumption option with the paid-up option exercised at the end of year \(\tau \) and the resumption option at the end of year \(\nu \), where \(0<\tau <\nu \le T\), or \(\tau =\nu =T\): Compared to the single paid-up option, the extra resumption option allows policyholders to resume the premium payments after exercising the paid-up option. Again, the benefit will then be adjusted to \(\gamma ^R_{\tau ,\nu }\).
3.
\(\vartheta ^{S}_{\xi }\), PV of a single surrender option with the surrender option exercised at the end of year \(\xi \), where \(0<\xi \le T\): This surrender option allows policyholders to terminate the policy and receive a surrender amount.
4.
\(\vartheta ^{PS}_{\tau ,\xi }\): PV of a combined paid-up and surrender option with the paid-up option exercised at the end of year \(\tau \) and the surrender option at the end of year \(\xi \), where either \(0<\xi \le \tau = T\) or \(0<\tau <\xi \le T\). With this combined option, policyholders can first exercise the paid-up option and then terminate the contract with the surrender option.
5.
\(\vartheta ^{PRS}_{\tau ,\nu ,\xi }\): PV of a combined paid-up, resumption, and surrender option with the paid-up option exercised at the end of year \(\tau \), the resumption option at the end of year \(\nu \), and the surrender option at the end of year \(\xi \), where \(\tau \), \(\nu \), and \(\xi \) fall in one of the three conditions:
(1)
\(0<\tau<\nu <\xi \le T\);
(2)
\(0<\tau <\xi \le T\) and \(\nu =T\);
(3)
\(0<\xi <T\) and \(\tau =\nu =T\).
All of these options permit policyholders to change their premium payments in different ways.
The NPV of the contract with or without premium-payment options can be generalised as:
Thereby, \(0<\tau \le T\), \(0<\nu \le T\), and \(0<\xi \le T\) denote the exercise points for a paid-up option, a resumption option and a surrender option respectively. If none of these options are exercised, \(\tau =\nu =\xi =T\). A fee, denoted by Fee, with \(0\le Fee<1\), is charged as a percentage of the policy account when any of the options is exercised to cover, e.g., the administration costs of the insurer. With Fee considered, the policy account value becomes \({{\hat{A}}}_t\). \({{\hat{A}}}_t=A_t, \) for \(t<\min \{ \nu , \tau , \xi \}\) and \({{\hat{A}}}_{t+1}=\{{{\hat{A}}}_{t}+{\,}_tp_xB_t^A(\max (g,\alpha (S_{t+1}/S_t-1)+1)\) for \(t>\min \{ \nu , \tau , \xi \}\) and \(t\not \in \{ \nu , \tau , \xi \}\), as no fee is applied when none of the premium-payment options is exercised. For \(t\in \{ \nu , \tau , \xi \}\) and \(t<T\), \({{\hat{A}}}_{t}=({{\hat{A}}}_{t-1}+{\,}_{t-1}p_{x}B_{t-1}^A)\cdot \Big (\max \big (g, \alpha ( S_{t}/ S_{t-1}-1)\big )+1\Big )\cdot (1-Fee).\) Beside exercise fees, we do not take other frictions into account that may influence the policyholder’s exercise behavior. For instance, in insurance practice, different tax treatments when using premium payment options can have an impact on the exercise strategy.
For a basic contract without any premium-payment options, \(\vartheta ^\emptyset \), \(\xi =T\), \(B_t=B^*\), and \(\gamma _{t+1}=\gamma ^*\). Equation (8) is, therefore, a special case of Eq. (9).
Single paid-up option
\(\vartheta ^P_\tau \) denotes the PV of a single paid-up option exercised at the end of year \(\tau \) with \(0<\tau \le T\).
With Eq. (9), \(\xi =T\), as no surrender option is offered. If \(\tau =T\), this contract matures without exercising the paid-up option. Thus, \(\vartheta ^P_T=\vartheta ^\emptyset =0\).
Otherwise, if \(\tau <T\), \(B_t=B^*\), \(\gamma _{t+1}=\gamma ^*\), when \(t<\tau \). When \(t\ge \tau \), \(B_t=0\) and \(\gamma _{t+1}=\gamma ^P_\tau \), with:
\(\vartheta ^{PR}_{\tau ,\nu }\) signifies the PV of a combined option with the paid-up option exercised at the end of year \(\tau \) and the resumption option at the end of year \(\nu \), where \(0<\tau <\nu \le T\) or \(\tau =\nu =T\). In the latter case, no options are exercised and \(\vartheta ^{PR}_{T,T}=\vartheta ^\emptyset =0\).
Without a surrender option offered, \(\xi =T\). If \(\nu =T\), this contract matures without exercising the resumption option and \(\vartheta ^{PR}_{\tau ,T}=\vartheta ^P_\tau \).
If \(\nu <T\), when \(t<\tau \), \(B_t=B^*\) and \(\gamma _{t+1}=\gamma ^*\). For \(\tau \le t<\nu \), \(B_t=0\) and \(\gamma _{t+1}=\gamma ^P_\tau \). For \(t\ge \nu \), \(B_t=B^*\) and \(\gamma _{t+1}=\gamma ^{R}_{\tau ,\nu }\) with:
\(\vartheta ^{S}_{\xi }\) denotes the PV of a single surrender option exercised at the end of year \(\xi \) with \(0<\xi \le T\). Without paid-up or resumption options, \(B_t=B^*\) and \(\gamma _{t+1}=\gamma ^*\). If \(\xi =T\), this contract matures without exercising the option, and hence, \(\vartheta ^{S}_{T}=\vartheta ^\emptyset =0\).
Combined paid-up and surrender option
\(\vartheta ^{PS}_{\tau ,\xi }\) signifies the PV of a combined option with the paid-up option exercised at the end of year \(\tau \) and the surrender option at the end of year \(\xi, \) where \(0<\tau <\xi \le T\) or \(0<\xi \le \tau =T\). If \(\tau =\xi =T\), no options are exercised and \(\vartheta ^{PS}_{T,T}=\vartheta ^\emptyset =0\). If \(\xi =T\), this contract matures without exercising the surrender option. In such a case, \(\vartheta ^{PS}_{\tau ,T}=\vartheta ^P_\tau \). If \(\tau =T\), this contract matures without exercising the paid-up option and \(\vartheta ^{PS}_{T,\xi }=\vartheta ^S_\xi \).
If \(0<\tau<\xi <T\), for \(t<\tau \), \(B_t=B^*\) and \(\gamma _{t+1}=\gamma ^*\). For \(\tau \le t\), \(B_t=0\) and \(\gamma _{t+1}=\gamma ^P_\tau \).
Combined paid-up, resumption, and surrender option
\(\vartheta ^{PRS}_{\tau ,\nu ,\xi }\) denotes the PV of the combined option including all three options: a paid-up option, a resumption option, and a surrender option exercised at the end of year \(\tau, \)\(\nu, \) and \(\xi \).
If \(\tau =\xi =\nu =T\), none of the options are exercised and \(\vartheta ^{PRS}_{T,T,T}=\vartheta ^\emptyset =0\).
If \(0<\tau <\nu =\xi =T\), only the paid-up option is exercised at the end of year \(\tau \), thus \(\vartheta ^{PRS}_{\tau ,T,T}=\vartheta ^{P}_\tau \).
If \(0<\tau<\nu <\xi =T\), the surrender option has not been exercised when the contract matures. Thus, \(\vartheta ^{PRS}_{\tau ,\nu ,T}=\vartheta ^{PR}_{\tau ,\nu }\).
If \(0<\xi <\tau =\nu =T\), neither the paid-up option nor the resumption option is exercised. Thus, \(\vartheta ^{PRS}_{T,T,\xi }=\vartheta ^{S}_{\xi }\).
If \(0<\tau<\xi <\nu =T\), the contract matures without the resumption option being exercised. Thus, \(\vartheta ^{PRS}_{\tau ,T,\xi }=\vartheta ^{PS}_{\tau ,\xi }\).
If \(0<\tau<\nu<\xi < T\), \(B_t=B^*\) and \(\gamma _{t+1}=\gamma ^*\) with \(t<\tau \). When \(\tau \le t<\nu \), \(B_t=0\) and \(\gamma _{t+1}=\gamma ^P_\tau \). When \(\nu \le t\), \(B_t=B^*\) and \(\gamma _{t+1}=\gamma ^R_{\tau ,\nu }\).
Interpretation
The basic contract contains regular premium payments, possesses a net present value of zero, and encloses no premium-payment options. The present values of the different forms of premium-payment options can be interpreted as the fair price the policyholder would need to pay at the beginning of the contract (\(t=0\)) in addition to the regular premium payments of the basic contract. For a rational policyholder in a complete and frictionless capital market, this price cannot be negative. In any case, the present value of premium-payment options depends strongly on the policyholder’s exercise strategy.
Typically, policyholders provide constant premium payments for participating life insurance contracts with premium-payment options. In the context of our paper, the fair price of premium-payment options can be transferred in a constant add-up on the regular premium payments for the basic contract (i.e., the contract without premium-payment options). Thereby, the term structure, mortality probabilities, and the exercise strategy in respect to the premium-payment options must be taken into account. Another way to obtain constant premium payments is to adjust the participation rate derived in the basic contract in a way that the insurance contract with premium-payment options has a net present value of zero. Again, the adjustment depends very much on the exercise strategy used by the policyholders. However, in order to derive results that are easy to compare for different scenarios and contract lengths, we decided to focus on the present values of premium-payment options only.
4 Valuation of premium-payment options
The assumed policyholder’s exercise strategy is central for option valuation. We begin by calculating the upper bound of the premium-payment options. This method indicates the options’ value range and demonstrates the worst-case scenario from the insurers’ viewpoint. However, it is not an accessible value since such a procedure requires information on the future development of the two stochastic sources. Hence, it is not a feasible strategy for policyholders. In the second step, we develop an adjusted LSMC strategy as an approximation of an optimal and feasible exercise approach. With this approach, we assume the policyholder to be rational in the following sense: The policyholder exercises the premium-payment option at its optimal value. It demonstrates the worst-case scenario from the insurers’ viewpoint.10
4.1 upper bound of the option value (\(^{UP}{{\overline{\vartheta }}}\))
Kling et al. (2006), Gatzert and Schmeiser (2008), and Schmeiser and Wagner (2011) calculate an upper bound for premium-payment options and discuss its economic interpretation in detail. Assuming policyholders know future developments, they would exercise the premium-payment option only if it is in the money (ITM if \(\vartheta >0)\) and at its maximum value for the whole contract period. Using Monte Carlo simulation with \(n=1...N\) paths, we have:
where \( ^{n}\vartheta _t \) denotes the different option values if exercised at the end of year t for the \(n^{th}\) simulation path. Options are non-negative, as policyholders do not exercise these options if they are out of the money (OTM) for the whole contract period. The upper bound can also be referred to as the PV of the option given perfect information about the future. Although such information clearly does not exist for a human being, the concept still provides useful insight as a reference for the upper bound of the option—or maximum loss from the insurer’s viewpoint.
4.2 Option valuation via the least square Monte Carlo strategy (\(^{LSMC}{{\overline{\vartheta }}}\))
The LSMC method was first presented by Longstaff and Schwartz (2001) to price American options. It has been used to value a single surrender option in life insurance contracts (cf. Andreatta and Corradin 2003; Nordahl 2008, and summarised by Bauer et al. 2010) with a single premium payment paid at the beginning of the contract.
The LSMC approach aims to find an optimal exercise point \(t^*\) with accessible information only. For different points in time t, two values are compared: exercise value and continuation value. Exercise value denotes the value if the option is exercised at t, while continuation value represents the value if the policyholder does not exercise the option and the contract goes forward. Following this strategy, policyholders exercise an option if its exercise value is larger than the continuation value and \(t^*=t\).
The original LSMC determines the exercise value as the PV of a defined and deterministic cash flow when the option is exercised. The continuation value is the PV of the future cash flows if the options are not exercised immediately. However, except for the surrender option, exercising premium-payment options does not always cause a defined immediate cash flow. Adjusting the original approach, we define both the exercise value and the continuation value as the PV of future cash flows under the condition that the premium-payment option is exercised or is not exercised. For the special case of surrender options, we compare both the original and the adjusted LSMC in the appendices. Our numerical examples suggest that the optimal option value with the adjusted LSMC strategy is slightly better than that of the original LSMC as the expected future interest development is taken into account. In the following, unless stated otherwise, LSMC refers to the adjusted LSMC strategy.
Future cash flows are stochastic. The original algorithm contains two approximations to estimate the optimal option value (cf. Clément et al. 2002). First, the continuation value at the end of year t denoted by \(_tC(\vartheta )\) is approximated by a combination of finite value functions with accessible relevant information. The second approximation determines the value functions via a least square regression. Two approximations are further added for the exercise value, \(\vartheta _t\), i.e., the option value when the option is immediately exercised at the end of year t. All of the values are discounted to the beginning of the contract for the purpose of convenient comparison.
\(_tC(\vartheta )\) is approximated by \(f({x_t^{1}...x_t^{J}})\), where \( x_t^{1}...x_t^{J}\) denotes all accessible information at the end of year t. As the option exercise decision is influenced by the financial market condition, we include all relevant variables at the end of year t, such as interest rate, \({{\hat{r}}}_{t}\), the asset value, \(S_{t}\), and the benefit, \(\gamma _{t+1}\).11
The second approximation includes K sets of basis functions, \(\upsilon ^{k} \) with \(k=1...K\), to approximate \(f({x^{1}_t...x^{J}_t})\) with K constant coefficients, \(\vec {a}_t=(a_t^1...a_t^K)\). Here, \(\upsilon ^{k} \) are weighted Laguerre polynomials suggested in Longstaff and Schwartz (2001):
The coefficients \(\vec {a}_t\) are unknown so far. With Monte Carlo simulation paths \(n=1...N\), we estimate \( \vec {a}_t \) via a least square linear regression. In Longstaff and Schwartz (2001), these estimators are based solely on in-the-money paths to reduce computation effort. However, in our case, all paths should be considered since the option we focus on is not standard (cf. Andreatta and Corradin 2003). The estimator for \( \vec {a}_t \) is provided by:
where \(_t^{n}{\hat{C}}(\vartheta )\), \(^{n}_tC(\vartheta )\), and \(^nx^j_t\) denote \(_t{\hat{C}}(\vartheta )\), \(_tC(\vartheta )\), and \(x^j_t\) in the \(n^{th}\) simulation path.
As explained above, the exercise value is unknown until maturity (\(t=T\)). Therefore, another two approximations are required to estimate \({\vartheta }_t \) as \(^{n}{\hat{\vartheta }}_t\) with accessible information \({\,}^nx_t^{1}...{\,}^nx_t^{J}\) per each simulation n:
This paper exams five different cases: two single premium-payment options; two double premium-payment options; and one triple premium-payment option. The single premium-payment options include the single paid-up option and the single surrender option. The double premium-payment options are the combined paid-up and resumption option and the combined paid-up and surrender option. The triple premium-payment option combines all three paid-up, resumption, and surrender option. The algorithm to approximate these optimal exercise strategies using the LSMC are shown in details in the Appendix.
5 Numerical results
This section discusses the key results of our numerical example. In particular, we focus on the impact of interest rate volatility \(\sigma _I\) on the value of the different embedded options. For ease of comparison, the parameters are chosen on the basis of those given in Schmeiser and Wagner (2011). Thus, the parameterisation used is not based on empirical evidence, but serves to illustrate our theoretical model. The cases considered in Schmeiser and Wagner (2011) are reflected in our calculations for the case \(\sigma _I=0\). The paper by Schmeiser and Wagner (2011) also includes sensitivity analyses. The authors show that the value of premium payment options increases ceteris paribus when the guaranteed interest rate or asset volatility increases. However, the value of premium payment options decreases as the contract term or the riskless rate of return increases. Based on these results for one risk source, we show the significant impact of interest rate risk on the value of premium payment options and present some sensitivity analyses. In particular, due to the current low interest rate environment, we perform a further sensitivity test to take into account the actual term structure. We argue that ignoring this second source of risk leads to a significant underestimation of the true value of the various embedded options. In addition, the stochastic interest rate process has a significant impact on the optimal exercise strategy. We also show that, in contrast to the results presented in Schmeiser and Wagner (2011), the risk of premium-payment options cannot be eliminated by introducing fees for the exercise of a premium-payment option.
Unless stated otherwise, the numerical results are gathered using a Monte Carlo simulation with \(N = 10^4\) and the LSMC method is employed with \(k=4\).12
5.1 Basic contract
We consider a basic contract with the following parameters: A 30-year-old female policyholder enters into a 10-year life insurance contract.13 The annual premium is 1200 currency units, with the yearly interest rate guaranteed at \(3\%\). The annual investment return rate combines both the interest and the asset processes laid down in Eq. (7). The asset volatility is fixed to \( \sigma _A=20\% \). The correlation coefficient between the asset development and the interest rate is fixed to \(\rho =0.05 \). Under the Vasicek model, to obtain \({{\hat{r}}}_t\), we use the parameters \( \kappa =8\% \), \( {{\hat{r}}}_{0}=4\% \), \(\theta =4\%\), and \(\lambda =0\). Based on these assumptions and using Eq. (2), the death benefit is 14,091 currency units. Table 2 summarises the initial parameters.
Table 2
Parameter table for the base case
\(B^*\)
Constant annual premium payment
1200
\(\gamma ^*\)
Constant death benefit
14,091
g
Guaranteed interest rate
3\(\%\)
T
Time to maturity
10
x
Initial age
30
\(\kappa \)
Interest rate reversion speed
8\(\%\)
\(\sigma _A\)
Asset volatility
\(20\%\)
\(\rho \)
Correlation coefficient between the asset development and the interest rate risks
5\(\%\)
\({{\hat{r}}}_0\)
Initial interest rate
4\(\%\)
\(\theta \)
Long-term interest mean
4\(\%\)
\(\lambda \)
Market price of risk
0\(\%\)
Figure 1 shows the relationship between the participation rate, \(\alpha \), and interest rate volatility, \(\sigma _I\), under different conditions. The participation rate, \(\alpha \), is derived such that the basic contract is fair at \(t = 0\) (i.e. \(\vartheta ^\emptyset =0\) in Eq. 8). A clear trend is shown whereby a higher volatility of interest rate results in a lower participation rate. For the same contract with the same interest rate guaranteed, insurers face higher risk and thus must lower the participation rate to ensure a risk-adequate return for the shareholders. In addition, the curves with different \(\theta \) move in parallel for \(\sigma _I\) from \(0.01\%\) to \(0.2\%\). This supports the conclusion drawn in Schmeiser and Wagner (2011), where \(\sigma _I\) is assumed to be 0: Given a fixed guaranteed interest rate, policyholders demand a higher participation rate as the interest rate is higher. As can be seen in Fig. 1b, with \(g=0\), the participation rate is pushed upward in parallel because the value offered by the guaranteed rate decreases. The longer duration emphasizes the \(\sigma _I\) impact on \(\alpha \). If the duration is prolonged to 40 years, Fig. 1c shows a stronger negative interrelation between \(\alpha \) and \(\sigma _I\). When \(\sigma _I\) is larger than \(1.1\%\) for \(\theta =3.5\%\) and \(1.4\%\) for \(\theta =4\%\), there exists no \(\alpha \) with \(0\le \alpha \le 1\) that satisfies the fairness condition.
Fig. 1
Participation rate (\(\alpha \)) for different combination of long-term interest mean (\(\theta \)) and volatility (\(\sigma _I)\) with various time to maturity (T) and interest guaranteed rate (g)
5.2.1 Triple premium-payment option: paid-up, resumption, and surrender option
In the initial setup, we assume that no fee is applied (\(Fee=0\)). Figure 2 demonstrates the results of the triple option (\(^{Up}{{\overline{\vartheta }}}^{PRS}\), \(^{LSMC}{{\overline{\vartheta }}}^{PRS}\)), consisting of paid-up, resumption, and surrender options. This option value is fairly small when \(\sigma _I\) is small, even if policyholders follow the LSMC strategy. However, the option value increases as \(\sigma _I\) enlarges. While \(\sigma _I\) is small, the options’ upper-bound value is higher with a higher mean of the interest rate (\(\theta \)). However, the option value with lower \(\theta \) grows much faster as \(\sigma _I\) increases. With the guaranteed rate closer to \(\theta \), the impact of the interest volatility on the fairness situation becomes much stronger. This influence of \(\theta \) and \(\sigma _I\) on the option value is captured if policyholders follow the LSMC strategy. Though the option value based on the LSMC strategy is much lower than the upper bound, it shows a similar structure: when \(\sigma _I\) is small, \(^{LSMC}{{\overline{\vartheta }}}^{PRS}\) with \(\theta =6\%\) is slightly higher than the option value with \(\theta =4\%\) or \(\theta =3.5\%\). As \(\sigma _I\) increases, the option value with smaller \(\theta \) increases faster. However, \(\theta \)’s impact on \(^{LSMC}{{\overline{\vartheta }}}^{PRS}\) is fairly small. As seen in Fig. 2b, the guaranteed value decreases with small g and hence \({\,}^{Up}{{\overline{\vartheta }}}^{PRS}\) increases especially when \(\sigma _I\) is small. This low guaranteed value and thus the high participation rate result in a low contract value variation. Thereby, the optimal exercise strategy becomes less efficient with g much smaller than \(\theta \) and the difference between \(^{LSMC}{{\overline{\vartheta }}}^{PRS}\) and \(^{Up}{{\overline{\vartheta }}}^{PRS}\) enlarges. Therefore, \(^{LSMC}{{\overline{\vartheta }}}^{PRS}\) with \(g=0\%\) is generally lower than the value if \(g=3\%\). Figure 2c shows that with the contract duration prolonged to 40 years, the option value increases substantially and faster with larger \(\sigma _I\) due to an even higher uncertainty (\(^{LSMC}{{\overline{\vartheta }}}^{PRS}\simeq 5570\) with \(T=40\) compared to \(^{LSMC}{{\overline{\vartheta }}}^{PRS}\simeq 570\) with \(T=10\) for \(\sigma _I=2\%\), \(\theta =6\%\)).
Fig. 2
Value of the triple option, \(^{LSMC}{{\overline{\vartheta }}}^{PRS}\) and \(^{Up}{{\overline{\vartheta }}}^{PRS}\) for different values of long-term interest mean (\(\theta \)), time to maturity (T), and the interest guaranteed rate (g)
To avoid a high premium for policyholders and to cover administrative costs arising on the insurers’ side when the premium-payment option is exercised, insurers usually charge a certain fee when the option is used. In policyholders’ perspective, Fee can be considered as market friction. Figure 3 demonstrates the option value with \(Fee\ge 0\). The fee’s impact on \(^{LSMC}{{\overline{\vartheta }}}^{PRS}\) enlarges with growing \(\sigma _I\). When \(\sigma _I\) is small, \(^{LSMC}{{\overline{\vartheta }}}^{PRS}\) has little value even if no fee is charged. When \(Fee> 0\), it is less likely to exercise a premium-payment option under the LSMC strategy. However, even with a fee equal to \(10\%\) of the policy account value, the premium-payment option value increases to more than 250 (almost a quarter of the annual premium) with \(T=10\) if \(\sigma _I\) exceeds 2\(\%\). The triple option is OTM, or zero intrinsic value most of the time. Nevertheless, with large \(\sigma _I\), certain extreme cases may occur, where policyholders benefit greatly from the embedded premium-payment options.
Fig. 3
Fee structure within \(^{LSMC}{{\overline{\vartheta }}}^{PRS}\) with \(\theta =4\%\), \(T=10\), and \(g=3\%\)
Figure 4 shows the discrepancy between \(^{LSMC}{{\overline{\vartheta }}}^{PRS}\) and \(^{LSMC}{{\overline{\vartheta }}}^{PS}\), or the value of the extra resumption option compared to the combined paid-up and surrender option. Though the upper bound of this resumption option’s value is non-negative and more than 2000 if the maturity is long (\(T=40\)), it is not possible to exercise this option efficiently with a feasible exercise strategy.
Fig. 4
Value of the extra resumption option as the difference between \(^{LSMC}{{\overline{\vartheta }}}^{PRS}\) and \(^{LSMC}{{\overline{\vartheta }}}^{PS}\) for \(T=10\) and \(T=40\)
Figure 5 shows cases where the policyholder exercised the paid-up option in the year 1, 3 or 5. This exercise of the paid-up option does not follow an optimal exercise strategy but may be due to a personal circumstance of the policyholder (e.g., a financial squeeze). Let us consider the case of \(\tau = 1\): After exercising the paid-up option in year 1, the policyholder could now start paying premiums again in year 2 or later. She could also surrender the policy after exercising the resumption option or not exercising the resumption at all (in which case we have \(\nu =10\)). Such an approach would typically result in a negative option value in the triple digits. If the policyholder proceeds in this way, the insurer does not appear to need to charge the option price for the three embedded premium payment options in the first place. Or the provider would generally be able to make a profit on these options if the option price were charged. However, relying on this (or similar) assumption is certainly risky for the insurer: As shown in this section, an optimal exercise strategy typically results in a significant option price. If policyholders increasingly adopt such a strategy, e.g., through online advice tools, the provider may face problems if the premium payment options have not been rightly priced. In this sense, the insurer must always hedge against optimal policyholder behavior, which means that the contract price should always reflect the option price in the way it is shown in chapter 4.2.
Fig. 5
The triple option value with respect to different exercise points \(\tau \), \(\nu \), and \(\xi \) with \(\sigma _I=2\%\), \(\theta =4\%\), and \(T=10\)
5.2.2 Double premium-payment option: paid-up and resumption option and paid-up and surrender option
Figure 6 demonstrates the value of the double option: the combined paid-up and resumption (\(^{LSMC}{{\overline{\vartheta }}}^{PR}\)) and the combined paid-up and surrender (\(^{LSMC}{{\overline{\vartheta }}}^{PS}\)). \(^{LSMC}{{\overline{\vartheta }}}^{PS}\) possesses much higher value compared to \(^{LSMC}{{\overline{\vartheta }}}^{PR}\), especially when \(\sigma _I\) is large. Figure 7 compares the double and the single option: \(^{LSMC}{{\overline{\vartheta }}}^{PR}\) with \(^{LSMC}{{\overline{\vartheta }}}^{P}\) and \(^{LSMC}{{\overline{\vartheta }}}^{PS}\) with \(^{LSMC}{{\overline{\vartheta }}}^{S}\). The extra resumption option generated by the difference between \(^{LSMC}{{\overline{\vartheta }}}^{PR}\) and \(^{LSMC}{{\overline{\vartheta }}}^{P}\) is only slightly positive even when \(\sigma _I\) is fairly large. As a call option, the value of the resumption option is small when the paid-up option as a put option is in the money in the previous contract years. The extra paid-up option generated by the difference between \(^{LSMC}{{\overline{\vartheta }}}^{PS}\) and \(^{LSMC}{{\overline{\vartheta }}}^{S}\) is negligible regardless of \(\sigma _I\). It is thus concluded that the double option value comes dominantly from a single option.
Fig. 6
Value of the double option: \(^{LSMC}{{\overline{\vartheta }}}^{PR}\) and \(^{LSMC}{{\overline{\vartheta }}}^{PS}\) with \(T=10\) while the LSMC strategy is followed
The double option values with different exercise points, \({{\overline{\vartheta }}}^{PR}_{\tau ,\nu }=\dfrac{1}{N}\Sigma ^N_{n=1}({\,}^n \vartheta ^{PR}_{\tau ,\nu })\) and \({{\overline{\vartheta }}}^{PS}_{\tau ,\xi }=\dfrac{1}{N}\Sigma ^N_{n=1}({\,}^n \vartheta ^{PS}_{\tau ,\xi })\) are presented in Fig. 8. Without following an optimal exercise strategy, both double option values are negative most of the time. In addition, the highest value occurs when only one option is used (\({{\overline{\vartheta }}}^{PR}_{\tau ,T}\) and \({{\overline{\vartheta }}}^{PS}_{\tau ,T}\)).
Fig. 8
The double option value with respect to different exercise points \(\tau \), \(\nu \), and \(\xi \) with \(\sigma _I=2\%\), \(\theta =4\%\), and \(T=10\)
Figure 8a shows that the combination of paid-up and resumption option without a surrender possibility is, for example, slightly positive if the first premium is paid in t = 0, the second premium is not paid, but premiums are paid from t = 2 or t = 3 until the end of the contract (in this case we have \(\tau =1\) and \(\nu =2\) or \(\nu =3\)). Cases of a sub-optimal use of a combination of paid up and surrender options as presented in Fig. 8b. Typically such a behaviour leads to negative option values—and hence to a profit for the insurer if the embedded premium payment options are not charged. The option value only slightly gets positive if the paid-up option is used in \(\tau = 6\), 7, 8, or 9, while not surrendering policy at all afterwards (\(\xi = 10\)).
5.2.3 Single premium-payment option: paid-up option and surrender option
Figure 9 demonstrates the single option value: the paid-up option and the surrender option. These two options possess put option features while the surrender option influences a larger cash flow. Hence, both \(^{LSMC}{{\overline{\vartheta }}}^{S}\) and \(^{Up}{{\overline{\vartheta }}}^{S}\) are much greater than \(^{LSMC}{{\overline{\vartheta }}}^{P}\) and \(^{Up}{{\overline{\vartheta }}}^{P}\), especially when \(\sigma _I\) is fairly large. Thus, while the triple option value is dominated by the double option, \(^{LSMC}{{\overline{\vartheta }}}^{PS}\), \(^{LSMC}{{\overline{\vartheta }}}^{PS}\) provides little extra value to the single option \(^{LSMC}{{\overline{\vartheta }}}^{S}\). Specifically, even if policyholders following an optimal exercise strategy, insurers do not need to charge extra when offering additional options.
Following an optimal strategy may seem a rather strong assumption, as policyholders are typically subject to several constraints (e.g., liquidity constraint) or other frictions.14 Additionally, the value of the premium-payment option is not the only concern when policyholders decide whether to exercise their options or not. Instead, this decision can be influenced by not only the external features (such as financial market developments) but also certain personal reasons. We therefore introduce a concept denoted as “Average”. Assuming that the options are equally likely to be exercised throughout the contract period, the option value is the average of the yearly option valued at t with \(0<t<T\). In formal terms, the option value under this measurement is calculated by:
In Fig. 9, the option value in “Average” is around zero and decreases mildly as \(\sigma _I\) increases. Exercising the paid-up or surrender option means to give up partial (for the paid-up option) or entire (for the surrender option) value provided by the guaranteed rate (g). Without following a certain exercise strategy, this positive guaranteed value increases as \(\sigma _I\) increases. Hence, \(^{Average}{{\overline{\vartheta }}}\) decreases with larger \(\sigma _I\). In addition, the guaranteed value is larger with \(\theta \) closer to g. Hence, as concluded if the LSMC strategy is followed, on average, lower \(\theta \) leads to a lower premium-payment option value.
Two things are important to consider in the context of our “average behaviour case”: (a) In this case, an average behaviour is assumed for all policyholders in the portfolio. Therefore, such a (sub-optimal) exercise procedure is not a feasible strategy for an individual policyholder. (b) Assuming such an “average behaviour case” is risky from the insurer’s point of view if policyholders increasingly move towards an optimal exercise strategy, e.g., through online advice tools, and the premium payment options have not been priced.
The previous section shows that the triple option has a fairly large value even if a \(10\%\) fee is charged when the option is exercised under the LSMC strategy. As the triple option values almost the same as the single surrender option, Fig. 10 shows the single surrender option value with \(Fee\ge 0\) assuming that policyholders exercise their options according to the LSMC strategy (\(^{LSMC}{{\overline{\vartheta }}}^S\) as a solid line) or equally likely throughout the contract years (\(^{Average}{{\overline{\vartheta }}}^S\) as a dotted line). \(^{Average}{{\overline{\vartheta }}}^S\) drops significantly, while the impact of the fee on \(^{LSMC}{{\overline{\vartheta }}}^S\) is relatively small, especially when \(\sigma _I\) is low. Following the LSMC strategy, it is less likely that the options will be exercised if a fee is charged as shown in Fig. 11. As it is not always optimal to exercise the option, the option fee charged when exercising the option does not significantly influence the option price in the LSMC setting. Especially when \(\sigma _I\) is large, extreme cases may occur, in which the surrender option goes deep ITM and becomes fairly valuable even with a \(10\%\) fee.
Fig. 10
\(^{LSMC}{{\overline{\vartheta }}}^{S}\)and \(^{Average}{{\overline{\vartheta }}}^{S}\) with \(Fee\ge 0\)
Adapting to a case of a low-interest rate environment, we reset our parameters with the guaranteed rate \(g=0\%\), the initial interest rate \({{\hat{r}}}_0=0.5\%\), and the long-term interest rate mean \(\theta \in \{0.5\%,\,1\%,\,1.5\%\}\), as described in Table 3.
The results show that as long as there is \(\alpha \) with \(0\le \alpha \le 1\) such that \(\vartheta ^\emptyset =0\) (i.e., it is possible to offer a fair contract), the relation between \(\sigma _I\) and the option value, and the relation among different option values remain mostly unchanged.
Figure 12 shows the participation rate, \(\alpha \), with various long term interest means. Larger \(\sigma _I\) leads to a lower participation rate, while the participation rate is higher with a larger \(\theta \). With a guaranteed rate much closer to \(\theta \), \(\alpha \) is fairly small compared to the previous example. For \(\theta =0.5\%\), there exists no \(\alpha \), with \(0\le \alpha \le 1\) when \(\sigma _I\) reaches \(2\%\). The interest impact on the option values is trivial if LSMC is assumed followed. As concluded in the previous example, even if following an optimal exercise strategy, an extra resumption option and an extra paid-up option contribute little to the triple option. Specifically, the triple option possesses around the same value as the single surrender option. Regarding the single surrender option, \(^{Average}{{\overline{\vartheta }}}^{S}\) is negative especially when \(\sigma _I\) is large and \(\theta \) is small.
Table 3
Parameter table for the low-interest rate case
\(B^*\)
Constant annual premium payment
1200
\(\gamma ^*\)
Constant death benefit
11944.96
g
Guaranteed interest rate
0\(\%\)
\({{\hat{r}}}_0\)
Initial interest rate
0.5\(\%\)
\(\theta \)
Long-term interest mean
\(0.5\%\)/1\(\%\)/\(1.5\%\)
Fig. 12
\(\alpha \)/\(\sigma _I\) combinations for different \(\theta \) resulting in a fair contract condition
The numerical results show that, if stochastic interest rates are taken into account and the policyholder follows an exercise strategy based on LSMC, the values of premium-payment options can be substantial. For instance, even with a short contract duration \(T=10\), a surrender option values up to two thirds of the annual premium if interest becomes rather volatile. Hence, insurers will need to charge extra for the options provided in order to finance adequate risk management measures. In addition, we demonstrate that the necessity of charging for premium-payment options cannot be eliminated by a fee structure dependent on the account value.
Beside the risk sources introduced in our model, life insurance companies typically face substantial model and parameter risk regarding the valuation of embedded options. Additionally, Swishchuk (2004) suggests that the risk is underestimated whenever the volatility is assumed to be constant in time. Considering these factors, insurers may need to charge even higher premiums than those proposed in our model setup.
If an optimal exercise strategy is followed, the multiple premium-payment option values only little more than a single option. The valuation of a single option, however, strongly depends on the policyholders’ exercise behaviour. Real insurance markets may be incomplete and face frictions such as taxation and information asymmetries (cf. Bauer and Moenig 2023). These kinds of effects may hinder policyholders’ attempts to take full advantage of future market developments—which are assumed to take place via a LSMC strategy—by using premium-payment options. In addition, policyholders may not act in a fully rational way when it comes to exercising options. To approximate such a behaviour, we assume in the paper that a fixed fraction of policyholders would exercise their single option each year. In such a case, the average option value is around zero or even negative if the interest rate is rather volatile. Hence, if such an exercise behaviour takes place, paid-up and surrender options could be offered free of charge. However, in the future, policyholders could be better advised on how to use embedded premium-payment options or the market efficiency may alter. Hence, if insurance companies base their pricing purely on the empirical exercise behaviour of policyholders (or any other assumption of a suboptimal exercise strategy), they can face a considerable risk of underpricing.
In most cases, insurance companies are not free to choose whether to offer premium-payment options or not. For instance, a participating life insurance contract must have at least a surrender option by law in all insurance markets we know. As pointed out, in order to provide these options, insurers must charge, in addition to the savings premium and the term life premium, a substantial price to finance adequate risk management measures. Given a financial market with alternative products in the field of old-age provision, such as fixed-income mutual funds (cf. e.g., Koijen et al. 2009), this additional charge may reduce the attractiveness of participating life insurance contracts.
Insurance companies typically charge policyholders a fee each time a premium payment option is used. The general idea behind such a fee model is that only those policyholders who use the premium-payment option should pay. This fee structure may discourage policyholders from exercising their options. In such a case, the introduction of a fee model can be beneficial to both insurers and those policyholders who do not or cannot follow an optimal exercise strategy: On the one hand, insurers face less uncertainty about the policyholders’ exercise behaviour. On the other hand, since, average option values are often negative, especially when interest rate volatility is high, policyholders are better off if they exercise their options less often. As we have shown in the paper, for policyholders using an LSMC strategy (as an approximation to an optimal exercise strategy), the fee model only slightly reduces the price of premium-payment options. Certainly, under fair pricing conditions, the situation of these policyholders remains unchanged—the NPV of the contract is zero with or without fees. In general, however, there will be fewer exercises.
At first glance, another approach might be to introduce a lock-up period during which policyholders are not allowed to exercise premium-payment options. A lock-up period shortens the period during which premium-payment options can be exercised. Therefore, the value of these options could generally be reduced. However, in the insurance markets with which we are familiar, the policyholder always has the right to surrender or to make the contract paid-up before maturity. Consequently, the agreement of such a lock-up period would therefore be contrary to the provisions of insurance contract law.
An alternative approach to addressing the issue raised in this paper would be to repay only the market value if a premium-payment option is exercised before maturity. This would eliminate the insurer’s exposure to premium-payment options and eliminate the need to charge any additional premium, as the option would have no value under any exercise assumption. It is evident that the insurer would be unable to guarantee a fixed payback at specific points in time, as is currently the practice in participating life insurance contracts with cliquet-style investment guarantees. In general, it is unlikely that such a procedure would be accepted by policyholders and/or regulatory authorities. It would certainly also be worth considering not guaranteeing minimum repayments in the first couple of years, but only paying out the market value of the assets to the customer if the premium payment option is utilized early. However, our numerical examples show that in most cases where an (approximately) optimal exercise strategy is used, the premium payment options are rarely utilized in the first three years. Therefore, such a contract amendment would not substantially reduce the price of the premium payment options or the provider’s potential risk (if an underpricing took place) from this kind of options.
In conclusion, the life insurance company must either accept risks associated with premium payment options (even if fees are included) or, to mitigate risk, calculate high premiums based on an optimal exercise strategy. An alternative is to offer primarily unit-linked life insurance products that offer no or only limited guarantees.
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Single premium-payment option case, \( ^{LSMC}{\overline{\vartheta }}^P\, \& ^{LSMC}{\overline{\vartheta }}^S\)
For single options, namely \(\vartheta _t^P\) and \(\vartheta _t^S\), we aim to find one optimal exercise point, \(^nt^*\), that maximises the option value in each path, n, by using accessible information (\(^nx_t^j).\) At the end of each year, policyholders decide whether to exercise the option or not. The option should be exercised if the exercise value exceeds the continuation value.
For the backward iteration \(m_1\) with \(m_1=T...1\), the single-option valuation procedure can be formally described as follows:
Step 1: Initialisation: Start with \(m_1=T\) and set all \(^nt^*=m_1=T\):
The option value is zero as at \(m_1=T\), the contract matures without exercising the option. The optimal option value is given by \(^n\vartheta _{^nt^*}=\)\(\vartheta _{T}=0\).
Step 2: One year backward:
One year backward at \(m_1=T-1\), the continuation value is set zero as \(^n_{T-1}C(\vartheta )=\vartheta _{T}=0\). If \(^n{{\hat{\vartheta }}}_{T-1}\) is positive and hence exceeds the continuation value, the option should be exercised and the optimal exercise point becomes \(^nt^*=m_1=T-1\). Otherwise, the contract continues and \(^nt^*\) remains unchanged. In formal terms, we have \(^nt^*= m_1 \text{, } \text{ if } ^n{{\hat{\vartheta }}}_{m_1}>{\,}^n_{m_1}C(\vartheta )\), with \(^n_{m_1}C(\vartheta )=0\) and \(^n{\hat{\vartheta }}_{m_1}\) derived from Eq. (15). Otherwise, \(^nt^*\) remains T.
Approximate the continuation value \(\,^n_{m_1}C(\vartheta )\) and the exercise value \(^n\vartheta _{m_1}\) with Eqs. (14) and (15).
(3)
If \(^n{\hat{\vartheta }}_{m_1}>{\,}^n_{m_1}{{\hat{C}}}(\vartheta )\), \(\,^nt^*=m_1\). Otherwise, \(^nt^*\) remains unchanged.
With the algorithm above, the optimal option value equals the average of each path’s option value exercised at its respective optimal point, \(^nt^*\):
\(^{LSMC}{\overline{\vartheta }}^P=\frac{1}{N}\sum _{n=1}^{N}(^n\vartheta ^P_{^nt^*})\) for the paid-up option and \(^{LSMC}{\overline{\vartheta }}^S=\frac{1}{N}\sum _{n=1}^{N}(^n\vartheta ^S_{^nt^*})\) for the surrender option.
The double premium-payment option includes the combined paid-up and resumption option, \(\vartheta ^{PR}_{\tau ,\nu }\), and the combined paid-up and surrender option \(\vartheta ^{PS}_{\tau ,\xi }\). The main difference between these two options is that the resumption option can only be exercised if the paid-up option has been exercised, i.e., \(\tau <\nu \), if \(\nu <T\). However, policyholders can exercise the surrender option independently, even if the paid-up option has not yet been exercised.
Combined paid-up and resumption option \(^{LSMC}{\overline{\vartheta }}^{PR}\)
The double option contains two optimal exercise points. Hence, its value cannot be estimated when deciding whether to exercise the first option, as the second exercise point has not been determined yet. However, for \(\vartheta ^{PR}_{\tau ,\nu }\), while the first paid-up option can be seen as a put option, the resumption option has the call option feature. Therefore, at the end of year t, if it is optimal to exercise the paid-up option, the intrinsic value of the resumption option is 0. Thus, we can first consider the paid-up option only. On the condition that the paid-up option is exercised, we then determine the second optimal exercise point for the resumption option.
First we determine \(^n \tau ^*={\,}^nt^*\) with the single paid-up option process described in the previous section. If \(^n \tau ^*=T\), the optimal strategy is not to exercise the paid-up option. In such a case, the resumption option will also expire and \(^n\nu ^*=T\), \({\,}^n\vartheta ^{PR}_{T,T}=0\). On the condition that \(^n\tau ^*<T\), we find the second optimal exercise point \(^n \nu ^*\) to maximise the second option. The second option value, or the resumption value, is the remaining value derived from \(^n\vartheta ^R_{^n\tau ^*,\nu }={\,}^n\vartheta ^{PR}_{^n\tau ^*,\nu }-{\,}^n\vartheta ^P_{^n\tau ^*}\). This second optimal exercise point \(^n\nu ^*\) can be computed by considering \(^n\vartheta ^R_{^n\tau ^*,\nu }\) as a single option with iteration \(m_2=T...^n \tau ^*+1\).
The optimal value can then be approximated by \(^{LSMC}{\overline{\vartheta }}^{PR}=\frac{1}{N}\sum _{n=1}^{N}(^n\vartheta ^{PR}_{^n \tau ^*,^n \nu ^*}).\)
Combined paid-up and surrender option\(^{LSMC}{\overline{\vartheta }}^{PS}\)
For the combination of paid-up and surrender, both options are considered as put options and can be exercised independently. More specifically, the surrender option as the second option can be exercised even if the first paid-up option has not yet been exercised. In formal terms, we have:
At the end of every year t, two put options are compared to determine whether to exercise one of the two options. We begin by finding the optimal exercise point for one of these two options (\(^n \tau ^*\) for \(^n\vartheta ^P_{^n\tau ^*}\) and \(^n \xi ^*\) for \(^n\vartheta ^S_{^n\xi ^*}\)) with the procedure described as follows:
With iteration \(m_3=T...1\):
Step 1. Initialisation:
For \(m_3=T\) with \(^n \tau ^*={\,}^n \xi ^*=T\), \(^n\vartheta _{^n\tau ^*}^P={\,}^n\vartheta _{^n\xi ^*}^S=0\).
Step 2. One year backward comparison:
At the end of year \(m_3=T-1\), the option value is \(^n{{\hat{\vartheta }}}^{PS}_{^n \tau ^*,^n \xi ^*}=\max \big (^n_{T-1}C(\vartheta )\), \({\,}^n{\hat{\vartheta }}^P_{T-1},^n{\hat{\vartheta }}^S_{T-1}\big )\). \(^n{\hat{\vartheta }}^P_{T-1}\) and \(^n{\hat{\vartheta }}^{{S}}_{T-1}\) are two estimators of \(^n{\vartheta }^P_{T-1}\) and \(^n{\vartheta }^S_{T-1}\), while the continuation value \(^n_{T-1}C(\vartheta )\) equals zero.
Three scenarios are considered:
(1)
If \(\max \big (^n_{T-1}C(\vartheta ),^n{\hat{\vartheta }}^P_{T-1},^n{\hat{\vartheta }}^S_{T-1}\big )=\)\(^n{\hat{\vartheta }}^P_{T-1}\): The best strategy is to exercise the paid-up option. Hence \(^n \tau ^*=m_3=T-1\) and \(^n \xi ^*=T\).
(2)
If \(\max \big (^n_{T-1}C(\vartheta ),^n{\hat{\vartheta }}^P_{T-1},^n{\hat{\vartheta }}^S_{T-1}\big )=\)\(^n{\hat{\vartheta }}^S_{T-1}\): The best strategy is to exercise the surrender option. Hence, \(^n \tau ^*=T\) and \(^n \xi ^*=m_3=T-1\).
(3)
If \(\max \big (^n_{T-1}C(\vartheta ),^n{\hat{\vartheta }}^P_{T-1},^n{\hat{\vartheta }}^S_{T-1}\big )=\)\(^n_{T-1}C(\vartheta )=0\): In this case, policyholders should exercise neither of the options, and \(^n \tau ^*={\,}^n \xi ^*=T\).
Step 3. Backward iteration: For \(m_3=T-2...1\):
Approximate the continuation value \(\,^n_{m_3}C(\vartheta )\) with \(^n_{m_3}{C}(\vartheta )=\)\(^n\vartheta _{^n \tau ^*,^n \xi ^*}\) and the exercise value \(^n\vartheta ^P _{m_3}\), \(^n\vartheta ^S _{m_3}\) with Eqs. (14) and (15).
After the iteration \(m_3\), we find two optimal exercise points, \(^n \tau ^*\) and \(^n \xi ^*\). For \(^n \tau ^*=T,\) the best strategy is either to exercise the surrender option only (\(^n \xi ^*<T\)), or not to exercise any of the options (\(^n \xi ^*=T\)). For the paths where \(^n \tau ^*<T\), the optimal strategy is to exercise the paid-up option at the end of year \(^n \tau ^*\). Given \(^n \tau ^*<T\), we update the second optimal exercise point, \(^n \xi ^*\), to maximise the remaining surrender option value, \(^n\vartheta '^{S}_{^n \tau ^*,\xi }={\,}^n\vartheta ^{PS}_{^n \tau ^*,\xi }-\,^n\vartheta ^P_{^n \tau ^*}\). This second optimal exercise point can be determined as the single option with another iteration procedure \(m_4=T...^n \tau ^*+1\).
This combined option’s optimal value is given by \( ^{{LSMC}} \bar{\vartheta }^{{PS}} = \frac{1}{N} \)\( \sum\limits_{{n = 1}}^{N} {\left( {^{n} \vartheta _{{n\tau ^{*} ,^{n} \xi ^{*} }}^{{PS}} } \right)} \)
When it comes to extending the algorithm with the combined paid-up, resumption, and surrender option, the decision process becomes complex: The paid-up and surrender options are both put options. Hence, for their combination, which option to exercise depends on whichever value is larger. For the triple option, the surrender option is compared with the combined paid-up and resumption option. The intrinsic value of the resumption option is zero whenever it is optimal to exercise either the surrender or the paid-up option. However, the time values of the resumption option is non-zero if the paid-up option is exercised. Specifically, when policyholders exercise the paid-up option, they can still use the resumption option to readjust the reduced benefit. On the contrary, once the surrender option is exercised, the contract terminates and the time value of all other options become zero. Hence, for \(\vartheta ^{PRS}_{\tau ,\nu ,\xi }\), the resumption option’s time value attached to the paid-up option should be considered when determining whether to exercise the paid-up option or the surrender option.
Step 1:
With an iteration \(m_4=T...1\), we determine whether to exercise the surrender or the paid-up option based on the comparison between \(^n{{\hat{\vartheta }}}^S_{m_4}\) and \(^n{{\hat{\vartheta }}}^{PR}_{m_4,^n\nu _{m_4}^*}\), where \(^n{{\hat{\vartheta }}}^{PR}_{m_4,^n\nu _{m_4}^*}\) denotes the expected optimal value of the combined paid-up and resumption option with the paid-up option exercised at the end of year \(m_4\). This expected optimal value is computed with another optimal exercise point \(^n\nu _{m_4}^*\), determined by another LSMC algorithm with \(m_5=T...m_4+1\). The first step determines two optimal exercise points for \(^n\tau ^*\) and \(^n\xi ^*\).
Step 2:
With an iteration \(m_6=T...^n\tau ^*+1\) for the paths that \(0<{\,}^n\tau ^*<T\), we decide either to exercise the resumption option or the surrender option. This decision is based on the two remaining option values:
On the condition that it is optimal to exercise the resumption option, we run another iteration \(m_7=T...^n\nu ^*+1\) for the paths that \(0<{\,}^n\tau ^*<{\,}^n\nu ^*<T\) to determine whether to exercise the remaining surrender option with its value \(^n\vartheta ''^S_{^n \tau ^*,^n \nu ^*,m_7}={\,}^n\vartheta ^{PRS}_{^n\tau ^*,^n\nu ^*,m_7}- {\,}^n\vartheta ^{PR}_{^n\tau ^*,^n\nu ^*}\).
With all three optimal exercise points determined, the optimal value of this triple option is given by:
Figure 18 compares the LSMC strategy discussed in this paper, termed “adjusted LSMC”, with the Longstaff and Schwartz (2001) method, termed “original LSMC”. For our numerical example, we demonstrate that the adjusted LSMC, leading to a slightly higher option value, is more efficient than the original LSMC strategy.
From another perspective, if many policyholders use their option optimally, it could lead to a massive simultaneous lapse behavior, causing instability to the life insurance sector.
One can argue that focusing on the pricing of surrender option in this form of participating life insurance contracts is in general sufficient and has been provided in the literature already. However, we think it is an important finding, showing that other premium payment options typically cannot be exercised optimally in the realistic case of multiple premium payment options and hence possess—in contrast to an isolated valuation—no substantial value.
In our case, the surrender value under the adjusted LSMC method is slightly better than the value under the original LSMC method (cf. appendices of this paper). Whether this adjusted LSMC method provides a better result and improves the downward bias in general may be a subject for future research.
For the case of the valuation of the surrender option only, Cheng and Li (2018) takes the insurer’s default risk into account. In addition, the authors include solvency rules: Whenever a certain solvency level is hit, the insurer is not allowed to sign new contracts but a run-off has to take place. Both effects—default risk and particular run-off rules—influence the policyholders’ cash-flows and hence the surrender option’s value.
Generally, policyholders can choose to (1) receive the surplus as cash each year, (2) purchase extra insurance and increase the death benefit amount, or (3) keep the surplus in the policy account and earn the return guaranteed with g. In such a case, the survival benefit is increased. In what follows, we assume that policyholders keep their surplus in the policy account.
As mentioned earlier, we use the contract design proposed in Schmeiser and Wagner (2011). However, in their setting, there are no administration costs c. Under the data set given in Schmeiser and Wagner (2011), this leads to a \(NPV = - c < 0\) (even though the authors argue that the NPV is zero in their numerical examples). We still want to use the setting and the data set of Schmeiser and Wagner (2011) to show the influence of a stochastic term structure on the results in a transparent way and allow a comparison under the different model settings. Hence, we introduced administration costs c to account for the aspect mentioned. Thereby, c depends on the policyholder’s survival rate.
If only rational decision makers are assumed in a complete and frictionless capital market, option values are non-negative as option holders will not exercise an option that is out of the money.
It is argued that policyholders neither behave rationally nor have access to the hedging tools to manage their interest-rate risk. However, with growing life settlement market, policyholders can sell their contracts to financial institutes, whose surrender behavior is rational and depended on the surrender option value (cf. Braun and Xu 2020).
In our model, both interest rate and asset risks are simulated based on Wiener processes. With the Markov structure, the future state of the financial market in \(t+1\) depends on the its current state in t and the stochastic behaviour of the Wiener process in the time interval (t, \(t+1\)).
When taking different N and k, our numerical results stabilise as N reaches \(10^4\) and \(k=4\). We must admit that due to the complexity of solving the underlying optimal stopping problem, we were not able to calculate more than \(N=10^4\) iterations. Therefore, one limitation of our numerical sample calculations is that we cannot make general statements on the stability and convergence of the calculated option prices.
For the case of the valuation of the surrender option only, Li and Szimayer (2014) shows that the rationality of the policyholders significantly influences the option value.
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