Interaction effects between dynamic hybrid products and traditional deferred annuities in the German life insurance market

We define a traditional deferred annuity as a textbook insurance policy that promises the policyholder a lump-sum payment at contract termination in case of survival in exchange for a single premium paid at contract start.¹ The lump-sum payment depends on the actuarial interest rate the insurer promises to the policyholder and the number of deaths among the policyholder cohort during the contract duration. A dynamic hybrid product is an insurance policy designed to combine potentially higher returns of investments in equity funds with a certain degree of security required by policyholders. This is achieved by a dynamic investment strategy that shifts money between different investment options. These “pots” are generally the insurer’s policy reserve (PR) and one or two equity funds. The dynamic aspect of the product is its periodic monitoring and reallocation between the pots. Ideally, the insurer invests the money in risky equity funds in order to generate higher returns and shifts money to the PR as soon as the guaranteed insurance benefit is endangered by an adverse market development, at which point the money is “parked” in the insurer’s policy reserve, where the money of the TDA is invested. The term “parked” is chosen deliberately to emphasise that the money is taking shelter for a short term and only waiting to be reinvested in riskier assets in order to generate higher returns. A DHP could be structured as represented in Fig. 1.

For this exemplary product, the policyholder’s money is split and allocated to three pots: the insurer’s PR, a guarantee fund (GF) with a downside protection that limits losses from 1 month to the next to \(\lambda \%\) (e.g. through short-term put options on the assets the GF is based on), and an equity fund (EF) with no protection and without the hedging costs of the GF. A worst case scenario for the single DHP components from 1 month to the next is shown in Fig. 2.

The money allocated to the PR is subject to the actuarial interest rate \(i_{G}\). \(i_G\) corresponds to the maximal interest rate allowed to discount future benefits of contracts stemming from a specific underwriting period. The insurer takes on the risk associated with the investment of the money—just as he does with the money of the TDA policyholders. However, the risk of investment in the GF and EF is borne by the DHP policyholders. Therefore, the worst case scenario for the money allocated to the GF is a loss of \(\lambda \%\), due to the downside protection. For the EF, the worst case scenario is a total loss of the money allocated to it. The upside of this arrangement is that the totality of profits goes to the policyholder in the event the two funds develop well.

The German life insurance market is regulated in the interest of making it a safe investment for consumers. Art.5 §1 of the “Deckungsrückstellungsverordnung” (DeckRV) (Regulation on the Principles Underlying the Calculation of the Premium Reserve) requires the insurer to employ prudent estimates of death and interest rates as well as future costs, especially when calculating the PR; Art.138 of the “Versicherungsaufsichtsgesetz” (VAG) (Insurance Supervision Act) requires the insurer to set premiums sufficiently high to cover resulting liabilities. The so-called first-order calculation bases (FOCB), necessary to comply with Art.5 §1 DeckRV, include a safety margin (as opposed to the so-called second-order calculation bases (SOCB) which do not). In general, calculations based on the FOCB entail premiums for the single insurance policies that are higher than what is actually needed for the promised payout. Insurers are thus forced by law to have a surplus at the end of each year. Art.153 §1 “Versicherungsvertragsgesetz” (VVG) (Insurance Contract Act) grants policyholders a legal claim to surplus participation. This participation usually occurs in two ways: first, through direct credits, or second, through the so-called provision for premium refunds (PPR). Since direct credits played only a subordinate role when DHPs were introduced in 2007,² we focus on the PPR. The PPR is a liability item in the insurer’s economic balance sheet conceived to smooth potential fluctuations in surpluses over the years. It does so by allowing different in- and outflows to it in the same year. For example, in years with lower surpluses, the outflows will generally be higher than the inflows and vice versa.³ The inflows are thereby a function of the surpluses, and the outflows are the bonuses allocated to the policyholders. For the insurer, the PPR is a useful cushion in times of sinking long-term asset returns. In contrast to the PR, a liability item that grows with the actuarial interest rate \(i_{G}\), the PPR only grows with inflows stemming from surpluses. Therefore, the insurer has an interest in keeping as much as possible in the PPR when interest rates and profits decrease. However, the insurer is not allowed to stock money in the PPR infinitely.⁴ Hence, bonuses are allocated to the policyholders, generally with a delay of a few years. The different approaches employed for distributing bonuses are described by Führer and Grimmer (2010, pp. 160–161). One option common for pension insurances with a single up-front premium is to treat the bonuses like additional single premiums the policyholder invests in new insurance policies with the same range of services and date of maturity as the original contract, thus increasing the policyholder’s lump-sum payment at the end of contract. We use this way of allocating bonuses in our model framework. We now explain how the first interaction effect between TDA and DHP takes place when the insurer calculates the inflows to the PPR. The second interaction takes place when it determines the outflows from the PPR.

In the following section, we describe the insurance company model. It is inspired by the model used by Bohnert and Gatzert (2014) and uses parts of the modelling strategies proposed by Kochanski and Karnarski (2011). Our contribution primarily lies in proposing algorithms for model inflows to the PPR and for the bonus distribution, two mechanisms not included in previous models but constituting the aforementioned interaction effect. The observed time periods will be \(t \epsilon \{0,...,T\}, \quad T= 12\cdot n +1\), where n is the contract duration of the policies sold by the insurer. For our standard contract duration of 30 years, \(T=361\), giving one period for each month in the 30 years plus one for the initial values (\(t=0\)) and one for the final values (\(t=361\)). Let \(Y:= \{t= 12\cdot i|i \epsilon \{1,...,n\} \} \subset \{0,\ldots ,T\}\) denote the set of end-of-year periods. This will come in handy since some variables change only at the end of a year. The balance sheet of the insurer is illustrated in Table 2.

On the asset side of the balance sheet, we have assets that are long-term investments \(A^{\mathrm{lt}}\) and others that are invested short term \(A^{\mathrm{st}}\). We further see the items \(\mathrm{GF}^{A}\) and \(\mathrm{EF}^{A}\) standing for investment in the DHP’s guarantee and equity funds. Since the policyholders bear the risk for the investment in the GF and EF, they are always equivalent to their counterparts on the liability side \(\mathrm{GF}^L\) and \(\mathrm{EF}^L\). On the liability side, we find the insurer’s policy reserves PR, split between those “belonging” to the two subgroups of policyholders. The provisions for premium refunds PPR and the equity capital EC complete the insurer’s liabilities. The EC is artificially split into a part that is invested in long-term assets (\(\mathrm{EC}^{\mathrm{lt}}\)) and a liquidity cushion (\(\mathrm{EC}^{\mathrm{st}}\)). Table 3 enumerates the variables used in the model.

The number of living policyholders for \(j \epsilon \{\mathrm{TDA},\mathrm{DHP}\}\) is assumed to evolve in the following way:where \(l^{\mathrm{SOCB}}_{m}\) are second-order calculation bases for the number of m year old men taken from the mortality table DAV2004R (without the security-margin imposed by law).⁶ We assume that each policyholder buys only one contract. The number of expected living (i.e. the number used by the insurer in period \(t-1\) to calculate their policy reserves in period t) is assumed to evolve as follows:where \(l^{\mathrm{FOCB}}_{m}\) are first-order calculation bases for the number of m-year-old men taken from the mortality table DAV2004R (with security margin). Note that we do not calculate this variable for DHP since we assume this product to pay a death benefit.⁷ A measure closely connected to the calculation bases and used at later points is the so-called number of discounted living at age m \(D_{m}\).⁸ It is defined asThe TDA’s “temporary final payout”, assumed to occur as lump-sum payment LSP, is given bywhere SP denotes the single premium paid up front by policyholders, h the policyholder’s age at the conclusion of contract, and n the contract duration in years. Note that, in general, \(\mathrm{LSP}_{t}\) denotes the expected payout at contract termination, disregarding bonus participation occurring after period t. It does not denote a payment occurring in period t. Let us now consider the development of the balance sheet items.

The transition mechanism of the long-term assets is given byThe short-term assets are calculated as the sum of the policy reserves and the equity capital invested short term. Since the short-term interest rate is zero, the development of short-term assets is given byThe development of the PR is given throughwhere multiplication with \(\gamma ^{\mathrm{TDA}}_{t} := L_{t}^{\mathrm{TDA}} \cdot (L_{t-1}^{\mathrm{TDA}})^{-1}\) reduces policy reserves to the amount needed to ensure payout for the remaining contracts. (Note that the risk result is taken into account in the bonus section and that for the DHP’s funds in the policy reserve, deaths are taken into account in the management decision in Eq. (19).)

If \(\mathrm{PR}_{t}^{\mathrm{DHP}} > 0 \), this liability item grows with the guaranteed interest rate \(i_{G}\). Thus, should funds be shifted back from the \(\mathrm{PR}^{\mathrm{DHP}}\) to the GF or EF, more funds need to be shifted back than what was taken earlier. Hence, the insurer needs assets that may be liquidated when this reallocation occurs. To take this into account, a growth through interest gains of the \(\mathrm{PR}^{\mathrm{DHP}}\) goes hand in hand with an equivalent decrease of \(\mathrm{EC}^{\mathrm{st}}\). Turning our attention to the development of \(\mathrm{EC}^{\mathrm{st}}\), note that for \(t \epsilon Y\), \(\mathrm{SP}^{\mathrm{DHP}}\) the single premium paid by DHP policyholders, and \(x\cdot \mathrm{SP}^{\mathrm{DHP}}\) the guaranteed benefit at the end of the contract term of the DHP, the amount of short-term liquidity needed to cover the DHPs’ interest gains is bound by the interest gains a single DHP policy would realise in the last year of contract term if all its funds were shifted to the insurer’s policy reserve (in that event, the DHP would pay only its guaranteed benefit and the interest gains would be smaller than \(i_{G}\cdot x\cdot \mathrm{SP}_{\mathrm{DHP}}\)) multiplied by the current number of DHP policies held by the insurer at the beginning of the previous year (\(L_{t-12}^{\mathrm{DHP}}\)), giving the rough estimation of maximum liquidity neededFor the sake of simplicity, we assume that the insurer makes its investment decisions at the beginning of each year and ensures that the liquidity cushion contains this amount.If the insurer needs to add money to the liquidity buffer, it does so by liquidating long-term assets at a cost \(\Theta \ge 0\). If the buffer exceeds the needed amount, the surplus is invested in long-term assets. Furthermore, a dividend \(\delta \) as percentage of the equity capital is paid out to the shareholders. The long-term assets are thus adjusted in each period \(t \epsilon Y\) toThe GF and EF are assumed to evolve following two GBMs, given by the stochastic differential equationsfor the GF andfor the EF, with constant drift coefficient \(\mu ^{i}\), volatility coefficient \(\sigma ^{i}\), two Wiener processes \(W^{i}(t)\) with correlation \(\rho \), and initial value \(S^{i}(0) = 1\), \(i \epsilon \{\mathrm{GF}, \mathrm{EF} \} \). Inspired by Kochanski and Karnarski (2011), the development of the GF and the EF is given byandwhere \(\nu \) denotes a yearly management fee, \(\lambda \) denotes the maximum loss in per cent that the downside protection of the GF allows, and \( P_{t,1/12}^{\mathrm{CP}}\) is a parameter from the Black–Scholes formula for option pricing.⁹ In a second step, the funds of the DHP are reallocated following a management decision \(MD\left( \mathrm{AV}_{t},t\right) \) inspired by Kochanski and Karnarski (2011). We model a product that is monitored on a monthly basis.¹⁰ The bonus the DHP holders are entitled to is added to the current account value (AV) of the DHP policies before funds are reallocated at the beginning of each period (note that \(\mathrm{Bonus}^{\mathrm{DHP}}_{t} = 0, \text { for } t \notin Y \)):The once again aggregated values for PR, GF, and EF are\(G_{t}\) is defined as the amount needed in period t to ensure the guaranteed benefit \(x\cdot \mathrm{SP}^{\mathrm{DHP}}\) via the actuarial interest rate at contract terminationDividing by \(L_{t}^{\mathrm{DHP}}\) in Eq. (18) and multiplying by \(L_{t+1}^{\mathrm{DHP}}\) in Eqs. (19), (20), and (21), we eliminate the death benefit paid out to deceased DHP policyholders. The development of the provision for premium refunds is given byThe insurer’s long-term equity capital is given by the residual valueBelow, special care is given to describing the modelling of inflows to the PPR and the bonus allocation mechanism, the stage of the interaction effect.

The input parameters for the model are displayed in Table 4 and are intended to represent a German life insurance company after the introduction of DHP in 2007. The maximum loss of the GF \(\lambda \) and the management fees \(\nu \) are taken from Kochanski and Karnarski (2011) whose model of the DHP inspired us. The drift parameters \(\mu ^{\mathrm{GF}}, \mu ^{\mathrm{EF}}\) and the volatility parameters \(\sigma ^{\mathrm{GF}}, \sigma ^{\mathrm{EF}}\) are adaptations to the values proposed by Kochanski and Karnarski (2011), taking into account different long-term interest rates and differing levels of risk and upward potential for GF and EF. Taking two correlated GBMs instead of one is a deliberate decision that in our view is closer to the actual investment praxis. A robustness check is performed on this decision. The interest rates \(i_{A}^{\mathrm{lt}}\), \(i_{A}^{\mathrm{st}}\), and \(i_{G}\) were chosen to match typical pre 2008 values. We want to stress that we assume constant interest rates and a normal yield curve. The constant interest rates (as opposed to stochastically modelled interest rates) are a modelling convenience. Since our main interest is the TDA’s payout and the investments in the insurers’ PR are assumed to be long term, we think that taking stochastic interest rates boils down to taking their long-term means. As for the inverted yield curve (i.e. for the case that \(i_{A}^{\mathrm{st}} > i_{A}^{\mathrm{lt}}\)), we abstract from a scenario where TDA policyholders potentially profit from a reallocation of DHP funds into the insurer’s policy reserves. However, since an inverted yield curve is an exceptional event that does not necessarily accompany a shift of DHP funds, we are confident that we capture the essential long-run dynamics despite leaving it out of the model. We model a homogeneous cohort of policyholders aged \(h=37\) at the conclusion of the contract. All products are bought for the same single premium SP = 100 and have the same contract duration \(n=30\). Like in Bohnert et al. (2014), the guaranteed benefit of the DHP is equal to the single premium paid by policyholders. The ratio of DHP policies to TDA policies is 1/2 but is varied during our analysis. The equity capital (\(\mathrm{EC}_{0}^{\mathrm{st}}+\mathrm{EC}_{0}^{\mathrm{lt}}\)) and the initial provision for premium refunds \(\mathrm{PPR}_{0}\) are 1.5% and 7.8% of the balance sum, respectively, thus matching the ratio over all German life insurers in 2007 according to the primary insurers’ statistics by the Federal Financial Supervisory Authority (“Erstversicherungsstatistik der BaFin”). A yearly dividend \(\delta \) is paid out to the shareholders. For simplicity, we assume the dividend to be equal to the long-term interest rate \(i_{A}^{\mathrm{lt}}\). The waiting time parameter d is set to 5. This reflects the insurer’s choice to maximise the funds in the PPR. Furthermore, we decided to make the liquidation of long-term assets costly. The parameter \(\Theta \) makes this assumption explicit and is set to \(\Theta = 0.05\).

This section presents the results obtained via our Monte Carlo simulation. We first illustrate the interaction mechanism using three exemplary paths from our simulation with standard parameters. Second, we discuss the influence of the initial number of DHP contracts \(N^{\mathrm{DHP}}_{0}\) on the surplus participation of the TDA policyholders. Third, we provide some sensitivity analysis of our results by performing robustness checks on input parameters. An overview of some variables that are not core to our analysis is given in Table 6 in “Appendix”. Figure 3 shows three developments in the account value of the DHP and its decomposition over time in three different scenarios: one scenario corresponds to the median account value of the DHP as well as a good and a bad scenario (corresponding to the paths closest to the 95% and 5% quantiles of the DHP’s account value).

The starting allocation per individual contract in \(t=0\) is computed by the management decision function and given by \(\mathrm{PR}^{\mathrm{DHP},\mathrm{Ind}}_{0}=0\), \(\mathrm{GF}^{\mathrm{Ind}}_{0}=55.4\), and \(\mathrm{EF}^{\mathrm{Ind}}_{0}=44.6\). In subfigures (a) and (b) of Fig. 3, we see two scenarios in which the GF’s and EF’s development is good enough to sustain the DHP without funds being shifted to the insurer’s PR. The payout of roughly 300 and 150 in the 95% quantile path and the median path, respectively, is not a brilliant performance compared to a payout of more than 250 (without surplus participation) for the TDA. However, given that, unlike the TDA the DHP contains a death benefit equal to the account value at the time of death, this extra guarantee needs to be taken into account when comparing product performance. This is beyond the scope of this article.

Shifting our attention to the in- and outflow to the PPR and thus to the stage of the interaction, Figs. 3, 4, and 5 show that the bonus payments vary significantly from the benchmark case if DHP funds are shifted to the insurer’s policy reserves. In path (c), bonuses to TDA policyholders drop to a quarter of the benchmark bonus payments in the worst years and stay low as long as DHP funds remain in the PR. This is not compensated by the higher bonus payments occurring in years 5 and 13. The spike in year 5 (common to all paths) is caused by lower inflows to the PPR in the first five years. To avoid a PPR that is higher than the last five inflows to it (see Eq. (35)), the insurer has to pay out a bigger bonus than in the benchmark case. The spike in year 13 is caused by the sudden drop in PPR inflows in the same years, dramatically lowering the sum of the PPR inflows of the last five years. Again, the insurer awards a big bonus in order to keep the PPR below its legal cap. Note that the PPR inflows and the bonuses are below the benchmark value, even if no DHP funds are shifted to the PR. This phenomenon is readily explained by the lower interest gains caused by the presence of a buffer invested in short-term assets. Equation (26) shows that while in the benchmark case, \(A^{\mathrm{st}}=0\), the insurer’s interest gains are smaller as soon as \(N^{\mathrm{DHP}}>0\) and therewith \(A^{\mathrm{st}}>0\). Of course, this crucially depends on the assumption \(i^{\mathrm{st}}_{A}< i^{\mathrm{lt}}_{A}\) and less so on the assumption that short-term interest is equal to zero, an assumption we make for the sake of simplicity. The presence of such a buffer is intended to capture what Bettels et al. (2011) call “liquidity risk” arising from DHP policies.

To what extent does the ratio of DHP to TDA policies drive results? We discuss this question not only for our standard parameter set but for two variations of it: one representing a wider profit margin for the enterprise (\(i^{\mathrm{lt}}_{A}=0.04\), ceteris paribus), one representing lower interest rates, keeping the profit margin constant (\(i_{G}=0.009\), \(i^{\mathrm{lt}}_{A}=0.014\), ceteris paribus). In the following paragraphs, we measure the adverse effect on the TDA policyholders by the percentage loss of the bonus share BS of the final TDA payout so as to have an easily interpretable measure that is comparable across parameter specifications. This bonus share is the part of the final payout that comes from bonus payments reinvested in the insurer’s policy reserve when taken from the PPR. The benchmark bonus share \(\overline{\mathrm{BS}}\) (calculated for each parameter set) is defined aswhere \(\overline{\mathrm{LSP}}_{n}\) is the final LSP in the corresponding benchmark scenario (i.e. \(N^{\mathrm{DHP}}_{0}=0\), all other input parameters being equal), SP is the single premium, D is the number of discounted living from Eq. (5), and h and n are the initial age of the policyholders and the contract duration, respectively. The percentage deviation of the bonus share for any given path is defined asOne general insight is common to all the three scenarios: the higher the ratio of DHP to TDA policies, the higher the mean percentage loss of the bonus share \(\Delta \left( \mathrm{BS}\right) \) and its variance, as shown in Fig. 6.

The results are summarised in Table 5 as results of a regression of \(\Delta \left( \mathrm{BS} \right) \) on \(N_{0}^{\mathrm{DHP}}\).¹⁹ The results can be interpreted as follows: for our standard parameter set, an increase of \(N^{\mathrm{DHP}}_{0}\) of 1000 (given \(N^{\mathrm{TDA}}_{0}=10,000\)) reduces the bonus share by 1.143 percentage points as compared to \(N^{\mathrm{DHP}}_{0}=0\). This effect is smaller for a bigger profit margin and bigger for lower interest rates. The variance of results increases in both cases, as indicated by the variance measure \(1-R^{2}\).

A higher profit margin mitigates the negative result for the TDA. In our simulation study, this can be traced back to PPR inflows and, consequently, bonus payments being much closer to their benchmark values. This in turn can be easily understood by noticing that short-term assets, stemming from DHP funds shifted to the insurer’s PR and the liquidity buffer, are relatively less important for higher long-term interest rates when calculating the net interest surplus (see Eq. (26)). Lowering the guaranteed interest rate \(i_{G}\) and the long-term interest rate \(i_{A}^{\mathrm{lt}}\) drastically worsens the results for the TDA. Mirroring the effect described for higher \(i_{A}^{\mathrm{lt}}\), a lower long-term interest rate increases the influence of the liquidity cushion and the \(\mathrm{PR}^{\mathrm{DHP}}\) on the net interest surplus in Eq. (26). For positive \(N^{\mathrm{DHP}}_{0}\), PPR inflows and bonuses are more distant from the benchmark case (\(N^{\mathrm{DHP}}_{0}=0\)). Moreover, a lower actuarial interest rate increases the amount needed in every period t to ensure the guaranteed benefit \(G_{t}\) of the DHP (see Eq. (22)). This influences the management decision (see Eq. (19)) and makes shifts to the insurer’s PR more likely and to occur earlier than in the other scenarios. More interaction effects lead to a stronger final effect on the TDA’s surplus participation. On the one hand, this should not be taken as indicative of the current real-world situation. Indeed, these days, most insurers would be glad if they could earn 1.4% interest on safe long-term investments. Furthermore, the initial values of the PPR and equity capital were set to values typical for the pre-crisis period, further complicating a transfer of the results to the present day. On the other hand, the result should urge policymakers to investigate whether the adverse effects described in this paper may have gained momentum in recent years.

The results are robust to differences in the waiting time d, the liquidation cost parameter \(\Theta \), the dividend parameter \(\delta \), and the management fees \(\nu \). If we model the DHP’s GF and EF based on one geometric Brownian motion instead of two correlated GBM, we obtain a slightly lower mean \(\Delta \left( \mathrm{BS} \right) \) and variance. Lowering the guaranteed benefit of the DHP to 90% instead of 100% of the single premium paid up-front as well as increasing the drift of the GF and EF mitigates the negative effects on \(\Delta (\mathrm{BS})\). In line with Bettels et al. (2011), increasing the volatility of the GF and EF exacerbates the effects on \(\Delta (\mathrm{BS})\) and thus increases the cost for the TDA. These last three results are consistent with the conclusion that negative interaction effects occur whenever the DHP’s funds are shifted to the insurer’s PR: while a lower guaranteed benefit as well as higher drifts of the funds lead to fewer instances of shifted funds, a higher volatility of the funds acts in the opposite direction. For more detail, the results of the robustness checks are displayed in Table 11 and illustrated in Fig. 7 in “Appendix”.