The Impact of Product-Dependent Policyholder Risk Sensitivities in Life Insurance: Insights from Experiments and Model-Based Simulation Analyses

3.1 Methodology: Choice-Based Conjoint (CBC) Analysis

To estimate the consumers’ willingness to pay, we perform choice-based conjoint (CBC) analyses, where the survey participants are asked to imagine a realistic purchase situation and must choose the most preferred product profile from a set of alternatives multiple times. In each set of alternatives, the product profiles differ in a fixed number of attributes K, where each attribute k can take one of M _k levels (see Louviere and Woodworth 1983). Assuming that the survey participant i has a linear additive utility function in the observed attributes, the participant’s i deterministic utility of alternative a can be given as

where the vector of dummy variables

describes the design of alternative a and the vector β _i describes the unknown part-worth utilities for individual i (see, e.g. Steiner and Meißner 2018). By extending the deterministic utilities v _ia by adding independent Gumbel-distributed error terms ɛ _ia , i.e. V _ia = v _ia + ɛ _ia , the fundamental equation of a CBC analysis can be derived by random utility theory, which is known as the multinomial logit model (see, e.g. McFadden 1974). It is given as

where P y i = a | A denotes the probability of individual i choosing alternative a from a set of alternatives A (see Louviere and Woodworth 1983).

When estimating the part-worth utilities β _i on an aggregate level, i.e. all individuals i share the same vector β _i = β, classic regression methods can be utilized, where the left side of Equation (2) is replaced by the observed frequency of respondents choosing alternative a from the set of alternatives A (see Louviere and Woodworth 1983). However, Markov chain Monte Carlo hierarchical Bayes methods make it possible to estimate the part-worth utilities β _i on an individual level, which is superior as it allows for modeling heterogeneity within the population of the survey participants (see, e.g. Lenk et al. 1996). Given Equation (1), the estimated single entries β _ikm of the vector β _i then describe the utility for individual i, when the product’s feature k equals level m. Therefore, the part-worth utilities directly provide insights into the customers’ product preferences. Furthermore, if the attribute k = 1 represents the product’s price levels p r i c e 1 , … , p r i c e M 1 , the part-worth utilities can be used to calculate more sophisticated metrics, like the marginal willingness to pay (MWTP) for changing the level of a non-price attribute, as in, e.g. Braun, Schmeiser, and Schreiber (2016). In this case, to compute the marginal willingness to pay M W T P i k h , l of an individual i for changing a non-price attribute k ≠ 1 from some level l to level h, the utility gain β _ikh − β _ikl is divided by a price coefficient V _price, i.e.

where the price coefficient

defines the increase in utility when decreasing the product’s price by one unit (see Braun, Schmeiser, and Schreiber 2016). The idea behind employing Equations (3) and (4) to compute the marginal willingness to pay is that the change in price should be equal to the change in utility divided by the utility gain for decreasing the price by a single unit.

The main advantage of using CBC analysis to indirectly compute the customers’ willingness to pay over direct approaches, where the survey participants must directly state their willingness to pay for certain product profiles, is that in a CBC analysis a more realistic purchase situation is provided based on intuitive and simple selection tasks (see DeSarbo, Ramaswamy, and Cohen 1995).^[5] Furthermore, a CBC analysis is very suitable for life insurance products in general, as research by Miller et al. (2011) has shown its advantages over other approaches in the case of higher-priced and less frequently purchased product categories with existing market competition. In life insurance in particular, the product design is more complex and contracts run for several years. This makes it more difficult for customers to evaluate default probabilities without any anchor points. Therefore, in contrast to previous research in non-life insurance (e.g. Wakker, Thaler, and Tversky 1997; Zimmer et al. 2018), we offer survey participants different decision alternatives rather than ask them directly for their willingness to pay.

Therefore, while incentive-compatible experiments, where the respondents’ decisions have an actual effect in the real world,^[6] are known to provide more accurate results (see, e.g. Miller et al. 2011; Zimmer et al. 2018), the CBC analysis is sufficiently suitable for our purposes. First, a market research company ensures the generation of high-quality survey responses from a large and balanced survey panel by confirming personal information during the recruiting process, as well as providing incentives such as monetary payouts, in addition to an ongoing monitoring process regarding the quality of the panel responses. Second, while we estimate the marginal willingness to pay with hierarchical Bayes methods on an individual level (see Equation (3)), we use the median of the single values to calibrate the functional relationship between the communicated default probability and the policyholders’ willingness to pay in the model, which we then employ in our simulation analysis. This makes the approach relatively robust against outliers, e.g. single respondents who may not state their true preferences. Finally, we do not have to solely rely on the “accuracy” of the experimental results, but can run various sensitivity analyses in the simulation analysis based on the experimentally observed values.

3.2 Experimental Design

To investigate the impact of a reported shortfall probability on policyholders’ willingness to pay and whether they are product-specific, we design two experiments for CBC analyses for annuities and term life insurances, respectively. To ensure that the survey participants understand the insurance products, we consider immediate annuities without bequest, which are sold against single premiums. In the case of term life insurance, we follow the setting in Braun, Schmeiser, and Schreiber (2016), where a monthly premium is considered.

At the beginning of both surveys, a specific situation in life is described, which is intended to place the survey participant in a realistic purchase situation for the given type of contract. For example, in the case of the immediate annuity, the following text guides the participant into the questionnaire, which is similar to the setting in Fuino, Maichel-Guggemoos, and Wagner (2020):^[7]

“Imagine that you are 65 years old, you are about to retire and you would like to invest 100,000 € in an immediate annuity, which from now on pays you a monthly annuity payment for a certain period of time. The period of time and the amount of the monthly annuity payment depend on the specific product design. In each of the following 12 scenarios, please select the product you most prefer.”

The age of 65 years is motivated by the average retirement age of Germans, which is normally slightly below the statutory retirement age of 67 years, and the amount of 100,000 € has previously been employed in a CBC analysis for immediate annuities in Shu, Zeithammer, and Payne (2016). For the description of a realistic purchase situation for a term life insurance, we build on the empirical findings by Swiss Re (2013) as well as the characteristics in Schreiber (2017), i.e. a 40-year-old, married person with children in a stable relationship, who is making more money than their partner. Furthermore, in the case of term life insurance, we used the same amount of 100,000 € for the sum insured as in Braun, Schmeiser, and Schreiber (2016).

The survey participants have to complete 12 selection tasks, where each time three different product profiles are randomly shown based on a fractional factorial choice design in order to optimize the balance and overlap of the shown attribute levels. We thereby follow the recommendations in Johnson and Orme (2003) to keep the number of attributes and levels in the CBC selection tasks as small as possible, i.e. three attributes with at most five levels, as shown in Table 1.

First, we include the contract term as a key component for the two insurance products in our analysis, where we use the same three levels of 10, 15 and 20 years for the term life insurance as in Braun, Schmeiser, and Schreiber (2016). To allow for comparability between the two contract types, we employ the same number of three levels and the distance of five years between the levels for the contract terms of the annuity, but postponed by 10 years to better fit the described purchase situation, i.e. for the annuity we utilize the contract terms of 20, 25 and 30 years.^[8]

As our experimental research aims to investigate the impact of an insurer’s reported default probability on its customers’ willingness to pay, we include the one-year default probability as a second attribute. Here we use five equally distanced levels with a relatively wide range from 0 % to 4 %. While we are aware that one-year default probabilities of more than 1 % are an unrealistic scenario in practice, we aim to provide clearly separated levels, so that the survey participants can better differentiate between the displayed choices. Furthermore, in reality it is more likely that policyholders access the insurers’ financial safety level through verbal ratings instead of a numerical expression (e.g. through comparison sites like www.quotacy.com in the US or www.check24.de in Germany as well as financial advisors), and experimental research by Zimmer et al. (2018) has shown that individuals strongly overestimate default probabilities in the case of verbal ratings.

For the third attribute we include the price of insurance, taking into account the communicated setting in the survey. The price for the term life insurance for a 40-year-old policyholder is given by the monthly premiums based on the sum insured of 100,000 € and the respective contract term. In the case of the immediate annuity, the communicated initial single premium of 100,000 € for a 65-year-old person and the specified contract term determines the monthly annuity payments. To obtain comparability between the two different contract types and to derive the marginal willingness to pay, we compute the insurance prices based on the actuarially fair values with five equally distanced price steps for the two product lines. The formulas for the actuarially fair values are shown in Section 4.

For both contract types, price steps of 10 % are used, in which the actuarially fair premium for term life insurances is increased and the actuarially fair annuity is decreased, i.e. the prices are given by 100 %, 110 %, 120 %, 130 % or 140 % of the actuarially fair premium for term life insurances and 100 %, 90 %, 80 %, 70 % or 60 % of the actuarially fair annuity. For the computation of the actuarially fair values (without considering the default risk), the actuarial interest rate is set to 0.25 %. The death probabilities are based on the first-order mortality tables “DAV 2008 T” and “DAV 2004 R” of the German Actuarial Association, and we employ different death probabilities for men and women in order to present more individual prices. For this, the survey participants’ gender is asked directly before the description of the purchase situation and the gender-specific prices are used for the CBC selection tasks. Similarly, in the case of term life insurance we further differentiate between prices for smokers and non-smokers, as is undertaken in practice and in Braun, Schmeiser, and Schreiber (2016).

For all three attributes (contract term, one-year default probability and price), a short explanation is provided at the bottom of each of the 12 selection tasks in order to ensure that all respondents understand the mechanics of the respective life insurance product (see Figure 1 for an illustration).^[9] Furthermore, in each of the 12 selection tasks we include a “no-buy-option”, allowing the respondents to refrain from selecting one of the three given alternatives. The 12 selection tasks are followed by two control questions, asking on a scale of 1–7 how realistically the introductory described purchase situation was perceived as well as how understandable the selection tasks were. This allows us to further increase the data quality. After the selection tasks, demographic questions about the respondents’ age, size (as a control question), education, job, wage and previous experience with life insurance products are asked, using the wording from Unger, Steul-Fischer, and Gatzert (2022).

Figure 1:

Example of a single CBC task (translated from German).

3.3 Sample Description and Results

For both CBC analyses, we have used the all-in-one survey research platform Conjointly^[10] to create and evaluate the surveys. Furthermore, we have employed the paid service of Conjointly to recruit the survey participants in order to access a balanced and high-quality survey panel in Germany, in which 218 (220) participants completed the survey about annuities (term life insurances). We have excluded five (six) respondents due to fraudulent behavior^[11] and six (four) because they had the same Internet protocol (IP) address. Finally, we have restricted our analysis to those respondents who answered both control questions with a score of at least 3 out of 7, which led to the exclusion of an additional 11 (22) participants. As a result n ^R = 196 (51.0 % female, average age 47.2 years) participants are included in the analysis of annuities and n ^S = 191 participants (40.8 % female, average age 49.3 years) in the analysis of term life insurances. In the survey about annuities (term life insurances) 30.1 % (28.1 %) of the respondents stated that they own one or more related life insurance products and additionally 26 % (18.8 %) of the respondents stated that they thought about buying one. In the survey about annuities the “no-buy-option” was selected in 12 % of the selection tasks and in 19 % in the case of the term life insurances.

Based on the respondents’ choices, two separate multinomial logit models as described by Equation (2) have been fitted, whereby Conjointly uses a Markov chain Monte Carlo hierarchical Bayes method to estimate the part-worth utility vectors β _i on an individual level. For both surveys, the multinomial logit models have yielded a strong fit with McFadden’s pseudo R² of 65.4 %.

As a result, Figure 2 shows the average part-worth utilities β ̄ k m = 1 / n ⋅ ∑ i = 1 n β ikm for annuities and term life contracts for the different attribute levels, which were transformed to zero-centered for each attribute and standardized such that the utility ranges (i.e. the absolute difference between average part-worth utilities of the best level and worst level per attribute) for the three different attributes sum up to 100 %. One can see that longer contract terms yield lower average utilities for both products, i.e. the average customer is not willing to pay the actuarially fair price increase for longer contract terms of 20/30 years, especially in the case of the term life insurance. One reason for this might be that policyholders do not understand why the monthly premium for the term life insurance increases for longer contract terms, as they do not associate a longer contract term with a higher likelihood of receiving a payment due to their increasing age and therefore increasing death probabilities. Regarding the reported one-year default probability and the product’s price, Figure 2 shows the expected order, as the part-worth utilities decrease for the increasing reported one-year default probability as well as for higher price levels.

Figure 2:

Average part-worth utilities for the different attribute levels (described in Table 1) for annuities and term life insurances based on the CBC analyses. Notes: The figure displays the average part-worth utilities β ̄ k m = 1 / n ⋅ ∑ i = 1 n β ikm , which are transformed to zero-centered by subtracting the average attribute utility β ̄ k = 1 / M k ⋅ ∑ m = 1 M k β ̄ k m and standardized by division by the maximum utility gain G max = ∑ k = 1 K max β ̄ k m − min β ̄ k m (with 95 % confidence intervals, see whiskers).

Besides the part-worth utilities, we evaluate the marginal willingness to pay when increasing the reported one-year default probability δ from 0 % to the higher levels ranging from 1 % to 4 %, using the median of the individual values given in Equation (3).^[12] As our experimental design defines the price for insurance indirectly, where the actuarially fair values are changed in 10 % increments, the marginal willingness to pay can be directly expressed as the percentage change compared to the actuarially fair premium in the case of term life, whereby the values for the annuities are also transformed accordingly.

The results for the marginal willingness to pay depending on the increase of the reported one-year default probability δ are displayed in Figure 3. For example, in the case of the term life insurance, the value of −2.55 means that the customers would (in terms of median) decrease their willingness to pay by 2.55 %, if the insurer increases the reported one-year default probability from 0 % to 1 %. One can see that for both types of insurance, the customer’s willingness to pay strictly decreases for an increasing reported one-year default probability, which is in line with the previous experiments in non-life insurance (e.g. Zimmer, Schade, and Gründl 2009; Zimmer et al. 2018). In contrast to previous research, however, we observe only a comparably small jump of the marginal willingness to pay if the default probability is increased from δ = 0 % to the first level of δ = 1 %. One explanation for this is that in previous research, the survey participants had to directly state their willingness to pay for a given default probability without any reference point. This might cause the decision to be based on a (binary) comparison between no default and a possible default. In our experimental setting, multiple options with different default probabilities are shown, which allows the respondents to better compare and rank the different levels of default probability.

Figure 3:

Median marginal willingness to pay for annuities and term life insurances depending on the reported one-year default probability δ based on the CBC analyses (reduction in median MWTP as compared to δ = 0 %).

Furthermore, this shows that the median marginal willingness to pay is indeed product-dependent, as already indicated by empirical research in different contexts (e.g. Eling and Schmit 2012; Phillips, Cummins, and Allen 1998). For reported one-year default probabilities of 1 % and 2 %, the decrease in policyholders’ willingness to pay is more pronounced for the annuity than for the term life insurance. This could be caused by the customers’ age, i.e. an immediate annuity is bought at an older age and a term life insurance is purchased at a younger age. While at an older age, it is impossible to compensate a default by working harder or working more, it is typically still possible to react to a default in younger ages. Another explanation could be the higher one-off premium of the immediate annuity, which might sensitize policyholders to think more about an even very unlikely potential default. Furthermore, it is often observed that annuities are unappealing for many individuals despite being theoretically beneficial in terms of expected utility, which is also known as the “annuity puzzle” and which may be even more present in case of a potential default.

However, for a default probability of 3 % and 4 %, this effect is reversed, i.e. the decrease in policyholders’ willingness to pay is more pronounced for the term life insurance than for the annuity. One explanation for this behavior could be that policyholders are more sensitized toward a potential reduction or default of their pension, possibilities that are often discussed publicly in many countries, which might carry over to the private sector, and thus react even for (comparably) lower default probabilities. For term life insurances, this sensitization might not exist to the same extent, and thus customers respond rather to higher default probabilities. The reaction might then be stronger compared to the annuity, as a default in the case of a term life insurance not only harms one’s own financial situation, but also that of relatives such as children or spouses. A more technical explanation would be that a term life insurance covers a low frequency but high severity risk, which is reversed for an annuity. As a result, it is more likely that a potential default of the life insurer negatively affects an owner of an annuity, but if the default affects an owner of a term life insurance, the economic consequences would be then more severe. Therefore, the default probabilities might seem smaller when buying a term life insurance, but individuals react rather for (comparably) higher default probabilities but then more vigorously. However, the results of our CBC analysis only provide information about how policyholders react (as a central input and starting point for the following model and simulation study with sensitivity analyses) and not why. For this, further research is required, where our findings could serve as a starting point.

4.1 The Product Mix and Corresponding Liabilities

We consider a fixed time horizon of T years, where cash flows only arise at the beginning or ending of each year. For any year t ∈ {1, …, T} we denote with t ⁻ the ending of the previous year and for t ∈ {0, …, T} the beginning of the current year with t ⁺. At time t = 0⁺ a total of N insurance contracts are taken out, consisting of N ^R temporary annuities and N ^S temporary term life insurances, i.e. N = N ^R + N ^S .

Both products are sold against single premiums P ^R and P ^S at time t = 0⁺ and have a contract term of T years. While monthly premiums would be more common in the case of the term life product as also used in the experimental setting (Section 3), comparability between the two products is improved and portfolio effects can be better identified if we assume single premiums for both. The annual annuity payment is denoted with R _t and in the case of term life insurance, the sum insured paid out if the policyholder dies in year t is denoted with S _t . Both payments can vary over time because of surplus distribution, as explained later, whereby we also study the case without surplus participation as in the experimental setting in a sensitivity analysis. The single premiums are further decomposed into the actuarially fair premium without default risk and a loading. The respective actuarial premiums are given by

with a x : T ̄ = ∑ t = 1 T v t t p x a n d A x | T = ∑ t = 0 T − 1 v t + 1 ⋅ p x t ⋅ q x + t , where v = ( 1 + r G ) − 1 denotes the discount factor with an actuarial interest rate r ^G , p x t represents the probability of an x-year old surviving t years, and q _x+t is the probability of dying within one year for a person of age x + t.

Adding a loading λ, which accounts for administrative and acquisition costs, the single premiums are given by

At a later point, these single premiums will be further adjusted to account for the policyholders’ willingness to pay.

The book value of liabilities at the end of each year is given by the actuarial reserves. The actuarial reserve for the pool of N ^R sold annuities at time t ⁻ is given by

where d i R denotes the number of deaths in year i from policyholders with annuities and V x R t = R t + 1 ⋅ a x + t : T − t ̄ (see Bohnert, Gatzert, and Jørgensen 2015). Analogously, the actuarial reserve for the pool of N ^S sold term life insurances at time t ⁻ is given by

where d i S denotes the number of deaths in year i from policyholders with term life insurances and V x S t = S t + 1 ⋅ | T − t A x + t . Therefore, as the overall actuarial reserve we get

4.2 Development of Assets and Liabilities

At the beginning of the contract term, shareholders make an initial contribution E ₀, which together with the premiums results in an initial investment in assets of

We assume that assets follow a geometric Brownian motion, i.e.

with constant drift μ, volatility σ and (W _t ) a standard Brownian motion. The solution to this equation is given by

for some initial value I ₀, and with r _t denoting the continuous one-period return. The asset development can thus be described by

The liabilities are given by the actuarial policy reserves P R t − . The generated surplus is first transferred to a buffer account defined by B t − = A t − − P R t − − E t , t = 1 , … , T , where equity capital (E _t ) is assumed to be constant over time. For the transition from time t ⁻ to t ⁺ three different cases can be distinguished (see Bohnert, Gatzert, and Jørgensen 2015). First, if the buffer account is large enough to pay out a constant fraction ξ of the shareholders’ initial contribution, dividends are paid out, i.e. D t = ξ ⋅ E 0 i f B t − ≥ ξ ⋅ E 0 , t = 1 , … , T . In this case, the buffer account is adjusted by B t + = B t − − D t , t = 1 , … , T and the assets at the beginning of year t + 1 are given by A t + = A t − − D t , t = 1 , … , T . Second, if the buffer account is positive, but not large enough to pay the fraction ξE ₀ as a dividend, then the amount of buffer account is paid as a partial dividend, set to zero, and the assets are adjusted, i.e. D t = B t − , B t + = 0 and A t + = A t − − D t , t = 1 , … , T . Furthermore, if the buffer account is negative, but equity capital can cover the losses, i.e. B t − < 0 a n d B t − + E t ≥ 0 , then no dividends are paid, the buffer account is set to zero and the assets stay unchanged. Formally this is described by D _t = 0, B t + = 0 and A t + = A t − , t = 1 , … , T . Third, if the buffer account is negative and equity capital is not sufficiently high enough to cover the losses, i.e. B t − + E t < 0 , the insurer is insolvent and is liquidated prematurely. In this case, the current assets are reduced by a bankruptcy/liquidation costs coefficient c and the remaining capital ( 1 − c ) ⋅ A ( t − 1 ) ⋅ exp r t is distributed to the policyholders with open contracts based on their reserves.

4.3 Surplus Distribution Scheme

In addition to their guaranteed sums insured, policyholders receive a share in the insurer’s surplus. For this purpose, the policy interest rate r t P that includes the guaranteed interest rate as well as surplus is calculated as

with a target buffer ratio γ, surplus distribution ratio α and an initial buffer account B 0 + (see Bohnert, Gatzert, and Jørgensen 2015; Grosen and Jørgensen 2000). We assume that the policy interest rate that exceeds the guaranteed interest rate r ^G is paid on the contract’s book values and that it is annuitized over the remaining contract term, thus increasing the guaranteed annuity payment and death benefit payment to

and

4.4 Policyholders’ Willingness to Pay

To integrate the policyholders’ willingness to pay in our model, at the beginning of the contract term the insurer reports an upper bound δ ∈ [0, 1) for its one-year default probability that for simplicity reasons is assumed to be constant over the entire time horizon.^[13] As this reported default probability will generally influence the paid amount of single premiums at time t = 0⁺, the premiums P ^R and P ^S in Equation (5) are multiplied by a factor ρ δ ; z that depends on the communicated one-year default probability δ and a scaling factor z. Here we use the same formula as in Eckert and Gatzert (2018), i.e.

where PR denotes the premium reduction, which will be calibrated based on the experimental results, as will be explained later. The scaling factor z takes the customers’ search costs into account, as the default probability may not be directly given to the policyholder, and also allows for running sensitivity analyses with respect to the policyholders’ risk sensitivity. To model the experimental observation that customers’ risk sensitivity to the reported one-year default probability can vary between different product lines, we assume different functional forms for the premium reduction PR ^R of annuities and PR ^S of term life insurances as well as scaling factors z ^R and z ^S , yielding two potentially different factors ρ R δ ; z R = max 1 − z R ⋅ P R R δ ; 0 and ρ S δ ; z S = max 1 − z S ⋅ P R S δ . Assuming that P R R 0 = P R S 0 = 0 , the loadings 1 + λ R and 1 + λ S in Equation (5) can now be interpreted as the maximum loading that policyholders would accept in the case without default risk, as was also argued in Gatzert and Kellner (2013).

4.5 Fair Valuation from the Policyholders’ and the Shareholders’ Perspective

During the contract term, the true shortfall probability should not exceed the reported one-year default probability. The underlying shortfall probability over the entire contract term is thereby defined as

where T s = inf t = 1 , … T : A t − < P R t − denotes the stopping time for the first occurrence of default. To account for the fact that the insurer typically only reports one-year default probabilities as, e.g. the case with Solvency II, the overall shortfall probability must be decomposed. For this purpose, we define the conditional annual shortfall probability as

describing the situation that the insurer becomes insolvent in year t, given that the insurer was solvent until year t − 1.

To ensure that the reported one-year default probability δ is not exceeded for each contract year, the maximum one-year default probability

must satisfy the inequality

For example, this could be achieved by calibrating the initial contribution E ₀ by

This approach is closely related to the estimation of the multistep value at risk measure in Wong, Sherris, and Stevens (2017). Other possibilities for the insurer are, for example, to choose a risk-reducing product composition of term life insurances and annuities or to invest in less risky assets (see Gründl, Post, and Schulze 2006).

To ensure a fair situation for the shareholders, the dividend rate ξ is calibrated by means of risk-neutral valuation such that the initial contribution of the shareholders equals the discounted (with the risk-free interest rate r _f ) expected value of dividends and the final payment under the risk-neutral pricing measure Q, i.e.

(see Bohnert, Gatzert, and Jørgensen 2015), where under the risk-neutral pricing measure Q the drift μ in Equation (6) is replaced with the risk-free interest rate r _f and the standard Brownian motion (W _t ) by a Q-standard Brownian motion W t Q . Since actuarial premiums are computed based on mortality tables with safety loadings, mortality risk has already been considered and is thus neglected here.

5.1 Input Parameters

We assume a time horizon of T = 30 years, where at the beginning a total of N = 100,000 contracts are taken out. All annuities are calibrated for x ^R = 65-year-old males and term life insurances for x ^S = 40-year-old males. The actuarially fair premiums are computed based on first-order mortality tables “DAV 2008 T” and “DAV 2004 R,” respectively, of the German Actuarial Association, in which a static life table is used for annuities. The actuarial interest rate r ^G is set to 0.25 %. We assume an initial annuity payout of R ₁ = 1, which results in an actuarially fair single premium of P a R = 15.10 . To obtain better comparability between different product mixes, the initial sum insured is calibrated to S ₁ = 108.81, which results in the same price P a R = P a S = 15.10 . The equityholders’ initial contribution E ₀ is set to 8 % of the total initial premiums without loadings, yielding E ₀ = 131, 304. The cost loadings are set to λ ^R = λ ^S = 10 %. Regarding the target buffer ratio, surplus distribution ratio and liquidation costs, the same parameters as employed in Bohnert, Gatzert, and Jørgensen (2015) are used, i.e. γ = 10 %, α = 70 % and c = 20 %. With respect to the assets, we utilize a drift μ = 6 % and a volatility σ = 8 %. These values represent a portfolio weighting of low- and high-risk investments, considering the development of the German market over the last 30 years, and result in a continuous expected one-year return of E r t = 5.68 % . The risk-free interest rate r _f is set to 0.5 %. The relevant parameters were all subject to a sensitivity analysis, since we are interested in general interaction effects. The dividend rate ξ is calibrated such that Equation (11) is satisfied and can be found in the Appendix (see Figures A.1 and A.2).

For risk measurement purposes, when simulating the actual (“real-world”) number of deaths d t R and d t S , in contrast to pricing, we use the corresponding second-order mortality tables without safety loadings. Regarding the functional forms of the premium reduction functions PR ^R and PR ^S , which depend on the reported one-year default probability δ, we consider the three scenarios in Figure 4 to gain insight in the impact of a product-dependent willingness to pay in life insurance on risk measures. In scenario 1, we assume that policyholders are “risk-neutral” and indifferent toward default risk, i.e. P R R δ = P R S δ = 0 , indicated by the horizontal solid line in Figure 4. In scenario 2, we assume that the premium income is strictly reduced for an increasing reported one-year default probability δ due to a decreasing willingness to pay, but without product-dependent risk sensitivities, i.e. we assume that the reduction is equal for both product lines. For this scenario, we fit an exponential function based on the combined data of our two surveys, as shown by the dashed line in Figure 4, which results in P R R δ = P R S δ = 0.02 ⋅ exp 59.12 ⋅ δ . Finally, in scenario 3 we assume that policyholders exhibit product-specific risk sensitivities in line with the observations in our experiment, i.e. (see Figure 3 for the values and the circles/crosses in Figure 4):

Figure 4:

Considered scenarios regarding the premium reduction function for annuities PR ^R and term life contracts PR ^S depending on the one-year default probability δ.

To take into account that in the experiment the survey participants were sensitized toward the probability of default,^[14] we first use a scaling factor of z ^R = z ^S = 0.5 in the following simulation analysis to adjust the premium reduction values (see Equation (7)), which we then vary in different settings.

5.2 The Impact of Policyholders’ Willingness to Pay on a Life Insurer’s Risk Situation

Figure 5 displays the impact of different levels of the reported one-year default probability δ (1 % and 4 %) in the previously defined three scenarios on the insurer’s actual shortfall probability under various portfolio compositions.^[15] The upper graphs show the overall shortfall probability over the entire contract term, and the bottom graphs the maximum annual shortfall probability (which should not exceed δ).

Figure 5:

The impact of different levels of the reported default probability δ in the three scenarios (see Figure 4 for a description) on the insurer’s actual overall and maximum annual shortfall probability (see Equations (8) and (9)) under various portfolio compositions with scaling factors z ^R = z ^S = 0.5.

When comparing scenarios 1 and 2, as an initial result one can see that taking into account the policyholders’ willingness to pay increases the overall shortfall probability over the entire contract term (upper graphs) for all portfolio compositions by approximately 0.5 (9.5) percentage points for a reported δ of 1 % (4 %). In contrast to this, the increase in S P max annual (lower graphs) strongly depends on the portfolio composition. Here, portfolios with a higher fraction of annuities are more affected by taking into account the policyholders’ willingness to pay, which can be explained by the product-specific cash flow structures (see also Gatzert and Wesker 2012). While the payouts for term life insurances increase over time, they decline for annuities. As a result, the risk of default is highest at the beginning of the contract term in the case of annuities, and at the end in the case of term life insurances. Since a reduction in the policyholders’ willingness to pay reduces the initial premium income, and thus mainly affects the shortfall probabilities in the first years, portfolio compositions with higher fractions of annuities are more affected.

In contrast to previous findings (see, e.g. Gatzert and Wesker 2012; Wong, Sherris, and Stevens 2017; both, however, have somewhat different model set-ups^[16]), we do not observe a portfolio composition with a minimum default risk value in scenarios 1 and 2, which can arise for mixed portfolios due to smoother cash flow structures. In scenario 1 as well as scenario 2, both SP ^overall and S P max annual strictly decrease for an increasing portion of annuities, as Figure 5 shows.^[17] One reason for this might be the policyholders’ surplus distribution, as the bonus system exponentially increases the annuities as well as the sums insured over time due to cliquet-style interest rate effects (see Bohnert, Gatzert, and Jørgensen 2015). Thus, later payments are more affected, which are more pronounced in the case of term life insurances. However, the calibrated fair dividends in Figure A.1 suggest that there are still some portfolio effects, as fair dividends are minimal for a portfolio composition consisting of only 90 or 80 percent annuities.

While in Figure 5 specific portfolio effects in regard to (e.g. risk-minimizing) shortfall probabilities cannot be seen for the first two scenarios, such an effect is clearly visible in the third and – for our research purpose, most relevant and new – scenario, where based on our experimental findings for the lower reported one-year default probability δ = 1 %, the policyholders’ risk sensitivities are higher for customers purchasing an annuity than for term life contracts, whereby for the higher δ = 4 %, this effect is reversed (see Figure 4). As a result, for δ = 1 % the initial premium income for portfolios with a higher fraction of annuities is reduced, and for δ = 4 % it is increased. In the case of the lower reported one-year default probability (δ = 1 %), risk-minimizing portfolio effects arise for S P max annual , as can be seen in the lower left graph in Figure 5. Increasing the fraction of annuities first strictly reduces S P max annual until a minimum of approximately 0.95 % is reached, before rising again. While S P max annual slightly increases for higher fractions of annuities in scenario 2, in the case of the higher reported one-year default probability (δ = 4 %) in scenario 3 it strictly decreases, because of the lower premium reduction for annuities as compared to term life contracts. Regarding the overall shortfall probability, the upper two graphs of Figure 5 show that in scenario 3, SP ^overall still only declines for an increasing fraction of annuities, but for δ = 1 % with a smaller slope and for δ = 4 % with a higher slope in comparison to scenario 2.

The existence of a risk-minimizing portfolio of annuities and term life insurance policies in the case of δ = 1 % in scenario 3 (see spades in the lower left of Figure 5) shows that natural hedging effects can also occur in our model set-up, but depending on the respective cash flow structures. Since in scenario 3, the policyholders’ willingness to pay for annuities is stronger reduced compared to the other two scenarios, there is a relative advantage of term life insurances over annuities at the beginning of the time horizon due to the declining cash flows of annuities over time compared to the increasing cash flows for term life insurances. As a result, adding term life contracts to the portfolio mix decreases the one-year default probabilities at the beginning of the time horizon, but with higher one-year default probabilities in later years. In this setting, for a portfolio consisting of approximately 50 % annuities and 50 % term life insurances this natural hedging effect is best balanced, with the lowest maximum one-year default probability S P max annual among all portfolio compositions. Therefore, a product-dependent policyholders’ willingness to pay can have a large impact on the existence of potential natural hedging effects caused by mixed product portfolios, which is also confirmed by the further analyses in the next section.

Furthermore, the numerical results indicate that in all three scenarios both SP ^overall and S P max annual can be substantially lowered by the “right” portfolio composition. Additionally, the lower left part of Figure 5 illustrates that product portfolio management is an important tool for life insurers to ensure that the maximum one-year default probability does not exceed the reported one at contract inception. In the example, only portfolio compositions which lie below the dotted line satisfy Equation (10) and are thus valid.^[18]

5.3 The Impact of Larger Deviations Between Product-Dependent Risk Sensitivities

In this section, we build on the previous observation in scenario 3, where in the considered setting, portfolio effects with a minimum shortfall probability could only be observed in the case of a larger premium reduction for annuities compared to term life insurance, i.e. we investigate the situation of the lower reported one-year default probability (δ = 1 %) in scenario 3. Therefore, for additional sensitivity analysis we first set the scaling factors from the previous z ^R = z ^S = 0.5 to z ^R = z ^S = 1, i.e. we use the (higher) premium reductions as observed in our experiment. In a second analysis, we artificially increase the differences between the premium reductions for annuities and term life insurances by setting the scaling factors to z ^R = 2 and z ^S = 0.5, where the experimentally observed premium reduction for annuities PR ^R is multiplied by two and the premium reduction PR ^S for term life insurances is divided by two.

Figure 6 shows that increasing the differences between the premium reductions in scenario 3 (with δ = 1 %) strongly influences the risk-reducing portfolio composition, where in the case of the highest difference (z ^R = 2; z ^S = 0.5) additional portfolio effects arise on the level of the overall shortfall probability SP ^overall. Thus, product-dependent risk sensitivities can imply new or at least strengthen portfolio effects in a life insurer’s product portfolio, depending on the respective differences.

Figure 6:

The impact of different deviations between product-dependent risk sensitivities given by the respective scaling factors z ^R and z ^S on the insurer’s actual overall and maximum annual shortfall probability (see Equations (8) and (9)) under various portfolio compositions in scenario 3 with a reported one-year default probability δ = 1 %. Notes: The product-dependent premium reduction value (see scenario 3 in Figure 4) for annuities is multiplied with z ^R and for term life with z ^S .

The left graph of Figure 6 shows that the risk-reducing portfolios regarding S P max annual contain a decreasing fraction of annuities when the deviation between the product-dependent risk sensitivities is increased. While in the case of the lowest deviation (z ^R = z ^S = 0.5) the risk-reducing portfolio consists of approximately 50 % annuities, it consists of only approximately 10 % in the other two cases. Furthermore, the right graph of Figure 6 shows that in the case of higher deviations between risk sensitivities (z ^R = 2; z ^S = 0.5), different portfolio effects arise, as it would be optimal to sell a portfolio consisting of 80 % annuities in order to reduce the overall shortfall probability SP ^overall. Therefore, minimizing the overall shortfall probability does not automatically imply that the annual maximum shortfall probability is minimized as well. This indicates that solely aiming to satisfy one-year regulatory requirements may not necessarily provide suitable incentives for a long-run risk management perspective.

5.4 Further Sensitivity Analyses

To investigate the robustness of our numerical results, we perform various sensitivity analyses using different model parameters in the setting of Figure 6 with the following results. First, portfolio compositions with higher fractions of term life insurances are more affected by an increasing asset volatility σ in the present setting (with a stronger emphasis on later payouts, where volatility plays a particularly important role). Therefore, in the case of the highest deviation between the premium reductions (z ^R = 2; z ^S = 0.5), the SP ^overall-reducing portfolio consists of a higher fraction of annuities when volatility is increased. Similarly, the maximum one-year default probability S P max annual is substantially higher for an increased volatility, but with similar risk-reducing portfolio compositions for S P max annual for all volatilities.

In contrast to this, a decreasing initial contribution E ₀ by the equityholders implies an upward shift of the overall shortfall probability SP ^overall for all portfolio compositions. But the increase in the maximum one-year default probability S P max annual is more pronounced for portfolios with higher fractions of annuities, as here a default mainly occurs during the first years, as it is more affected by a decreasing initial contribution.

A strong influence can also be found when varying the surplus distribution rate α, where we especially investigate the situation without surplus in line with our experiment. An increasing surplus distribution rate α thereby leads to higher (cliquet-style) interest rate guarantees for policyholders and thus generally results in higher shortfall probabilities. Since later payments are more affected, varying α has a stronger impact on portfolios with higher fractions of term life insurances with higher payouts in later contract years. As a result, we observe similar (but more pronounced) effects as in the case of the asset volatility σ. Overall, the surplus distribution mechanism increases the differences between the product-specific cash flow structures and is thus important for portfolio effects in the present setting in the sense of the possibility of smoothing cash flows of mixed portfolios, but possibly (due to higher guarantees) at higher shortfall levels.

We also study the situation of a higher risk-free rate r _f and actuarial interest rate r ^G , which becomes relevant in a high interest-rate environment. Since the risk-free rate r _f only affects the dividend rate ξ given in Equation (11) in the present setting, the impact on the life insurer’s risk situation is comparatively small, where we observe a slightly reduced dividend rate for an increasing risk-free rate and lower default probabilities of the life insurer. While a higher risk-free rate in Equation (11) results in more pronounced discounting, this effect is outweighed by a reduced default probability of the life insurer under the risk-neutral pricing measure Q due to the assets’ higher drift. A stronger influence can be found when the actuarial interest rate r ^G increases, which increases the guarantees and thus results in a higher risk for the life insurer. In this case, depending on the respective risk measure and assumptions regarding product-dependent risk sensitivities, portfolio effects become more visible, as a smoothing of cash flows through mixing different products may result in lower shortfall probabilities as compared to portfolios that only consist of annuities or only term life insurances.