Innovative Equity-Linked Life Insurance Based on Traditional Products: The Case of Select-Products

Innovative Equity-Linked Life Insurance Based on Traditional Products: The Case of Select-Products Maria Alexandrova, Alexander Bohnert, Nadine Gatzert, Jochen Russ Working Paper Department of Insurance Economics and Risk Management Friedrich-Alexander University Erlangen-Nürnberg (FAU) Version: March 2015

INNOVATIVE EQUITY-LINKED LIFE INSURANCE BASED ON TRADITIONAL PRODUCTS: THE CASE OF SELECT-PRODUCTS Maria Alexandrova, Alexander Bohnert, Nadine Gatzert, Jochen Russ This version: March 27, 2015 ABSTRACT Select-Products are equity-linked life insurances with investment guarantee in the German market which in contrast to typical guaranteed equity-linked products are constructed by using a traditional life insurance contract and suitably leveraging the annual surplus distribution. In this paper, we describe the unique features of traditional life insurance (particularly the collective savings process) and analyze how these features contribute to such products. We present a model framework for the most prominent type of Select-Products and compare the product design when offered by a bank or an insurer. Our analysis emphasizes that the current attractiveness of such products arises from the unique features of traditional life insurance by pooling risks as well as the utilization of the balance sheet in the current low interest rate environment. We discuss these aspects in detail and further address benefits as well as detriments of these products depending on the market conditions. We also explain how such products with alternative guarantees interact with an insurer s book of business and can help reduce the risks resulting from old guarantees. Keywords: Equity-indexed annuities, IndexSelect, collective policy reserves, collective saving, smoothing mechanism 1. INTRODUCTION Against the background of the demographic development, volatile capital markets and low interest rates, the development of innovative and attractive insurance product designs in the life and pensions segment has become increasingly important. In particular, life insurance products with new forms of guarantees that are supposed to be less risky for the insurer and still attractive to the policyholder are being discussed in many markets. In this context, in 2007 Allianz has introduced a new deferred annuity product called IndexSelect in the German Maria Alexandrova, Alexander Bohnert and Nadine Gatzert are at the Friedrich-Alexander University Erlangen-Nürnberg (FAU), Department of Insurance Economics and Risk Management, Lange Gasse 20, 90403 Nuremberg, Germany, maria.alexandrova@fau.de, alexander.bohnert@fau.de, nadine.gatzert@fau.de; Jochen Russ is at the Institut für Finanz- und Aktuarwissenschaften and at Ulm University, Lise-Meitner-Str. 14, 89081 Ulm, Germany, j.russ@ifa-ulm.de.

2 market, which explicitly utilizes the unique features of traditional life insurance, particularly a balance-sheet-based smoothing scheme and a collective savings process, at the same time providing certain features of equity-indexed annuity products. Several other insurers followed by offering similar products which may, however, be different in certain details. In the remainder of this paper, we consider the product design of the first such product: IndexSelect. During the accumulation phase, this product offers the policyholder an annual choice between a) the participation in an equity index, where the payoff is calculated as the maximum of 0% return and the sum of the monthly returns of the EURO STOXX 50 with a monthly cap that is reset annually, and b) a so-called safe interest rate, which essentially coincides with the interest rates for traditional products. The safe interest rate is set and communicated in advance and is the result of German insurance law and regulation demanding that at least 90% of the insurer s book value returns on the general assets has to be given to the policyholders as a so-called total annual interest rate, which is the sum of the guaranteed rate plus surplus. 1 In addition, the product offers a money-back guarantee: The benefit at maturity is at least the single premium or in case of regular premium payment the sum of the premiums paid. Typically, Select-Products are traditional products and the participation in the equity index is only a special form of surplus participation. In particular, the policy design does not involve an investment of the policyholders funds in the relevant equity index or a derivative on this index. The policy funds are rather kept in the collective policy reserves for traditional contracts throughout the entire contract duration, independent of the policyholders choice between the safe interest rate and the index participation. Hence, the concept of the product makes use of the insurer s balance sheet and the collective savings features (pooling and smoothing as well as the long-term investment horizon). In case the policyholder chooses the equity index participation, only the next year s safe interest rate (which is set and communicated in advance) is used to finance a hedging strategy, which essentially means buying a suitable derivative on the index from a bank. This derivative generates the equity-linked pay- 1 In Germany, this is regulated in the Mindestzuführungsverordnung (MindZV). In particular, the total annual interest rate ( Gesamtverzinsung ) consists of investment returns (i.e. realized book value returns), differences between charges and expenses, and the mortality result. The latter arises due to prudent actuarial premium calculations (assuming e.g. higher than best estimate mortality rates when pricing products with death benefit and lower rates when pricing annuities). Policyholders have to be credited at least 90% of the book value returns (but no less than the guaranteed interest rate), 90% of the mortality result, and 50% of the remaining surplus which essentially results from differences between charges and expenses. As mentioned before, this total annual interest rate plays the role of the safe interest rate in Select-Products. In the remainder of this paper, we will also only use the term safe interest rate when we refer to traditional products without index participation.

3 off at the end of the year, which will then also be invested in the collective policy reserve and increases the policyholder s policy value accordingly. The aim of this paper is to analyze the pricing of such products and study how they utilize the unique features of traditional life insurance, i.e. the use of the balance sheet and smoothing schemes. We further explain how the products would differ if they were designed without using a traditional life insurance product (and hence the insurer s balance sheet) as a basis, i.e. as unit-linked insurance products or, equivalently, as a bank product. In this context, we emphasize the role of life insurers and their special abilities as providers of life and pension policies as compared to other financial institutions, which may be more or less attractive, depending on the economic conditions and the consumers preferences. In the literature, products with equity-linked benefits have received considerable attention, especially so-called equity-indexed annuities (EIAs) in the US market. A detailed description of such products and the fundamental concepts can be found in Palmer (2006), for instance, while advantages and disadvantages of EIAs are addressed by VanderPal (2004), and their performance is analyzed in Kuhlemeyer (2001), Reichenstein (2009), and VanderPal, Marrion, and Babbel (2011). However, most of the attention focuses on pricing and / or hedging of EIAs as is done in, e.g., Brennan and Schwartz (1976), Boyle and Schwartz (1977), Boyle and Hardy (1997), Tiong (2000), Gerber and Shiu (2003), Lee (2003), Young (2003), Jaimungal and Young (2005), Bernard, Le Courtois, and Quittard-Pinon (2006), and Gaillardetz and Lakhmiri (2011). Specific aspects of regular equity-indexed products that arise under German legislation, in particular with respect to accounting and the resulting risks, can be found in Russ (1999). This considerable amount of research regarding EIA policies cannot, however, be directly applied to the new Select-Products since these in contrast to previous equity-indexed annuities make use of the unique features of the collective saving process and various smoothing mechanisms of traditional life insurance. The latter, as one characteristic of collective savings, has also been extensively studied, but without the index-participation feature. Smoothing mechanisms have been analyzed by, e.g., Guillen, Joergensen, and Nielsen (2006) and Maurer et al. (2014), while the benefits of collective saving have been discussed in, e.g., Bovenberg et al. (2007), Hoevenaars and Ponds (2008), and Goecke (2013). Behavioral aspects associated with Select-Products, in particular the multi-period decision behavior of policyholders, are considered in Koranda and Post (2014). To the best of our knowledge, however, there exists no literature on the construction and the economics of such products.

4 This paper contributes to the literature by presenting a model framework for pricing and evaluating the Select-Products offered in the German market and by explicitly discussing the use of the unique features of traditional life insurance and the resulting difference to similar banking products, which is also of high relevance for other markets. The payoff structure that is predominant in the German market comes with a monthly cap. We analyze Select-Products using this payoff structure and conduct a Monte Carlo simulation to obtain numerical results for fair cap values of the monthly index returns and study their sensitivities to different model parameters. Finally, we discuss merits and detriments of the new deferred annuity product from different perspectives, namely insurers, owners of old traditional policies and new clients, who purchase such a product. The remainder of the paper is organized as follows. Section 2 explains the design of Select- Products and discusses pricing aspects of the option to participate in an equity index. The unique features of traditional life insurance products such as the balance-sheet-based smoothing and their role in the design of Select-Products are presented and discussed in Section 3. Numerical results obtained by means of Monte Carlo simulation are given in Section 4, and Section 5 concludes. 2. INNOVATIVE LIFE INSURANCE PRODUCTS IN GERMANY: SELECT-PRODUCTS To illustrate the central product features of Select-Products, in what follows we focus on the concrete design of the product IndexSelect offered by Allianz Life Insurance in Germany, even though after its introduction in 2007 a variety of similar products has been offered in the German market, whereby certain product features may differ. 2 Also, for the sake of simplicity, in the remainder of this paper we only consider the case of a single premium product although typically these products also allow for regular premium payments. Product features IndexSelect is a deferred annuity product that allows the policyholders in the accumulation phase to annually choose between a safe return (i.e. the interest rate that will be credited to the collective policy reserves at the end of the year which is, however, set and communicated at 2 For example, AXA offers Relax Rente that not only allows for participation in an excess-return and volatility-controlled version of the EURO STOXX 50, but also for a direct investment of a portion of the policy value in different investment funds; IndexGarant by SparkassenVersicherung offers a choice between VolaIndexPerform (a volatility-controlled version of the EURO STOXX 50) and the EURO STOXX 50 as the underlying index. Also, some products use a different participation formula, invest only a portion of the annual total interest rate in the derivative on the index, or generate the maturity guarantee differently than described in this paper.

5 the beginning of the year 3 ) and a participation in the EURO STOXX 50 according to a participation formula explained below. Each year, the client can make the choice for the following year after the safe interest rate and all parameters of the participation formula have been fixed for the following year. A combination of the two choices, i.e. receiving the safe return on a portion of the policy value and the index participation on the remaining part, is also possible in steps of 25%. If the client chooses to participate in the index, the respective index return is calculated as follows: The monthly equity index returns r, 1 = I, 1 I, 1 are subject to a monthly cap t τ + t τ + t τ cap t, where I t,τ is the value of the equity (price) index at the end of month τ in year t, τ = 0, 1,, 11 and t = 1, 2,..., T with T being the last year of the accumulation phase of the contract. The cap can be changed each year for the following year. The annual equity-linked return of 0%, i.e. r is then given by the sum of the capped monthly index returns and has a floor equity linked t 11 equity linked rt ( capt ) = max min ( rt, τ + 1, capt ),0, with t = 1, 2,..., T. τ = 0 After the announcement of both, the cap and the safe interest rate, the policyholder can choose between the safe interest rate, a participation in the index or a combination of both. In case the policyholder participates in year t with some fraction s t of the policy value F t-1 in the safe return safe r t, the policy value at the end of the accumulation phase F T can be described by T t= 1 combined ( t ) FT = F0 1 + r, (1) where F 0 is the single up-front savings premium, which is the gross premium P 0 minus upfront charges c 0 including, e.g., distribution charges and a risk premium for the death benefit, and ( ) ( 1 ) ( ) r = s r c + s r cap, combined safe equity linked t t t t t t t 3 The fact that next year s surplus participation is set and communicated in advance is a special feature of traditional products e.g. in the German market. At the beginning of the year, the management of a life insurance company decides on the surplus distribution for the upcoming year and thus on the total interest that is credited to the policy reserves of all policies on the policy anniversary date that lies in the respective year.

6 is the combined annual return combined rt where for t > 0, c t represents ongoing charges at time t. 4 safe In this product design, the policy value cannot decline since both, rt and r equity linked t cannot become negative. However, a decline of the policy value might occur for different product designs due to charges, whereas the product design analyzed in this paper comes with a charging structure where a decline is not possible. In addition, the product comes with a moneyback guarantee of the paid premium, i.e. at maturity at least the gross premium P 0 >F 0 is paid back to the policyholder. We will explain below how this guarantee is generated. Independent of the choice between index participation and safe interest rate, the policyholders funds are always invested in the insurer s general assets that cover the existing pool of traditional business. Hence, from the insurer s perspective, a Select-Product is essentially a traditional participating deferred annuity policy with two guarantees: a year-by-year guarantee of 0% on the policyholder s policy value and a maturity-only guarantee to pay back the gross premium P 0. The maturity-only guarantee is much less risky for the insurer than typical cliquet-style guarantees (see, e.g., Reuss, Russ, and Wieland, 2015). Finally, the product contains further features, such as mortality benefits and a surrender option, and it also provides a certain degree of flexibility and several options, which we will not consider in our model. Generation of the equity-linked payoff We next focus on the generation of the equity-linked payoff if the index participation is selected. The savings premium F 0 is invested in the insurer s general assets together with the premiums of the existing traditional policies. The asset base is invested according to the in- safe surer s overall asset management strategy. Once the safe interest rate r t for year t is known and communicated at the beginning of year t, the monthly cap t is determined for the upcoming year. Then, the policyholder is given a choice between this safe interest rate and the index participation. In case the policyholder chooses the safe return for year t, no action needs to be taken as the policy reserves (and hence the value of the policy) will earn the safe return (minus charges c t ) similar to the traditional policies. In case the index participation is chosen, the insurer uses the safe interest rate (after deduction of charges c t ) and invests the amount 4 Note that typically the safe interest rate that is communicated to clients is already the value after deduction of ongoing charges. However, for the sake of comparability to the surplus distribution of traditional policies, we denote the value before deduction of ongoing charges as the safe interest rate. Furthermore, we would like to mention that some insurers use a safe interest rate that deviates from the surplus distribution of traditional policies (see also Figure 2).

7 safe ( ) t 1 t t F r c in a derivative on the index that generates the payoff in Equation (1). Typically, this derivative is a suitable option which is purchased from bank and the cap is set by the bank such that the price of this derivative coincides with the invested amount. In what follows, we present how the cap can be determined in a simple model. We assume that the index It, τ follows a geometric Brownian motion. IndexSelect uses the price index of the EURO STOXX 50, i.e. dividends are not reinvested in the index. Assuming a continuous dividend yield d, the EURO STOXX 50 under the risk-neutral measure Q follows f ( ) di = r d I dt + σ I dw, Q t t t t where r f Q is the constant risk-free interest rate, σ is the volatility and W t is a standard Brownian motion under the risk-neutral measure Q (see Glasserman, 2010, p. 32). safe We assume (as described in Allianz, 2013a, p. 9-10) that the amount Ft 1 ( rt ct ) is invested by the insurer at the beginning of the year. We further assume that the bank s charges for hedging the payoff in Equation (1) also have to be paid at the beginning of the year. Hence, the annual cap must be chosen such that the expected discounted annual guaranteed payoff under the risk-neutral measure Q is equal to the invested amount, i.e. 11! exp ( r f ) F Q 1 1 max min (, 1, ),0 1 ( safe bank t Et rt τ + capt = Ft rt ct ct ) τ = 0, (2) where value. bank c t refers to the bank s (explicit or implicit) charges as a percentage of the policy One can see from Equation (2) that the safe interest rate is one major driver in the determination of the cap since it determines the amount of money available to purchase the derivative that generates the equity-linked payoff. Generation of the maturity-only guarantee The insurer s strategy for ensuring the maturity-only guarantee of the single premium is exhibited in Figure 1. First, up-front charges c 0 are subtracted from the gross premium P 0. The F = P 1 c represents the savings premium and thus the initial policy remaining amount ( ) 0 0 0 value at time t = 0. This policy value is then either compounded with the safe interest rate or the index return.

8 Under typical interpretation of local German GAAP, when a guarantee at maturity is given, the insurer needs to make sure that a prospective (minimum) policy reserve is met, which amounts to the terminal value of the guarantee discounted using an actuarial interest rate. 5 If the policy value is lower than this prospective (minimum) policy reserve, the insurer has to set up an additional policy reserve on top of the policy value. Of course, the insurer wants to avoid this situation. Therefore, if at some point in time, the policy value F t gets so close to this reserve that it would fall below this reserve after a bad index year with 0% return, the insurer no longer offers the choice to participate in the index. Instead, the policyholder will receive the safe interest rate. This event is denoted cash-lock in Figure 1. In this case, the actuarial interest rate used for calculating the prospective reserve becomes a guaranteed rate for the next year. This ensures that the contract value never falls below the prospective reserve and hence the contract value at maturity F T will amount at least to the premium paid at inception of the contract. Figure 1: The insurer s strategy for ensuring the terminal money-back guarantee on gross premiums at maturity in case of a single up-front premium P 0 P 0 F 0 cash-lock d P0 0 index It, τ policy value guarantee P 0 F t discounted guarantee d P t time T We would like to stress again that due to this construction, Select-Products not only come with a different level of guarantee but also with a different type of guarantee than usually offered in traditional products. Besides the cliquet-style year-by-year guaranteed annual interest rate of 0% on the policy value, the product has an additional money-back guarantee at maturi- 5 This prospective minimum reserve was first mentioned by Herde (1997) and the financial consequences for certain guaranteed equity-linked products were analyzed in detail in Russ (1999). Note that the actual reserve is given by the maximum of the surrender value and this prospective policy reserve.

9 ty that corresponds to a certain interest rate that exceeds 0%. However, the latter guarantee becomes effective only if the policy value approaches the prospective reserve for this guarantee. This is equivalent to the alternative 1 guarantee analyzed in Reuss, Russ, and Wieland (2015). They look at three different types of guarantees in traditional life insurance products and find that the way of generating a terminal guarantee in their alternative 1 product (and hence also in IndexSelect) is less risky and hence more capital efficient from an insurer s perspective when compared to a cliquet-style guarantee. 3. THE ROLE OF THE COLLECTIVE POLICY RESERVES AND THE INSURER S BALANCE SHEET An insurer s collective policy reserves along with the collective saving process with longterm investments and the use of the balance sheet in general are unique to life insurance companies and cannot be found in other financial institutions. While a Select-Product could basically be offered by both, insurers and banks, the main advantage (at least in the current market environment) when offered by an insurance company arises from the use of these unique features. In particular, even though the Select-Product offers a participation in equity index returns, the policy value is not invested in the index or in a derivative on the index, but rather remains invested in the general assets covering the traditional policies. Only the annual surplus is used for generating the required index returns. In what follows, we discuss the features of the collective savings process, which are also of high relevance when assessing whether the business model of life insurers as pension providers can still be attractive for consumers in the future and to what extent these features contribute to creating value for policyholders. 6 The safe interest rate is currently (and has been for several years) higher than the one-year market interest rate, as can be seen in Figure 2, where the safe interest rates of Allianz traditional polices (for the time period 1980-2014) and IndexSelect (for the time period 2007-2014) are displayed along with the one-year and ten-year market interest rates (represented by the respective REX-indices taken from the DataStream database) and the legally maximum possible guaranteed interest rate. Note that the safe interest rate for traditional policies is generally slightly higher than the safe interest rate for IndexSelect. Figure 2 further shows that the safe interest rate of Allianz has been relatively stable over time (which is similar for most other life insurers). For example, in 2014 it amounts to 4.2% for traditional polices and 3.9% for IndexSelect, whereas the interest rate for a one-year (or even 6 See e.g. Hieber, Korn, and Scherer (2014). They analyze the attractiveness of traditional policies for different generations of policyholders in a case where all assets are pooled and in a case where assets covering the different generations are segregated.

10 a ten-year) government bond is considerably lower. The one-year interest rate is the budget a bank would have at its disposal to generate the equity-linked return in a Select-Product. The portfolio of the insurer consists of investments made periodically during decades, and therefore still contains bonds with coupons that are considerably above today s interest rates. When interest rates fall, the return of this asset portfolio declines only with significant delay. In addition, certain smoothing mechanisms explained below create stable returns for the policyholders and, hence, also a rather stable cap in IndexSelect policies (see Allianz, 2014a), whereas the cap of a comparable product offered by a bank would be much more sensitive to the changes in market interest rates. However, if market interest rates would increase sharply and quickly, the short-term interest rate could exceed the safe interest rate offered by an insurer. Figure 2 shows that this was the case e.g. in the early 1990s. In such a scenario, a bank could use the high short-term interest rates to provide a higher cap than an insurer and thus offer more attractive conditions in this year. Figure 2: Development of investment returns over time (including the safe interest rates of Allianz for traditional and IndexSelect policies (since 2007), see Allianz, 2010a, 2010b, 2012, 2013b, 2014a) yield (p.a.) 12% 10% 8% safe interest rate (traditional) safe interest rate (IndexSelect) guaranteed interest rate REX (sub-index 10-year) REX (sub-index 1-year) 6% 4% 2% 0% 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 time We will now have a closer look at the collective savings process which is essentially a smoothing scheme and makes use of the insurer s balance sheet in three main ways: 1) Long-term collective savings On the asset side, investment returns are based on the collective savings process, as the savings premiums paid by the policyholders of Select-Products and other traditional products are

11 pooled and jointly invested in the capital market. This is in contrast to a bank, where each customer typically has an individual account. Therefore, the total return of an insurer s asset portfolio in any given year results from all securities in the portfolio and (as long as this return exceeds the guaranteed rates of all products) all insurance contracts more or less equally participate in this return. As life and pension policies usually have long-term contract durations, it is possible for an insurance company to invest in long-term (and to a certain degree also in less liquid) securities that typically offer higher returns than short-term investments. Also, this implies lower overall transaction costs. Currently, around 90% of the insurers assets are invested in longterm fixed-interest bonds, where older bonds that were bought several years ago provide higher coupon payments, which especially in the recent low interest rate environment implies higher asset returns as compared to new bond investments. Hence, insurance companies are still able to provide stable investment returns that are higher than what could be achieved directly in the market. 7 However, these long-term bonds with high coupons will expire gradually over time and insurers have to invest in bonds with lower interest rate payments, which induces a considerable reinvestment risk. 2) Smoothing account provision for premium refunds (PPR) On the liability side, not all surplus resulting from prudent premium calculation (from asset returns as well as from mortality and charges/expenses see also Footnote 1) is immediately credited to the policy reserves. Instead, it is transferred to the so-called account for uncommitted provision for premium refunds (PPR), 8 which serves as a buffer account and is thus of high relevance for the smoothing of surplus over time and hence for the creation of stable returns for the policyholder. As a result, a traditional life insurance contract s value is typically allocated to two accounts on the liability side of the insurer s balance sheet: the actuarial reserves and the provision for premium refunds. The first policyholder account on the liability side of the insurer s balance sheet therefore is the actuarial reserves. They have to be compounded at least with a contractually defined guaranteed interest rate. An upper limit for this guaranteed rate is given by regulation and has been as high as 4% in the past, when market interest rates were higher (see Figure 2). In the 7 8 The situation can, however, reverse when market interest rates increase. In this case the earlier investments with lower returns may delay the increase in returns of the insurer s portfolio as a whole. In German Rückstellung für Beitragsrückerstattung (RfB). Note that the Danish formula-based smoothed investment-linked annuity product TimePension makes use of a similar feature to exploit the smoothing effect of traditional contracts (see Jørgensen and Linnemann, 2011; Gatzert and Schmeiser, 2013).

12 current market conditions, fulfilling these obligations becomes highly challenging and insurers started to develop new contracts with lower guarantees or different types of guarantees. Since the pool of the insurer s portfolio consists of contracts with different guaranteed interest rates, this allows for intergenerational cross-subsidization effects (see Døskeland and Nordahl, 2008). Therefore, new contracts with lower guaranteed interest rates can reduce the pressure on the insurer. Since from the insurer s perspective, Select-Products are very similar to traditional products with a guaranteed interest rate of 0%, these products are particularly suitable for this type of risk reduction. The second policyholder account on the liability side of the insurer s balance sheet is the provision for premium refunds. The provision for premium refunds is built from surplus distribution as described above. Funds can remain in the account for up to three years until they are transferred to the actuarial reserves. Even if the amounts flowing into the PPR may fluctuate from year to year, the transfer to the actuarial reserve can be performed in a rather stable way, which reduces volatility of the policy value. 3) Creating and dissolving hidden reserves Finally, the determination of the asset return 90% of which has to be credited to the policyholders as surplus is based on book values. This gives the insurer some discretion in particular with respect to the timing of dissolving hidden reserves (difference between market and book values of assets) (see e.g. Reuss, Russ, and Wieland, 2015). Summary of main features and balance sheet effects In summary, the provision for premium refunds together with hidden reserves and long-term investments play an important role in smoothing surplus allocations over time. Therefore, surplus credited to the traditional policies is less volatile than market interest rates as also shown in Albrecht (2010). 9 The stable safe interest rates resulting from these mechanisms serves as a basis for the product design of Select-Products. From the policyholders perspective, the product combines the upside potential of an equity participation with safety (through the 0% floor in the equity participation and the option to take the safe return). Moreover, when they feel pessimistic about 9 Albrecht (2010) performs a comparison of the average performance of German life insurers with the German equity index (DAX) and German bond market index (REXP) for the time period 1980-2009 and concludes that the former exhibits the lowest volatility and the highest risk adjusted return.

13 equity markets, they can always choose the safe return and thus essentially switch to a traditional policy. From the insurer s point of view, such a product corresponds to a traditional policy with a year-to-year guaranteed interest rate of 0% (except in a situation of a cash-lock as shown in Figure 1) and is therefore considerably less risky and hence less capital intensive than products with typical minimum interest rate guarantees. Furthermore, since in some countries, e.g. in Germany, different generations of traditional products with different guaranteed interest rates are pooled, this helps reducing the risks and hence the capital requirement of the existing traditional portfolio by lowering the overall guarantee level. It can thus help reduce the pressure resulting from old traditional products with high interest rate guarantees. Overall, stable returns are valued by risk-averse policyholders, whereas insurers may benefit from improved stability and profitability, hence, the smoothing can be beneficial for both counterparties (see also Maurer et al., 2014). 4. NUMERICAL ANALYSIS In this section, we provide numerical examples to illustrate the pricing of the product presented in Section 2 under various market conditions. Furthermore, we compare the positions of a bank and an insurance company as potential providers of such policies to study the advantages and disadvantages of life insurers making use of the balance sheet and the smoothing schemes. Input parameters To price the annual index participation, Equation (2) must be solved for the relevant parameter, i.e. the cap t. The value of the cap t, determines the attractiveness of the product to policyholders, as c.p. higher cap means potentially higher annual equity-linked return. We assume that the new index year begins on 11/01/2014 (as offered by Allianz, see Allianz, 2014b). The policyholders are informed about the new cap and the safe interest rate three weeks before the new index year begins (see Allianz, 2014c); therefore, the parameter values are taken on 10/10/2014 to reflect the pricing under the respective market conditions. The risk-free interest rate r f is based on the yield to maturity of one-year German government bonds on 10/10/2014 using the respective REX index. For the EURO STOXX 50, the implied p.a. volatility is chosen, since it reflects the expectations about the future volatility of the index. For this purpose, we use the VSTOXX 360 index on 10/10/2014. As a proxy for the expected dividend yield, we use the historic average calculated based on the monthly historical data for the EURO STOXX 50 price and performance indices for the period 12/31/1986 to 12/31/2013.

14 We use the safe interest rate by Allianz of 3.9%, which is applicable for the index year starting on 11/01/2014 of all IndexSelect contracts (see Allianz, 2014b). Annual charges of 0.5% of the policy value are being used and the up-front charges are omitted in this calculation bank since they do not impact the annual cap. The bank s charges c t are not publicly available and are assumed to be 0.5% and will be subject to variation similar to the other parameters. The numerical results are derived based on Monte Carlo simulation with 500,000 latin hypercube samples. To ensure the comparability of the results, the random numbers are fixed for all simulation runs and different sets of random numbers are tested to ensure the stability of the results. The values of the input parameters are given in Table 1. Table 1: Input parameters for the numerical analyses Parameter Notation Value Risk-free interest rate f r -0.0555% Implied volatility of the EURO STOXX 50 σ 21.39% Dividend yield d 2.60% Safe interest rate for IndexSelect Annual ongoing charges by the insurer Bank s charges safe r t 3.90% c 0.50% t bank t c 0.50% Impact of different market conditions on the sensitivity of the monthly cap Under the assumptions in Table 1 (base case), the cap would amount to 5.33%. This is considerably higher than the cap offered by Allianz for the respective year, which is 3.7% (see Allianz, 2014b). The main reason for this deviation is probably the fact that banks selling such options use more complex asset models with more risk drivers. However, our general conclusions, in particular the sensitivity of the cap with respect to different input parameters, the interaction of different effects, as well as the difference between caps that can be offered within an insurance product and caps that can be offered within a bank product should be similar under different models. Figure 3 illustrates sensitivities of the cap for various levels of the risk-free interest rate r f, different values for the EURO STOXX 50 s volatilities, the insurer s safe interest rate, the bank s charges, the insurer s ongoing charges and the annual dividend yield. It can be seen in Figure 3 (upper left graph) that the cap decreases with an increasing risk-free interest. Furthermore, the cap strongly decreases when the bank s charges increase. At a value of around 1.95%, the cap amounts to 3.7% (see right graph in second row in Figure 3).

15 Figure 3: Sensitivity of the cap to changes in different parameters of the model cap cap cap 8% 7% 6% 5% 4% 3% 2% 1% 0% 2% 4% 6% 8% risk-free rate 8% 7% 6% 5% 4% 3% 2% 1% 2% 3% 4% 5% 6% safe interest rate 8% 7% 6% 5% 4% 3% 2% 1% 0.0% 0.2% 0.4% 0.6% 0.8% 1.0% ongoing charges cap cap cap 8% 7% 6% 5% 4% 3% 2% 1% 12.5% 17.5% 22.5% 27.5% volatility 8% 7% 6% 5% 4% 3% 2% 1% 0% 1% 2% 3% bank's charges 8% 7% 6% 5% 4% 3% 2% 1% 0% 1% 2% 3% 4% 5% dividend yield

16 Regarding the index volatility, it might be surprising at first glance that for volatilities above 15%, an increase in the volatility of the underlying equity index leads to an increase in the cap in the present setting. This is due to the fact that the structure from Equation (2) approximately corresponds to twelve short calls and one long put. With increasing volatility, the short calls become more expensive (making the option cheaper c.p., i.e. allowing for a higher cap for a given option price). But at the same time the long put also becomes more expensive making the option c.p. more expensive. The upper right graph in Figure 3 shows that for volatilities up to about 15%, the second effect dominates while for higher volatilities, the first effect is stronger. Further analyses showed that the extent of this effect also depends on the level of the bank s charges. It can further be seen that the cap strongly increases with an increase in the safe interest rate, as more capital is available for generating the index participation, i.e. a more expensive option (with a higher cap) can be afforded. In line with this, the insurer s annual fees have the opposite effect since they are simply deducted from and hence reduce the safe interest rate that can be used to generate the index participation. Lastly, the results show the impact of the dividends on the cap. As the equity-linked payoff becomes cheaper for higher dividend yields, the insurer is able to offer higher caps. Comparing the position of insurers and banks as providers of Select-Products Besides insurance companies, banks or investment funds may also act as providers of pension products. Moreover, a bank provides the hedging for the Select-Products to the insurer. In what follows, we thus use back-testing to study the setting if a bank would offer a comparable product for the capital accumulation phase and analyze in which situations this would be more or less attractive for the consumer. We focus on the time period from 1980 to 2013. Since a history of the VSTOXX indices is not available, we use the average historical volatility of index returns between 12/31/1986 and 12/31/2013 for all points in time, which amounts to 19.05%. Note that the volatility affects the insurance and the bank product in a similar way. Therefore, while the absolute values calculated below would differ if the historical implied volatilities were used, the difference between the insurance and the bank product would be very similar as it is mainly driven by the difference between the one-year risk-free interest rate and the insurer s safe interest rate. The historical one-year REX sub-index values are used as the (time dependent) risk-free rate r f bank and the bank s charges c t are assumed to be 0.5% as before. In case of the insurer, the historical safe interest rates of Allianz for traditional policies from Figure 2 are taken as an approximation for the IndexSelect safe interest rates for the period from 1980 until 2006 (since the product was first introduced in 2007), whereas from

17 2007 on, the actual safe interest rates for IndexSelect as also shown in Figure 2 are used. Moreover, we assume that when the bank offers the product, it has the same ongoing charges c t for administration as the insurer, which need to be subtracted from the safe interest rate (i.e. from r f in case of the bank) in addition to the bank s charges for hedging. The annual cap values when solving Equation (2) under these assumptions are displayed in Figure 4. The findings illustrate that around 1980 and around 1990 in times of high risk-free interest rates, a bank could have offered such a product with a higher cap when compared to an insurer. During all other time periods in the past, the insurer could have offered (sometimes considerably) better and also more stable conditions. In addition, from 2009 on, except for one point in time in 2011, a bank could not have offered the product at all, since the one-year interest rate would not have been sufficient to pay the bank s charges and the 0% minimum annual return, even for a cap of 0%, i.e. without any index participation. Therefore Equation (2) cannot be solved for these four years. On the other hand, if risk-free interest rates would be increasing in the future and exceed the safe annual interest rate generated by the insurers as was the case around 1980 and 1990, banks could (temporarily) offer more favorable conditions (caps) than insurers. Figure 4: Annual cap values for Select-Products when offered by a bank and an insurer 14% 12% 10% insurer bank cap (p.a.) 8% 6% 4% 2% 0% 1980 1983 1986 1989 1992 1995 1998 2001 2004 2007 2010 2013 time

18 5. SUMMARY In this paper, we describe the unique features of traditional life insurance (particularly the collective savings process) and analyze how these features contribute to Select-Products, which are innovative life insurance products with alternative guarantees resembling equityindexed annuities. In its accumulation phase, these deferred annuity products offer the policyholders an annual choice between a safe return (essentially the surplus distribution of traditional products) and an equity-linked return through participation in the EURO STOXX 50 index. While to consumers the product resembles an equity-indexed annuity product, it is constructed as a traditional product with a different type of guarantee and a different form of surplus participation. Therefore, from the insurer s perspective, it is essentially a traditional product with an annual 0% interest rate guarantee and a money-back guarantee at maturity, which considerably reduces the insurer s risk and hence capital requirement. In this paper, we introduce a model framework for Select-Products reflecting specific characteristics of the German market. We also discuss to what extent the product makes use of the smoothing mechanisms of traditional life insurance, which can only be performed on a life insurer s balance sheet and can therefore not be replicated by other pension product providers. In addition, we present a pricing approach that can be used for determining the monthly cap in the index participation formula. As the safe interest rate is used for financing the hedging strategy, we comprehensively discuss the role of the collective savings process and the use of the balance sheet for smoothing returns over time since these features help stabilize this interest rate. In a numerical analysis, we then calculate fair values for the cap and examine the cap s sensitivity to changes in various input parameters. We show that the safe interest rate as well as (explicit or implicit) charges by the bank selling the option to the insurer are major drivers for the cap. Our findings also show that a bank could offer similar products with more attractive conditions than an insurer only under conditions that could rarely be observed in the past (i.e. when short term market interest rates exceed the insurer s safe interest rate). In the current market environment such products are clearly more attractive to policyholders when offered by a life insurer. These products interact with an insurer s existing book of business in various ways. On the one hand, policyholders buying such a product now can profit from the fact that the insurer currently distributes surplus rates that considerably exceed market rates. This is largely a result of old bonds with high coupons that were bought many years ago. Since new policyholders funds are not invested separately, they, too, profit from these high coupons. On the other hand, since the lower and less valuable guarantees of these products are pooled with the insurer s existing business, they can help reduce the average guaranteed rate in the insurer s

19 block of business moving forward, making it easier for the insurer to fulfill the guarantees. So under certain circumstances and in particular under current market conditions, these products benefit from the existing business but under certain adverse future scenarios the existing business might benefit from these new products as well. Under risk-based capital regimes like Solvency II or the Swiss Solvency Test, such adverse future scenarios are the driver of solvency capital. Therefore, Select-Products and other products with alternative guarantees can reduce the capital requirement resulting from old traditional products with high interest rate guarantees even if the relevant adverse scenarios never happen. It would be interesting to analyze these effects and fairness issues between different generations of policyholders in more detail. Also, it is worth noting that many different product designs exist. In particular, policies with different underlying indices are offered in Germany, and products with different participation formulas (e.g. essentially a plain vanilla one-year call or formula based on Asian options) are available in Germany and Switzerland. Also, in some Swiss products, the policyholder is not allowed to choose between the safe rate and the index participation but rather receives the index participation every year. Furthermore, a much larger variety of products than currently observed would be possible. In general, one could use a traditional product as a basis to generate the desired guarantees and then suitably leverage the surplus distribution to adjust the risk-return profile of the product to specific needs. These product design issues give ample room for future research. In summary, in this paper we not only present a model framework and a pricing approach for setting the annual cap for Select-Products, but also discuss the unique beneficial features of traditional life insurance (especially the balance sheet), which strongly contribute to the attractiveness of these products and which will also be of high relevance in the future when designing new products with alternative guarantee concepts. However, more research is necessary in regard to the portfolio effects and possible substitution effects that arise from introducing a new type of product and adding the new policies to the existing pool of participating deferred life and pension products.

20 REFERENCES Albrecht, P., 2010, 30 Jahre Kapitalanlageperformance der Deutschen Lebensversicherer, Zeitschrift Versicherungswirtschaft 65(12): 880 884. Allianz, 2010a, Allianz Gruppe Status und Ausblick, https://www.allianz.com/v_1339507360000/media/press/document/results/bpk_2010/diek mann_bpk_2010_deutsch.pdf, accessed 11/25/2014. Allianz, 2010b, Finanzstärke als Wettbewerbsfaktor, https://www.allianzdeutschland.de/index.php/download_file/473/172/, accessed 11/25/2014. Allianz, 2012, Allianz Lebensversicherung bietet 2013 gesamte Verzinsung von 4.2%, https://www.allianzdeutschland.de/news/news-2012/05-12-12-allianz-leben-verzinsung- 2013, accessed on 11/25/2014. Allianz, 2013a, Kundenerwartungen erfolgreich managen: Sicherheit plus Chance, http://www.bca-onlive.de/fileadmin/user_upload/apm/2013-10- 09_BCA_OnLive_Interview_IndexSelect.pdf, accessed on 12/15/2014. Allianz, 2013b, Allianz Leben hält 2014 gesamte Verzinsung von 4,2 Prozent stabil, https://www.allianzdeutschland.de/news/news-2013/04-12-13-allianz-leben-haelt-2014- gesamte-verzinsung-von-42-proz, accessed 11/24/2014. Allianz, 2014a, Wertentwicklung Vorsorgekonzept IndexSelect (Rententarife), https://makler.allianz.de/static-resources/amisonline.allianz.de/vorsorge/privat/produkte /altersvorsorge-privat/privatrente_indexselect_riu2/konditionen_und_wertentwicklung/ v_1415015196000/2014-11-01_az_indexselect_wertentwicklung_komplett.pdf, accessed 11/24/2014. Allianz, 2014b, Konditionen und Wertentwicklung Renten IndexSelect, https://makler.allianz.de/res/produkte-beratung/leben_privat/produkte/kinder vorsorge/kinderpolice_indexselect_riu2/konditionen_und_wertentwicklung/link-pr-a-39179.html, accessed 11/24/2014. Allianz, 2014c, Allianz IndexSelect Produktbeschreibung, https://makler.allianz.de/res/produkte-beratung/leben_privat/produkte/altersvorsorgeprivat/privatrente_indexselect_riu2/produktbeschreibung/35777.html, accessed 12/15/2014. Bernard, C. L., O. A. Le Courtois, and F. M. Quittard-Pinon, 2006, Development and Pricing of a New Participating Contract, North American Actuarial Journal 10(4): 179 195. Bovenberg, L., R. Koijen, T. Nijman, and C. Teulings, 2007, Saving and Investing over the Life Cycle and the Role of Collective Pension Funds, Economist 155(4): 347 415. Boyle, P. P. and M. Hardy, 1997, Reserving for Maturity Guarantees: Two Approaches, Insurance: Mathematics and Economics 21(2): 113 127.

21 Boyle, P. P. and E. S. Schwartz, 1977, Equilibrium Prices of Guarantees under Equity-Linked Contracts, Journal of Risk and Insurance 44(4): 639 660. Brennan, M. J. and E. S. Schwartz, 1976, The Pricing of Equity-Linked Life Insurance Policies with an Asset Value Guarantee, Journal of Financial Economics 3(3): 195 213. Døskeland, T. M. and H. A. Nordahl, 2008, Intergenerational Effects of Guaranteed Pension Contracts, Geneva Risk and Insurance Review 33(1): 19 46. Gaillardetz, P. and J. Y. Lakhmiri, 2011, A New Premium Principle for Equity-Indexed Annuities, Journal of Risk and Insurance 78(1): 245 265. Gatzert, N. and H. Schmeiser, 2013, New Life Insurance Financial Products, In G. Dionne (Ed.), Handbook of Insurance (2 Ed.) (pp. 1061 1095). New York, NJ: Springer. Gerber, H. U. and E. S. Shiu, 2003, Pricing Perpetual Fund Protection with Withdrawal Option, North American Actuarial Journal 7(2): 60 77. Glasserman, P., 2010, Monte Carlo Methods in Financial Engineering. New York, NJ: Springer. Goecke, O., 2013, Pension Saving Schemes with Return Smoothing Mechanism, Insurance: Mathematics and Economics 53(3): 678 689. Guillén, M., P. L. Jørgensen, and J. P. Nielsen, 2006, Return Smoothing Mechanisms in Life and Pension Insurance: Path-Dependent Contingent Claims, Insurance: Mathematics and Economics 38(2): 229 252. Herde, A., 1997, Indexgebundene Lebensversicherung: Was noch zu klären ist, Versicherungswirtschaft 52(1997): 1354 1356. Hieber, P., R. Korn, and M. Scherer, 2014, Analyzing the Effect of Low Interest Rates on the Surplus Participation of Life Insurance Policies with Different Annual Interest Rate Guarantees, European Actuarial Journal (forthcoming). Hoevenaars, R. P. M. M. and E. H. M. Ponds, 2008, Valuation of Intergenerational Transfers in Funded Collective Pension Schemes, Insurance: Mathematics and Economics 42(2): 578 593. Jaimungal, S. and V. R. Young, 2005, Pricing Equity-Linked Pure Endowments with Risky Assets that Follow Lévy Processes, Insurance: Mathematics and Economics 36(3): 329 346. Jørgensen, P. L. and P. Linnemann, 2011, A Comparison of Three Different Pension Savings Products with Special Emphasis on the Payout Phase, Annals of Actuarial Science 6(1): 137 152. Korada, A. and S. Post, 2014, Mehrperiodiges Entscheidungsverhalten bei Lebensversicherungsverträgen mit Indexpartizipation unter Unsicherheit, Zeitschrift für die gesamte Versicherungswissenschaft 103(3): 225 242. Kuhlemeyer, G. A., 2001, The Equity Index Annuity: An Examination of Performance and Regulatory Concerns, Financial Services Review 9(4): 327 342.

22 Lee, H., 2003, Pricing Equity-Indexed Annuities with Path-Dependent Options, Insurance: Mathematics and Economics 33(3): 677 690. Maurer, R., O. S. Mitchell, R. Rogalla, and I. Siegelin, 2014, Accounting and Actuarial Smoothing of Retirement Payouts in Participating Life Annuities, Working Paper, National Bureau of Economic Research. Palmer, B. A., 2006, Equity-Indexed Annuities : Fundamental Concepts and Issues, Working Paper, Information Insurance Institute, www.iii.org/assets/docs/pdf/eia_paper.pdf, accessed 06/23/2014. Reichenstein, W., 2009, Financial Analysis of Equity-Indexed Annuities, Financial Services Review 18(4): 292 311. Reuss, A., J. Russ, and J. Wieland, 2015, Participating Life Insurance Contracts under Risk Based Solvency Frameworks: How to Increase Capital Efficiency by Product Design, Innovations in Quantitative Risk Management, Springer Proceedings in Mathematics & Statistics, Vol. 99 (forthcoming). Russ, J., 1999, Die aktienindexgebundene Lebensversicherung mit garantierter Mindestverzinsung in Deutschland. Ulm: Ifa-Verlag. Tiong, S., 2000, Valuing Equity-Indexed Annuities, North American Actuarial Journal 4(4): 149 163. VanderPal, G., 2004, The Advantages and Disadvantages of Equity Index Annuities, Journal of Financial Planning 17(1): 56 63. VanderPal, G., J. Marrion, and D. Babbel, 2011, Real-World Index Annuity Returns, Journal of Financial Planning 24(3): 50 59. Young, V. R., 2003, Equity-Indexed Life Insurance: Pricing and Reserving Using the Principle of Equivalent Utility, North American Actuarial Journal 7(1): 68 86.