# ارزش تضمین های نرخ بهره در مشارکت قراردادهای بیمه عمر: وضعیت موجود و طراحی محصول جایگزین

کد مقاله | سال انتشار | مقاله انگلیسی | ترجمه فارسی | تعداد کلمات |
---|---|---|---|---|

24374 | 2013 | 13 صفحه PDF | سفارش دهید | محاسبه نشده |

**Publisher :** Elsevier - Science Direct (الزویر - ساینس دایرکت)

**Journal :** Insurance: Mathematics and Economics, Volume 53, Issue 3, November 2013, Pages 491–503

#### چکیده انگلیسی

We compare cliquet-style interest rate guarantees used in German participating life insurance contracts across different economic environments. These guarantees are proportional to the average market interest rate at contract inception and typically set at 60% of the 10-year rolling average of government bond yields. Currently, however, in the face of prolonged low interest rates and stricter solvency regulation, the continued viability of this type of product is in question. A discussion of alternative guarantee designs is thus highly relevant. To this end, we perform a comparative analysis of contracts sold in different interest rate environments with regard to the guarantee value and show that the current practice of proportional guarantees leads to higher guarantee values the lower the market interest rate. We also observe an increased interest rate sensitivity. Additionally, alternative product designs that mitigate the interest rate dependency of the guarantee value are illustrated and assessed from the policyholder perspective.

#### مقدمه انگلیسی

Interest rate guarantees are a common feature of traditional life insurance products. We focus on so-called cliquet-style guarantees, found in the German market, under which the insurance company promises to credit the policyholder’s account with at least a guaranteed rate of return every year. As the policyholder’s account also contains previous years’ surplus, the guarantee is implicitly applied to the distributed surplus as well. The guaranteed interest rate is proportional to the current average market interest rate at contract inception, typically 60% of the 10-year rolling average of government bond yields. Hence, we refer to it as the 60% rule. 2 Over the last decade, the life insurance industry’s situation has deteriorated due to substantial changes in both the economic and regulatory environment. Under the upcoming Solvency II regime, the industry’s solvency requirements will be fundamentally reformed and lead to higher capital requirements for these traditional interest rate guarantees. At the same time, insurers’ earnings have been adversely affected by the sustained decline of returns on low-risk fixed-income assets. In Germany, life insurers are under additional pressure due to peculiarities of national regulation, such as the participation of policyholders in asset valuation reserves upon contract termination (see Section 153 German Insurance Contract Act, VVG) as this accelerates the replacement of old higher yielding bonds with new lower yielding bonds (Fromme, 2011). Stocks and other asset classes cannot be used to compensate for low earnings from fixed-income assets due to the significantly increased volatility of capital markets. The former strategy of buying low-risk bonds is thus no longer possible to the extent traditional interest rate guarantees would require. Unfavorable development of the capital market and shortcomings in risk management have led to life insurer default in several countries.3 Therefore, participating life insurance contracts and their embedded options, such as interest rate guarantees, have received a great deal of attention and analysis from academia over the last decade.4 There are two prevailing approaches to analyzing financial guarantees in life insurance: the actuarial and the financial. Consequently, most of the literature on the topic can be divided into two groups; some, however, such as Barbarin and Devolder (2005), Gatzert and Kling (2007), and Graf et al. (2011), combines both approaches. The actuarial approach focuses on analyzing the risk of different contract specifications and surplus distribution schemes under an objective probability measure. Participating contracts are analyzed by Bartels and Veselčić (2009), Cummins et al. (2007), Kling et al., 2007a and Kling et al., 2007b, and Rymaszewski (2011). Cummins et al. (2007) provide an empirical comparison of life insurance contracts typical of several European markets and those common in the United States by computing risk–return profiles. Different surplus distribution schemes and their interaction with the guaranteed rate with respect to the insurer’s shortfall risk are analyzed by Kling et al., 2007a and Kling et al., 2007b. Bartels and Veselčić (2009) extend this model with a jump process asset framework and dynamic asset allocation strategies in order to quantify the model risk. Rymaszewski (2011) also considers the risk arising from interest rate guarantees and quantifies the diversification effect caused by pooling undistributed surplus among inhomogeneous policyholder groups. The financial approach is primarily concerned with fair pricing of contracts and the options embedded therein. Many scholars analyze participating contracts, including Bacinello, 2001 and Bacinello, 2003, Bauer et al. (2006), Büsing (2005), Grosen and Jørgensen, 2000 and Grosen and Jørgensen, 2002, Hansen and Miltersen (2002), Zaglauer and Bauer (2008), and Zemp (2011). Bacinello, 2001 and Bacinello, 2003 show how to decompose a fair participating Italian contract into three parts (basic contract, participation, and surrender option), which can be priced separately. For the case of Denmark, Grosen and Jørgensen (2000) find the fair contract value to depend significantly on the bonus policy applied and the spread between market rate and guaranteed rate. Hansen and Miltersen (2002) demonstrate that collecting an annual fee as compensation for providing interest rate guarantees allows greater contract variety compared to receiving a share of the distributed surplus. Bauer et al. (2006) analyze cliquet-style guarantees typical of German contracts and find fair contract values to be sensitive to several model parameters, including the risk-free rate. Zaglauer and Bauer (2008) provide an extension with respect to stochastic interest rates and show that the value of the embedded options changes significantly, whereas the total contract value is only moderately affected. Zemp (2011) compares the British, Danish, German, and Italian bonus distribution system with regard to risk valuation and shows that the Italian system is most sensitive to changes in asset volatility. The extant literature tends to focus on pricing existing contracts in different economic environments. Thus the guaranteed rate is typically considered as fixed and independent of the economic environment. We take a step forward and analyze the 60% rule under different economic environments (high/low interest rates). A second contribution of this paper is to analyze alternative designs for the guaranteed rate in traditional products. To our knowledge, this has not yet been done in the academic literature, although it is an issue of high interest among practitioners. To date, the solutions most frequently suggested are temporary and reduced guarantees (see, e.g., Goecke, 2011, Heinen, 2011 and Pohl, 2011). This article presents a comparative analysis of the 60% rule and alternative product designs in different interest rate environments with respect to the fair guarantee value. The analysis is designed as a ceteris paribus analysis and we consider a typical German participating life insurance contract where the guaranteed rate depends on the long-term average of interest rates. We adopt the valuation framework presented in Bauer et al. (2006) and its extension for stochastic interest rates by Zaglauer and Bauer (2008). Their methodology allows us to decompose the contract into its components and hence price the interest rate guarantee separately. To compare contracts sold in different economic environments, we calibrate the surplus-related parameters so that the compared contracts have a net present value of zero under the risk-neutral pricing measure. We also assess the policyholder utility of the proposed alternative designs. We find that the current practice of setting the guaranteed rate leads to significantly higher guarantee values in times of low interest rates and to an increased sensitivity to interest rates. However, alternative products can be designed to mitigate the interest rate dependency of the guarantee value. Our findings also show that from the policyholder perspective there does not appear to be a substantial difference between the different guarantee types. These results contribute to the ongoing discussion of how to reform insurance regulation and design products so as to cope with the pressures arising from low interest rates. Risk managers and regulators will particularly benefit from this analysis as we identify the shortcomings of proportional cliquet-style interest rate guarantee schemes. In our analysis we consider a typical German contract, but the model is sufficiently flexible so that our analysis can easily be extended to accommodate other regulatory regimes with cliquet-style guarantees. The remainder of this paper is structured as follows. In Sections 2 and 3 we briefly introduce the general modeling framework and valuation methodology used by Bauer et al. (2006) and Zaglauer and Bauer (2008). A description of the different product designs is presented in Section 4. We also discuss adjustments to the existing model that are necessary to incorporate the new guarantee types and the utility analysis. Numerical results for both the 60% rule and its alternatives are given in Section 5. In Section 6 we conclude and identify areas for further research.

#### نتیجه گیری انگلیسی

In this section, we analyze the fair guarantee value in different interest rate environments, examine how assumptions about the regulatory parameters yy and δδ impact the different guarantee types, and quantify the model risk with respect to asset volatility. In a last step, we compare the policyholder utility of the alternative designs. It can happen that the calibration procedure stops prematurely at a local minimum, thus not implying a fair contract. In some of these cases, we can manually find optimal parameter combinations. However, since this approach is rather arbitrary, we do not include those results in the subsequent analysis. 5.1. Guarantee value Fig. 1 shows the fair guarantee value for contracts sold in different economic environments under the current 60% rule and the alternative designs introduced in Section 4.1. The optimal surplus parameters and the corresponding decomposition of the fair contract value are shown in Table A.1, Table A.2, Table A.3, Table A.4, Table A.5 and Table A.6 in the Appendix.Fig. 1(a) illustrates how different economic environments ϑϑ and initial reserve levels x0x0 affect the fair guarantee value C0C0 under the 60% rule. In low interest rate environments, the guarantee increases significantly in value and is highly sensitive to further changes in the average market rate. According to the fair contract condition (Eq. (8)), any increase in the guarantee value has to be compensated by an equal change of View the MathML sourceD0+ΔR0. In contrast to the value of dividend payments, D0D0, which decreases in low interest environments, the expected change in reserves, View the MathML sourceΔR0, strongly increases. This implies that the shareholders receive less compensation for their investment and more money is kept as a reserve for future generations of policyholders. For higher market rates, we observe a notably lower and less sensitive guarantee value. Under both variations of the reduced guarantee, as well as under the money-back and temporary guarantee, the guarantee value is generally lower but shows characteristics similar to the 60% rule. The increase for low interest rate levels, however, is more pronounced (see Fig. 1(b)–(e)). For the reduced guarantee, we observe the guarantee value to decrease with the portion ππ of the average market rate passed on as guarantee. Thus, for π→0π→0, this guarantee type converges to the money-back guarantee. Similar results are found for the temporary guarantee: for τ1→0τ1→0 it converges to the money-back guarantee, for τ1→Tτ1→T to the 60% rule. For the fixed safety margin guarantee (illustrated in Fig. 1(f)), we observe some fluctuations of the otherwise rather stable guarantee value for x0=0x0=0 and x0=0.1x0=0.1. However, when compared with results obtained by using different initial values for the calibration, we find these fluctuations to be random. Thus, the fixed safety margin guarantee produces guarantee values independent of economic environment. In general, higher safety margins lead to lower guarantee values. However, due to the restriction g≥0g≥0, this guarantee type is equal to the money-back guarantee in low interest rate environments (in fact, these two guarantee types coincide for all average market rates below the chosen safety margin). This observation identifies the spread between average market rate and guaranteed rate to be one of the main determinants of the guarantee value. In the case of no initial reserves, the guarantee accounts for up to 6% of the contract value. For a higher initial reserve level x0x0, the guarantee value is lower for all designs and all average market rates as adverse portfolio returns can be compensated by releasing reserves. The values shown in Table A.1, Table A.2, Table A.3, Table A.4, Table A.5 and Table A.6 suggest that the surplus parameters leading to fair contracts are relatively smooth in ϑϑ. For all designs except the fixed safety margin guarantee we observe the lower bound aa of the reserve corridor to be decreasing in ϑϑ—meaning the company is crediting the target rate more often and thus is retaining less surplus to build reserves. The upper bound bb is in most cases only slightly decreasing in ϑϑ, whereas the factor zfzf for the target rate increases notably. All these effects are less pronounced for increasing initial reserves. For the fixed safety margin guarantee, there is no clear trend in how the surplus parameters depend on the average market rate; they are rather stable, with only minor fluctuations. This smooth behavior can be used in cases where the numerical calibration method fails. For example, in the case of the fixed safety margin guarantee (Table A.6), we show that even though the calibration procedure fails for (ϑ,x0)=(0.09,0.05)(ϑ,x0)=(0.09,0.05), the parameters (a,b,zf)=(0.0390,0.2693,1.2035)(a,b,zf)=(0.0390,0.2693,1.2035) lead to a fair contract with the decomposition C0=207.12,D0=87.86C0=207.12,D0=87.86 and View the MathML sourceΔR0=119.26. 5.2. Impact of the regulatory parameters yy and δδ Analyzing the impact of the parameters yy and δδ on the fair guarantee value is particularly interesting as these parameters are subject to regulatory decisions. The regulator might modify accounting rules or minimum surplus requirements, leading to changes of yy and δδ. A different value for yy means that insurers will have to show a lower or higher portion of their market value earnings as book value earnings. As the minimum surplus is based on book value earnings, yy directly influences the amount of surplus distributed and thus the guarantee value. Changing δδ also directly affects the minimum surplus. For the case of yy we consider two scenarios: in one the approximation factor is changed to y=0.3y=0.3, in the other to y=0.7y=0.7. The higher the value for yy, the more restrictions in asset valuation apply and book value earnings are closer to market value earnings. As the movement toward IFRS leads to more market-value-based accounting, the results for y>0.5y>0.5 are of special importance. For δδ we consider the assumption that the regulator suspends the minimum surplus requirements, that is, sets δ=0δ=0. Our results are inconclusive as to the impact of yy on the guarantee value. That is, we cannot identify a clear trend as to whether changing yy increases or decreases the fair guarantee value in general. Since the calibration procedure usually does not yield a unique solution, the impact of yy might vary depending on the chosen solution. Thus we show in Fig. 2 the absolute value of the guarantee value’s deviation relative to the base case y=0.5y=0.5. When yyincreases (more restrictions on determining book values), the guarantee value deviates more strongly from the base case compared to when yy decreases. In general, the higher the average market rate, the lower the deviation of the guarantee value. It is difficult to definitely state which guarantee design is most sensitive to changes in asset valuation restrictions. For y=0.3y=0.3 it seems to be the money-back and fixed safety margin guarantees whereas for y=0.7y=0.7 the numbers are ambiguous. An analysis of the influence of δδ yields results comparable to the case of y=0.3y=0.3 which can be explained by the functional form of the minimum surplus requirement.5.3. Model risk For a fixed initial reserve quota of x0=0.05x0=0.05, we assume the asset volatility σAσA is lower/higher than initially anticipated. Fig. 3 shows the relative deviation of the guarantee value when the asset volatility is overestimated by 50 bps (dash–dot line), or underestimated by 50 bps (solid line), meaning that σAσA is not 0.036, but in fact 0.031 (overestimation) or 0.041 (underestimation). Again, cases where no fair contract was found are omitted (i.e., no marker for these is shown in Fig. 3).Fig. 3 clarifies that underestimating asset volatility generally results in a larger relative deviation compared to overestimation. Except for the fixed safety margin guarantee, the deviation increases with the average level of interest rates. This effect is most pronounced for the money-back guarantee, where the relative deviation more than triples. The fixed safety margin guarantee, in contrast, shows a rather stable, slightly decreasing deviation. Of all the compared guarantee designs, the money-back guarantee is the most sensitive to a misspecification of asset volatility, thus exhibiting the highest model risk. For the chosen safety margin of 150 bps, the fixed safety margin guarantee has the lowest model risk. 5.4. Policyholder utility To conclude the analysis, we compare the different guarantee designs for fair contracts by assessing their utility from the policyholder perspective. To this end, we again fix the initial reserve quota at x0=0.05x0=0.05 and assume policyholders with different degrees of risk aversion, represented by γ=2γ=2, 5, and 10 in Eq. (17). The value γ=2γ=2 implies a very low degree of risk aversion (cp. Broeders et al., 2011), whereas γ=5γ=5 and 10 correspond to higher degrees of risk aversion. To simplify matters, we compare the corresponding certainty equivalents, which are illustrated in Table 3. For each pair of (γ,ϑ)(γ,ϑ), the highest value is printed in bold. We find that the certainty equivalent, and thus the expected policyholder utility, increases with the average market rate since this, ceteris paribus, increases the payoff. We also see that the expected utility decreases with increasing risk aversion, which can be explained by the functional form of the utility function. Compared to the certainty equivalent variation between the market rates and the risk-aversion levels, the variation observed between the different guarantee types appears small. That is, we observe only relatively small deviations when comparing the certainty equivalent for a given combination of market rate and risk-aversion level (the maximum reduction is 1.38% with the money-back guarantee, 10% average market interest rate, and the highest risk aversion). It thus seems that for the policyholder, differences in the expected utility between different guarantee types are not very substantial. A possible explanation for these rather small deviations is that the payoff is mainly determined by the reference portfolio’s performance and not so much by the actual guarantee type. For very low average market rates, the money-back and fixed safety margin guarantees yield the highest utility, independent of the degree of risk aversion.15 One reason for this result might be that from the policyholder perspective, a lower guarantee can lead to a higher expected utility as it imposes less restrictions on the insurer’s asset allocation (cp. Wagner and Schmeiser, 2012). Our results allow no clear conclusion for the case of higher average market rates, but the fixed safety margin guarantee does appear to prevail for higher degrees of risk aversion. An additional sensitivity analysis reveals that for all guarantee types the certainty equivalent is positively correlated with the equity risk premium λAλA as it drives the reference portfolio’s return. The correlation with λrλr is also positive, albeit to a much lower extent. For different sets of optimal surplus parameters, the results are similar.