The E ects of Load Factors and Minimum Size Restrictions on Annuity Market Participation

Transcription

1 The E ects of Load Factors and Minimum Size Restrictions on Annuity Market Participation Paula Lopes 1 November Department of Finance, London School of Economics. p.v.lopes-cocco@lse.ac.uk. I am very grateful for the helpful comments given by participants at the First Workshop of the RTN Project on Financing Retirement in Europe and at the UBS Seminar Series, including Ron Anderson, David Blake, Helmuth Cremer, Otto van Hemert, Marco Pagano, Pierre Pestieau, Guillaume Plantin, Rafael Repullo and the seminar discussant Ernesto Villanueva. I am also grateful to Joao Cocco, Joachim Inkmann, Alex Michaelides, James Poterba, David Webb and anonymous referees for helpful suggestions. Financial support of the RTN Project on Financing Retirement in Europe and the UBS Pensions Research Programme is gratefully acknowledge.

2 Abstract This paper solves an empirically parameterized model of households optimal demand for nominal and in ation-indexed annuities. The model incorporates preexisting annuitized wealth, load factors and a minimum size restriction to buy an annuity. The sources of uncertainty are mortality, in ation, and real interest rate risk. From the model s policy functions we construct the threshold load factor which makes consumers choose not to participate in the annuity market, at each point of the wealth distribution. The threshold loads for in ation-indexed annuities are considerably higher then for nominal annuities, re ecting consumers welfare gains from having access to the in ation-indexed annuities even at a higher price. Under current market conditions, the model generates no participation in the annuity market for the US and very limited participation in the UK. However, participation is expected to increase substantially once pensions systems moved from de ned bene t to de ned contribution Moreover, model simulations show that the restriction to the minimum size to an annuity can account for a decrease in market participation of up to 40 Keywords: Annuities, In ation Risk, Real Interest Rate Risk, Mortality Risk JEL classi cation codes: E21, G11

3 1 Introduction In most social security systems, the government provides each retiree with a state pension for as long as he lives. This way the government provides insurance against longevity risk. Moreover, since nominal state pensions are usually indexed to a measure of consumer prices it also insures retirees against in ation risk. 2 However, given the observed increase in life expectancies and decrease in birth rates these, mostly unfunded, pension systems have become largely unsustainable. As a result, many countries are starting change their pension systems from unfunded de ned bene t (DB) to funded de ned contribution (DC). Apart from any possible changes to the level of retirees bene ts, such a shift has important implications on the retirees nancial problem. This is because, in a funded DC system, each person accumulates savings in a retirement account which will become liquid at retirement age. However, accumulated nancial wealth that is run down at retirement may not be a good alternative to state pensions, because it lacks some of the insurance features of the latter. A portfolio of nancial assets does not necessarily provide insurance against in ation risk and longevity risk. Whereas in certain countries retirees may invest their nancial portfolio in in ation-indexed bonds and in this way insure themselves against in ation risk, it may be harder to obtain insurance against longevity risk. In theory, as demonstrated in the pioneering work of Yaari (1965), the optimal choice of a non-altruistic individual is to hold all his assets on actuarial notes, whenever the probability of death is positive. This result is driven by the fact that the annuity system reduces the ine ciency created by the uncertainty of the time of death, as well as the fact that actuarial notes pay a higher interest (market interest rate plus a mortality premium) conditional of being alive. More recently Davido, 2 However, insurance against in ation risk is not complete because state pensions are usually indexed to expected in ation which may di er from realized in ation, and because the basket of goods of a typical retiree may be di erent than the basket of goods used to compute the measure of consumer prices. 1

4 Brown and Diamond (2005) show that even under less restricted assumptions than those used by Yaari, full annuitization is still optimal. Applying this result, one alternative is for individuals to use their accumulated nancial wealth to purchase an annuity upon retirement. If the annuity is in ationindexed it will insure the retiree against in ation movements in addition to longevity risk. Annuities are o ered in the private marketplace in many countries, including the UK and US. Thus, it seems, the features of state pensions may be replicated in the marketplace, at some cost, for households who wish to do so. One concern, though, is that the interest by retirees in annuities in the private marketplace has, to this date, been limited. 3 Literature on annuity demand has been trying to reconcile this fact with the theory by relaxing some of the Yaari s model restrictive assumptions. One line of reasoning is based on the imperfect information of the annuity market which results in self selection. This externality could drive households to insure themselves within the family units instead of relying on a private annuity market, as shown in Kotliko and Spivak (1981) and Brown and Poterba (2000). However, despite the self selection in annuities markets, empirical evidence shows that annuities are not unfairly priced for the annuitant. For the individual with average life expectancy, Mitchell, Poterba, Warshawsky and Brown (1999) nd that the expected present discounted value of payo s, relative to the initial cost of the annuity, has been increasing over time in the US. Similarly, Finkelstein and Poterba (2002) and Cannon and Tonks (2002) present evidence that annuity rates are not unreasonably low in the UK. Another line of reasoning is to introduce a risky asset as means of investment. This can improve the budget constraint in the future and therefore create a real option value of delaying the decision to annuitize. This is the path taken in Milevsky and Young (2002), in order to show that it might be optimal for the individual to delay annuitization. Bernheim (1991) and more recently Dushi and Webb (2004) present empirical evidence that strongly suggests that state pensions provide insurance for most longevity risk that 3 This empirical regularity has been well documented in previous studies. See Friedman and Warshawsky (1990), Poterba and Wise (1996) and Brown (1999), among others. 2

5 retirees require. Once that is not longer the case the demand for annuities will increase. The e ect of uncertain medical expenditures on annuity demand has also been the focus of attention in previous studies, namely Sinclair and Smetters (2004), Turra and Mitchell (2004) and Webb (2005). These studies o er potential explanations for the fact that full annuitization might not be the optimal choice for retirees. However, there is still the open question of why most retirees choose not to annuitize any of their wealth, and therefore choose not to participate in the annuity market. In the present paper we o er new evidence which could explain the lack of participation in the voluntary annuity market. In order to do so we build on existing literature by solving a realistically parameterized model of consumer demand for annuities incorporating features already analyzed in previous studies. These include load factors, pre-existing annuitized wealth in the form of real pension income and real interest rate risk. In order to replicate the retirees problem as realistically as possible we add new elements to the model. These are in ation risk and a minimum size restriction to buy an annuity. Finally, the economy s wealth distribution is used to obtain simulated annuity demands. We simulate annuity demands for two economies, the UK and the US, within each economy we distinguish between nominal and in ation-indexed annuities, and in the UK we also distinguish between compulsory and voluntary markets. From the optimal annuity demands, welfare gain measures and the simulation results we are able to assess how the di erent elements can account for the observed lack of annuity market participation. Signi cant contributions to the existing literature are drawn from our model. First, we obtain for each level of nancial wealth at retirement the threshold load factor at which it becomes uneconomical for the consumer to participate in the annuity market. This function provides valuable insight on the maximum load factor consumers are willing to accept. Comparing these threshold loads with actual loads charged by the insurance companies we learn that in fact it is optimal for the american consumer, with average life expectancy, not to participate in the annuity market, 3

6 both for nominal and for in ation-indexed annuities. In the UK the model predicts some annuity market participation for nominal annuities, for the consumer with average life expectancy. These results give us some indication that in fact, for the current level of pre-existing pension bene ts, the load factors charged by insurance companies are too high and might explain why only individuals with higher than average life expectancy nd it optimal to participate in the annuity market. This result is also consistent with previous studies that nd evidence for self selection in the annuities market. Second, we are able to draw predictions on how the relation between load factors and annuity market participation is likely to change when the observed trend in shift from de ned bene t to de ned contribution pension systems is realized. In that case, pre-existing annuitized wealth decreases and the individual decides how much to annuitize from his accumulated pension wealth. Comparative statics on the level of pre-existing annuitized wealth show that a decrease in pension bene ts will result in an increase in the threshold load factor. As a result, other things equal, annuity market participation is predicted to increase. Third, in surveying the major insurance companies operating in the US and UK s annuity market we learned that there is a non-neglectable restriction on the minimum size consumers face when they wish to buy an annuity. By simulating optimal annuity demands over the nancial wealth distributions, from both economies, we are able to quantify how binding is these restrictions on the minimum size for an annuity are. In particular, we show that this minimum size restrictions alone can account for up to 40% decrease in participation when comparing to a case with no restrictions. Moreover, the decrease in participation due to this restriction is higher for lower load factors. Finally, additional contributions are on the e ect of load factors on consumers portfolio choice and on the literature debate on the presence of adverse selection in annuities markets. The evidence on the presence of adverse selection in annuities markets is usually based on the fact that the mortality rate of the annuitant population 4

7 is lower than the mortality rate of the general population. However, mortality rate is also negatively correlated with wealth accumulation (see Attanasio and Hoynes, 2000). Therefore it becomes unclear whether the fact that annuitants are longer lived results from active selection or passive selection, as described in Finkelstein and Poterba (2002) and Murthi, Orzag and Orzag (1999). Distinguishing between the two types of selection is important because they have di erent policy implications. The present model provides an example of passive selection. Only wealthier individuals participate in the annuity market. The remainder of the paper is organized as follows. In Section 2, model, solution technique and parameterization are presented. In Section 3, welfare gains and the policy functions for the demand for annuities are presented and discussed. Section 4 reports empirical evidence, and Section 5 concludes. 2 The Model 2.1 Model speci cation Time parameters and preferences We model the consumption and asset choices of an household from retirement onwards. 4 The household may either be an individual or a couple in which case the household only dies when both individuals die. The household lives for a maximum of T periods. We let each period in the model correspond to one year, and we allow for uncertainty in the age of death in the manner of Hubbard, Skinner and Zeldes (1995). Let p t denote the probability that the household is alive at date t, conditional 4 The dynamic model is in the line of the precautionary savings literature models proposed by Carroll (1997) and Deaton (1991). 5

8 on being alive at date t 1. Of course, p 1 1. Then, household i s preferences are described by the time-separable power utility function: E 1 TX Y t 1 t 1 t=1 j=1 p j " p t C 1 it 1 + (1 p W 1 i;t t)b i 1 # ; (1) where is the time discount factor and is the coe cient of relative risk aversion. With probability (1 p t ) the household dies and leaves his wealth W t as a bequest. The parameter B measures the intensity of the bequest motive. In each period t, t = 1; :::; T, the household chooses consumption C t. The term structure of nominal and real interest rates Nominal and in ation-indexed annuities di er because nominal interest rates are variable over time. This variability comes from movements in both the expected in ation rate and the ex-ante real interest rate. Following Campbell and Cocco (2003) we use a model that captures variability in both these components of the short-term nominal interest rate, and allows for some predictability of interest rate movements. Thus in the model there will be periods when households can rationally anticipate declining or increasing short-term nominal interest rates. We assume that expected in ation follows a rst-order autoregressive process. That is, log one-period expected in ation, 1t = log(1 + 1t ), follows the process: 1t = (1 ) + 1;t 1 + t ; (2) where t is a normally distributed white noise shock with mean zero and variance 2. The expected log real return on a one-period bond, r 1t = log(1 + R 1t ), is given by: r 1t = r + t ; (3) 6

9 where r is the mean log real interest rate and t is a normally distributed white noise shock with mean zero and variance 2. The log yield on a one-period nominal bond, y 1t = log(1 + Y 1t ), is equal to the log real return on a one-period bond plus expected in ation: y 1t = r 1t + 1t : (4) To model long-term nominal interest rates, we assume that the log pure expectations hypothesis holds for both nominal and real interest rates. 5 That is, we assume that the log yield on a long-term n-period nominal bond is equal to the expected sum of successive log yields on one-period nominal bonds which are rolled over for n periods: Xn 1 y nt = (1=n) E t [y 1;t+i ]: (5) i=0 This model implies that excess returns on long-term bonds over short-term bonds are unpredictable, even though changes in nominal short rates are partially predictable. Available nominal and in ation-indexed annuity contracts We study both nominal and in ation-indexed annuity contracts. If at retirement age the interest rate on a nominal bond with maturity t is Y t;1, and the household purchases a nominal annuity that makes an annual nominal payment of A N;N (where the double superscript refers to the nominal payout of a nominal annuity), the expected present discounted value (EPDV) of the annuity payouts is given by: 5 For a textbook exposition and summary of the empirical evidence on this model, see Campbell, Lo, and MacKinlay (1997), Chapter 10. 7

10 EP DV N 1 = TX j=1 A N;NQ j k=1 p k (1 + Y j;1 ) j : (6) This EPDV can be compared with the premium cost of the annuity to obtain a measure of the money s worth of the annuity. The di erence between EP DV N 1 and the premium cost of the annuity covers expenses and other administrative costs associated with the sales of annuities and normal pro ts of insurance companies. The annuity premium (P ) and the money s worth are therefore de ned as: AnnuityCost = (1 + P ) EP DV Money 0 sw orth = EP DV=AnnuityCost Nominal annuity payments are xed at contract initiation, so that real annuity payments A R;N t are inversely proportional to the price level P t : A R;N t = AN;N P t (7) This implies that a nominal annuity contract is a risky contract because its real capital value is highly sensitive to in ation. Similarly, if at retirement age the interest rate on a in ation-indexed bond with maturity t is R t;1, and the household purchases an in ation-indexed annuity that makes an annual real payment of A R;R (where the double superscript refers to the real payout of a real annuity), the expected present discounted value (EPDV) of the annuity payouts is given by: EP DV R 1 = TX j=1 A R;RQ j k=1 p k (1 + R j;1 ) j : (8) 8

11 Real annuity payments are xed at contract initiation, and nominal payments increase in proportion to the price level P t. Thus, unlike a nominal annuity, the real capital value of an in ation-indexed annuity is not sensitive to in ation. Retirement Wealth and Pension Income At retirement age the household has nancial wealth W 1, which he can use to purchase an annuity. In addition, the household is endowed with gross real pension income in each period, L. As usual we use a lower case letter to denote the natural log of the variable, i.e., l log(l). Household j s real pension income is exogenous and is given by: l j = f(w j1 ); (9) where f(w j1 ) is a deterministic function household j s nancial wealth at retirement. This is due to the fact that pension income is likely to be positively correlated to nancial wealth at retirement W j1 (accumulated assets plus nal salary). This is true not only for those covered by a de ned bene t plan, in which case pension income is related to nal salary but also for those contributing to a de ned contribution plan. Contributions (and therefore bene ts) are usually related to salary. Summary of the household s optimization problem In summary, the household s control variables are fc t g T t=1 ; which is the optimal consumption stream during retirement, and at t = 1 how much wealth to annuitize. n The vector of state variables can be written as X t = t; y 1t ; W t ; P t ; 1t ; A R;i t o T t=1 where W t is real liquid wealth or cash-on-hand, P t is the date t price level and A R;i t, i = N; R is the date t real annuity on the nominal/real annuity purchased at retirement. The equation describing the evolution of real cash-on-hand can be written as: 9

12 W t+1 = (W t C t )(1 + R 1;t+1 ) + L t+1 + A R;i t+1; (10) where i = N; R. in ation-indexed bonds. This equation assumes that savings are invested in one-period Solution technique This problem cannot be solved analytically. Given the nite nature of the problem a solution exists and can be obtained by backward induction. We discretize the state space and the choice variables using equally spaced grids in the log scale. The density functions for the random variables will be approximated using Gaussian quadrature methods to perform numerical integration (Tauchen and Hussey 1991). 2.2 Parameterization An important component of any quantitatively focused study is the parameterization of the model. With this respect, there are several issues that the reader should keep in mind. First, we take one period in the model to represent one year and use annual data throughout. Second, to the extent that in ation and interest rate processes, pre-existing annuitized wealth, wealth distribution or longevity risk di er across countries, the bene ts of annuities will also di er. In order to explore these di erences we calibrate the model for two countries, namely the US and the UK In ation and Interest Rate Processes In order to estimate the in ation rate process we used data for the US Consumer Price Index from 1962 to The estimated parameters are shown in Table 1. The 10

13 mean log in ation over this period was 4:6%, with a standard deviation of 3:9%, and a rst-order autoregression coe cient equal to 0:754. For the in ation rate process for the UK we used data from the Retail Price Index between 1985 and The mean and variances are considerably lower, this being partly due to the di erence in the sample period. In ation indexed bonds where only available in the UK after In order to be consistent with the real interest rate estimation, the sample period for the in ation process also starts after In order to estimate the parameters for the real interest rate process we constructed a measure of real interest rate by subtracting the logarithm of in ation to the log yield on one-year US Treasury bonds. The mean annual log real yield is equal to 2%, and it is fairly stable. For the UK we used the yield on Index-Linked British Government Securities from 1985 to Mean log yield was 4.6% and also fairly stable. Results are reported in Table Baseline Parameters We study the problem of retirees, therefore, period one in our model is taken to be age 65, the most common retirement age. Conditional survival probabilities for male population is taken from the US census and the UK Government Actuary s Department, from ages 65 to 99. To simplify, we assume that the individual dies at age 100 with certainty, if he is still alive then. Several studies nd supporting evidence for the fact that the annuitant survival probabilities are higher than the survival probabilities for the whole population. However, we use the latter because the aim of the paper is to explain why it is optimal not to participate in the annuity market for the retiree with average life expectancy. In the baseline case we use a coe cient or relative risk aversion equal to 3 and a discount factor equal to 0:98. Following Brown (2001), who nds no evidence that bequest motives are an mportant factor in making marginal annuity decisions, the bequest parameter is set to zero. Table 2 summarizes these results. 11

14 2.2.3 Annuity Contracts Annuity Premium In order to assess the impact of the load factors on annuity demand we need to have some reference parameters of the prices charged in the annuity markets. While obtaining these load factors we have to take into account that for each country we observe di erent economic scenarios and di erent market structures which a ect insurance companies pricing policies. For example, in the UK there is a compulsory and a voluntary market for annuities, whereas in the US there is no mandatory annuitization. Also, within a certain annuity market individuals have access to di erent annuity contracts, for example with di erent payout schedules. In addition we observe, for the same contract, di erences as high as 10% between the highest and the lowest industry payout. 6 Several studies have analyzed annuity markets pricing and report industry averages for speci c contracts. Therefore, we will calculate the value for the annuity premium P using the reported M oney0sw orth: These M oney0sw orth are calculated using average annuity payouts available to a 65 year old male, discounted using risk free interest rates and the population survival probabilities described above. For the US market, we set Money0sW orth equal to for nominal annuities and 0:75 8 for real annuities. The UK annuity market is divided in two segments: -compulsory annuity market, where tax-quali ed retirement funds are annuitized; - and a voluntary market, for all other annuitizations. Although some studies nd that M oney0sw orth do not di er substantially between the compulsory and the voluntary market, see Murthi, Orszag and Orszag (1999b), others do nd that prices di er in the two markets. For nominal annuities we used the estimates of Finkelstein and Poterba (2002), who report values of 0.87 and 0.9 for the voluntary and compulsory markets, respectively. 6 See Brown, Mitchell and Poterba (2001). 7 As reported by Poterba & Warshawsky (2000). 8 Following Brown, Mitchell and Poterba (2000). For real 12

15 annuities, and for the compulsory market, we have set Money0sW orth equal to 0.85, following Brown, Mitchell and Poterba (2001). M oney0sw orth for real annuities in the voluntary market is set equal to 0.8, following James and Vittas (1999). Table 3 summarizes these results. There are substantial di erences in the price charged for nominal and in ationindexed annuities. One possible explanation would be the presence of self-selection in the choice of annuity type, see Finkelstein and Poterba (2002, 2004), in particular, individuals with higher life expectancy are more likely to choose back-loaded annuities, and as a result insurance companies charge a higher premium. Other explanations include: (i) Insurance companies are more restricted in the set of instruments they are able to invest while hedging themselves against in ation risk, primarily government bonds, and therefore have to forego the higher yields of alternative assets, which do not o er in ation protection. James and Vittas (1999) show evidence that in countries with more developed in ation-indexed markets, the di erences in load factors between real and nominal annuities is smaller; (ii) A small total size from in ationindexed annuities markets could lead insurance companies to charge higher markups; (iii) A risk averse individual is willing to pay a higher price for a real annuity which o ers insurance for both the mortality risk and for the in ation risk, as opposed to the nominal annuity which only protects consumers from the mortality risk. Minimum Size Restrictions Existing studies on annuities tend to overlook a signi cant feature of these markets, which is the minimum size restriction that households face when they wish to buy an annuity. In order to have a better understanding of the magnitude of this participation restrictions we surveyed the most important insurance companies listed in insurance industry rankings for both the UK and in the US. 9 In the UK, for the voluntary market, there were only two companies that would sell an annuity for less then 5,000. Since the annuity rates for this limit 9 Major insurance companies operating in the annuities market are ranked in the Life Insurers Fact Book published by the American Council of Life Insurers and in the Annual Statistics reported by the Association of British Insurers. 13

16 are very low and not representative of the rest of the annuity market, and for consistency purposes we will use the restriction limit reported by the great majority which is 10,000. For the US none of the insurance companies reported a minimum size restriction of less than $10,000. Table 3 summarizes these annuity markets features. The most plausible explanation for the fact that insurance companies avoid small annuities amounts is that the administrative costs of setting up an account, payments processing, etc, is too high and not compensated by the potential passive selection which could exist in this segment of the market. This selection results from the fact that the individuals buying small amount annuities are poorer and therefore more likely to live less. In fact Murthi, Orszag and Orszag (1999) show that there is a decrease in the premium of 2 to 3% when comparing an annuity of 10,000 and 100,000, which means that the selection e ect is more than o set by the administrative cost reduction Wealth Distribution For the UK, we use data from the Family Resources Survey (FRS), which is a continuous rotating survey with an annual target sample size of 24,000 households. This survey is sponsored by the Department of Work and Pensions and gives information on a wide range of questions about households circumstances including income, - nancial assets and demographics. We pooled three waves of the survey, from 1999 to 2002, controlling for year e ects. For the US we used the 1998 Survey of Consumer Finances (SCF), which o ers a detailed description of US families nancial condition. In order to match the de nition of wealth in our model to the one in the data we add two variables, namely: income (which represents households gross income of the previous year) and nancial wealth (which represents households accumulated wealth in nancial assets, excluding retirement wealth and housing wealth). Also, we considered individuals aged between 55 and 70 because these are the ones approaching retirement or just retired and therefore representative for the demand of annuities. 14

17 2.2.5 Pension income In our model pension income is a function of wealth at retirement. However, in the data we do not observe these two variables at the same time. Therefore, we estimate pension income (where pension income is reported state pension and personal/company pension) for those already retired and aged less then 75, as a function of their nancial wealth (de ned in above section). In order to be consistent with the model and to avoid endogeneity problems we removed from both samples (FSR and SCF) individuals reporting positive annuity income. In this way the model predicts potential annuity demand for households who have not yet bought an annuity. We estimate pension income as a linear function of wealth (W j ), l j = a + bw j + " j ; with; W j = assets j + l j : (11) however in order to avoid an estimation bias, we rst estimate pension income (l j ) as a function of assets l j = a=(1 b) + (b=(1 b))assets j + j : (12) and then recover the desired estimates from equation (11). The results for the estimated parameters are shown on Table Welfare Metric Consumer welfare gains from having access to the annuity market are computed using the wealth equivalence measure W E. The model is solved for two di erent scenarios: one where consumers have access to annuities and another where consumers have no access to annuities. For each scenario we compute the corresponding value functions V 1 (W 1 ) and V 0 (W 0 ) as a function of retirement wealth. For each level of wealth, in 15

18 the case of no access to annuities, (W 0 ) we compute the level of wealth (W 1 ) that equates V 0 (W 0 ) = V 1 (W 1 ): This means that the wealth equivalence measure is de ned as: W E = W 1 W 0 The wealth equivalence is less than one whenever annuities are welfare improving to consumers. 3 Policy Functions, Welfare Analysis and Simulation Results In this section we study the optimal demand for annuities and the welfare gains resulting from having access to the annuity market. Figure 1 shows the optimal demands as a function of nancial wealth at retirement, both for nominal and in ation-indexed annuities that are actuarially fair priced. Demands are shown for the US and UK parameterizations. The vertical axis shows the dollar/pound amount the annuity contract pays each year. The positive slope of the demand functions comes from the fact that for precautionary saving consumers saving is an increasing function of wealth. Whenever there is no bequest motive and annuities are actuarially fair priced it is optimal for consumers to fully annuitize their savings. The di erences in the US and UK annuity demands is mainly due to di erences in pre-existing in ation-indexed annuities which in our case are in the form of real pension income. Lower pre-existing pension income increases annuity demand. Within one economy di erences between nominal and in ation-indexed annuity demands are due to the in ation process. Welfare gains to consumers resulting from having access to annuity markets are measured with the wealth equivalence measure (W E). Figure 2 illustrates W E measures for the demand functions shown in Figure 1. The lower the value the higher the welfare gain. Welfare gain is increasing in wealth because saving rate, hence annuitization rate, is also increasing in wealth. For each level of wealth the welfare gains 16

19 of real annuities are higher than of nominal annuities. This is because real annuities o er an extra insurance component, i.e., against in ation risk. 3.1 The E ect of Load Factors We now introduce load factors to the annuity contracts and analyze how it a ects annuity demand. We analyze the e ects of the load factors on annuity demand from di erent perspectives, rst we predict annuity demand for speci c load factors (i.e., the ones currently charged by insurers), then we look at how portfolio allocation is a ected by load factors. We also analyze how the load factors a ect annuity market participation. In order to do so we follow a more general approach and calculate, for each point in the wealth distribution, the load factor at which consumers choose not to participate in the market. If we apply the load factors charged in di erent markets, i.e., UK and US, and within the UK compulsory and voluntary market, we observe the following impact on annuity demands: i) for the US, annuity demand both for nominal and in ationindexed annuities becomes zero, at all levels of wealth; ii) for the UK, annuity demands are lower but still positive for the compulsory market and positive for the voluntary market for nominal annuities, however there is no annuity demand in the voluntary market for in ation-indexed annuities. Figure 3 shows the demand for nominal annuities, for the UK, using di erent load factors, in particular the actuarially fair, the compulsory and the voluntary market ones. As expected as the load premium increases the demand for annuities decreases. We now analyze the e ect of the load factors on consumer portfolio allocation. As shown in the previous section, when annuities are actuarially fair priced it is optimal for consumers to fully annuitize their savings, however as the load factor increases, annuities become less attractive and the optimal portfolio allocation changes towards higher proportions of savings allocated to bonds. In Figure 4 we can see how higher load factors a ect consumers optimal portfolio choice. With P = 1:0 optimal portfo- 17

20 lio allocation is to annuitize all savings, for all levels of wealth above 23,000$, this is the level at which consumers start saving. However, as P increases full annuitization is no longer optimal. In fact, for P = 1:1 although saving starts at a level of wealth of 23,000$ annuitization only starts at 42,000$ and full annuitization is only optimal for wealth levels higher than 180,000$. One of the main goals of the present paper is to determine what a ects consumers decision to participate or not in the annuity market. Having this in mind we analyze the e ect of the load factors on market participation from a broader perspective by asking the question of what is the threshold load factor that makes the consumer choose not to by an annuity. We calculate this threshold level for each point in the wealth distribution. The resulting function is presented in Figure 5. The dashed columns represent the actual nancial wealth distribution for the US economy. It gives us some indication of how a speci c load factor might a ect annuity market participation, however in order to quantify this impact we have to simulate annuity demands over this wealth distribution. This is performed in the following section. Comparing the load factors currently charged by insurance companies, and described in Table 3, with the threshold loads gives us some insight of how annuity demand predicted by the model is zero. The threshold load is an increasing function of wealth, in other words consumers with higher nancial wealth at retirement are willing to accept higher load factors. Also, the slope of the threshold load function is decreasing in wealth, being very steep for the lower end of the nancial wealth distribution. This comes from the fact that for lower levels of wealth the insurance o ered by annuities is very valuable, hence, consumers are willing to accept higher increases in the load factor without changing the decision to participate. The threshold load is closely related with the welfare gains from having access to the annuities market. Therefore, threshold loads for in ation-indexed annuities are higher than for nominal annuities, and this di erence is increasing in wealth. The di erence in the threshold loads between nominal and in ation-indexed annuities is a potential explanation for the observed di erence in the loads charged by insurers, however it does not explain everything because the 18

21 latter is still higher than the former. Figure 6 shows the threshold load factor for the UK and the US economies. Di erences between the two economies re ect the di erent levels of pre-existing annuitized wealth, in the form of pension income, which is higher in the US than in the UK. The di erence between the in ation-indexed and nominal threshold load functions is sightly higher for the UK than for the US. One important question is then how will this threshold load function change with the potential shift from DB to DC pension systems. To get some valuable insight on this question we calculated the threshold load in a context of no pre-existing annuities, i.e. no pre-existing pension income. Figure 7 illustrates this case. Without preexisting annuitized wealth annuities are very valuable and consequently consumers are willing to accept higher load factors for all levels of wealth. This gives an indication that, other things equal, a shift in the pension systems from DB to DC results in a higher annuity market participation. We also performed some sensitivity analysis on other parameters. Figure 7 also illustrates the case with a lower risk aversion parameter, i.e. =2. Willingness to accept load factors decreases for all levels of wealth, as consumers are now less risk averse and therefore annuities are less valuable. 3.2 Reconciling Results With Existing Literature In the previous section we illustrated how using the actual load factors charged in by US insures optimal demand for both nominal and in ation-indexed annuities became zero, for all levels of wealth. If that is the case how can we reconcile the nding of zero annuity demand in the US market with the fact that in reality, although small, there is a positive demand for annuities? In interpreting the model s predictions we have to keep in mind that the life expectancies used in the dynamic programing are the survival probabilities for the whole population, however, as previous research shows, the M oney0sw orth for the same annuity payouts calculated using the annuitants mortality table are considerably higher. In Brown, Mitchell and Poterba (2000) the 19

22 reported M oney0sw orth for 65 year old men, using the annuitants mortality table, are and for nominal and in ation indexed annuities respectively. If we apply a premium of 1.15, which corresponds to a Money0sW orth of 0.87, optimal annuity demand in the US is positive, for both nominal and in ation-indexed annuities. The literature on the e ect of the load factors on annuity market participation has not been conclusive. On one hand, the study by Friedman and Warshawsky (1990) found that for a 65 year old male the market yield di erential, between annuities payouts and other means of savings, is su cient to explain the observed lack of participation in the individual life annuity market. However, Mitchell et al. (1999) conclude that with an WE of 0.70 between annuitized at fair value and non-annuitized assets individuals are still better o purchasing an annuity that is priced at 70% below its fair value. Which would mean that the load factors charged by insurance companies would not be su cient to explain little annuity demand. However, in order to make this statement the authors are assuming the optimal portfolio allocation with load factor of 1.42 is also full annuitization, which is not the case. Figure 4 shows optimal portfolio allocation for di erent annuity premia. Higher annuity premia result in portfolio allocation with less than 100% annuitization. Consequently the welfare gains from having access to the annuities market are also reduced. 3.3 The E ect of the Minimum Size Restriction and Participation Simulations We now turn to the second question of this paper which has to do with the e ect of the restriction on the minimum size to buy an annuity on consumers market participation. In analyzing this question we take the following approach. We simulate annuity demand, for both countries, over a sample of 1000 individuals. Individuals di er from each other in their accumulated nancial wealth at age 65. The distribution of wealth of the simulated data are the actual distributions for the US and the UK, respectively. We perform this simulation for di erent cases: -load factor equal to 20

23 0% and 15%; -and the cases with and without minimum size restriction. In the case with minimum size restriction we use the actual values presented in Table 3. The average demands and participation rates for nominal and in ation-indexed annuities are shown in Tables 5 and 6. As the results illustrate, the e ects of the minimum size restriction are substantial and can account for a decrease in participation of up to 40%. The decrease in participation due to this restriction is higher for lower load factors. In general, whenever annuity demand increases the minimum size restriction becomes more binding. This results from the combination of two factors: - on one hand, shifts in the annuity demand function to the left (see Figure3 for an example), increases the potential participation of low wealth individuals; - on the other hand, wealth distributions tend to be skewed to the right, great proportion of individuals on the lower end of the wealth distribution. These two factors combined result in more potential annuitants beeing left out as a consequence of the minimum size restriction. In particular a shift from DB to DC pension systems, which results in less pre-existing annuitized wealth and therefore higher annuity demand, is likely to make the minimum size restriction more binding, even though the consumers will have higher levels of non-annuitized wealth. These simulations also show how load factors together with minimum size restriction can account for a decrease in annuity market participation of 85%. By assuming a load factor of 15% we are implicitly looking at the consumers who actually are buying annuities, because these are the ones with higher life expectancies and therefore lower implicit load factors. 3.4 Survey Evidence In order to access how the model s predictions compare to survey data we report summary statistics on annuity demand and nancial wealth of voluntary annuitants from the FRS. We observe that the mean nancial wealth of voluntary annuity holders 21

24 is 81,600 compared to mean nancial wealth of 30,800 for non-annuity holders. Therefore, there appears to be some evidence for the presence of liquidity constraints in the annuity market. We also observe that 65% of annuity holders receive an annual annuity income which lies above the minimum size restriction. The annual annuity income that corresponds the minimum size restriction of 10,000 was calculated using the annuity factor obtained from the interest rate and in ation processes used in the model s parameterization, and a load factor of 15%, which is the observed annuity factor for nominal annuities in the UK s voluntary annuity market. In terms of the model s predictions for annuity demand in the UK compared with observed annuity demand we have: (i) predicted participation is 9% of the population, whereas in the FRS there is only 2% participation; (ii) predicted average demand for annuity holders is 13,000 compared to a much lower observed 3,000. Clearly the model predicts higher annuity demands than we observe in the survey data, however, as mentioned in Banks and Emmerson (1999), there is strong evidence of underreporting in the survey annuity income data when compared with aggregate data. 4 Conclusion This paper solves a dynamic model of households demands for nominal and in ationindexed annuities, in the context of uncertainty. Uncertainty comes from di erent sources: in ation, real interest rate and time of death. In order to draw realistic predictions we also introduce pre-existing annuitized wealth in the from of pension income, load factors and a restriction on the minimum size of an annuity. We solve the consumers optimal annuity demands as a function of the level of nancial wealth at retirement and then simulate these policy functions over actual wealth distributions. Signi cant contributions are made to the existing literature on annuity demand. First, we develop a measure of the load factor which makes individuals choose not to participate in the annuity market for each level of wealth. Comparing these threshold loads with loads currently charged by annuity providers we are able to account for 22

25 the lack of participation in the US annuity market, for the consumer with average life expectancy. In the UK the model predicts positive but small participation. Second, in addition to account for the current lack of participation in the annuity market the model draws valuable predictions on the e ects on market participation of a shift in pension systems from DB to DC. In particular, simulations show that a decrease in the pre-existing level of annuitization in the form of real pension income will result in a substantial increase in annuity market participation. Third, by introducing an empirically parameterized restriction on the minimum size to buy an annuity in consumer s optimization problem, we show how biding this restriction can be. In fact, simulations show that the e ects of the minimum size restriction are substantial and can account for a decrease in participation of up to 40%. 23

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