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Is There a Role for Government Intervention in the Annuity Market?

Simon Power,a and Peter G.C. Townleyb

aDepartment of Economics, Carleton University, Ottawa, Ontario, CANADA K1S 5B6

bDepartment of Economics, Acadia University, Wolfville, Nova Scotia,
CANADA B0P 1X0


Abstract

The private annuity market is plagued by the problem of adverse selection leading to market failure. Potential remedies include government-run compulsory or voluntary life-annuity plans. A compulsory plan, while simple to design, has the drawback that it could actually decrease welfare. In contrast, a voluntary plan is always welfare-increasing, but would seem difficult, if not impossible, to design because it requires the use of private information. This paper uses a simulated optimal control model to demonstrate that, despite this seemingly insuperable problem, the government could actually design such a voluntary plan with considerable accuracy.

Keywords: optimal control; simulation; asymmetric information; annuity market

JEL Classification: C60, D82, H55

*Corresponding Author: Professor Simon Power, Department of Economics, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario, CANADA K1S 5B6.


INTRODUCTION

The private annuity market is plagued by the problem of adverse selection leading to market failure and inefficient equilibria with the result that, in many jurisdictions, it barely exists at all (Eckstein, Eichenbaum, and Peled, 1985; Townley and Boadway, 1988). The fundamental problem is one of asymmetric information: Retirees inevitably possess private information concerning their past health-care experiences and lifestyle choices which impacts upon their further life expectancy. One potential solution is government intervention in the form of a compulsory life-annuity plan (Eckstein, Eichenbaum, and Peled, 1985), but such a plan may in fact decrease welfare and, moreover, such an adverse outcome would not be signaled to policy-makers even after implementation (Townley, 1990). An alternative form of government intervention, which must by its very nature be welfare-increasing, is an actuarially sound, voluntary, life-annuity plan. Such a plan would have the additional advantage of being able to mimic the properties of a competitive market in contrast to the typically oligopolistic nature of the private annuity market that currently obtains. Furthermore, we could expect a government to be risk-neutral whereas private-sector insurers are risk-averse. At first glance, however, implementation of such a voluntary plan would seem to be frustrated by the fact that much of the information necessary for plan construction lies outside the public domain. This paper contributes to the literature by using a simulated optimal control model to demonstrate that, despite this seemingly insuperable problem, the government could indeed design and implement such a voluntary plan with considerable accuracy. This opens up the possibility of an unambiguously positive role for government intervention in this market.

The optimal control model we consider below focuses on a particular cohort of retirees and incorporates the major determinants of the demand for annuities, namely, retirement wealth, attitudes toward risk-taking, perceptions of life expectancy, and the magnitude of expected social security payments. By simulating this model under a variety of fairly extreme assumptions, we show that although a government does not have access to all the information regarding the demand for annuities, it could nevertheless design and implement an actuarially sound, voluntary, life-annuity plan – due to the fact that the "equilibrium annuity price" is found to be relatively insensitive to information outside the public domain.

The institutional framework that we adopt assumes a common retirement age at which all retirees in the cohort have to make a once-and-for-all, irrevocable, portfolio decision as to what proportion of their retirement wealth to spend on the purchase of a life-annuity and what proportion to hold as non-annuity wealth. This particular setup is consistent with legislation in Canada and other jurisdictions where retirees have a significant tax incentive to convert specifically designated retirement savings into annuities early in their retirement and before a specified age. Notice that in the case of the United States, for example, such an incentive does not exist – designated retirement savings may be converted into annuities at any age without either tax penalties or advantage. This difference is key both for how an individual retiree’s portfolio decision should be modeled and for the properties of competitive equilibria. The optimal control methods that we employ are appropriate when a single decision must be made at a particular age conditional on information available at that age. In contrast, dynamic programming methods would be more appropriate in the case of a United States-type environment where retirees can either spread their annuity purchases over time or purchase one-year annuities year after year, all the while garnering updated information that may affect their perceptions of their life expectancy. This latter approach has been explored by Friedman and Warshawsky (1990).

The organization of the paper is as follows: Section 2 introduces and motivates the model, Section 3 describes the mortality assumptions, Section 4 outlines the simulation results, and Section 5 concludes.

THE MODEL

Individuals are assumed to retire at the common age of 65, , and we focus on one such cohort of retirees.1 Upon retirement, retirees are assumed to have accumulated wealth of dollars and to expect, with certainty, to receive an annual social security payment of z dollars for their remaining lifetime.2 Each retiree must plan his consumption over his remaining lifetime so as to maximize his expected utility. We denote instantaneous utility by , where is consumption at age t. It is assumed that the retiree derives utility only from his own consumption and that is a well-behaved utility function that is strictly concave in its argument. Throughout, it is assumed that retirees do not discount future consumption for any reason other than risk aversion (implicitly) and that no interest-bearing savings instruments exist.3

The only financial-insurance instrument available to help with retirement-income planning is a life-annuity (a straight-life, single-premium, immediate annuity) and a once-and-for-all, irrevocable, portfolio decision must be made at t = 0 by each retiree as to how much of their accumulated wealth to spend on an annuity and how much to hold as non-annuity wealth. For convenience, we denote non-annuity wealth by and annuity wealth by . Each dollar spent on an annuity pays q dollars per year beginning at t = 0 and continuing until the individual dies, so 1/q may be interpreted as the price of the annuity. Thus, at any age t, a retiree’s consumption would be based on instantaneous income derived from at most three sources: z dollars of social security income, dollars of annuity income, and whatever amount the retiree wishes to spend out of any remaining non-annuity wealth. It is assumed that retirees cannot borrow against future annuity payments and that annuity contracts contain no provision for survivors’ benefits.

What makes the retiree’s problem at all difficult is that he does not know the exact duration of his remaining life-span at t = 0, the point at which his crucial portfolio decision has to be made. Rather, he simply knows the probability distribution of his remaining life-span and the fact that he will die with certainty before some maximum possible life-span denoted by T. This knowledge is based upon the retiree’s private information concerning his past health-care experiences and lifestyle choices – information that is not available to the government. This probability distribution is assumed to have the following properties:

;

;

;

; , .

Formally then, the retiree faces the following maximization problem:

(1)

and ;

where a dot over a variable indicates its time derivative.4 Below, we present the essentials of the solution to this maximization problem. A more complete derivation of the key equations appears in the Appendix.

If we define age M to be the earliest age at which the retiree exhausts all the wealth he chooses not to spend on an annuity, then M will be the solution to the following equation:

. (2)

Moreover, M will be unique for all values of q when . Notice that if the price of the annuity, 1/q, is equal to , then the annuity will be actuarially fair. Further, if such an annuity were to be offered, then from (2), M = 0 indicating that the retiree would spend all of his wealth on the annuity, thus fully insuring himself. His planned consumption path would be flat and he would consume at the rate of dollars per period. This gives us a more intuitive interpretation of the price 1/q. If an annuity were to be offered such that 1/q = n, then it would be actuarially fair with respect to an expected duration of retirement of n years. Thus, annuity prices in this model are measured in years: The lower the number of years, the more attractive the annuity becomes to the retiree.

At the other extreme, if the price of the annuity is such that the retiree chooses not to purchase an annuity (), then his planned consumption path would be everywhere decreasing with zero consumption planned for age T. Along this path, his expected marginal utility of consumption would be constant.

For annuity prices between these extremes, the retiree would choose to spend some, but not all, of his wealth on an annuity. His planned consumption path would be decreasing until age M (0 < M < T) and constant at the rate of dollars thereafter.

In all cases, the retiree’s demand for annuities function is given by:

(3)

where denotes the inverse of the retiree’s elasticity of marginal utility of income or index of relative risk aversion. We shall confine our discussion and simulation exercises to utility functions where this index, denoted by RRA, and equal to , is constant (for a given and T combination). Notice that for , and for . Also, as one would expect, the retiree’s demand for annuities is increasing in relative risk aversion, q, and life expectancy, and decreasing in z. Moreover, when z = 0 the demand for annuities is homogeneous of degree one in wealth.

At this point, the individual retiree’s problem is essentially solved. Equations (2) and (3) can be used to fully describe his portfolio decision at t = 0 and his entire planned consumption path thereafter.

These results for the individual retiree’s problem can be used to help obtain the solution for the overall cohort-wide equilibrium annuity price. Clearly, if the government were to offer life-annuities to a cohort of retirees, actuarial soundness would require that payments into the scheme equal expected annuity payments out of the scheme. Thus, after dividing the cohort into N risk classes indexed by i and substituting appropriately subscripted versions of equation (3), the equilibrium annuity price, 1/q, will be the solution to:

(4)

where is the known (to the members of the risk class) maximum possible duration of retirement of the members of risk class i, is the probability of a member of risk class i being alive at age t, and is the proportion of the cohort in risk class i.

Due to the fact that this is a highly non-linear problem with no closed-form solution, it must be solved numerically. For this purpose, we employed a modification of the Brent root-finding algorithm (Press, Flannery, Tuekolsky, and Vetterling, 1986).

3. MORTALITY ASSUMPTIONS

To simulate this model – to solve for the equilibrium annuity price – we must assign specific characteristics to the members of each of the N risk classes in the cohort. Thus for each risk class i, we must specify the values of , , , , and . In this section, we discuss the mortality assumptions, namely, the specifications of , , and . These will be common across all simulations.

First, we need to choose a specific probability distribution in order to relate the probability of retirees in a given risk class being alive at age t, , to their maximum possible duration of remaining life-span, . For computational simplicity, we use Champernowne’s (1969) probability distribution, such that for , which implies inter alia that the life expectancy of a member of risk class i as of retirement, t = 0, is years.

Second, we need to choose the proportions of the cohort in each risk class. For this purpose, we arbitrarily use Statistics Canada combined sexes mortality data from 1990-92 to give us the proportions of a cohort of 65 year-olds that lie in 42 risk classes with increasing life expectancies as of age 65 ranging from 1 through 42 years (at which point the Statistics Canada series is truncated) (Statistics Canada, 1995). These data are presented in Table 1.

By way of illustration, consider a retiree in risk class 1, where . This retiree would have a life expectancy of one year as of age 65. Thus, on average, he would receive one annuity payment, but would die before age 66 – and his second annuity payment. A retiree in risk class 9 on the other hand, where , would have a life expectancy of 9 years as of age 65. On average, he would receive 9 annuity payments, but would die before reaching age 74 – and his tenth annuity payment.

It should be stressed that this combination of mortality assumptions, while computationally convenient, generates an empirically implausible range of maximum possible life-spans. This is absolutely true, but only serves to strengthen our conclusions below regarding the robustness of the equilibrium annuity price to changes in the underlying assumptions of the model, for more "realistic" mortality assumptions would only serve to further reduce the variation in the equilibrium annuity price.

Notice also that the use of "combined sexes" data in our analysis implies that the government would not discriminate between the genders in the annuity market. Obviously, given the requirement for actuarial soundness, this would imply that women would gain and men would lose, since women, on average, have a longer life expectancy than do men. If discrimination between the genders were to be practiced, then we would have in effect two completely separate annuity markets, each of which could be analyzed independently along similar lines (see Power and Townley, 1993).

4. SIMULATION RESULTS

Having specified the common mortality assumptions, we begin our simulations with a base case, Case 1. Here we assume that wealth and life expectancy are not related across risk classes – all retirees possess retirement wealth of $100,000 – and, moreover, that risk preferences are identical across all retirees.5 We then simulate the model for a wide range of values of the index of relative risk aversion (RRA), , from 2 to 7 and for a wide range of annual social security payments, z, from $0 to an implausibly high $200,000. These results are given in Table 2. For any given level of z, we can see that the equilibrium annuity price increases as RRA decreases, i.e., as retirees become less averse to risk-taking. This makes sense: If all retirees were extremely risk-averse, , then each retiree would spend his entire wealth on an annuity – which would be equivalent to having a compulsory plan – and the result would be an equilibrium annuity actuarially fair with respect to the life expectancy of an average-lived person in the cohort.

Reading down any column, the impact of ceteris paribus changes in the level of social security payments can be seen. From equation (3), the demand for annuities is a decreasing function of z. The results here reveal the relative impact of changes in z on different risk classes. As z increases, the equilibrium annuity price increases, indicating that short-lived retirees are more sensitive to changes in the level of social security payments than their long-lived counterparts.

Davies (1981) suggests that RRAs ranging from 3 to 5 are realistic. For this range and beyond, equilibrium annuity prices would not seem to be very sensitive to RRA given any particular level of z. Although specification of RRA would appear to matter little here, any such general conclusion would be premature, because the present assumption that RRA is unrelated to life expectancy and/or retirement wealth would seem to be heroic.

For our second and third cases, Cases 2 and 3, we make the far more plausible assumption that wealth and life expectancy are positively related. This assumption makes intuitive sense. The relatively wealthy are able to afford better living conditions and higher quality health-care. It is also consistent with rational economic behavior because we would expect a retiree with a relatively long life expectancy to save more for retirement than a retiree with a relatively short life expectancy. Certainly, the opposite assumption that wealth and life expectancy are negatively related seems untenable, especially in view of our earlier assumption that the utility of a retiree is entirely a function of his own consumption, which implies inter alia that there is no bequest motive.

Specifically, we assume that the form of the positive relationship between wealth and life expectancy is such that the retirement wealth of a retiree in risk class i is given by:

(5)

so that, for example, the retirement wealth of a (non-existent) retiree with a maximum possible duration of retirement of 43 years is $100,000.

The distinction between Cases 2 and 3 lies in the assumptions concerning the relationship between life expectancy and RRA. It is not at all obvious a priori what would constitute the most plausible form of such a relationship, so we consider two extremes: Case 2 assumes that preferences exhibit increasing RRA and Case 3 assumes that preferences exhibit decreasing RRA. These assumptions are implemented in the following way: To a (non-existent) retiree with a maximum possible duration of retirement of 43 years – to whom we would assign a retirement wealth of $100,000 – we assign various reference levels of RRA equal to those used for Case 1 above, adding to or subtracting from this level for other members of the cohort depending on whether RRA is assumed to be increasing or decreasing in wealth. Then, for each reference level of RRA, we conduct a series of simulations, each using a different distribution of indices about this reference level.

To illustrate this process, suppose we take an example from Case 2, where wealth is increasing in life expectancy and preferences are characterized by increasing RRA, and we choose an initial reference RRA of 3. We also assume (initially) that RRA differs by increments of 0.01 between risk classes. The first three columns of Table 3 show the various combinations of retirement wealth and RRA that would describe retirees in each risk class of this simulation cohort. Using this same reference level of RRA, we then also consider alternative RRA increments of 0.05 and 0.10, shown in the fourth and fifth columns of Table 3. In Table 4 we then show the equilibrium annuity prices associated with each of these simulation cohorts for different possible levels of social security payments.

It should be noted that for all reference levels of RRA the choice of increment levels to use is limited to those consistent with all members of a simulation cohort being risk-averse. The precise range of increment levels used for each reference level of RRA is given in Table 5. This gives a total of 34 simulation cohorts for Case 2 and 34 simulation cohorts for Case 3. The only difference between the specifications of the simulation cohorts for Case 2 and Case 3 being that for Case 2 the RRA levels increment with increasing life expectancy of the risk class, while for Case 3 they decrement.

To save space in the reporting of the equilibrium annuity prices for Cases 2 and 3, we condense the results of these simulations in Tables 6 and 7, respectively. Beginning with Case 2, the left hand-side of Table 6 presents a summary of the results for the entire range of reference levels of RRA from 2 through 7 showing, for each level of social security payment, z, the minimum and maximum equilibrium annuity prices (for all 34 simulation cohorts), together with their range. As can readily be seen, the range of equilibrium annuity prices is relatively small even for relatively high levels of social security payments. The right hand-side of Table 6 shows similar calculations for the case where the range of reference levels of RRA is restricted to that suggested by Davies as being realistic. (In this case, the various calculations are based on 16 rather than 34 simulation cohorts). Notice that now the distribution of results narrows considerably with the range of equilibrium annuity prices for each level of social security payment diminishing by approximately one-half. In short, especially for the more empirically realistic 3 to 5 range of RRA reference levels, the equilibrium annuity price would not appear to be particularly sensitive to the reference level of RRA.

Turning to Table 7, we can see that a similar story holds for Case 3. The only significant difference in this case being that the range of equilibrium annuity prices at each level of social security payment tends to be slightly larger than for Case 2, but is still very narrow, especially for the more empirically realistic 3 to 5 range of RRA reference levels.

Overall, it would appear from the results for Cases 2 and 3 that a plan designer’s ignorance of whether retirees’ preferences truly exhibit increasing or decreasing RRA is not crucial to the determination of the equilibrium annuity price. Of course, both Cases 2 and 3 are based on the assumption that retirement wealth and life expectancy are positively related and, moreover, that the form of the assumed relationship involves a reasonably wide distribution of wealth across risk classes – from $4,651 for a retiree in the shortest life expectancy risk class to $195,348 for a retiree in the longest life expectancy risk class. Assuming that retirement wealth and life expectancy are indeed positively related, the question remains as to whether or not the actual form of the relationship matters. To answer this question, perhaps the best comparison to make is that between the relationship assumed for Cases 2 and 3 and the relationship assumed for Case 1, where in fact no relationship exists – retirees in all risk classes possess $100,000 of retirement wealth regardless of life expectancy. Table 8 shows this comparison for the empirically more realistic 3 to 5 range of RRA reference levels. For each level of social security payment, the average equilibrium annuity price was calculated for each of Cases 1, 2, and 3. For example, for Case 1, when z = $0, the entry 22.46 in Table 8 indicates the average of the equilibrium annuity prices 22.53, 22.49, 22.43, and 22.38 in Table 1, while for Case 2, when z = $0, 24.63 is the average of equilibrium annuity prices from all 16 simulation cohorts, and similarly for Case 3. The comparison in Table 8, therefore, is between a perfectly even distribution of retirement wealth, Case 1, and one where the wealthiest (longest-lived) retiree possesses 42 times more wealth than the poorest (shortest-lived) retiree, Cases 2 and 3. Looking at the results in Table 8, it seems clear that although the equilibrium annuity prices for Case 1 are consistently lower than those for Cases 2 and 3, the differences are not great, the range declining from a high of 2.17 years for z = $0 to a low of 0.70 years for z = $200,000.

Altogether, if one is prepared to accept the notion of a non-negative relationship between life expectancy and retirement wealth, the precise nature of this relationship seems to be relatively unimportant for the determination of the equilibrium annuity price. In practice, of course, it may well be possible for policy-makers to determine more accurately what the true relationship is, perhaps from estate and inheritance data. Furthermore, it should be noted that it is the distribution of expenditures on annuities per risk class that matters, not the distribution of wealth per se. After all, the government could always place a limit on annuity purchases, thus narrowing the distribution of expenditures on annuities.

5. CONCLUSION

This paper has used a simulated optimal control model to examine whether or not it would be possible for the government to design and implement an actuarially sound, voluntary, life-annuity plan despite the fact that much of the information required to construct such a plan lies outside the public domain. To this end, we have executed a variety of simulation exercises based on a fairly extreme range of assumptions as to the parameters of this private information and found that equilibrium annuity prices are not particularly sensitive. Thus, a government plan-builder’s ignorance of the precise nature of this private information would not be of great practical concern.

More specifically, our results indicate that the most important determinants of annuity demand are the actual distribution of life expectancies of a cohort of retirees and the amount of social security received and these data are in the public domain. Thus, at least within the context of our model, it would seem that the government could indeed design and implement an actuarially sound, voluntary, life-annuity plan with considerable accuracy, thus opening up the possibility of an unambiguously positive role for government intervention in this market.

APPENDIX: Derivation of the Key Equations in the Individual Retiree’s Maximization Problem

Beginning with the formal statement of the retiree’s maximization problem (1), the relevant Hamiltonian is given by:

(A-1)

where and are the co-state variable and the Lagrange multiplier associated with non-annuity wealth at age t, respectively. The first-order conditions for this maximization problem are:

(A-2)

; (A-3)

; ; ; (A-4)

and

. (A-5)

The transversality condition, , also arises from equation (A-2) because by assumption.

Manipulating equation (A-5), first take the partial derivative of with respect to :

(A-6)

and then integrate over the range to yielding:

(A-7)

From (A-3):

(A-8)

and, performing the integration of the last term:

. (A-9)

The first-order condition (A-5) states that at the optimum this last equation, (A-9), must be equal to 0 and hence, using first-order condition (A-3), together with the transversality condition, , it follows that at the optimum:

(A-10)

In order to solve the above equations for the retiree’s demand for annuities and planned consumption path, one starting point is to note that if at any age r, , then for any age s, s > r, That is, the retiree will spend his non-annuity wealth earlier rather than later during his retirement, and, once expended, annuity payments and social security payments will be his only sources of income. To see this, assume the converse, that is that . Given the fact that the utility function is assumed to be strictly concave, this implies that and, because , that From (A-2), this would imply that and, from (A-3), that However, if this is so, then equation (A-4) would require that But, with all non-annuity wealth spent by age r, consumption at any age older than r, such as s, would be limited to thus contradicting our initial assumption that cr < cs.

We define age M to be the earliest age at which all non-annuity wealth is exhausted. With this in mind, it is useful to rearrange equation (A-10), substituting from equation (A-2), as:

(A-11)

Notice that during the first interval, the period from t = 0 through t = M, the retiree’s stock of non-annuity wealth is positive. Thus by (A-4), by (A-3), is constant; and, by (A-2), is constant. Continuity of the path requires that which implies that the first term on the right hand-side of (A-11) is equal to During the second interval, the period from t = M through t = T, planned consumption is constant and equal to which implies that the second term on the right hand-side of (A-11) may also be simplified. Therefore, (A-11) may be rewritten as:

(A-12)

Solving this equation for q then yields equation (2) in the text:

(2)

Given the way that we have defined age M, it follows that:

(A-13)

Consider now equation (A-2), which, when differentiated with respect to time, yields:

(A-14)

Because when equation (A-14) may be rearranged as:

; (A-15)

where is the inverse of the retiree’s elasticity of marginal utility of income or index of relative risk aversion.

Solving (A-15), for Thus we can calculate:

(A-16)

Finally, equating (A-13) with (A-16) yields the demand for life-annuities function:

(3)

which is equation (3) in the text.

FOOTNOTES

The authors wish to acknowledge the support of the Social Sciences and Humanities Research Council of Canada. They also wish to thank Lars Osberg for helpful comments on an earlier draft.

[1] This model builds on earlier work by one of the authors (Townley, 1990).

[2] Obviously, in the real world the level of social security payments does vary across retirees, but limits on contributions tend to make these payments more uniform than they would be otherwise.

[3] These assumptions do not affect the results obtained if retirees and the government earn the same rate of interest on savings.

[4] Implicit in this formulation is the assumption that utility is additively separable over time and age-independent, and that the probability of being alive at any age is independent of consumption at that or any other age.

[5] Simulation results are quite insensitive to the choice of a reference level of retirement wealth, especially at low levels of social security payments, given the income elasticity of demand for annuities.

REFERENCES

Champernowne, D.G. (1969) Uncertainty and Estimation in Economics: Vol. 3. London: Oliver and Boyd.

Davies, J.B. (1981) Uncertain Lifetime, Consumption, and Dissaving in Retirement. Journal of Political Economy 89:561-577.

Eckstein, Z., Eichenbaum, M., and Peled, D. (1985) Uncertain Lifetimes and the Welfare Enhancing Properties of Annuity Markets and Social Security. Journal of Public Economics 26:303-326.

Friedman, B.M. and Warshawsky, M.J. (1990) The Cost of Annuities: Implications for Saving Behavior and Bequests. Quarterly Journal of Economics 105:135-154.

Power, S. and Townley, P.G.C. (1993) The Impact of Government Social Security Payments on the Annuity Market. Insurance: Mathematics and Economics 12:47-56.

Press, W.H., Flannery, B.P., Tuekolsky, S.A., and Vetterling, W.T. (1986) Numerical Recipes: The Art of Scientific Computing. New York: Cambridge University Press.

Statistics Canada (1995) Life Tables: Canada and the Provinces 1990-1992. Catalogue #84-537 Occasional. Ottawa: Statistics Canada.

Townley, P.G.C. (1990) Life-Insured Annuities: Market Failure and Policy Dilemma. Canadian Journal of Economics 23:546-562.

Townley, P.G.C. and Boadway, R.W. (1988) Social Security and the Failure of Annuity Markets. Journal of Public Economics 35:75-96.


Table 1: Proportions of the Cohort in Each Risk Class

Risk Class (i)

Ti

Proportion (vi)

1

2

0.015254770

2

4

0.016411333

3

6

0.017668725

4

8

0.018973565

5

10

0.020272474

6

12

0.021648488

7

14

0.023125330

8

16

0.024732656

9

18

0.026417086

10

20

0.028059999

11

22

0.029714774

12

24

0.031411066

13

26

0.033142945

14

28

0.034809582

15

30

0.036292355

16

32

0.037609058

17

34

0.038759690

18

36

0.039726458

19

38

0.040384809

20

40

0.040663571

21

42

0.040550880

22

44

0.040058599

23

46

0.039180797

24

48

0.037875957

25

50

0.036120354

26

52

0.033955505

27

54

0.031482239

28

56

0.028730212

29

58

0.025764666

30

60

0.022662705

31

62

0.019519226

32

64

0.016446920

33

66

0.013540684

34

68

0.010847968

35

70

0.008445876

36

72

0.006364062

37

74

0.004638114

38

76

0.003262101

39

78

0.002188573

40

80

0.001417531

41

82

0.000854077

42

84

0.001014217


Table 2: Equilibrium Annuity Prices: Case 1

z($)

RRA

 

2

2.5

3

3.33

4

5

6

7

0

22.73

22.61

22.53

22.49

22.43

22.38

22.34

22.31

5000

23.71

23.40

23.22

23.08

22.86

22.69

22.58

22.51

10000

24.49

24.07

23.83

23.64

23.38

23.18

22.93

22.77

15000

25.11

24.65

24.36

24.12

23.85

23.50

23.28

23.16

20000

25.65

25.14

24.78

24.56

24.24

23.87

23.59

23.37

25000

26.13

25.59

25.16

24.96

24.57

24.17

23.88

23.66

30000

26.59

26.01

25.54

25.30

24.93

24.46

24.13

23.89

35000

26.93

26.32

25.88

25.61

25.19

24.73

24.40

24.10

40000

27.27

26.66

26.16

25.94

25.49

24.98

24.60

24.35

45000

27.64

26.95

26.48

26.17

25.71

25.20

24.87

24.51

50000

27.88

27.23

26.70

26.45

26.00

25.46

25.02

24.71

55000

28.20

27.47

26.96

26.67

26.17

25.62

25.21

24.92

60000

28.42

27.75

27.20

26.88

26.40

25.84

25.44

25.05

65000

28.65

27.93

27.39

27.14

26.61

26.04

25.57

25.22

70000

28.93

28.21

27.66

27.29

26.77

26.18

25.73

25.43

75000

29.09

28.39

27.81

27.49

26.98

26.36

25.93

25.54

80000

29.31

28.55

27.97

27.72

27.17

26.56

26.06

25.66

85000

29.56

28.79

28.21

27.84

27.29

26.68

26.18

25.82

90000

29.68

28.96

28.36

28.00

27.45

26.81

26.33

25.99

95000

29.85

29.09

28.49

28.22

27.68

26.99

26.53

26.08

100000

30.15

29.25

28.65

28.36

27.78

27.15

26.62

26.19

110000

30.34

29.60

28.98

28.60

28.03

27.36

26.84

26.47

120000

30.79

29.85

29.22

28.94

28.34

27.69

27.13

26.66

130000

30.94

30.20

29.56

29.13

28.53

27.85

27.30

26.86

140000

31.20

30.37

29.73

29.46

28.84

28.07

27.53

27.12

150000

31.52

30.69

30.02

29.63

29.01

28.32

27.75

27.26

160000

31.67

30.89

30.23

29.81

29.19

28.47

27.89

27.44

170000

31.93

31.04

30.39

30.15

29.50

28.66

28.09

27.69

180000

32.20

31.34

30.64

30.25

29.61

28.91

28.31

27.80

190000

32.31

31.53

30.86

30.40

29.75

29.02

28.43

27.93

200000

32.49

31.65

30.97

30.63

29.96

29.17

28.57

28.11


Table 3: Sample Cohort Specification: Case 2: Reference RRA = 3

i

($)

RRA Increment

 
 
   

0.01

0.05

0.10

1

4651

2.795

1.975

0.950

2

9302

2.805

2.025

1.050

3

13953

2.815

2.075

1.150

4

18605

2.825

2.125

1.250

5

23256

2.835

2.175

1.350

6

27907

2.845

2.225

1.450

7

32558

2.855

2.275

1.550

8

37209

2.865

2.325

1.650

9

41860

2.875

2.375

1.750

10

46512

2.885

2.425

1.850

11

51163

2.895

2.475

1.950

12

55814

2.905

2.525

2.050

13

60465

2.915

2.575

2.150

14

65116

2.925

2.625

2.250

15

69767

2.935

2.675

2.350

16

74419

2.945

2.725

2.450

17

79070

2.955

2.775

2.550

18

83721

2.965

2.825

2.650

19

88372

2.975

2.875

2.750

20

93023

2.985

2.925

2.850

21

97674

2.995

2.975

2.950

22

102326

3.005

3.025

3.050

23

106977

3.015

3.075

3.150

24

111628

3.025

3.125

3.250

25

116279

3.035

3.175

3.350

26

120930

3.045

3.225

3.450

27

125581

3.055

3.275

3.550

28

130233

3.065

3.325

3.650

29

134884

3.075

3.375

3.750

30

139535

3.085

3.425

3.850

31

144186

3.095

3.475

3.950

32

148837

3.105

3.525

4.050

33

153488

3.115

3.575

4.150

34

158140

3.125

3.625

4.250

35

162791

3.135

3.675

4.350

36

167442

3.145

3.725

4.450

37

172093

3.155

3.775

4.550

38

176744

3.165

3.825

4.650

39

181395

3.175

3.875

4.750

40

186047

3.185

3.925

4.850

41

190698

3.195

3.975

4.950

42

195349

3.205

4.025

5.050


Table 4: Equilibrium Annuity Prices: Case 2: Reference RRA = 3

z($)

RRA Increment

0.01

0.05

0.10

 

0

24.66

24.70

24.75

5000

25.42

25.52

25.61

10000

26.00

26.05

26.14

15000

26.42

26.47

26.54

20000

26.75

26.84

26.89

25000

27.04

27.09

27.17

30000

27.35

27.38

27.41

35000

27.55

27.59

27.63

40000

27.82

27.83

27.84

45000

27.98

27.99

28.01

50000

28.17

28.19

28.21

55000

28.37

28.37

28.36

60000

28.50

28.49

28.48

65000

28.66

28.65

28.63

70000

28.86

28.84

28.82

75000

28.96

28.93

28.90

80000

29.07

29.04

29.00

85000

29.21

29.16

29.11

90000

29.39

29.32

29.25

95000

29.47

29.43

29.38

100000

29.56

29.51

29.45

110000

29.78

29.69

29.61

120000

30.02

29.96

29.83

130000

30.15

30.07

30.00

140000

30.33

30.22

30.11

150000

30.57

30.42

30.26

160000

30.67

30.59

30.47

170000

30.80

30.68

30.58

180000

30.96

30.80

30.67

190000

31.17

30.96

30.78

200000

31.25

31.15

30.91


Table 5: Distribution of Risk Preferences Used for Cases 2 and 3

Reference RRA

Increments Used

2.00

0.01,

0.05

         

2.50

0.01,

0.05,

0.10

       

3.00

0.01,

0.05,

0.10

       

3.33

0.01,

0.05,

0.10,

0.15

     

4.00

0.01,

0.05,

0.10,

0.15

     

5.00

0.01,

0.05,

0.10,

0.15,

0.20

   

6.00

0.01,

0.05,

0.10,

0.15,

0.20,

0.25

 

7.00

0.01,

0.05,

0.10,

0.15,

0.20,

0.25,

0.30


Table 6: Equilibrium Annuity Prices: Case 2

z($)

RRA 2 - 7

RRA 3 - 5

 

min

max

range

min

max

range

0

24.47

24.92

0.46

24.52

24.75

0.23

5000

24.78

26.00

1.23

25.02

25.61

0.59

10000

25.14

26.61

1.47

25.43

26.14

0.71

15000

25.43

27.08

1.65

25.75

26.54

0.79

20000

25.66

27.46

1.80

26.05

26.89

0.84

25000

25.90

27.82

1.92

26.31

27.17

0.86

30000

26.08

28.07

1.99

26.52

27.41

0.89

35000

26.26

28.36

2.11

26.73

27.63

0.91

40000

26.44

28.56

2.13

26.92

27.84

0.92

45000

26.57

28.85

2.28

27.08

28.01

0.93

50000

26.72

29.01

2.29

27.26

28.21

0.95

55000

26.88

29.20

2.32

27.41

28.37

0.97

60000

26.98

29.43

2.45

27.53

28.50

0.97

65000

27.09

29.55

2.46

27.68

28.66

0.99

70000

27.22

29.70

2.48

27.84

28.86

1.03

75000

27.35

29.91

2.56

27.93

28.96

1.03

80000

27.43

30.04

2.61

28.03

29.07

1.05

85000

27.52

30.14

2.62

28.15

29.21

1.06

90000

27.62

30.27

2.65

28.29

29.39

1.10

95000

27.73

30.44

2.71

28.36

29.47

1.11

100000

27.84

30.59

2.75

28.43

29.56

1.13

110000

27.97

30.75

2.78

28.60

29.78

1.17

120000

28.14

31.00

2.86

28.81

30.02

1.21

130000

28.32

31.22

2.90

28.91

30.15

1.24

140000

28.42

31.35

2.94

29.03

30.33

1.30

150000

28.53

31.55

3.02

29.18

30.57

1.40

160000

28.66

31.80

3.14

29.35

30.67

1.32

170000

28.81

31.89

3.09

29.43

30.80

1.36

180000

28.88

32.01

3.13

29.52

30.96

1.44

190000

28.96

32.17

3.22

29.62

31.17

1.56

200000

29.04

32.43

3.38

29.74

31.25

1.51


Table 7: Equilibrium Annuity Prices: Case 3

RRA 2 - 7

RRA 3 - 5

z($)

min

max

range

min

max

range

0

24.43

24.81

0.37

24.48

24.65

0.17

5000

24.66

25.80

1.15

24.82

25.38

0.56

10000

24.95

26.48

1.53

25.22

25.95

0.73

15000

25.21

26.98

1.77

25.57

26.38

0.81

20000

25.45

27.40

1.96

25.82

26.72

0.89

25000

25.66

27.75

2.09

26.09

27.02

0.93

30000

25.85

28.04

2.19

26.32

27.33

1.01

35000

26.05

28.37

2.32

26.54

27.54

1.00

40000

26.20

28.60

2.40

26.73

27.80

1.07

45000

26.37

28.91

2.54

26.95

27.97

1.03

50000

26.53

29.09

2.57

27.09

28.17

1.07

55000

26.65

29.34

2.68

27.27

28.39

1.13

60000

26.80

29.55

2.75

27.45

28.54

1.09

65000

26.95

29.71

2.76

27.57

28.72

1.15

70000

27.05

29.95

2.90

27.72

28.94

1.23

75000

27.17

30.11

2.94

27.90

29.06

1.16

80000

27.30

30.25

2.95

28.00

29.20

1.20

85000

27.43

30.43

3.00

28.11

29.40

1.28

90000

27.52

30.64

3.12

28.25

29.54

1.29

95000

27.61

30.73

3.12

28.39

29.64

1.26

100000

27.72

30.86

3.14

28.46

29.78

1.31

110000

27.95

31.22

3.28

28.64

30.09

1.44

120000

28.10

31.39

3.29

28.88

30.27

1.39

130000

28.30

31.67

3.37

29.00

30.58

1.58

140000

28.44

31.91

3.46

29.15

30.74

1.60

150000

28.56

32.06

3.50

29.34

30.92

1.58

160000

28.71

32.34

3.63

29.48

31.23

1.75

170000

28.88

32.53

3.65

29.58

31.34

1.76

180000

28.96

32.65

3.69

29.71

31.49

1.78

190000

29.06

32.83

3.77

29.88

31.73

1.85

200000

29.17

33.14

3.96

30.02

31.90

1.89


Table 8: Average Equilibrium Annuity Prices: RRA 3 – 5

z($)

Case 1

Case 2

Case 3

0

22.46

24.63

24.55

5000

22.96

25.31

25.10

10000

23.51

25.80

25.57

15000

23.96

26.17

25.95

20000

24.36

26.48

26.28

25000

24.72

26.75

26.56

30000

25.06

26.99

26.82

35000

25.35

27.20

27.05

40000

25.64

27.40

27.27

45000

25.89

27.57

27.47

50000

26.15

27.74

27.65

55000

26.36

27.90

27.84

60000

26.58

28.03

28.00

65000

26.79

28.17

28.14

70000

26.98

28.32

28.31

75000

27.16

28.43

28.46

80000

27.36

28.54

28.58

85000

27.51

28.65

28.70

90000

27.66

28.78

28.84

95000

27.84

28.88

28.98

100000

27.98

28.97

29.09

110000

28.24

29.14

29.29

120000

28.55

29.35

29.54

130000

28.76

29.50

29.71

140000

29.02

29.64

29.90

150000

29.24

29.80

30.10

160000

29.42

29.96

30.26

170000

29.67

30.08

30.41

180000

29.86

30.19

30.58

190000

30.01

30.32

30.74

200000

30.18

30.45

30.88



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