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A Fuzzy Approach to Draw Down Plans

A Fuzzy Approach to Draw Down Plans. İrini Dimitriyadis Bahçeşehir University C. Kahraman, A. Beşkese, T. Bozbura. Pension System. Social Security Occupational Pensions Individual Pension Plans The so called three pillar system Our interest is in Individual Pension Plans.

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A Fuzzy Approach to Draw Down Plans

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  1. A Fuzzy Approach to Draw Down Plans İrini Dimitriyadis Bahçeşehir University C. Kahraman, A. Beşkese, T. Bozbura

  2. Pension System • Social Security • Occupational Pensions • Individual Pension Plans The so called three pillar system Our interest is in Individual Pension Plans

  3. Two Stages in IPP • Accumulation Stage How much you will accumulate depends on Years of contribution Yearly contributed dollars Investment plan chosen

  4. Accumulation Stage Internal rate of return of your investments is sensitive to • Tax advantages provided • Fees charged (fund management fees, management fees etc)

  5. Decumulation Stage How should I manage my accumulated funds? 1- Take it as lump sum? 2- Buy an annuity at retirement or a deferred annuity? 3- Choose some other draw- down plan?

  6. Decisions Decision at point of retirement depends • On the expectation of life of the retiree • On his bequest motives • Utility he places on consumption

  7. Main Problem: Longevity The main problem in the decisions of the insureds is their perception of “future lifetime” With increasing longevity their probability of depleting their capital before death increases.

  8. Longevity Another problem that has not yet been treated sufficiently is that people of today ( the group of 30 or 35 of today) might have to face still another problem. They might have to fund Their own retirement + Subsidize their families

  9. Annuities Annuities as protection against longevity Advantage: Mortality credits B(t) B(t) = Disadvantage:No bequest High charges Not liquid (needs in case of unexpected expenses)

  10. Comperative performance ofdrawdown plans(literature review) • Constant draw-down Here you draw out a constant percentage of the remaining fund at each period. • Expected time to death [1/T] Here T may be defined as the last age of a mortality table or a “last age” as perceived by the retiree.

  11. Drawdown plans • The Expected Future Lifetime [1/E(T)] This implies the expected future lifetime “at the attained age”. • Mixed Plans Some kind of drawdown plan + annuity at some age. The question then is when to annuitize?

  12. Which is best when? • Studies in this field have taken as benchmark an annuity bought at 65. • They define “shortfall probability” as the probability of the drawdown plan giving a payment below that of the annuity. • The comparison involves the early ages of retirement as well where the annuity always provides larger yearly payment.

  13. Which is best when? The results come out as follows: (Reference: I.Dus, R. Maurer, O.S. Mitchell, “Betting on Death and Capital Markets in Retirement: a shortfall Analysis of Life Annuities versus Phased Withdrawal Plans” ) Constant drawdown : The consecutive payments decrease may run out of money Expected time to death [1/T] Smaller payouts until age 80, and then an abrupt rise. I

  14. Which is best when? • Expected future lifetime[1/E(T)] Startsat a level of 85% of the annuity payment and increases to 100% at age 70. Reaches a maximum of 150% at age 83, and reaches the benchmark annuity value at 91. Then starts to decrease risky if you live too long !!

  15. What if we would like to optimize choices over a utility function? When we optimize we have to assume a stepwise decision process whereby the retiree at each step gives a decision about his consumption level, his investment strategy and the weight he places to his bequest motives.

  16. The Turkish system and statement of our problem • The system has started to accept contributions in 2004. • Participators are eligible to take draw out their funds at the end of minimum 10 years staying in the system+attaining age 56. Early age retirement

  17. Possible behavioral aspects Early retirees will probably continue working so they will not need a high complementary income at least up to 65 years of age. So we assume thatbuying an annuity at 56 will not be preferred.

  18. Behavioral aspects Turkish society: The bequestmotive might behigherthan most other western countries + Obligationstowardsfamilyhigher + No employer sponsored pension plans

  19. Assumptions We assume that contributors are 35 year of age and they contribute to the system for 20 years until they are eligible for a pension at 56. We have chosen age 35 since it is the most representative age of our portfolios.

  20. Accumulations We may classify the contribution ranges as low, medium and high and the maximum and minimum possible accumulations are as follows Low contributors (13,873 -109,686 NTL)* Medium contributors (41,619 - 219,371 NTL) High level contributors(83,238-548,428 NTL) *Reference: Bilge Önder, An Analysis of the Turkish Individual Pension System, ongoing MsC thesis, Bahçeşehir University

  21. Statement of the problem Our objective is to determine when is it best to annuitize given that the person desires and states a certain minimal amount he wants to get as a pension for his later ages.

  22. Minimum annuity desired We define the minimum guarantee that the retiree would want to have as a pension in his later years as: • The minimum wage in real terms of a worker [3600 NTL/year] • Twice the minimumwage[7200 NTL/Year] • High value income of a social pensioner [12000 NTL/Year]

  23. Decumulation Stage • We assume that the retiree starts to draw out money in line with the 1/E(T) rule. • The starting fund may or may not be sufficient for buying an annuity of the defined amount at retirement age (This is for example the case with minimum contributions).

  24. Decumulation Stage We follow the evolution of the fund as the person continues drawing out 1/E(T) and mark thefirst point(age)where the accumulated fund after retirement equals the present value of an annuity of the desired amountat the attained age if that was not possible at the beginning. Basic assumption is thatinterest rate of fund > guaranteed interestof the annuity.

  25. Decumulation Stage • We also mark the last point in time(maximum age) for which this is possible. • At each point our outputs provide information for the amount of bequest. • We also trace down the amount of the actual value of the annuity that could have been bought at each age.

  26. Steps of the problem • Is the fund sufficient to buy an annuity for life of the specified amount at the attained age? Is Vt >at t>56 • For what range of time (for which ages)is this satisfied assuming person continues the 1/E(T) rule in the meantime? • What is the value of bequests in the range where the above is satisfied?

  27. Results from scenario analysis The low earner group : If they wish to have an annuity of 3600NTL yearly,and with an interets rate of 4.35% they can only annuitize at 70 (not enough funds at 56), assuming they draw down funds complying to the 1/E(T) rule. The bequest advantage is minimal even if they invests at the highest rate. (Annuity interest: 4%)

  28. Low contributor group Assuming that they invest their accumulation in a fund providing 5% per annum in real terms then the time range they canannuitizeis between65-82 years of age and thebequest takes amaximumvalue at 74. This implies that whenever they annuitize in this period they supply themselves with an income of 3600 yearly for life.

  29. A sample outcome Assume interest is 5% on funds; Minimum contribution band

  30. Medium Contributors Medium Contributors have sufficient funds for an annuity of 3600 from the very beginning. So if they stick to this value bequest is very high. The same is true for an annuity of 7200. 1/E(T) provides them with a lower value than the annuity at each age until over 86.

  31. NTL Years after retirement

  32. Double controls Since bequests are so high in this case one may ask for the possibility of getting an annuity that might increase in steps. That is an optimisation problem to be solved under a specific utility function.

  33. Why fuzzify 1- It is important to see what the newly established system can provide for the future of the contributors. 2- Rather than talking about a certain value or rather than considering all possible values of participations we prefer to model the accumulated funds at the end of the accumulation period as fuzzy values

  34. Fuzzification We thus get two different outlooks Outlook I: We consider the system at the point where the contributions start.This outlook assumesfuzzy accumulation of funds + fuzzy mortality after retirement +fuzzy investment income after retirement.

  35. Fuzzification Outlook II: We assume a known(crisp) accumulation at the start of retirement and look forward to retirement with fuzzy mortality and fuzzy interest rates. This is exactly what the person at 56 will be expecting to face.

  36. Fuzzification • We consider the1/[E(T)]rule. • We take the expected lifetime from a mortality table and fuzzify it. We assume a triangular fuzzy number The minimum expected lifetime is taken from CSO 1980, the maximumvalue expected future lifetime from CSO 2000, and the middle value the average of the two.

  37. Fuzzification of mortality A more sophisticated system of fuzzification can be assumed in the future where a basic mortality law, for example Makeham’s Law may be fitted to a CSO 1980 table and the parameters may then be fuzzified.

  38. 1.0 i 0.04 0.032 0.05 0.04 0.06 0.048 Fuzzification of interest rate We also fuzzify the interest rate around two values; 5% and 4% Around 5% Around 4% The inverse functions Vi1 and Vi2 are: For the interest around %5 Vi1= 0,04 +0,01y Vi2= 0,06 – 0,01y For the interest around 4% Vi1 =0,032 + 0,008y Vi2= 0,048 – 0,008y

  39. What information? • We look at the funds accumulated under the drawdown plan chosen at five year intervals. We also note the amount that can be drawn. • This can give a broader idea about what groups of people can do under varying market conditions and under differing lifelengths.

  40. Low Contributor Group • Consider the low contributor group under two possible investment decisions (4%, 5%). Note the value of the accumulated fund may reach under the 1/E(T) rule. You look at the values at the end of 60, 65, 70,75,80, 85 years of age.

  41. How to do it? You form the following cashflow: Vt: Fund at beginning of year t (crisp or fuzzy) • [1/E(T)]Vt:Amount withdrawn by the retiree. • Vt+1=(Vt – [1/E(T)]Vt) (1 + rt) the fund at the end of the first year or the beginning of the second year. • The second year drawdown is Vt+1[1/(E(T+1)] • Starting t; t=56, the retirement age.

  42. Scenarios We evaluate the results under the following combinations: 1)Fuzzy initial fund,fuzzy mortality,fuzzy interest 2)Fuzzy initial fund, fuzzy mortality, crisp interest 3)Crisp initial fund, fuzzy mortality,fuzzy interest

  43. 1.0 Age 80-5% Age 80 4% V(t)*w 0 1680 5075 20462 62122 First Outlook(Accumulation fuzzy)/Minandmax attainable ______: Fund that may be drawn at 80 with low contribution and 4% investment income ______: Fund that may be drawn at 80 with high contribution and 5% investment income Fuzzy draw-outs at 80 for high and low contributions

  44. 1.0 Age 80 4% Age 80-5% V(t)*w 1680 1696 20,462 0 20,704 Fuzzy Capital versus crisp capitalLow contribution Crisp Capital;45,702 Interest:4% Draw-downs: (0, 2405, 16449) Fuzzy draw-outs at age 80 for 4% and 5% fuzzy interest

  45. Comparisons for draw-downs Everything crisp; Draw-downs at 80 for indicated interest rates. (2,856; 4.3%) ; (2,990; 4.5%) ; (3,375; 5%) Crispcapital; mortality and interest fuzzy(4%) Draw-downs at 80:(0 2,405 16,449) Crisp interest: capital and mortality fuzzy Draw-downs at 80 : ( 0 1,680 16,817) Everything fuzzy : Draw-downs at 80 : (0 1,680 20,462)

  46. 1.0 Age90-4% Age 90- 5% V(t)*w 0 2122 2163 172035 176149 Draw downs at age 90/Fuzzy capital _______: Draw down at age 90 with high contribution and 3% investment income Note that effect of interest gets smaller _______: Draw down at age 90 with low contribution and 4% investment income _______: Draw down at age 90 with low contribution and 5% investment income Fuzzy draw-outs at specified ages

  47. 1.0 1.0 Age 80 8 Age 60 2 Age 55 1 0 0 7362 5277 16598 18145 31924 31844 45702 46876 54687 54283 V(t) V(t) Fund atDecumulation Low contribution; interest crisp 4% Most possible values: Age 60: 31,002 Age 80: 13,289 Low contribution; interest rate 4%; both fuzzy

  48. Money’s worth of annuity under fuzzy investment return Definition: Money’s worth is defined as: where: EPDV = F0 is the accumulated fund A is the annuity payment

  49. 1.0 i 0.86 0.96 1.09 Fuzzy money’s worth Money’s worth of an annuity bought at 56, with fuzzy interest rate around4%

  50. Bibliography • Bayraktar, E., Young, V.R., “Optimal Deferred Life Annuities to Minimize theProbability of Lifetime Ruin, 2007 2. Brown, J. R., “Private Pensions, mortality risk and the decision to annuitize, Journal of Public Economics, Volume 82, Issue 1,Ocober 2001. 3. Dus, I., Maurer, R., Mitchell, O., “Betting on Death and Capital Markets in Retirement: A Sortfall Risk Analysis of Life Annuities versus Phased Withdrawl Plans, Working Paper,WP2003-063 University of Michigan, Retirement Research Center. 4. Hornef, W.J, Maurer,R., Stamos, M, “Optimal Gradual Annuitization: Quantifying the Costs of Switcihing Annuities,

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