PII: 1057-0810(94)90016-7 FINANCIAL SERVICES REVIEW, 3(2): 93-108 Copyright 0 1994 by JAI Press Inc. ISSN: 10.57-0810 All rights of reproduction in any form reserved. An Optimization Model for Scheduling Withdrawals from Tax-Deferred Retirement Accounts Cliff T. Ragsdale Andrew F. Seila Philip L. Little As a growing number of Americans reach refiremen? age, more and more people are facing important decisions about how to withdraw savingsfrom tax-deferred retirement accounts (TDRAs). These decisions are complicated by the Federal Tax Code which imposes a number of rules and regulations on these withdrawals. Since these decisions collectively involve billions of dollars, the potential lossfrom even slightly suboptimal decision making is very large. In this paper, we present a mathematical programming model that can be used to assist retirees ami/or their advisors in determining the optimal schedule of withdrawals from TDRAs. I. INTRODUCTION As the general population of the United States ages, more and more Americans are having to make important decisions about how to withdraw money from tax-deferred retirement accounts (TDRAs). TDRAs include individual retirement accounts (IRAs), qualified corporate retirement plans, and tax-deferred salary reduction plans covered by Section 403(b) of the Internal Revenue Service (IRS) code. Generally speaking, TDRAs are an attractive investment alternative for two primary reasons: 1) they offer a way for some taxpayers to reduce their current tax liability; and 2) interest earnings on these investments are sheltered from taxes until they are withdrawn. Typically the argument is made that people should invest in TDRAs so their investment will grow faster (i.e., money is being com- pounded rather than taxed) and likely will be subjected to a lower tax rate at the taxpayer’s retirement. Cliff T. Ragsdale l Department of Management Science, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0235; Andrew F. Seila l Department of Insurance, Real Estate, Legal Studies and Management Science, Terry College of Business Administration, University of Georgia, Athens, GA 30602-6255; Philip L. Little l Department of Accounting, Western Carolina University, Cullowhee, NC 28723. 94 FINANCIAL SERVICES REVIEW, 3(2) 1994 American Demographics (1989) reports that investors have recently put as much as $100 billion annually into IRAs alone. Thus, the amounts and timing of the withdrawals from TDRAs can have a significant impact on the amount of taxes one pays and the accumulation of personal wealth However, this decision is complicated by the Federal Tax Code which imposes a myriad of rules and regulations on these withdrawals. In addition to the normal income tax that applies to withdrawals from TDRAs, there are also penalty taxes associated with withdrawing too much or too little and making withdrawals too soon or too Zate. Coupled with these rules are a number of more subtle investment issues which further complicate the process of determining the optimum amount to withdraw each year. In this paper we first briefly review the rules regarding distributions from TDRAs in order to demonstrate and motivate the need for an optimization model that can assist investors (or their advisors) in determining how to withdraw money from these accounts. Next, a mathematical programming model for this problem is proposed and described in detail. Finally, an example is provided to demonstrate the potential merit of the proposed model versus a number of heuristics that have been suggested in the literature. II. THE PROBLEM A complete description of the regulations regarding TDRA distributions is available in IRS publication 590 (1992) or less technical summary articles (e.g., Abramson, 1989; Katz, 1990; McCommbe, 1989). For the reader’s convenience and reference, we have also summarized these rules in Table 1. It goes without saying that there are some minor exceptions to these rules. Premature Distributions Tax Congress has established various rules concerning when withdrawals can be made from TDRAs. Generally speaking, the tax law is written with the intention that withdrawals from TDRAs begin no earlier than age 59.5. With few exceptions, any money withdrawn from TDRAs before age 59.5 is subject to a 10 percent Premature Distribution Tax penalty in addition to regular income taxes. TABLE 1. Summary of TDRA Tax Rules (na = not applicable) Age When Withdrawal is Made Rule Premature Distribution Tax’ Minimum Distribution Tax2 Excess Distribution Tax3 Before 59.5 From 59.5 to 70.5 10% na 5z 1% After 70.5 GO 15% Notes: ‘Applies to the total amount withdrawn in any year. 2Applies to the amount by which the minimum legally required withdrawal in any year exceeds the actual amount withdrawn. 3Applies to amounts in excess of $150,000 withdrawn in any year. This tax also has an impact on the determination of estate taxes at the taxpayer’s death. Withdrawal from Tax-Deferred Retirement Accounts 95 Minimum Distribution Tax As the name implies, TDRAs were originally intended to provide income to retirees- not estates for their heirs. Thus, additional rules require that minimum withdrawals be made from TDRAs on an annual basis beginning no later than the year in which the taxpayer reaches age 70.5. The actual withdrawal for this first “required” year (the year in which age 70.5 is reached) may be deferred as late as April 1 of the following year if desired. This deferral provision can be advantageous since it also defers the taxes on the first “required” year’s withdrawal for a year. However, since the taxpayer must also make a withdrawal for the second “required” year (the year in which age 7 1.5 is reached), effectively making two withdrawals in this second year could place some income into a higher tax bracket which might more than offset the benefit of deferring the taxes on the first “required” year’s withdrawal (e.g., Katz, 1990, p. 56). At any rate, annual withdrawals must also be made in each of the following years (where ages 72.5,73.5, . . . are reached). The actual minimum amount that must be withdrawn in each year beginning at age 70.5 is determined as follows. For each separate TDRA the taxpayer owns, a theoretical minimum figure is calculated by dividing the balance in the account at the beginning of the year by the joint life expectancy of the taxpayer and the designated beneficiary for the account (e.g., Johnson, 1990). A schedule of life expectancy factors is supplied in IRS publication 590 (1992) to assist in determining this minimum amount. The minimum figure for each account is theoretical in that the IRS does not actually require the money to be withdrawn from this particular account, provided the sum of the actual withdrawals is at least as much as the sum of the theoretical figures (e.g., Geller, 1988; Solbee, 1988). For instance, Table 2 shows the calculations for a taxpayer with two TDRAs where the minimum required withdrawal is $72,172. This taxpayer may satisfy the IRS requirements by with- drawing uf least $72,172 from either account or in any combination between accounts. Of course, the manner in which the taxpayer elects to split the withdrawal between these accounts will determine the amounts left in the accounts and, therefore, also affects the minimum required withdrawals in subsequent years. For retirees who are age 70.5 or older and fail to make the minimum required withdrawal, a Minimum Distribution Tax penalty of 50 percent applies to the difference between the actual amount withdrawn and the minimum required withdrawal. For instance, TABLE 2. Minimum Distribution Tax Example Account I Account 2 Age Balance 70 $7OO,ooo 71 ? 72 ? 73 ? 74 ? Joint Life Expectancy’ 26.2 25.3 24.4 23.5 22.7 Balance $900,000 ? ? ? ? Join? Life ExpectancyZ 19.8 19.0 18.2 17.3 16.5 Minimum Required Withdrawa (at age 70) = w+y=$72,172 Notes: lAssuming 40 year old beneficiary *Assuming 72 year old beneficiary 96 FINANCIAL SERVICES REVIEW, 3(2) 1994 if the minimum required withdrawal is $100,000 and the taxpayer only withdraws $50,000, a nondeductible penalty tax of $25,000 (i.e., 0.50(100,000 - 50,000)) must be paid (e.g., McCommbe, 1989). Excess Distributions Tax In keeping with the philosophy that TDRAs should be used to provide retirement income, the Excess Distributions Tax is intended to discourage people from using TDRAs to amass or receive excessive retirement benefits. This tax imposes a penalty on the amount by which the total withdrawal from all TDRAs exceed $150,000 in any year. For any such “excess distribution” made prior to age 59.5 the tax is effectively five percent of the amount exceeding $150,000 (and the 10 percent Premature Distribution Tax applies to the entire amount withdrawn). For “excess distributions” made at or after age 59.5 the tax is 15 percent of the amount exceeding $150,000. Note that this tax applies even to those who are 70.5 or older and are required to make a “minimum withdrawal” (as discussed above) in excess of $150,000. For instance, if the minimum required withdrawal is $175,000 and the taxpayer withdraws this amount, he or she must pay an Excess Distribution Tax penalty of $3,750 (i.e., 0.15(175,000 - 150,000)) in addition to the normal income taxes which apply. Of course, if this taxpayer tries to avoid the Excess Distribution Tax by withdrawing only $150,000 the 50 percent Minimum Distribution Tax would levy a penalty of $12,500 (i.e., 0.50(175,000 - 150,000)). Estate Taxes An estate tax return must be filed if a taxpayer’s gross estate exceeds $600,000 at the time of death. Generally, the balances in TDRAs are included in the valuation of one’s estate at the time of death and are subject to normal estate taxes. However, the law allows special exclusions which effectively eliminate normal estate taxes on accounts where one’s spouse is the beneficiary or where the remaining balances are left to qualified charitable organiza- tions (e.g., IRS Publication 448, 1992). The Excess Distributions Tax described above also plays a role in determining the taxes due on one’s estate. Again, since TDRAs are only intended to provide retirement income for a given taxpayer, the tax law maintains that there should not be an “excessive accumulation” of funds left in these accounts at the taxpayer’s death. Congress has decided that a “reasonable” amount to have in TDRAs at one’s death should total no more than the present value of a $150,000 annuity for the remaining actuarial life expectancy of the taxpayer at the time of their death. Thus, one’s estate must pay a 15 percent penalty tax on the amount by which the total value of the deceased’s TDRAs exceed this “reasonable” amount. This “excess accumulation” tax applies to all TDRAs regardless of beneficiary designations. III. A DIFFICULT DECISION From the previous discussion it is clear that the question of how one should go about making withdrawals from TDRAs can be difficult. For many, it is probably challenging enough to make withdrawals that are within the IRS guidelines. However, if one attempts to make withdrawals that are not only “legal” but also maximize the value of one’s benefits, the Withdrawals from Ta-Deferred Retirement Accounts 97 problem enters a new realm of difficulty. In either case, a number of practical questions must be addressed such as: 1) When should withdrawals begin? 2) How long should they continue? 3) How much should be withdrawn each year? 4) From which accounts should withdrawals be made? Clearly, the Premature Distributions Tax can be avoided by not making withdrawals before age 59.5 and the Minimum Distribution Tax can be avoided by making the required minimum withdrawal beginning at age 70.5. Thus, with respect to the first question above we can generally say that withdrawals should begin no sooner than age 59.5 and no later than 70.5. A more specific answer would require consideration of an individual taxpayer’s income needs. Similarly, with respect to the third question above we can generally say that the taxpayer should make at least the minimum required withdrawal beginning at age 70.5 in order to avoid the onerous Minimum Distribution Tax penalty. However, in some instances it may be necessary and/or wise to withdraw more than the minimum required amount. The “best” or optimal answer to all of the questions listed above requires one to consider the simultaneous impacts of a number of subtle factors over a period of years. For instance, when deciding from which accounts to actually withdraw money one must consider the rates of return on the various accounts and the schedule of life expectancy factors which will apply to future balances in the accounts. One might intuitively sense that the optimal withdrawal policy would first involve making withdrawals from the accounts with the lowest rate of return. However, it is possible that the beneficiary designations on higher yielding accounts will, in subsequent years, impose higher minimum required withdrawals which might force more taxable income into higher tax brackets. For instance, let us suppose that the rate of return on Account 1 in Table 2 is less than that the return on Account 2. If the required withdrawal for Year 1 is made from Account 1 this will obviously cause more money to accumulate in Account 2 which, in turn, will cause the minimum required withdrawal in subsequent years to be higher (due to the smaller joint life expectancy value on this account). Part of this higher minimum required withdrawal may fall into a higher tax bracket that may more than offset the higher earnings on this account. Thus, in some situations it might actually be best to withdraw money from accounts earning the highest rates of return. Similarly, it is easy to believe that one should avoid making withdrawals for as long as possible (if not needed as current income) as this allows the investment to continue to earn interest and defer taxes. However, it is possible that in some cases withdrawals should begin before age 70.5 to help avoid exposure to the Excess Distribution Tax. Simultaneously evaluating all these factors affecting the decision can quickly over- whelm us and prompt many to adopt heuristic withdrawal policies (e.g., Gould, 1988; McIntosh & Hollinrake, 1988; Quinn, 1988; Saftner & Fink, 1990; Sage, 1988; TIAA- CREF, 1991; Tritch, 1988). Two such policies are summarized below: Minimal Withdrawal Policy: Withdraw the maximum of the minimum required by the IRS or the minimum needed to reach the desired level of retirement income. The actual withdrawal is made from the account(s) paying the lowest rate(s) of interest. 98 FINANCIAL SERVICES REVIEW, 3(2) 1994 Proportional Withdrawal Policy: Same as above except the actual withdrawal is made from all the accounts in proportion to their balances at the beginning of the year. (Note: This is the default withdrawal policy used by TIAA-CREF.) Notice that these policies ensure that the taxpayer withdraws at least as much as required by the IRS to avoid the 50 percent Minimum Distribution Tax penalty. While such policies may provide good “rules of thumb” for the average investor to follow, specific individuals can lose thousands of dollars by making suboptimal withdrawals using these heuristics (e.g., Katz, 1990; Ragsdale, Seila, & Little, 1993). If one considers the number of individuals with TDRA investments and the amount of money deposited in these accounts, the total potential loss to individual taxpayers as a result of poor or even slightly suboptimal decision making is considerable. Thus, even heuristics that generally work well still may leave specific individuals with very suboptimal withdrawal schedules. IV. AN OPTIMIZATION MODEL A taxpayer facing the decisions described above might be interested in determining the schedule of withdrawals that maximizes the net (after tax) present value (NPV) of the withdrawals made over their life expectancy plus the NPV of the remaining TDRA balances passing to their beneficiaries (all within IRS regulations). In this section, we present a mathematical programming model that can be solved to determine the schedule of withdraw- als that achieves this objective. Assumptions Since our model considers a series of withdrawals over a number of years, it is clearly unrealistic to assume that the tax law will not change during this time. On the other hand, it is also clearly impossible to foresee what these changes will be-particularly those of a political or economic nature. However, other changes do seem reasonably certain. For instance, it is reasonable to expect the tax rate schedules to change every year to account for inflation. Today’s tax rate schedules will almost certainly not apply 10 years from now. But we might expect the current schedules adjusted for inflation may reasonably estimate what may exist 10 years hence. Similarly, the $150,000 limit involved in today’s Excess Distri- bution Tax and Estate Tax will almost certainly be adjusted for inflation in the future. Thus, our model accounts for inflationary changes in the tax law wherever appropriate. However, it does not account for unforeseen structural changes in the tax law. Such changes would have to be incorporated into the model as they occur. A number of additional assumptions are also reflected in this model. 1. We assume that withdrawals are made at the end of each year to allow the taxpayer to accumulate as much tax-deferred income as possible. The formulation can easily be modified so that withdrawals are assumed to occur at the beginning of each year or on some other periodic basis, if so desired. 2. We assume that no contributions have been made to any accounts on a non-taxable basis. Additional variables and constraints, currently omitted for simplicity, can be added to our model to accommodate non-taxable contributions. 100 FINANCIAL SERVICES REVIEW, 3(Z) 1994 withdrawals are determined by dividing the balance in each account at the beginning of the year (b$ by the joint life expectancy factor (a$ supplied by the IRS. C Cwij + Wn:j - by/Uo) 2 0, i = n, (4) j=l i(Wo-bijIa,)>O, i=n,+I,...,n (5) j=l As mentioned in Section 2.3, amounts withdrawn from TDRAs in excess of the “reasonable withdrawal limit” (presently $150,000) in any year are subject to a 15 percent Excess Distributions Tax. Constraint equations (6) and (7) force the variable ei to equal the amount by which withdrawals exceed the inflation-adjusted “reasonable withdrawal limit” (_‘$$?J in year i. Any such “excessive” withdrawals are then subjected to the 15 percent Excess Distribution Tax as shown in equation (1). ~Wij-eiS~y i=l,..., n,,n,+2 ,..., n (6) j=l “I C(Wu+W,;j)-eiISi, i=n,+, (7) j=l In equations (8) and (9) the total taxable income from withdrawals (wij) and other sources (oi) is allocated to the variables that represent the total income falling into each of the three different personal income tax brackets each year (i.e., xii, xi2, and xi3). Given the current personal tax rate structure, the objective function in equation (1) will ensure that taxable income will be first allocated to Xi, and then to xi2 and Xi3 since the tax rates for each of these income brackets (represented by rfh ) are monotonically increasing (i.e., t’; < t$ c t$ ). Inflation-adjusted upper bounds (rnyk ) for the income brackets are given in equation (10). ~Xi~-~Wy-Oi=O, i= 1,. . ,n,,n,+P,. . . ,n (8) k=l j=l i xik- i (Wg + dnj) - Oi = 0, i=n,+, (9) k=l j=l XikIm$ i=l,...,n;k=l,2 (10) The constraints in equations (11) through (13) indicate that each account’s balance at the beginning of each year (b$ should equal the prior year’s beginning balance plus the interest earned, less any amount withdrawn from the account. Constraint equations (12) and Withdrawals from Tar-Deferred Retirement Accounts 101 (13) apply, respectively, to the years in which the taxpayer reaches age 70.5 (n,) and 71.5 (n,,). These constraints make the necessary adjustments to the beginning balances for years ++I and 4+2 if the deferral provision described in Section 2.2 is utilized. Notice that if the deferral provision is not used (i.e., if all W,:j = 0) constraint equations (12) and (13) assume the same form as equation (11). bi+I,j- (l+rii)bii+Wii=O,i=l ,..., n,._l,n,+2, . . . . n;j=l,...,nl (11) bi+l,j- (1 +ru)by+wy+w$ =O,i=n,;j=l,. . . ,nl (12) bi+l,j- (l+r~)b~+w~-~~~Wn~j =O,i=n,+i;j=l,...,nI (13) The remaining constraints in the model have to do with estate taxes. As mentioned earlier, an estate tax return does not have to be filed if the value of a taxpayer’s estate does not exceed a certain cutoff point (presently $600,000). We use Y to represent this cutoff point (adjusted for inflation) and B to represent the estimated future value of the taxpayer’s estate excluding TDRAs. Since more than one beneficiary can be designated for a given TDRA, it is also necessary to consider the percentage bj) of each TDRA which passes to the taxpayer’s spouse or to a qualified charitable organization (which are both exempt from estate taxes) in determining the taxable value of an estate. In equation (14) the surplus variable s will equal the amount by which the taxpayer’s gross taxable estate exceeds Y. Similarly, if estate taxes must be paid the optimal value for the binary variable h will be one in equation (15) (where M represents a very large number). “1 C(l-~j)bn+lj+8-~I~ j=l (14) slM?L (15) From equations (14) and (15) we know that if estate taxes must be paid, h = 1 and the taxable value of the estate is Y+s. In this case, it is necessary to allocate the taxable value of the estate to variables in equation (1) which represented the different estate income tax brackets (i.e., the yk). This is accomplished in equation (16). Upper bounds for the different estate income tax brackets are given in equation (17). Also notice that since the yk are penalized in equation (1) an optimizer will attempt to sets = 0 and h = 0 whenever possible. Thus, in equations (14) and (15), s and h will assume strictly positive values if and only if estate taxes must be paid. 19 Cy,-s-m=0 k=l (16) Yked Example) PV of Withdrawals PV of Income Taxes on Withdrawals NPV of Withdrawals PV of Ending Balance PV of Estate Taxes on Ending Balance NPV of Ending Balance Total NPV Minimal $2,892,660 (1,003,087) $1,889,573 1,653,690 (143,998) $1,509,692 $3,399,265 Withdrawal Policy Proporrional Optimal $2,884,784 $2,892,660 (999,464) (1,003,087) $1,885,320 $1,889,573 1,636,203 1,653,690 (141,375) (143,998) $1,494,828 $1.509.692 $3,380,148 $3,399,265 Second, this example also highlights a number of withdrawal policy matters where one’s intuition can fail. For instance, our example dramatically demonstrates how undesir- able it can be to leave money in a TDRA with a non-spouse beneficiary. This might lead one to believe that, given the option, it is best to first make withdrawals from accounts with non-spouse beneficiaries. However, the schedule of withdrawals in Table 5 clearly indicates that this is not always the case since the optimal policy here involves first making withdraw- als from Account 1 where the spouse is the beneficiary. Similarly, our example might lead one to believe that it is best to make one’s spouse the beneficiary on all TDRAs so as to avoid the payment of normal estate taxes at one’s death. (Note that this alternative may not always be possible since not all taxpayers are married and, even if they are, they may be unable to change the beneficiary designations on certain accounts.) The results for this scenario are presented in Table 7. In Table 7 we see that if the spouse is the beneficiary on both accounts, the minimal and optimal withdrawal policies are identical while the proportional policy is only slightly suboptimal. Notice, however, that the NPV of the withdrawals under this scenario are larger than for the example in Table 4. This is due to the fact that the joint life expectancy of the taxpayer and his or her spouse is smaller than the joint life expectancy of the taxpayer and his or her child. This, in turn, forces the required minimum withdrawals that begin at age 70.5 to be larger. Thus, under this scenario the taxpayer receives $82,053 more in NPV during their lifetime via withdrawals, but the NPV being left to his or her spouse is reduced by $139,188. Thus, the overall effect of making the spouse the beneficiary on both accounts is to decrease the NPV of the taxpayer’s total benefits by about $57,000 compared to the optimal solution in the original example. Of course, there might be some who would prefer the solution offered by this second scenario even though the total NPV of the benefits are smaller. This illustrates yet another important aspect of the model. By allowing a person to easily play out such “What if?” scenarios, our model can determine not only the optimal withdrawal policy for a given set of conditions, but also allow the user to explore how changes in these conditions impact the solution. This can lead to the discovery of “better” (higher NPV) solutions. It can also lead to the discovery of solutions which have a smaller NPV but greater utility to individual decision makers. In either case, the model can provide the user with a greater understanding and sharper intuition about the problem they face and an objective means for assessing the trade-offs among the various possible decisions. Withdrawals from Tax-Deferred Retirement Accounts 1W VII. CONCLUSIONS In this paper we presented a mathematical programming model for assisting retirees (or their advisors) in determining how to make withdrawals from TDRAs. This model can be used to determine the withdrawal schedule that maximizes the NPV of the taxpayer’s retirement benefits. The potential benefits of this model were illustrated relative to a number of heuristic withdrawal policies likely to be used in practice. It is important to note that the use of this model is not intended to be a one-time occurrence. While the model determines the optimal schedule of withdrawals over a number of years, the taxpayer really is only immediately interested in what action they should take in the current year. Thus, the model can and should be updated on an annual basis to reflect changes in investment returns, life expectancies, beneficiary designations and, of course, structural changes in the tax law. When used in this manner our model offers the taxpayer the assurance of knowing what the best possible withdrawal decision is made each year based on the information at hand. Finally, as mentioned at the outset, there are some exceptions to the tax rules embodied in our model that could have significant impacts on individual taxpayers. Thus, our model should not be used as a replacement for tax advisors but in tandem with qualified financial planning professionals. APPENDIX Glossary of Terms in the Model aij bij 4 6 E: f 3 ei 2 n 4 nl Oi Pi Pvi life expectancy factor at year i for investmentj (from IRS [ 1991b] tables). balance at beginning of year i in TDRA investmentj. desired minimum total taxable income from all sources in Year i. the “risk free” discount rate. estimated total taxable future value of the taxpayer’s estate (excluding TDRAs). the estimated rate of inflation. the “reasonable” withdrawal limit adjusted for inflation (i.e., ._!Zi = $150,000 x (1 +ni>. amount in excess of the “reasonable” withdrawal limit Zi withdrawn from TDRAs in Year i. an arbitrarily large positive number. a constant representing the maximum amount of personal income allowed in personal tax bracket k in Year i (adjusted of inflation). a constant representing the maximum amount of estate income allowed in estate tax bracket k (adjusted for inflation). the life expectancy of the taxpayer at distribution Year 1 (or the year in which the first withdrawal is made). the distribution year in which the tax payer turns age 70.5. the number of TDRA investments. other (non-TDRA) taxable income in Year i. the percentage of the balance in TDRA j which at the taxpayer’s death passes to his or her spouse or to a qualified charitable organization. present value interest factor at year i (i.e., pvi = (1 + 6)-Q. 10s FINANCIAL SERVICES REVIEW, 3(2) 1994 rij 3% Yj W' "rj xik Yk the expected rate of return in Year i on TDRA investmentj. the “reasonable” limit on the value of the taxpayer’s TDRAs at the time of their death (i.e. 9 = %n x (1 - (1 + S)-“d)/S where nd is the remaining actuarial life expectancy at the taxpayer’s death.) amount by which the final value of TDRAs exceed .% . the maximum amount of gross estate value allowed without having to file an estate tax return adjusted for inflation (i.e., y= 600,000 x (1 +a”). the tih marginal personal income tax rate. the Ich marginal estate tax rate. withdrawal in Year i from TDRA investmentj. withdrawal for year n, from TDRA investment i made in year n, + 1. amount of total taxable income in Year i subject to tax rate tfl amount of total estate subject to tax rate ti REFERENCES Abramson, E.M. 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