Abstract

This paper describes a model of the tax-deferred and after-tax account balances and money flows during an individual’s retirement years. The focus is on tax-deferred (Keogh, IRA, 401(k), 403(b)) savings and IRS constraints on withdrawals. The model is formulated and solved as a Linear Programming (LP) problem to establish an optimal withdrawal level. Computational results are included.

Introduction

A comfortable retirement requires sufficient funds to enjoy the fruits of not working for income. For years, the financial press trumpeted tax-deferred, defined contribution, qualified, savings plans (for example, IRAs, Keogh, 401(k), 403(b), and SEP plans) as vehicles for funding retirement. The Taxpayers’ Relief Act of 1997 added a new saving vehicle, the Roth IRA.

Well-planned withdrawals from retirement savings will assure that sufficient funds are available for retirement living without making sacrifices that end up benefiting only the tax collector. Three scenarios illustrate what can occur during retirement:

• If withdrawals are made at too high of a level then the money runs out with the retiree still alive.
• If withdrawals are made at too low of a level then investment return will compound and a large estate will remain, much of which will go to estate taxes. The retirement standard of living has been sacrificed to enrich the tax collector.
• An optimal withdrawal level will show an investment account that grows in the early retirement years due to the compounding of investment returns. In the later years, capital is withdrawn and spent. At the end the account is brought to $1,000,000, the amount that is passed to heirs without estate taxes. The retiree, not the tax collector, enjoys the benefits of retirement savings.

An optimal retirement plan does these things:

• Establishes the level of withdrawals from retirement savings.
• Maximizes the total amount of money available for retirement spending over the entire retirement period.
• Regulates the flow of funds from tax-deferred savings to after-tax reinvestment and retirement spending.

When the time for retirement arrives, distributions from tax-deferred savings should be planned to stay within the constraints imposed by the IRS. One strategy is to take a lump-sum distribution from tax-deferred savings. Taxes are paid according to IRS dictated 5 year forward averaging. The proceeds are invested in stable, after-tax assets, to assure a steady income stream.

Financial planners often advise leaving the tax-deferred savings alone, drawing down after-tax investments first, and then withdrawing from the tax-deferred savings [1, 17].

This paper presents the results of a mathematical analysis aimed at determining the optimal pattern of annual withdrawals from tax-deferred and after-tax savings. The results of the analysis confirm the financial planners’ advice to draw down assets on which taxes have been paid before withdrawing from the tax-deferred assets.

To perform this analysis a mathematical model (called ORP for Optimal Retirement Planning) was developed and implemented as computer software. The model begins with the initial balances of tax-deferred and after-tax assets, assumptions about inflation and tax rates, the length of retirement, and the desired estate balance. The model is solved for an optimal retirement plan that provides for a maximum yearly withdrawal while meeting IRS requirements and minimizing penalties. For each year of retirement the optimal retirement plan includes:

• The amount withdrawn from tax-deferred savings.
• The amount withdrawn from after-tax savings.
• The amount withdrawn from a Roth IRA.
• The amount to be transferred from tax-deferred savings to after-tax savings.

The most interesting model result is the after-tax amount, in today’s dollars, that is available in each year of retirement for living expenses and gift giving.

Prior Work

Recently, articles in the journal of the American Association of Individual Investors (AAII) address the problem of distributions from tax-deferred savings. In the June 1995 edition, Hopewell[8] developed the idea of making an estimate of life expectancy and devising a distribution plan that draws down assets to zero over that term. The plan incorporates continued asset growth, taxes, and inflation. In the same June issue, Blackman and McAuliffe[17] describe the excess distribution penalty and its impact on tax-deferred distributed planning. In the August 1995 edition, Scott[10] published a Retirement Spending Worksheet for estimating the yearly withdrawals from tax-deferred and after-tax savings, in today’s dollars.

The model presented here builds on these ideas by formulating the problem as a linear programming model [2]. The model is time dynamic in that it computes the flow of money from the current year, usually pre-retirement, through the end of life expectancy and allows for leaving an estate. The formulation includes tax-deferred savings, Roth IRA, after-tax savings, taxes, penalties, and inflation. The model is implemented by personal computer software that is called Optimal Retirement Planning (ORP).

Assumptions

Retirement financial planning begins with these precepts:

• You are going to die.
• You can’t take it with you.
• Inflation is not going away. Even small rates of inflation erode capital.
• Taxes must be paid.
• Assets grow through the compounding of their returns until they are withdrawn.

Every retiree has an actuarial life expectancy. The optimal strategy is to provide for a comfortable lifestyle during retirement and to provide a cushion in case the term is exceeded. The exact form of the cushion and how it is transferred to the heirs is estate planning and outside the scope of this model.

Some positive, average return on investment can be achieved over the years, although there will be yearly fluctuations. Historically, common stocks have returned 10% per year while bonds returned somewhat less[8]. How this average return is achieved is investment strategy and outside the scope of this model.

Social Security payments and pensions are omitted from the model for computational simplicity and because they do not change ORP’s results. The fundamental assumption is that the retiree wishes to maximize the value of his or her plan and other sources of income can be added with little or no impact. Outside income should be considered when estimating the personal income tax rate.

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IRS Rules For Tax-deferred Plans

The IRS has several rules that apply to the distribution of funds from a tax-deferred plan:

• All deferred-tax distributions are taxed at ordinary income rates. Capital gains tax rates do not apply to tax-deferred distributions.
• Upon reaching the age of 70 1/2 the retiree is required to begin minimum withdrawals from tax-deferred savings. The amount of the minimum is computed by one of two methods, the term certain and recalculation methods. Both methods depend on the account balance, the martial status of the retiree, and the life expectancy of the retiree. If the minimum distribution is not met then a 50% penalty is assessed on the difference between the actual and required distribution [6].
• The penalty for pre retirement distributions is 10%.

The Roth IRA

A Roth IRA is a savings account that is funded with after-tax dollars but returns are accumulate and are withdrawn without paying any Federal Income Tax. There are restrictions:

• Contributions are limited to $2,000 per year for a single person, $4,000 for a married couple.
• Contributions phase out at $95,000 income for a single person, $150,000 for a married couple.
• Distributions are penalized if taken before age 59 ½.
• Distributions cannot begin until five years after the first contribution..
• An existing IRA can be rolled over into a Roth IRA for wage earners of $100,000 or less.

Analytically, there is little difference between a regular IRA and a Roth IRA, all other things being equal:

V = b(1+r^y) (1-t) is the distribution value of a regular IRA at year y, earning at rate r, starting with before-tax balance b and paying tax rate t.

R = (1-T)b(1+r^y) is the value of a Roth IRA at year y, earning at rate r, starting with before-tax balance b and paying tax rate T at the time of the contribution.

There are 3 cases:

If t = T then V = R and whole thing is a wash. This case occurs when the distribution tax rate (t) is the same as the pre retirement personal income tax rate (T).

If t < T then V > R the Tax-deferred Account provides more retirement funds than does the Roth IRA. This case occurs when income tax rate during retirement (t) is less than during the working years (T). This is the normal retirement situation.

If t > T then V<R and the Roth IRA is a better deal. This case occurs when the distribution tax rate (t) is larger than the tax rate before retirement (T). This is the situation for very young wage earners, graduate students, and the like. Roth IRA savings during this period in a persons working life can have a big payoff.

No more can be said about this without getting into the area of financial planning.

Description of ORP

This section provides an overview of ORP. ORP computes the optimal retirement plan that maximizes the total amount of money available for retirement and shows the amount available for living expenses for each year of retirement.

For purposes of this description the following terminology has been adopted: Money taken out of a qualified retirement plan for spending or reinvestment is said to be distributed. Withdrawn money leaves the model, to be spent on living expenses or gifts, and is no longer available for investment.

Figure 1 shows the components of ORP:

• The Tax-deferred Account contains the money invested in one or more IRS approved tax-deferred plans. The Tax-deferred Account balance is computed for each year. The balance will vary according to the beginning balance, pre-retirement contributions, investment return, and the level of distributions.
• The Roth IRA account contains contributions on which taxes have been paid and on which no further federal taxes are due.
• The After-tax Account contains money on which tax has been paid and on which taxes are paid on each year’s investment returns. The After-tax Account balance is computed for each year. The balance will vary according to the beginning balance, pre-retirement contributions, after-tax investment return, transfers from the Tax-deferred Account, and the level of withdrawals.
• The distribution component takes money from the Tax-deferred Account to transfer it to the After-tax Account, use it to meet living expenses and fund the estate, or to pay taxes and IRS penalties. The distribution component also withdraws money from the After-tax Account to meet living expenses and to contribute to the estate requirements. Beginning at age 70 ½, the IRS requires a minimum distribution from the Tax-deferred Account.
• The estate constraint forces the desired amount to be available at the conclusion of the life expectancy term. Because of the stiff taxes applied to estates above $1,000,000, the size of the estate will tend to be the larger of $1,000,000 or the amount specified as the desired estate.
• ORP’s objective function is the sum of the after-tax withdrawals, available for spending, over time from the Tax-deferred Account and After-tax Account. Penalties and taxes are not included in the objective function. The model maximizes the objective function.

Money enters the Tax-deferred Account from the initial balance, annual pre retirement contributions, and investment return compounding. Distributions leave the account for taxes, spending, and transfers to the After-tax Account. Transfers to the After-tax account occur when IRS regulations require that money be distributed from the Tax-deferred Account after the age of 70 ½.

Money enters the After-tax Account from the initial balance, investments, transfers in from the Tax-deferred account, and investment return compounding. Distributions are used for spending and to pay taxes.

Money enters the Roth IRA Account from the initial balance, investments and investment return compounding. ORP does not model the transferring of money from the Tax-deferred account to the Roth IRA because there is no economic incentive to do so (see above) and there are IRS restrictions that limit such transfers. Money is withdrawn only for spending. ORP does not model the transferring of money from the Roth IRA to the other two accounts because there are no restrictions to leaving the money in the Roth IRA and once the money is in the account it will give the best net return of the three.

ORP Model Parameters

The parameters that drive the model and their nominal values for the computation results shown later are shown in Table 1. Appendix B describes each parameter in detail.

The values shown in Table 1 are for a married couple, ages 60 and 57. The balance in the Tax-deferred Account is $900,000. The After-tax Account balance is $100,000. Since the principle wage earner has five years until retirement, contributions to both accounts are planned until retirement. Each year’s account contributions are increased by the rate of inflation. A return of 10% is assumed since that is the average annual return for large company common stocks for the past 50 years [10]. The anticipated average inflation rate is optimistically assumed to be 3.5%.

The model results are organized as follows:

• The Retirement Plan Summary contains:
  • The withdrawal level of the plan is the amount of money, in today’s after-tax dollars, that can be withdrawn for living expenses in each year of retirement. If inflation rate is zero then this amount would be withdrawn in every year throughout the life expectancy term.
  • The Account Balance at Retirement is sum of all accounts at the year of retirement. Taxes have not been paid on the tax-deferred component of this amount.
  • The value of the plan is the sum of all withdrawals.

• The Tax-deferred Account Report shows the yearly activity, in inflation adjusted, thousands of dollars, for the Tax-deferred Account. The columns of this report are:
  • Age of the retiree.
  • Account balance at the beginning of each year.
  • Pre-retirement contributions to the account.
  • Before tax distributions from the account.
  • After-tax distributions applied to living expenses.
  • Penalties paid for early distributions.
  • Transfers from the Tax-deferred Account to the After-tax Account.
  • The minimum required level of distribution after the age of 70.

• The After-tax Account Report shows the yearly activity, in inflation adjusted, thousands of dollars, for the After-tax Account. The columns of this report are:
  • Age of the retiree.
  • Account balance at the beginning of each year.
  • Pre-retirement contributions to the account.
  • Transfers in from the Tax-deferred Account.
  • After-tax Account withdrawals.

• The Roth IRA Report shows the yearly activity, in inflated, thousands of dollars, for the Roth IRA. The columns of this report are:
  • Age of the retiree
  • Account balance at the beginning of each year
  • Pre-retirement contributions to the account
  • After-tax Account withdrawals.

ORP is a linear programming (LP) model [2], formulated using the commercial personal computer software Visual Math Programming [5], and solved by the BdLP optimizer [3]. The model is a system of linear constraints that ORP interactively solves in such a way as to maximize the withdrawals. ORP consists of the VMP and BdLP runtime modules, a data preprocessor, a post-solution report generator, and an interactive user interface.

Computational Results

ORP solved several scenarios to gain an understanding of the results of various retirement strategies. Two scenarios characterize the flow of money during retirement for a married couple and an unmarried person. The lump-sum withdrawal strategy is compared to leaving tax-deferred saving in place. Two scenarios show the impact of inflation and the rate of investment return on the model’s results. The usefulness of the Roth IRA is then examined. Another scenario compares ORP’s result to those demonstrated by Scott’s worksheet method [10]. The consequences of planning to out live the IRS life expectancy is then addressed. The final scenario addresses the choice of tax-deferred contribution levels for a young person just entering the work force.

Joint Life Expectancy

The parameter values shown earlier in Table 1, representing a married couple with one retirement plan, are used to establish the base case from which the effects of parameter changes can be compared.

ORP’s results are that this couple will have $74,000, in current, after-tax, dollars, available annually for 28 years of retirement living. The value of their estate will be $1,00,000 in inflated dollars. Their account balances at retirement (age 65) will total $1,714,400. The total dollars to be withdrawn will be $4,499,000, after inflation.

Table 2 is the Withdrawal Report for the joint life expectancy scenario. The Spending column shows maximum withdrawal level computed by ORP, including adjustments for inflation. At retirement, the After-tax Account is spent first and then withdrawals are begun from the Tax-deferred Account, following the advice of professional financial advisors [1]. The rationale for this result is well known to financial advisors [9,16] and is demonstrated in Appendix A. The amounts shown in the TaxDef column of Table 2 are after taxes have been paid on the tax-deferred distribution.

Table 3 is the Tax-deferred Account Report for the joint life expectancy scenario. $15,000 per year is added to the Tax-deferred Account until retirement. Although retirement begins at the age of 65, tax-deferred distributions do not begin until the age of 67, when the After-tax Account is depleted. The account balance continues to rise until the age of 80 at which point distributions catch up with account investment returns. At age 78, distributions are forced to the minimum distribution level. The amount beyond the living expense level is transferred into the After-tax Account where it will continue to earn returns. The amount transferred is the surplus of the required minimum distribution above the computed living expenses.

Table 4 shows the After-tax Account Report for the married couple scenario. $5,000, adjusted for inflation, is added to the account each year until retirement. After-tax investments are for savings above the limit set by the IRS for contributions to various qualified tax-deferred plans. The After-tax Account funds retirement spending for the first three years, until the account is exhausted.

Single Life Expectancy

The single life expectancy scenario assumes that the money a married couple invested in their children’s’ college education has been aggressively invested in the After-tax Account. There are two differences between the joint life expectancy scenario and the single life expectancy scenario:

• The single’s After-tax Account contains $1,000,000 at the age of 60.
• The single person has a shorter (16 years) life expectancy than a married couple for purposes of computing the minimum distribution that begins at the age of 70.

ORP’s results are that the unmarried person will have $155,000, in current dollars, available yearly for 16 years of retirement. The value of the estate will be $1,000,000. The total dollars withdrawn will be $6,573,300.

Table 5, The Withdrawal Report, shows that withdrawals from the After-tax Account begin immediately upon retirement and continue until age 81. The minimum Tax-deferred Account distribution begins at the age of 70 and substantially reduces the level of withdrawals from the After-tax Account.

Table 6 shows the activity for the Tax-deferred Account for the single person scenario.

The single retiree may feel $155,000 annual withdrawal is more than actual needs and invest in assets having smaller returns to trade a reduced withdrawal level for reduced portfolio volatility. That could be achieved by switching from stocks to bonds. If the investment return is set to 7% then the annual withdrawal for the single retiree will be $99,000 on a total plan value of $4,567,700.

Lump-sum Distribution

The introduction mentioned the intuitively appealing retirement strategy of making a lump-sum distribution from the Tax-deferred Account and managing it solely as an After-tax Account. To compare this strategy with the advice of financial planners to draw down the After-tax Account first, a lump-sum scenario was solved by ORP. The results are compared to the Joint Life Expectancy base case results shown earlier. The base case parameters are as shown previously in Table 1, with three exceptions. First, the plan starts at age 65. Second, there is no beginning Tax-deferred Account balance. Third, the after-tax Account beginning balance is set to $1,173,000, the sum of the After-tax Account balance at age 65 plus the age 65 Tax-deferred Account balance after taxes have been paid on the lump sum distribution.

Table 1c compares the Joint Life Expectancy base case with the LumpSum case.

These results support the advice of professional financial advisors to draw down the After-tax Account first and leave the Tax-deferred Account alone as long as possible.

Effects of Inflation

The anticipated average rate of inflation is an important assumption for the model. Although the 1996 rate of inflation is low, the inflation rate from 1980 through 1991 averaged 4.66% per year [8]. Table 1a demonstrates the effect of different inflation rates on the withdrawal level of the model. All parameters are the default values shown in Table 1 except for variations in the rate of inflation. The Withdrawal values shown are the annual after-tax spending amounts as measured in today’s dollars. The value column contains the total amount of money available throughout retirement.

Inflation effects the levels of after-tax withdrawal as measured in current dollars. The higher the inflation rate the less money there is to spend each year. There is no change in how the retirement plan is managed, only in the annual withdrawal level. The total value of the retirement plan goes up as inflation increases, but it buys less goods and services.

Adjusting the Investment Return Rate

The expected return on investments on both accounts also has a significant impact on the withdrawal level as measured in today’s dollars. These results are shown in Table 1b. All parameters are the default values shown in Table 1 except for variations in the rate of investment return.

The rate of inflation is not something that the retiree can control. Skillful management of the portfolio can influence the retiree’s portfolio rate of return and that in turn will show up in the amount of money available each year for living expenses.

Roth IRA

The Roth IRA is an alternate saving-for-retirement vehicle that needs to be evaluated by most taxpayers. Because no taxes are paid on returns from Roth IRA investments the Roth IRA falls somewhere between a Tax-deferred Account and an After-tax Account. Table 1d shows three Roth IRA options for the base case given above.

All cases assume that the $15,000 annual contribution to the Tax-deferred Account continues to age 65. The tax rate has been adjusted according to amount in the Roth IRA. Since the Roth IRA return is not taxable its contribution to spending will lower the effective tax rate.

The Supplement case replaces $4,000 of After-tax Account contributions with an equal amount of Roth IRA contributions, assuming that other restrictions have been met. Five years of Roth IRA contributions doesn’t make that much difference.

The Partial Rollover case distributes $450,000 of the Tax-deferred Account to the Roth IRA. $167,000 of taxes re paid, spread over four years, leaving a Roth IRA initial balance of $293,000.

The Total Rollover case distributes the full Tax-deferred Account balance to the Roth IRA, paying $315,000 in taxes on the way.

Table 7 shows the Withdrawal Report for the Supplement case. As before the After-tax Account is drawn down first, followed by distributions from the Roth Ira. At age 70 the minimum Tax-deferred Account minimum distribution requirement reduces the distributions from the Roth IRA. The Roth IRA is exhausted at age 85 and Tax-deferred Accounts funds are used exclusively thereafter.

Table 1d indicates that the Supplement case provides for the highest retirement spending. The reduced tax rate on the Tax-deferred Account distributions is the reason. Money has been saved in the Tax-deferred Account at higher tax rate than is being paid during distribution. The funds remaining the Tax-deferred Account after roll over continue to compound before taxes are paid at distribution. The roll over amount depends on each taxpayer’s unique situation, particularly on how much taxable income is present beyond that available from the Tax-deferred Account.

Scott’s Scenario

Since ORP and Scott’s Retirement Spending Worksheet [10] are addressing the same problem it is to be expected that they should produce similar results. Scott presents an example with the parameter values shown in Table 1c.

Scott computes a before tax, annual distribution of $33,062, as measured in today’s dollars. The Estimated Taxes On Savings section of Scott’s worksheet was applied to this amount using a tax rate of 35%. No distinction was between the tax rates for personal income and capital gains. Withdrawals from each account were assumed to be proportional to the original balances of the two accounts. The computed after-tax available amount was $18,171.

ORP computes an after-tax annual spending amount of $25,000 for this scenario. The difference occurs because ORP realizes a higher total withdrawal value because it withdraws from the After-tax Account before withdrawing from the Tax-deferred Account. Appendix A shows discuses why this is true. Table 8 shows the Withdrawal Report for this scenario. For the first five years of retirement, money is withdrawn only from the After-tax Account. At age 70, distributions begin from the Tax-deferred Account, and continue at the minimum determined by the Term Certain Method of minimal distribution computation. The remainder needed for living expenses in each year is withdrawn from the After-tax Account.

Living to a Hundred

Some retirees may feel, for a variety of reasons, that they have a reasonable probability of outliving the IRS life expectancy estimate. Extending the term of the plan to age 100 reduces the withdrawal rate to $70,000 as compare to $74,000 for the Joint Life Expectancy scenario that ends at age 92. Table 9 shows the Withdrawal Report for a life expectancy term of age 100 for the married couple scenario. As before, the After-tax account is depleted first. Withdrawals begin on the Tax-deferred Account at age 67. The After-tax Account supplements the Tax-deferred Account beginning at age 95. At age 100 the estate balance is $1,000,000.

Table 10 shows the Tax-deferred Account activity. At age 82, the required minimum distributions exceed the computed spending requirements. The excess is transferred into the After-tax Account. Since the ReCalc method is being used to determine the minimum distribution requirement distributions from the Tax-deferred Account will continue to the end of the plan. Toward the end of the plan, age 95 and beyond, the money transferred into the After-tax Account is used to supplement the withdrawals from the Tax-deferred Account. The estate balance of $1,000,000 is sufficient to cover less than four additional years, should the retiree choose to live longer.

Young Tax-deferred Investors

It is sometimes difficult to persuade younger employees of the virtues of investing some of their before-tax wages in their company’s 401(k) plan. One personnel director of an accounting and data entry firm revealed that most the firm’s younger employees did not participate in the company’s 401(k) program. Those that did contributed an average of only 5% of their wages. Many younger employees appear not to appreciate the power of compounding.

To justify ORP’s view of compounding a pair of scenarios was solved using the following assumptions:

• Current age is 26
• No beginning balance in either account
• No After-tax Account contributions
• Single life expectancy of 16 years at the age of 70
• Current year’s salary is $25,000

All other parameters were at their default values.

The first scenario assumes a 5% Tax-deferred Account contribution ($1,250) and the second has a 15% Tax-deferred Account contribution ($3,750). The out of pocket difference for the second scenario is actually $2170 because taxes are reduced by $375. Contributions were increased each year by the rate of inflation.

The results are that the 5% contributor can expect a yearly after-tax retirement withdrawal of $9,000, while the 15% contributor will be enjoying a $31,000 yearly withdrawal. Thus, for 2.7 times the pre tax contribution in the present, the future after-tax reward is increased by a factor of 3.44.

Conclusion

The results shown above support the advice of many financial advisors regarding the management of retirement funds, before and during retirement:

• Determine a schedule of Tax-deferred Account distributions to begin by age seventy and one half. The level of distribution should be sufficient to meet minimum IRS requirements, to provide for inflation adjusted living expenses, and to draw down assets to the point where inheritance taxes can be minimized at termination of the plan while leaving sufficient funds if life expectancy is exceeded.
• Invest sensibly even after retirement. A seventy year old unmarried person has a sixteen year life expectancy and can think in terms of long term investments. The continued compounding of returns after retirement is the key to a comfortable old age.
• Continue contributing into the Tax-deferred Account for the entire working life. Assets in the Tax-deferred Account will compound at a tax-advantaged rate both before and after retirement.
• Spend the After-tax Account first because its rate of compounding is less than that of the Tax-deferred Account.
• Sometime in the mid to late seventies, begin shifting money from the Tax-deferred Account to the After-tax Account to reduce the 15% excise tax on excess withdrawals.

Some retirees are concerned about the consequences of dying before the end of the term of the plan, leaving a large Tax-deferred Account balance. Although this issue is one of estate planning and outside the scope of this paper, two observations may be made:

• If the retiree lives to his or her life expectancy then the estate balance of $1,000,000 will be distributed to the heirs. Early death will leave a much larger estate balance and more than $1,000,000 will be distributed, even after taxes. The heirs will inherit more money, sooner.
• The model’s IRS life expectancy and term parameters are used to plan for married couples in which the wife will probably out live the husband. Estate taxes are not an issue because the surviving spouse inherits both the Tax-deferred Account and the After-tax Account without paying estate taxes.

Use of ORP to plan an individual’s retirement can begin before retirement to determine if the computed optimum level of distribution is sufficient for a comfortable retirement. The result may affect savings plans before retirement and influence the choice of retirement age. Large account could call for early retirement if that is an otherwise desirable option. A lower than desired level of plan disbursements may cause the potential retiree to stay in the work force past the age of 65.

ORP assumes a fixed average return. In real life any particular year’s return will vary significantly. The timing of these variations will effect the account balances through out the term of the plan because capital is being distributed each year. If the stock market is down during the early years of retirement then the value of the plan will be less over the term of the plan than if the market gains during the early years. This topic is discussed fully in a recent AAII Journal article[19].

During retirement, the model could be run annually, with appropriate parameter adjustments, to determine how the year’s living expenses are to be funded and to determine transfers from the Tax-deferred Account to the After-tax Account. Actual investment returns will affect the account balances for a particular year. Unplanned disbursements or the lack thereof, will also affect account balances thereof. Forecasted changes in the inflation rate and changes in the anticipated rate of investment return should be included in the model. All changes in parameters will cause incremental changes in planned withdrawals for later years.

Appendix A

The rationale for leaving money in a Tax-deferred Account is the same as that used to justify using a tax-deferred plan in the first place. The Tax-deferred Account distribution, after-taxes are paid, is larger than the capital plus after-tax returns accumulating in the After-tax Account. Table A1 shows how this works.

Table A1 shows 20 years of compounding of returns. Column 1 is the year. Column 2 (Deferred) shows the 10% rate of compounding on $100 left in a Tax-deferred Account. Column 3 (Value) shows the money realized by distributing the account balance and paying taxes on it.

Column 4 (Taxed) shows the after-tax compounding that occurs if the $100 were taken out of the Tax-deferred Account, taxes are paid yielding $65, and the proceeds are invested in an After-tax Account.

Column 5 shows the difference between the two accounts as a percentage of the After-tax Account balance, for each year.

Table A1 assumes a tax rate of 35% and an investment rate of return of 10%.

A realistic refinement to this example is to assume the investment of the lump-sum distribution in growth stocks that pay no dividends. The value of growth stocks will compound without paying taxes. At the time of sale the capital gains tax is 28% of the difference between appreciated value and the original cost.

Table A2 demonstrates this example. As in Table A1, the Deferred column shows the compounding of the Tax-deferred Account and the Value column shows the after-tax value of the Tax-deferred Account. The Growth column shows the compounding of the After-tax Account without taxes. (Elementary algebra will show that the Value column and the Growth column are the same value.) The Withdrawal column shows the value of the After-tax Account should it be withdrawn and capital gains taxes paid. The percentage column shows that the Tax-deferred Account out performs the After-tax Account invested for growth, although not by as much as shown previously in Table A1.

Appendix B

The parameters that drive ORP are described in this appendix.

Account Balances

Name Default Value Description

AfterTax 100 Current After-tax Account Balance ($000)

TaxDef 900 Current Tax-deferred Account balance ($000)

RothIRA 0 Roth IRA Account Balance ($000)

Estate 1000 Desired estate size ($000)

Account Contributions

Name Default Value Description

ContDef 15 Planned contribution to Tax-deferred Account ($000)

ContSave 5 Planned yearly contribution to After-tax Account($000)

ContRoth 0 Contribution to the Roth IRA

Ages

Name Default Value Description

CurrAge 60 Current age

RetAge 65 Anticipated retirement age

Spouse 22 Age of retiree’s spouse.

Term Age at which the plan is to end.

Parameters

Name Default Value Description

Inflation 3.5 % average inflation rate.

Return 10 Tax-deferred Account % average investment return

ReturnI 10 After-tax Account % average investment return

TaxRate 35 Tax-deferred Account % average anticipated tax rate

TaxRateI 35 After-tax Account % average anticipated tax rate

Glossary

An After-tax Account is an account in which taxes are paid on all investments before making the investment, and taxes are paid on the returns as they are realized.

A distribution is the money taken out of the Tax-deferred Account to meet withdrawals, pay taxes or to be transferred to the After-tax Account.

An early distribution penalty of 10% is assessed by the IRS for distributions from tax-deferred savings that occur before the of age of 59 ½.

Estate planning and investment strategies are two topics that are outside the scope of this paper. The are represented in the ORP model by two parameters, set by the user but they are not implemented in the mathematics of the model.

Life expectancy is an individual matter but for purposes of modeling the IRS life expectancy tables is sufficient [11]

Linear programming [2] is a popular management science modeling technique used as a decision support tool in a number of different fields, ranging from running oil refineries, to locating warehouses, formulating chicken feed, and even extending into the financial world to perform portfolio optimization [3]. Linear means that the model is described as a system of linear equations. Programming means that the output of the model is a program to be followed to obtain the computed optimal result. There are several commercial, personal computer software systems available to solve linear programming models.

Living expense is the amount of money that the ORP computes, in today’s dollars, that are available for spending in retirement.

Math programming is the field of constrained optimization that includes models with discrete and non-linear elements. Linear programming is one part of math programming.

The IRS requires a minimum distribution from tax-deferred savings either after the age of 70 ½ or retirement, which ever comes later. This rule does not apply to Roth IRAs.

The model is the mathematical description of the behavior, over time, of the Tax-deferred Account and the After-tax Account.

The objective of the linear programming model is to compute a program of disbursements, from the Tax-deferred and After-tax Accounts, that maximizes the money available to be spent during retirement or left in the retiree’s estate. The program is subject to the constraint of providing for inflation adjusted living expenses and the constraints imposed by the IRS.

The term optimal appears in title of this paper because the results described herein were obtained by formulating the retirement problem as an linear programming model. Values are assigned to policy parameters and the model is solved for the optimal retirement plan. The optimal solution shows the maximum objective that satisfies all constraints.

Optimal Retirement Planning (ORP) is computer software that implements the model.

The optimal retirement plan is the schedule of account disbursements and withdrawals that maximizes the objective.

The recalculation method of computing the minimum distribution at age 70 ½ is as follows: Each year the minimum distribution is computed by determining a new life expectancy value from the IRS actuarial table and dividing that value into the Tax-deferred Account balance. Since you will always have a life expectancy larger than one, the Tax-deferred Account balance never goes to zero.

A Roth IRA is an investment account in which taxes have been paid on all contributions and all returns are tax exempt.

A Tax-deferred Account is an account in which taxes have not been paid on either the contributions or the investment returns. Taxes are paid when money is distributed from the account. Individual Retirement Accounts (IRA), 401(k) plans, 403(b) plans and profit sharing plans (Keogh) are examples of Tax-deferred Accounts.

The Term Certain method of computing the minimum tax deferred distribution amount is where the IRS provides a life expectancy figure for a person at the age of 70 and that value is reduced by one for each year thereafter. The minimum distribution is then the amount in the Tax-deferred Account divided by the life expectancy for a particular year.

The retirement plan’s value is the total amount of money withdrawn from the plan plus the account balances at the end of the life expectancy, i.e. the estate.

A withdrawal is money taken out of the retirement accounts for living expenses, gifts, or left in the estate.

References

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Retirees Fret About Stretching Nest Eggs, New York Times, Oct. 8, 1995

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Acknowledgments

The author would like to thank Fred Chen, CPA, for his assistance in interpreting the US Tax Code and his help with the vocabulary of accounting and tax planning. However, all errors contained herein are solely the responsibility of the author. Thanks go also to Rodney Wildermuth for his continuing contributions to the ORP web site: http://www.i-orp.com

About the Author

James S. Welch is president of Sundown Software Systems, Inc., a firm that specializes in developing software tools for the math programming model formulator. Mr. Welch has 38 years of computer programming experience, concentrating on math programming and database management.