The Mathematics of a Conversion of a Traditional IRA to a Roth IRA. Floyd Vest

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1 The Mathematics of a Conversion of a itional IRA to a IRA Floyd Vest Based on the following (corrected) table comparing a IRA conversion with a itional IRA, a cursory examination may suggest that a IRA pays better after taxes than a itional IRA. In the table, the 2019 after tax return on the conversion is reported to be $204,232 while for the itional IRA, it is only $195,813. The conversion appears to pay $8419 more. Conversion in 2010 itional IRA Year Before-Tax Value Tax Due After-Tax Value* Before-Tax Value Tax Due After-Tax Value* 2006 $100,000 $100, $108,000 $108, $116,640 $116, $125,971 $125, #136,049 $136, $19,000 $127,933 $146,933 $41,141 $105, $19,000 $119,168 $158,688 $44,433 $114, $128,701 $171,383 $47,987 $123, $138,997 $185,093 $51,826 $133, $150,117 $199,901 $55,972 $143, $162,126 $215,893 $60,450 $155, $175,100 $233,164 $65,286 $167, $189,104 $251,817 $70,509 $181, $204,232 $271,963 $76,150 $195,013 *Assumes value less tax paid from funds outside IRA and 401(k) Source: However, see the following theorem. Theorem: Other things being equal, a IRA and a itional IRA pay the same after tax rate of return. Proof: Assume as in the table, a tax rate of 28% (or x if you prefer). Consider a itional IRA funded with $5000 in pretax dollars. In n years, at the rate r, the itional IRA pays after taxes S = 5000(1 + r) n (1 0.28). Consider the equivalent after tax amount invested in a IRA. This is 5000(1 0.28) = After n years at the rate r the IRA invests a tax equivalent 5000(1 0.28) and pays S = 5000(1 0.28)(1 + r) n with no taxes owed at the end. Notice that S = S, as stated in the theorem. (See the Compound Interest Formula in the Side Bar Notes.) One advantage of the IRA. An advantage of the is that at a 28% marginal tax bracket, it allows $5000 after tax dollars to be invested, which is tax equivalent to The Mathematics of a Conversion of a itional IRA to a IRA Spring,

2 5000 = $ pretax dollars ($ more in pretax dollars). $5000 in pretax 1! 0.28 dollars invested in a itional IRA is tax equivalent to $3600 in after tax dollars. Thus the allows $1400 more in after tax dollars. The itional IRA allows $5000 in pretax dollars, which is equivalent to $3600 after taxes. So the allows a greater deposit in after tax dollars than the itional IRA. Correcting the table. In the discussion of the table, the author mentions converting $136,000 from a itional IRA to a IRA. The IRS rule for 2010 conversions allows half the tax owed to be paid in 2011 and the other half in Thus the author calculates taxes to be 0.28(136,000) = $38,000, and half of $38,000 gives $19,000 to be paid in each of two years. You see the $19,000 in paid taxes entered in the table. However, the table converts $136,049. This is shown by the table for a 2010 balance of $136,049. The end of year 2011 balance is 136,049( ) = 146, So 146, ,000 = $127,933 which is the After Tax Value in the for 2011 given by the table. The table assumes an 8% return on the investments. But taxes on $136,049 at 28% are $38, with half being $19,046.86, not $19,000. Thus in the table, less is paid in taxes than should have been. This would give a final after tax return of the conversion greater than what is correct. Demonstration of the Theorem. In the table, we will assume that all of the taxes are paid in In the following discussion we will use the $38, in taxes. If paid on the 2010 balance, then 136,049 38, = $97, remains. Continuing for the investment, 97, ( ) 9 = $195, Note that this is the same after tax amount as for the itional IRA at the end of Of course, this is less than the 2019 balance for the conversion given in the table. Thus, under this treatment of the tax, the itional IRA paid the same after tax as the. One advantage of the IRS rule for 2010 conversions. An advantage of the rule allowing for delaying payment of taxes and distributing them over two years is that the unpaid taxes earns interest that is invested in the conversion. We will examine this feature as demonstrated in the table and use their $19,000 for half the taxes for purpose of discussion. For 2011, the calculation is 136,049( ) 19,000 = $127,933 for the end of year 2011 balance in the table. The taxes can be paid by April 15, For 2012, the calculation is 127,933( ) 19,000 = $119,168 for the end of year 2012 balance in the table. The taxes can be paid by April 15, The table shows that instead of paying all the taxes for 2010, by April 15, 2011, it is assumed than $38,000 earns interest in 2011, and $19,000 earns interest in Calculating the interest earned on the delayed payment of taxes, 38,000(0.08) = $3040 for This money continues to be invested at 8% in the IRA for eight years giving 3040( ) 8 = $ at the end of The Mathematics of a Conversion of a itional IRA to a IRA Spring,

3 For 2012, 19,000(0.08) = $1520. This money continues to be invested at 8% in the IRA for seven years giving 1520( ) 7 = $ at the end of year This should make the balance for the end of 2019 larger than the after tax balance in the itional IRA by = $ The table value is larger than it should be because less than required taxes were reported as paid. But the table illustrated the advantage of postponing payment of taxes and drawing interest on taxes for two years and reinvesting the interest. We can use the $ in the following calculations to reproduce the results in the table for the conversion at end of year For 2010, we have 136,049 38,000 = $98,049. Then 98048(1.08) 9 = 196, Adding 196, = $204, Note that this figure is the same as that given in the table for the end of 2019 balance for the conversion. Completing the Correction of the table. The balance for end of year 2011 should read 136,049( ) = $127,886. The balance for the end of year 2012 should read 127,886( ) = $119,070. Then 119,070( ) 7 = $204,065 instead of $204,232 as in the table. Example 1 If Mr. and Mrs. TM had $5000 deducted from their paycheck before taxes and invested it in a itional IRA and were in the 28% marginal income tax bracket, $1400 would be unpaid customary income taxes, and $3600 would be the customary after tax dollars. If the $5000 was part of a paycheck, the paycheck would be only $3600 less than usual. But with a itional IRA, all of any money withdrawn is taxed at the marginal rate. If they invested $5000 after tax dollars in a IRA, the $5000 after tax is equivalent to 5000 = $ before tax dollars. But, the maximum contribution for a is 1! 0.28 $5000 after tax dollars. The allows an amount equivalent to $ before tax dollars while the itional IRA allows only $5000 before tax dollars. Since after tax dollars are invested in the, there is no tax on money withdrawn from the. Example 2 If $10, was invested in a IRA for twenty years at 7.92% and a 25% income tax rate, the after tax return would be 10,019.30( ) 20 = $46, For a itional IRA, a tax equivalent before tax dollars would be $13, and invested gives 13,359.07( ) 20 (1 0.25) = $46, Notice that when tax equivalent sums are invested, both the IRA and itional IRA pay the same after taxes, other things being equal. The Mathematics of a Conversion of a itional IRA to a IRA Spring,

4 Exercises 1. Following Example 1, give a derivation showing that the after tax return on a IRA is the same as for a itional IRA, when investing tax equivalent dollars. Invest for 20 years at 7.9%, with a 25% marginal income tax rate. 2. Mr. and Mrs. TM have a taxable income on their 1040 income tax form of $97, According to the 2010 Tax Rate Schedule in the Side Bar Notes, a) What is their marginal (highest) income tax rate? b) How much money can they convert from a itional IRA to a IRA without encountering a higher tax bracket? They have to pay taxes on the conversion. c) How much would they pay in income taxes on the conversion? d) Assuming a rate of return of 7.92% on the $40, in the itional IRA, what is the after tax recovery after 20 years at the 25% tax rate? (For simplicity, we are assuming a withdrawal in one year although withdrawals are likely spread over several years thus avoiding a higher tax bracket.) e) If taxes are paid with the IRA money, how much could be deposited in the IRA? f) If the proceeds of $30, are invested in the for twenty years at 7.92%, and withdrawn, how much is the after tax recovery? How was this answer different from the answer to (d)? Discuss. g) It is recommended that the $10, in taxes on the conversion be paid with money that is taxed yearly instead of IRA money, thus investing the total $40, in the. These taxes and their future value are referred to as Opportunity Costs (the money you invest in taxes for the conversion and no longer have it to make you more money.) If the conversion is done at Age 45, and the money would earn 7.92% taxed each year at 25%, what is the future value of the $10, after taxes at Age 65? h) If all of the $40, is invested in the, what is its after tax value at Age 65? i) At the 25% income tax rate, considering at Age 65 the after tax value of the itional IRA, the IRA, and the Opportunity Cost, how much profit did Mr. and Mrs. TM make on the conversion? Why was there a profit? j) Draw a cash flow timeline in terms of money going out and money coming in, to analyze the 25% marginal tax rate conversion, and taxes paid from the annually taxed fund. k) If the $10, in taxes stayed in the itional IRA at 7.92% until Age 65, and was withdrawn at a tax rate of 25%, what was its after tax value? l) If the after tax equivalent of the pretax $10, was invested in a, what was the after tax value. How do the answers to k and l compare? m) One advantage of Conversion to a is to protect from anticipated increases in income tax rates. If by Age 65, they are in the 28% bracket, how much did Mr. and Mrs. TM profit by the conversion? n) If at Age 65, their marginal tax rate is 15%, how much did they lose from the conversion? (For simplicity, we are assuming a withdrawal in one year, although withdrawals could be made over several years resulting in a lower marginal tax rate.) The Mathematics of a Conversion of a itional IRA to a IRA Spring,

5 3. For the following problems, let r be the rate of return on the investments, t be the annual income tax rate, and n the number of years. Derive a formula for S, for the after tax return from investment of P after tax dollars in a IRA. 4. Derive a formula for S, for the after tax return from an investment of pretax dollars equivalent to P after tax dollars, in a itional IRA. 5. Derive a formula for S for an investment of P after tax dollars, and the investment is Tax taxed at the rate t each year. 6. There are investments called Tax Deferred where after tax dollars are invested and tax on the interest is postponed until they are cashed. Examples are Single Premium Tax Deferred Annuities, EE bonds, and I bonds. Derive a formula for the after tax return S Def for a tax-deferred investment. 7. Apply the formulas S, S, S and Tax S Def to $10,000 after tax dollars invested for twenty year at 6% with t = 0.25, when investing tax equivalent dollars. What comparisons can be made? 8. If they invested the $10,000 for 20 years in a Single Premium Tax Deferred Annuity that had an annual insurance security fee of 0.5% subtracted from the 6%, at a 25% tax rate, what was the after return? 9. To make themselves a home-made $100,000 indexed annuity, Mr. and Mrs. TM invested enough in a 20-year 5% zero coupon bond (the interest accumulates in the bond) in their IRA. a) How much did they invest in the bond to get their $100,000 back in 20 years? b) They invested the rest in a stock index fund that averaged 8%. How much did they invest in the stock index fund? c) What was their average rate of return? d) Do the research on what 20-year zero-coupon bonds or Treasury Stips pay. See and click bonds and look at Treasury Strips. e) Check with an insurance salesman about their indexed annuities. 10. In the Conversion Evaluator at they used a 7.9% before inflation return and 5.6% as the after inflation return. What average inflation rate did they assume? Which method did they use to calculate it? The long-term average inflation rate is approximately 3.2%. What do you notice about their number? 11. In the spreadsheet discussed at the beginning of this article, the method of paying the income taxes on the conversion from a itional IRA to a IRA is questionable. Why? The Mathematics of a Conversion of a itional IRA to a IRA Spring,

6 Side Bar Notes The Compound Interest Formula is used extensively in this article: S = P(1 + i) n, where i is the interest rate per compounding period, n is the number of compounding periods, P is the Present Value, e.g., amount invested, S is the Sum, e.g., Future Value of an investment. Paying taxes on a conversion from a itional IRA to a IRA. You can pay the taxes on the conversion with money from outside an IRA or from IRA money. Using outside money to pay your tax bill effectively lets you shift money from a taxable account into a tax shelter and lets you keep more in the IRA s. (Kiplinger s Personal Finance, 11/2010, p. 70) New money must be in a for five years to avoid an early withdrawal penalty. Opportunity costs. When making an investment with your money, consider that you are losing the money, and the return that you would otherwise receive on the money, as an additional cost of the investment. For conversions to a IRA, opportunity costs are often calculated by assuming that assets in a non-retirement, taxable account are used to pay the conversion taxes. Conversion of a itional IRA to a IRA can be done within the institution that holds the qualified (IRA) account. You simply get their advice, complete the forms, and pay the taxes. There are several advantages to a IRA. See the References. You can contribute to a 401k or 403b. IRS 2010 Tax Rate Schedule (See Married filing jointly. Taxable Income Over But not over The tax is Of the amount over 0 16,750 10% $0 16,750 68, % 16,750 68, ,300 9, % 68, , ,250 26, % 137, , ,650 46, % 209, , , % 373,650 It is often said that ten percent of the people pay ninety percent of the income taxes. The Mathematics of a Conversion of a itional IRA to a IRA Spring,

7 References Vest, Floyd. Does TM Need a? HiMAP Pull-Out, Consortium 72. Vest, Floyd. Comparing Taxed, Tax-Deferred, and Tax Sheltered Investments, HiMAP Pull-Out, Consortium 50. Luttman, Frederick W. Selected Applications of Mathematics to Finance and Investment UMAP Module (51 pages of the basics of mathematics of finance, 24 formulas, derivations, examples, exercises, model test, and answers.) Nievergelt, Yves, A Graphic Introduction to Functions, The Federal Income-Tax Law UMAP Module (Considers Average tax rate, Average rate of change, Marginal tax rate, Tax brackets, slope, tangent.) See fidelity.com. Search for conversion to a. See the Conversion Evaluator. See IRS Publication 590, Individual Retirement Arrangements (IRAs), and Instructions for Form 8606 at The Mathematics of a Conversion of a itional IRA to a IRA Spring,

8 Answers to Exercises 1. For a itional IRA with $5000 pretax dollars invested for 20 years a 7.9%, the after tax withdrawal is 5000( ) 20 (1 0.25) = $17, For the IRA, an after tax equivalent of $3750 is invested for an after tax withdrawal of 3750( ) 20 = $17, In terms of tax equivalency both yield $17, after taxes. 2, (a) The marginal tax rate is 25%. (b) They can convert 137,300 97, = $40, (c) Taxes on the conversion = 0.25(40,077.25) = $10, (d) 40,077.25( ) 20 (1 0.25) = $138, (e) 40, , = $30, (f) 30,057.94( ) 20 = $138, This is the same as in (d). (g) FV = 10, = $31, (h) 40,077.25( ) 20 = $184, (i) Profit = 184, , , = $14, There was a profit because taxes were paid with yearly taxed money. (j) Age 45 Age 65 $184, $31, $138, (k) S = 10,019.31( ) 20 (1 0.25) = $34, (l) S = 10,019.31(1 0.25)( ) 20 = $34, (m) S = 40,077.22( ) 20 (1 0.28) = $132, , , , = $19, profit. (n) S = 40,077.22( ) 20 (1 0.15) = $156, , , , = $ They lost $ on the conversion. 3. S = P(1 + r) n where P is after tax dollars. S = $184, P n P S = (1 + r) (1 t), where 1 t 1 t 4. S = P[ r t ] 5. Tax 1 + (1 ) n. is before tax and tax equivalent to P after tax. n S = P (1 + r) (1 t) + t. 6. Def The Mathematics of a Conversion of a itional IRA to a IRA Spring,

9 7. S = 10,000( ) 20 = $32, S = 10,000 1! 0.25 ( )20 (1! 0.25) = $32, S = 10,000 " (1! 0.25) Tax # $ %20 = $24, S Def = 10,000 " # ( ) 20 (1! 0.25) $ % = $26, S = 10,000 " # ( ) 20 (!0.25) $ % = $24, Def 9. (a) P( ) 20 = 100,000. P = $37, (b) This leaves $62, invested in a stock index fund. The future value of the stock investment = 62,311.05( ) 20 = $290, The total = 390, (c) The average rate of return r is such that 100,000(1+ r) 20 = 390, r = = 7.05% average annual return before taxes. r I 10. = inflation adjusted rate of return ! I = I = = 2.18%. 1 + I 1+ I This figure is small since the-long term average rate of inflation is approximately 3.24%. (See The Mathematics of a Conversion of a itional IRA to a IRA Spring,

10 Teachers Notes For the basics of comparing taxed investments, itional IRA investments, and IRA investments, see Does TM Need a? and Comparing Taxed, Tax-Deferred, and Tax Sheltered Investments in the References and in this course. For the basics of Mathematics of Finance see Luttman or Kasting in this course. Students studying mathematics should learn to tolerate and master long multi-step problems. This article presents an example in financial mathematics of a long drawn out problem with precise attention to details. Little two sentence word problems don t give the desired training. Black Boxes. To many people, the spreadsheet discussed in this article is a black box. They don t know the required financial mathematics to analyze it, and often draw the wrong conclusions. Hopefully with financial mathematics education, black boxes will turn into gray boxes or even white boxes. Black boxes are increasingly a problem with common access to computer computation. The Mathematics of a Conversion of a itional IRA to a IRA Spring,