$IR A. Does T. M. Need a Roth?

HIGH SCHOOL MATHEMATICS AND ITS APPLICATIONS CONSORTIUM HIGH SCHOOL 72 MATHEMATICS AND ITS APPLICATIONS HIGH SCHOOL MATHEMATICS ROTH HIGH SCHOOL MATHEMATICS AND ITS APPLICATIONS HIGH SCHOOL MATHEMATICS AND ITS APPLICATIONS HIGH SCHOOL MATHEMATICS HIGH SCHOOL MATHEMATICS AND ITS APPLICATIONS HIGH SCHOOL MATHEMATICS AND ITS APPLICATIONS HIGH SCHOOLDoes MATHEMATICS T. M. AND ITS APPLICATIONS HIGH SCHOOL MATHEMATICS Need aand ITS APPLICATIONS HIGH SCHOOL MATHEMATICS AND ITSRoth? APPLICATIONS HIGH SCHOOL MATHEMATICS AND ITS APPLICATIONS HIGH SCHOOL MATHEMATICS AND ITS APPLICATIONS HIGH SCHOOL MATHEMATICS Floyd Vest APPLICATIONS HIGH SCHOOL MATHEMATICS AND ITS APPLICATIONS $IR A HIGH SCHOOL MATHEMATICS AND ITS APPLICATIONS HIGH SCHOOL Does T. M. Need a Roth? Dr. Floyd Vest is Professor of MATHEMATICS AND ITS APPLICATIONS HIGH SCHOOL MATHEMATICS Mathematics and Education at the University of North Texas. He years and enjoys applications of HIGH SCHOOL MATHEMATICS AND ITS APPLICATIONS Pull-Out HIGH Section, SCHOOL write to him at: Dr. Floyd Vest, MATHEMATICS AND ITS APPLICATIONS HIGH SCHOOL MATHEMATICS Dept. of Mathematics, NOTE TO THE TEACHER. University of North Texas, AND ITS APPLICATIONS HIGH SCHOOL MATHEMATICenton, AND TX 76203-5116. ITS Your students may need a review of the Compound Interest Formula before they do this HiMAP Pull-Out. For additional activities, have students bring to class and apply mathematics to articles in publications such as The Wall Street Journal, Money Magazine, or other has been teaching math for 35 secondary school mathematics. To submit ideas for the HiMAP HIGH SCHOOL publications MATHEMATICS advertising different types of AND IRAs. Invite ITS a speaker APPLICATIONS to talk HIGH SCHOOL to your class on finance. If you would like more lessons on financial MATHEMATICS mathematics, ANDconsider ITSHigh APPLICATIONS School Lessons in Mathematical Applications, HIGH SCHOOL MATHEMATICS COMAP, 800-772-6627. (Contains 27 HiMAP Pull-Outs with exercises, AND ITS APPLICATIONS including 11 on financial mathematics). HIGH SCHOOL MATHEMATICS AND ITS HIGH SCHOOL MATHEMATICS AND ITS APPLICATIONS HIGH SCHOOL MATHEMATICS AND ITS APPLICATIONS HIGH SCHOOL MATHEMATICS

2 Dallas Morning News, February 4, 1993 Dear Scott Burns: I am 17, in high school and have a part-time job.... I have been able to save $250 a month in my credit union. My dad says that if I put $2000 into an IRA for 10 years, it will be worth $1 million by the time I retire at age 65. Sincerely, T. M., Garland, Texas. Dear T. M.: Your dad is right.... These figures make the assumption that your money can grow tax-free or tax-deferred. Sincerely, Scott Burns, Financial Columnist. IRAS ON THE INTERNET. See the following internet sites (3/98): http://funds.scudder.com Click on Investment Resourses, then IRA Center, then Roth. www.troweprice.com This site offers interactive work sheets that help you make educated decisions about IRAs. http://www.vanguard.com Click on Roth IRA Navigational Guide. This guide has work sheets and downloadable forms. http://www32.fidelity.com.80/ Click on Learn how the new Roth IRA could benefit you. This page compares the after tax accumulation of the Roth IRA with that for other types of IRAs. This site offers interactive work sheets that help you project how much you could have at retirement if you contribute to a Roth IRA vs. a Traditional IRA. See IRA Evaluator. http://www.worth/com Click on Search. Type in Roth IRA. www.strong-funds.com Click on Roth IRA. This site provides interactive comparisons of the returns from different IRA s. See if you can verify the figures given by these various sites. In the letter shown to the left from the 17 year-old T. M., he discusses putting $2000 into an IRA. The Taxpayer Relief Act of 1997 created new investment IRAs for T. M. and others. As a result of this new law, T. M. is now faced with more choices of IRA investing. Roth is the hot new name in IRAs. T. M. can choose between a traditional deductible IRA and a Roth IRA. What do these two terms mean? (See the HiMAP Pull-Out Section, Comparing Taxed, Tax Deferred, and Tax Sheltered Investments, Consortium, Summer 1994, for discussion of more types of IRAs.) Traditional tax deductible IRA. Our friend T. M. might consider a tax deductible IRA because of an impressive advantage for savings. The trick is that T. M. would be allowed to invest yearly up to 2000 before-tax dollars. Let us discuss after-tax dollars and before-tax dollars. When most people get their paycheck, it amounts to after-tax dollars. The income tax has been taken out of their earned income. For example, if T. M. earned $100 before-tax dollars, and his income tax rate is 15%, then 15% would normally be paid in taxes, and $85 would be the after-tax dollars left after paying $15 in taxes. If T. M. invests the entire $100, which includes taxes and after-tax dollars, he has invested before-tax dollars. In addition for a paycheck from which 100 before-tax dollars is deducted and invested, the paycheck would only be $85 less than usual, even though $100 dollars was invested. All of the $100 would earn interest and grow in a tax deductible IRA. So both the $85 and $15 are working for the investor. However, as you would expect, when a tax deductible IRA is cashed, you have to pay income taxes on all of the money withdrawn. If P/(1 r) before-tax dollars are invested in a tax deductible program where the current income tax rate is r, then this is equivalent to investing P after-tax dollars. To see this, let the tax rate r = 0.15, P = 85 after-tax dollars, and x = before-tax dollars. Then, P = 85 after-tax dollars = (before-tax dollars) r(before-tax dollars) = x rx = x(1 r). HiMAP Pull-Out Section: Winter 1999 (1) So, P = x (1 r) and x = P/(1 r) for P after-tax dollars, for x = P/(1 r) before-tax dollars, and a tax rate of r. In this example, where r = 0.15 and P = 85 after-tax dollars, the equivalent before-tax dollars is x = 85/(1 0.15) = $100. Investing 100 before-tax dollars is equivalent to investing 85 after-tax dollars because a take-home paycheck is depleted the same in both cases.

3 HiMAP Pull-Out Section: Winter 1999 For the derivation of a formula for the after-tax accumulation of a tax-deferred IRA, consider investing P/(1 r) before-tax dollars (which is the before-tax amount equivalent to an after-tax amount of P dollars) in a tax deductible IRA for n years at the interest rate i D per year. By the Compound Interest Formula, the accumulation P in the fund would be (1 + i D ) n. However, upon cashing the 1 r investment, taxes would be paid at some rate r on all of the accumulation. Therefore, the after-tax value would be P = (1 + i D ) n P r[ (1 + i D ) n ] so that 1 r (2) = P (1 + i D ) n for P after-tax dollars, P/(1 r) before-tax dollars, a tax rate of r, a before-tax rate of return i D, and for n years. For example, if T. M. invests P/(1 r) = 2000 before-tax dollars (equivalent to 1700 after-tax dollars), for 48 years at the annual rate of 7%, with the tax rate at deposit and withdrawal of 15%, then 1700 = (1 + 0.07) 48 0.15 1700 1 0.15 1 0.15 1 + 0.07 accumulation taxes = 1700(1 + 0.07) 48 1 r ( )48 SENATOR ROTH Senator Roth (R-Del.) was a principal sponsor of the Roth IRA provisions of the Taxpayer Relief Act of 1997 and chairs the Senate Finance Committee. He has served five terms in the Senate. People are beginning to understand the need for savings in America. It s critically important to the family and it s critically important to the nation, says Roth. EDUCATION IRAS. This new type of IRA allows nondeductible annual contributions up to $500 per beneficiary who is under age 18. Earnings accumulate tax-free and can be withdrawn tax free if used solely for the beneficiary s qualified higher education expenses. Call 1-800-TAX-FORM (1-800-829-3676) and ask for a free copy of IRS Publication 590 Individual Retirement Arrangements. Every investor should obtain an updated copy of this owners manual for IRAs. = $43,739.14. (At this stage, some readers may be concerned that the tax rate at time of deposit may not equal the tax rate at the time of withdrawal. For this and more advanced considerations, they may see the following You Try Its or consider constructing their own models.) Roth IRA. This new type of IRA was available starting January 1, 1998. A Roth IRA is purchased with after-tax money. But the money grows taxfree and no taxes are charged against withdrawals (if T. M. s Roth IRA has been established for at least five years and T. M. is past 59 years of age). T. M. would be interested in comparing these two types of IRAs on two counts: (1) Which will maximize the ultimate after-tax pay out? (2) Which yields the greatest after-tax rate of return on the investment? For a derivation of the after-tax accumulation for the Roth IRA, let r = tax rate, and P = the amount of after-tax dollars invested, at the rate of return i R. Then by the compound interest formula (3) S R = P (1 + i R ) n is the accumulation after n years on which no taxes are owed. Comparing Formula (2) for the traditional tax deductible IRA and Formula (3) for the Roth IRA, we see that if i D = i R, then they both produce the same after tax accumulation for P after-tax dollars invested in the Roth and a tax equivalent P/(1 r) before-tax dollars invested in the traditional tax deductible IRA. For example,

4 HiMAP Pull-Out Section: Winter 1999 $2000 before tax invested in the deductible IRA is equivalent to $1700 after tax invested in the Roth IRA so that WHO CAN INVEST IN A ROTH IRA? IRA stands for Individual Retirement Account. Beginning January 1, 1998, most people with income can invest in a Roth IRA. A Roth IRA can even be purchased for a non-working spouse. Funds may be withdrawn without penalty for post secondary education expenses, or beginning at age 59 1/2, and under certain other conditions. (See IRS Publication 590, Individual Retirement Arrangements. ) THE COMPOUND INTEREST FORMULA. For P dollars invested at the interest rate i per year for n years, the accumulated principal and interest is S = P(1 + i) n. This formula is taught in mathematics classes from Grade 7 up. See if you can prove the compound interest formula. KIDDIE IRA (CNN 3/12/98). Have your child deposit $2000 per year in an IRA, from age 15 to 21, and at this time stop adding deposits. By age 65 and at 8%, they will have $569,000. Do the calculations to confirm these figures. INVESTING FOR KIDS 8 TO 17. The First Start Growth Fund from USAA (800-531-0553) features a low minimum investment of $250. Plus it provides newsletters and interactive games for young people. Their e-mail shows that youth have money, and are interested in saving (Mutual Funds Magazine, March 1998, p. 20). = 2000(1 + 0.07) 48 0.15(2000)(1 + 0.07) 48 = 1700(1 + 0.07) 48 = $43,739.14; and S R = 1700(1 + 0.07) 48 = $43,739.14. If T. M. invests 2000 after-tax dollars in a Roth IRA, the $2000 after tax is equivalent to 2000/(1 0.15) = 2352.94 before-tax dollars. But, this last observation indicates an important consideration. Since the maximum allowable yearly IRA investment is $2000 for any type of IRA, and since $1700 after-tax dollars is equivalent to $2000 beforetax dollars, the Roth IRA allows T. M. to maximize his final aftertax accumulation. For example, using maximum allowable investment dollars, and i D = i R = 0.07, with A R and A D denoting after-tax recoveries, A R = 2000(1 + 0.07) 48 = $51,457.81, from Formula (3), and A D = 2000(1 + 0.07) 48 0.15(2000)(1 + 0.07) 48 = 1700(1 + 0.07) 48 = $43,739.14. In this example, by using mathematics and the Roth IRA, T. M. can increase his after-tax accumulation for a single year s IRA by $7718.67, or by 17.6% over the traditional deductible IRA when compared on investing maximum allowable but not tax equivalent dollars. Do the following You Try Its. Write complete sentences. Label all answers. You Try It #1 Show how a $300 paycheck (before deductions) is depleted the same by a $100 before-tax deduction as by a $85 after-tax deduction. Use a tax rate of r = 0.15. What number is equivalent to what other number and in what way? You Try It #2 In the examples in this HiMAP Pull-Out, we calculated = 1700(1 + 0.07) 48 = $43,739.14 for a single year s tax deductible IRA of $2000 before taxes. Show a derivation using $2000 before taxes and subtracting taxes that gets the same answer. You Try It #3 In the examples in this HiMAP Pull-Out, we calculated = 1700(1 + 0.07) 48 = $43,739.14 for a single year s tax deductible IRA of $2000 before-tax dollars. (a) Can more money be invested in that year s IRA in order to increase the final accumulation? (b) Given the same time, tax rate, and rate of return, is there any way a tax deductible IRA can earn more than a $2000 Roth?

5 HiMAP Pull-Out Section: Winter 1999 You Try It #4 The following is a quote from a magazine article about the Roth IRA. Show the calculations and explain why this is correct. The Roth is a better deal for eligible savers because it permits what are, in effect, larger pre-tax contributions. For example, for a tax payer in the 28% bracket, a $2000 nondeductible contribution to a Roth IRA represents $2778 of pre-tax earnings. (Mutual Funds Magazine, November 1997, p. 28.) You Try It #5 (a) Thousands of investors have read the following quote. Do the calculations and explain why this is correct. Say you... are in the 28% bracket.... If you make a single $2000 deductible contribution to a traditional IRA that appreciates at 10% a year for twenty years, your account will grow to $13,455. After paying 28% tax on the ultimate distribution of that sum, you are left with $9688 after taxes. To fairly compare that with a Roth IRA, you must assume that you start with the same $2000 of earned income and pay taxes at the 28% rate, leaving you with $1440 after taxes for investing in the Roth. After twenty years of 1 10% growth your $1440 grows to the same $9688, which you can distribute to yourself tax-free after age 59. 2 (Mutual Funds Magazine, January 1998, p. 52.) (b) Use a graphing calculator or a mathematical software package to graph the growth of these two investments over the 20 years. Label and explain your graph. You Try It #6 Do the calculations and explain how the following is correct: This subtle distinction between a conventional IRA and a Roth IRA can make a big difference. If you make a $2000 deductible contribution to an ordinary IRA at 10% for 20 years, the account will grow to $13,455. After paying taxes at 28%, you are left with $9688. On the other hand if you contribute $2000 after taxes to a Roth IRA at 10% for 20 years, the account will grow to $13,455 on which no taxes are owed. Thus the Roth allows greater after-tax accumulation. You Try It #7. Show the calculations and explain why the following quote is correct: On the other hand if you are in a lower tax bracket during retirement, the traditional IRA will generally be better.... For example, if your tax rate is 28%, and if you drop to a 15% tax bracket during retirement, your $2000 deductible contributions to a traditional IRA at 10% after 20 years will net you $11,437 after taxes nearly $2000 more than the equivalent $1440 at 10% after 20 years yielding $9688 from a Roth. (Mutual Funds Magazine, January 1998, p. 52.) You Try It #8 Do the calculations and explain that the following quote is correct: This subtle distinction between a conventional IRA and a Roth IRA can make a big difference over time. If you make a $2000 deductible contribution to an ordinary IRA at the beginning of each year for thirty years and grow your account 10% a year, you will retire with a $361,887 nest egg. After paying 28% taxes on that amount when it is distributed to you, you are left with $260,559. By contrast thirty $2000 nondeductible contributions each year to a Roth IRA also grows to $361,887, but that sum can be distributed to you entirely tax free. (Mutual Funds Magazine, November 1997, p. 28.)

6 HiMAP Pull-Out Section: Winter 1999 You Try It #9 Prove the following three theorems: For tax equivalent amounts invested, and for K before-tax dollars, and for the tax rate at time of withdrawal r w, where 1 > r w > 0, and for the tax rate at the time of deposit r d, where 1 > r d > 0, n > 0, and i = i D = i R, 0 < i < 1, then for the after-tax accumulations A D and A R, (a)a D = K (1 + i) n (1 r w ), (b) A R = K (1 r d )(1 + i) n and (c) if r w < r d, then A R < A D. (d)write interpretation of part (c). You Try It #10 (a) Prove the following theorem: For a single year s Roth IRA and a single year s deductible IRA, both invested at the rate i, 0 < i < 1, with r w = tax rate at the time of withdrawal, r d = tax rate at the time of deposit, and maximum (not tax equivalent) dollars invested in each, for any r w > 0, the after-tax accumulation for the Roth IRA is greater than that for the deductible IRA. (b) Write an interpretation of this theorem. You Try It #11 Devise and demonstrate a realistic method of comparing a single year s Roth IRA and a tax deductible IRA where (a) maximum dollars are invested in the Roth, (b) the comparison includes an investment program with a tax deductible IRA and a taxable investment to supplement it where the total dollars invested in the two parts are tax equivalent to those in the Roth. There are two methods that the author has constructed. He discovered that the above procedure was published in http://personal32.fidelity.com/. It involves supplementary savings invested and taxed annually, the other involves finding an additional program in which to invest the supplemental funds in a tax deductible and tax deferred mode. You Try It #12 For the following examples based on formulas in You Try It #9, use a three-dimensional graphing package to graph the following planes on the r w, r d, A R A D axes. For each plane, identify the intersection with the r w, A plane, the r d, A plane, and the r w, r d plane: (a) A D = 2000(1 + 0.07) 48 (1 r w ), (b) A R = 2000(1 + 0.07) 48 (1 r d ). (c) Then graph both planes on the same axes. Discuss the relationships between the planes when r w = r d, r w < r d, and r w > r d. Identify the intersection of the planes. Discuss the financial significance of these relationships.

7 Activity Answers HiMAP Pull-Out Section: Winter 1999 1 2 3 a b 4 5 a 6 7 The $100 before-tax deduction gives $300 100 = $200. Then, they pay tax at 15% on the remainder. So the after-tax amount is 200(1 0.15) = $170. The paycheck is depleted by $130. For the $85 after-tax deduction: They pay 15% taxes on the $300 leaving 300(1 0.15) = $255. Deduct $85 after tax to get 255 85 = $170. The paycheck is depleted by $130 in both cases. The 100 before-tax dollars is equivalent to the 85 after-tax dollars. Investing $2000 before tax dollars gives = 2000(1 + 0.07) 48 0.15[2000(1 + 0.07) 48 ] = 2000(1 + 0.07) 48 (1 0.15) = $43,739.14. This is the same answer as in the examples, but by using $2000 before taxes and subtracting the income tax paid at withdrawal. No. This calculation assumes a before-tax investment of $2000 in the tax deductible IRA, which is the maximum the government will allow for a single year s IRA. Formula (2), = P (1 + i D ) n, is based on P after-tax dollars equivalent to P/(1 r) before-tax dollars. So, if P/(1 r) = $2000 before-tax dollars is invested in the tax deductible IRA, then P = $1700 in this example. No. For a 28% bracket, given $2778 pre-tax, you pay 28% tax, which is 0.28(2778) = $777.84. So $2778 777.84 = $2000.16 is the after-tax amount or nondeductible amount, which is equivalent to $2778 pre-tax or the deductible amount. But the maximum allowable contribution for a deductible IRA is $2000 pre tax. So in effect, the Roth permits a contribution ($2000 after tax) that is equivalent to a pre-tax amount ($2778), which is larger than the maximum allowed for the deductible IRA. For the 28% bracket, after 20 years, you have 2000(1 + 0.10) 20 = $13,455 for the deductible IRA. After paying taxes, you have 13,455(1 0.28) = $9688. To fairly compare the Roth, $2000 pre tax is equivalent to 2000(1 0.28) = $1440 after taxes. After 20 years, you have 1440(1 + 0.10) 20 = $9688 that is tax free. For tax equivalent dollars, both IRAs yield the same after-tax accumulation in 20 years. Thus they both yield the same after-tax rate of return. If you make a $2000 deductible contribution to an ordinary IRA earning 10% for 20 years, you will have 2000(1 + 0.10) 20 = $13,455. At 28%, you have 13,455(1 0.28) = $9688 after taxes. For the Roth, contributing $2000 after taxes yields 2000(1 + 0.10) 20 = $13,455 on which no taxes are owed. The Roth allows what is effectively a greater contribution (when compared on tax equivalent dollars) and thus a greater after-tax accumulation in 20 years. Your $2000 deductible contribution to a traditional IRA grows to 2000(1 + 0.10) 20 = $13,455. Paying 15% taxes at withdrawal gives 13,455(1 0.15) = $11,437. The tax equivalent $1440 for 20 years at 10% in the Roth gives 1440(1 + 0.10) 20 = $9688. The difference is 11,437 9688 = $1749 greater for the traditional under the circumstance of a lower tax rate at time of withdrawal of 15%. For the tax deductible IRA, the writer refers to the $1749 more as nearly $2000 more.

8 Activity Answers HiMAP Pull-Out Section: Winter 1999 8 9 a b c d 10 a b 12 c If you make a $2000 deductible contribution to an ordinary IRA at the beginning of each year for 30 years at 10%, you have 2000 ( 1.10) 30 1 (1.10) = $361,886.85, 0.10 using the formula for the sum of a geometric sequence or the formula for the sum of an annuity due. After 28% taxes, you have 361,886.85(1 0.28) = $260,558.53. By contrast, thirty $2000 nondeductible contributions for 30 years at 10% to a Roth gives 2000 ( 1.10) 30 1 (1.10) = $361,886.85 0.10 with no taxes owed. The Roth yields 361,886.85 260,558.53 = $101,328.32 more. Proof: Let K = amount of before-tax dollars. For the deductible IRA, invested at the rate i for n years where the tax rate at time of withdrawal is r w, A D = K(1 + i) n (r w )K (1 + i) n, so A D = K(1 + i) n (1 r w ). Proof: For the Roth IRA invested at the rate i for n years where the tax rate at time of deposit is r d, K (1 r d ) after-tax dollars is tax equivalent to K before-tax dollars, so that A R = K (1 r d )(1 + i) n is the after-tax accumulation. Proof: Given r w < r d. Then r w > r d, 1 r w > 1 r d, K(1 + i) n > 0, K(1 + i) n (1 r w ) > K(1 + i) n (1 r d ). Therefore A R < A D for tax equivalent dollars invested. (See if you can justify each step in this proof.) If tax rates drop at the time of withdrawal, when compared on tax equivalent dollars, the tax deductible IRA is the best deal. For the deductible IRA, the after-tax accumulation A D = 2000(1 + i) n (1 r w ). For the Roth IRA, the after-tax accumulation A R = 2000(1 + i) n. For r w > 0, r w < 0, 1 r w < 1, 1 > 1 r w, 2000(1 + i) n > 0, 2000(1 + i) n > 2000(1 + i) n (1 r w ). So A R > A D. Therefore for any r w > 0, where maximum (not tax equivalent) dollars are invested in both, the after-tax accumulation for the Roth IRA is greater than that for the deductible IRA. The r d does not play a role in this result. (See if you can justify each step in this proof.) Even if tax rates at the time of withdrawal drop, when compared on maximum possible deposits, the Roth IRA is the best deal. If r w = r d, then the A D and A R planes intersect and both IRAs yield the same after-tax return. If r w > r d then the A D plane is under the A R plane and the Roth IRA is the best deal. See the Maple code and graph below. > restart: with (plots): > planel := plot3d(2000*1.07^48*(1 rw), rw=0.. 1, rd=0.. 1, axes=normal, orientation=[19,45]): > plane2 := plot3d(2000*1.07^48*(1 rd), rw=0.. 1, rd=0.. 1, axes=normal, orientation=[19,45]): > line := spacecurve ([t, t, 2000*1.07^48*(1 t)], t=0.. 1, axes=normal, color=black): > display ({plane1, plane2, line});