# Present Value Tax Expenditure Estimate of Tax Assistance for Retirement Saving

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1 Preset Value Tax Expediture Estimate of Tax Assistace for Retiremet Savig Tax Policy Brach Departmet of Fiace Jue 30, 1998

4 4 I the first case, cosider the beefit of deferrig oe dollar of icome from age M to N assumig a costat margial tax rate ad costat pre-tax rate of retur o ivestmet. The et proceeds i a future year of savig the dollar i a tax-assisted pla are give by: NP TA = (1 N-M (1 - t) where: NP TA = the et (or after-tax) proceeds of tax-assisted savig i = ivestmet retur N= age whe the proceeds are withdraw M= age whe dollar is cotributed, ad t= margial tax rate. The factor (1 N-M reflects the fact that o tax is paid o the dollar of icome cotributed i year M ad o tax is paid o ivestmet icome, i, durig the years of accumulatio. The factor (1-t) reflects the fact that the gross proceeds are subject to tax whe withdraw from the pla i year N. I cotrast, the et proceeds i a future year of savig the after-tax amout from oe dollar of pre-tax icome ad payig tax o the ivestmet icome each year are give by: NP NTA = (1- t)(1+ (1- t)i) N-M where NP NTA is the et proceeds of o-tax-assisted savig. Here, the factor (1-t) reflects the fact that, out of \$1 of pre-tax icome, oly a et amout of (1-t) is available to be saved after icome tax is paid. The factor (1+ (1- t)i) N-M reflects the fact that the ivestmet retur, i, is subject to tax each year. To arrive at the beefit i the dollars of the year the cotributio is made, i.e., the year correspodig to age M, the future beefit must be discouted by the appropriate discout rate. This raises a iterestig issue, as the primary effect of the program is to chage the rate of retur faced by the household. With access to tax assistace, cotributors face the pre-tax rate. Without access, they ca oly ear the after-tax rate of retur. Which of these discout rates should be used to discout the future beefits is ot iitially clear. Here the equivalece betwee the future beefit to cotributors ad the govermet s tax expediture (see Appedix A) proves useful. The beefit calculated as the differece betwee NP TA ad NP NTA is equal to the value of the govermet s tax expediture expressed i the dollars of the future year whe the cotributor is age N. To fid the preset value of this cost, it must be discouted at the rate of retur faced by the govermet. Sice the govermet borrows ad leds at the pre-tax rate of retur at the margi, this is the appropriate rate at which to discout its cash flows. This provides the aswer to the questio: How much must the govermet borrow today to provide idividuals with a future tax beefit equal to that provided by the tax assistace program? So, the preset value tax expediture is give by:

5 5 = (NP TA - NP NTA )/ (1 N-M or: (a) = (1-t){1-[(1+(1-t)i) N-M /(1+i) N-M ]} To illustrate, cosider the after-tax et proceeds from a oe dollar cotributio out of pretax icome made at age 50 ad withdraw at age 60, assumig a 7% yield o pla fuds ad a federal margial tax rate of 25%: NP TA = \$1(1.07) 10 (.75) = \$1.48 Compare this to the et proceeds of oe dollar of pre-tax icome which is saved for 10 years i a o-tax-assisted accout yieldig 7%. Assume that the after-tax amout of ivestmet icome is re-ivested each year ad that the federal tax rate is 25% i all periods: NP NTA = \$1(.75)(1+ (.75)(.07)) 10 = \$1.25 The preset value beefit that the govermet provides (the tax expediture) of savig i a tax-sheltered istrumet is give by the differece i the et proceeds of the two cases above discouted by the pre-tax rate of iterest: = (NP TA - NP NTA )/ (1 N-M = (\$ \$1.25)/1.97 = \$0.12 I other words, uder these assumptios, \$1 ivested today i a retiremet savigs pla will be worth \$0.23 (= \$ \$1.25) more tha if it were ivested i a o-tax-assisted istrumet. The preset value of that future advatage is \$0.12. Therefore, i the simplified example above, the preset value of the beefit the federal govermet is providig is equal to \$0.12 of each dollar cotributed to a retiremet savigs pla. To illustrate how varies with N-M, the of oe dollar of cotributio is estimated below for differet values of N-M assumig a costat federal margial tax rate of 25%: N-M (\$)

9 9 30 Percet 25 Cotributios Adjusted Payouts Uadjusted Payouts Age (mid-poit of five-year age groups) As the chart shows, the effect of the adjustmet is to weight the age distributio of payouts to higher ages to correct for the bias toward payouts at youger ages cotaied i the curret year s distributio. Assumptio (c) raises a importat procedural questio: should the preset value tax expediture estimates i various years be based o a uchagig set of assumptios about future rates of retur, tax rates ad payout distributios (eve if these may seem urealistic i light of curret experiece), or should they reflect chages i these variables? Our view is that the estimates should reflect chages i average margial tax rates, payout profiles, past cotributio growth rates ad average rates of retur. May of these parameters will chage oly slowly ayway, but the idea would be to base the updated estimate for each successive year o the movig average value of these parameters. 4. Presetatio of the Estimates (a) Estimates of the of the tax assistace for retiremet savig provided by the federal govermet for 1993, 1994 ad 1995 are foud i Table 1. The assumptios used for these calculatios are also show. AMTR <65 is the average federal margial tax rate o cotributios to RPPs ad RRSPs. AMTR 65+ is the average federal margial tax rate applied to icome received from those plas. The preset value tax expediture estimates are show i the shaded colum. The fial colum shows the cash-flow estimate for compariso.

10 10 Table 1 Preset Value Tax Expediture Estimates Year Average rate of retur AMTR <65 AMTR 65+ RRSP RPP Total Cash-flow (%) (\$ billio) Dividig the estimate by the aggregate level of cotributios provides a measure of the percetage of the cotributios that is subsidised through the tax system. This is show below i Table 2. To measure the total subsidy to the taxpayer, provicial icome tax must be icluded i the calculatio. Takig 50% as represetative of the provicial tax rate, the percetage subsidies for the three years rise oly slightly. This happes because the icrease i expected future tax reveues that will be collected whe fuds are withdraw partially offsets the icrease i curret forgoe taxes. Table 2 Tax Expediture as Percetage of Cotributio Year Prov&Fed Cotributios Federal Subsidy Combied Subsidy (\$ Billios) (%) (b) To assess the sesitivity of the to the rate of retur, the estimates were recalculated usig differet rates of retur. The results are preseted below i Table 3.

11 11 Table 3 Sesitivity Aalysis of to Chages i Rates of Retur Year 6% 7% 8% (\$ billio) 1996 These estimates show that the estimate is quite sesitive to the assumed rate of retur. A oe percetage poit icrease i the rate of retur icreases the by roughly 8%. (c) The estimates may also be sesitive to chages i the tax rate assumptios. Preseted below i Table 4 is a sesitivity aalysis of chages i the tax rates for I all cases, oly the oted parameter has bee chaged. All others maitai their bechmark settigs. Table 4 Sesitivity of 1996 Estimates to Tax Rates Parameter chaged Magitude of chage (%) (\$billio) Bechmark ** AMTR <65 +1 AMTR <65-1 AMTR AMTR (c) I order to check the sesitivity of the results to the profile of payouts, it will be ecessary to recall the aalysis o page 5. That aalysis showed the tax expediture per dollar of cotributio for various differet deferral periods. For all years, the average deferral period was close to 20 years, weightig by cotributios. The calculatios from page 5 are recreated here, usig 19, 20, ad 21 year average deferral periods i order to idicate the sesitivity of the estimates to the patter of payouts.

12 12 Table 5 Sesitivity to Average Deferral Period Average Deferral Chage From Base Case (years) (cets) (%) (e) Oe of the key assumptios uderlyig both the cash-flow ad preset value tax expediture estimates preseted i this report is that omial icome would remai subject to tax i the bechmark system. I order to gauge the sesitivity of the preset value estimate to this assumptio, the was re-calculated assumig that oly real icome was subject to tax. The preset value formula that icorporates a bechmark system where oly real icome is taxed is as follows: N = (1- t N ) - [(1-t M ) Π (1+ (1-t j (r j /i j ))i j ) / Π (1+ i j )] j=m+1 j=m+1 N where: i j = the omial rate of retur i period j, as before, ad r j = the real rate of retur i period j. Based o a historical aalysis of real versus omial rates of retur from 1944 to 1993, the ratio of r j / i j is assumed to be The results usig a real iterest taxatio bechmark system are preseted i Table 5. Table 6 Usig Real Iterest Taxatio Bechmark omial real Federal subsidy Combied subsidy 1996

13 13 5. Relatig Preset Value Estimates to Cash Flow Estimates The preset value estimates are smaller tha cash flow estimates. There are two reasos uderlyig this discrepacy. The first reaso relates to the curret state of maturity i the Caadia system, as discussed o page 1. To uderstad the differece of preset value ad cash flow approaches, cosider a case i which the followig coditios hold: a) age-specific cotributio patters are fixed b) populatio growth plus icome growth are costat ad equal to the discout rate. c) tax rates are costat I this case, cotributios i each year would be equal to withdrawals. This meas that accoutig for cotributios ad withdrawals usig either method would deliver the same result. Presetly i Caada, these coditios fail to hold. First, age-specific cotributio patters have chaged through the years. Cotributio levels of today s 40 year-olds are higher tha the 40 year-olds of a few decades ago. Secod, the baby boom geeratio is curretly i its peak cotributio years. This bulge i populatio meas that curret cotributios are higher tha they will be i a few years whe the baby boom geeratio begis to retire. So, both of these deviatios from the coditios outlied above poit to a overstatemet of the tax expediture usig the cash flow method. The secod reaso for the differece betwee cash flow ad preset value tax expediture estimates relates to the treatmet of the tax o ivestmet icome. I the cash flow method, the tax expediture is calculated by applyig the tax rate to the iterest icome that would have bee eared o the existig stock of assets i tax-assisted savigs accouts. This stock of assets has bee growig o a tax-deferred basis. I cotrast, the preset value tax expediture approach looks at the forgoe tax reveue o iterest icome durig the duratio of a particular cotributio. I calculatig what tax would have bee collected without tax assistace, the cotributio would be growig o a after-tax basis. The overall size of the base o which tax is applied grows more slowly. Because the base grows more slowly, the preset value approach delivers a lower estimate of tax expediture tha the cash flow method. This meas that, eve i a mature system with stable demographics, the cash flow estimate will always exceed the preset value estimate, give positive tax rates. This differece will grow with the size of the tax rate.

14 14 Appedix A: Proof of Equivalece of Beefit ad Forgoe Tax Approaches The value of the beefit of tax assistace to a cotributio of oe dollar years i the future uder costat tax rates was show i Sectio 2 to be: (A1) NP TA NPNTA = ( 1 t)(1 (1 t)[1 + (1 t) i] The tax cost to the govermet of providig tax assistace is preseted below i Table A-1. For each iitial dollar of cotributio, the govermet must give up t dollars i reveue. The future value of this amout periods hece is foud by multiplyig the amout t by (1+i), sice the govermet borrows ad leds at the pre-tax rate of iterest. At the ed of the first period, the cotributor would have eared taxable ivestmet icome of i o the pricipal of (1-t). The forgoe tax reveue for period 1 is t times this taxable iterest icome, which leads to the term that appears for period 1 i the table. This is the brought forward to period dollars by multiplyig by (1+i) -1. The pricipal to start the secod period would be (1-t)[1+(1-t)i]. Agai, it ears iterest i by the ed of the period ad would have bee taxed at the rate t. This patter repeats itself util reachig period. I period, the tax expediture must accout for both the forgoe iterest paymet, as well as the receipt of taxes upo withdrawal from the tax assisted ivestmet. Accordigly, the tax receipt (i period * i the chart) is subtracted from the measure of cost. Table A-1 The Curret ad Future Values of Forgoe Tax Reveues Period Curret value Future value 0 t t(1+i) 1 ( 1 t) it 1 2 ( 1 t)[1 + (1 t) i] it 2 3 ( 1 t)[1 + (1 t) i] it (1 t ) it(1 1 1 (1 t )[1 + (1 t) i] it(1 2 (1 t )[1 + (1 t) i] it( (1 t)[1 + (1 t) i] 1 it 1 ( 1 t)[1 + (1 t) i] it(1 + i ) 0 * t ( 1+ i) t ( 1+ i) The iitial forgoe reveue i period 0 ad the fial tax receipt i period * cacel with each other. This leaves the forgoe reveue o ivestmet icome i periods 1 through

15 15 to fully accout for the tax expediture. Fortuately, this sum takes the form of a geometric series, which simplifies the aalysis. The sum of a geometric series with k terms is give by: (A2) S k = a(1 g 1 g k ), where a is the iitial term for the series, g is the rate of growth, ad k is the umber of terms. From ispectio of Table A-1, it ca be see that a = (1 t) it(1 1, 1+ (1 t) i g = 1 i +, k =. Substitutig for a, g, ad k i (A2) gives (A3) S k = [(1 t) it( (1 t) i ] 1 (1. 1+ (1 t) i 1 1+ i The deomiator simplifies to a simple term that cacels with parts of the a term i the umerator, leavig: S k = [(1 t) it( (1 t) i ] 1 (1 it 1+ i = (1 t)(1 [1 + (1 t) i] 1 (1 Expadig what remais through the large brackets leads to (A4) S ] k = ( 1 t)(1 (1 t)[1 + (1 t) i. The expressio for S k i (A4) is exactly the same as the calculated future beefit to the cosumer i (A1). This meas that calculatios usig the future beefit ad those usig forgoe tax reveues ca be substituted freely for each other wheever coveiet.

16 16 Appedix B: Estimates Usig After-tax Discout Rate Below are preseted the preset value estimates usig the after tax rate of retur to discout the future beefit to cotributors. This estimate measures the preset value beefit to idividuals of the tax assistace program, as opposed to its cost to the govermet. It aswers the questio: How much would idividuals have to ivest at the after-tax (or o-sheltered) rate of retur to provide themselves a future beefit equal to the beefit provided by the existece of the program? The methodology for calculatig this estimate is exactly the same as that for measurig the govermet s cost, except that the after-tax discout rate (1+(1-t)i) N-M is used to calculate the preset value of the beefit to idividuals. Table B-1 Estimates Usig After-tax Discoutig Year (after-tax discoutig) (after-tax discoutig, real taxatio bechmark) (\$ billio) (\$ billio)

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