Present Value Tax Expenditure Estimate of Tax Assistance for Retirement Saving


 Phoebe Stevens
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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
2 2 Preset Value Tax Expediture Estimate of Tax Assistace for Retiremet Savig This paper outlies a method for derivig preset value estimates of tax assistace for retiremet savig ad provides estimates based o alterative assumptios. The paper is divided ito four sectios. Sectio 1 discusses the ratioale for providig a preset value estimate of the tax expediture associated with the tax treatmet of retiremet savig. Sectio 2 elaborates a methodology for such a estimate. Sectio 3 documets the key assumptios used i applyig the methodology to Caadia data. Fially, sectio 4 provides, for the years 1993 to 1996, base estimates as well as alterative estimates that reflect differet assumptios about iterest rates, tax rates, ad the bechmark tax system. 1. Ratioale for a Preset Value Estimate As oted i recet Tax Expeditures reports, tax expediture cocepts are arbitrary i several respects ad must be treated with cautio. For example, the estimates deped o the bechmark tax system agaist which the removal of tax provisios is measured. Tax expediture reports assume that the bechmark tax system is oe i which omial icome is taxed. It is also assumed that there is o behavioural respose to the removal of tax provisios. Tax expediture estimates would be reduced if they were based o a bechmark tax system that assumes oly the oiflatio compoet of ivestmet icome (i.e., real icome) were taxed or if behavioural resposes were take ito accout. Additioal cosideratios arise where provisios allow for tax deferrals  as with tax assistace for retiremet savig (TARS)  rather tha a permaet tax exemptio. The cashflow tax expediture estimate of TARS preseted i Tax Expeditures reports is obtaied by calculatig: the reveue that would be gaied if the cotributios made i the year to retiremet savigs plas, ad the iterest eared i the year i those plas, were taxable, mius the reveue that would be lost if the beefits paid out from retiremet plas i the year were ot taxable. I additio to the caveats regardig the bechmark tax system ad behavioural respose, this estimate has other shortcomigs. I particular, the size of the estimate depeds directly o the maturity of the retiremet savigs system ad the relative size of the workig ad retired populatios. For example, i the early years of a pesio system cotributios ted to be high relative to beefit payouts whereas uder a mature system total beefits will exceed total cotributios. Thus, cash flow estimates of tax assistace may be expected to declie (as a proportio of total earigs) as the pesio system matures.
3 3 Accordigly, it should be useful to supplemet the cashflow tax expediture estimate with estimates based o aother approach. The preset value approach focuses o the pesio ad RRSP cotributios made i a year ad estimates the amout of tax assistace associated with each dollar of those cotributios util it is withdraw. Thus it igores tax deferrals associated with cotributios made i past years; istead, it estimates the tax assistace associated with curret pesio ad RRSP cotributios i a way that explicitly takes ito accout the prospect of cotiuig the tax deferral o those cotributios i the future as well as their evetual taxatio as they (ad the associated ivestmet earigs) are paid out of the plas. The preset value approach starts with the followig questio: what is the reveue cost to the govermet of oe dollar of icome saved i a taxassisted pla compared to a dollar saved i a otaxassisted istrumet? I cotrast, the cashflow approach attempts to estimate the amout of taxassistace provided o retiremet savig i a give year. The preset value approach attempts to estimate the amout of tax assistace that will be provided o a give year s cotributios over the lifetime of those cotributios. The Auditor Geeral has oted the cotroversy about tax expediture estimates, particularly with respect to deferrals, ad i 1994 asked the Departmet of Fiace to develop a preset value estimate of the tax expediture associated with TARS. 2. Descriptio of the Estimatig Method This sectio develops a method for obtaiig a preset value estimate of tax assistace for retiremet savig i four steps: I step (a), the tax advatage of deferrig a dollar of icome from age M to age N is determied, assumig a costat margial tax rate ad costat pretax rate of retur o ivestmet over the period; I step (b), the result i step (a) is modified by recogisig the possibility of varyig tax rates ad ivestmet yields over the period; I step (c), the result i step (b) is elaborated by replacig the sigle payout i year N by a patter of payouts over the remaider of the cotributor s life; ad I step (d), a total cost estimate is obtaied by replicatig the results of step (c) for cotributios at differet ages ad the weightig the resultig perdollar estimates to reflect the actual patter of cotributios i the year for which the tax cost is beig estimated. This approach fids the future beefits of tax assistace to cotributors. As prove i Appedix A, this beefit is mathematically equivalet to the govermet s cost of providig tax assistace o the cotributios. While it is the govermet s cost of providig tax assistace that is of ultimate iterest, the beefit approach is used i the calculatios because it is computatioally more attractive.
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 pretax rate of retur o ivestmet. The et proceeds i a future year of savig the dollar i a taxassisted pla are give by: NP TA = (1 NM (1  t) where: NP TA = the et (or aftertax) proceeds of taxassisted 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 NM 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 (1t) 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 aftertax amout from oe dollar of pretax icome ad payig tax o the ivestmet icome each year are give by: NP NTA = (1 t)(1+ (1 t)i) NM where NP NTA is the et proceeds of otaxassisted savig. Here, the factor (1t) reflects the fact that, out of $1 of pretax icome, oly a et amout of (1t) is available to be saved after icome tax is paid. The factor (1+ (1 t)i) NM 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 pretax rate. Without access, they ca oly ear the aftertax 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 pretax 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 NM or: (a) = (1t){1[(1+(1t)i) NM /(1+i) NM ]} To illustrate, cosider the aftertax 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 pretax icome which is saved for 10 years i a otaxassisted accout yieldig 7%. Assume that the aftertax amout of ivestmet icome is reivested each year ad that the federal tax rate is 25% i all periods: NP NTA = $1(.75)(1+ (.75)(.07)) 10 = $1.25 The preset value beefit that the govermet provides (the tax expediture) of savig i a taxsheltered istrumet is give by the differece i the et proceeds of the two cases above discouted by the pretax rate of iterest: = (NP TA  NP NTA )/ (1 NM = ($ $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 otaxassisted 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 $0.12 of each dollar cotributed to a retiremet savigs pla. To illustrate how varies with NM, the of oe dollar of cotributio is estimated below for differet values of NM assumig a costat federal margial tax rate of 25%: NM ($)
6 Takig provicial taxes ito accout would raise the tax rate to about 40%. The tax assistace beefits i the above table would icrease to $0.14, $0.25, $0.33 ad $.39 for the four deferral periods. The result i (a) ca be modified to recogise the possibility of varyig tax rates ad ivestmet yields over the deferral period: (b) = P N = (1t N )  [(1t M ) Π (1+ (1t j )i j ) / Π (1+ i j )] j=m+1 j=m+1 N N where: i j = ivestmet retur i year j (j=1,...,n) t j = taxpayer's margial tax rate i year j Π = product operator, e.g.: 2 Π(1+ (1t j )i j ) = [1+ (1 t 1 )i 1 ][1+ (1t 2 )i 2 ] j=1 (To simplify the aalysis, the model used to calculate the assumes a costat rate of retur o ivestmet, oe average margial tax rate up to ad icludig age 64 ad oe average margial tax rate for age 65 ad over.) Because payouts from retiremet savigs plas are ormally received i a stream of paymets over the retiremet period, it is ecessary to allow for more tha a sigle payout at age N. Thus, the result i (b) is elaborated by replacig the sigle payout at age N by a patter of payouts over age M+1 to N*: (c) where: N* = Q M = Σ a k P k k=m+1 a k = the proportio of the cotributio made at age M that is paid out at age k, ad N* = the highest age at which a payout is made (age 99 i the model). Because of the differet life expectacies of males ad females, differet values of Q M may be determied for males ad females at age M. The distributio of payouts observed i the curret year for each age group is used to estimate the future payout patter for the curret year s cotributios.
7 7 The curret year s distributio of payouts by age is used to estimate probabilities for this year s cotributors of receivig payouts at various future ages. Fially, the result i (c) is aggregated across the cotributios made to retiremet plas i a give year by males ad females of differet ages: (d) where: M* = Σ c m Q m m=19 c m = the proportio of total cotributios made i the year by idividuals of age m, ad M* = the highest age at which cotributios ca be made. 3. Applicatio to Caadia Data Sice a preset value tax expediture estimate is obtaied by discoutig estimated future costs, the estimate depeds critically o assumptios made about future evets. The key assumptios used i applyig the estimatio method to Caadia data are set out below: (a) I the base estimate, the future rate of retur o ivestmets, i, is assumed to be 7% i all periods. To idicate the sesitivity of the results to this assumptio, estimates are also provided assumig rates of retur of 6% ad 8%. (b) (c) Preset ad future margial tax rates i priciple should reflect ot oly differet margial tax rates for differet taxpayers, but differet margial tax rates by age for each taxpayer. For example, if icome rises with age ad the falls i retiremet, margial tax rates may follow the same profile. To simplify the aalysis, oe average margial tax rate is used for ages up to ad icludig age 64 ad aother for ages 65 ad over. The tax rates used are effective average margial tax rates. They iclude the effect of refudable tax credits ad social beefit repaymets. The tax rate used as the uder age 65 tax rate is calculated as the tax rate that would have bee paid o RPP ad RRSP cotributios, averaged across all cotributors (regardless of age). The rate used for those 65 ad over is calculated as the rate of tax actually paid o RPP ad RRSP icome, averaged across all icome recipiets (regardless of age). The distributio of future payouts over the remaiig lifetime of the taxpayer is critical to the aalysis. The loger the average holdig period, the greater the tax assistace (as demostrated o page 5). However, the distributio of payouts origiatig from a give year of cotributios caot be directly observed from payout data. For example, a particular payout i 1996 could derive from a cotributio made i 1985 or i Ad, eve if there were such evidece, it might ot be a good guide for future payout patters sice it would be based o factors which chage over time, such as the average age of retiremet ad the
8 8 regulatory eviromet affectig preretiremet payouts. The age profile of payouts i a recet year is ot a good idicator of the expected payout profile (eve assumig o future chage) because of major chages i cotributio levels ad profiles over time. For example, $1 of payout received i the curret year by a 75 year old may derive from a pesio cotributio made 40 years ago while $1 of payout received by a 35 year old must derive from a much more recet cotributio. Sice retiremet savigs pla cotributios have grow dramatically over time  RRSPs were first itroduced 40 years ago future payouts will exceed the payouts see at preset. The curret crosssectio of payouts will be overweighted toward cotributios at youg ages. To correct for this bias, the age profile of payouts is estimated by reweightig the payouts reported i 1993 by the factor: (1.135/1.07) age19 The factor grows curret year icome by 13.5% per year  the average aual growth rate of total RPP ad RRSP cotributios from 1966 to 1991 to accout for the fact that future withdrawals will exceed those see today. This growth factor is adjusted by 7%  the average aual rate of retur from 1944 to to accout for the beefit growth over time that is already take ito accout. This avoids double coutig. The icome is grow for the period age19 where age is the age of idividuals receivig RPP ad/or RRSP icome i The result of this correctio is a payout profile that ca be used to divide each dollar of cotributios ito portios destied to be paid out (with associated iterest) at various later ages. (d) Age 19 is assumed to be the first age at which cotributios are made ad payouts received. The actual cotributio ad payout profiles are based o five year aggregates, e.g., uder 20, 2024, The midpoits of the five year categories are used i assigig values to 'age' (e.g., 22, 27, 32...) except for the first ad last age groups, where age 19 is used as a proxy for the age of cotributors uder 20 ad age 77 is used as a proxy for the age of cotributors 75 ad over. Age 77 is assumed to be the latest age at which cotributios are made, which reflects the possibility of idividuals over age 69 cotributig to spousal RRSPs. Sigle year ages, as opposed to five year midpoits, are used to assig payouts to ages from 65 to 99 (e.g., 65, 66, 67...). Chart 1 sets out the age distributio of cotributios ad adjusted ad uadjusted payouts. Chart 1: Distributio of RPP ad RRSP Cotributios ad Payouts by Age
9 9 30 Percet 25 Cotributios Adjusted Payouts Uadjusted Payouts Age (midpoit of fiveyear 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 cashflow estimate for compariso.
10 10 Table 1 Preset Value Tax Expediture Estimates Year Average rate of retur AMTR <65 AMTR 65+ RRSP RPP Total Cashflow (%) ($ 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 <651 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 cashflow 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 recalculated 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 )  [(1t M ) Π (1+ (1t 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) agespecific 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, agespecific cotributio patters have chaged through the years. Cotributio levels of today s 40 yearolds are higher tha the 40 yearolds 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 taxassisted savigs accouts. This stock of assets has bee growig o a taxdeferred 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 aftertax 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 A1. 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 pretax rate of iterest. At the ed of the first period, the cotributor would have eared taxable ivestmet icome of i o the pricipal of (1t). 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 (1t)[1+(1t)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 A1 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 A1, 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 Aftertax 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 aftertax (or osheltered) 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 aftertax discout rate (1+(1t)i) NM is used to calculate the preset value of the beefit to idividuals. Table B1 Estimates Usig Aftertax Discoutig Year (aftertax discoutig) (aftertax discoutig, real taxatio bechmark) ($ billio) ($ billio)
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