# Appendix K Additional time value concepts.

Chapter 14 discusses basic time value concepts. This appendix discusses how to adjust time value concepts for inflation or growth and for taxes.ADJUSTING FOR INFLATION OR GROWTH FACTORS

Overview

Often investment analyses and other financial planning problems involve adjustments for inflation or for expected systematic increments or decrements of payments or cash flows over time. For example, when planning for how much one must accumulate for retirement it is common to assume that the amount needed each year in retirement will increase as a result of inflation. The annuity formulas presented earlier will compute the present value of a series of level payments for a specified number of years, but how does one compute the present value if the payments are assumed to be increasing at some constant rate rather than remaining level?

Actually, the formulas given in Chapter 32 are generally still perfectly applicable, with some slight modification, if one substitutes inflation-adjusted or growth-adjusted rates for nominal rates.

Inflation/Growth-Adjusted Rate of Return

The inflation- or growth-adjusted rate of return, p, is defined as follows, where r is the nominal rate of return and i is the inflation or growth factor:

Equation RIA

[rho] = (1 + r/1 + i) - 1 = (r - i/1 + I)

Example 1: Our client earns 12% on her investment for the year. However, inflation for the year is 4%. What is her real inflation-adjusted rate of return?

For each $100 invested, your client has $112 in nominal terms at the end of the year. However, the purchasing power of that $112 has declined by 4% as a result of inflation. Therefore, the real inflation-adjusted value of each dollar invested is only $112 / 1.04 = $107.69 (using equation PV1 from Chapter 32). So her real inflation-adjusted return is only 7.69%.

Applying equation RIA, one derives the same result:

[rho] = (12% - 4%) / (1.04) = 7.69%.

Inflation/Growth Adjusted ROR, PVs and FVs

Equation PV1 computes the present value of a future value. However, if the future value is expressed in current dollars and one expects inflation during the intervening period, one must first inflate the future value by the anticipated inflation using equation FV1 (from Chapter 32) before computing the present value investment required today to reach the future inflated value. In other words, one must compute the required present value in a two-step procedure.

However, the calculation can take just one step by using an inflation-adjusted rate of return where both the present value and future value are expressed in current dollars.

Example 2: Recall, your client's child will be attending college in 5 years and she asked you how much she will need to set aside today to pay the first year's tuition and fees. If the current tuition and fees are $36,000, and inflation for college costs averages 6% over the next five years, she will need to accumulate $48,176, not just $36,000 (using equation FV1). Assuming she earns 5% on the money she invests for this purpose, she will need to invest $37,747 today (using equation PV1), not just $28,207, to meet her child's first year college need.

However, the amount can be computed directly using just one equation with the inflation-adjusted rate of return of -0.9434% [(5% - 6%) / 1.06] and the $36,000 goal expressed in today's dollars.

PB = $36,000/[(1 - 0.9434%).sup.5] = $36,000/0.953712 = $37,747

Inflation/Growth Adjusted ROR and Annuities

The inflation/growth-adjusted rate becomes much more useful when dealing with calculations involving annuities. For inflation-adjusted ordinary annuities where the payment is assumed to grow at i% per period, the present value can be computed by adjusting equation pv4 (from Chapter 32) for the growth of payments as follows:

Equation PV4'

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Multiplying inside the bracket and dividing outside the bracket by (1 + i) one derives the following formula:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Substituting (1 + [rho]) for (1 + r) / (1 + i), the result is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

With p substituted for r, this formula is identical to the formula used to derive equation PV7 (from Chapter 32) for the present value of an ordinary annuity, except that it is multiplied by 1 / (1 + i). Therefore, the equation for computing the present value of an inflation-adjusted ordinary annuity with payments increasing at the rate of i% per period is:

Equation PV7'

PV = Pmt x (1 - [(1 + [rho].sup.-n]/[rho]) x [(1 + i).sup.-1], if [rho] [not equal to] 0

PV = Pmt x n x [(1 + i).sup.-1], if [rho] = 0

Through similar algebraic manipulations and substitutions of equation PV11, (from Chapter 32) the formula derived for the present value of an inflation-adjusted annuity due is:

Equation PV11'

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Example 3: Example 6 (from Chapter 32) computed the present value of your client's four beginning-of-year college cost payments at the time her daughter begins college, assuming that the annual cost would be $36,000. Example 2 showed that if college costs were assumed to inflate at the rate of 6% per year for the 5 years until your client's child starts college, the first-year cost would be $48,176, not just $36,000. It also computed the growth-adjusted rate of return, -0.9434%, assuming a 5% nominal rate of return and a 6% rate of inflation for college costs. Assuming that college costs continue to rise at a 6% rate after your client's daughter begins college, how much will your client have to accumulate by the beginning of her child's first year of college in 5 years to pay the 4-year cost if she can earn 5% on her money?

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Your client will need to accumulate a sum of almost $200,000 ($195,474) by the time her child begins college in 5 years, not just $134,000 as determined in Example 6 when inflation was ignored.

Example 4: What is the total amount your client would need to invest today to reach her goal of $195,474 in 5 years?

You could compute this amount by discounting the $195,474 target determined in Example 3 at 5% for 5 years using equation PV1. The result is $153,160.

Alternatively, you could compute the amount required today by realizing that the present value of a 4-year annuity due commencing at the end of 5 years is equal to the value of a 9-year annuity due commencing today less the value of a 5-year annuity due commencing today.

Using equation PV11 with n = 9, [rho] = -0.9434%, and Pmt=$36,000 (you would use the current college cost value since the analysis starts from today), the present value of a 9-year inflation-adjusted annuity is $336,621. Similarly, using equation PV11 with n = 5, [rho] = -0.9434, and Pmt = $36,000, the present value of a 5-year inflation-adjusted annuity is $183,461. The difference is $153,160.

This 2-step calculation derives exactly the same result as was determined by using the 3-step process of (1) inflating the $36,000 payment for 5 years of inflation at 6% to derive the first-year college cost figure, (2) calculating the present value in 5 years of a 4-year inflation-adjusted annuity due, and (3) computing the present value today (5 years earlier) of the amount determined in step 2.

The formulas for computing the future value of inflation-adjusted annuities due and ordinary annuities can be derived by similar manipulations and adjustments of equations FV2 and FV4 (from Chapter 32). The future value of an inflation-adjusted ordinary annuity is:

Equation FV2'

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The future value of an inflation-adjusted annuity due can be determined by simply multiplying the future value of the ordinary annuity by (1 + r):

Equation FV4'

FV = Pmt x [(1 + i).sup.n] x [(1 + [rho]).sup.n] - 1/[rho] x (1 + r), if [rho] [not equal to] 0

FV = Pmt x n x [(1 + i).sup.n-1] x (1 + r), if ][rho] = 0

An alternative form is sometimes preferred, expressed only in terms of [rho] and i, by multiplying [(1 + i).sup.n-1] in front of the equation by (1 + i) and dividing (1 + r) at the end of the equation by (1 + i) to derive:

Equation FV4"

FV = Pmt x [(1 + i).sup.n] x ([(1 + [rho]).sup.n]-1/[rho]) x (1 + [rho]), if [rho] [not equal to] 0

FV = Pmt x n x [(1 + i).sup.n], if [rho] = 0

The formulas to compute Pmt amounts for inflation-adjusted annuities can be derived from equations PV7, PV11, FV2, and FV4 (from Chapter 32). Equation Pmtl' shows the formula for computing the initial payment amount for an inflation-adjusted ordinary annuity based upon a given present value:

Equation Pmt1'

Pmt x ([rho]/1 - [(1 + [rho]).sup.-n] x (1 + i), if [rho] [not equal to] 0

Pmt = PV/n x (1 + i), if [rho] = 0

Equation Pmt3' shows the formula for computing the initial payment amount for an inflation-adjusted annuity due based upon a present value:

Equation Pmt3'

Pmt = PV x ([rho]/1 - [(1 + [rho]).sup.-n] x 1/(1 + [rho]), if [rho] [not equal to] 0

Pmt = PV/n, if [rho] = 0

Equations Pmt5' and Pmt7' show the corresponding formulas for computing the initial payment for inflation adjusted ordinary annuities and annuities due based upon a future value:

Equation Pmt5'

Pmt = FV x (1 / [(1 + i).sup.n-1] x [([rho]/(1 + [rho]).sup.n]-1, if [rho][not equal to] 0

Pmt = FV/n x (1/[(1 + i).sup.n-1], if [rho] [not equal to] 0

Equation Pmt7'

Pmt = FV x (1 / [(1 + i).sup.n]) x ([rho] / [(1 + [rho]).sup.n]-1) x 1 /(1 + [rho]),

Pmt = FV/n x (1/[(1 + i).sup.n]), if [rho] = 0

Example 5: Example 4 determined that your client needs $153,160 invested today to meet her child's 4-year education costs starting in 5 years assuming your client invests to earn 5% and that college costs increase at 6% per year. Your client already has set aside $60,000 for this purpose, so she is currently $93,160 short. She anticipates that her income will increase at 4% per year. She wishes to know how much she would need to save at the end of each year until the beginning of her child's last year of college (8 years), assuming that she increases the amount saved by 4% each year. (This would permit her to keep the yearly payment equal to a fixed percentage of her growing earnings.)

PV = $93,160

r = 5%

I = 4%

n = 8 years

[rho] = (5% - 4%) f (1 + 4%) = 0.96154%

Equation Pmt1' calculates the initial yearly amount:

Pmt = $93,160 x 0.96154% x (1 + 4%)/1 - [(1 + 0.96154%).sup.-8]

mt = $93,160 x 0.0096154 x 1.04/0.0736986 = $12,641

Beginning with $60,000 and adding subsequent investments starting at $12,641 at the end of the first year that increase each year by 4% (i.e. $12,641 x (1.04) = $13,147 the second year, etc.), your client will be able to finance her child's education over the period until the child commences her senior year in college.

ADJUSTING FOR TAXES

Overview

Bottom line: what investors get to keep after tax is what matters. Virtually all financial planning issues and investment choices involve tax considerations. The real value of any financial planning strategy or tactic or investment choice relates to the real spendable dollars the strategy or tactic or investment choice provides relative to the alternatives. For instance, nominal yields on taxable bonds are virtually uniformly higher than returns on tax-free municipals of comparable maturity. However, for some taxpayers in high tax brackets, the tax-free yields of municipal bonds are higher than the after-tax yields from taxable bonds, so the tax-free municipals are a preferable investment.

If it is assumed that the investment return is entirely currently taxable, then the after-tax return is simply equal to the before-tax return less the taxes on the return. Specifically, if the tax rate is assumed to be t and the before-tax rate of return is r, then the after-tax return, [r.sub.at], is:

[r.sub.at] = r x (1 - t)

For example, investors earning 6% before tax, whose tax rate on that income is 30%, will earn 4.2% after tax [6% x (1 - 30%) = 4.2%].

Although this formula is frequently used to compute after-tax returns, it is often not an accurate measure. Only a relatively small class of investments, such as money market funds, bank accounts, and the like, provide investment returns that are entirely currently taxable. Other investments, such as stocks, real estate, and the like provide some combination of currently taxable income and income, return, or gain on which tax is deferred. In addition, a whole host of vehicles, such as qualified retirement plans, commercial annuities, life insurance, IRC Section 529 plans, and the like, provide unique tax incentives that cannot be accounted for using the simple formula above.

In many financial planning and investment situations, not only the level of taxation but also the timing of taxation is a critical factor. Between two investments providing identical before-tax returns and identical total tax burdens, the one that defers some or all of the taxation to a later date is preferable.

For instance, the tax incentives associated with qualified plans and IRAs are hard to beat. With tax-deductible contributions and tax-deferred accumulations, it adds up fast. But qualified plans, IRC Section 401(k) plans, SEP IRAs, traditional IRAs, Roth IRAs, IRC Section 403(b) TDAs, and SIMPLE IRAs are not the only tax-preferred accumulation vehicles, nor are they necessarily the best for all circumstances. These tax-favored plans have their limitations as well. In order to assess whether one tax-favored investment is superior to another, advisers and investors need quantitative tools to measure the tax effects, and to weigh the trade-offs.

The most prevalent type of tax-deferral vehicle is not qualified plans, IRAs, and the like, but rather appreciating assets. Under our current tax laws, the tax on capital gains is deferred until the gain is recognized, usually when the asset is sold. Also, long-term gains are usually taxed at a lower rate than ordinary income.

Furthermore, techniques may be employed to defer full recognition of capital gains even beyond the time of disposal including the use of installment sales, IRC Section 1031 like-kind exchanges, private annuities, and, in the case of personal residences, the lifetime gain exclusion provisions. Section 1035 provides similar nonrecognition treatment for qualifying life insurance exchanges. Certain other transfers of life insurance policies between related parties also avoid recognition. In the area of corporate securities, there is an assortment of provisions allowing nonrecognition treatment, including recapitalizations, reorganizations, mergers, and acquisitions.

Life insurance and capital gains are special cases. In the case of life insurance, if the proceeds are paid in the form of death benefits, in effect the tax loan associated with the earnings on the cash value is generally completely forgiven. If amounts are withdrawn from the cash value during the insured-owner's lifetime through policy loans, there is generally no income taxation on amounts withdrawn, even if the amounts withdrawn are attributable to investment earnings inside the policy. If amounts are withdrawn directly, not through policy loans, then the earnings will generally be subject to tax.

In the case of appreciated assets that are held at death, the step-up in basis effectively cancels the tax loan. (1) Also, in the case of charitable gifts of appreciated property where the appreciation qualifies as long-term capital gain, the donor is generally able to take a deduction for the entire amount without recognizing or paying tax on the gain. Once again, the tax loan is essentially forgiven.

The optimal use of tax-advantaged tools and techniques is a major wealth-accumulation principle and objective. Tax advantage in the form of deferred taxes helps to finance wealth accumulation and to reduce the rate at which wealth is depleted. The use of various tax-preferred tools and techniques should not proceed without careful consideration of the trends in tax policy. What looks like a favorable arrangement today can quickly change with changes in the tax laws.

The following sections will explain the concept of tax leverage and describe the tax-adjusted time value tools financial advisers must understand to properly serve their clients' financial planning needs. The following sections discuss the tools necessary to account for the 5 most-prevalent types of tax leverage, ranging from nondeductible fully currently taxable vehicles, such as T-bills, on one end of the spectrum to tax-deductible, fully tax-deferred vehicles, such as IRC Section 401(k) plans, on the other end of the spectrum.

Tax Leverage

Any strategy or technique that defers taxes that would otherwise be paid currently creates what is called tax leverage. Tax leverage is similar to the concept of financial leverage, where an investor borrows money to help finance an investment. When financial leverage is successfully employed, the rate of return earned from the investment exceeds the cost of borrowing. The borrowing works like a lever to increase the earning power of the investor's equity. The return on equity rises above the rate actually paid by the investment because the return includes not only the amount earned on the investment, but also the differential between the investment return and the borrowing rate on the portion of the investment that was financed.

Example 6: If $1,000 is invested for one year at a fully taxable 10% rate of return, an investor in a 50% 2 combined federal, state, and local tax bracket will have $1,050 after tax at the end of the year. In other words, his after-tax rate of return is 5%.

Suppose the investor can borrow $1,000 at 6%, which is equivalent to 3% after tax assuming the interest is deductible. This $2,000 total investment will earn $200 before tax. The investor must pay $1,000 plus $60 back to the lender, so the investor is left with $1,140 before tax, or $1,070 after tax. The investor has increased the after-tax return on his equity from 5% to 7% through the use of financial leverage.

The problem with financial leverage is the risk the investment will earn less than the cost of borrowing. This negative leverage reduces the investor's return below what it would have been without borrowing. Of course, if the loan were interest-free, the risk of negative leverage would be very small. Any positive return at all from the investment would produce positive leverage. But who would lend at a 0% rate? The U.S. government, for one.

Example 7: The investor in the previous example now borrows $1,000 at 0% interest, rather than 6%. At the end of the year he now will have $1,200 before tax after paying off the debt. At his 50% tax rate, he will take home $1,100. His after-tax rate of return on equity doubles from 5% to 10%.

How does the government get into the 0% lending business? The government does it by providing tax incentives for employing various tools and techniques that allow taxpayers to defer the payment of taxes. The deferred taxes are the equivalent of a loan. Generally, if tax rates remain constant and the taxpayer remains in the same tax bracket, the taxpayer will pay the deferred taxes in full at a later date, but with no increase or adjustment for the lapse in time. In other words, the ability to defer the payment of tax is essentially an interest-free loan from the government.

Example 8: Assume the government institutes a new program permitting taxpayers to contribute before-tax dollars to accounts to fund Special Pre-Olympic Recruiting and Training Schools (SPORTS) for athletically gifted children. Under this program, taxes on both contributions and on earnings are deferred until money is withdrawn to pay for the approved school's tuition. When the money is withdrawn to pay tuition, the amount withdrawn is then subject to tax.

Assume the taxpayer from our previous examples elects to contribute $2,000 before tax to a SPORTS account earning a 10% rate of return. At the end of the first year, the taxpayer withdraws the entire balance of $2,200 to pay the taxpayer's oldest child's tuition at one of the government-approved SPORTS schools during the child's summer vacation. At a 50% tax rate, the taxpayer is left with $1,100 after tax.

Had the taxpayer not invested in the SPORTS account, the taxpayer would have had only $1,000 after tax available for investment outside the SPORTS account and would have accumulated only $1,050 after tax. By using the SPORTS account, the results are exactly the same as in the previous example where the investor borrowed $1,000 interest-free. The taxpayer has essentially received an interest-free loan of $1,000 from the government and has raised his after-tax rate of return, as before, from 5% to 10%.

Tax-favored investments that permit tax-deductible contributions and defer tax on earnings, such as pension plans, profit-sharing plans, and deductible IRAs, are essentially equivalent to nondeductible investments whose earnings are received tax-free. In the SPORTS account example above, the investment in the SPORTS account is equivalent to investing the taxpayer's $1,000 after-tax amount at 10% tax-free.

Of course, this relationship holds only as long as the taxpayer's tax rate remains the same. In some situations, the amount of tax that ultimately must be repaid may be greater than the amount originally deferred. For instance, this may occur as a result of tax penalties for early withdrawals or an increase in tax rates generally. But even in this case, the client may benefit. The tax leverage involved is then still essentially equivalent to a subsidized or below-market discount loan rather than a fully interest-free loan. In some other cases, the amount of tax paid in the future may be less than the amount of tax originally deferred. This is essentially equivalent to an interest-free loan where part of the debt is forgiven. Some of these types of situations are discussed later.

Since tax leverage may arise from up-front deductions, tax-deferred earnings, or both, there are several variations of the time value formulas for computing the future value of single payments in tax-leveraged situations. The principal factors in the analysis are:

r = The assumed before-tax rate of return

[t.sub.c] = The client's current marginal tax rate

[t.sub.f] = The client's assumed future marginal tax rate

n = The number of years in the planning horizon

P = The amount available for tax-leveraged investment

Up-Front Deductions or Exclusions

Assume the situation being investigated involves up-front deductions or exclusions, such as would be the case if one were comparing investments outside of an IRC Section 401(k) plan with voluntary tax-excludable salary-reduction contributions to an IRC Section 401(k) plan. In this case, your client would have only (1-[t.sub.c]) dollars to invest outside the plan for each dollar that could be invested inside the plan. In other words, by foregoing the elective deferrals, your client would only have the opportunity to invest his after-tax dollars outside the IRC Section 401(k) plan. By electing to defer salary into the IRC Section 401(k) plan, he would have a greater number of before-tax dollars earning money inside the IRC Section 401(k) plan.

In contrast, if your client were comparing nondeductible contributions to an IRA to an investment outside the IRA, he would have $1 available after tax to invest outside the nondeductible IRA for each $1 available after tax for investment inside the nondeductible IRA. In either case, your client would still have only (1-[t.sub.c]) dollars to invest for each $1 that otherwise would be available, for example, to invest in a deductible IRA or IRC Section 401(k) plan.

Tax Status of Amount Invested Initial Investment Inside Plan Outside Plan Tax Deductible $P $P x (1 - [t.sub.c]) Non-Tax Deductible $P x (1 - [t.sub.c]) $P x (1 - [t.sub.c])

Example 9: Your client, who is in a 28% combined state and federal tax bracket (3), is considering increasing IRC Section 401(k) contributions by $1,000. Since contributions are tax deductible, the entire $1,000 will go into the plan. In contrast, if the client does not make the $1,000 elective deferral into the plan, the amount is taxable. The client will have only $720 after tax of the $1,000 before tax remaining ($1,000 x 0.28 = $280 tax) to invest outside the plan.

Tax-Deferred Earnings or Benefits

A tax-leveraged vehicle may or may not provide tax-deferred earnings, but most do. If the earnings are entirely tax-deferred, the total before-tax amount accumulated within the plan is simply equal to the future value of PV dollars compounded using the before-tax return:

Equation FVbt1

FV(before tax) = PV(before tax) x [(1 + r).sup.n]

Example 10: Your client, described in the previous example, is earning 10% on amounts invested in the IRC Section 401(k) plan. If your client continues to earn 10%, how much will your client accumulate before tax in 8 years? First, calculate that [(1 + .10).sup.8] equals 2.143589. Therefore, the total before-tax accumulation will be $2,143.59 [$1,000 x 2.143589].

If the initial contribution to the plan was tax deductible or excludable, the entire accumulation (initial contribution plus accumulated earnings) will be subject to tax when the amounts are withdrawn. In this case, the total after-tax accumulation for PV dollars of contributions is determined by subtracting the tax payable on the total amount from the total before-tax accumulation:

Equation FVat1

FV (after tax) = PV (before tax) x [(1 + r).sup.n] x (1 - [t.sub.f])

Example 11: Your client, described in the previous example expects to be in a 35% tax bracket in eight years. Disregarding any penalty taxes that may apply if your client withdraws the funds at that time, your client's total after-tax accumulations will be (1 - 0.35) x $2,143.59 = $1,393.33.

The tax-deductible amount one needs to invest in a tax-deferred plan to reach a specified future after-tax value is computed by simply rearranging equation FVat1 to isolate PV on the left-hand side of the equation:

Equation PVat1

PV (before tax) = FV (after tax)/[(1 + r).sup.n] x (1 - [t.sub.f])

The formula for the future after-tax value of a series of level end-of-period tax-deductible contributions to a plan where investment earnings are tax deferred is:

Equation FVAEat1

FV (after tax)1 = Pmt (before tax) x ([(1 + r).sup.n] - 1/r) x (1 = [t.sub.f]),

If r [not equal to] 0

FV (after tax) = Pmt (before tax) x n x (1 - tf), if r = 0

The formula for the future after-tax value of before-tax annuity due payments is computed simply by multiplying the right-hand side of equation FVAEat1 by (1 + r):

Equation FVABatl

(1 + r)n - 1

FV (after tax) = Pmt (before tax) x ([(1 + r).sup.n] - 1/r)

x (1 + r) x (1 - [t.sub.f]), if r [not equal to] 0

FV (after tax) = Pmt (before tax) x (1 - [t.sub.f]), if r = 0

The formula for determining the Pmt(before tax) that one needs to invest each period to reach a specified future after-tax value, is simply the desired FV(after tax) divided by the multiplier of Pmt(before tax) in equation FVAEat1 or equation FVABat1, as appropriate.

The formula to determine the present before-tax value one needs to invest in a tax-deductible, tax-deferred plan to generate after-tax ordinary annuity payments is:

Equation PVAEat1

(l - (1 + r) A

PV (before tax) = Pmt (after tax) x (1 - [(1 + r).sup.-n]/r x (1 - [t.sub.f])), if r [not equal to] 0

PV (before tax) = Pmt (after tax) x n x n/(1 - [t.sub.f]), if r = 0

The present before-tax value one needs to invest to generate level after-tax annuity due payments is equal to equation PVAEat1 multiplied by (1 + r):

Equation PVABat1

PV (before tax) = Pmt (after tax) x (1 - [(1 + r).sup.-n]/r x(1 - [t.sub.f]) x (1 + r),

PV (before tax) = Pmt (after tax) x n/(1 - [t.sub.f], if r = 0

Once again, the formula for determining the Pmt(after tax) that can be supported by a specified present before-tax value is simply PV(before tax) divided by the multiplier of Pmt(after tax) in equations PVAEat1 or PVABat1, as appropriate.

Nondeductible Contributions and Tax-Deferred Earnings

Some vehicles permit investors to make after-tax payments to tax-deferred accounts, such as non-deductible contributions to nondeductible IRAs and to commercial annuities. If the initial contribution to the plan is not tax deductible, tax must be applied only to the accumulated earnings (that is, the growth and income), not the entire accumulation, when the funds are withdrawn.

The total before-tax earnings for each $1 contribution are equal to the total accumulated value less the initial investment of $1: [[(1 + r).sup.n] - 1]. The total after-tax return on each dollar of earnings accumulated within the plan is computed by subtracting the amount of tax that is due from the total before-tax earnings: [[(1 + r).sup.n] - 1] x (1 - [t.sub.f]). The total after-tax accumulation for each $1 contribution is the total after-tax return plus $1: [[(1 + r).sup.n] - 1] x (1 - [t.sub.f]) + 1. Rearranging and simplifying terms we derive the formula for the future after-tax value of a lump sum after-tax present value investment:

Equation FVat2

FV (after tax) = PV(after tax) x {[[(1 + r).sup.n] x (1 - [t.sub.f])] + [t.sub.f]}, if r [not equal to] 0

FV (after tax) = PV(after tax), if r = 0

Example 12: Assume your client, described in the previous three examples, was considering a nondeductible investment in an IRA rather than the tax-deductible contribution to an IRC Section 401(k) plan. Since the IRA contribution is not tax deductible, your client will have only $720 to invest from $1,000 of pretax income (assuming a 28% tax rate at time of contribution). The total net amount your client would accumulate in 8 years, assuming a 10% annual rate of return and a 35% tax rate at the time of withdrawal, is $720 x [[((1 + .10).sup.8] x (1 - 0.35)) + 0.35] = $1,255.20.

The present after-tax value one needs to invest to accumulate a specifiedfuture after-tax value in a nondeductible, tax-deferred plan is:

Equation PVat2

PV (after tax) = FV (after tax)/[[(1 + r).sup.n] x (1 - [t.sub.f])] + [t.sub.f]

PV (after tax) = FV (after tax), if r = 0

The formula for the future after-tax value of a series of end-of-period nondeductible contributions to a plan where investment earnings are tax deferred is the same as equation FVAEat1, substituting Pmt (after tax) for Pmt (before tax) and adding the term n x [t.sub.f]:

Equation FVAEat2

FV (after tax) = Pmt (after tax) x

[[(1 + r).sup.n]-1/r) x (1 - [t.sub.f]) + (n x [t.sub.f]), if r [not equal to] 0

FV (after tax) = Pmt (after tax) x n, if r = 0

The formula for the future after-tax value of beginning-of-period nondeductible contributions to a plan where investment earnings are tax deferred is the same as equation FVBEat1, substituting Pmt (after tax) for Pmt (before tax) and adding the term n x [t.sub.f]:

Equation FVABat2

FV (after tax) = Pmt (after tax) x

[[(1 + r).sup.n] - 1/r) x (1 + r) x (1 - [t.sub.f]) + (n x [t.sub.f]), if r [not equal to] 0

FV (after tax) = Pmt (after tax) x n, if r = 0

The formula for determining the Pmt(after tax) that one needs to invest each period to accumulate a specified future after-tax value is simply FV(after tax) divided by the multiplier of Pmt(after tax) in equations FVAEat2 or FVABat2, as appropriate.

The present after-tax value one needs to invest in a nondeductible, tax-deferred investment to generate a series of periodic level after-tax ordinary annuity payments cannot be simplified to a single reduced-form equation similar to the future after-tax value formula. Instead, one must calculate the present after-tax value of each separate periodic payment using equation PVat2 with Pmt(after tax) replacing FV(after tax) and then summing these separate present values to derive the total:

Equation PVAEat2

PV (after tax) = Pmt (after tax) x

[n.summation over (j=1] 1/[[(1 + r).sup.j] x (1 - [t.sub.f])] + [t.sub.f], if r [not equal to] 0

PV (after tax) = Pmt (after tax) x n, if r = 0

Similar to earlier formulations, the formula for determining the Pmt(after tax) that can be supported by a specified present after-tax value is simply PV(after tax) divided by the multiplier of Pmt(after tax) in equations PVAEat2 or PVABat2, as appropriate.

The future after-tax value that would be accumulated by investing a present after-tax value in a fully taxable investment (no tax deferral on earnings) is determined by assuming the income is subject to tax when earned. That is, for each r dollars of return, the after-tax return is r x (1 - t) (ignoring, for the time being, any capital gain component). Therefore, the future after-tax value of a present after-tax value invested in a fully taxable investment is:

FVat3: FV(after tax) = PV(after tax) x [[1 + r x (1 - [t.sub.f])].sup.n]

Example 13: If your client invested $720 in a fully taxable investment earning 10% before tax (assuming your client stays in the 28% tax bracket until the end of the eighth year), in 8 years your client will have $1,255.71 [$720 x {1 + (0.10 x [(1 - 0.28))}.sup.8]].

The present after-tax value one needs to invest to accumulate a specifiedfuture after-tax value in a nondeductible, fully taxable investment is:

Equation PVat3

PV (after tax) = FV (after tax)/[[1 + r x (1 - [t.sub.f])].sup.n]

The formula for the future after-tax value of a series of end-of-period nondeductible contributions to a plan where investment earnings are currently taxable is:

Equation FVAEat3

FV (after tax) = Pmt (after tax) x ([[1 + r x (1 - [t.sub.f])].sup.n] - 1/r x (1 - [t.sub.f]), if r x (1 - [t.sub.f]) [not equal to] 0

FV (after tax) = Pmt (after tax) x n, if r = 0

The formula for the future after-tax value of a series of beginning-of-period nondeductible contributions to a plan where investment earnings are currently taxable is:

FV (after tax) = Pmt (after tax) x ([[1 + r x (1 - [t.sub.f])f].sup.n] - 1 / r x (1 - [t.sub.f])

x [1 + r x (1 - [t.sub.f]], if r x (1 - [t.sub.f]) [not equal to] 0

FV (after tax) = Pmt (after tax) x n, if r = 0

The formula for determining the Pmt(after tax) that one needs to invest each period to accumulate a specified future after-tax value is simply FV(after tax) divided by the multiplier of Pmt(after tax) in equations FVAEat3 or FVABat3, as appropriate.

The formula to determine the present before-tax value one needs to invest in a nondeductible currently fully taxable investment to generate after-tax ordinary annuity payments is:

Equation PVAEat3

PV (after tax) = Pmt (after tax) x (1 - [[1 + r x (1 - [t.sub.f]).sup.-n]/r x (1 - [t.sub.f]).

If r [not equal to] 0

PV (after tax) = Pmt (after tax) x n, if r = 0

The formula to determine the present before-tax value one needs to invest in a nondeductible currently fully taxable investment to generate level after-tax annuity due payments is:

Equation PVABat3

PV (after tax) = Pmt (after tax) x V (1 - [[1 + r x (1 - [t.sub.f])].sup.n]/r x (1 - [t.sub.f])

X[1 + r x (1 - [t.sub.f])], if r [not equal to] 0

PV (after tax) = Pmt (after tax) x n, if r = 0

Similar to earlier formulations, the formula for determining the Pmt(after tax) that can be supported by a specified present after-tax value is simply PV(after tax) divided by the multiplier of Pmt(after tax) in equations PVAEat3 or PVABat3, as appropriate.

Applying the Formulas

Example 14: Let us suppose your client wishes to know how much better off he would be if he increased his IRC Section 401(k) contribution by $1,000. Assuming an 8% interest rate, a 28% tax bracket both now and in the future, and a 15-year planning horizon, the after-tax amounts inside and outside the plan are as follows:

Total after tax from plan P x (1 - [t.sub.f]) x [(1 + r).sup.n] $1,000 x (1 - 0.28) x [(1.08).sup.15] $1,000 x 0.72 x 3.17217 = $2,283.96

Total after tax outside plan P x (1 - [t.sub.c]) x [{1 + [r x (1 - [t.sub.f])f]}.sup.n] $1,000 x (1 - 0.28) x {1 + [0.08 x [(1 - 0.28)]}.sup.15] $1,000 x 0.72 x {1 + 0.0576}15 $1,000 x 0.72 x 2.31644 = $1,667.84 Difference = $616.12

Your client will have about $616 more after tax by increasing the contribution to the IRC Section 401(k) plan rather than investing outside the plan. This represents about a 37% after-tax increase over the fully taxable outside investment.

Example 15: Your client now wishes to know if it would still be better to invest the extra amount in the plan if the plan balance is taxed at a higher rate when he withdraws it. Assuming your client's tax rate increases to 35% and assuming a worst-case scenario that it will increase only just before he withdraws the money from the plan (that is, the earnings on the outside investment enjoy the lower 28% tax rate throughout the period), the comparison is:

Total after tax from plan P x (1 - [t.sub.f]) x [(1 + r).sup.n] $1,000 x (1 - 0.35) x (1 + 0.08)15 $1,000 x 0.65 x 3.17217 = $2,061.91

Total after tax outside plan (same as previous example) = $1,667.84 Difference = $394.07

Even if the tax rate increases from 28% to 35%, your client is still almost $395 better off investing inside the plan. This represents almost a 24% advantage, despite the increase in tax rates.

Example 16: Your client complains that investment options and potential rates of return are limited within the investment options available from the IRC Section 401(k) plan (presently yielding 8%) and feels that he could earn 10% if he had more control over the investments. You suggest that making contributions to a nondeductible traditional IRA with a different custodian might be better than investing in the IRC Section 401(k). However, compared with each dollar invested in the IRC Section 401(k), your client would have only 72 cents (1 - 0.28 tax rate) to invest in the IRA. The question is whether the tax-deferred accumulation at the higher 10% rate within the IRA would more than compensate for the loss of deductibility. Assuming the tax rate in 15 years remains at 28%, the after-tax amounts are as follows:

Total after tax from IRC Section 401(k) plan (from Example 30) = $2,283.96

Total after tax in IRA plan / P x (1 - [t.sub.c]) x {[[(1 + r).sup.n] x (1 - [t.sub.f])] + [t.sub.f]} $1,000 x 0.72 x {[[(1.10).sup.15] x 0.72] + 0.28} $1,000 x 0.72 x [(4.17725 x 0.72) + 0.28] $1,000 x 0.72 x (3.00762 + 0.28) $1,000 x 0.72 x 3.28762 = $2,367.09 Difference = ($83.13)

If your client can earn two% more per year by investing in the nondeductible traditional IRA rather than the IRC Section 401(k), he will be slightly better off investing the money in the IRA, even though the contribution to the IRA is nondeductible. The results would favor the IRA even slightly more if tax rates increased during or at the end of the 15-year period.

Example 17: Your client now wants to know if he should make all contributions to the nondeductible traditional IRA rather than to the IRC Section 401(k) from now until retirement (15 years).

At some point, further contributions to the IRA would be unproductive. In some future year, the compound value of the additional 2% return for the fewer remaining years would not be sufficient to compensate for the loss of the tax deductibility of the contributions. Your client asks you to determine in what year he should shift back to the IRC Section 401(k). You could recalculate the values for each year and determine in which year the difference turns positive.

Adding Capital Gains or Partially Tax-Deferred Income

Assets that promise substantial capital appreciation, such as growth stocks and real estate, or that permit tax on some portion of investment earnings to be deferred, are themselves tax-leveraged investments. Gains generally are not taxed until they are recognized, usually at the time of sale or liquidation.

Ironically, given the widespread use of these types of assets, the tax-leverage associated with capital gains is largely ignored when evaluating and comparing alternative strategies. Perhaps it is ignored because of the difficulty of prospectively measuring the possible gains. However, the tax-leverage of capital gains can provide very sizable benefits. Failure to account properly for the tax-advantage of potential capital gains or other forms of tax-deferred investment earnings may lead to comparative overvaluation of alternative tax-leveraged techniques and lead to improper decisions. For instance, if you proceed as if the outside (taxable) investments are immediately taxable when comparing the benefit of additional contributions to a IRC Section 401(k) plan, IRA, or the like, you will overestimate the benefit of the tax-free build-up type investment. You must consider the tax-deferred capital appreciation on the outside investments when comparing the investment alternatives.

Ignoring the capital gain component of returns could be especially critical in the current tax environment. Under current law, the long-term capital gain tax rate for most investment assets generally is capped at 15%. Even if the current legislation sunsets and the tax law reverts to prior rules, the maximum long-term capital gain tax rate on most investment assets will be 20% or less. We may even see lower capital gains rates in future years.

A lower capital gains tax rate would significantly enhance tax benefits since, in effect, part of the tax loan would be forgiven. In addition, with many of the other tax-leveraged tools and techniques, capital appreciation is irrelevant since all income and gains are taxed alike. For example, generally any capital appreciation on assets in qualified plans, IRAs, and IRC Section 401(k) plans, life insurance, annuities, and the like is ultimately taxed at ordinary rates (even if there is a preferential capital gains rate for gains on investments held outside these vehicles).

Although the formula for projecting the after-tax return from capital gain assets is somewhat more complicated than those previously presented, it is not incomprehensible. The following equation calculates the future after-tax value of a PV dollar investment in a vehicle with a tax-deferred capital gain or tax-deferred return component of total return:

Equation FVat4

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where: ar = The accumulation rate after tax on the currently taxable component of return (usually ordinary income), but before tax on the accumulating and tax-deferred capital gains or tax-deferred income component of return

= (g x r) + (1 - [t.sub.o]) x (1 - g) x r

= r x (1 - (1 - g) x [t.sub.o])

r = The total before-tax rate of return

g = The proportion of total before-tax return, r, attributable to tax-deferred capital gain (or tax-deferred income)

[t.sub.o] = The tax rate on the currently taxable ordinary income component of return

[t.sub.d] = The tax rate on the long-term capital gain (or the tax-deferred) component of investment return

b = The built-in gain or tax-deferred return proportion of the investment, PV, or the periodic payment, Pmt, if any

The first part of equation FVat4 is familiar in form. It is the same as the before-tax future value of an entirely tax-deferred investment earning ar% instead of r%, as shown earlier in equation FVbt1.

Equation FVbt4

FV (before cap - gain tax) = PV x [(1 + ar).sup.n]

The accumulation rate, ar, is composed of two parts. The first part, (g x r), is the capital gain (or tax-deferred) component of the total return, r. This is the part of the return that is not subject to tax until the asset is sold or liquidated.

The second part of the accumulation rate, ar, is the after-tax income component of the total return. The portion of the total return, r, that is subject to current taxation is (1 - g) x r. The amount left after paying tax on the currently taxable return component of total return is determined by multiplying this component of return by (1 - [t.sub.o]). Therefore, the first part of equation FVat4 is the total accumulated value of an investment of PV dollars in n years (assuming reinvestment of after-tax income) before taxation of the accumulated capital gain or tax-deferred return component of total return.

The second part of equation FVat4 is the tax on the accumulated capital gain or tax-deferred return component of total return. Equation FVCG shows the total accumulated gain or total tax-deferred return component of total return for a lump sum investment of PV dollars. Multiplying the accumulated gains or deferred return component by the tax rate on deferred return or long-term capital gains, [t.sub.d], gives the total tax on the tax-deferred return or capital gains at the time of sale or liquidation of the investment. Finally, [t.sub.d] x b is the tax paid upon sale or liquidation on the built-in gain or total tax-deferred return per initial dollar of PV or Pmt. For example, if one is computing the future after-tax value of an investment of $100,000 in a stock portfolio, where $40,000 of the current balance is attributable to as yet untaxed capital appreciation, then b equals 40% ($40,000 ?*? $100,000).

Equation FVCG

Total tax - deferred return

= PV x [g x r x([(1 + ar).sup.n] -1/ar) + b], if ar [not equal to] 0

Total tax - deferred return = PV x b, if ar = 0

Equation FVat4 is really a generic formula that can be used in virtually any tax-leveraged analysis.

For example, suppose a client wishes to determine the after-tax future value of a fully tax-deferred investment such as salary reduction contributions to a SEP IRA plan. In this case, set g equal to 1, since the entire return is tax deferred, set [t.sub.d] equal to [t.sub.f], since the tax-deferred return is taxed at the investor's future ordinary income tax rate, and set b = 1, since the entire present value is a before-tax amount subject to tax upon liquidation. As a result of these assumptions, the accumulation rate, ar, equals the before-tax rate of return, r. When these values are substituted into equation FVat4, the resulting equation is the same as equation FVat2.

In contrast, assume the investment vehicle is one where the return is fully currently taxable and where PV is an after-tax contribution. In this case, no portion of the return is attributable to appreciation or is tax deferred, so g and b are set to 0, and ar then equals r x (1 - [t.sub.f])f. The result is the same as shown in equation FVat3 for the total future after-tax value from a fully taxable investment.

Example 18: In Example 16, we found that your client would be better off investing in a nondeductible IRA rather than in an IRC Section 401(k) plan if that client can earn 2% more by directing investments in the IRA. Would your client enjoy even greater success if he instead invests the funds outside the IRA in growth stocks earning a 10% total return that is composed of 2.5% dividend yield and 7.5% capital growth?

No, not generally. The IRA and the growth stock each earn 10% total return, but the IRA's return is entirely tax-deferred. Only 75% of the stock's return is tax-deferred.

However, what if the tax rate on capital gains is reduced and the tax rate on ordinary income increased? Assuming the tax rate on capital gains is capped at 20% and your client's tax rate on ordinary income increases to 36%, would the stock investment outside the IRA be preferable?

To put the stock investment in the least favorable light possible, assume the tax changes take effect right after the investment is made. In this case, the income portion of the stock investment return is immediately subjected to the higher tax rate.

Total after tax in IRA plan

P x (1 - [t.sub.c]) x {[(1 - [t.sub.f])f x [(1 + r).sup.n]] + [t.sub.f]} $1,000 x 0.72 x {[0.64 x [(1.10).sup.15]] + 0.36} $720 x [(0.64 x 4.17724816942) + 0.36] $720 x (2.67344 + 0.36) $720 x 3.03344 = $2,184.08

Total after tax in stock

ar = (g x r) + ((1 - [t.sub.c]) x (1 - g) x r) ar = (0.75 x 0.10) + (0.64 x 0.25 x 0.1) ar = 0.09100

[(1 + ar).sup.n] = [1.09100.sup.15] = 3.69293 [t.sub.d] = 0.20

$720 x (3.69293 - (0.20 x 0.075 x (2.69293 / 0.09100))) $720 x (3.69293 - 0.44389) $720 x 3.2490 = $2,339.31 Difference = ($155.23)

In this case, your client is better off investing in the stock outside the IRA. The additional after-tax gain is about 21.6% of the $720 initial investment or about 7% more than the after-tax accumulation using the IRA.

Example 19: Your client's balance in a stock account is $25,000. His basis is $15,000, so the built-in gain proportion, b, is $10,000 / $25,000, or 40%. Assume the stocks are expected to return an average of 11% (r) per year on a compound basis and that about 80% (g) of this return is expected to be attributable to capital appreciation. Assume also that your client's marginal tax rate on ordinary income is expected to be 35% and the tax rate on long-term capital gains is expected to be 20%. If your client plans to liquidate the stock account in 12 years to pay college tuitions for his children, how much money after tax should he expect to accumulate?

b = $10,000 f $25,000 = 0.4

ar = (0.8 x 0.11) + ((1 - 0.35) x (1 - 0.8) x 0.11)

ar = 0.088 + (0.65 x 0.2 x 0.11) = 0.088 + 0.0143

ar = 0.1023

[(1 + ar).sup.n] = [(1.1023).sup.12] = 3.2180863

ATFV$1 = 3.2180863 (0.2 x (0.4 + 0.8 x 0.11 x (2.2180863 / 0.1023)))

ATFV$1 = 2.75648

FV after tax of $25,000 = $25,000 x 2.75648 = $68,912

Now suppose instead that the income return on the stocks is qualified dividend income taxed at a maximum rate of 15% and that the capital gains will also be taxed at a maximum rate of 15%. In this case the future after-tax value is determined as follows:

b = $10,000 f $25,000 = 0.4

ar = (0.8 x 0.11) + ((1 - 0.15) x (1 - 0.8) x 0.11)

ar = 0.088 + (0.85 x 0.2 x 0.11) = 0.088 + 0.0187

ar = 0.1067

[(1 + ar).sup.n] = [(1.1067).sup.12] = 3.375662

ATFV$1 = 3.375662 (0.15 x (0.4 + 0.8 x 0.11 x (2.375662 f 0.1067)))

ATFV$1 = 3.0218

FV after tax of $25,000 = $25,000 x 3.0218 = $75,544

Now, suppose further that you advise your client that the 15% tax rate on qualifying dividends and capital gains is scheduled to expire after 4 years, with the tax rates reverting to the old rules. In the event this occurs, what would be the after-tax accumulation if the 15% tax regime applies in the first 4 years and the old tax regime where dividend income is taxed at ordinary income tax rates (35%) and the 20% (18% 5 year holding period) applies to capital gains? You advise your client that essentially all of his long-term gains should qualify for the 18% 5-year-holding-period rate.

From above, the accumulation rate, ar, for the first 4 years is 10.67%, so using equation FVat4, the accumulation over the first 4 years before paying the capital gains tax is

FV 4 yrs. before CG tax = $25,000 x [(1.1067).sup.4] = $25,000 x 1.5001 = $37,502

The accumulated but as yet untaxed capital gains at the end of year 4 are computed using equation FVCG:

Capital gains yr. 4

= $25,000 x (0.4 + 0.8 x 0.11 x (0.5001 /.f 0.1067)) = $25,000 x 0.812452 = $20,311.30

Therefore, the built-in gain, b, after 4 years is $20,311.30 f $37,502 = 54.16%.

To find the final after-tax accumulation In 12 years, the balance after 4 years, $37,502, must be accumulated for another 8 years applying the 35% rate for investment income and the 18% rate for capital gains using equation FVat4. As was shown above, the accumulation rate, ar, under this regime is 10.23%.

The future value before capital gain tax = $37,502 x [(1.1023).sup.8] = $37,502 x 2.17971 = $81,743

The accumulated gain

= $37,502 x (0.5416 + 0.8 x 0.11 x (1.17971 f 0.1023)) = $37,502 x 1.5564 = $58,368

The tax on long-term gain = $58,368 x 0.18 = $10,506

The after-tax accumulation = $81,743 - $10,506 = $71,237

Not surprisingly, the value falls between the $68,9l2 computed applying the old tax rules over the entire period and the $75,544 computed applying the new tax rules over the entire period.

Example 20: In Example 19, what is the effective after-tax rate of return (or tax-free-equivalent rate of return) earned by your client over the 12 year period in the case (1) where investment income is taxed at 35% and capital gains are taxed at 20 and he accumulates $68,912 after tax and (2) in the case where investment income and capital gains are taxed at 15% and he accumulates $75,544 after tax?

Using equation Rl [r = [(FV / PV).sup.1/n] - 1] to compute the rates of return, the results for cases (1) and (2) are:

(1) r = [($68,912 f $25,000).sup.1/12] - 1 = [2.7565.sup.1/12] - 1 = 1.0882 - 1 = 8.82%;

(2) r = [($75,544 f $25,000).sup.1/12] - 1 = [3.0218.sup.1/12] - 1 = 1.0965 -1 = 9.65%

Investing under the tax assumptions of scenario (1) or (2) are equivalent to investing in tax-free investments paying 8.82% or 9.65%, respectively. __

Present Value Formula

Equation PVat4 presents the after-tax, future value formula for an investment where both ordinary and capital gain or tax-deferred income elements comprise the total return. In the same fashion as any other future value formula, the inverse of equation FVat4 is the present value formula, that is PV formula = 1 / FV formula. Equation PVat4 is the present value formula for finding the amount that would have to be invested today to accumulate some specified desired after-tax future value.

Equation PVat4

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Example 20: Your client, described in Example 19, has determined that he will probably need about $120,000 to finance his children's education in 12 years. As shown in Example 19, (assuming a 35% tax rate on ordinary income and 20% tax rate on capital gains) he can expect the $25,000 he has set aside already to accumulate to $68,912 after tax in 12 years. How much would he have to add to his stock account today to accumulate the extra $51,088 after tax in 12 years?

Since the additional new investment would not have any built-in gain when invested today, b should be set at zero. From Example 20, ar is equal to 10.23% and [(1.1023).sup.12] is equal to 3.2180863, so the after-tax future value factor for a $1 investment, ATFV$1, is:

ATFV$1 = 3.2180863 (0.2 x (0.8 x 0.11 x (2.2180863 f 0.1023)))

ATFV$1 = 2.8365

PV to reach $51,088 after tax = $51,088 f 2.8365 = $18,011

The additional amount your client would have to set aside today to meet the additional college-financing need 12 years from now would be about $18,000.

Future Value of Annuity Formulas

Similar to any other annuity formula, the formula for the future value after tax of an annuity is simply the sum of the future after-tax values of each of the level periodic investments or payments. The future after-tax value of ordinary annuity payments is:

FVAEat4

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

FV (after tax) = Pmt x n x (1 - [t.sub.d] x b), if ar = 0

The built-in gain (b) in this case is measured with respect to the periodic payments, Pmt. In most cases, b will be equal to 0, since amounts invested periodically to accumulate some future value are usually after-tax dollars. However, if one is calculating the amount one would accumulate after tax by contributing $10,000 before-tax each year to an IRC Section 401(k) plan, for instance, b would be set equal to 1, [t.sub.d] and [t.sub.b] would be set equal to the investor's anticipated future tax rate on ordinary income, and ar would be set equal to the before-tax rate of return r.

This formula appears daunting but becomes less so when broken into its component parts. The future value before capital gain tax of annuity due payments can be broken into its component parts as follows. The future value before paying the tax on capital gains or tax-deferred returns of ordinary annuity payments is:

Equation FVAEbt4

FV (before tax) = Pmt x [(1 + ar).sup.n] - 1 ^, if ar [not equal to] 0

FV (before tax) = Pmt x n, if ar = 0

The total accumulated gains or total tax-deferred return is:

Equation FVAECG

Total tax - deferred return =

Pmt x {g x r /ar [([(1 + ar).sup.n] - 1/ar) - n x b}, if ar [not equal to] 0

Total tax - deferred return = Pmt x n x b, if ar = 0

Therefore, the tax on the accumulated gains or taxdeferred return is:

Equation FVAECGtax

Total tax on deferred return =

Pmt x [t.sub.d] x {g x r /ar x[([(1 + aarr).sup.n] - 1/ar)- n] + n x b}, if ar [not equal to] 0

Total tax on deferred return = Pmt x n x [t.sub.d], x b, if ar = 0

The future after-tax value of level annuity due payments is:

Equation FVABat4

FV (after tax) = Pmt x

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

FV (after tax) = Pmt x n x (1 - [t.sub.d] x b), if ar = 0

The future value before paying the tax on capital gains or tax-deferred returns of annuity due payments is:

Equation FVABbt4

FV (before tax) = Pmt x ((1 at ar).sup.n] -1]/ar) x (1 + ar), if ar [not equal to] 0

FV (before tax) = Pmt x n, if ar = 0

The total accumulated gains or total tax-deferred return is:

Equation FVABCG

Total tax - deferred return = Pmt x

{g x r /ar x [([(1 + ar).sup.n] - 1 /ar) x (1 + ar) - n] + n x b 1, if ar [not equal to] 0

Total tax - deferred return = Pmt x n x b, if ar = 0

Therefore, the tax on the accumulated gains or tax-deferred return is:

Equation FVABCGtax

Total tax on deferred return = Pmt x [t.sub.d] x

{g x r /ar x [([(1 + ar).sup.n] - 1 /ar) x (1 + ar) - n] + n x b 1, if ar [not equal to] 0

Total tax on deferred return = Pmt x n x [t.sub.d] x b, if ar = 0

Just as in prior cases, the formula for determining the periodic payment, Pmt, is derived by dividing FV(after tax) by the multiplier of Pmt in equations FVAEat4 or FVABat4, as appropriate.

Present Value of Annuity Formulas

The formulas for determining the present value amount one needs to invest today to generate after-tax ordinary annuity payments cannot be simplified to a reduced form similar to the future value annuity equations. In order to compute the present value of a series of periodic after-tax payments, one must use equation PVat4 to compute the present value of each periodic payment separately and then sum them to derive the total amount one needs to invest today to generate the given series of payments. Therefore, the present value of a series of ordinary annuity payments is:

Equation PVAEat4

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The present value of annuity due payments is:

Equation PVABat4

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Once again, the formula for determining the periodic payment, Pmt, is derived by dividing PV by the multiplier of Pmt(after tax) in equations PVAEat4 or PVABat4, as appropriate.

Inflation/Growth Adjusted Formulas

Inflation/Growth Adjusted Rate of Return

The tax-adjusted time value formulas can be modified to include inflation or growth adjustments in a manner analogous to the discussion earlier. Simply substituting ar for r in equation RIA, one derives the inflation- or growth-adjusted accumulation rate, [rho]. That is, where all variables are as defined earlier:

[rho] = 1 + ar/1 + I = ar - 1/1 + i

Inflation/Growth-Adjusted Payment Streams

Investment and financial planning applications often involve payment streams that are adjusted for anticipated inflation or growth. For instance, when ascertaining how much a person needs to accumulate for retirement, the analysis usually makes some assumption about how much the cost of living will increase each year during retirement and increases the required annual income accordingly. Analogously, when determining how much a person needs to save each year for retirement, the analysis often assumes that payments will increase each year in relation to the anticipated growth in the investor's salary.

The future after-tax value of a series of ordinary annuity payments increasing (or decreasing) each year at i% (i [not equal to] 0) per year is:

Equation FVAEat5

FV (after tax) = Pmt x

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Similarly, the future after-tax value of annuity due payments increasing (or decreasing) each year at i% (i [not equal to] 0) per year is:

Equation FVABat5

FV (after tax) = Pmt x

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Similar to equations PVAEat4 and PVABat4, the present value of the investment necessary to generate inflation/growth-adjusted after-tax annuity payments cannot be expressed in a simple reduced-form equation. Rather, the present value of the total investment is equal to the sum of the present values of each inflation/growth-adjusted after-tax payment computed separately. The ordinary annuity payment formula is:

Equation PVAEat5

PV = Pmt (after tax) x

[n-1.summation over (j=1)] [(1 + i).sup.j-1] x {[(1 + ar).sup.j] - [t.sub.d] x [g x r x ((1 + ar).sup.i] - 1/ar])-b]}

if ar [not equal to] 0

PV = Pmt (after tax)/(1 - [t.sub.d] x b) x ([(1 + i).sup.n]-1/i), if ar = 0

The annuity due payment formula is:

Equation PVABat5

PV = Pmt (after tax) x

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

CHAPTER ENDNOTES

(1.) Under current law, the step-up in basis at death is scheduled to be largely eliminated in 2010. However, that same legislation is also scheduled to "sunset" with estate tax and step-up basis rules reverting to 2001 law after 2010. At this time it is impossible to be sure whether the law will be permitted to expire or whether it will be extended beyond 2010. Most experts expect that at least some of the current law rules will be extended beyond 2010, but have doubts about whether the step-up basis rules will be largely phased-out.

(2.) A 50% tax rate is used to simplify the numbers and better illustrate the concepts. Under the current tax regime, a 50% tax rate would be rare even if one combined the highest federal, state, and local tax rates.

(3.) Under the 2003 tax act, the tax rate changes scheduled for future years have been accelerated into 2003. In addition, the maximum tax rate on certain dividend income and on long-term capital gains for most common investment instruments has been cut to 15%. The examples in this chapter were calculated using tax rates that may or may not be actual tax rates at any point in time. However, the general principles and concepts presented in these examples do not change with changes in tax rates, although the relative after-tax returns certainly do. For instance, under new law, many equity investments qualify for a 15% tax on dividend income and for a maximum long-term capital gain rate of 15% and so these investments become relatively better after-tax investments than they were previously. However, these tax benefits still do not generally outweigh or overcome the tax advantages of making either after-tax contributions to a tax-free investment, such as a Roth IRA, or before-tax contributions to a tax-deferred investment, such as a 401(k) plan.

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Title Annotation: | Appendices |
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Publication: | Tools & Techniques of Investment Planning, 2nd ed. |

Date: | Jan 1, 2006 |

Words: | 11029 |

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