19 45 137,952 1.56 20,693 308,859 75,264

29 55 271,372 2.09 40,706 943,477 110,167

39 65 533,829 2.81 80,074 2,457,518 161,257

40 Total 7,445,673 1,116,851 Real Annuity 49,668

A B C D E F

1 Retirement Years Income Growth Rate of Inflation Savings Rate ROR rROR

2 25 0.07 0.03 0.15 0.06 =(E2-C2)/(1+C2)

3 Age Income Deflator Savings Cumulative Savings rConsumption

4 30 50000 1 =B4*$D$2 =D4 =(B4-D4)/C4

5 31 =B4*(1+$B$2) =C4*(1+$C$2) =B5*$D$2 =E4*(1+$E$2)+D5 =(B5-D5)/C5

39 65 =B38*(1+$B$2) =C38*(1+$C$2) =B39*$D$2 =E38*(1+$E$2)+D39 =(B39-D39)/C39

40 Total =SUM(B4:B39) =SUM(D4:D39) Real Annuity =PMT($F$2,$A$2,-$E$39/$C$39,0,0)

Bodie’Kane’Marcus: VI. Active Investment 18. Taxes, Inflation, and © The McGraw’Hill

Essentials of Investments, Management Investment Strategy Companies, 2003

Fifth Edition

629

18 Taxes, Inflation, and Investment Strategy

In our initial plan (Spreadsheet 18.1), we envisioned consuming a level, nominal annuity

for the retirement years. This is an inappropriate goal once we account for inflation, since it

would imply a declining standard of living starting at age 65. Its purchasing power at age 65

in terms of current dollars would be $64,542 (i.e., $181,362/2.81), and at age 90 only $30,792.

(Check this!)

It is tempting to contemplate solving the problem of an inadequate retirement annuity by

increasing the assumed rate of return on investments. However, this can only be accomplished

by putting your savings at risk. Much of this text elaborates on how to do so efficiently; yet it

also emphasizes that while taking on risk will give you an expectation for a better retirement,

it implies as well a nonzero probability of doing a lot worse. At the age of 30, you should be

able to tolerate some risk to the retirement annuity for the simple reason that if things go

wrong, you can change course, increase your savings rate, and work harder. As you get older,

this option progressively fades, and increasing risk becomes less of a viable option. If you do

choose to increase risk, you can set a “safety-first target” (i.e., a minimum acceptable goal) for

the retirement annuity and continuously monitor your risky portfolio. If the portfolio does

poorly and approaches the safety-first target, you progressively shift into risk-free bonds”you

may recognize this strategy as a version of dynamic hedging.

The difficulty with this strategy is twofold: First it requires monitoring, which is time-

consuming and may be nerve-racking as well. Second, when decision time comes, it may be

psychologically hard to withdraw. By shifting out of the risky portfolio if and when your port-

folio is hammered, you give up any hope of recovery. This is hard to do and many investors

fail the test. For these investors, therefore, the right approach is to stick with the safe, lower

ROR and make the effort to balance standard of living before and after retirement. Avoiding

sleepless nights is ample reward.

Therefore, the only variable we leave under your control in this spreadsheet is the rate of

saving. To improve retirement life style relative to the preretirement years, without jeopar-

dizing its safety, you will have to lower consumption during the saving years”there is no

free lunch.

<

2. If you project a rate of inflation of 4%, what nominal ROR on investments would Concept

you need to maintain the same real retirement annuity as in Spreadsheet 18.2?

CHECK

An Alternative Savings Plan

In Spreadsheet 18.2, we saved a constant fraction of income. But since real income grows over

time (nominal income grows at 7% while inflation is only 3%), we might consider deferring

our savings toward future years when our real income is higher. By applying a higher savings

rate to our future (higher) real income, we can afford to reduce the current savings rate. In

Spreadsheet 18.3, we use a base savings rate of 10% (lower than the savings rate in the previ-

ous spreadsheet), but we increase the savings target by 3% per year. Saving in each year there-

fore equals a fixed savings rate times annual income (column B), times 1.03t. By saving a

larger fraction of income in later years, when real income is larger, you create a smoother pro-

file of real consumption.

Spreadsheet 18.3 shows that with an initial savings rate of 10%, compared with the un-

changing 15% rate in the previous spreadsheet, you can achieve a retirement annuity of

$59,918, larger than the $49,668 annuity in the previous plan.

Notice that real consumption in the early years is greater than with the previous plan. What

you have done is to postpone saving until your income is much higher. At first blush, this plan

is preferable: It allows for a more comfortable consumption of 90% of income at the outset, a

consistent increase in standard of living during your earning years, all without significantly af-

fecting the retirement annuity. But this program has one serious downside: By postponing the

Bodie’Kane’Marcus: VI. Active Investment 18. Taxes, Inflation, and © The McGraw’Hill

Essentials of Investments, Management Investment Strategy Companies, 2003

Fifth Edition

630 Part SIX Active Investment Management

S P R E A D S H E E T 18.3

Saving from real income

A B C D E F

1 Retirement Years Income Growth Rate of Inflation Savings Rate ROR rROR

2 25 0.07 0.03 0.1 0.06 0.0291

3 Age Income Deflator Savings Cumulative Savings rConsumption

4 30 50,000 1.00 5,000 5,000 45,000

5 31 53,500 1.03 5,511 10,811 46,592

9 35 70,128 1.16 8,130 44,351 53,480

19 45 137,952 1.56 21,492 260,927 74,751

29 55 271,372 2.09 56,819 947,114 102,471

39 65 533,829 2.81 150,212 2,964,669 136,331

40 Total 7,445,673 1,572,466 Real Annuity 59,918

A B C D E F

1 Retirement Years Income Growth Rate of Inflation Savings Rate ROR rROR

2 25 0.07 0.03 0.1 0.06 =(E2-C2)/(1+C2)

3 Age Income Deflator Savings Cumulative Savings rConsumption

4 30 50000 1 =B4*C4*$D$2 =D4 =(B4-D4)/C4

5 31 =B4*(1+$B$2) =C4*(1+$C$2) =B5*C5*$D$2 =E4*(1+$E$2)+D5 =(B5-D5)/C5

39 65 =B38*(1+$B$2) =C38*(1+$C$2) =B39*C39*$D$2 =E38*(1+$E$2)+D39 =(B39-D39)/C39

40 Total =SUM(B4:B39) =SUM(D4:D39) Real Annuity =PMT($F$2,$A$2,-$E$39/$C$39,0,0)

bulk of your savings to a later age, you come to depend on your health, longevity, and, more

ominously (and without possibility of insurance), on a successful future career. Put differently,

this plan achieves comfort by increasing risk, making this choice a matter of risk tolerance.

>

3. Suppose you like the plan of tilting savings toward later years, but worry about the

Concept

increased risk of postponing the bulk of your savings to later years. Is there any-

CHECK thing you can do to mitigate the risk?

18.3 ACCOUNTING FOR TAXES

To initiate a discussion of taxes, let™s assume that you are subject to a flat tax rate of 25% on

flat tax

taxable income less one exemption of $15,000. This is similar to several proposals for a sim-

A tax code that

plified U.S. tax code that have been floated by one presidential candidate or another prior to

taxes all income

elections”at least when you add state taxes to the proposed flat rate. An important feature of

above some

exemption at this (and the existing) tax code is that the tax rate is levied on nominal income and applies as

a fixed rate. well to investment income. (This is the concept of double taxation”you pay taxes when you

earn income and then you pay taxes again when your savings earn interest). Some relief from

the effect of taxing nominal dollars both in this proposal and the current U.S. code is provided

by raising the exemption, annually, by the rate of inflation. To adapt our spreadsheet to this

simple tax code, we must add columns for taxes and after-tax income. The tax-adjusted plan

is shown in Spreadsheet 18.4. It adapts the savings plan of Spreadsheet 18.2.

The top panel of the sheet deals with the earning years. Column D adjusts the exemption

(D2) by the price level (column C). Column E applies the tax rate (cell E2) to taxable income

(column B column D). The savings rate (F2) is applied to after-tax income (column B

column E), allowing us to calculate cumulative savings (column G) and real consumption

(column H). The formula view shows the detailed construction.

As you might have expected, real consumption is lower in the presence of taxes, as are sav-

ings and the retirement fund. The retirement fund provides for a real, before-tax annuity of

only $37,882, compared with $49,668 absent taxes in Spreadsheet 18.2.

The bottom panel of the sheet shows the further reduction in real consumption due to

taxes paid during the retirement years. While you do not pay taxes on the cumulative savings

in the retirement plan (you did that already as the savings accrued interest), you do pay taxes

on interest earned by the fund while you are drawing it down. These taxes are quite signifi-

cant and further deplete the fund and its net-of-tax earning power. For this reason, your

Bodie’Kane’Marcus: VI. Active Investment 18. Taxes, Inflation, and © The McGraw’Hill

Essentials of Investments, Management Investment Strategy Companies, 2003

Fifth Edition

631

18 Taxes, Inflation, and Investment Strategy

S P R E A D S H E E T 18.4

Saving with a simple tax code

A B C D E F G H

1 Retirement Years Income Growth Rate of Inflation Exemption Now Tax Rate Savings Rate ROR rROR

2 25 0.07 0.03 15000 0.25 0.15 0.06 0.0291

3 Age Income Deflator Exemption Taxes Savings Cumulative Savings rConsumption

4 30 50,000 1.00 15,000 8,750 6,188 6,188 35,063

5 31 53,500 1.03 15,450 9,605 6,584 13,143 36,224

9 35 70,128 1.16 17,389 13,775 8,453 50,188 41,319

19 45 137,952 1.56 23,370 31,892 15,909 245,334 57,864

29 55 271,372 2.09 31,407 69,943 30,214 733,467 81,773

39 65 533,829 2.81 42,208 148,611 57,783 1,874,346 116,365

40 Total 1,884,163 834,226 Real Annuity= 37,882

RETIREMENT

41

42 Age Nom Withdraw Deflator Exemption Taxes Funds Left rConsumption

43 66 109,792 2.90 43,474 17,247 1,877,014 31,931

47 70 123,572 3.26 48,931 15,743 1,853,382 33,056

52 75 143,254 3.78 56,724 12,200 1,721,015 34,656

57 80 166,071 4.38 65,759 6,047 1,422,954 36,503

62 85 192,521 5.08 76,232 0 883,895 37,882

67 90 223,185 5.89 88,374 0 0 37,882

68 Total 4,002,944 203,199

A B C D E F G H

1 Retirement Years Income Growth Rate of Inflation Exemption Now Tax Rate Savings Rate ROR rROR

2 25 0.07 0.03 15000 0.25 0.15 0.06 =(G2-C2)/(1+C2)

3 Age Income Deflator Exemption Taxes Savings Cumulative Savings rConsumption

4 30 50000 1 =$D$2*C4 =(B4-D4)*$E$2 =(B4-E4)*$F$2 =F4 =(B4-E4-F4)/C4

5 31 =B4*(1+$B$2) =C4*(1+$C$2) =$D$2*C5 =(B5-D5+G4*$G$2)*$E$2 =(B5-E5)*$F$2 =G4*(1+$G$2)+F5 =(B5-E5-F5)/C5

39 65 =B38*(1+$B$2) =C38*(1+$C$2) =$D$2*C39 =(B39-D39+G38*$G$2)*$E$2 =(B39-E39)*$F$2 =G38*(1+$G$2)+F39 =(B39-E39-F39)/C39

40 Total =SUM(E4:E39) =SUM(F4:F39) Real Annuity =PMT($H$2,$A$2,-$G$39/$C$39,0,0)

41 RETIREMENT

42 Age Nom Withdraw Deflator Exemption Taxes Funds Left rConsumption

43 66 =$H$40*C43 =C39*(1+$C$2) =$D$2*C43 =MAX(0,(G39*$G$2-D43)*$E$2) =G39*(1+$G$2)-B43 =(B43-E43)/C43

44 67 =$H$40*C44 =C43*(1+$C$2) =$D$2*C44 =MAX(0,(G43*$G$2-D44)*$E$2) =G43*(1+$G$2)-B44 =(B44-E44)/C44

67 90 =$H$40*C67 =C66*(1+$C$2) =$D$2*C67 =MAX(0,(G66*$G$2-D67)*$E$2) =G66*(1+$G$2)-B67 =(B67-E67)/C67

68 Total =SUM(B43:B67) =SUM(E43:E67)

consumption annuity is lower in the early years when your fund has not yet been depleted and

earns quite a bit.

In the end, despite a handsome income that grows at a real rate of almost 4%, an aggressive

savings rate of 15%, a modest rate of inflation, and a modest tax, you will only be able to

achieve a modest (but at least low-risk) real retirement income. This is a reality with which

most people must struggle. Whether to sacrifice more of today™s standard of living through an

increased rate of saving, or take some risk in the form of saving a real annuity and/or invest in

a risky portfolio with a higher expected return, is a question of preference and risk tolerance.

One often hears complaints about the double taxation resulting from taxing income earned

on savings from dollars on which taxes were already paid. It is interesting to see what effec-

tive tax rate is imposed on your lifetime earnings by double taxation. To do so, we use Spread-

sheet 18.4 to set up your lifetime earnings, exemptions, and taxes:

Income

Labor income $7,445,673

Total exemptions during working years 949,139

(i) Lifetime taxable income $6,496,534

Taxes

During labor years 1,884,163

During retirement 203,199

(ii) Lifetime taxes $2,087,362

Lifetime tax rate (ii)/(i) 32.13%

Thus, double taxation is equivalent to raising the effective tax rate on long-term savers from

the statutory rate of 25% to an effective rate of over 32%.

<

4. Would a 1% increase in the exemption compensate you for a 1% increase in the Concept

tax rate?

CHECK

Bodie’Kane’Marcus: VI. Active Investment 18. Taxes, Inflation, and © The McGraw’Hill

Essentials of Investments, Management Investment Strategy Companies, 2003

Fifth Edition

632 Part SIX Active Investment Management

18.4 THE ECONOMICS OF TAX SHELTERS

Tax shelters range from the simple to the mind-bogglingly complex, yet they all have one

tax shelters

common objective: to postpone payment of tax liabilities for as long as possible. We know

Means by which to

already that this isn™t small fry. Postponement implies a smaller present value of tax payment,

postpone payment of

and a tax paid with a long delay can have present value near zero. However, delay is neces-

tax liabilities for as

long as possible. sarily beneficial only when the tax rate doesn™t increase over time. If the tax rate on retirement

income is higher than during earning years, the value of a tax deferral may be questionable; if

the tax rate will decline, deferral is even more preferable.

A Benchmark Tax Shelter

Postponing tax payments is the only attainable (legal) objective since, whenever you have tax-

able income, a tax liability is created that can (almost) never be erased.4 For this reason, a

benchmark tax shelter postpones all taxes on savings and the income on those savings. In this

case, your entire savings account is liable to taxation and will be paid upon retirement, as you

draw down the retirement fund. This sort of shelter is actually equivalent to the tax treatment