Number

of $1 a year for each of t of Years 5% 6% 7% 8% 9% 10%

years

1 .952 .943 .935 .926 .917 .909

2 1.859 1.833 1.808 1.783 1.759 1.736

3 2.723 2.673 2.624 2.577 2.531 2.487

4 3.546 3.465 3.387 3.312 3.240 3.170

5 4.329 4.212 4.100 3.993 3.890 3.791

10 7.722 7.360 7.024 6.710 6.418 6.145

20 12.462 11.470 10.594 9.818 9.129 8.514

30 15.372 13.765 12.409 11.258 10.274 9.427

54 SECTION ONE

PV = $465,000 — 20-year annuity factor

[ ]

1 1

= $465,000 — “

r r(1 + r)20

At an interest rate of 8 percent, the annuity factor is

[ ]

1 1

“ = 9.818

.08 .08(1.08)20

(We also could look up the annuity factor in either Table 1.8 or Table A.3.) The present

value of the $465,000 annuity is $465,000 — 9.818 = $4,565,000. That “$9.3 million

prize” has a true value of about $4.6 million.

This present value is the price which investors would be prepared to offer for the se-

ries of cash flows. For example, the gambling casino might arrange for an insurance

company to actually make the payments to the lucky winner. In this case, the company

would charge a bit under $4.6 million to take over the obligation. With this amount in

hand today, it could generate enough interest income to make the 20 payments before

running its “account” down to zero.

ANNUITIES DUE

The perpetuity and annuity formulas assume that the first payment occurs at the end of

the period. They tell you the value of a stream of cash payments starting one period

hence.

However, streams of cash payments often start immediately. For example, Kangaroo

Autos in Example 8 might have required three annual payments of $4,000 starting im-

mediately. A level stream of payments starting immediately is known as an annuity

Level

ANNUITY DUE

due.

stream of cash flows starting

If Kangaroo™s loan were paid as an annuity due, you could think of the three pay-

immediately.

ments as equivalent to an immediate payment of $4,000 plus an ordinary annuity of

$4,000 for the remaining 2 years. This is made clear in Figure 1.12, which compares the

cash-flow stream of the Kangaroo Autos loan treating the three payments as an annuity

(panel a) and as an annuity due (panel b).

In general, the present value of an annuity due of t payments of $1 a year is the same

as $1 plus the present value of an ordinary annuity providing the remaining t “ 1 pay-

ments. The present value of an annuity due of $1 for t years is therefore

PV annuity due = 1 + PV ordinary annuity of t “ 1 payments

[ ]

1 1

= 1+ “

r r (1 + r)t“1

By comparing the two panels of Figure 1.12, you can see that each of the three cash

flows in the annuity due comes one period earlier than the corresponding cash flow of

the ordinary annuity. Therefore, the present value of an annuity due is (1 + r) times the

present value of an annuity.2 Figure 1.12 shows that the effect of bringing the Kangaroo

loan payments forward by 1 year was to increase their value from $9,947.41 (as an an-

nuity) to $10,942.15 (as an annuity due). Notice that $10,942.15 = $9,947.41 — 1.10.

2Your financial calculator is equipped to handle annuities due. You simply need to put the calculator in

“begin” mode, and the stream of cash flows will be interpreted as starting immediately. The begin key is la-

beled BGN or BEG/END. Each time you press the key, the calculator will toggle between ordinary annuity

versus annuity due mode.

The Time Value of Money 55

FIGURE 1.12

Annuity versus annuity due. 3-year ordinary annuity

(a) Three-year ordinary

annuity. (b) Three-year $4,000 $4,000 $4,000

annuity due.

Year

0 1 2 3

Present value

4,000

$3,636.36

1.10

4,000

$3,305.79

(1.10)2

4,000

$3,005.26

(1.10)3

(a)

Total $9,947.41

Immediate

2-year ordinary annuity

payment

$4,000 $4,000 $4,000

Year

0 1 2 3

Present value

$4,000.00

4,000

$3,636.36

1.10

4,000

$3,305.79

(1.10)2

Total $10,942.15 (b)

When calculating the value of the slot machine winnings in Example 9, we assumed

Self-Test 8

that the first of the 20 payments occurs at the end of 1 year. However, the payment was

probably made immediately, with the remaining payments spread over the following 19

years. What is the present value of the $9.3 million prize?

Home Mortgages

EXAMPLE 10

Sometimes you may need to find the series of cash payments that would provide a given

value today. For example, home purchasers typically borrow the bulk of the house price

from a lender. The most common loan arrangement is a 30-year loan that is repaid in

56 SECTION ONE

equal monthly installments. Suppose that a house costs $125,000, and that the buyer

puts down 20 percent of the purchase price, or $25,000, in cash, borrowing the remain-

ing $100,000 from a mortgage lender such as the local savings bank. What is the ap-

propriate monthly mortgage payment?

The borrower repays the loan by making monthly payments over the next 30 years

(360 months). The savings bank needs to set these monthly payments so that they have

a present value of $100,000. Thus

Present value = mortgage payment — 360-month annuity factor

= $100,000

Mortgage payment = $100,000

360-month annuity factor

Suppose that the interest rate is 1 percent a month. Then

Mortgage payment = $100,000

[ ]

1 1

“

. .01 .01(1.01)360

= $100,000

97.218

= $1,028.61

This type of loan, in which the monthly payment is fixed over the life of the mort-

gage, is called an amortizing loan. “Amortizing” means that part of the monthly pay-

ment is used to pay interest on the loan and part is used to reduce the amount of the

loan. For example, the interest that accrues after 1 month on this loan will be 1 percent

of $100,000, or $1,000. So $1,000 of your first monthly payment is used to pay inter-

est on the loan and the balance of $28.61 is used to reduce the amount of the loan to

$99,971.39. The $28.61 is called the amortization on the loan in that month.

Next month, there will be an interest charge of 1 percent of $99,971.39 = $999.71.

So $999.71 of your second monthly payment is absorbed by the interest charge and the

remaining $28.90 of your monthly payment ($1,028.61 “ $999.71 = $28.90) is used to

reduce the amount of your loan. Amortization in the second month is higher than in the

first month because the amount of the loan has declined, and therefore less of the pay-

ment is taken up in interest. This procedure continues each month until the last month,

when the amortization is just enough to reduce the outstanding amount on the loan to

zero, and the loan is paid off.

Because the loan is progressively paid off, the fraction of the monthly payment de-

voted to interest steadily falls, while the fraction used to reduce the loan (the amortiza-

tion) steadily increases. Thus the reduction in the size of the loan is much more rapid in

the later years of the mortgage. Figure 1.13 illustrates how in the early years almost all

of the mortgage payment is for interest. Even after 15 years, the bulk of the monthly

payment is interest.

What will be the monthly payment if you take out a $100,000 fifteen-year mortgage at

Self-Test 9

an interest rate of 1 percent per month? How much of the first payment is interest and

how much is amortization?

The Time Value of Money 57

FIGURE 1.13

Mortgage amortization. This

Amortization Interest Paid

figure shows the breakdown 14,000

of mortgage payments

between interest and 12,000

amortization. Monthly

10,000

payments within each year

are summed, so the figure

8,000

Dollars

shows the annual payment on

the mortgage.

6,000

4,000

2,000

0

1 4 7 10 13 16 19 22 25 28

Year

How Much Luxury and Excitement

EXAMPLE 11

Can $96 Billion Buy?

Bill Gates is reputedly the world™s richest person, with wealth estimated in mid-1999 at

$96 billion. We haven™t yet met Mr. Gates, and so cannot fill you in on his plans for al-

locating the $96 billion between charitable good works and the cost of a life of luxury

and excitement (L&E). So to keep things simple, we will just ask the following entirely

hypothetical question: How much could Mr. Gates spend yearly on 40 more years of

L&E if he were to devote the entire $96 billion to those purposes? Assume that his

money is invested at 9 percent interest.

The 40-year, 9 percent annuity factor is 10.757. Thus

Present value = annual spending — annuity factor

$96,000,000,000 = annual spending — 10.757

Annual spending = $8,924,000,000

Warning to Mr. Gates: We haven™t considered inflation. The cost of buying L&E will

increase, so $8.9 billion won™t buy as much L&E in 40 years as it will today. More on

that later.

Suppose you retire at age 70. You expect to live 20 more years and to spend $55,000 a

Self-Test 10

year during your retirement. How much money do you need to save by age 70 to sup-

port this consumption plan? Assume an interest rate of 7 percent.

FUTURE VALUE OF AN ANNUITY

You are back in savings mode again. This time you are setting aside $3,000 at the end

of every year in order to buy a car. If your savings earn interest of 8 percent a year, how

58 SECTION ONE

FIGURE 1.14

Future value of an annuity $3,000 $3,000 $3,000 $3,000