Year

0 1 2 3 4

Future value in Year 4

$3,000 3,000

$3,240 3,000 1.08

(1.08)2

$3,499 3,000

(1.08)3

$3,799 3,000

$13,518

much will they be worth at the end of 4 years? We can answer this question with the

help of the time line in Figure 1.14. Your first year™s savings will earn interest for 3

years, the second will earn interest for 2 years, the third will earn interest for 1 year, and

the final savings in Year 4 will earn no interest. The sum of the future values of the four

payments is

($3,000 — 1.083) + ($3,000 — 1.082) + ($3,000 — 1.08) + $3,000 = $13,518

But wait a minute! We are looking here at a level stream of cash flows”an annuity.

We have seen that there is a short-cut formula to calculate the present value of an an-

nuity. So there ought to be a similar formula for calculating the future value of a level

stream of cash flows.

Think first how much your stream of savings is worth today. You are setting aside

$3,000 in each of the next 4 years. The present value of this 4-year annuity is therefore

equal to

PV = $3,000 — 4-year annuity factor

[ ]

1 1

= $3,000 — “ = $9,936

.08 .08(1.08)4

Now think how much you would have after 4 years if you invested $9,936 today. Sim-

ple! Just multiply by (1.08)4:

Value at end of Year 4 = $9,936 — 1.084 = $13,518

We calculated the future value of the annuity by first calculating the present value and

then multiplying by (1 + r)t. The general formula for the future value of a stream of cash

flows of $1 a year for each of t years is therefore

Future value of annuity of $1 a year = present value of annuity

of $1 a year (1 + r)t

[ ]

1 1

(1 + r)t

= “

r r(1 + r)t

(1 + r)t “ 1

=

r

If you need to find the future value of just four cash flows as in our example, it is a

toss up whether it is quicker to calculate the future value of each cash flow separately

The Time Value of Money 59

TABLE 1.9

Interest Rate per Year

Future value of a $1 annuity

Number

of Years 5% 6% 7% 8% 9% 10%

1 1.000 1.000 1.000 1.000 1.000 1.000

2 2.050 2.060 2.070 2.080 2.090 2.100

3 3.153 3.184 3.215 3.246 3.278 3.310

4 4.310 4.375 4.440 4.506 4.573 4.641

5 5.526 5.637 5.751 5.867 5.985 6.105

10 12.578 13.181 13.816 14.487 15.193 15.937

20 33.066 36.786 40.995 45.762 51.160 57.275

30 66.439 79.058 94.461 113.283 136.308 164.494

(as we did in Figure 1.14) or to use the annuity formula. If you are faced with a stream

of 10 or 20 cash flows, there is no contest.

You can find a table of the future value of an annuity in Table 1.9, or the more exten-

sive Table A.4 at the end of the material. You can see that in the row corresponding to

t = 4 and the column corresponding to r = 8%, the future value of an annuity of $1 a year

is $4.506. Therefore, the future value of the $3,000 annuity is $3,000 — 4.506 = $13,518.

Remember that all our annuity formulas assume that the first cash flow does not

occur until the end of the first period. If the first cash flow comes immediately, the fu-

ture value of the cash-flow stream is greater, since each flow has an extra year to earn

interest. For example, at an interest rate of 8 percent, the future value of an annuity start-

ing with an immediate payment would be exactly 8 percent greater than the figure given

by our formula.

Saving for Retirement

EXAMPLE 12

In only 50 more years, you will retire. (That™s right”by the time you retire, the retire-

ment age will be around 70 years. Longevity is not an unmixed blessing.) Have you

started saving yet? Suppose you believe you will need to accumulate $500,000 by your

retirement date in order to support your desired standard of living. How much must you

save each year between now and your retirement to meet that future goal? Let™s say that

the interest rate is 10 percent per year. You need to find how large the annuity in the fol-

lowing figure must be to provide a future value of $500,000:

$500,000

0 1 2 3 4 • • • • 48 49 •

Level savings (cash inflows) in years

1“50 result in a future accumulated

value of $500,000

FINANCIAL CALCULATOR

Solving Annuity Problems

Using a Financial Calculator

The formulas for both the present value and future value What about the balance left on the mortgage after 10

of an annuity are also built into your financial calculator. years have passed? This is easy: the monthly payment is

Again, we can input all but one of the five financial keys, still PMT = “1,028.61, and we continue to use i = 1 and

and let the calculator solve for the remaining variable. In FV = 0. The only change is that the number of monthly

payments remaining has fallen from 360 to 240 (20 years

these applications, the PMT key is used to either enter

or solve for the value of an annuity. are left on the loan). So enter n = 240 and compute PV as

93,417.76. This is the balance remaining on the mortgage.

Solving for an Annuity Future Value of an Annuity

In Example 3.12, we determined the savings stream In Figure 3.12, we showed that a 4-year annuity of $3,000

that would provide a retirement goal of $500,000 after invested at 8 percent would accumulate to a future value

50 years of saving at an interest rate of 10 percent. To of $13,518. To solve this on your calculator, enter n = 4, i

find the required savings each year, enter n = 50, i = 10, = 8, PMT = “3,000 (we enter the annuity paid by the in-

FV = 500,000, and PV = 0 (because your “savings ac- vestor to her savings account as a negative number since

count” currently is empty). Compute PMT and find that it is a cash outflow), and PV = 0 (the account starts with

it is “$429.59. Again, your calculator is likely to display no funds). Compute FV to find that the future value of the

the solution as “429.59, since the positive $500,000 savings account after 3 years is $13,518.

cash value in 50 years will require 50 cash payments

Calculator Self-Test Review (answers follow)

(outflows) of $429.59.

1. Turn back to Kangaroo Autos in Example 3.8. Can you

The sequence of key strokes on three popular cal-

now solve for the present value of the three installment

culators necessary to solve this problem is as follows:

payments using your financial calculator? What key

Hewlett-Packard Sharpe Texas Instruments strokes must you use?

HP-10B EL-733A BA II Plus 2. Now use your calculator to solve for the present value of

PV PV PV

0 0 0 the three installment payments if the first payment comes

immediately, that is, as an annuity due.

n n n

50 50 50

3. Find the annual spending available to Bill Gates using the

I/YR i I/Y

10 10 10

data in Example 3.11 and your financial calculator.

FV FV FV

500,000 500,000 500,000

PMT COMP PMT CPT PMT

Solutions to Calculator Self-Test Review Questions

1. Inputs are n = 3, i = 10, FV = 0, and PMT = 4,000. Com-

Your calculator displays a negative number, as the 50

pute PV to find the present value of the cash flows as

cash outflows of $429.59 are necessary to provide for

$9,947.41.

the $500,000 cash value at retirement.

2. If you put your calculator in BEGIN mode and recalcu-

late PV using the same inputs, you will find that PV has

Present Value of an Annuity

increased by 10 percent to $10,942.15. Alternatively, as

depicted in Figure 3.10, you can calculate the value of the

In Example 3.10 we considered a 30-year mortgage

$4,000 immediate payment plus the value of a 2-year an-

with monthly payments of $1,028.61 and an interest

nuity of $4,000. Inputs for the 2-year annuity are n = 2, i

rate of 1 percent. Suppose we didn™t know the amount

= 10, FV = 0, and PMT = 4,000. Compute PV to find the

of the mortgage loan. Enter n = 360 (months), i = 1, PMT

present value of the cash flows as $6,942.15. This amount

= “1,028.61 (we enter the annuity level paid by the bor-

plus the immediate $4,000 payment results in the same

rower to the lender as a negative number since it is a

total present value: $10,942.15.

cash outflow), and FV = 0 (the mortgage is wholly paid

3. Inputs are n = 40, i = 9, FV = 0, PV = “96,000 million.

off after 30 years; there are no final future payments be-

Compute PMT to find that the 40-year annuity with pres-

yond the normal monthly payment). Compute PV to find

ent value of $96 billion is $8,924 million.

that the value of the loan is $100,000.

60

The Time Value of Money 61

We know that if you were to save $1 each year your funds would accumulate to

(1 + r)t “ 1 (1.10)50 “ 1

Future value of annuity of $1 a year = =

r .10

= $1,163.91

(Rather than compute the future value formula directly, you could look up the future

value annuity factor in Table 1.9 or Table A.4. Alternatively, you can use a financial

calculator as we describe in the nearby box.) Therefore, if we save an amount of $C each

SEE BOX

year, we will accumulate $C — 1,163.91.

We need to choose C to ensure that $C — 1,163.91 = $500,000. Thus C =

$500,000/1,163.91 = $429.59. This appears to be surprisingly good news. Saving

$429.59 a year does not seem to be an extremely demanding savings program. Don™t

celebrate yet, however. The news will get worse when we consider the impact of

inflation.

What is the required savings level if the interest rate is only 5 percent? Why has the

Self-Test 11

amount increased?

Inflation and the Time Value of Money

When a bank offers to pay 6 percent on a savings account, it promises to pay interest of

$60 for every $1,000 you deposit. The bank fixes the number of dollars that it pays, but

it doesn™t provide any assurance of how much those dollars will buy. If the value of your

investment increases by 6 percent, while the prices of goods and services increase by

10 percent, you actually lose ground in terms of the goods you can buy.

REAL VERSUS NOMINAL CASH FLOWS

Prices of goods and services continually change. Textbooks may become more expen-

sive (sorry) while computers become cheaper. An overall general rise in prices is known

as inflation. If the inflation rate is 5 percent per year, then goods that cost $1.00 a year

Rate at

INFLATION

ago typically cost $1.05 this year. The increase in the general level of prices means that

which prices as a whole are

the purchasing power of money has eroded. If a dollar bill bought one loaf of bread last

increasing.

year, the same dollar this year buys only part of a loaf.

Economists track the general level of prices using several different price indexes.

The best known of these is the consumer price index, or CPI. This measures the num-

ber of dollars that it takes to buy a specified basket of goods and services that is sup-

posed to represent the typical family™s purchases.3 Thus the percentage increase in the

CPI from one year to the next measures the rate of inflation.

Figure 1.15 graphs the CPI since 1947. We have set the index for the end of 1947 to

100, so the graph shows the price level in each year as a percentage of 1947 prices. For

example, the index in 1948 was 103. This means that on average $103 in 1948 would

62 SECTION ONE

FIGURE 1.15

Consumer Price Index 700

600

100)

500

Consumer Price Index (1947

400

300

200