ñòð. 16 |

0

1947 1951 1955 1959 1963 1967 1971 1975 1979 1983 1987 1991 1995 1998

Year

have bought the same quantity of goods and services as $100 in 1947. The inflation rate

REAL VALUE OF $1 between 1947 and 1948 was therefore 3 percent. By the end of 1998, the index was 699,

Purchasing power-adjusted

meaning that 1998 prices were 6.99 times as high as 1947 prices.4

value of a dollar.

The purchasing power of money fell by a factor of 6.99 between 1947 and 1998. A

dollar in 1998 would buy only 14 percent of the goods it could buy in 1947 (1/6.99 =

.14). In this case, we would say that the real value of $1 declined by 100 â€“ 14 = 86 per-

cent from 1947 to 1998.

As we write this in the fall of 1999, all is quiet on the inflation front. In the United

States inflation is running at little more than 2 percent a year and a few countries are

even experiencing falling prices, or deflation.5 This has led some economists to argue

that inflation is dead; others are less sure.

Talk Is Cheap

EXAMPLE 13

Suppose that in 1975 a telephone call to your Aunt Hilda in London cost $10, while the

price to airmail a letter was $.50. By 1999 the price of the phone call had fallen to $3,

while that of the airmail letter had risen to $1.00. What was the change in the real cost

of communicating with your aunt?

In 1999 the consumer price index was 3.02 times its level in 1975. If the price of tele-

phone calls had risen in line with inflation, they would have cost 3.02 Ã— $10 = $30.20

in 1999. That was the cost of a phone call measured in terms of 1999 dollars rather than

1975 dollars. Thus over the 24 years the real cost of an international phone call declined

from $30.20 to $3, a fall of over 90 percent.

The Time Value of Money 63

What about the cost of sending a letter? If the price of an airmail letter had kept pace

with inflation, it would have been 3.02 Ã— $.50 = $1.51 in 1999. The actual price was

only $1.00. So the real cost of letter writing also has declined.

Consider a telephone call to London that currently would cost $5. If the real price of

Self-Test 12

telephone calls does not change in the future, how much will it cost you to make a call

to London in 50 years if the inflation rate is 5 percent (roughly its average over the past

25 years)? What if inflation is 10 percent?

Economists sometimes talk about current or nominal dollars versus constant

or real dollars. Current or nominal dollars refer to the actual number of

dollars of the day; constant or real dollars refer to the amount of purchasing

power.

Some expenditures are fixed in nominal terms, and therefore decline in real terms.

Suppose you took out a 30-year house mortgage in 1988. The monthly payment was

$800. It was still $800 in 1998, even though the CPI increased by a factor of 1.36 over

those years.

Whatâ€™s the monthly payment for 1998 expressed in real 1988 dollars? The answer is

$800/1.36, or $588.24 per month. The real burden of paying the mortgage was much

less in 1998 than in 1988.

The price index in 1980 was 370. If a family spent $250 a week on their typical pur-

Self-Test 13

chases in 1947, how much would those purchases have cost in 1980? If your salary in

1980 was $30,000 a year, what would be the real value of that salary in terms of 1947

dollars?

INFLATION AND INTEREST RATES

Whenever anyone quotes an interest rate, you can be fairly sure that it is a nominal, not

a real rate. It sets the actual number of dollars you will be paid with no offset for future

inflation.

If you deposit $1,000 in the bank at a nominal interest rate of 6 percent, you will

NOMINAL INTEREST

RATE Rate at which have $1,060 at the end of the year. But this does not mean you are 6 percent better off.

money invested grows. Suppose that the inflation rate during the year is also 6 percent. Then the goods that cost

$1,000 last year will now cost $1,000 Ã— 1.06 = $1,060, so youâ€™ve gained nothing:

$1,000 Ã— (1 + nominal interest rate)

Real future value of investment =

(1 + inflation rate)

$1,000 Ã— 1.06

= = $1,000

REAL INTEREST RATE

1.06

Rate at which the purchasing

In this example, the nominal rate of interest is 6 percent, but the real interest rate

power of an investment

is zero.

increases.

64 SECTION ONE

The real rate of interest is calculated by

1 + nominal interest rate

1 + real interest rate =

1 + inflation rate

In our example both the nominal interest rate and the inflation rate were 6 percent. So

1.06

1 + real interest rate = =1

1.06

real interest rate = 0

What if the nominal interest rate is 6 percent but the inflation rate is only 2 percent?

In that case the real interest rate is 1.06/1.02 â€“ 1 = .039, or 3.9 percent. Imagine that

the price of a loaf of bread is $1, so that $1,000 would buy 1,000 loaves today. If you

invest that $1,000 at a nominal interest rate of 6 percent, you will have $1,060 at the

end of the year. However, if the price of loaves has risen in the meantime to $1.02, then

your money will buy you only 1,060/1.02 = 1,039 loaves. The real rate of interest is 3.9

percent.

a. Suppose that you invest your funds at an interest rate of 8 percent. What will be your

Self-Test 14

real rate of interest if the inflation rate is zero? What if it is 5 percent?

b. Suppose that you demand a real rate of interest of 3 percent on your investments.

What nominal interest rate do you need to earn if the inflation rate is zero? If it is 5

percent?

Here is a useful approximation. The real rate approximately equals the difference be-

tween the nominal rate and the inflation rate:6

Real interest rate â‰ˆ nominal interest rate â€“ inflation rate

Our example used a nominal interest rate of 6 percent, an inflation rate of 2 percent,

and a real rate of 3.9 percent. If we round to 4 percent, the approximation gives the same

answer:

Real interest rate â‰ˆ nominal interest rate â€“ inflation rate

â‰ˆ 6 â€“ 2 = 4%

The approximation works best when both the inflation rate and the real rate are small.7

When they are not small, throw the approximation away and do it right.

Real and Nominal Rates

EXAMPLE 14

In the United States in 1999, the interest rate on 1-year government borrowing was

about 5.0 percent. The inflation rate was 2.2 percent. Therefore, the real rate can be

found by computing

6 The squiggle (â‰ˆ) means â€œapproximately equal to.â€

7 When the interest and inflation rates are expressed as decimals (rather than percentages), the approximation

error equals the product (real interest rate Ã— inflation rate).

The Time Value of Money 65

1 + nominal interest rate

1 + real interest rate =

1 + inflation rate

1.050

= = 1.027

1.022

real interest rate = .027, or 2.7%

The approximation rule gives a similar value of 5.0 â€“ 2.2 = 2.8 percent. But the ap-

proximation would not have worked in the German hyperinflation of 1922â€“1923, when

the inflation rate was well over 100 percent per month (at one point you needed 1 mil-

lion marks to mail a letter), or in Peru in 1990, when prices increased by nearly 7,500

percent.

VALUING REAL CASH PAYMENTS

Think again about how to value future cash payments. Earlier you learned how to value

payments in current dollars by discounting at the nominal interest rate. For example,

suppose that the nominal interest rate is 10 percent. How much do you need to invest

now to produce $100 in a yearâ€™s time? Easy! Calculate the present value of $100 by dis-

counting by 10 percent:

$100

PV = = $90.91

1.10

You get exactly the same result if you discount the real payment by the real interest

rate. For example, assume that you expect inflation of 7 percent over the next year. The

real value of that $100 is therefore only $100/1.07 = $93.46. In one yearâ€™s time your

$100 will buy only as much as $93.46 today. Also with a 7 percent inflation rate the real

rate of interest is only about 3 percent. We can calculate it exactly from the formula

1 + nominal interest rate

(1 + real interest rate) =

1 + inflation rate

1.10

= = 1.028

1.07

real interest rate = .028, or 2.8%

If we now discount the $93.46 real payment by the 2.8 percent real interest rate, we

have a present value of $90.91, just as before:

$93.46

PV = = $90.91

1.028

The two methods should always give the same answer.8

8 If

they donâ€™t there must be an error in your calculations. All we have done in the second calculation is to di-

vide both the numerator (the cash payment) and the denominator (one plus the nominal interest rate) by the

same number (one plus the inflation rate):

payment in current dollars

PV =

1 + nominal interest rate

(payment in current dollars)/(1 + inflation rate)

= (1 + nominal interest rate)/(1 + inflation rate)

payment in constant dollars

=

1 + real interest rate

66 SECTION ONE

Remember:

Current dollar cash flows must be discounted by the nominal interest rate;

real cash flows must be discounted by the real interest rate.

Mixing up nominal cash flows and real discount rates (or real rates and nominal flows)

is an unforgivable sin. It is surprising how many sinners one finds.

You are owed $5,000 by a relative who will pay back in 1 year. The nominal interest rate

Self-Test 15

is 8 percent and the inflation rate is 5 percent. What is the present value of your rela-

tiveâ€™s IOU? Show that you get the same answer (a) discounting the nominal payment at

the nominal rate and (b) discounting the real payment at the real rate.

How Inflation Might Affect Bill Gates

EXAMPLE 15

We showed earlier (Example 11) that at an interest rate of 9 percent Bill Gates could, if

he wished, turn his $96 billion wealth into a 40-year annuity of $8.9 billion per year of

luxury and excitement (L&E). Unfortunately L&E expenses inflate just like gasoline

and groceries. Thus Mr. Gates would find the purchasing power of that $8.9 billion

steadily declining. If he wants the same luxuries in 2040 as in 2000, heâ€™ll have to spend

less in 2000, and then increase expenditures in line with inflation. How much should he

spend in 2000? Assume the long-run inflation rate is 5 percent.

Mr. Gates needs to calculate a 40-year real annuity. The real interest rate is a little

less than 4 percent:

1 + nominal interest rate

1 + real interest rate =

1 + inflation rate

1.09

= = 1.038

1.05

so the real rate is 3.8 percent. The 40-year annuity factor at 3.8 percent is 20.4. There-

fore, annual spending (in 2000 dollars) should be chosen so that

$96,000,000,000 = annual spending Ã— 20.4

annual spending = $4,706,000,000

Mr. Gates could spend that amount on L&E in 2000 and 5 percent more (in line with

inflation) in each subsequent year. This is only about half the value we calculated when

we ignored inflation. Life has many disappointments, even for tycoons.

You have reached age 60 with a modest fortune of $3 million and are considering early

Self-Test 16

retirement. How much can you spend each year for the next 30 years? Assume that

spending is stable in real terms. The nominal interest rate is 10 percent and the inflation

rate is 5 percent.

The Time Value of Money 67

REAL OR NOMINAL?

Any present value calculation done in nominal terms can also be done in real terms, and

vice versa. Most financial analysts forecast in nominal terms and discount at nominal

rates. However, in some cases real cash flows are easier to deal with. In our example of

Bill Gates, the real expenditures were fixed. In this case, it was easiest to use real quan-

tities. On the other hand, if the cash-flow stream is fixed in nominal terms (for exam-

ñòð. 16 |