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Here's
the basic point about
discounting future income, in the form of a question:
Which is worth more
to you, according to economic theory:
$200 given to you
today, or $200 given to you one year from now?
Suppose
that there is no risk.
You absolutely, positively, will get the money at the time you choose.
Also
suppose that there is no inflation. $200 in one year will have the same
buying
power as $200 does today.
Quite obviously, $200 today is
worth more than $200 a year from now, because if you get $200
now, you can put
it in the bank. In a year you'll have
your $200, plus the interest it will earn.
Time
Preference:
In
my reply to the above
question, I emphasized the existence of interest-paying bank accounts. There's
a more fundamental reason why present income is worth more than future
income:
time preference.
Time
preference is preferring
income today to getting the same income in the future. Economists
assume that
pretty much everybody has time preference, and here is why:
Life
is short. Suppose you're
broke (for many students, that's not too hard to imagine) and you need
a car
today to be able to drive to the job you want. Working and saving to
buy a car
someday may not be your best option. If the job you want pays better,
you'll be
better off borrowing money to buy a car now, even though you'll have to
pay
interest to the lender. Because there are always people in this
circumstance,
for whom borrowing is a good idea, there is a market for loanable
funds, and
that's why there are bank accounts that pay interest. The existence of
these
bank accounts in turn means that even if you don't have a pressing need
for
money now, you're still better off getting it now than getting it
later.
(One exception to the time
preference rule is that some people like to have their future money
held for
them so they don't spend it foolishly now. Here at USC, some faculty
who get
paid only from August to May asked the payroll office to take a slice
out of
each paycheck and hold it, then pay it out during the following summer.
These
faculty didn't trust themselves to save for the summer on their own. At
first,
the University paid no interest on the deferred income. Even so, many
faculty
signed up. Only some years later did the University offer a plan that
paid
interest on this deferred salary.)
Bank
Account Math:
Let's
go over the math of bank
accounts:
Suppose we put $200
in a bank
account and leave it there for a year. The bank account pays 5%
interest at the
end of each full year. After one year, after the 5% interest is paid,
how much
will be in the account?
The answer is: $210.
That's
the $200 we started
with, plus 5% of $200, which is $10.
$200
×(1.05)
|
=
$210 |
Present
Value ×( 1 + Interest Rate )
|
=
Future Value in One Year |
Multiplying
$200 by 1.05 is
mathematically equivalent to adding 5% to it.
Let's go to two
years. If we
leave all the money in the bank for two years, how much will we have at
the
end?
The
answer is $ 220.5 after the
second year; After one year you have $210.
After
the second year, you get
5% of $210, which is $10.50, in interest. Your
new total
is $210 + $10.50 =
$220.50.
We
can formally express it like
this:
$200
×(1.05)²
|
=
$220.50 |
Present
Value ×( 1+Interest Rate )²
|
=
Future Value in Two Years |
To
calculate how much we'll
have in two years, we multiply by 1.05 twice, once for the first year
and once
for the second year.
Now,
let's do three years. If
we leave all the money in the bank for three years, we have:
$200
×(1.05)³
|
=
$231.52 |
Present
Value ×( 1+Interest Rate )³
|
=
Future Value in Three Years |
To
calculate how much we'll
have in three years, we multiply by 1.05 three times, once for the
first year,
once for the second year, and once for the third year.
By
now, you can probably
imagine the general formula for any number of years:
$200
×1.05ª
|
. |
Present
Value ×( 1+Interest Rate )ª
|
=
Future Value in a Years |
To
calculate how much we'll
have in a years, we multiply by 1.05 a times, once for each year.
If
interest is paid and
compounded more frequently than once a year, the formula gets more
complicated,
but the basic idea is the same.
Our
formula, again, is Future
Value = Present Value ×( 1 + Interest Rate )ª,
where a is the
number of years in the future.
Using
that, we can construct this table, based on a present value of $200 and
an annual interest rate of 5%:
| . |
Years
in the future (a) |
| 0
|
1
|
2
|
3
|
4
|
5
|
6
|
| $200
|
$210
|
$220.50
|
$231.52
|
$243.10
|
$255.26
|
$268.02
|
| .
|
How $200 grows at 5% interest per year,
compounded annually: ($200×1.05ª)
|
Now, let's use the same
reasoning, except in reverse, to answer this question:
How much would you
need today to have $200 in one year? Assume that your only
possible investment is this 5% bank account.
| . |
Years
in the future (a) |
| 0
|
1
|
2
|
3
|
4
|
5
|
6
|
| ????
|
$200
|
. |
.
|
. |
. |
. |
The answer is $190.48;
You
want the amount that will grow to $200
in one year. You want X such that
X
× (1.05) = $200. Divide both sides
of this by 1.05, to get:
X
= $200/1.05, which calculates
to
X = $190.48.
Present Value:
$200
divided by 1.05 equals
$190.48 (rounded to the nearest penny). $190.48 is the present value of $200
one year from now, if putting money in a 5% bank account is our
best
investment. Under that circumstance, we are equally well off getting
$190.48
now or $200 in one year. I say that we're equally well off, because
either way
gives us the same amount of money next year.
The present value of a
future
income amount is the amount that, if we had it today, we could invest
and have
it grow to equal the future income amount.
What is the present
value of $200 two years from now?
| . |
Years
in the future (a) |
| 0
|
1
|
2
|
3
|
4
|
5
|
6
|
| ????
|
$190.48
|
$200 |
. |
. |
. |
. |
The answer is $181.41; You want
the amount that will grow to $200 in two years.
You want X such that
X
× 1.05² = $200. Solve for X,
and you get:
X = $200/(1.05)², which equals
$181.41.
We need the amount of money
that will grow to $200 in two years at 5% interest. This is the amount
X such
that
X×1.05² = $200.
Divide both sides of that by (1.05)²
to solve for X:
X = $200/1.05² = $181.41
To calculate the present value of $200 two
years in the future, we divide by
1.05 twice.
Notice, by the way, the present
value of $200 in two years ($181.41) is less than the present value of
$200 in
one year ($190.48).
To
calculate the present value of $200 three years in the future, how many
times
do you divide by 1.05?
The
answer is: 3; To find the
present value of an income amount 3 years in the future, divide by 1.05
three
times.
| . |
Years
in the future (a) |
| 0
|
1
|
2
|
3
|
4
|
5
|
6
|
| $172.77
|
$181.41
|
$190.48 |
$200 |
. |
. |
. |
The general formula for the
present value of a future income amount a years in the future is:
Present Value = (Future Value) / ( 1 + Interest Rate )ª
Notice
that this is equivalent to the formula given earlier for the Future
Value:
Future Value = (Present Value) × ( 1
+ Interest Rate )ª
The
Discount Rate = The Interest Rate Used in Reverse
When an interest rate is used
in reverse like this, to calculate how much you need now to have a
certain
amount later, economists conventionally use the term discount rate
rather than
interest rate.
The
two terms mean the same thing. A reason for using the term
"discount rate" when you calculate a present value is that you are
taking a larger number, the future value, and calculating from it a
smaller
number, the present value.
Our
formula may be restated as:
Present Value = (Future Value) / ( 1 +
Discount Rate )ª.
An
alternative definition of the discount rate, used in some textbooks, is
Discount rate = 1/(1 + interest rate).
If
the interest rate is 5%, the discount rate, by this definition, is
about
0.9524, what 1/1.05 equals. As you see, this alternative definition is
awkward
to use. The concept is really the same as in my preferred definition. Either
way, the discount rate is measuring the opportunity cost of capital. It
is
measuring how much interest you could earn on your money if you put
that money
away.
| . |
Years
in the future (a)
In
this table's upper row, the a numbers are in
descending order.
|
| 6
|
5
|
4
|
3
|
2
|
1
|
0
|
| $149.24
|
$156.71
|
$164.54 |
$172.77 |
$181.41 |
$190.48 |
$200 |
|
The
numbers in the row just above show the present
value of $200 in a years, at a 5% discount rate.
$200
/ (1 + .05)ª |
Imagine
that there is for sale
a $200 zero-coupon bond that matures in five years. That means the bond
pays
$200 in 5 years. If the discount rate is 5%, how much will the
bond sell
for today? (Ignore sales expenses like the broker's commission.)
The
table below shows values for a, the number of years in the future, from
6
down to 0.
| . |
Years
in the future (a)
|
| 6
|
5
|
4
|
3
|
2
|
1
|
0
|
| $149.24
|
$156.71
|
$164.54 |
$172.77 |
$181.41 |
$190.48 |
$200 |
The answer is $200 / (1 +
.05)5 = $156.71.
The bond's price will be the
amount that will grow to $200 in 5 years at 5% interest.
Suppose you buy the
bond. Two
years go by, and you decide to sell the bond. If the discount rate is
still 5%,
how much should you get for selling your bond, which now has three
years left
to maturity? (Ignore sales expenses, such as the broker's
commission.)
The answer is $200 / (1 + .05)3 =
$172.77.
The bond's value grows 5% each year, until
the day it matures, when
the value reaches the full $200.
How Present
Value Changes When the Discount
Rate Changes?
So
far, we've done everything
with a discount rate of 5%. Now let's see how the changes in the
discount rate
affect the present value.
Our
formula is Present Value =
(Future Value) / ( 1 + Discount Rate )ª,
where a is the number of years in the
future that the future value will be
received.
Dust
off your high school
algebra and tell me what happens to the Present Value in this formula
if the
Discount Rate goes up. (Assume that the Future Value and a stay the
same, and
that a is bigger than or equal to 0.)
Obviously, the present value
goes down when the discount rate goes up. If the Discount Rate goes up, the
denominator gets bigger, so the whole fraction gets smaller.
Summary
The
key concepts of this
interactive tutorial are:
Income received in the future
is worth less now than income received now.
That's because income you get
now can earn interest and grow.
The future value of an amount
you get now is
Future Value = Present Value ×( 1 + Interest Rate )ª, where
a is the number of
years it grows.
Therefore, the present value of
a future income amount a years in the future is:
Present Value = (Future Value) / ( 1 + Interest Rate )ª
The discount rate is another
name for the interest rate, so
Present Value = (Future Value) / ( 1 + Discount Rate )ª
When
the discount rate goes up,
present values go down. When the discount rate goes down, present
values go up.
***That's all for now. Thanks for
participating!***
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