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.Which is worth more?
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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.)
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?
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We can formally express it like this:
In case your browser won't render
the multiplication and exponent symbols, this print will tell you what
each formula is supposed to say.
At 5% interest, $200 in the bank today will grow to $210 in one year.
| $200 ×(1.05)
$200 times 1.05 |
= $210 |
| Present Value ×( 1 + Interest Rate )
Present Value times (1 + Interest Rate) |
= Future Value in One Year |
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?
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We can formally express it like this:
| $200 ×(1.05)²
$200 times (1.05 squared) |
= $220.50 |
| Present Value ×( 1+Interest Rate )²
Present Value times ( 1+Interest Rate ) squared |
= Future Value in Two Years |
Now, let's do three years. If we leave all the money in the bank for three years, we have:
| $200 ×(1.05)³
$200 times (1.05 cubed) |
= $231.52 |
| Present Value ×( 1+Interest Rate )³
Present Value times ( 1+Interest Rate ) cubed |
= Future Value in Three Years |
By now, you can probably imagine the general formula for any number of years:
| $200 ×1.05ª
$200 times (1.05 to the a power) |
|
| Present Value ×( 1+Interest Rate )ª
Present Value times ( 1+Interest Rate ) to the a power |
= Future Value in a Years |
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,
( 1 + Interest Rate ) to the a power
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)
($200 times 1.05 to the a power) |
||||||
| Years in the future | ||||||
| 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| ???? | $200 | |||||
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 | ||||||
| 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| ???? | $190.48 | $200 | ||||
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.
X times (1.05 squared)
Divide both sides of that by (1.05)² to solve for X:
X = $200/1.05² = $181.41
$200/(1.05 squared)
To calculate the present value of $200 two years in the future, we
divide by 1.05 twice.
If this still seems a bit mysterious, please go back up to the question above and try the wrong answers. My comments on those answers should help clarify why the only correct calculation method is to 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?
| Years in the future | ||||||
| 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| $172.77 | $181.41 | $190.48 | $200 | |||
Our formula may be restated as Present Value = (Future Value) / ( 1
+ Discount Rate )ª.
(1 + Discount Rate) to the a power
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 $200/(1 + .05) to the a power |
||||||
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.
The second row shows corresponding values of $200/1.05ª. 1.05
to the a power
Click on the value that equals what the bond would sell for.
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.)
Our formula is Present Value = (Future Value) / ( 1 + Discount Rate
)ª,
( 1 + Discount Rate ) to the a power
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.)
The applet below shows how present values change as the discount rate changes. Move the slider left and right to make the discount rate lower or higher. Click on an end arrow to change the discount rate by one percentage point. (On some Macs, you may only see the end arrows.) Click in the space between the slider and an end to change the discount rate by ten percentage points. At any slider position, representing an interest rate, the applet will show you the present value of $200 for 6, 5, 4, 3, 2, 1, and 0 years in the future for that interest rate. 0 years in the future is now, of course, so the value under the 0 is always $200.
In the table below, "a" is the number of years in the future. The expression under the a at the left end shows the formula for calculating each present value. It is $200 divided by (1 plus the interest rate raised to the a power).
As you move the slider left and right, imagine that you chair the Federal Reserve Board. Your Board has the power to change the discount rate in the U.S. Doing so makes bond prices go up and down just like the present values in the applet above. Stock prices go up and down the same way, because stocks, like bonds, represent promises to pay amounts of money in the future. Move the slider to the right and watch the market crash! What fun!
That's all for now. Thanks for participating!
