Economics Basics: Lesson 4

Discounting Future Income and Present Value

ConcreteBasics.org thanks Dr. Samuel L. Baker, Ph.D., Department of Health Services Policy and Management,University of South Carolina, U.S.A for granting the permission to publish this article. Click Here to access complete list of articles under “Economics Basics for Civil Engineers” 

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
Amount $200 $210 $220.50 $231.52 $243.10 $255.26 $268.02

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
Amount ???? $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
Amount ???? $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
Amount $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) 6 5 4 3 2 1 0
Amount $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
Amount $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!***

Copyright © 1985-2000 Samuel L. Baker

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