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University of South Carolina, Norman J. Arnold School of Public Health, Dept. of Health Administration, Economics Interactice Tutorials, January 28, 2008

Economics Interactive Tutorial (Instructions)
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Discounting Future Income and Present Value

Copyright © 1997-2002 Samuel L. Baker
Bonus for you:
Calculate your own net present values with the spreadsheet template available for you to copy at the end of this tutorial!

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

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?

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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
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?

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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
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 
$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
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 
$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
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,
                                                                                                           ( 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)

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
0 1 2 3 4 5 6
???? $200

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
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
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
                                                                    (1 + Interest Rate) to the a power
Notice that this is equivalent to the formula given earlier for the Future Value:
Future Value = (Present Value) × ( 1 + Interest Rate )ª
                                                             times  (1 + Interest Rate) to the a power
 

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.
                                                                                                                                    (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
To digress for a moment, this table also shows how prices are figured for zero-coupon bonds. (A bond is an I.O.U., a promise to pay a certain amount at a certain time in the future. A zero coupon bond pays only at the end of the time, with no payments along the way. U.S. Savings Bonds are examples of zero-coupon bonds, because they pay no monthly or annual interest. You get all your money the day you cash them in.)

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.)

 
 

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,
                                                                                                 ( 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!

Spreadsheet for calculating net present value

The box below this paragraph has a spreadsheet set-up for calculating the net present value of an income stream. To use it, select the contents of the box (right-click in the box and chose Select All, in most browsers). Copy, then paste to the upper left corner of a blank worksheet in Microsoft Excel or other spreadsheet program. The discount rate and income stream numbers are samples, of course, that you can change. Row 5 has two sum formulas, one for Excel and one for Lotus and Quattro Pro. To extend the spreadsheet for a longer income stream, copy E2:E4 and paste across row 2 as far as needed.

Summary

The key concepts of this interactive tutorial are:

Epilogue

You get a fortune cookie with this fortune inside:
It is better to have a hen tomorrow than an egg today.
Do you agree?

That's all for now. Thanks for participating! 


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