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?
$200 today is worth more.
$200 in one year is worth more.
No difference. The two are worth the same, as long as I am protected against price inflation.
Leaving space until the above question is answered correctly.
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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 carelessly now. Here at USC, years ago, some faculty who got paid only from August to May asked the payroll office to take a slice out of each paycheck and hold it until 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.
Nowadays, 1% would be more realistic, but I'll use 5% to get more nice round numbers in the calculations.After one year, after the 5% interest is paid, how much will be in the account?
5% is 1/20, by the way, so 5% of 200 is 200/20.
Leaving space until the above question is answered correctly.
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We can do the calculation this way:
Adding 5% is the same as multiplying by 1.05.
$200 ×(1.05) | = $210 |
Present Value ×( 1 + Interest Rate ) | = Future Value in One Year |
Let's go to two years. If we leave in the bank all the money, that is, the original $200 plus the interest we earn, how much will we have at the end of two years?
$220.00
$220.50
Leaving space until the above question is answered correctly.
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We can write the calculation like this:
$200 ×(1.05)² | = $220.50 |
Present Value ×( 1+Interest Rate )² | = Future Value in Two Years |
Now, let's do three years. If we leave all the money in the bank for three years, we multiply by (1 + Interest Rate) how many times?
Leaving space until the above question is answered correctly.
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By now, you can probably guess the general formula for any number of years:
$200 ×1.05ⁿ | |
Present Value ×( 1+Interest Rate )ⁿ | = Future Value in n 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.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 (n) | ||||||
0 | 1 | 2 | 3 | 4 | 5 | 6 |
$200 | $210 | $220.50 | $231.52 | $243.10 | $255.26 | $268.02 |
What $200 grows to at 5% interest per year, compounded annually. ($200×1.05ⁿ) |
That is a lot to take in if you have not seen it before.
You might need to take a break before continuing with the next part --
-- because we are going to run that reasoning backwards to answer this next question:
How much would you need today to have $200 in your bank account in one year?
Years in the future | ||||||
0 | 1 | 2 | 3 | 4 | 5 | 6 |
???? | $200 |
Leaving space until the above question is answered correctly.
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if putting money in a 5% bank account is our best investment.Thanks to the bank, we are equally well off getting $190.48 now or $200 in one year. Either gives us the same amount of money next year.
In general, the present value of an amount at a certain future time is the amount that, if we had it today, would grow to equal that amount at that future time.
In our example, $200 is the "future value" and $190.48 is the "present value."
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 |
Leaving space until the above question is answered correctly.
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Notice that 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?
Leaving space until the above question is answered correctly.
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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 n years in the future is:
Present Value = (Future Value) / ( 1 + Interest Rate )ⁿ
Notice how this correspondents with the formula given earlier for the future value of an amount in the present:
Future Value = (Present Value) × ( 1 + Interest Rate )ⁿ
In the literature, I have found two ways to define "discount rate."
One way is to just call the interest rate the discount rate.
That makes our general formula Present Value = (Future Value) / ( 1
+ Discount Rate )ⁿ.
The U.S.'s central bank, the Federal Reserve, says on its web site, "The discount rate is the interest rate charged to commercial banks and other depository institutions on loans they receive from their regional Federal Reserve Bank's lending facility--the discount window." That is the first definition of "discount rate."
The alternative definition of the discount rate, used in some textbooks, is Discount rate = 1/(1 + interest rate).
This makes the general formula Present Value = (Future Value) × (Discount Rate)ⁿ.
If the interest rate is 5%, the discount rate, by this second definition, is about 0.9524, what 1/1.05 equals.
The two definitions lead to exactly the same results. I think the first definition is more intuitive.
It is usually easy to tell which definition is being used from the number that they quote. If the number is a little more than 0, they are using the first definition. If the number is a little less than 1, they are using the second definition. For example, the Federal Reserve's discount rate for Primary Credit is 0.75% (at this writing). That confirms that they are using the first definition. By the second definition, that would be a discount rate of 99.256%.
Either way, the discount rate measures the opportunity cost of capital -- how much interest you could earn if you put your money in an interest-bearing safe investment.
Bond values use present value calculations.
A bond is an I.O.U., a promise to pay a certain amount at a certain time in the future.The simplest bonds are zero coupon bonds. These pay only at the end of the time, at "maturity," with no payments along the way. U.S. Savings Bonds are examples of zero-coupon bonds, because they pay no monthly or annual interest. If you have one, you get all your money the day you cash it 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 your discount rate is 5%, how valuable is that bond to you today? (Ignore sales expenses like a broker's commission. Also, assume that there is no risk that the bond will not be paid. I have another tutorial about risk, which discusses the effect of risk on a bond price.)
In the table below, click the button for the most that you would be willing to pay for a $200 zero-coupon bond that matures in five years when your discount rate is 5%.
Years in the future
(n)
In this table's upper row, the n numbers are in descending order. |
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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 $200 / (1 + .05)ⁿ, the present value of $200 in n years, at a 5% discount rate. |
Suppose you bought the bond two years ago. The bond now has three years to maturity. If everybody's discount rate is 5%, how much should you expect to get for selling your bond? (Ignore sales expenses, such as the broker's commission.) Click a button in the table below.
Years in the future
(n)
In this table's upper row, the n numbers are in descending order. |
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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 $200 / (1 + .05)ⁿ, the present value of $200 in n years, at a 5% discount rate. |
For variety, let us use a $1000 face-value bond with a maturity of 10 years and no risk that it won't be paid. (You can actually buy a U.S. Treasury bond like that. The only risk is that someday Republicans in Congress may block the Treasury from paying out money.)
If the discount rate for bond-buying people and institutions is 5%, the present value of that bond will be
$613.91 = $1000 / ( 1.05 )^{10}. The bond will sell for $613.91 on the day it is newly issued.
Try some different discount rates.
0%
2.75%
5%
15.8%
Suppose you have an investment that pays off on several occasions, not just once at the end. To get the total present value of such an "income stream," to use the economics jargon, you calculate the present value of each payment and then add them up:
Present value =
(Income in year 0) / ( 1 + discount rate )^{0}
+ (Income in year 1) / ( 1 + discount rate )^{1}
+ (Income in year 2) / ( 1 + discount rate )^{2} + ... and so forth.
If Income in year 0 includes your initial cost, as a negative,
(For that matter, all of your yearly income numbers should include any outlays during those years.)then you have:
Net Present value =
(Net income in year 0) / ( 1 + discount rate )^{0}
+ (Net income in year 1) / ( 1 + discount rate )^{1}
+ (Net income in year 2) / ( 1 + discount rate )^{2} + ... and so forth.
If your incomes and expenses come at other intervals, monthy, for example, then you would use monthly discount rates rather than annual.
The net present value is used for deciding whether to do an investment or not.
For example, if a 10-year zero-coupon $1000 Treasury bond is selling for $762.40, the net present value is:
-762.40 / ( 1 + discount rate )^{0} + 10000 / ( 1 + discount rate )^{10}
Plug your discount rate into that formula. If the net present value is positive, buy the bond. If it is negative, don't.
If you have cash lying around, your discount rate is the interest rate you could get by lending the cash to somebody. If you do not have spare cash, your discount rate is interest rate that you would pay to borrow from somebody.
Do you agree?
I agree. I'll wait for the hen.
I disagree. Give me the egg!
Maybe. It depends on my discount rate and my rate of time preference.
It depends on the lucky numbers on the back of the fortune.
That's all for now. Thanks for participating!
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