1. 24 Module The Time Value of Money KRUGMAN'S MACROECONOMICS for AP*
Margaret Ray and David Anderson

2. What you will learn in this Module:
Why a dollar today is worth more than a dollar a year from now
How the concept of present value can help you make decisions when costs or benefits come in the future

3. Veruca Salt appreciates the value of having things in the present
Veruca Salt appreciates the value of having things in the present. She wanted the “golden-egg-laying-goose” NOW!

4. Borrowing, Lending, and Interest
Suppose you could have $1,000 today or $1,000 next year, which would you prefer?
Why?
You need money today.
You could put it in the bank and in a year you would have more than $1,000
$1,000 today is worth more than a $1,000 tomorrow
Ask the students: “Suppose you could have $1000 today or $1000 next year? Which would you choose?”
$1000 today! Of course, but why?
It would allow me the satisfaction of buying or saving today, rather than waiting.
For example, if I need to buy food or pay my rent, I can’t wait a year to get my hands on that money.
The other reason is that if you had the money today, you could put it in the bank and in a year you would have more than $1000.
So for both reasons, $1000 today is worth more than waiting a year to get $1000.
Note: it can be useful to get the students to think about lending money to someone for a year. They have a self-interest to start thinking about the benefits of receiving interest rather than paying interest in a borrowing example.
Example:
You are going to lend your friend $100, and he is going to pay you back in one year.
Assume no inflation, you agree to a 10% interest rate, the going rate you could receive if you had simply saved the money. Why do you need to receive interest on this loan?
The opportunity cost of lending your friend $100 is the interest you could have earned, $10, after a year had passed.
So the interest rate measures the cost to you of forgoing the use of that $100.
Rather than saving it, you could have spent $100 on clothing right now that would have provided immediate benefit to you.
Repayment received on lending $100 for one year = $100 + $100*.10 = $100*(1+.10)
What if you were going to lend your friend the money for two years?
Repayment in two years = $100(1.10)*(1.10) = $121
Generalization:
Your friend, as a borrower, must pay you $21 to compensate you for the fact that he has your $100 for a period of two years.
You, as a saver, could put the $100 in the bank today, two years from now you would have $121 to spend on goods and services.
This implies that you would be completely indifferent between having $100 in your pocket today or $121 two years from today.
They are equivalent measures of purchasing power, just measured at two different points in time, and it is the interest rate that equates the two.

5. Borrowing, Lending, and Interest
When you lend money you earn interest.
Why do you need to earn interest?
The interest payment covers the cost to you of not having the money today
You could have saved it and earned interest.
You could have spent it and received an immediate benefit from it today.
Ask the students: “Suppose you could have $1000 today or $1000 next year? Which would you choose?”
$1000 today! Of course, but why?
It would allow me the satisfaction of buying or saving today, rather than waiting.
For example, if I need to buy food or pay my rent, I can’t wait a year to get my hands on that money.
The other reason is that if you had the money today, you could put it in the bank and in a year you would have more than $1000.
So for both reasons, $1000 today is worth more than waiting a year to get $1000.
Note: it can be useful to get the students to think about lending money to someone for a year. They have a self-interest to start thinking about the benefits of receiving interest rather than paying interest in a borrowing example.
Example:
You are going to lend your friend $100, and he is going to pay you back in one year.
Assume no inflation, you agree to a 10% interest rate, the going rate you could receive if you had simply saved the money. Why do you need to receive interest on this loan?
The opportunity cost of lending your friend $100 is the interest you could have earned, $10, after a year had passed.
So the interest rate measures the cost to you of forgoing the use of that $100.
Rather than saving it, you could have spent $100 on clothing right now that would have provided immediate benefit to you.
Repayment received on lending $100 for one year = $100 + $100*.10 = $100*(1+.10)
What if you were going to lend your friend the money for two years?
Repayment in two years = $100(1.10)*(1.10) = $121
Generalization:
Your friend, as a borrower, must pay you $21 to compensate you for the fact that he has your $100 for a period of two years.
You, as a saver, could put the $100 in the bank today, two years from now you would have $121 to spend on goods and services.
This implies that you would be completely indifferent between having $100 in your pocket today or $121 two years from today.
They are equivalent measures of purchasing power, just measured at two different points in time, and it is the interest rate that equates the two.

6. Borrowing, Lending and Interest
Example:
You are going to lend your friend $100, and he is going to pay you back in one year.
Assume no inflation, you agree to a 10% interest rate, the going rate you could receive if you had simply saved the money. Why do you need to receive interest on this loan?
The opportunity cost of lending your friend $100 is the interest you could have earned, $10, after a year had passed.
So the interest rate measures the cost to you of forgoing the use of that $100.
Rather than saving it, you could have spent $100 on clothing right now that would have provided immediate benefit to you.
Repayment received on lending $100 for one year =
$100 + $100*.10 = $100*(1+.10) = $110
As the above examples demonstrate, there is a difference between dollars received today and dollars received in the future. We will provide some more specifics to this relationship.
Generalization:
To see the relationship between dollars today (present value PV) and dollars 1 year from now (future value FV), a simple equation is applied:
Future Payment, or FV = PV*(1+r) or, using our example,
FV = $100*(1.10) = $110
In other words, one year into the future, $100 in the present will be worth $110.
This is true whether you saved it or lent it to your friend.
We can also rearrange our equation and solve for the present value PV:
PV = FV/(1+r)
Using our example again,
PV = $110/(1.10) = $100
This tells us that $110 received a year from now is worth $100 in today’s dollars.
Now let’s look again at the decision to lend the money for a period of t=2 years:
Repayment in two years = $100(1.10)*(1.10) = $121
FV = PV(1+r)(1+r) = PV(1+r)t Or
PV = FV/(1+r)t
Money today is more valuable than the same amount of money in the future.
The present value of $1 received one year from now is $1/(1+r).
The future value of $1 invested today is $1*(1+r).
Interest paid on savings and interest charged on borrowing is designed to equate the value of dollars today with the value of future dollars.

7. Defining Present Value
When evaluating a course of action, time must be taken into account because:
$1 paid to you today is worth more than $1 paid to you tomorrow
$1 that you must pay today is more burdensome than $1 that you must pay tomorrow
There is a difference between dollars received today and dollars received in the future.
Interest rates can be used to convert future benefits and costs into present value
As the above examples demonstrate, there is a difference between dollars received today and dollars received in the future. We will provide some more specifics to this relationship.
Generalization:
To see the relationship between dollars today (present value PV) and dollars 1 year from now (future value FV), a simple equation is applied:
Future Payment, or FV = PV*(1+r) or, using our example,
FV = $100*(1.10) = $110
In other words, one year into the future, $100 in the present will be worth $110.
This is true whether you saved it or lent it to your friend.
We can also rearrange our equation and solve for the present value PV:
PV = FV/(1+r)
Using our example again,
PV = $110/(1.10) = $100
This tells us that $110 received a year from now is worth $100 in today’s dollars.
Now let’s look again at the decision to lend the money for a period of t=2 years:
Repayment in two years = $100(1.10)*(1.10) = $121
FV = PV(1+r)(1+r) = PV(1+r)t Or
PV = FV/(1+r)t
Money today is more valuable than the same amount of money in the future.
The present value of $1 received one year from now is $1/(1+r).
The future value of $1 invested today is $1*(1+r).
Interest paid on savings and interest charged on borrowing is designed to equate the value of dollars today with the value of future dollars.

8. Defining Present Value
Let FV = future value of $
PV = present value of $
r = real interest rate
n = # of years
The Simple Interest Formula
FV = PV + (PV x r) $110 = (100 x .10) or
FV = PV x ( 1 + r )n $110 = 100 x ( )1
What this tells us is that a dollar today will generate a future value that is larger.
As the above examples demonstrate, there is a difference between dollars received today and dollars received in the future. We will provide some more specifics to this relationship.
Generalization:
To see the relationship between dollars today (present value PV) and dollars 1 year from now (future value FV), a simple equation is applied:
Future Payment, or FV = PV*(1+r) or, using our example,
FV = $100*(1.10) = $110
In other words, one year into the future, $100 in the present will be worth $110.
This is true whether you saved it or lent it to your friend.
We can also rearrange our equation and solve for the present value PV:
PV = FV/(1+r)
Using our example again,
PV = $110/(1.10) = $100
This tells us that $110 received a year from now is worth $100 in today’s dollars.
Now let’s look again at the decision to lend the money for a period of t=2 years:
Repayment in two years = $100(1.10)*(1.10) = $121
FV = PV(1+r)(1+r) = PV(1+r)t Or
PV = FV/(1+r)t
Money today is more valuable than the same amount of money in the future.
The present value of $1 received one year from now is $1/(1+r).
The future value of $1 invested today is $1*(1+r).
Interest paid on savings and interest charged on borrowing is designed to equate the value of dollars today with the value of future dollars.

9. Defining Present Value
We can also use this formula to calculate what a future payment would be worth today
The Simple Interest Formula
PV = FV / (1 + r)n PV = $110/(1.10) = $100
This tells us that $110 received a year from now is worth $100 in today’s dollars.
As the above examples demonstrate, there is a difference between dollars received today and dollars received in the future. We will provide some more specifics to this relationship.
Generalization:
To see the relationship between dollars today (present value PV) and dollars 1 year from now (future value FV), a simple equation is applied:
Future Payment, or FV = PV*(1+r) or, using our example,
FV = $100*(1.10) = $110
In other words, one year into the future, $100 in the present will be worth $110.
This is true whether you saved it or lent it to your friend.
We can also rearrange our equation and solve for the present value PV:
PV = FV/(1+r)
Using our example again,
PV = $110/(1.10) = $100
This tells us that $110 received a year from now is worth $100 in today’s dollars.
Now let’s look again at the decision to lend the money for a period of t=2 years:
Repayment in two years = $100(1.10)*(1.10) = $121
FV = PV(1+r)(1+r) = PV(1+r)t Or
PV = FV/(1+r)t
Money today is more valuable than the same amount of money in the future.
The present value of $1 received one year from now is $1/(1+r).
The future value of $1 invested today is $1*(1+r).
Interest paid on savings and interest charged on borrowing is designed to equate the value of dollars today with the value of future dollars.

10. Defining Present Value
Money today is more valuable than the same amount of money in the future.
The present value of $X received one year from now is $X/(1+r).
The future value of $X invested today is $X*(1+r).
Interest paid on savings and interest charged on borrowing is designed to equate the value of dollars today with the value of future dollars.
As the above examples demonstrate, there is a difference between dollars received today and dollars received in the future. We will provide some more specifics to this relationship.
Generalization:
To see the relationship between dollars today (present value PV) and dollars 1 year from now (future value FV), a simple equation is applied:
Future Payment, or FV = PV*(1+r) or, using our example,
FV = $100*(1.10) = $110
In other words, one year into the future, $100 in the present will be worth $110.
This is true whether you saved it or lent it to your friend.
We can also rearrange our equation and solve for the present value PV:
PV = FV/(1+r)
Using our example again,
PV = $110/(1.10) = $100
This tells us that $110 received a year from now is worth $100 in today’s dollars.
Now let’s look again at the decision to lend the money for a period of t=2 years:
Repayment in two years = $100(1.10)*(1.10) = $121
FV = PV(1+r)(1+r) = PV(1+r)t Or
PV = FV/(1+r)t
Money today is more valuable than the same amount of money in the future.
The present value of $1 received one year from now is $1/(1+r).
The future value of $1 invested today is $1*(1+r).
Interest paid on savings and interest charged on borrowing is designed to equate the value of dollars today with the value of future dollars.

11. Using Present Value
Decisions often involve dollars spent, or received, at different points in time. We can use the concept of FV to evaluate whether we should commit to a project (or choose between projects) today when benefits may not be enjoyed for several years.
Example:
What if you could invest $10,000 now and receive a guaranteed (after inflation) $20,000 later? Good deal? Maybe.
As the above examples demonstrate, there is a difference between dollars received today and dollars received in the future. We will provide some more specifics to this relationship.
Generalization:
To see the relationship between dollars today (present value PV) and dollars 1 year from now (future value FV), a simple equation is applied:
Future Payment, or FV = PV*(1+r) or, using our example,
FV = $100*(1.10) = $110
In other words, one year into the future, $100 in the present will be worth $110.
This is true whether you saved it or lent it to your friend.
We can also rearrange our equation and solve for the present value PV:
PV = FV/(1+r)
Using our example again,
PV = $110/(1.10) = $100
This tells us that $110 received a year from now is worth $100 in today’s dollars.
Now let’s look again at the decision to lend the money for a period of t=2 years:
Repayment in two years = $100(1.10)*(1.10) = $121
FV = PV(1+r)(1+r) = PV(1+r)t Or
PV = FV/(1+r)t
Money today is more valuable than the same amount of money in the future.
The present value of $1 received one year from now is $1/(1+r).
The future value of $1 invested today is $1*(1+r).
Interest paid on savings and interest charged on borrowing is designed to equate the value of dollars today with the value of future dollars.

12. Using Present Value
What if you had to wait 10 years to receive your $20,000?
If I put my $10,000 in an alternative investment earning 8%:
FV is $21, = 10,000 x (1.08)10
What is $20,000 in 10 years worth today?
PV is $9, = 20,000 / (1.08)10
As the above examples demonstrate, there is a difference between dollars received today and dollars received in the future. We will provide some more specifics to this relationship.
Generalization:
To see the relationship between dollars today (present value PV) and dollars 1 year from now (future value FV), a simple equation is applied:
Future Payment, or FV = PV*(1+r) or, using our example,
FV = $100*(1.10) = $110
In other words, one year into the future, $100 in the present will be worth $110.
This is true whether you saved it or lent it to your friend.
We can also rearrange our equation and solve for the present value PV:
PV = FV/(1+r)
Using our example again,
PV = $110/(1.10) = $100
This tells us that $110 received a year from now is worth $100 in today’s dollars.
Now let’s look again at the decision to lend the money for a period of t=2 years:
Repayment in two years = $100(1.10)*(1.10) = $121
FV = PV(1+r)(1+r) = PV(1+r)t Or
PV = FV/(1+r)t
Money today is more valuable than the same amount of money in the future.
The present value of $1 received one year from now is $1/(1+r).
The future value of $1 invested today is $1*(1+r).
Interest paid on savings and interest charged on borrowing is designed to equate the value of dollars today with the value of future dollars.

13. Using Present Value
If your grandma said you could have $5,000 when you graduate from college in four years, what is that money worth today?
PV = FV / (1 + r)n
PV = 5,000 / (1 + r)4
It depends on the interest rate
If r = 3%, PV = $4,442.43
If r = 5%, PV =
If r = 10%, PV =
As the above examples demonstrate, there is a difference between dollars received today and dollars received in the future. We will provide some more specifics to this relationship.
Generalization:
To see the relationship between dollars today (present value PV) and dollars 1 year from now (future value FV), a simple equation is applied:
Future Payment, or FV = PV*(1+r) or, using our example,
FV = $100*(1.10) = $110
In other words, one year into the future, $100 in the present will be worth $110.
This is true whether you saved it or lent it to your friend.
We can also rearrange our equation and solve for the present value PV:
PV = FV/(1+r)
Using our example again,
PV = $110/(1.10) = $100
This tells us that $110 received a year from now is worth $100 in today’s dollars.
Now let’s look again at the decision to lend the money for a period of t=2 years:
Repayment in two years = $100(1.10)*(1.10) = $121
FV = PV(1+r)(1+r) = PV(1+r)t Or
PV = FV/(1+r)t
Money today is more valuable than the same amount of money in the future.
The present value of $1 received one year from now is $1/(1+r).
The future value of $1 invested today is $1*(1+r).
Interest paid on savings and interest charged on borrowing is designed to equate the value of dollars today with the value of future dollars.