Pump Primer : Why do you think a dollar today is worth more than a dollar a year from now?

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

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

Biblical Integration : Making decisions in your life can become very stressful and complicated as you move into adulthood. It is for that reason why we should lean on God for strength and wisdom in making the right choices. (Prov 3: 13-18)

The Concept of Present Value “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.

Borrowing, Lending, and Interest 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.

Borrowing, Lending, and Interest Borrowing, Lending, and Interest 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

Borrowing, Lending, and Interest Borrowing, Lending, and Interest 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.

Defining Present Value Defining Present Value Let fv = future value of $ pv = present value of $ r = real interest rate n = # of years The Simple Interest Formula fv = ( 1 + r ) n * pv pv = fv / (1 + r) n

Defining Present Value Defining 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

Defining Present Value Defining Present Value 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.

Defining Present Value Defining Present Value 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 Generalization: FV = PV(1+r)(1+r) = PV(1+r)t Or PV = FV/(1+r)t

Defining Present Value Defining Present Value 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.

Using Present Value 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? What if you had to wait 10 years to receive your $20,000?

Using Present Value Using Present Value If I put my $10,000 in an alternative investment earning 8%: FV = 10,000*(1.08)10 = $21, What is the $20,000 in 10 years worth today? PV = 20,000/(1.08)10 = $ So, you would only have to invest $ to get $20,000 in 10 years, rather than the aforementioned $10,000. Either way, you’re wise to pass on this investment opportunity. 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? What if you had to wait 10 years to receive your $20,000?

Using Present Value Using Present Value $18,200 $20,200 $ 2,000 $ 4,000 $ (14,000) $ 4,000 Cash Outlays for Investing in Lighting YEAR Options01234TOTAL A. CFL’s$15,000$800 B. Incandescents$1,000$4,800 B – A: Savings from CFL’s

Using Present Value Using Present Value Should you do it? At first it seems like a no-brainer. You should switch the bulbs. But, you could have invested the $14,000 in something else, or just put it in the bank. Let’s look at the PV of the savings from the initial expenditure to year four. PV = -14,000/(1+r) /(1+r) /(1+r) /(1+r) /(1+r) 4 If PV > 0, it makes sense to switch the bulbs.

Using Present Value Using Present Value If I put my $10,000 in an alternative investment earning 8%: FV = 10,000*(1.08)10 = $21, What is the $20,000 in 10 years worth today? PV = 20,000/(1.08)10 = $ So, you would only have to invest $ to get $20,000 in 10 years, rather than the aforementioned $10,000. Either way, you’re wise to pass on this investment opportunity. $2,000 $ 185 $ $ 4,000 $ 3810 $ 3636 $ 3628 $ 3305 $ 3455 $ 3005 PV of Savings from CFL’s at 2 Discount Rates (PV = FV/(1+r) YEAR Interest r01234TOTAL 0%($14,000) 5%($14,000) 10%($14,000) $ 4,000 $ 3290 $ /1.05 =

Using Present Value Using Present Value The higher the interest rate, the less value is placed upon future dollars (they’re more heavily discounted) and more emphasis is placed upon current dollars. A higher interest rate makes an alternative (like a simple savings account) more attractive