Barry Haworth
University of Louisville
Department of Economics
Economics 201

## Finding Present Value

A present value calculation tells us how much future money is worth today. It is the opposite of asking what a sum of money is worth in a later period, after that money collects interest in an interest-bearing account. That is, a present value calculation is the inverse of a future value calculation.

Assume throughout this handout that interest is compounded annually (i.e. at the end of each year, \$X becomes \$X(1 + r)). Under this assumption, we calculate present value in one of two ways:

 (a) PV of \$X = \$X/(1 + r)t (b) PV = [\$X1(1 + r)1] + [\$X2/(1 + r)2] + ... + [\$XN/(1 + r)N]

(a) is used when we are receiving \$X in one future period. That is, (a) gives us the present value of \$X collected after t years.

(b) is used when we receive a series of payments in different years. That is, (b) gives us the present value of collecting \$X1 after 1 year, then \$X2 after 2 years, etc.

Example I: Show me the money!
Bill has just learned that he has just inherited money in a time deposit that will provide him with \$100 million when the account matures in two years. He is considering whether to withdraw this money today. The interest rate on the account is 8%. If there were no early withdrawal penalty, then how much would Bill get after withdrawing this money today?

PV of \$100m = \$100m/(1 + 0.08)2
PV of \$100m = \$85.7m

Bill would receive \$85.7 million, if he makes an early withdrawal.

Example II: Who wants to be a "thousand-aire"?
Joan is a winning contestant on a game show and she must choose between three prizes. The first choice involves getting a \$10,000 payment, but not until four years from now. The second involves receiving three \$3,000 payments, one per year for the next three years. The third choice is that she receives \$8,000 today. If all interest rates were equal to 10%, what should she choose?

In order to compare these three options, we must calculate the present value of each choice (note on choice 3 that the present value of receiving \$8000 today is \$8000).

Choice 1:
PV1 = \$10000/(1 + 0.1)4
PV1 = \$6830.14

Choice 2:
PV2 = [\$3000/(1 + 0.1)1] + [\$3000/(1 + 0.1)2] + [\$3000/(1 + 0.1)3]
PV2 = \$7460.56

Choice 3:
PV3 = \$8000

Joan should choose to receive \$8000 today.