University of Louisville

Department of Economics

Economics 201

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 = [$X_{1}(1 + r)^{1}] + [$X_{2}/(1 + r)^{2}] + ... + [$X_{N}/(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 $X_{1} after 1 year, then
$X_{2} 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:*

PV_{1} = $10000/(1 + 0.1)^{4}

PV_{1} = $6830.14

*Choice 2:*

PV_{2} = [$3000/(1 + 0.1)^{1}] + [$3000/(1 + 0.1)^{2}] + [$3000/(1 + 0.1)^{3}]

PV_{2} = $7460.56

*Choice 3:*

PV_{3} = $8000

Joan should choose to receive $8000 today.