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.
Bill would receive $85.7 million, if he makes an early
withdrawal.
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).
Joan should choose to receive $8000 today.
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?
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?
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