Lincoln Memorial hospital has just been informed that private donor is willing to contribute $5 million per year at the beginning of each year for fifteen years. What is the current dollar value of this contribution if the discount rate is 9 percent?
We have two main annuity formulas
1. PV of annuity = payment [ 1 - (1+i)^-n ]/i
and
2. Amount of annutiy = payment [ (1+i)^n - 1 ]/i
both of these assume that the payments are made at the end of each interest period.
So for this question, we have to separate the first payment from the remaining 14.
PV = 5 million + 5million( 1 - 1.09^-14)/.09
= 5 million + 5million( 7.78615...)
= $43,930,751.94
To find the current value of the $5 million annual contribution for fifteen years, we need to discount the future cash flows to their present value. The present value represents the current dollar value of future cash flows, taking into account the time value of money.
To calculate the current value, we can use the formula for the present value of an annuity:
PV = R * (1 - (1 + r)^(-n)) / r
Where:
PV = present value
R = annual payment
r = discount rate
n = number of years
In this case, the annual payment (R) is $5 million, the discount rate (r) is 9% (0.09 as a decimal), and the number of years (n) is fifteen.
Using the formula, we can calculate the present value:
PV = $5,000,000 * (1 - (1 + 0.09)^(-15)) / 0.09
Calculating this equation will give us the current dollar value of the contribution.