The Derr-McGee Manufacturing Company plans to build a new $50,000 warehouse seven years from now. They plan to accumulate the $50,000 in an account before beginning construction. If money is worth 7% compounded annually, how much must each year?s deposit be in order to accumulate $50,000 at the end of the seventh year?
To find out how much must be deposited each year in order to accumulate $50,000 at the end of the seventh year, we will use the formula for future value of a series of deposits:
FV = P * ((1 + r)^n - 1) / r
Where:
FV is the future value
P is the annual deposit
r is the interest rate per year (as a decimal)
n is the number of years
In this case, we want to accumulate $50,000 at the end of the seventh year, so FV = $50,000, the interest rate is 7% or 0.07, and the number of years is 7.
Using the formula, we can rearrange it to solve for the annual deposit (P):
P = FV * r / ((1 + r)^n - 1)
Plugging in the values:
P = $50,000 * 0.07 / ((1 + 0.07)^7 - 1)
Simplifying:
P = $50,000 * 0.07 / (1.07^7 - 1)
P = $50,000 * 0.07 / (1.5386 - 1)
P = $50,000 * 0.07 / 0.5386
P ≈ $6,500.87
Therefore, each year's deposit must be approximately $6,500.87 in order to accumulate $50,000 at the end of the seventh year.
To find out how much must be deposited each year, we can use the concept of future value of a series of deposits.
The formula for the future value of a series of deposits is:
FV = P * [(1 + r) ^ n - 1] / r
Where:
FV is the future value we want to accumulate (in this case, $50,000)
P is the annual deposit
r is the interest rate per compounding period (in this case, 7% or 0.07)
n is the number of compounding periods (in this case, 7 years)
We can rearrange this formula to solve for the annual deposit amount (P):
P = FV * (r / [(1 + r) ^ n - 1])
Substituting the given values into the formula, we have:
P = $50,000 * (0.07 / [(1 + 0.07) ^ 7 - 1])
Calculating this equation will give us the annual deposit amount.