Determine the principal that must be invested at 7% compounded monthly, so that $500,000 will be available for retirement in 49 years.
Answer in units of dollars
x*((1 + .07/12)^(49*12)) = 500,000
Solve for x
16356
To find the principal amount that must be invested, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
where:
A = final amount ($500,000 in this case)
P = principal amount
r = annual interest rate (7% or 0.07 in decimal form)
n = number of times interest is compounded per year (monthly compounding, so n = 12)
t = number of years (49 years in this case)
Rearranging the formula to solve for P:
P = A / (1 + r/n)^(nt)
Now we can substitute the known values into the formula and calculate the principal amount.
P = 500,000 / (1 + 0.07/12)^(12*49)
First, let's calculate the term in the parentheses:
(1 + 0.07/12)^(12*49) = 1.00583^588 = 132.16089
Now, divide the initial amount by this result:
P = 500,000 / 132.16089 ≈ $3,783.35
Therefore, the principal amount that must be invested at 7% compounded monthly, so that $500,000 will be available for retirement in 49 years, is approximately $3,783.35.