Grandparents plan to open an account on their grandchild’s birthday and contribute each month until she goes to college. How much must they contribute at the beginning of each month in an investment that pays 13%, compounded monthly, if they want the balance to be $150,000 at the end of 18 years? (Round your answer to the nearest cent.)\
To calculate how much the grandparents must contribute at the beginning of each month to reach a balance of $150,000 at the end of 18 years in an investment that pays 13% interest compounded monthly, we can use the formula for future value in compound interest:
\[FV = P \left(1 + \frac{r}{n}\right)^{nt}\]
Where:
– \(FV\) is the future value we want to reach, which is $150,000.
– \(P\) is the initial monthly contribution we want to find.
– \(r\) is the annual interest rate in decimal form, which is 13% or 0.13.
– \(n\) is the number of times the interest is compounded per year, which is 12 for monthly compounding.
– \(t\) is the number of years the money is invested for, which is 18 in this case.
We need to solve for \(P\), so we’ll rearrange the formula:
\[P = \frac{FV}{\left(1 + \frac{r}{n}\right)^{nt}}\]
Now, plug in the values:
\[P = \frac{150,000}{\left(1 + \frac{0.13}{12}\right)^{12 \cdot 18}}\]
Let’s calculate this:
\[P = \frac{150,000}{\left(1 + \frac{0.013}{12}\right)^{216}}\]
Now, calculate the denominator first:
\[P = \frac{150,000}{\left(1.01083333333\right)^{216}}\]
Raise the denominator to the power of 216:
\[P = \frac{150,000}{4.31755}\]
Now, divide to find the initial monthly contribution:
\[P \approx 34,775.27\]
So, the grandparents must contribute approximately $34,775.27 at the beginning of each month in the investment that pays 13%, compounded monthly, in order to have a balance of $150,000 at the end of 18 years.
In summary, to ensure that their grandchild’s college fund reaches $150,000 in 18 years with a 13% annual interest rate compounded monthly, the grandparents should contribute approximately $34,775.27 at the start of each month. This financial planning will help them achieve their savings goal and provide for their grandchild’s education.
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