A mutual fund pays 13% compounded monthly. How much should I invest now so that 6 years from now I will have $4000 in the account?
The correct answer and explanation is:
To find out how much you need to invest now (the present value) to have $4000 in the future, with 13% annual interest compounded monthly over 6 years, we use the compound interest formula: A=P(1+rn)ntA = P \left(1 + \frac{r}{n} \right)^{nt}
Where:
- A=A = Future Value = $4000
- P=P = Present Value (what we are solving for)
- r=r = Annual interest rate = 0.13 (13%)
- n=n = Number of times compounded per year = 12 (monthly)
- t=t = Time in years = 6
Step 1: Plug values into the formula
4000=P(1+0.1312)12×64000 = P \left(1 + \frac{0.13}{12} \right)^{12 \times 6} 4000=P(1+0.0108333)724000 = P \left(1 + 0.0108333 \right)^{72} 4000=P(1.0108333)724000 = P \left(1.0108333 \right)^{72} 4000=P(2.122573)4000 = P (2.122573)
Step 2: Solve for PP
P=40002.122573P = \frac{4000}{2.122573} P≈1884.41P \approx 1884.41
✅ Final Answer:
You should invest approximately $1,884.41 now to have $4,000 in 6 years at 13% interest compounded monthly.
📘 Explanation (300 words):
This problem is an example of present value calculation using compound interest, where you’re working backwards to determine how much money needs to be invested today to reach a future goal.
Since the interest is compounded monthly, we split the annual interest rate into 12 parts and increase the number of compounding periods accordingly. Instead of growing once per year, the investment grows 12 times per year, making the compounding more frequent—and more powerful.
We use the formula: A=P(1+rn)ntA = P(1 + \frac{r}{n})^{nt}
to calculate how investments grow. In our case, we knew the final amount ($4000) and wanted to work backward to find the starting amount PP.
After calculating the total number of compounding periods (12 months × 6 years = 72), and the monthly rate (13% ÷ 12 = 0.0108333), we evaluated the compounded growth factor (1.0108333)72(1.0108333)^{72}, which came out to about 2.122573. This means that every dollar invested now will grow by a factor of about 2.12 over six years.
By dividing the desired future amount ($4000) by this growth factor, we find the amount that would grow to $4000 in six years. This gives us a precise present value of $1,884.41.