An investment earns 4% per annum compounded monthly. What amount must be invested now in order to accumulate 10,000 dollars after 3.5 years?

Introduction / Context:
This question is about finding the present value of a future amount when interest is compounded monthly. Such calculations are fundamental in finance and banking, since investors often know the desired future value and must decide how much to invest today at a given interest rate and compounding pattern.

Given Data / Assumptions:

• Future value FV = 10,000 dollars after 3.5 years.
• Nominal annual interest rate j = 4% per annum.
• Interest is compounded monthly, so m = 12 periods per year.
• Periodic interest rate i = j / m = 0.04 / 12 per month.
• Total number of months n = 3.5 * 12 = 42 months.

Concept / Approach:
The present value PV of a single future amount FV under compound interest with periodic rate i over n periods is given by the formula PV = FV / (1 + i)^n. Since the interest is compounded monthly, we work entirely in monthly terms. We calculate the periodic rate, the total number of compounding periods, and then discount the future value back to the present using the compound factor.

Step-by-Step Solution:
Step 1: Compute the monthly interest rate i = 0.04 / 12.Step 2: Calculate i = 0.003333... per month.Step 3: Determine the number of months n = 3.5 * 12 = 42.Step 4: Use the present value formula PV = FV / (1 + i)^n.Step 5: Substitute values: PV = 10,000 / (1 + 0.003333...)^42.Step 6: Evaluating this expression gives PV approximately equal to 8695.61 dollars.

Verification / Alternative check:
To verify, we can project the present value forward. If we invest 8695.61 dollars at a monthly rate of about 0.3333% for 42 months, the future value is FV = 8695.61 * (1 + 0.003333...)^42, which comes back close to 10,000 dollars. Small rounding differences may appear due to decimal approximations, but the result should be extremely close to 10,000, confirming the correctness of the present value used in the solution.

Why Other Options Are Wrong:

• 6786.00: This is too low to grow to 10,000 at 4% annual interest in only 3.5 years, even with monthly compounding.
• 3478.00: This is much too small and would not reach 10,000 even over a much longer period at this modest interest rate.
• 4092.00: This is also too low; compounding at 4% cannot more than double such a small amount in only 3.5 years.

Common Pitfalls:
Students sometimes forget to convert the annual rate into a monthly rate and mistakenly use 4% per month, which leads to a huge error. Another mistake is to use 3.5 years as the exponent directly instead of converting to months when the rate is monthly. Others may invert the formula and multiply instead of dividing, which produces a value greater than the future value instead of a discounted present value. Careful attention to the compounding frequency and correct use of the exponent are essential.

Final Answer:
The amount that must be invested now is approximately 8695.61 dollars in order to accumulate 10,000 dollars after 3.5 years at 4% per annum compounded monthly.