Difficulty: Medium
Correct Answer: 8,870.97 dollars
Explanation:
Introduction / Context:
This problem involves finding the present value of a single future amount when interest is compounded monthly. It tests your ability to handle both discounting and the conversion from an annual nominal rate to a monthly periodic rate.
Given Data / Assumptions:
Concept / Approach:
The present value of a single future amount when compounding is periodic is:
PV = FV / (1 + i)^n where i is the periodic rate and n is the total number of periods. Here, we first find the monthly rate and then apply the formula.
Step-by-Step Solution:
Step 1: Compute the monthly rate i = 0.04 / 12 ≈ 0.003333. Step 2: Number of months n = 36. Step 3: Compute the growth factor (1 + i)^n = (1.003333)^36. Step 4: Compute PV = 10000 / (1.003333^36). Step 5: Evaluating gives PV ≈ 8,870.97 dollars. Thus, you must invest about 8,870.97 dollars today to have 10,000 dollars after 3 years.
Verification / Alternative Check:
As a rough check, if you invested 9,000 dollars at 4% compounded annually for 3 years, the future value would be roughly 9,000 * 1.1249 ≈ 10,124 dollars. Since monthly compounding gives slightly more growth, a present value just under 9,000 dollars makes sense for a 10,000 dollar target.
Why Other Options Are Wrong:
Option B (8,695.61 dollars) and option C (7,695.00 dollars) are too low, meaning they would not grow enough to reach 10,000 dollars in 3 years at 4% monthly compounding.
Option D (9,092.00 dollars) and option E (9,500.00 dollars) are higher than necessary; investing these amounts would produce more than 10,000 dollars.
Common Pitfalls:
Common mistakes include using the annual rate directly instead of dividing by 12, or using the wrong exponent (such as 3 instead of 36). Another error is to use simple interest formulas, which do not capture the effect of monthly compounding.
Final Answer:
You must invest approximately 8,870.97 dollars now to accumulate 10,000 dollars after 3 years at 4% compounded monthly.
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