If money can be invested at 8% annual interest compounded quarterly, what is the present value today of $3800 to be received 5 years from now?

Introduction / Context:
This question uses the time value of money concept to find the present value of a future payment when interest is compounded quarterly. Instead of asking how much a current investment will grow to, we reverse the process and discount a known future amount back to today. This is a key idea in finance used for valuing bonds, comparing different payment streams, and making investment decisions.

Given Data / Assumptions:

• Future amount A = 3800 dollars
• Nominal annual interest rate r = 8% per annum
• Compounding frequency m = 4 times per year (quarterly)
• Time t = 5 years until the payment is received
• We must find the present value PV today

Concept / Approach:
For nominal rate r compounded m times per year, the periodic rate per quarter is r / m. The relationship between present value PV and future value A over t years is A = PV * (1 + r / m)^(m * t). To find PV, we rearrange this to PV = A / (1 + r / m)^(m * t). Since the compounding is quarterly, we must use the quarterly rate and the total number of quarters in the exponent. The result will be less than the future value because we are discounting back to the present.

Step-by-Step Solution:
A = 3800, r = 8% = 0.08, m = 4, t = 5 years Periodic quarterly rate = r / m = 0.08 / 4 = 0.02 Total number of quarters = m * t = 4 * 5 = 20 Present value formula: PV = A / (1 + 0.02)^20 Compute the growth factor: (1.02)^20 Using a calculator, (1.02)^20 ≈ 1.48595 PV ≈ 3800 / 1.48595 ≈ 2557.3 So PV is approximately 2557.29 dollars

Verification / Alternative check:
We can verify by checking that if we invest about 2557.29 dollars today at 8% compounded quarterly for 5 years, we should get approximately 3800 dollars. Compute A = 2557.29 * (1.02)^20. Using the same growth factor, A ≈ 2557.29 * 1.48595 ≈ 3800. This confirms that the present value is consistent with the given future value and interest rate.

Why Other Options Are Wrong:
1557.29 is far too low; investing that amount at 8% compounded quarterly for 5 years would not grow to 3800. Values like 2457 and 2567 are near the correct range but do not match the precise calculation using the correct factor. Only 2557.29 closely matches the exact present value obtained from the formula.

Common Pitfalls:
Common errors include using the annual compounding formula instead of quarterly, ignoring the compounding frequency entirely, or accidentally multiplying by the growth factor instead of dividing to find present value. Another mistake is to use t instead of m * t in the exponent. Carefully identifying the periodic rate and the total number of periods is critical for getting the right answer.

Final Answer:
The present value of 3800 dollars due in 5 years at 8 percent interest compounded quarterly is approximately $2557.29 today.