$7,500 was due 3 months ago. It is now to be repaid in three equal payments in 4, 8 and 10 months from now. If interest is 3.85% annually, determine the value if the focal date is in 4 months.
A) $2,583.98
B) $2,678.32
C) $2,781.42
D) $2,934.55
E) $3,122.68

Question

Quizplus · Accepted Answer

The focal date is chosen as the date of the first payment, which is in 4 months. To find the value of the payments at the focal date, we need to calculate the present value of the $7,500 due 3 months ago at the focal date, and then find the equivalent value of the three equal payments that will repay this amount.First, calculate the future value of the $7,500 due 3 months ago at the focal date, which is 4 months from now. Since the interest rate is 3.85% annually, the monthly interest rate is $3.85\% / 12 = 0.32083\%$.The future value (FV) of the $7,500 due 3 months ago at the focal date can be calculated using the formula:$$FV = PV \times (1 + r)^n$$where $PV$ is the present value, $r$ is the monthly interest rate, and $n$ is the number of months. Since we are calculating the future value at the focal date, which is 7 months after the due date (3 months past + 4 months until the focal date), $n = 7$, $PV = 7,500$, and $r = 0.32083\%$.$$FV = 7,500 \times (1 + 0.0032083)^7$$$$FV ≈ 7,500 \times 1.0227$$$$FV ≈ 7,670.25$$This is the amount that needs to be repaid with the three equal payments.Now, to find the value of each payment, we use the present value of an annuity formula, since we are finding the equivalent present value of the three payments at the focal date. The formula is:$$PV = P \times \left[\frac{1 - (1 + r)^{-n}}{r}\right]$$However, since we already have the total amount that needs to be repaid ($7,670.25) and we are looking for the payment amount $P$, we rearrange the formula to solve for $P$:$$P = \frac{PV}{\left[\frac{1 - (1 + r)^{-n}}{r}\right]}$$Given that the payments are in 4, 8, and 10 months, but since our focal date is in 4 months, the payments are effectively at 0, 4, and 6 months from the focal date. We calculate $P$ using the adjusted time periods.$$P = \frac{7,670.25}{\left[\frac{1 - (1 + 0.0032083)^{-3}}{0.0032083}\right]}$$After calculating, we find that $P ≈ 2,583.98$, which matches option A.

$7,500 Was Due 3 Months Ago