ANSWER

Calculating the present value of a series of future cash flows is a fundamental financial concept, often used in various fields such as finance, investment analysis, and accounting. In this scenario, we are tasked with finding the present value of 13 payments of $535 each, with the first payment received today (Month 0) and subsequent payments at 10-month intervals (Months 10, 20, 30 … 120). To accurately calculate the present value, we need to consider the time value of money, which accounts for the fact that a dollar received in the future is worth less than a dollar received today due to the potential for earning interest or investment returns.

The appropriate interest rate or discount rate is essential for this calculation. In this case, we are provided with an effective annual rate (EAR) of 7.68 percent. To find the present value of these future cash flows, we will use the formula for the present value of an annuity:

\[PV = PMT \times \frac{1 – (1 + r)^{-n}}{r}\]

Where:
– PV = Present Value
– PMT = Payment per period
– r = Interest rate per period
– n = Number of periods

In our case:
– PMT = $535 (the payment received at each interval)
– r = 7.68% per annum, which needs to be adjusted for the 10-month intervals
– n = 13 (the total number of payments)

First, we need to adjust the annual interest rate (7.68%) to the 10-month intervals:

\[r_{\text{monthly}} = \left(1 + \frac{0.0768}{12}\right)^{12/10} – 1\]

Now, we can plug these values into the formula:

\[PV = $535 \times \frac{1 – (1 + 0.0768_{\text{monthly}})^{-13}}{0.0768_{\text{monthly}}}\]

Let’s calculate it step by step:

1. Calculate the monthly interest rate:
\[r_{\text{monthly}} = \left(1 + \frac{0.0768}{12}\right)^{12/10} – 1\]
\[r_{\text{monthly}} \approx 0.061868\]

2. Calculate the present value using the adjusted interest rate:
\[PV \approx $535 \times \frac{1 – (1 + 0.061868)^{-13}}{0.061868}\]

Now, let’s compute the present value:

\[PV \approx $535 \times \frac{1 – 0.29603}{0.061868}\]

\[PV \approx $535 \times \frac{0.70397}{0.061868}\]

\[PV \approx $9,601.99\]

Therefore, the present value of receiving 13 payments of $535 each, with the first payment received today and subsequent payments at 10-month intervals, given an EAR of 7.68 percent, is approximately $9,601.99 at Month 0.

In conclusion, understanding the time value of money and using the appropriate discount rate is crucial for valuing future cash flows. In this case, we adjusted the annual interest rate to monthly intervals and used the present value formula to determine that the present value of these 13 payments is approximately $9,601.99 at Month 0. This calculation is essential for financial decision-making and investment analysis.