Final answer:
To find the implied interest rate of Matthew's loan, we need to determine the yearly interest rate that equates the present value of his three $950 repayments to the initial loan amount of $2,587.09. We use the present value formula for multiple cash flows and may require numerical methods or financial calculators to solve the non-linear equation.
Step-by-step explanation:
To calculate the implied interest rate in this agreement, we need to use the present value formula for multiple cash flows. In this case, Matthew borrowed $2,587.09 and agreed to repay the loan in three equal installments of $950 at the end of each year. We will find the interest rate that equates the present value of these payments to the original loan amount.
Let the yearly interest rate be r. The present value (PV) of the repayments can be calculated as follows:
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- PV of first $950 payment at the end of year 1: $950 / (1 + r)^1
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- PV of second $950 payment at the end of year 2: $950 / (1 + r)^2
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- PV of third $950 payment at the end of year 3: $950 / (1 + r)^3
The sum of these present values must equal the amount borrowed, $2,587.09. Hence, the equation is:
950 / (1 + r) + 950 / (1 + r)^2 + 950 / (1 + r)^3 = 2,587.09
This is a non-linear equation, which we typically solve using numerical methods, such as the Newton-Raphson method or using financial calculators or spreadsheet tools that include built-in functions for solving for the interest rate.
Once r is calculated, it represents the implied yearly interest rate for Matthew's loan agreement.