Final answer:
Kaitlyn must make two equal payments to clear a $15,000 loan with 3% monthly interest, using three months as the focal date.
Step-by-step explanation:
Kaitlyn has to make two equal payments to settle a $15,000 loan with an interest rate of 3.00% per month. To calculate the size of the payments, we will use the present value of annuities formula because the payments are equal and regularly spaced. Since the focal date is in three months, the first payment has no interest, and the second payment will accumulate interest for three months.
To find the present value (PV) of the second payment, we'll use the present value formula for a future sum:
Present Value (PV) = Future Value (FV) / (1 + i)^n
Where:
- i is the monthly interest rate (3%)
- n is the number of periods until payment (3 months)
- FV is the future value of the payment
Since the total debt is $15,000 and there are two payments, each payment in today's terms would be $7,500. However, when we account for interest, the present value of the second payment will be less than $7,500, meaning the first payment will need to be greater than half to compensate.
Let's calculate the present value of the second payment:
PV = $7,500 / (1 + 0.03)^3
PV ≈ $7,500 / (1.093)
PV ≈ $6,864.37
Now, we know the present value that the second payment should represent today. The sum of the first payment and the present value of the second payment should equal the total debt:
First payment + $6,864.37 = $15,000
First payment = $15,000 - $6,864.37
First payment ≈ $8,135.63
Therefore, Kaitlyn's first payment today should be approximately $8,135.63, and the second payment, in three months, will be $7,500.