Questions: In 6 years, Mrs. Folkers must pay off a note with a face value of 13,000, and interest of 12% per year, compounded semiannually. Find the future value of the note. Then find the amount that the holder of the note should accept as complete payment today if money can be invested at 8% per year, compounded quarterly. What is the maturity value of the note? 26158.55 (Round to the nearest cent as needed.) How much money should the holder of the note accept as complete payment today? (Round to the nearest cent as needed.)

Solution Steps

To solve this problem, we need to follow these steps:

1. Calculate the future value of the note: Use the compound interest formula to find the future value of the note after 6 years with an interest rate of 12% per year, compounded semiannually.
2. Calculate the present value: Use the present value formula to find out how much the holder of the note should accept as complete payment today if the money can be invested at 8% per year, compounded quarterly.

1. Future Value Calculation:

   • Use the formula for compound interest: \( A = P \left(1 + \frac{r}{n}\right)^{nt} \)
   • Here, \( P = 13000 \), \( r = 0.12 \), \( n = 2 \) (since it's compounded semiannually), and \( t = 6 \).

2. Present Value Calculation:

   • Use the formula for present value: \( PV = \frac{FV}{\left(1 + \frac{r}{n}\right)^{nt}} \)
   • Here, \( FV \) is the future value calculated in step 1, \( r = 0.08 \), \( n = 4 \) (since it's compounded quarterly), and \( t = 6 \).

To find the future value \( A \) of the note, we use the compound interest formula: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]

Given:

• \( P = 13000 \)
• \( r = 0.12 \)
• \( n = 2 \) (compounded semiannually)
• \( t = 6 \)

Substitute the values: \[ A = 13000 \left(1 + \frac{0.12}{2}\right)^{2 \times 6} \] \[ A = 13000 \left(1 + 0.06\right)^{12} \] \[ A = 13000 \left(1.06\right)^{12} \] \[ A \approx 26158.55 \]

To find the present value \( PV \), we use the present value formula: \[ PV = \frac{FV}{\left(1 + \frac{r}{n}\right)^{nt}} \]

Given:

• \( FV = 26158.55 \)
• \( r = 0.08 \)
• \( n = 4 \) (compounded quarterly)
• \( t = 6 \)

Substitute the values: \[ PV = \frac{26158.55}{\left(1 + \frac{0.08}{4}\right)^{4 \times 6}} \] \[ PV = \frac{26158.55}{\left(1 + 0.02\right)^{24}} \] \[ PV = \frac{26158.55}{\left(1.02\right)^{24}} \] \[ PV \approx 16263.34 \]

Final Answer