To determine the proceeds of a non-interest-bearing promissory note for [tex]$7231.56, discounted 55 months before its due date at an interest rate of 9.9% compounded semi-annually, we need to follow the step-by-step process below:
1. Note Amount:
- The face value (or maturity value) of the note is $[/tex]7231.56.
2. Discount Rate and Compounding:
- The nominal annual interest rate is 9.9% (expressed as 0.099 in decimal form).
- Interest is compounded semi-annually, which means there are 2 compounding periods per year.
3. Time to Maturity:
- The time to maturity is 55 months.
- Since interest is compounded semi-annually, we need to convert months to years. There are 12 months in a year, so:
[tex]\[
\text{Years to maturity} = \frac{55 \text{ months}}{12 \text{ months/year}} = 4.5833\overline{3} \approx 4.5833 \text{ years}
\][/tex]
4. Present Value Factor:
- The present value factor for a discount rate compounded semi-annually is calculated using the formula:
[tex]\[
\text{Present value factor} = \left(1 + \frac{\text{annual rate}}{\text{periods per year}}\right)^{-\text{periods per year} \times \text{years to maturity}}
\][/tex]
- Substituting the given values:
[tex]\[
\text{Present value factor} = \left(1 + \frac{0.099}{2}\right)^{-2 \times 4.5833}
\][/tex]
- Simplifying, we get:
[tex]\[
\text{Present value factor} \approx 0.6422
\][/tex]
5. Calculate Proceeds:
- The proceeds, or the present value of the note, are obtained by multiplying the note amount by the present value factor:
[tex]\[
\text{Proceeds} = \text{Note amount} \times \text{Present value factor}
\][/tex]
- Using the given note amount and present value factor:
[tex]\[
\text{Proceeds} = 7231.56 \times 0.6422 \approx 4644.01
\][/tex]
In conclusion, the proceeds of the non-interest-bearing promissory note discounted 55 months before its due date at 9.9% compounded semi-annually is approximately $4644.01.
