Sure, let's go through the steps to find the value of the equipment recorded at the time of purchase:
1. Terms and Rates Given:
- Note Payable: [tex]\( \$300,000 \)[/tex]
- Note Term: 5 years
- Coupon Rate: 3.00%
- Market Rate: 10.379%
2. Calculate the Present Value of the Principal:
- The principal amount that needs to be paid at the end of the 5-year term is [tex]\( \$300,000 \)[/tex].
- We need to discount this amount back to its present value using the market rate (10.379%).
- The formula for the present value of the principal is:
[tex]\[
\text{PV(Principal)} = \frac{\$300,000}{(1 + \text{market rate})^{\text{note term}}}
\][/tex]
- Substituting in the values:
[tex]\[
\text{PV(Principal)} = \frac{\$300,000}{(1 + 0.10379)^5}
\][/tex]
- The present value of the principal is calculated to be approximately:
[tex]\[
\text{PV(Principal)} \approx \$183,100.27
\][/tex]
3. Calculate the Present Value of the Interest Payments (Annuities):
- The annual interest payment is determined by the coupon rate:
[tex]\[
\text{Annual Interest Payment} = \$300,000 \times 0.03 = \$9,000
\][/tex]
- We need to find the present value of these annual [tex]\( \$9,000 \)[/tex] payments over the 5-year term, discounted at the market rate.
- The formula for the present value of an annuity is:
[tex]\[
\text{PV(Interest)} = \sum_{i=1}^{5} \frac{\$9,000}{(1 + \text{market rate})^i}
\][/tex]
- Calculating the sum of these discounted payments gives us the present value of the interest payments to be approximately:
[tex]\[
\text{PV(Interest)} \approx \$33,789.31
\][/tex]
4. Calculate the Total Present Value of the Note (Value of the Equipment):
- The total present value of the note, which is the value of the equipment, is the sum of the present value of the principal and the present value of the interest payments.
- Thus,
[tex]\[
\text{Equipment Value} = \text{PV(Principal)} + \text{PV(Interest)}
\][/tex]
[tex]\[
\text{Equipment Value} \approx \$183,100.27 + \$33,789.31
\][/tex]
- The value of the equipment recorded at the time of purchase is approximately:
[tex]\[
\text{Equipment Value} \approx \$216,889.57
\][/tex]
Therefore, the value of the equipment recorded at the time of purchase is [tex]\( \$216,889.57 \)[/tex].
