Certainly! To determine the bond proceeds (i.e., the present value (PV) of the bonds), we'll calculate the present value of both the coupon payments and the face value of the bond. Here is a step-by-step solution:
### Given Information:
- Face Value (FV): [tex]$1,200,000
- Coupon Rate (C): 8% per year
- Market Rate (r): 6% per year
- Years to Maturity (n): 5 years
- Payments per Year: 2 (semiannual payments)
### Step-by-Step Solution:
1. Determine the semiannual coupon payment:
\[
\text{Coupon Payment} = \frac{\text{Face Value} \times \text{Coupon Rate}}{\text{Payments per Year}}
\]
\[
\text{Coupon Payment} = \frac{1,200,000 \times 0.08}{2} = 48,000
\]
2. Determine the total number of coupon payments:
\[
\text{Total Periods} = \text{Years to Maturity} \times \text{Payments per Year}
\]
\[
\text{Total Periods} = 5 \times 2 = 10
\]
3. Calculate the present value of the annuity (coupon payments):
Using the formula for the present value of an annuity:
\[
PV_{\text{Coups}} = \text{Coupon Payment} \times \left[ \frac{1 - (1 + r/n)^{ -nt }}{ r/n } \right]
\]
where \( r \) is the market rate per period, \( n \) is the number of periods per year, and \( t \) is the number of years.
\[
PV_{\text{Coups}} = 48,000 \times \left[ \frac{1 - (1 + 0.03)^{-10}}{0.03} \right]
\]
\[
PV_{\text{Coups}} = 48,000 \times 8.5302 \approx 409,449.6
\]
4. Calculate the present value of the face value due at maturity:
\[
PV_{\text{FV}} = \frac{\text{Face Value}}{(1 + r/n)^{nt}}
\]
\[
PV_{\text{FV}} = \frac{1,200,000}{(1 + 0.03)^{10}}
\]
\[
PV_{\text{FV}} = \frac{1,200,000}{1.3439} \approx 893,357.5
\]
5. Calculate the bond proceeds (sum of the present values of the coupon payments and the face value):
\[
\text{Bond Proceeds} = PV_{\text{Coups}} + PV_{\text{FV}}
\]
\[
\text{Bond Proceeds} = 409,449.6 + 893,357.5 \approx 1,302,807.1
\]
### Final Answer:
The bond proceeds, or the present value of the bonds, is approximately $[/tex]1,302,807.1.
