VIPSolutionsGive step-by-step solution with explanation and final answer: Carla Corporation issues $530,000 of 9% bonds, due in 10 years, with interest payable semiannually. At the time of issue, the market rate for such bonds is 10% . Click here to view factor tables. Compute the issue price of the bonds, (Round present value factor calculations to 5 decimal places, eg. 1.251.24 and the final answer to 0 decimal ploces, eg, 58,971) Issue price of the bonds
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Give step-by-step solution with explanation and final answer: Carla Corporation issues $530,000 of 9% bonds, due in 10 years, with interest payable semiannually. At the time of issue, the market rate for such bonds is 10% . Click here to view factor tables. Compute the issue price of the bonds, (Round present value factor calculations to 5 decimal places, eg. 1.251.24 and the final answer to 0 decimal ploces, eg, 58,971) Issue price of the bonds
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Asked by: Nishant Saini
Created at: 2025-09-27 22:08:58
Asked by: Nishant Saini
Created at: 2025-09-27 22:08:58
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# Step-by-Step Solution: Bond Issue Price Calculation
Carla Corporation issues $530,000 of 9% bonds, due in 10 years, interest payable **semiannually**.
**Market rate** at issuance: **10%** (annual).
We are to compute the **issue price** of the bonds.
---
## 1. Identify Cash Flows
- **Face Value (FV):** $530,000 (paid at maturity)
- **Coupon Rate:** 9% annual ⇒ 4.5% semiannual
- **Market Rate:** 10% annual ⇒ 5% semiannual
- **Periods:** 10 years × 2 = **20 periods**
### Semiannual Interest Payment
\[
\text{Interest Payment} = \text{Face Value} \times \text{Coupon Rate per period}
\]
\[
= \$530,000 \times 4.5\% = \$530,000 \times .045 = \$23,850
\]
---
## 2. Present Value Calculations
### a. Present Value of Interest Payments (Annuity)
\[
\text{PV}_{\text{interest}} = \text{Payment} \times \text{PVIFA}(r, n)
\]
Where:
- \( r = 5\% \)
- \( n = 20 \)
- \(\text{Payment} = \$23,850\)
Look up or compute the Present Value Interest Factor of an Annuity (PVIFA) for 5%, 20 periods.
#### PVIFA Formula
\[
\text{PVIFA}(r, n) = \frac{1 - (1 + r)^{-n}}{r}
\]
Plug in values:
\[
\text{PVIFA}(5\%, 20) = \frac{1 - (1 + .05)^{-20}}{.05}
\]
\[
(1 + .05)^{-20} = (1.05)^{-20} \approx .37689
\]
\[
\text{PVIFA} = \frac{1 - .37689}{.05} \approx \frac{.62311}{.05} = 12.46220
\]
So,
\[
\text{PV}_{\text{interest}} = \$23,850 \times 12.46220 = \$297,205.47
\]
---
### b. Present Value of Face Value (Single Sum)
\[
\text{PV}_{\text{principal}} = \text{Face Value} \times \text{PVIF}(r, n)
\]
- \(\text{PVIF}(5\%, 20) = (1 + .05)^{-20} = .37689\)
\[
\text{PV}_{\text{principal}} = \$530,000 \times .37689 = \$199,751.70
\]
---
## 3. Add Both Present Values
\[
\text{Issue Price} = \text{PV}_{\text{interest}} + \text{PV}_{\text{principal}}
\]
\[
= \$297,205.47 + \$199,751.70 = \$496,957.17
\]
---
## 4. Final Answer (Rounded)
\[
\boxed{\$496,957}
\]
---
## Summary Table
| Component | Calculation | Present Value |
|-------------------|-----------------------------|-----------------|
| Interest Payments | $23,850 × 12.46220 | $297,205 |
| Principal | $530,000 × .37689 | $199,752 |
| **Total** | | **$496,957** |
---
**Final Answer:**
**The issue price of the bonds is** $\boxed{496,957}$.
---
**Note:**
- Present value factors rounded to 5 decimal places as required.
- Final answer rounded to decimal places.
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