On December​ 31, 2025​, when the market interest rate is 14​%, Benson Realty issues $ 500,000 of 11.25​%, 10​-year bonds payable. The bonds pay interest semiannually. Determine the present value of the bonds at issuance. ​(Round all currency amounts to the nearest whole​ dollar.)

To determine the present value of the bonds at issuance, we'll calculate the present value of the bond's future cash flows, which includes both the present value of the interest payments and the present value of the principal repayment.

Given:
- **Face value of the bonds** ($F$) = $500,000
- **Coupon rate** = 11.25%
- **Market interest rate** = 14%
- **Time to maturity** ($n$) = 10 years
- **Payment frequency** = semiannual

### Step 1: Calculate the semiannual coupon payment
The bonds pay interest semiannually, so we divide the annual coupon rate by 2.

\[
\text{Semiannual coupon rate} = \frac{11.25\%}{2} = 5.625\%
\]

\[
\text{Semiannual coupon payment} = 500{,}000 \times 5.625\% = 28{,}125
\]

### Step 2: Calculate the number of periods and the semiannual market interest rate
Since the bonds pay interest semiannually and have a 10-year maturity:

\[
\text{Number of periods} = 10 \times 2 = 20
\]

\[
\text{Semiannual market interest rate} = \frac{14\%}{2} = 7\%
\]

### Step 3: Calculate the present value of the interest payments (Annuity)
The present value of an annuity formula is:

\[
PV_{\text{annuity}} = C \times \left(\frac{1 - (1 + r)^{-n}}{r}\right)
\]

Where:
- $C$ is the semiannual coupon payment
- $r$ is the semiannual market interest rate
- $n$ is the number of periods

Substituting the values:

\[
PV_{\text{annuity}} = 28{,}125 \times \left(\frac{1 - (1 + 0.07)^{-20}}{0.07}\right)
\]

### Step 4: Calculate the present value of the principal (Single sum)
The present value of the principal is calculated using the formula:

\[
PV_{\text{principal}} = F \times (1 + r)^{-n}
\]

Substituting the values:

\[
PV_{\text{principal}} = 500{,}000 \times (1 + 0.07)^{-20}
\]

### Step 5: Calculate the total present value of the bond
The total present value of the bond is the sum of the present values of the interest payments and the principal.

Let me calculate this for you.The present value of the bonds at issuance is **$427,166**.

Do you want further details or have any questions? Here are some related questions:

1. How does the market interest rate affect the present value of bonds?
2. What is the significance of the coupon rate in bond valuation?
3. How would the present value change if the bonds were issued at a different market interest rate?
4. What are the implications of issuing bonds at a discount?
5. How do you calculate the yield to maturity (YTM) on a bond?

**Tip:** When the market interest rate is higher than the coupon rate, the bond will be issued at a discount, meaning its present value is less than its face value.

Learn how to calculate the present value of bonds issued by Benson Realty at a 14% market interest rate. Detailed step-by-step solution including semiannual coupon payments and principal repayment. Present value calculation: $427,166. Suitable for finance professionals and students.

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