​Fitzgerald's 20​-year bonds pay 9 percent interest annually on a ​$1 comma 000 par value.  If the bonds sell at $ 925​, what is the​ bond's yield to​ maturity?  What would be the yield to maturity if the bonds paid interest​ semiannually?  Explain the difference. round to three decimal places

To calculate the bond's Yield to Maturity (YTM), we can use the following formula:

\[
YTM = \frac{C + \frac{{F - P}}{n}}{\frac{{F + P}}{2}}
\]

Where:
- $C$ is the annual coupon payment.
- $F$ is the face value of the bond (par value).
- $P$ is the current price of the bond.
- $n$ is the number of years to maturity.

For the given problem:
- $C = 9\% \times 1000 = 90$ dollars
- $F = 1000$ dollars
- $P = 925$ dollars
- $n = 20$ years

### 1. Yield to Maturity (YTM) for Annual Payments

Using the above formula, let's calculate the approximate YTM.

### 2. Yield to Maturity (YTM) for Semiannual Payments

For semiannual payments, the number of periods doubles, and the coupon payment is halved:

- Semiannual coupon payment = $\frac{90}{2} = 45$ dollars
- Number of periods = $20 \times 2 = 40$ periods

The YTM calculation involves solving the following equation:

\[
P = \sum_{t=1}^{n} \frac{C/2}{(1 + YTM/2)^t} + \frac{F}{(1 + YTM/2)^n}
\]

I will calculate the YTM for both scenarios.### Results:

1. **Yield to Maturity (YTM) for Annual Payments**: 
   \[
   YTM \approx 9.740\% 
   \]

2. **Yield to Maturity (YTM) for Semiannual Payments**:
   \[
   YTM \approx 19.732\%
   \]

### Explanation of the Difference:
The YTM is higher when calculated with semiannual payments because the bondholder receives payments more frequently, and this additional compounding effect increases the effective yield. In the semiannual scenario, the bond's interest is compounded more often, leading to a higher effective yield compared to annual compounding.

Would you like more details or have any questions?

### Related Questions:
1. How does the current bond price affect the YTM?
2. What happens to the YTM if the bond price increases?
3. How does the frequency of interest payments influence the effective yield?
4. What is the relationship between bond maturity and YTM?
5. How would the YTM change if the coupon rate were different?

### Tip:
Always consider the compounding frequency when comparing yields on bonds, as it can significantly affect the overall return.

Learn how to calculate the Yield to Maturity (YTM) for Fitzgerald's 20-year bonds paying 9% annual interest on a $1,000 par value. Understand the difference in YTM if interest were paid semiannually. Round answers to three decimal places.

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