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Suppose the following bond quote for IOU Corporation appears in the financial page of today's newspaper. Assume the bond has a face value of [tex]$\$2{,}000[tex]$[/tex], and the current date is April 19, 2023.

\begin{tabular}{|c|c|c|c|c|c|}
\hline
Company (Ticker) & Coupon & Maturity & Last Price & Last Yield & \begin{tabular}{l}
Estimated Volume \\
(e00s)
\end{tabular} \\
\hline
IOU (IO) & 7.50 & April 19, 2040 & 92.962 & $[/tex]? ?$ & 99 \\
\hline
\end{tabular}

a. What is the yield to maturity of the bond?
Note: Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.

b. What is the current yield?
Note: Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.

\begin{tabular}{|l|l|l|}
\hline
a. Yield to maturity & & \% \\
\hline
b. Current yield & & \% \\
\hline
\end{tabular}



Answer :

GinnyAnswer
Let's break down the solution step-by-step to find the yield to maturity (YTM) and the current yield for the IOU Corporation bond.

### Parameters and Definitions
- Face Value of the Bond: \$2,000
- Coupon Rate: 7.5%
- Maturity Year: 2040
- Current Date: 2022
- Last Price: 92.962% of face value

### Step-by-Step Solution

#### a. Yield to Maturity (YTM)
To find the YTM, we use the given formula:
[tex]\[ \text{YTM} \approx \left(\frac{\text{Coupon Payment} + \left(\frac{\text{Face Value} - \text{Current Price}}{\text{Years to Maturity}}\right)}{\frac{\text{Face Value} + \text{Current Price}}{2}}\right) \times 100 \][/tex]

Step 1: Calculate the annual coupon payment.
[tex]\[ \text{Annual Coupon Payment} = \text{Face Value} \times \text{Coupon Rate} \][/tex]
[tex]\[ \text{Annual Coupon Payment} = 2000 \times 0.075 = 150 \][/tex]

Step 2: Calculate the current price in dollars.
[tex]\[ \text{Current Price} = \frac{92.962}{100} \times 2000 = 1859.24 \][/tex]

Step 3: Calculate the years to maturity.
[tex]\[ \text{Years to Maturity} = 2040 - 2022 = 18 \text{ years} \][/tex]

Step 4: Calculate the YTM using the formula.
[tex]\[ \text{YTM} \approx \left(\frac{150 + \left(\frac{2000 - 1859.24}{18}\right)}{\frac{2000 + 1859.24}{2}}\right) \times 100 \][/tex]
[tex]\[ \text{YTM} \approx \left(\frac{150 + \left(\frac{140.76}{18}\right)}{1929.62}\right) \times 100 \][/tex]
[tex]\[ \text{YTM} \approx \left(\frac{150 + 7.82}{1929.62}\right) \times 100 \][/tex]
[tex]\[ \text{YTM} \approx \left(\frac{157.82}{1929.62}\right) \times 100 \][/tex]
[tex]\[ \text{YTM} \approx 8.18\% \][/tex]

The yield to maturity (YTM) for the IOU Corporation bond is approximately 8.18%.

#### b. Current Yield
To calculate the current yield, we use the formula:
[tex]\[ \text{Current Yield} = \left(\frac{\text{Annual Coupon Payment}}{\text{Current Price}}\right) \times 100 \][/tex]

Step 1: Use the annual coupon payment calculated earlier.
[tex]\[ \text{Annual Coupon Payment} = 150 \][/tex]

Step 2: Divide the annual coupon payment by the current price.
[tex]\[ \text{Current Yield} = \left(\frac{150}{1859.24}\right) \times 100 \][/tex]
[tex]\[ \text{Current Yield} \approx 8.07\% \][/tex]

The current yield for the IOU Corporation bond is approximately 8.07%.

### Final Answers
[tex]\[ \begin{tabular}{|l|l|l|} \hline a. Yield to maturity & & 8.18\% \\ \hline b. Current yield & & 8.07\% \\ \hline \end{tabular} \][/tex]

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