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Sagot :
To determine the yield to maturity (YTM) and the current yield for the IOU Corporation bond, we need to go through a detailed process:
### Part a: Yield to Maturity (YTM)
#### Step-by-Step Solution:
1. Given Data:
- Face Value (FV): [tex]$2000 - Coupon Rate: 7.65% - Maturity Date: April 19, 2040 - Current Date: April 19, 2022 - Last Price: 93.013% of the face value 2. Calculate Annual Coupon Payment: The annual coupon payment (C) is calculated using the coupon rate: \[ C = \text{Face Value} \times \text{Coupon Rate} = 2000 \times \frac{7.65}{100} = \$[/tex]153
\]
3. Calculate Last Price in Dollars:
The last price (P) is expressed as a percentage of the face value:
[tex]\[ P = \frac{93.013}{100} \times 2000 = \$1860.26 \][/tex]
4. Calculate Years to Maturity:
The years to maturity (T) is the difference between the maturity year and the current year:
[tex]\[ T = 2040 - 2022 = 18 \text{ years} \][/tex]
5. Approximation Formula for YTM:
The approximation formula for YTM is:
[tex]\[ \text{YTM} \approx \frac{C + \frac{FV - P}{T}}{\frac{FV + P}{2}} \times 100 \][/tex]
Substituting the values:
[tex]\[ \text{Numerator} = C + \frac{FV - P}{T} = 153 + \frac{2000 - 1860.26}{18} = 153 + 7.764 = 160.764 \][/tex]
[tex]\[ \text{Denominator} = \frac{FV + P}{2} = \frac{2000 + 1860.26}{2} = 1930.13 \][/tex]
[tex]\[ \text{YTM} \approx \frac{160.764}{1930.13} \times 100 \approx 8.33\% \][/tex]
### Part b: Current Yield
#### Step-by-Step Solution:
1. Given Data: (Same as above)
- Face Value (FV): [tex]$2000 - Coupon Rate: 7.65% - Last Price: 93.013% of the face value 2. Calculate Annual Coupon Payment: (Already calculated above) \[ C = \$[/tex]153
\]
3. Calculate Last Price in Dollars:
(Already calculated above)
[tex]\[ P = \$1860.26 \][/tex]
4. Calculate Current Yield:
The current yield is given by:
[tex]\[ \text{Current Yield} = \frac{C}{P} \times 100 \][/tex]
Substituting the values:
[tex]\[ \text{Current Yield} = \frac{153}{1860.26} \times 100 \approx 8.22\% \][/tex]
### Summary:
- a. Yield to maturity: 8.33%
- b. Current yield: 8.22%
So, we have the results for the IOU Corporation bond:
\begin{tabular}{|l|l|l|}
\hline a. Yield to maturity & 8.33 & \% \\
\hline b. Current yield & 8.22 & \% \\
\hline
\end{tabular}
### Part a: Yield to Maturity (YTM)
#### Step-by-Step Solution:
1. Given Data:
- Face Value (FV): [tex]$2000 - Coupon Rate: 7.65% - Maturity Date: April 19, 2040 - Current Date: April 19, 2022 - Last Price: 93.013% of the face value 2. Calculate Annual Coupon Payment: The annual coupon payment (C) is calculated using the coupon rate: \[ C = \text{Face Value} \times \text{Coupon Rate} = 2000 \times \frac{7.65}{100} = \$[/tex]153
\]
3. Calculate Last Price in Dollars:
The last price (P) is expressed as a percentage of the face value:
[tex]\[ P = \frac{93.013}{100} \times 2000 = \$1860.26 \][/tex]
4. Calculate Years to Maturity:
The years to maturity (T) is the difference between the maturity year and the current year:
[tex]\[ T = 2040 - 2022 = 18 \text{ years} \][/tex]
5. Approximation Formula for YTM:
The approximation formula for YTM is:
[tex]\[ \text{YTM} \approx \frac{C + \frac{FV - P}{T}}{\frac{FV + P}{2}} \times 100 \][/tex]
Substituting the values:
[tex]\[ \text{Numerator} = C + \frac{FV - P}{T} = 153 + \frac{2000 - 1860.26}{18} = 153 + 7.764 = 160.764 \][/tex]
[tex]\[ \text{Denominator} = \frac{FV + P}{2} = \frac{2000 + 1860.26}{2} = 1930.13 \][/tex]
[tex]\[ \text{YTM} \approx \frac{160.764}{1930.13} \times 100 \approx 8.33\% \][/tex]
### Part b: Current Yield
#### Step-by-Step Solution:
1. Given Data: (Same as above)
- Face Value (FV): [tex]$2000 - Coupon Rate: 7.65% - Last Price: 93.013% of the face value 2. Calculate Annual Coupon Payment: (Already calculated above) \[ C = \$[/tex]153
\]
3. Calculate Last Price in Dollars:
(Already calculated above)
[tex]\[ P = \$1860.26 \][/tex]
4. Calculate Current Yield:
The current yield is given by:
[tex]\[ \text{Current Yield} = \frac{C}{P} \times 100 \][/tex]
Substituting the values:
[tex]\[ \text{Current Yield} = \frac{153}{1860.26} \times 100 \approx 8.22\% \][/tex]
### Summary:
- a. Yield to maturity: 8.33%
- b. Current yield: 8.22%
So, we have the results for the IOU Corporation bond:
\begin{tabular}{|l|l|l|}
\hline a. Yield to maturity & 8.33 & \% \\
\hline b. Current yield & 8.22 & \% \\
\hline
\end{tabular}
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