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Using Probability Distributions [LO3] Suppose the returns on long-term corporate bonds and T-bills are
a. What is the probability that in any given year, the return on long-term corporate bonds will be greater than 10 percent? Less than 0 percent?
b. What is the probability that in any given year, the return on T-bills will be greater than 10 percent? Less than 0 percent?
c. In 1979, the return on long-term corporate bonds was −2.76 percent. How likely is it that such a low return will recur at some point in the future? T-bills had a return of 10.56 percent in this same year. How likely is it that such a high return on T-bills will recur at some point in the future?
a)
To determine: The probability of earning more than 10 percent on long-term corporate bonds.
Introduction:
The Normal distribution curve is a bell-shaped curve formed based on the frequency distribution of the observations The mean or average of the observations and their standard deviation define the normal distribution curve.
Standard deviation refers to the variation in the actual observations from the average. Z-Score helps to know how many numbers of standard deviations is the raw score or outcome away from the average or mean.
Answer to Problem 24QP
The probability of earning more than 10 percent on long-term corporate bonds is 32.98 percent.
Explanation of Solution
Given information:
Assume that the returns of long-term corporate bonds have a normal distribution. The average return or mean of long-term corporate bonds is 6.3 percent, and the standard deviation is 8.4 percent (Refer to Figure 12.10 in the text).
Determine the probability of having a return greater than 10 percent on long-term government bonds:
Follow the common steps from Step 1 to Step 3 given below. Next, proceed with the Step 4.
The common steps to be followed to use the “NORM.DIST” function in Excel:
Step 1:
Open an Excel worksheet.
Step 2:
Place the cursor in cell A1.
Step 3:
Select the “Formulas” tab, and go to “More functions” in the ribbon. Under “More functions”, select “Statistical”. Under the drop-down menu of “Statistical”, select “NORM.DIST” function.
After clicking the “NORM.DIST” function, a popup window named “Function arguments” appears.
Step 4:
Enter the values. “X” represents the raw score or outcome. Here, it is necessary to test the probability of having more than 10 percent returns. Hence, “X” equals 10 percent. The mean or average return is 6.3 percent. The standard deviation is 8.4 percent. The cumulative distribution function provides the probability of the area to the left of Z. Hence, enter “TRUE” in the “Cumulative” column.
Press “OK” after providing the inputs. The probability of the area to the left of Z is as follows:
The probability of 0.670203873 represents the area to the left of Z. The area to the left of Z is the probability of getting less than 10 percent return. The area to the right of Z is the probability of getting a return of 10 percent or more.
The total area represented by the normal distribution curve has a probability of “1”. The area to the left of Z has a probability of 0.670203873. Hence, the probability of the area to the right of Z is “1” minus the probability of the area to the left of Z. Hence, the probability of getting 10 percent return or more is 0.329796127 or 32.98 percent
To determine: The probability of earning less than 0 percent on long-term corporate bonds.
Answer to Problem 24QP
The probability of earning less than 0 percent on long-term corporate bonds is 0.226627352 or 22.66 percent.
Explanation of Solution
Given information:
Assume that the returns of long-term corporate bonds have a normal distribution. The average return or mean of long-term corporate bonds is 6.3 percent, and the standard deviation is 8.4 percent (Refer to Figure 12.10 in the text).
Determine the probability of having a return less than 0 percent on long-term government bonds:
Follow the common steps from Step 1 to Step 3 given below. Next, proceed with the Step 4.
The common steps to be followed to use the “NORM.DIST” function in Excel:
Step 1:
Open an Excel worksheet.
Step 2:
Place the cursor in cell A1.
Step 3:
Select the “Formulas” tab, and go to “More functions” in the ribbon. Under “More functions”, select “Statistical”. Under the drop-down menu of “Statistical”, select “NORM.DIST” function.
After clicking the “NORM.DIST” function, a popup window named “Function arguments” appears.
Step 4:
Enter the values. “X” represents the raw score or outcome. Here, it is necessary to test the probability of having (0 percent) return or less. Hence, “X” equals (0 percent). The mean or average return is 6.3 percent. The standard deviation is 8.4 percent. The cumulative distribution function provides the probability of the area to the left of Z. Hence, enter “TRUE” in the “Cumulative” column.
Press “OK” after providing the inputs. The probability of the area to the left of Z is as follows:
The probability of 0.226627352 represents the area to the left of Z. The area to the left of Z refers to the probability of getting (0 percent) return or less because the left-hand side of the normal distribution curve indicates negative returns. Hence, the probability of earning less than 0 percent is 0.226627352 or 22.66 percent.
b)
To determine: The probability of earning more than 10 percent on Treasury bills.
Answer to Problem 24QP
The probability of earning more than 10 percent on Treasury bills is 0.018006785 or 1.80 percent.
Explanation of Solution
Given information:
Assume that the returns of Treasury bills have a normal distribution. The average return or mean of Treasury bills is 3.5 percent, and the standard deviation is 3.1 percent (Refer to Figure 12.10 in the text).
Determine the probability of having a return greater than 10 percent on Treasury bills:
Follow the common steps from Step 1 to Step 3 given below. Next, proceed with the Step 4.
The common steps to be followed to use the “NORM.DIST” function in Excel:
Step 1:
Open an Excel worksheet.
Step 2:
Place the cursor in cell A1.
Step 3:
Select the “Formulas” tab, and go to “More functions” in the ribbon. Under “More functions”, select “Statistical”. Under the drop-down menu of “Statistical”, select “NORM.DIST” function.
After clicking the “NORM.DIST” function, a popup window named “Function arguments” appears.
Step 4:
Enter the values. “X” represents the raw score or outcome. Here, it is necessary to test the probability of having more than 10 percent returns. Hence, “X” equals 10 percent. The mean or average return is 3.5 percent. The standard deviation is 3.1 percent. The cumulative distribution function provides the probability of the area to the left of Z. Hence, enter “TRUE” in the “Cumulative” column.
Press “OK” after providing the inputs. The probability of the area to the left of Z is as follows:
The probability of 0.981993215 represents the area to the left of Z. The area to the left of Z is the probability of getting less than 10 percent return. The area to the right of Z is the probability of getting a return of 10 percent or more.
The total area represented by the normal distribution curve has a probability of “1”. The area to the left of Z has a probability of 0.981993215. Hence, the probability of the area to the right of Z is “1” minus the probability of the area to the left of Z. Hence, the probability of getting 10 percent return or more is 0.018006785 or 1.80 percent
To determine: The probability of earning less than 0 percent on Treasury bills.
Answer to Problem 24QP
The probability of earning less than 0 percent on Treasury bills is 0.129442113 or 12.94 percent.
Explanation of Solution
Given information:
Assume that the returns of Treasury bills have a normal distribution. The average return or mean of Treasury bills is 3.5 percent, and the standard deviation is 3.1 percent (Refer to Figure 12.10 in the text).
Determine the probability of having a return less than 0 percent on Treasury bills:
Follow the common steps from Step 1 to Step 3 given below. Next, proceed with the Step 4.
The common steps to be followed to use the “NORM.DIST” function in Excel:
Step 1:
Open an Excel worksheet.
Step 2:
Place the cursor in cell A1.
Step 3:
Select the “Formulas” tab, and go to “More functions” in the ribbon. Under “More functions”, select “Statistical”. Under the drop-down menu of “Statistical”, select “NORM.DIST” function.
After clicking the “NORM.DIST” function, a popup window named “Function arguments” appears.
Step 4:
Enter the values. “X” represents the raw score or outcome. Here, it is necessary to test the probability of having (0 percent) return or less. Hence, “X” equals (0 percent). The mean or average return is 3.5 percent. The standard deviation is 3.1 percent. The cumulative distribution function provides the probability of the area to the left of Z. Hence, enter “TRUE” in the “Cumulative” column.
Press “OK” after providing the inputs. The probability of the area to the left of Z is as follows:
The probability of 0.129442113 represents the area to the left of Z. The area to the left of Z refers to the probability of getting (0 percent) return or less because the left-hand side of the normal distribution curve indicates negative returns. Hence, the probability of earning less than 0 percent is 0.129442113 or 12.94 percent.
c)
To determine: The probability of earning (2.76 percent) on long-term corporate bonds.
Answer to Problem 24QP
The probability of earning (2.76 percent) on long-term corporate bonds is 0.140389412 or 14.04 percent.
Explanation of Solution
Given information:
Assume that the returns of long-term corporate bonds have a normal distribution. The average return or mean of long-term corporate bonds is 6.3 percent, and the standard deviation is 8.4 percent (Refer to Figure 12.10 in the text).
Determine the probability of having (2.76 percent) on long-term government bonds:
Follow the common steps from Step 1 to Step 3 given below. Next, proceed with the Step 4.
The common steps to be followed to use the “NORM.DIST” function in Excel:
Step 1:
Open an Excel worksheet.
Step 2:
Place the cursor in cell A1.
Step 3:
Select the “Formulas” tab, and go to “More functions” in the ribbon. Under “More functions”, select “Statistical”. Under the drop-down menu of “Statistical”, select “NORM.DIST” function.
After clicking the “NORM.DIST” function, a popup window named “Function arguments” appears.
Step 4:
Enter the values. “X” represents the raw score or outcome. Here, it is necessary to test the probability of having (2.76 percent) return or less. Hence, “X” equals (2.76 percent). The mean or average return is 6.3 percent. The standard deviation is 8.4 percent. The cumulative distribution function provides the probability of the area to the left of Z. Hence, enter “TRUE” in the “Cumulative” column.
Press “OK” after providing the inputs. The probability of the area to the left of Z is as follows:
The probability of 0.140389412 represents the area to the left of Z. The area to the left of Z refers to the probability of getting (2.76 percent) return or less because the left-hand side of the normal distribution curve indicates negative returns. Hence, the probability of earning (2.76 percent) is 0.140389412 or 14.04 percent.
To determine: The probability of earning 10.56 percent on Treasury bills.
Answer to Problem 24QP
The probability of earning 10.56 percent on Treasury bills is 0.011380598 or 1.14 percent.
Explanation of Solution
Given information:
Assume that the returns of Treasury bills have a normal distribution. The average return or mean of Treasury bills is 3.5 percent, and the standard deviation is 3.1 percent (Refer to Figure 12.10 in the text).
Determine the probability of having a return of 10.56 percent on Treasury bills:
Follow the common steps from Step 1 to Step 3 given below. Next, proceed with the Step 4.
The common steps to be followed to use the “NORM.DIST” function in Excel:
Step 1:
Open an Excel worksheet.
Step 2:
Place the cursor in cell A1.
Step 3:
Select the “Formulas” tab, and go to “More functions” in the ribbon. Under “More functions”, select “Statistical”. Under the drop-down menu of “Statistical”, select “NORM.DIST” function.
After clicking the “NORM.DIST” function, a popup window named “Function arguments” appears.
Step 4:
Enter the values. “X” represents the raw score or outcome. Here, it is necessary to test the probability of having 10.56 percent returns. Hence, “X” equals 10.56 percent. The mean or average return is 3.5 percent. The standard deviation is 3.1 percent. The cumulative distribution function provides the probability of the area to the left of Z. Hence, enter “TRUE” in the “Cumulative” column.
Press “OK” after providing the inputs. The probability of the area to the left of Z is as follows:
The probability of 0.988619402 represents the area to the left of Z. The area to the left of Z is the probability of getting less than 10.56 percent return. The area to the right of Z is the probability of getting a return of 10.56 percent or more.
The total area represented by the normal distribution curve has a probability of “1”. The area to the left of Z has a probability of 0.988619402. Hence, the probability of the area to the right of Z is “1” minus the probability of the area to the left of Z. Hence, the probability of getting 10.56 percent return or more is 0.011380598 or 1.14 percent
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Chapter 12 Solutions
FUND.OF CORPORATE FINANCE(LL)
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