Financial Risk Manager Part 1
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The returns generated by a sample of five stocks from the Karachi Stock Exchange are given in the exhibit below.
| Stock | Return |
|---|---|
| A | 12% |
| B | 13% |
| C | 5% |
| D | 4% |
| E | 20% |
What is the standard deviation of this sample?
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A
6.53%
B
5.84%
C
7.53%
D
8.34%
Explanation:
Calculation Explanation
Step 1: Calculate the Mean [ \text{Mean} = \frac{(0.12 + 0.13 + 0.05 + 0.04 + 0.20)}{5} = \frac{0.54}{5} = 0.108 \text{ or } 10.8% ]
Step 2: Calculate Deviations and Squared Deviations
| Stock | Return | X – Mean | (X – Mean)² |
|---|---|---|---|
| A | 12% | 1.2% | 0.000144 |
| B | 13% | 2.2% | 0.000484 |
| C | 5% | –5.8% | 0.003364 |
| D | 4% | –6.8% | 0.004624 |
| E | 20% | 9.2% | 0.008464 |
| Total | 0.017080 |
Step 3: Calculate Sample Standard Deviation [ \text{Sample Standard Deviation} = \sqrt{\frac{\sum(X - \text{Mean})^2}{n - 1}} = \sqrt{\frac{0.017080}{4}} = \sqrt{0.00427} = 0.0653 \text{ or } 6.53% ]
Key Points:
- We use n-1 in the denominator because this is a sample standard deviation (not population)
- The sample standard deviation is an unbiased estimator of the population standard deviation
- The calculation follows the formula: ( s = \sqrt{\frac{\sum(x_i - \bar{x})^2}{n-1}} )_
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