Financial Risk Manager Part 1

Explanation

To compute the sample standard deviation, we use the formula for sample variance:

$s^2 = \frac{1}{(n-1)} \left[ \sum_{i=1}^{n} x^2 - n \bar{x}^2 \right]$

Where:

• $\sum x = 31,353$ (sum of all observations)
• $n = 100$ (sample size)
• $\sum x^2 = 10,687,041$ (sum of squared observations)

Step 1: Calculate the sample mean ($\bar{x}$)

$\bar{x} = \frac{\sum x}{n} = \frac{31,353}{100} = 313.53$

Step 2: Calculate the sample variance ($s^2$)

$s^2 = \frac{1}{(100-1)} \left[ 10,687,041 - 100 \times (313.53)^2 \right]$

First, calculate $\bar{x}^2 = (313.53)^2 = 98,301.0609$

Then, $100 \times 98,301.0609 = 9,830,106.09$

Now, $10,687,041 - 9,830,106.09 = 856,934.91$

Finally, $s^2 = \frac{856,934.91}{99} = 8,655.9082$

Step 3: Calculate the sample standard deviation ($s$)

$s = \sqrt{s^2} = \sqrt{8,655.9082} = 93.0371 \approx 93$

Key Points:

• The sample variance uses $(n-1)$ in the denominator (Bessel's correction) for unbiased estimation
• The sample standard deviation is the square root of the sample variance
• The result matches option B (93)