Question

You have the following stock price information of firm XYZ. (4pts)

Period (t)	XYZ Stock Price per share ($)
0	2.50
1	2.70
2	2.66
3	2.43
4	2.57
5	2.55
6	2.56
7	2.63
8	2.85
9	2.94
10	2.99

a) Calculate the standard deviation for the XYZ stock returns if t=0 indicates the XYZ’s initial public offering (IPO).

b) Calculate the standard deviation for the XYZ stock returns if the given periods (from t=0 to t=10) are some extracted periods from the population (i.e., random sampling).

Answer 1

First we need to find the returns from the share price, this can be found as:

Return of day t+1 = ( Price t+1 - Price t ) / Price t

Period	Stock price	Returns
0	2.5
1	2.7	0.08
2	2.66	-0.01481
3	2.43	-0.08647
4	2.57	0.057613
5	2.55	-0.00778
6	2.56	0.003922
7	2.63	0.027344
8	2.85	0.08365
9	2.94	0.031579
10	2.99	0.017007

Question a)

Standard deviation of returns, considering the given data as the population (all data points are given, because the first entry is considered the IPO price) :

Formula:

SD =

Finding the average:

= Sum of returns / 10

= 0.019205

SD = Sqrt ( (0.08 - 0.019205)^2 + (0.08 - 0.019205)^2 + (0.08 - 0.019205)^2 + (0.08 - 0.019205)^2 + (0.08 - 0.019205)^2 + (0.08 - 0.019205)^2 + (0.08 - 0.019205)^2 + (0.08 - 0.019205)^2 + (0.08 - 0.019205)^2 + (0.08 - 0.019205)^2 / 10)

Solving in excel:

Period	Stock price	Returns	Return - mean	(Return - mean)^2
0	2.5
1	2.7	0.08	0.060794865	0.003696016
2	2.66	-0.01481	-0.034019949	0.001157357
3	2.43	-0.08647	-0.1056713	0.011166424
4	2.57	0.057613	0.038408034	0.001475177
5	2.55	-0.00778	-0.026987236	0.000728311
6	2.56	0.003922	-0.015283566	0.000233587
7	2.63	0.027344	0.008138615	6.62371E-05
8	2.85	0.08365	0.064445055	0.004153165
9	2.94	0.031579	0.012373813	0.000153111
10	2.99	0.017007	-0.002198332	4.83266E-06
	Average	0.019205		0.022834218	Sum
				SD^2	0.002283

SD^2 is calculated by The sum value divided by 10. (As the formula given)

Thus,

SD = sqrt ( 0.002283) = 0.047785 or 4.78%

Question b)

Instead of considering this data set as the population, if we consider it as a random sample, we need to make a small change to the formula to calculate the SD.

In this case formula:

SD =

The rest of the steps are the same as above, so we will take the value of sum from the above part:

Sum = 0.022834218

We now divide this by 9, instead of 10 and get:

SD^2 = 0.002537

Thus,

SD = sqrt ( 0.002537) = 0.05037 or 5.037%

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