1 The standard deviation of daily returns of a stocks price

1) The standard deviation of daily returns of a stock’s price is used as a measure of the risk of that stock. Suppose that in a sample of 101 days, the standard deviation of a particular stock is 1.15%.

a) Find the 90% confidence interval of the population variance for this stock.        

b) In the past, the standard deviation of the daily returns of this stock has been 1.56%. Test the hypothesis at the 1% level of significance that the standard deviation has decreased from its previous level.

Solution

n = Sample size =101

std dev = 1.15%

a) For 90% z alpha/2 = 1.645

Hence confidence interval Confidence Interval Formula for 2 is as follows:
(n - 1)s2/2/2 < 2 < (n - 1)s2/21 - /2 where:
(n - 1) = Degrees of Freedom,=100

chi square = 124.3421

Hence confidence interval for variance

=( 0.9249 , 1.4757)

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b) Past std deviation =1.56

H0: sigma^2 = 1.56

Ha: sigma^2<1.56

Left tailed test for variances

Construct confidence interval for variance for 1.15 at 99%

20.005 = 140.1697

Hence confidence interval=(0.8204, 1.7081)

As this interval contains 1.56% we accept null hypothesis.

There is no evidence to show that std deviation has decreased from its previous level.