Question

A portfolio generates the following returns over the past 10 years:

Year Return (%)

1 -7

2 -15

3 20

4 21

5 9

6 -6

7 15

8 23

9 2

10 -5

Calculate the test statistic to test whether the standard deviation of this portfolio's return is different from the benchmark portfolio standard deviation of 22. Enter answer accurate to 3 decimal places. Bonus thinking question: can you reject the hypothesis that the two standard deviations are equal? (use the CHISQ.INV() spreadsheet function to find the rejection points for your desired test probability)

Answer 1

Solution:

Here, we have to use Chi square test for the population standard deviation.

The null and alternative hypotheses are given as below:

Null hypothesis: H0: σ = 22

Alternative hypothesis: Ha: σ ≠ 22

We assume level of significance as 5% or α = 0.05.

This is a two tailed test.

The test statistic formula is given as below:

Chi square = (n – 1)*S^2 / σ^2

We are given

n = 10

S = 13.7036

σ = 22

Chi square = (10 – 1)* 13.7036^2 / 22^2

Chi square = 9*13.7036^2 / 22^2

Chi square =3.491938

Test statistic = 3.492

Degrees of freedom = n – 1 = 10 – 1 = 9

α = 0.05

Lower critical value = 2.7004

Upper critical value = 19.0228

[By using excel commands =CHIINV(1- α/2, df) and =CHIINV(α/2, df)]

Test statistic value is lies between above critical values, so we do not reject the null hypothesis.

We cannot reject the hypothesis that the two standard deviations are equal.

There is sufficient evidence that the population standard deviation for the portfolio returns is 22.