To perform a hypothesis test to determine if the average return of a stock is different from zero, we will use the t-test. The t-test is appropriate in this case because we are dealing with a single sample and want to compare its mean (average return) to a known value (zero), and the population variance (σ2) is unknown.
First, we’ll describe the null and alternative hypotheses:
Null Hypothesis (H0): The average return of the stock is equal to zero. Mathematically, we can write this as µ = 0.
Alternative Hypothesis (H1): The average return of the stock is different from zero. Mathematically, we can write this as µ .
Before we move forward, let’s assume we have a random sample of stock returns for N periods, and we calculated the average return (x̄) and standard deviation (s) for this sample.
Now, we can perform the following steps to conduct the t-test:
1. Calculate the t-statistic:
Given that we are comparing the sample mean to a known value (zero), the t-statistic can be calculated using the following formula:
$$t = \frac{\bar{x} - \mu}{(s / \sqrt{N})}$$
In our case, µ = 0. So, the formula becomes:
$$t = \frac{\bar{x}}{(s / \sqrt{N})}$$
2. Choose a significance level:
The significance level (α) is the probability of rejecting the null hypothesis when it is true. Commonly used significance levels are 0.01, 0.05, and 0.10. For example, if we choose α = 0.05, we are willing to accept a 5% chance of committing a Type I error (rejecting the null hypothesis when it is true).
3. Determine the critical t-value:
Given the chosen significance level, we can find the critical t-value from a t-distribution table. Since our alternative hypothesis is μ ≠ 0, it is a two-tailed test.
The degrees of freedom (df) for the t-distribution will be N - 1. Find the t-value corresponding to the chosen α and df in the t-distribution table. Let’s denote this value as tcrit.
4. Compare t-statistic with critical t-value:
Finally, compare the calculated t-statistic with the critical t-value:
If |t| > tcrit, we reject the null hypothesis in favor of the alternative hypothesis, concluding that the average return of the stock is different from zero.
If |t| ≤ tcrit, we fail to reject the null hypothesis, meaning there is not enough evidence to conclude that the average return of the stock is different from zero.
As an example, let’s say our sample has N = 25 periods, with a mean return x̄ = 0.01 and a standard deviation s = 0.05.
1. Calculate the t-statistic:
$$t = \frac{0.01}{(0.05 / \sqrt{25})} = \frac{0.01}{0.01} = 1$$
2. Choose a significance level:
Let’s choose α = 0.05.
3. Determine the critical t-value:
With 24 degrees of freedom (25 - 1) and α = 0.05, we find a two-tailed critical t-value of tcrit = 2.064 from the t-distribution table.
4. Compare t-statistic with critical t-value:
Since |1| ≤ 2.064, we fail to reject the null hypothesis. There is insufficient evidence to conclude that the average return of the stock is different from zero.