A Classic Test of Statistical Significance Using T-Test to Determine Whether a Sharpe Ratio for a Given Trading Rule is Likely to Be Positive Given the Estimate of its Mean and Standard Deviation

The Sharpe ratio is a key metric used in finance to assess the risk-adjusted return of a trading strategy. It is defined as the ratio of the expected excess return (above the risk-free rate) to the standard deviation of returns. However, is the Sharpe ratio for a given trading rule likely to be positive? This is where statistical testing, specifically the T-test, can help determine if the Sharpe ratio is significantly different from zero.

What is the T-Test?

The T-test is a statistical test used to determine if there is a significant difference between the mean of a sample and a known value (like zero) or between the means of two groups. In this case, we can use the T-test to test whether the Sharpe ratio of a given trading rule is significantly greater than zero.

• Null hypothesis (H₀): The Sharpe ratio is equal to zero (no excess return).
• Alternative hypothesis (H₁): The Sharpe ratio is greater than zero (positive excess return).

The Sharpe Ratio Formula

To calculate the Sharpe ratio, we use the following formula:

$$ S = \frac{\mu}{\sigma} $$

Where: - S = Sharpe ratio - μ = Expected return (mean) - σ = Standard deviation of returns

In this context, we are interested in testing if μ > 0.

Assumptions for Using the T-Test

Before applying the T-test, certain assumptions must hold: 1. Independence of returns: The returns must be independent from one another. 2. Normality of returns: Returns should ideally be normally distributed for accurate T-test results. 3. Large sample size: Larger sample sizes improve the robustness of the T-test.

Given these assumptions, we can proceed to calculate the T-statistic.

Calculating the T-Statistic

The formula for the T-statistic is given by:

$$ T = \frac{\mu - 0}{\frac{\sigma}{\sqrt{n}}} $$

Where: - μ = Mean of the sample (expected return) - σ = Standard deviation of the sample - n = Number of observations

The T-statistic follows a t-distribution, and we compare the result against critical values from the t-table, typically at a 95% confidence level.

Example

Consider a hypothetical scenario where you have a trading rule with the following statistics: - Mean return (μ) = 0.05 (5% annual return) - Standard deviation (σ) = 0.15 (15% annual volatility) - Sample size (n) = 100 (trading days)

To calculate the T-statistic:

$$ T = \frac{0.05}{\frac{0.15}{\sqrt{100}}} = \frac{0.05}{0.015} = 3.33 $$

At a 95% confidence level, the critical T-value (for 99 degrees of freedom) is approximately 1.984. Since 3.33 > 1.984, we reject the null hypothesis and conclude that the Sharpe ratio is significantly greater than zero.

Python Code for Calculation

You can use Python to calculate the T-statistic and test the hypothesis easily:

```python import scipy.stats as stats import math

Define parameters

mean_return = 0.05 std_dev = 0.15 sample_size = 100

Calculate T-statistic

t_statistic = mean_return / (std_dev / math.sqrt(sample_size))

Degrees of freedom (n - 1)

df = sample_size - 1

Critical value for 95% confidence

critical_value = stats.t.ppf(0.95, df)

Print results

print(f”T-statistic: {t_statistic}”) print(f”Critical value: {critical_value}”)

Compare T-statistic with critical value

if t_statistic > critical_value: print(“Reject the null hypothesis: The Sharpe ratio is significantly greater than zero.”) else: print(“Fail to reject the null hypothesis: The Sharpe ratio is not significantly greater than zero.”)