# How do you calculate beta?

Learn from Mathematical Finance

Calculating beta is essential for understanding the risk and volatility of an investment compared to the market. Beta is a measure of a stock's volatility in relation to the overall market. Here's a step-by-step guide to calculate beta:

Step-by-Step Guide to Calculating Beta

1. Collect Data

To calculate beta, you need the following data:

- Historical prices of the stock.

- Historical prices of a market index (e.g., S&P 500).

- Corresponding dates for the stock and index prices.

2. Calculate Returns

Next, calculate the periodic returns for both the stock and the market index. Returns are typically calculated on a daily, weekly, or monthly basis using the formula:

\[ \text{Return} = \frac{\text{Current Price} - \text{Previous Price}}{\text{Previous Price}} \]

3. Determine Average Returns

Calculate the average return for the stock and the average return for the market index over the same period.

4. Calculate Covariance

Covariance measures how the returns of the stock and the market index move together. Use the formula:

\[ \text{Covariance} = \frac{\sum (R_{\text{stock},i} - \bar{R}_{\text{stock}}) \times (R_{\text{market},i} - \bar{R}_{\text{market}})}{n - 1} \]

Where:

- \( R_{\text{stock},i} \) is the return of the stock at time \( i \).

- \( \bar{R}_{\text{stock}} \) is the average return of the stock.

- \( R_{\text{market},i} \) is the return of the market index at time \( i \).

- \( \bar{R}_{\text{market}} \) is the average return of the market index.

- \( n \) is the number of observations.

5. Calculate Variance

Variance measures the dispersion of the market index returns. Use the formula:

\[ \text{Variance} = \frac{\sum (R_{\text{market},i} - \bar{R}_{\text{market}})^2}{n - 1} \]

6. Calculate Beta

Finally, calculate beta using the formula:

\[ \beta = \frac{\text{Covariance}}{\text{Variance}} \]

This formula can also be expressed as:

\[ \beta = \frac{\text{Cov}(R_{\text{stock}}, R_{\text{market}})}{\text{Var}(R_{\text{market}})} \]

Example Calculation

Assume you have the following returns for a stock and the market index over five periods:

- Stock returns: 0.02, 0.03, 0.01, -0.02, 0.04

- Market returns: 0.01, 0.02, 0.01, -0.01, 0.03

Calculate Average Returns

- Average stock return: \( (0.02 + 0.03 + 0.01 - 0.02 + 0.04) / 5 = 0.016 \)

- Average market return: \( (0.01 + 0.02 + 0.01 - 0.01 + 0.03) / 5 = 0.012 \)

Calculate Covariance

- Covariance: \( \frac{(0.02 - 0.016)(0.01 - 0.012) + (0.03 - 0.016)(0.02 - 0.012) + ... }{4} = 0.00018 \)

Calculate Variance

- Variance: \( \frac{(0.01 - 0.012)^2 + (0.02 - 0.012)^2 + ... }{4} = 0.00011 \)

Calculate Beta

- Beta: \( \frac{0.00018}{0.00011} = 1.636 \)

Conclusion

Beta is a crucial measure for investors looking to understand the risk profile of a stock relative to the market. By following these steps, you can accurately calculate the beta of a stock, aiding in making informed investment decisions.