To calculate the beta of a stock relative to an index using linear regression, we’ll need the historical price data for the stock and the index. Let’s assume we have the price data, and we’ll use the following notation:
- Pi: The price of the stock at time i
- Pi − 1: The price of the stock at time i − 1
- Ii: The price of the index at time i
- Ii − 1: The price of the index at time i − 1
We’ll begin by calculating the returns for both the stock and the index at each time point. The return at time i for the stock is given by:
$$r_{Pi} = \frac{P_i - P_{i-1}}{P_{i-1}}$$
Similarly, the return at time i for the index is given by:
$$r_{Ii} = \frac{I_i - I_{i-1}}{I_{i-1}}$$
Next, we’ll calculate the mean returns for both the stock and the index:
$$\bar{r}_P = \frac{1}{n}\sum_{i=1}^n r_{Pi} \qquad \bar{r}_I = \frac{1}{n}\sum_{i=1}^n r_{Ii}$$
Where n is the total number of data points.
Now, we’ll proceed with the linear regression. We are trying to find the relationship between the stock’s returns and the index’s returns. Mathematically, the relationship will be in the form:
rPi = α + βrIi + ϵi
Where ϵi is the error term representing factors not explained by the index’s returns. To find the beta β, we’ll use the formula:
$$\beta = \frac{\sum_{i=1}^n (r_{Ii} - \bar{r}_I)(r_{Pi} - \bar{r}_P)}{\sum_{i=1}^n (r_{Ii} - \bar{r}_I)^2}$$
First, let’s calculate the covariance of stock returns and index returns:
$$\text{Cov}(r_P, r_I) = \frac{1}{n}\sum_{i=1}^n (r_{Pi} - \bar{r}_P)(r_{Ii} - \bar{r}_I)$$
Next, let’s calculate the variance of index returns:
$$\text{Var}(r_I) = \frac{1}{n}\sum_{i=1}^n (r_{Ii} - \bar{r}_I)^2$$
Finally, we’ll calculate the beta:
$$\beta = \frac{\text{Cov}(r_P, r_I)}{\text{Var}(r_I)}$$
The β represents the sensitivity of the stock’s returns to the index’s returns. In other words, it represents how the stock price reacts to market movements. A β > 1 indicates that the stock tends to outperform the market in periods of market growth and underperform in periods of market decline, while a β < 1 indicates that the stock tends to have a lower risk and return profile compared to the market. A β close to 1 indicates that the stock is closely related to the overall market performance.
In summary, to calculate the beta of a stock relative to an index using linear regression, follow these steps:
1. Collect historical price data for the stock and the index.
2. Calculate the returns at each time point for both the stock and the index.
3. Compute the mean returns for the stock and the index.
4. Calculate the covariance between the stock returns and the index returns.
5. Calculate the variance of the index returns.
6. Compute the beta as the ratio of the covariance to the variance.
After calculating the beta, you can interpret the stock’s relationship with the market and make informed investment decisions.