In finance:
“Beta is seemed because the contrast among the market index and the historical volatility of a employer”
Basically, calculating the beta of a inventory allows finance specialists to estimate the go back for a positive risk they may take.
An organization has invested in certain shares of a company, and the returns of both the company and the market are as follows:
Company’s Return = 5, 10, 8, 12, 6, 9
Market’s Return = 6, 9, 7, 11, 5, 10
Now, we calculate the beta to estimate whether the stock's price will increase or decrease in relation to the market.
The following table presents the values:
| Obs. | rM (Market Return) | rS (Stock Return) |
| 1 | 6 | 5 |
| 2 | 9 | 10 |
| 3 | 7 | 8 |
| 4 | 11 | 12 |
| 5 | 5 | 6 |
| 6 | 10 | 9 |
Now we can calculate the regression coefficient:
| Obs. | rM (Market Return) | rS (Stock Return) | Xᵢ² (Square of rM) | Yᵢ² (Square of rS) | Xᵢ * Yᵢ (Product of rM and rS) |
| 1 | 6 | 5 | 36 | 25 | 30 |
| 2 | 9 | 10 | 81 | 100 | 90 |
| 3 | 7 | 8 | 49 | 64 | 56 |
| 4 | 11 | 12 | 121 | 144 | 132 |
| 5 | 5 | 6 | 25 | 36 | 30 |
| 6 | 10 | 9 | 100 | 81 | 90 |
| Sum = | 48 | 50 | 412 | 450 | 428 |
The following formulas will help us calculate the values:
\( SS_{XX} = \sum^n_{i=1} X_i^2 - \dfrac{1}{n} \left(\sum^n_{i=1} X_i \right)^2 \)
\( = 412 - \dfrac{1}{6} (48)^2 \)
\( = 412 - \dfrac{1}{6} (2304) = 412 - 384 = 28 \)
\( SS_{YY} = \sum^n_{i=1} Y_i^2 - \dfrac{1}{n} \left(\sum^n_{i=1} Y_i \right)^2 \)
\( = 450 - \dfrac{1}{6} (50)^2 \)
\( = 450 - \dfrac{1}{6} (2500) = 450 - 416.67 = 33.33 \)
\( SS_{XY} = \sum^n_{i=1} X_i Y_i - \dfrac{1}{n} \left(\sum^n_{i=1} X_i \right) \left(\sum^n_{i=1} Y_i \right) \)
\( = 428 - \dfrac{1}{6} (48)(50) = 428 - \dfrac{2400}{6} = 428 - 400 = 28 \)
Finally, we calculate the beta:
\( \hat{\beta}_1 = \dfrac{SS_{XY}}{SS_{XX}} = \dfrac{28}{28} = 1.0 \)
Since the beta value is 1.0, this means that the stock's price moves in direct proportion to the market's return.
A terrible beta cost suggests a clear difference with recognize to the market fee of an index. but it not often takes place.
From the source Wikipedia: Beta (finance), Interpretation of values, significance as hazard measure, Technical aspects, choice of marketplace portfolio and danger-free rate, Empirical estimation, Equilibrium use: fair praise for threat?
