BETA
DEFINITION
Beta refers to the volatility of a security or portfolio, compared to the market returns as a whole. It is a measure that indicates a security’s exposure to market risk. Market risk or systematic risk refers to the risk that affects the returns of the whole market. Beta, therefore, indicates whether the returns on a stock is more, less or as volatile as the overall market return. Here, market return refers to the return on any standard stock index such as SENSEX, NIFTY 50, S & P 500, NASDAQ 100, etc.
CALCULATOR

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Beta of the stock:

FORMULA AND DERIVATION
Beta is the variation in the returns of a security compared to the market returns, expressed as a proportion of the market returns. In simple terms, beta is equal to covariance of returns of a security and the market index, divided by the variance of the returns on the market index. This can be mathematically expressed as follows:-
${\beta }_i=\ \frac{{\sigma }_{im}}{{\sigma }^2_m}$
where,
${\beta }_i$ is the beta for security $$i$$
$${\sigma }_{im}$$ is the covariance between returns on security $$i$$ and returns on the market index
$${\sigma }^2_m$$ is the variance of the returns of the market index

In order to be able to calculate this, we need to define a few other measures as follows:-

Let $$r_i$$ and $$r_m$$ represent the returns on stock $$i$$ and market returns respectively, over n number of time periods.
Let $${\tilde{r}}_i$$ and $${\tilde{r}}_m$$ represent the mean or average of the returns on stock and returns of market respectively, for the same n number of time periods.
As defined above, $${\sigma }^2_m$$ is the variance of market returns, and can be calculated as follows:-
${\sigma }^2_m=\frac{\sum{r^2_{m\ }}-\ {n\tilde{r}}^2_m}{n-1}\$
As defined above, $${\sigma }_{im}$$ is the covariance between the returns on the security and market returns, and can be calculated as follows:-
${\sigma }_{im}=\ \frac{\sum{r_ir_m-}{n\tilde{r}}_i{\tilde{r}}_m}{n-1}$
Once these two values are calculated, beta can be can be calculated by dividing $${\sigma }_{im}$$ by $${\sigma }^2_m$$ as mentioned above.

Different values of beta for a security, indicate different degrees of volatility of returns compared to market returns, as follows:-

Value of Beta Interpretation
$$\beta >1$$ Beta greater than 1 indicates that the return on the security is more volatile than the market returns.
$$\beta =1$$ Beta equal to 1 indicates that the return on the security is as volatile as the market return.
$$\beta <1$$ Beta lesser than 1 indicates that the return on the security is less volatile than the market returns.
$$\beta < 0$$ Beta less than 0, i.e. negative beta, indicates that the return on the security moves in the opposite direction to that of market returns.

EXAMPLE
Example 1

A company needs to determine the beta of its shares to calculate its cost of equity. The returns on the company’s shares and the returns on the benchmark stock index for six months are recorded as follows:-

Months Returns on company's share Stock index returns
1 $$2.4\%$$ $$1.5\%$$
2 $$2.5\%$$ $$1.2\%$$
3 $$2\%$$ $$1.8\%$$
4 $$1.5\%$$ $$0.9\%$$
5 $$3.4\%$$ $$2.4\%$$
6 $$5.7\%$$ $$3.2\%$$

Beta for the share can be calculated as follows:-

$$r_i$$ $$r_m$$ $$r^2_m$$ $$r_ir_m$$
0.024 0.015 0.000225 0.00036
0.025 0.012 0.000144 0.0003
0.02 0.018 0.000324 0.00036
0.015 0.009 0.000081 0.000135
0.034 0.024 0.000576 0.000816
0.057 0.032 0.001024 0.001824
$\sum{r_i}=0.175$ $\sum{r_m}=0.11$ $\sum{r^2_m}=0.002374$ $\sum{r_ir_m}=0.003795$

${\tilde{r}}_i=\ \frac{\sum{r_i}}{n}=\ \frac{0.175}{6}=0.02916$
${\tilde{r}}_m=\frac{\sum{r_m}}{n}=\ \frac{0.11}{6}=0.0183$

${\sigma }_{im}=\frac{\sum{r_ir_m-\ {n\tilde{r}}_i{\tilde{r}}_m}}{n-1}$ $\ \ \ \ \ \ \ =\frac{0.003795-6\times 0.029167\times 0.0183}{5}$ $\ \ \ \ \ \ \ =\mathrm{0.00011733}$

${\sigma }^2_m=\ \frac{\sum{r^2_m-{n\tilde{r}}^2_m}}{n-1}$ $\ \ \ \ \ \ =\frac{0.002374-6\times {0.0183}^2}{5}$ $\ \ \ \ \ \ =0.000071466$

${\beta }_i=\ \frac{{\sigma }_{im}}{{\sigma }^2_m}$ $\ \ \ \ =\frac{0.00011733}{0.000071466}$ $\ \ \ \ =1.6418$