# Beta Estimation

The security ‘s beta, which measures the sensitivity of the security ‘s return to those of the market because beta captures the market risk of security as opposed to its diversifiable risk, it is the appropriate measure of risk or a wealth diversified investor.
Using historical returns
We would like to know a stock’s beta in the future that is how sensitive will its features returns to market risk. In practice, We estimate beta based on the stock’s historical sensitivity. This approach makes sense if a stock is a beta that remains relatively stable over time, which appears to be the case for most firms.
Many data sources provide estimates of beta based on historical data. Typically those data sources estimate.correlation and volatilities from two or five years of weekly or monthly returns.
Beta estimation has two method.

## Direct method

Beta is the measures of systematic risk and it is a ratio of covariance between market return variance.
$\beta&space;_{j}&space;=\frac{Covarj,m}{\sigma&space;^{2}m}$

=$\frac{\sigma&space;_{j}\sigma&space;_{m}corj,m}{\sigma&space;_{m}\times&space;\sigma&space;_{m}}=\frac{\sigma&space;_{j}}{\sigma&space;_{m}}\times&space;cor_{j,m}$

Let’s consider an example suppose that percent return on the market. Represented by the BSE Sensex(sensitivity index) and the share of ABC Infotech limited for recent five years.
Return on Sensex and ABC Infotech

 Year Market return(%) ABC infotech(%) 1 18.60 28.46 2 -16.50 -36.15 3 63.85 52.64 4 -20.65 -7.29 5 -17.80 -12.95

Beta estimation for ABC infotech limited

 year $r_{m}$ $r_{j}$ $(r_{m}-\bar{r}_{m})$ $(r_{j}-\bar{r}_{j})$ $(r_{m-}\bar{r}_{m}&space;)\times(r_{j}-\bar{r}_{j})$ $(r_{m}-\bar{r}_{m})^{2}$ 1 18.60 23.46 13.11 19.51 255.91 171.98 2 -16.50 -36.13 -21.98 -40.08 880.83 483.08 3 63.83 52.64 58.35 48.69 2841.35 3404.85 4 -20.65 -7.29 -26.13 -11.24 293.64 682.96 5 -17.87 -12.95 -23.35 -16.90 394.57 545.35 $\bar{r}_{m}=5.48$ $\bar{r}_{j}=3.95$ sum=4666.30 sum=5288.23

i)Average return on market
$cov_{m.j}=\frac{4666.30}{5}=933.26$

ii) Square deviations of market return
$\sigma&space;^{2}=\frac{5228.23}{5}=1057.65$

iii) Divide the covariance of market and ABC infotech by the market variance to get beta.
$\beta&space;_{j}=\frac{cov_{j,m}}{\sigma&space;^{2}m}=\frac{933.26}{1057.65}=0.88$

The intercept term is given by the following formula
$\alpha&space;_{j}=\bar{r}_{j}-\beta&space;_{j}\times\bar{r}_{_{m}}$
$3.95-0.88\times&space;5.48=-0.89$
The characteristic line of ABC Infotech p=0.89+0.88

### The market model or Index model

Another procedure of calculating beta is the use of market model.In the market model, we regress return on a security against returns of the market index.The market model is given by the following regression equation.
$R_{j}=\alpha+\beta&space;_{j}R_{m}+e_{j}$
where
$R_{j}$=expected market return
$\alpha$= intercept
$e_{j}$=error term
$\beta&space;_{j}$=rgression measures the variability of the security’s beta

Beta is the ratio of the covariance between the security returns and the market returns and it is the covariance between the security returns and the market returns to the variance of the market return.$\alpha$ indicates the return on a security when the market return is zero. It could be interpreted as the return on security on account of unsystematic risk. Over a along given the randomness of unsystematic risk .
The observed return on market and ABC share and a regression line. The regression line of the market model is called the characteristics line.
The characteristics line
The value of $\alpha$ is 0.89 and the value of $\beta$ is 0.88.
The value of $\beta$ and $\alpha$ in the regression equation are given by the following equations.

$\beta&space;=&space;\frac{N\Sigma&space;XY-(\Sigma&space;X)(\Sigma&space;Y)}{N\Sigma&space;X^{2}-(\Sigma&space;X^{})^{2}}$

$\beta&space;_{j}=\frac{(5)4,774.49)-(27.42)(19.73)}{(5)(5,438.58)-(27.42)^{2}}$

=$\frac{23,872.45-541.00}{27,192.90-751.86}$

=$\frac{23,331.45}{26,441.04}$   $=0.88$

Alpha=$\alpha=\bar{Y}-\beta&space;\bar{X}$
Alpha=$\alpha&space;_{j}=3.95-(0.88)(5.48)=&space;-0.89$

Estimation for the regression equation

 Year $X_{m}(X)$ $r_{m}(Y)$ XY $X^{2}$ $Y^{2}$ 1 18.60 23.46 436.30 345.88 550.37 2 -16.50 -36.13 595.99 272.10 1305.38 3 63.83 52.64 3360.26 4074.86 2770.97 4 -20.65 -7.29 150.54 426.42 53.14 5 -17.87 12.95 231.41 319.31 167.70 Sum $\Sigma&space;X=27.42$ $\Sigma&space;Y=19.73$ $\Sigma&space;XY=4774.49$ $\Sigma&space;X^{2^{}}=5438.58$ $\Sigma&space;Y^{2}=4847.56$ Average $\bar{X}=5.48$ $\bar{Y}=3.95$

Beta estimation in practice
In practice, the market portfolio is approximate by a well-diversified share price index. Portfolio should include all risky assets shares, bonds,gold, silver real estate art objects, etc.
In computing beta by regression .We need data on return market index and the security for which beta is estimating over a period of time.There are no theoretical determined time intervals for calculating beta. The time period and the time interval may vary. The returns may be measured on daily,weekly, or monthly basis.
The return on a share and market index may be calculated as a total return that is, divided yield plus capital gain.

$r=\frac{D_{t}+(P_{t-1})}{P_{t-1}}=\frac{D_{t}}{P_{t-1}}+[\frac{P_{t}}{P_{t-1}}-1]$

In practice, one may use capital gain/loss or price return that is  $p_{t}/p_{t-1}-1$ rather total return to estimate beta of the company’s share. A further modification may be made in calculating the return.