    # Beta Calculation Beta measures the risk or volatility of an individual asset relative to the market portfolio. It is the covariance of the asset's return with the market portfolio's return, divided by the variance of the market portfolio.

β = Covim

σ2 m

Systematic risk is measured by beta. A particular stock's beta can help one predict how much the security will go up or down, provided one knows which way the market will go, and therefore figure out risk and expected return. Under the Capital Asset Pricing Model, there is a relationship between a stock's expected or required return and its beta. The following formula is very helpful in determining a stock's expected return.

rj = rf + b(rm - rf)

or

Expected return = Risk free rate + Beta (Market risk premium)

Assets with beta less than one are called defensive assets. Assets with beta greater than one are called aggressive assets. Risk-free assets have a beta equal to zero. The beta of a portfolio is the weighted average of the betas of assets included in the portfolio. The weights are the relative share of the assets in the portfolio.

Example:
Two assets with beta values of 0.6 and 1.1 have been combined in the proportion of 1:3. So, the beta of the resultant portfolio will be

=> (0.6 x 0.25 + 1.1 x 0.75) => 0.975

If the standard deviation of the market portfolio is 25%, the standard deviation of the portfolio would be: => 0.975 x 25 => 24.38 %.
Example:

If the price of a stock on Jan 1 is \$50, the annual dividend received at the end of the year is \$2 and the year end price on Dec 31 is \$60, what is the rate of return?

Total return = Current return (yield) + Capital returns (gains/losses)

Unlevering Beta:
In an all-equity or unlevered firm, the value of equity beta is equal to the assets beta. This is referred to as the unlevered beta and determination of assets beta from the observed securities betas is termed as unlevering of beta.

β assets = β portfolio = (D/V) + β debt + (E/V) + β equity

Relevering Beta:
Relevering of beta involves determination of equity beta, with given assets beta, for the proposed financing structure, using the following equation:

β equity = β assets + (β assets - β debt) D/E

The above calculation is similar to the determination of "Cost of Equity" under net operating income approach.

Example:
The equity beta of Company A is 1.4 and debt beta is 0; it�s present debt-equity ratio is 1:2. A new Company, B plans to issue equity shares for the first time. The Company B is similar in size and risk of Company A and its debt-equity ratio is 1:1. The equity beta of Company B may be obtained by unlevering the equity beta of Company A and then relevering for the proposed financing structure of Company B as illustrated below:

The unlevered or assets beta is:

β assets = (1/3) x 0 + (2/3) x 1.4 = 0.93

For the proposed debt-equity ratio of 1:1, the equity beta would be:

β equity = 0.93 + (0.93-0) (1/1) = 1.86

Example:  Determination of beta

 Given: Standard deviation of stock of Z ltd (σz) = 11% Standard deviation of market portfolio (σm) = 9% Correlation of share with the market (rzm) = + 0.6

Beta of Z Ltd. Share = (σz σm rzm ) / σ2m
=> (11 x 9 x 0.6) / (9)2
=> 0.73

 Given: Standard deviation of the portfolio(σp) = 4% Standard deviation of the market portfolio (σm) = 2.5% Correlation of portfolio with the market (rpm) = + 0.8

Beta = (4 x 2.5 x 0.8)/(2.5)2
=> 1.28

 Beta What it means: 0 The security's return is independent of the market Example: Risk-free security like T-bills or Government bonds 0.5 The security is only half as responsive as the market 1.0 The security has the same responsive or risk as the market 2.0 The security is twice as responsive, or risky, as the market

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