Calculating Alpha And Beta Value Of A Security

The Alpha and Beta value for a security are key metrics in finance derived from the Capital Asset Pricing Model (CAPM).

Beta ( $\beta$ ) is calculated first, as it is a required input for calculating Alpha.

Alpha ( $\alpha$ ) is then calculated using the CAPM formula, which incorporates the Beta value.

1. Calculating Beta ( $\beta$ )

Beta ( $\beta$ ) is a measure of a security’s volatility (or systematic risk) in relation to the overall market (benchmark).

Beta Formula:

The most common way to calculate Beta is using historical returns through a regression analysis, which mathematically is the covariance of the security’s returns with the market’s returns, divided by the variance of the market’s returns:

$\beta = \frac{\text{Covariance}(R_e, R_m)}{\text{Variance}(R_m)}$

Where:

$R_e$ = The return on the individual security (e.g., stock).

$R_m$ = The return on the overall market (the benchmark index, like the S&P 500 or FTSE 100).

$\text{Covariance}(R_e, R_m)$ = Measures how the security’s returns move in relation to the market’s returns.

$\text{Variance}(R_m)$ = Measures how dispersed the market’s returns are from their mean.

Interpretation of Beta:

Beta Value	Interpretation
$\beta = 1.0$	The security moves exactly with the market. Its volatility is the same as the benchmark.
$\beta > 1.0$	The security is more volatile than the market. It tends to amplify market movements (e.g., a $\beta$ of $1.5$ means the stock is $50\%$ more volatile than the market).
$0 < \beta < 1.0$	The security is less volatile than the market. It moves in the same direction but less dramatically (e.g., a $\beta$ of $0.5$ means it’s $50\%$ less volatile).
$\beta = 0$	The security’s price movements are completely uncorrelated with the market.
$\beta < 0$	The security moves in the opposite direction to the market (i.e., when the market goes up, the security tends to go down).

Real Business Example:

A major, established utility company like E.ON (Germany/UK) often has a low Beta (e.g., $0.5$ to $0.8$ ) because its business is stable and less sensitive to overall economic cycles.

A high-growth technology company like NVIDIA (USA) often has a high Beta (e.g., $1.5$ or higher) because its stock is more volatile and sensitive to market sentiment and growth expectations.

2. Calculating Alpha ( $\alpha$ )

Alpha ( $\alpha$ ), specifically Jensen’s Alpha, is the excess return a security or portfolio generates compared to the return that was expected given its Beta (risk). It measures the value added by a security’s active management or performance unique to the company.

Alpha Formula:

Alpha is calculated by rearranging the CAPM formula to solve for the difference between the actual return and the expected return:

$\alpha = R_p - [R_f + \beta \times (R_m - R_f)]$

Where:

$\alpha$ = Alpha (the excess return).

$R_p$ = The actual return of the security or portfolio.

$R_f$ = The risk-free rate of return (typically the yield on a short-term government bond).

$\beta$ = The Beta of the security (calculated in step 1).

$R_m$ = The market return (the return of the benchmark index).

$[R_f + \beta \times (R_m - R_f)]$ = The expected return according to CAPM.

Interpretation of Alpha:

Alpha Value	Interpretation
$\alpha > 0$ (Positive)	Outperformance. The security or manager earned a return higher than expected given the level of risk ( $\beta$ ). This is generally sought after by investors.
$\alpha = 0$ (Zero)	In-line performance. The security’s return was exactly what was expected given its risk.
$\alpha < 0$ (Negative)	Underperformance. The security or manager earned a return lower than expected given the level of risk ( $\beta$ ).