# Portfolio Beta

Portfolio beta refers to the weighted-average beta coefficient of the individual investments in a portfolio. It is calculated as the sum of the products of weights of individual investments and beta coefficient of those investments. It is a measure of the systematic risk of the portfolio.

While variance and standard deviation of a portfolio is calculated using a complex formula which includes mutual correlations of returns on individual investments, beta coefficient of a portfolio is the straight weighted-average of individual beat coefficients. This is because beta coefficient represents the systematic risk which can’t be diversified away and hence less than perfect correlation between returns on individual investments doesn’t reduce overall systematic risk.

Portfolio beta is an important input in calculation of Treynor's measure.

## Formula

Portfolio beta can be calculated using the following formula:

$$ Portfolio\ (\beta_p)\ =w_A\times\beta_A+w_B\times\beta_B+...+w_n\times\beta_n $$

Where β_{p} is the portfolio beta coefficient, w_{A} is the weight of the first investment, β_{A} is the beta coefficient of first investment; w_{B} is the weight of the second investment, β_{B} is the beta coefficient of second investment; w_{n} is the weight of the nth investment, β_{n} is the beta coefficient of nth investment and so on.

See how the above formula is different from the formula for portfolio variance and portfolio standard deviation:

$$ {\sigma_P}^2={w_A}^2{\sigma_A}^2+{w_B}^2{\sigma_B}^2+2w_Aw_B\sigma_A\sigma_B\rho $$

$$ \sigma_P=\sqrt{{w_A}^2{\sigma_A}^2{+w_A}^2{\sigma_A}^2+2\times w_Aw_B\sigma_A\sigma_B\rho} $$

## Example

Let’s say we have a 2-asset portfolio. Their weights are 35% and 65%, their standard deviations are 2.3% and 3.5% and their betas are 0.9 and 1.2 respectively. Their mutual correlation coefficient is 0.5.

The portfolio beta in this case is 1.095:

$$ Portfolio\ (\beta_p)\ =35\%\times0.9+65\%\times1.2=1.095 $$

The standard deviation of the portfolio in this case is 2.77% which is lower than the weighted-average of the individual standard deviations which works out to 3.08%.

### Portfolio Beta vs Portfolio Standard Deviation

As we can see above, the portfolio standard deviation of 2.77% is lower than what we would get based on a weighted average i.e. 3.08%. The difference is attributable to diversification benefits. The decrease in portfolio standard deviation evident above is due to less than perfect correlation between returns on both assets. This is because the standard deviation is a measure of total risk of portfolio including both diversifiable and non-diversifiable risks. Including more than one asset in a portfolio has reduced the diversifiable risk and hence lower standard deviation.

A beta coefficient, on the other hand, is a measure of systematic risk which can’t be diversified. Adding more and more assets to a portfolio doesn’t reduce the portfolio beta.