# Portfolio Beta

Portfolio beta is a measure of the overall systematic risk of a portfolio of investments. It equals the weighted-average of the beta coefficient of all the individual stocks in a portfolio.

While variance and standard deviation of a portfolio are 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 beta coefficients. This is because beta coefficient represents the systematic risk which cannot be diversified away and hence less than perfect correlation between returns on individual investments does not reduce overall systematic risk.

Since the broad market has a beta coefficient of 1, a portfolio beta of less than 1 means that the portfolio has lower systematic risk than the market and vice versa.

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

## Formula

Portfolio beta equals the sum of products of individual investment weights and beta coefficient of those investments. It is a measure of the systematic risk of the portfolio.

βp = wA × βA + wB × βB + ... + wN × βN

Where βp is the portfolio beta coefficient, wA is the weight of the first investment, βA is the beta coefficient of first investment; wB is the weight of the second investment, βB is the beta coefficient of second investment; wn is the weight of the nth investment, βn is the beta coefficient of nth investment and so on.

## Example

Let us 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 = 35% ×: 0.9 + 65% ×: 1.2 = 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 lowered standard deviation.

A beta coefficient, on the other hand, is a measure of systematic risk which cannot be diversified. Adding more assets to a portfolio does not reduce the portfolio beta.