# Standard Deviation vs Beta

Beta coefficient is a measure of an investment’s systematic risk while the standard deviation is a measure of an investment’s total risk. In a portfolio of investments, beta coefficient is the appropriate risk measure because it only considers the undiversifiable risk. However, for standalone assets, standard deviation is the relevant measure of risk.

Risk inherent in an equity investment arises mainly from two sources: (a) from company specific factors such as loss of a major customer, loss of a legal battle, any major regulatory action, etc. and (b) from broad economy-wide shocks such as a change in central bank policy rate, change in taxes, war, earthquake, etc. Risk that results from company-specific factors is called unique risk while the risk that affects the whole market is called systematic risk.

This can be expressed using the following equation:

$$ Total\ Risk=Unique\ Risk+Systematic\ Risk $$

## Standard Deviation – a Measure of Total Risk

Standard deviation is a measure of the total variability of an investment or an investment portfolio regardless of its source. It includes both the unique risk and systematic risk.

Following is the equation for standard deviation of a portfolio:

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

σ_{P} = portfolio standard deviation

w_{A} = weight of asset A in the portfolio

w_{A} = weight of asset B in the portfolio

σ_{A} = standard deviation of asset A

σ_{B} = standard deviation of asset B

ρ = correlation coefficient between returns on asset A and asset B.

Standard deviation of two assets with correlation of less than 1 is less than the weighted average of the standard deviation of individual stocks. This is because in a portfolio context, risk that results from company-specific or unique factors can be eliminated by holding more and more investments. This is because a loss for one company is a win for another and holding a well-rounded mix of companies will cause the company-specific factors to cancel out such that there is no net-risk from unique factors. The unique risk is hence called diversifiable risk. Therefore, standard deviation is not a good measure of risk in a portfolio context because it includes certain a portion of risk which can be eliminated. However, it is useful when looking at an investment individually.

## Beta Coefficient – a Measure of Systematic Risk

Beta coefficient is a measure of sensitivity of an investment (when considered in a well-diversified portfolio) to the systematic risk factors.

The economy-wide factors affect all stocks in one way or the other. There is no way to escape this component of risk. Hence, it is called undiversifiable risk. It is also called systematic risk because it results from and affects the whole macroeconomic system. Some investments are affected more by the systematic risk and some less.

The following equation expresses the relationship between standard deviation and beta:

$$ Risk\ Captured\ by\ \sigma=Risk\ Captured\ by\ \beta\ +\ Unique\ Risk $$

Where σ stands for investment standard deviation while β refers to the investment’s beta coefficient.

In evaluating an investment in a portfolio context, the beta coefficient is relevant because the unique risk can be diversified away, and only undiversifiable risk should be priced. It is why Treynor’s ratio is considered a better measure of a portfolio’s return per unit of risk than the Sharpe ratio which is based on standard deviation.