Difference between Beta and Volatility (FRM Part 1, Part 2)
1. Context
In this swatch, we look at a very common question: the difference between beta and volatility (or standard deviation). Readings in FRM Part 1 and Part 2 repeatedly mention these two measures of risk in various contexts, but mostly in topics related to modern portfolio theory and performance evaluation. The details of the related readings are given below:
| Area | Foundations of Risk Management |
| Reading | Applying the CAPM to Performance Measurement |
| Reference | Noel Amenc and Veronique Le Sourd, Ch10. Applying the CAPM to Performance Measurement: Single-Index Performance Measurement Indicators. In Portfolio Theory and Performance Analysis, West Sussex, England: John Wiley & Sons, 2003. |
| Area | Investment Management |
| Reading | Portfolio Performance Evaluation |
| Reference | Zvi Bodie, Alex Kane, and Alan J. Marcus, Ch24. Portfolio Performance Evaluation. In Investments, 10th Edition, New York: McGraw-Hill, 2013. |
2. Volatility
Volatility or standard deviation of a security $i$ is a measure of dispersion of it’s return $R_i$, measured as the square root of average or expected squared deviation from the mean: $$\sigma =\sqrt{E(R_i-\overline{R_i})^2}$$ In financial literature, this measure is used to denote total risk, or the propensity of returns to deviate very quickly in either direction – although what matters more to investors is downside risk, i.e. the risk that risk of returns going negative. Statistically, volatility is measured using historical price data of a security and is therefore sensitive to sampling error related to the dataset chosen. Such an estimate based on historical data is backward looking and may or may not be reflective of the future. Note that $\sigma$ of a risk-free security is assumed to be zero (in practice, it isn’t zero, but much smaller than volatilities of stocks).
3. Beta
Beta of a security $i$ is a measure of it’s systematic risk and can be viewed as a measure of it’s exposure to the rate of return of the market $(R_m)$. This is to say that if the expected market return changes by $1\%$, the expected return of the security will change by $\beta_i\%$, which is to say that: $$\beta_i=\frac{E(\Delta R_i)}{\Delta R_m}$$ Statistically, $\beta$ is measured as the slope coefficient from a regression of security’s returns vs market returns $(R_m)$: $$R_{i,t}=\alpha_i + \beta_i R_{m,t} + \varepsilon_{i,t}$$ where, $\alpha_i$ denotes the intercept and $\varepsilon_i$ denotes the contribution of random errors and omitted factors. Based on this regression, one can calculate beta as: $$\beta=\rho_{im}\frac{\sigma_i}{\sigma_m}$$ where, $\rho_{im}$ is the correlation between returns of security $i$ and the market and $\sigma_i$ and $\sigma_m$ are their respective volatilities. A $\beta$ of 1 would mean that security $i$’s returns mirror the index. It doesn’t mean that security $i$ has the same risk as the index (as we will see in next section). It just means that, keeping other factors (specific to this security) constant, returns of security $i$ move in sync with the index. A bullish investor would want to place her portfolio as overweighted in high beta stocks, while a bearish investor would want to overweight her portfolio in low beta stocks (or even cash, which has a beta of zero).
4. Connecting Beta and Volatility
Going by our above regression and with the assumption that macroeconomic factors (that move the market) and firm-specific or idiosyncractic factors (that affect only security $i$) are uncorrelated, we can express the total variance of security $i$’s returns as: $$\sigma^2_{R_i}=\beta^2\sigma^2_{R_m}+\sigma^2_{\varepsilon_i}$$ which is to say that the total risk of a stock is composed of systematic risk (first term) and specific risk (second term). Higher the $\beta$, higher is the systematic component of risk and therefore, higher is the total risk. High total risk ($\sigma$) doesn’t necessarily imply a high systematic risk, as this risk may be coming from firm specific factors.
5. Usage in Practice
Beta (systematic risk) and volatility (total risk) have a number of models and performance measures based on them. In the total risk camp, we have measures like Sharpe Ratio, while in the beta camp, we have Treynor’s measure and Jensen’s alpha. We explore these performance measures in another swatch.