Stock-Specific Volatility as Residual Risk After Controlling for Beta
Summary
The document defines stock-specific, or idiosyncratic, volatility within a single-factor return model. A stock’s return is represented as a risk-free component, exposure to the market’s excess return through beta, and a residual term. The standard deviation of that residual is the stock-specific volatility: it describes variation left after accounting for the modeled market exposure.
Under the stated assumption that the residual is normally distributed with zero mean, the stock’s expected return follows from the risk-free rate and expected market excess return. Its total variance combines beta-squared times market variance with residual variance. This distinction clarifies why residual volatility is not the same as the stock’s marginal volatility and why conditioning on a benchmark involves assumptions about the model. The explanation is limited to this factor setup; it does not specify a benchmark volatility or provide enough inputs to calculate a particular conditional probability.
Key ideas
- Stock-specific volatility is the standard deviation of the return residual after controlling for market exposure through beta.
- The factor model separates a stock return into a risk-free component, market-driven return, and residual return.
- Under a zero-mean normal residual assumption, expected stock return depends on the risk-free rate and expected market excess return.
- Total stock return variance combines market variance scaled by beta squared with residual variance.
- Residual volatility differs from marginal volatility and alone does not determine every conditional probability.
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Full text
# stock specific volatility
# stock specific volatility
I was unsure about the precise definition of "stock specific volatility". Used in this question "A stock has beta of 2.0 and stock specific daily volatility of 0.02. Suppose that yesterday's closing price was 100 and today the market goes up by 1%. What's the probability of today's closing price being at least 103?"
- this questions appears in an existing thread (linked below) but I was unable to comment as I'm a new user and my answer post was removed because it was a question!
Probability of stock closing over a certain price
In the answer given in the thread above they assume that $\sigma^2$ = the conditional variance of the stock return given the benchmark
I would have thought that this would have been a more natural definition $\sigma^2$ = the marginal variance of the stock return
But if you then treat the stock and benchmark returns as a 2-variate normal and try to work out the conditional distribution for the stock given the benchmark then you don't have enough information. Need something like the benchmarks volatility as well. Can anyone clarify this for me please?
## Answer by Kevin (score 4)
https://quant.stackexchange.com/a/46654
The stock specific volatility (also known as idiosyncratic volatility) is the volatility that remains after controlling for beta. I suppose you have $$R_i = R_f + \beta_i \cdot \big(R_m-R_f\big) + \varepsilon_i.$$ Then, the standard deviation of epsilon is your stock specific volatility. One frequently assumes $\varepsilon_i\sim N(0,\sigma^2_{\varepsilon_i})$. Then, the returns $R_i$ are normally distributed and $\mathbb{E}[R_i]=R_f + \beta_i \cdot \big(\mathbb{E}[R_m]-R_f\big)$. The variance is given by $\mathbb{V}\text{ar}[R_i]=\beta^2_i \cdot \sigma_m^2+\sigma_{\varepsilon_i}^2$.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.