Calculating Total Risk and Idiosyncratic Risk for individual stocks?

Question

I am working on a research project but am having some trouble wrapping my head around how I need to go about replicating two of key dependent variables from (Serfling, 2014).

1. Total Risk, defined as: the annualized standard deviation of daily stock returns during the fiscal year.

2. Idiosyncratic Risk, defined as: the annualized standard deviation of the residuals from the regression of daily stock returns on the Fama & French three factors estimated over the fiscal year.

I think I already know for Total Risk: 1. change in daily closing price to get daily stock returns, 2. STDEV of daily stock returns 3. calculate variance, i.e. square STDEV of daily stock returns 4. make assumption on number of trading days, i.e. 250 or 252 5. annualized variance, i.e. variance * number of trading days and, 6. SQRT of annualized variance.

Can anyone confirm this?

How do I go about doing calculating the idiosyncratic risk? I will have a larger data set spanning at least from 1992 till 2016, so I would have to be able to calculate idiosyncratic risk for many firm-years in Stata and/or Excel.

The main source of my confusion is that some people state: Total Risk - Systematic Risk = Idiosyncratic Risk but there is also the approach in the Alpha Architect link. I am not sure if they are the same, but it does not appear to be the case.

Sources:

https://alphaarchitect.com/2014/12/19/a-quick-lesson-in-volatility-measures/

How to calculate unsystematic risk?

(Serfling, 2014) --> https://www.sciencedirect.com/science/article/pii/S092911991300148X

Answer 1

Total Volatility

Your approach for calculating Total Risk seems right in general. Let's be more precise:

The total volatility $Vol_i$ of stock $i$ can be calculated as $$Vol_i = 100 \sqrt{\frac{\sum^n_{t=1}{(R_{i,t}-\bar{R_i})^2}}{n-1}} \sqrt m$$

where $R_{i,t}$ is the return of stock $i$ during period $t$, $\bar{R_i}$ is the average return of stock $i$ taken over all periods used in the calculation of $Vol_i$, $n$ is the number of periods of data used in the calculation and $m$ is the number of periods in one year.

Idiosyncratic Volatility

The idiosyncratic volatility is measured as the residual standard error from a time-series regression of periodic excess stock returns on the returns of a factor model (e.g. Fama/French). So you have to run a regression $$r_{i,t} = \alpha_i + \beta_{i1} \mathit{RMRF}_t + \beta_{i2} \mathit{SMB}_t + \beta_{i3} \mathit{HML}_t + \epsilon_{it}$$ where $r_{i,t}$ is the excess return of stock $i$ (i.e. above the risk-free rate of return) and $RMRF$, $SMB$ and $HML$ are the corresponding factors. You can download them from Kenneth French's website. The residual standard error $RSE_i$ from this regression is calculated as $$RSE_i = \sqrt{\frac{\sum^n_{j=1}{\epsilon^2_{i,j}}}{n-k}} $$ where $n$ is the number of data points that are used to fit the regression and $k$ is the number of parameters estimated by the regression (which are four by the Fama/French model: three factor loadings and the intercept). The idiosyncratic volatility $IdioVol_i$ is then measured as

$$IdioVol_i = 100 \cdot RSE_i \cdot \sqrt{m}$$

with $m$ as the number of return periods in a year.

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Source: Bali/Engle/Murray (2016) , Empirical Asset Pricing: The cross section of stock returns, 1. ed.