Corporate Operating Risk Framework – Summary

1. Objective

Measure corporate operating risk using publicly available data by combining:

Revenue growth volatility (YoY)
Cost structure dynamics
Shock vs persistence decomposition (AR model)

2. Core Definitions

Quarterly log growth

$x_t = \log R_t - \log R_{t-1}$

Year-over-year (YoY) growth

$y_t = \log R_t - \log R_{t-4}$

Relationship: $y_t = x_t + x_{t-1} + x_{t-2} + x_{t-3}$

3. Volatility Measure

Primary operating risk measure: $\sigma_{\text{YoY}} = \text{StdDev}(y_t)$

4. Möbius Transformation

\[f(x) = \frac{x}{x + 2}\]

General form: $f_k(x) = \frac{x}{x + k}, \quad k > 1$

5. AR(1) Model

\[x_t = \phi x_{t-1} + \epsilon_t\]

6. Closed-Form Link

\[\sigma_x^2 = \frac{\sigma_\epsilon^2}{1 - \phi^2}\] \[\operatorname{Var}(y_t) = \frac{\sigma_\epsilon^2}{1-\phi^2} \left(4 + 6\phi + 4\phi^2 + 2\phi^3 \right)\] \[\sigma_{\text{YoY}} = \sigma_\epsilon \sqrt{ \frac{4 + 6\phi + 4\phi^2 + 2\phi^3}{1-\phi^2} }\]

7. Interpretation

$\phi > 0$: persistence
$\phi \approx 0$: shocks dominate
$\phi < 0$: reversal

8. Example Insight

For $\phi \approx 0: \sigma_{\text{YoY}} \approx 2\sigma_\epsilon$

9. Cost–Revenue Interaction

\[E = \frac{\Delta \text{Cost}}{\Delta \text{Revenue}}\] \[\rho = \text{Corr}(\Delta R, \Delta C)\]

10. Final Structure

Operating Risk =

YoY volatility
Shock volatility
Persistence
Cost flexibility

11. Stata Implementation

gen lrev = log(revenue)
gen x = D.lrev

arima x, ar(1)

scalar phi = _b[ARMA:L1.ar]

predict eps, resid
summ eps
scalar sigma_eps = r(sd)