EWMA and practical volatility — from forecast to position size

EWMA: GARCH's pragmatic little sibling

Take GARCH(1,1) from D1-07 and make two brutal simplifications: drop the floor ($\omega = 0$) and force persistence to exactly one ($\alpha + \beta = 1$, writing $\beta = \lambda$). What's left is the exponentially weighted moving average of squared returns:

$\sigma_t^2 = \lambda\, \sigma_{t-1}^2 + (1-\lambda)\, r_{t-1}^2$

One parameter instead of three, and no fitting: J.P. Morgan's RiskMetrics standardized $\lambda = 0.94$ for daily data in 1996 and much of the industry still runs it. Unrolling the recursion shows the name — today's variance is a weighted sum of all past squared returns with weights $(1-\lambda)\lambda^k$ decaying geometrically. Yesterday matters most; last month whispers; last year is gone.

What you give up versus full GARCH: because $\alpha + \beta = 1$, there is no unconditional variance ($\omega / (1 - \alpha - \beta)$ divides by zero — the answer to D1-07's closing prediction) and hence no mean reversion. EWMA's forecast for every future horizon is just today's estimate, whereas GARCH glides back to the long-run anchor. For 1-to-5-day risk that difference barely matters; for pricing a 1-year option it matters a lot. For a trading desk sizing tomorrow's positions, EWMA's robustness — no three-parameter fit to go wrong, no unstable $\omega$ — usually wins.

Effective lookback: about 17 days

A geometric decay has no hard cutoff, but its center of mass sits at roughly

$\text{effective lookback} \approx \frac{1}{1-\lambda} = \frac{1}{0.06} \approx 17 \text{ days}$

So RiskMetrics EWMA behaves like "a ~17-day window, but blurry at the edges." Raise $\lambda$ to 0.97 and you get a smoother, slower ~33-day memory; drop to 0.90 for a twitchy ~10-day one. One dial trades responsiveness against noise — the same tension as choosing a rolling window length, minus the cliff.

Rolling window vs EWMA: the step response

Think like an engineer and feed both estimators a single impulse — one crash day in an otherwise calm series:

• 20-day rolling std: vol jumps when the crash enters, stays flat at the elevated level for 20 days (the crash has equal weight whether it was yesterday or three weeks ago), then cliff-drops on day 21 when the crash exits. That second move — the echo — is pure artifact: your risk estimate, and thus position size, lurches on a day when nothing happened.
• EWMA: vol jumps immediately (a fresh observation carries weight $1-\lambda$, the largest of any single day — EWMA reacts faster), then decays smoothly by a factor $\lambda$ per day. No anniversary, no echo, no phantom resize.

Faster reaction and graceful forgetting for one parameter is why EWMA is the practical default. One upgrade worth knowing exists: realized volatility built from higher-frequency (e.g. 5-minute) returns estimates each single day's variance far more precisely and is the modern research standard — we only wave at it here, since daily bars are what you have from free data.

From forecast to position: vol targeting

Here is the whole practical recipe that D1-07 promised. Pick a target volatility for the strategy, then scale the position by

$w_t = \min\!\left(\frac{\sigma_{\text{target}}}{\hat{\sigma}_t},\; w_{\max}\right)$

Worked example: target 10% annualized, EWMA forecast says the asset currently runs 20% — hold half a unit. Forecast drops to 5% — the raw ratio says 2×, and the cap decides whether you take it. The cap is not decoration. Vol forecasts are smallest exactly when markets are eerily calm — often right before they aren't (volatility clusters, but regime breaks arrive unannounced; D1-10). An uncapped vol targeter maximizes leverage at the worst possible moment. Sized this way, dollar risk stays roughly constant across regimes, which stabilizes realized Sharpe and — critically — keeps one wild month from dominating a year's P&L.

Try it

Draw it

On paper, sketch the step response: x-axis is days, one crash on day 0 in an otherwise flat series. Draw two curves of estimated vol — a 20-day rolling std and an EWMA — through day 30. Mark three features: which jumps higher on day 1, the rolling estimate's cliff at day 20, and the EWMA's smooth decay. Label the echo.

Predict before the next lesson

Next is D1-09: cointegration and pairs trading — where we finally hunt mean reversion not in one series but in the gap between two. Predict: two stocks have daily return correlation 0.9. Does that guarantee the ratio of their prices stays bounded over ten years? Sketch why or why not before reading.