EWMA & RiskMetrics

GARCH is powerful, but fitting it means estimating three parameters by maximum likelihood — overkill when you just need a fast, robust volatility number for a risk report tonight. Enter the exponentially weighted moving average (EWMA): a one-parameter volatility estimator that captures the most important feature — recent moves matter more than old ones — with arithmetic simple enough to do in a spreadsheet. It became the heart of J.P. Morgan’s RiskMetrics system, the framework that brought Value-at-Risk to the whole industry. EWMA is the pragmatic volatility tool: not the most accurate, but the one a desk reaches for first.

The problem with equal-weighted windows

Analogy. A simple rolling-window volatility is like judging today’s weather by averaging the last 100 days equally — a hurricane 99 days ago counts exactly as much as yesterday, and then vanishes completely the moment it ages out to day 101. Two artifacts follow: stale data gets equal say, and the estimate jumps when a big old observation exits the window even though nothing happened today.

The fix. Common sense says recent returns should matter more than ancient ones, and influence should fade, not fall off a cliff. EWMA encodes exactly that: weights that decline geometrically into the past, with no hard cutoff.

EWMA: variance with fading memory

Definition. The EWMA variance update is a weighted blend of yesterday’s variance estimate and yesterday’s squared return: $\sigma_t^2 = \lambda\,\sigma_{t-1}^2 + (1 - \lambda)\,r_{t-1}^2,$ where $\lambda$ (lambda), the decay factor, sits between 0 and 1. Unrolling the recursion shows the geometric weighting explicitly: $\sigma_t^2 = (1 - \lambda)\sum_{k=0}^{\infty}\lambda^{k}\,r_{t-1-k}^2.$ The return from $k$ days ago carries weight $(1-\lambda)\lambda^k$ — a geometric decay. The weights sum to 1 (a proper weighted average), and the most recent return always gets the biggest slice, $(1-\lambda)$.

What $\lambda$ controls. $\lambda$ is the memory dial:

• High $\lambda$ (near 1, e.g. 0.97): weights decay slowly → long memory → smooth, slow-to-react estimate.
• Low $\lambda$ (e.g. 0.85): weights decay fast → short memory → jumpy, fast-reacting estimate that snaps to recent moves.

Worked example — one EWMA update. Take $\lambda = 0.94$, yesterday’s variance $\sigma_{t-1}^2 = 0.0004$ (vol $2\%$), and yesterday’s return $r_{t-1} = -0.03$ ($-3\%$ day, $r_{t-1}^2 = 0.0009$): $\sigma_t^2 = 0.94(0.0004) + 0.06(0.0009) = 0.000376 + 0.000054 = 0.00043.$ Today’s vol estimate is $\sqrt{0.00043} \approx 0.0207 = 2.07\%$ — the $-3\%$ day nudged it up, weighted by $(1-\lambda) = 6\%$.

The half-life: how long is EWMA’s memory?

A clean way to summarize $\lambda$ is its half-life — the number of days back at which the weight has decayed to half the most recent day’s. Solve $\lambda^h = 0.5$: $h = \frac{\ln 0.5}{\ln \lambda}.$

Worked examples.

• $\lambda = 0.94$ (RiskMetrics daily): $h = \ln(0.5)/\ln(0.94) = -0.693 / -0.0619 \approx 11.2$ days. Roughly the last two trading weeks hold half the weight.
• $\lambda = 0.97$ (RiskMetrics monthly): $h = -0.693/-0.0305 \approx 22.8$ days — a longer memory, smoother estimate.
• $\lambda = 0.85$: $h = -0.693/-0.1625 \approx 4.3$ days — very short memory, snaps to recent moves fast.

The half-life turns an abstract decay factor into something interpretable: “this estimate effectively remembers about two weeks of returns.”

EWMA is GARCH’s special case (IGARCH)

Here’s the elegant connection to the previous lesson. Compare the two variance equations:

• GARCH(1,1): $\sigma_t^2 = \omega + \alpha\,r_{t-1}^2 + \beta\,\sigma_{t-1}^2$.
• EWMA: $\sigma_t^2 = (1-\lambda)\,r_{t-1}^2 + \lambda\,\sigma_{t-1}^2$.

They are the same equation under three identifications: $\omega = 0$, $\alpha = 1 - \lambda$, and $\beta = \lambda$. So EWMA is exactly GARCH(1,1) with $\omega = 0$ and $\alpha + \beta = (1-\lambda) + \lambda = 1$ — the IGARCH (integrated GARCH) case from the GARCH lesson.

The consequence: because persistence $\alpha + \beta = 1$, EWMA has no mean reversion and no long-run variance (the formula $\omega/(1-\alpha-\beta)$ divides by zero). Volatility shocks are treated as permanent: EWMA never pulls its forecast back toward a long-run “climate” level the way a stationary GARCH does. Its multi-step forecast is just a flat line at the current estimate. That’s the price of simplicity — and the key behavioral difference from GARCH.

RiskMetrics and the famous λ = 0.94

In 1994, J.P. Morgan published RiskMetrics, a methodology (and free dataset) for measuring market risk that put Value-at-Risk (VaR) on every desk. Its volatility engine was EWMA, and it shipped with a now-iconic default:

• $\lambda = 0.94$ for daily data, and $\lambda = 0.97$ for monthly data.

These weren’t fitted per asset — they were fixed constants, chosen by RiskMetrics to work reasonably well across a huge cross-section of assets. That universality was the point: a single, simple, no-estimation-required recipe any institution could apply consistently to compute VaR.

How EWMA feeds VaR. A one-day 99% VaR under a normal assumption is roughly $\text{VaR} \approx 2.33 \times \sigma_t \times (\text{position value}),$ where $\sigma_t$ is the EWMA volatility estimate and $2.33$ is the 99% normal quantile. Because $\sigma_t$ updates daily with the EWMA recursion, the VaR breathes with the market — rising in turbulence, falling in calm — without re-fitting anything.

Worked example — daily VaR. A $10,000,000 position, EWMA daily vol $\sigma_t = 2\%$, 99% one-day VaR: $2.33 \times 0.02 \times 10\,000\,000 = 466\,000$ dollars. If a turbulent week pushes the EWMA estimate to $3\%$, the VaR rises to $2.33 \times 0.03 \times 10\,000\,000 = 699\,000$ dollars — automatically, just from the updated volatility.

EWMA vs GARCH: when to use which

The trade-off. EWMA wins on simplicity, speed, and robustness: no fitting, no estimation risk, transparent, reproducible across thousands of assets — ideal for production risk systems and short-horizon VaR. GARCH wins on accuracy and structure: it captures mean reversion, produces a proper volatility term structure for multi-horizon forecasts, and (with t-errors or asymmetry) fits the data better. The rule of thumb: EWMA for fast, uniform, short-horizon risk numbers; GARCH when forecast accuracy and the volatility term structure matter (options pricing, longer horizons, research).

Putting it together

EWMA estimates variance as a geometrically decaying weighted average of past squared returns, $\sigma_t^2 = \lambda\sigma_{t-1}^2 + (1-\lambda)r_{t-1}^2$, fixing the two flaws of equal-weighted windows: it weights recent returns more and lets old shocks fade smoothly rather than dropping off a cliff (the ghosting artifact). The single parameter $\lambda$ is the memory dial — higher means smoother and longer-memoried — and its half-life $\ln(0.5)/\ln\lambda$ makes that memory interpretable in days. Algebraically, EWMA is exactly GARCH(1,1) with $\omega = 0$ and $\alpha+\beta=1$ (IGARCH), so it has no mean reversion and no long-run variance — its multi-step forecast is a flat line. RiskMetrics built VaR on EWMA with the iconic fixed defaults $\lambda = 0.94$ (daily) and $0.97$ (monthly), prizing a single, no-fitting, firm-wide-consistent recipe. The trade-off is clean: EWMA for fast, robust, short-horizon, uniform risk numbers; GARCH when accuracy, mean reversion, and the volatility term structure matter.

Next up — backtesting pitfalls — we shift from building models to trusting them. Look-ahead bias, overfitting, and data-snooping are the traps that make a strategy look brilliant on history and fail live. The final lesson before the exam is about not fooling yourself.