EWMA Volatility and RiskMetrics

A simple 30-day rolling volatility estimate has an odd flaw: a single huge return from exactly 30 days ago counts with the same full weight as yesterday's return, right up until day 31, when it vanishes from the average entirely and the estimate can jump abruptly even though nothing new happened. Markets do not actually forget information on a hard 30-day cliff, recent shocks matter more than old ones, and that extra importance should fade gradually. The exponentially weighted moving average (EWMA), popularized by J.P. Morgan's RiskMetrics, replaces the hard cutoff with a smooth fade.

The analogy

Think of how a rumor's influence on your opinion decays. Something you heard this morning shapes your view strongly. Something you heard a week ago still nudges you a little. Something from a year ago barely registers, but it never disappears entirely, it is just discounted almost to nothing. A simple rolling average is like an office where gossip from exactly 30 days ago is discussed with full seriousness right up until the 31st day, when it is suddenly, completely forgotten and never mentioned again. EWMA is the more human version: everything fades continuously, nothing drops off a cliff.

Building the formula

Let $r_t$ be today's return and $\sigma_t^2$ yesterday's variance estimate. EWMA updates the variance estimate as

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

In words: today's variance estimate is a blend of yesterday's variance estimate and today's squared return, one fresh data point recalibrating an existing belief, not a fresh calculation from scratch over a whole window. $\lambda$ (the "decay factor," RiskMetrics famously used $\lambda = 0.94$ for daily data) controls how much weight stays on the past: closer to 1 means slow-moving, heavily-smoothed volatility; closer to 0 means the estimate reacts almost entirely to today's move and forgets yesterday almost completely.

Unrolling the recursion shows the "smooth fade" directly:

$\sigma_t^2 = (1-\lambda)\sum_{i=0}^{\infty} \lambda^i r_{t-i}^2.$

In words: today's variance is a weighted sum of every past squared return, with weights $(1-\lambda), (1-\lambda)\lambda, (1-\lambda)\lambda^2, \dots$ shrinking geometrically the further back you go. No return is ever fully forgotten, and none is dropped in one abrupt jump, unlike a rolling window's hard edge.

Generate a path and picture a shock, one unusually large move partway through. In a rolling-window estimate, that shock's effect appears suddenly and then disappears suddenly exactly $N$ days later, a visible step down. In an EWMA estimate, the same shock's effect appears immediately but decays smoothly afterward.

Worked example 1: updating the estimate by hand

Yesterday's EWMA variance was $\sigma_{t-1}^2 = 0.0001$ (a daily standard deviation of $1\%$). Today's return comes in unusually large at $r_t = 3\%$, so $r_t^2 = 0.0009$. Using $\lambda = 0.94$:

$\sigma_t^2 = 0.94 \times 0.0001 + 0.06 \times 0.0009 = 0.000094 + 0.000054 = 0.000148.$

New daily volatility: $\sqrt{0.000148} \approx 1.22\%$. A single 3% shock moved the volatility estimate from 1.0% to 1.22%, a meaningful but not overwhelming jump, exactly the "recent news matters, but does not dominate" behavior the decay factor is designed to produce. Tomorrow, absent another shock, this estimate will itself decay back toward calmer levels rather than staying frozen at 1.22% for a fixed window length.

Worked example 2: comparing decay factors

Same shock, $r_t^2 = 0.0009$, same starting point $\sigma_{t-1}^2 = 0.0001$, but now with $\lambda = 0.99$ (slow-moving, used for lower-frequency risk):

$\sigma_t^2 = 0.99 \times 0.0001 + 0.01 \times 0.0009 = 0.000099 + 0.000009 = 0.000108 \Rightarrow \sigma_t \approx 1.04\%.$

Versus $\lambda = 0.90$ (fast-reacting):

$\sigma_t^2 = 0.90 \times 0.0001 + 0.10 \times 0.0009 = 0.00009 + 0.00009 = 0.00018 \Rightarrow \sigma_t \approx 1.34\%.$

Same shock, same starting volatility, three quite different next-day estimates (1.04%, 1.22%, 1.34%) purely from the choice of $\lambda$. A risk system tuned with too high a $\lambda$ will be slow to flag a genuine regime change; one tuned too low will whipsaw its risk limits after every ordinary noisy day. There is no universally correct $\lambda$, only a trade-off a risk manager chooses deliberately.

What this means in practice

EWMA volatility feeds directly into daily VaR calculations at most banks (it is the core of the original RiskMetrics methodology), position sizing rules that scale exposure inversely with recent volatility, and stop-loss triggers that widen or tighten with market conditions. Its main appeal over full GARCH is that it needs no numerical estimation, $\lambda$ is chosen once and the recursion runs in real time with a single multiply-and-add per day.