Zero-mean EWMA variance

Calculation contract

v_(t+1) = lambda*v_t + (1 - lambda)*r_t^2

Method and interpretation

Each supplied return updates the variance for the next period. The output path therefore begins after the first observation, not at the initial variance. This convention avoids attaching an updated estimate to a decision made before the return was observed.

With initial variance 0.0001 and decay 0.94, the first 1% return leaves variance unchanged. The following -2% return raises it to 0.000118. A 1.5% third return then produces 0.00012442. The path and terminal value are both available for reconciliation.

Where the reference stops

A zero conditional mean is assumed. Lambda=1 freezes the initial value; lambda=0 uses only the latest squared return. No parameter fitting, expected return adjustment, missing-period decay or empirical forecast score is provided.

Published synthetic inputs

{
  "returns": [
    0.01,
    -0.02,
    0.015
  ],
  "initial_variance": 0.0001,
  "lambda": 0.94
}

Frozen calculation output

{
  "variance_path": [
    0.0001,
    0.00011800000000000002,
    0.00012442000000000003
  ],
  "next_period_variance": 0.00012442000000000003,
  "next_period_volatility": 0.01115437134042076
}

Zero conditional mean assumption; each observation updates variance for the following period. No fitted decay parameter or forecast validation.

Verification coverage

• Recursion independently replayed, including decay endpoints.
• Initial condition, ordering and negative variance validation covered.

Unit and independent-reference checks test arithmetic and rejection rules, not future investment performance.

Method references

• RiskMetrics technical document, fourth edition

Read the assumptions in context