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FRM Exam Part I · Measuring and Monitoring Volatility

EWMA Volatility Model and RiskMetrics Lambda 0.94

Updated 11 October 2026 · Fact-checked

The EWMA model estimates variance as a weighted average of past squared returns, with weights falling by a factor λ each period. You update it with σₙ² = λσₙ₋₁² + (1 − λ)rₙ₋₁². RiskMetrics sets λ = 0.94 for daily data. Square root the result to get volatility.

Understand EWMA Model and RiskMetrics

Volatility changes over time. An equally weighted average of past squared returns treats a shock from 100 days ago the same as one from yesterday. That makes the estimate slow to react, and it drops suddenly when an old large return leaves the window.

The exponentially weighted moving average (EWMA) model fixes this. It gives the most recent squared return the largest weight. Each older return gets a weight that is λ times the weight of the next more recent one. The decay factor λ lies between 0 and 1. Weights always sum to 1.

The weights form a geometric series, so the whole model collapses into a simple recursion. Today's variance estimate equals λ times yesterday's variance estimate plus (1 − λ) times yesterday's squared return. You only need the last variance and the last return. That is why it is easy to maintain in a risk system.

A lower λ means faster decay: the estimate reacts quickly but is noisier. A higher λ means slower decay: the estimate is smoother but slower to react. RiskMetrics uses λ = 0.94 for daily data (and 0.97 for monthly data). The model assumes the mean return is zero, which is reasonable for daily returns.

EWMA is a special case of GARCH(1,1) with ω = 0 and α + β = 1, where α = 1 − λ and β = λ. Because α + β = 1, EWMA has no long-run average variance. It has no mean reversion. GARCH(1,1) with α + β < 1 does revert to a long-run variance.

Key formulas to remember

EWMA recursive update
σₙ² = λσₙ₋₁² + (1 − λ)rₙ₋₁²
σₙ² is the variance forecast for day n, made at the end of day n−1. rₙ₋₁ is the return on day n−1. Assumes zero mean return.
EWMA weights
weight on rₙ₋ᵢ² = (1 − λ)λⁱ⁻¹, for i = 1, 2, 3, ...
Weights decline geometrically and sum to 1 over an infinite history.
Volatility from variance
σₙ = √(σₙ²)
Always take the square root last. Daily volatility scales to T days by multiplying by √T.
EWMA as GARCH(1,1)
ω = 0, α = 1 − λ, β = λ
Since α + β = 1, there is no long-run variance and no mean reversion.
Ratio of successive weights
wᵢ₊₁ ÷ wᵢ = λ
Useful for finding the weight on an older return from a recent one.

How to solve EWMA Model and RiskMetrics questions

Use this method for any EWMA question on the exam, whether it asks for variance, volatility, a weight or a model comparison.

  1. 1Identify λ. If the question says RiskMetrics and daily data, use 0.94. Otherwise use the value given.
  2. 2Write down the previous variance estimate σₙ₋₁². If only volatility is given, square it first.
  3. 3Write down the latest return rₙ₋₁ and square it. Keep the units consistent: 1% is 0.01, or use percent squared throughout.
  4. 4Apply σₙ² = λσₙ₋₁² + (1 − λ)rₙ₋₁².
  5. 5Take the square root if the question asks for volatility.
  6. 6If asked for a longer horizon, multiply daily volatility by √T.
  7. 7If the question asks about weights, use (1 − λ)λⁱ⁻¹ for the return i days back.
  8. 8Sanity check: the new variance must lie between the old variance and the squared return.

Quickest way: Weighted-average shortcut

When to use it: Use it for any single-step EWMA update when you have a calculator and little time.

  1. Treat the update as a weighted average: 94% on old variance, 6% on the new squared return.
  2. Work in percent, so a 1.5% volatility has variance 2.25 and a 2% return has squared value 4.
  3. Compute old variance × 0.94, then squared return × 0.06, and add.
  4. Check the answer lies between the two inputs. Then take the square root only at the end.
  5. Eliminate options that are above both inputs or below both.

Common mistakes in EWMA Model and RiskMetrics

  • Putting the weight λ on the squared return instead of on the old variance.

    Students remember (1 − λ) and λ but not which goes where.

    Fix: Remember that λ is persistence. A high λ means the old variance dominates, so λ multiplies σₙ₋₁².

  • Forgetting to square the return, or squaring volatility incorrectly.

    The question gives volatility and returns, and students mix them up.

    Fix: Convert everything to variance before applying the formula. Take the square root only at the end.

  • Reporting variance when the question asks for volatility.

    The calculation stops one step early under time pressure.

    Fix: Circle what the question asks for. Volatility is the square root of variance.

  • Saying EWMA has mean reversion to a long-run variance.

    Students blend EWMA with GARCH(1,1).

    Fix: EWMA has ω = 0 and α + β = 1, so there is no long-run variance. Only GARCH with α + β < 1 mean-reverts.

  • Thinking a higher λ makes the model react faster.

    Students link a larger number with a stronger response.

    Fix: A higher λ means slower decay and more weight on history. A lower λ reacts faster.

Worked examples

Example 1

Under RiskMetrics (λ = 0.94), yesterday's daily volatility estimate was 1.20%. Yesterday's return was −2.50%. What is today's daily volatility estimate?

Show the solution
  1. Old variance = 1.20² = 1.44 (in percent squared).
  2. Squared return = (−2.50)² = 6.25.
  3. σₙ² = 0.94 × 1.44 + 0.06 × 6.25.
  4. 0.94 × 1.44 = 1.3536. 0.06 × 6.25 = 0.375.
  5. σₙ² = 1.3536 + 0.375 = 1.7286.
  6. σₙ = √1.7286 ≈ 1.3148%.

Answer: About 1.31% per day.

Example 2

An analyst uses EWMA with λ = 0.90. The latest return squared has weight 0.10. What is the weight on the squared return from three days before the latest one (that is, i = 4), and what is the sum of the weights on the latest four returns?

Show the solution
  1. Weight on return i is (1 − λ)λⁱ⁻¹.
  2. For i = 4: 0.10 × 0.90³ = 0.10 × 0.729 = 0.0729.
  3. Weights for i = 1 to 4: 0.10, 0.09, 0.081, 0.0729.
  4. Sum = 0.10 + 0.09 + 0.081 + 0.0729 = 0.3439.
  5. Check: 1 − λ⁴ = 1 − 0.6561 = 0.3439.

Answer: The weight on i = 4 is 0.0729. The latest four weights sum to 0.3439.

Exam tips

  • Questions often give volatility, not variance. Square it first, then take the square root of your answer.
  • Expect conceptual questions comparing EWMA, equal weighting and GARCH(1,1). Know that EWMA has no long-run variance.
  • Remember RiskMetrics λ = 0.94 for daily data. Only use it when the question points to RiskMetrics or gives no other λ.
  • Use the weighted-average check: the new variance sits between the old variance and the squared return.
  • Watch units. Using 0.0125 in one place and 1.25 in another is a classic trap.

Practice questions from Measuring and Monitoring Volatility

EWMA Model and RiskMetrics: frequently asked questions

What is the EWMA volatility formula in FRM Part I?

The variance update is σₙ² = λσₙ₋₁² + (1 − λ)rₙ₋₁². Volatility is its square root. It assumes a zero mean return.

Why does RiskMetrics use λ = 0.94?

RiskMetrics found 0.94 gave a good fit for forecasting daily volatility across many markets. For FRM, just remember it as the daily-data value. It puts 6% weight on the latest squared return.

What is the difference between EWMA and GARCH(1,1)?

EWMA is GARCH(1,1) with ω = 0 and α + β = 1. It has no long-run variance and no mean reversion. GARCH(1,1) with α + β < 1 pulls variance back toward a long-run level.

How is EWMA different from equally weighted volatility?

Equal weighting gives every observation in the window the same weight. EWMA gives recent returns more weight and older ones geometrically less. So EWMA reacts faster and avoids sudden drops when old shocks leave the window.