FRM Exam Part I · Measuring and Monitoring Volatility

Equally Weighted Moving Average Volatility for FRM Part I

Updated 11 October 2026 · Fact-checked

Equally weighted (moving average) volatility estimates variance as the simple average of the last M squared returns, often assuming a zero mean. Every observation in the window gets weight 1/M. You take the square root for volatility. The method is simple but reacts slowly and shows ghost effects when old shocks drop out.

Understand Equally Weighted (Moving Average) Volatility

Volatility is the standard deviation of returns. You cannot see it directly, so you estimate it from past returns. The simplest way is a rolling window: take the last M daily returns, square them, and average.

Each return in the window has the same weight, 1/M. A return from yesterday counts as much as one from M days ago. Returns older than the window count for nothing. That is why the method is called equally weighted. As each day passes, you add the newest squared return and drop the oldest one, so the window moves forward.

In practice, for daily data the mean return is very small compared with the standard deviation. So risk managers often set the mean to zero and use the average of squared returns directly. This is the form GARP-style questions usually use.

The window length M is a trade-off. A long window uses more data, so the estimate is more stable and has less sampling error. But it reacts slowly to new conditions and may mix in old, irrelevant regimes. A short window reacts fast, but it is noisier because it uses few observations.

Two drawbacks matter most. First, ghost effects (ghost features): a big shock raises the estimate when it enters the window. M days later it drops out, and the estimate falls sharply even though nothing happened that day. Second, slow response: after a shock the estimate rises only by 1/M of the squared return, and the equal weights give no extra importance to recent data. EWMA and GARCH models were built to fix these issues.

Key formulas to remember

Equally weighted variance (zero mean): σ²ₙ = (1/M) × Σ rₙ₋ᵢ², for i = 1 to M; rₙ₋ᵢ is the return i days before day n. Each squared return has weight 1/M. Volatility = √σ²ₙ.
Equally weighted variance (with sample mean): σ̂² = [1/(M − 1)] × Σ (rᵢ − r̄)²; Unbiased sample variance. Use it only when the question asks for the mean to be estimated.
Rolling update of the estimate: σ²ₙ₊₁ = σ²ₙ + (1/M) × (r²ₙ − r²ₙ₋M); Add the newest squared return and drop the oldest one. Handy when only a change is asked.
Annualizing volatility: σ(annual) = σ(daily) × √T; T is the number of periods per year, commonly 252 trading days for daily data. This assumes independent returns.
Standard error of a variance estimate (normal returns): se(σ̂²) ≈ σ² × √(2/M); Shows why a longer window gives a more precise estimate.

How to solve Equally Weighted (Moving Average) Volatility questions

Use this method for any question on rolling-window volatility, window choice or ghost effects.

1Identify the window length M and list the M most recent returns. Ignore anything older.
2Check whether the question assumes a zero mean. If it does not say otherwise for daily data, follow the instruction given.
3Square each return in the window. Keep returns in the same units, such as decimals or percentages.
4Add the squared returns and divide by M (or by M − 1 if the sample mean is used). This is the variance.
5Take the square root to get daily volatility. Annualize with √252 or the stated factor only if asked.
6For conceptual questions, link the window to the trade-off: longer is stable but slow, shorter is responsive but noisy.
7For ghost-effect questions, find the day a large return leaves the window. The estimate drops by r²/M on that day.

Quickest way: Squared-return shortcut

When to use it: Use it when the question gives returns in percent and asks for a volatility or a change in the estimate after the window moves.

Work in percent-squared units and square each return mentally.
Sum the squares, divide by M, then take the square root last.
For a window update, compute only (new² − dropped²) ÷ M and add it to the old variance.
Eliminate options that put a ghost drop on the day of the shock instead of M days later.

Common mistakes in Equally Weighted (Moving Average) Volatility

Taking the square root too early or forgetting it at the end.
The formula gives variance, and options often list both variance and volatility values.
Fix: Read what is asked. Variance is the average of squared returns. Volatility is its square root.
Including returns outside the window or miscounting M.
Students use all data given in the table.
Fix: Use only the last M returns. Count them before you sum.
Saying the estimate drops when the large shock occurs.
Confusing when the shock enters with when it leaves.
Fix: The estimate jumps when the shock enters and falls M days later when it leaves. That later fall is the ghost effect.
Believing a longer window is always better.
More data lowers sampling error, so it feels safer.
Fix: A longer window is slower to reflect new conditions. The right length balances precision and responsiveness.
Annualizing by multiplying by 252 instead of √252.
Variance scales with time, volatility scales with the square root.
Fix: Multiply daily variance by 252, or daily volatility by √252.
Subtracting the mean when the question says to assume zero mean.
Habit from the standard sample variance formula.
Fix: Follow the stated assumption. With zero mean, just average the squared returns.

Worked examples

Example 1

The last five daily returns of a USD equity index are +1%, −2%, +3%, −1% and 0%. Using an equally weighted window of M = 5 and a zero mean, what is the daily volatility estimate?

Show the solution

Square the returns in percent: 1, 4, 9, 1, 0.
Sum: 1 + 4 + 9 + 1 + 0 = 15.
Variance = 15 ÷ 5 = 3 (in %²).
Volatility = √3 = 1.732%.

Answer: Daily volatility ≈ 1.73%.

Example 2

With M = 20, the current daily variance estimate is 1.50 (%²). Today's return is −4%. The return that drops out of the window is +1%. What is the new daily volatility estimate?

Show the solution

Use σ²ₙ₊₁ = σ²ₙ + (1/M)(r²new − r²old).
r²new = 16 and r²old = 1, so the difference is 15.
Change = 15 ÷ 20 = 0.75.
New variance = 1.50 + 0.75 = 2.25.
Volatility = √2.25 = 1.5%.

Answer: New daily volatility = 1.5%, up from √1.50 ≈ 1.22%.

Exam tips

Questions often give returns in percent. Keep percent-squared units until the final square root.
Know the ghost-effect story: the estimate jumps on entry of a shock and falls on exit, M days later.
For conceptual items, name both sides of the window-length trade-off: precision against responsiveness.
Remember the motivation for EWMA: it puts more weight on recent returns and avoids abrupt drops.
Check whether the answer options are variances or volatilities before choosing.

Practice questions from Measuring and Monitoring Volatility

Equally Weighted (Moving Average) Volatility: frequently asked questions

What is the formula for equally weighted moving average variance?

Variance equals the average of the last M squared returns: σ²ₙ = (1/M) Σ rₙ₋ᵢ². This assumes a zero mean, which is common for daily data. Take the square root for volatility.

How do I choose the window length for a volatility estimate?

It is a trade-off. A longer window gives a more stable estimate with less sampling error but responds slowly and may include outdated conditions. A shorter window responds fast but is noisier.

What are ghost features in rolling window volatility?

They are sudden jumps in the estimate that are not caused by current events. When a large return leaves the window after M days, the estimate falls sharply even though the market did nothing that day.

Why is equally weighted volatility criticized?

It gives old and new returns the same weight and drops data abruptly. It reacts slowly to new shocks and produces ghost effects. EWMA and GARCH models address these problems.