FRM Exam Part I · Measuring Return, Volatility, and Correlation

Variance, Volatility and Annualization of Returns

Updated 11 October 2026 · Fact-checked

Volatility is the standard deviation of returns. Compute the sample variance by summing squared deviations from the sample mean and dividing by n − 1, then take the square root. To annualize, multiply by √T, where T is the number of periods per year, for example √252 for daily returns. This assumes independent, identically distributed returns.

Understand Variance, Volatility and Annualization

Variance measures how widely returns spread around their average. You take each return, subtract the mean, square the gap, and average the squares. Squaring stops positive and negative gaps from cancelling out.

Volatility is the square root of variance. It is in the same units as returns, so a 2% daily volatility reads naturally. Variance is in squared units, so it is hard to interpret but easier to add up across assets and time.

When you estimate variance from a sample, you divide by n − 1, not n. You have already used the data to estimate the mean, which pulls the observations closer to the sample mean than to the true mean. Dividing by n would understate the true variance on average. Dividing by n − 1 gives an unbiased estimator of variance. The sample standard deviation is still slightly biased, because the square root is a nonlinear function. If you know the true mean, or you are describing a full population, divide by n.

To move across horizons, you use the square-root-of-time rule. If returns are independent and identically distributed (i.i.d.) and you use log returns, the variance of the T-period return is T times the one-period variance. Variance grows linearly with time, so volatility grows with √T. Daily to annual means × √252 (or √250 or √260, depending on the stated trading days). Monthly to annual means × √12.

The rule fails if returns are autocorrelated or if volatility changes over time. Positive autocorrelation makes the true long-horizon volatility larger than the rule says. Negative autocorrelation makes it smaller. Treat the rule as an approximation that the exam asks you to apply under stated assumptions.

Key formulas to remember

Sample mean: r̄ = (1 ÷ n) × Σ rᵢ; Sum all n returns and divide by n.
Sample variance: s² = Σ (rᵢ − r̄)² ÷ (n − 1); Unbiased estimator of variance. Use when the mean is estimated from the same data.
Population variance: σ² = Σ (rᵢ − μ)² ÷ N; Use when you have the whole population or the true mean μ is known.
Volatility (standard deviation): s = √s²; Same units as the returns.
Square-root-of-time rule: σ(T periods) = σ(1 period) × √T; Holds for i.i.d. returns with constant volatility. Variance scales by T.
Annualizing daily volatility: σ(annual) = σ(daily) × √252; Use the number of trading days given in the question; 252 is common.
Converting down: σ(daily) = σ(annual) ÷ √252; Divide by the square root, not by 252.

How to solve Variance, Volatility and Annualization questions

Use this routine for any question on sample variance, volatility or horizon scaling.

1Read the data and the units. Note whether returns are daily, weekly, monthly or annual, and whether they are in % or decimals.
2Decide whether the question wants sample or population variance. If it says sample, estimate, or gives no true mean, divide by n − 1.
3Compute the mean of the returns.
4Subtract the mean from each return, square each gap and add them up.
5Divide by n − 1 (sample) or n (population) to get variance. Take the square root for volatility.
6If the question asks for another horizon, find T as the ratio of the target horizon to the data horizon and multiply volatility by √T. Scale variance by T instead.
7Check that the answer is sensible: annual volatility should be larger than daily, and the units should match the question.

Quickest way: Shortcut with squared sums

When to use it: When you have a handful of returns and need the sample variance quickly, or the question gives Σr and Σr² directly.

Compute Σr and Σr².
Use s² = [Σr² − n × r̄²] ÷ (n − 1), where r̄ = Σr ÷ n.
Take the square root for volatility.
For scaling, multiply by √T. Memorize √252 ≈ 15.87, √12 ≈ 3.46, √52 ≈ 7.21 and √4 = 2.
For scaling a variance, multiply by T. Do not take a square root.
On the calculator, use the statistics mode and read the sample standard deviation (the n − 1 key), not the population one.

Common mistakes in Variance, Volatility and Annualization

Dividing by n instead of n − 1 for a sample.
Students remember the plain 'average of squares' idea.
Fix: If the mean is estimated from the data, divide by n − 1. Only use n when told it is a population or the true mean is known.
Multiplying daily volatility by 252 to annualize.
Daily returns add up over time, so students scale volatility linearly.
Fix: Variance scales by T, volatility by √T. Multiply daily volatility by √252.
Taking the square root when scaling variance, or forgetting to when scaling volatility.
Variance and volatility are mixed up under time pressure.
Fix: Write which quantity you hold. Variance × T. Volatility × √T.
Mixing percent and decimals.
Returns are given as 1.5% and then entered as 1.5 in some steps and 0.015 in others.
Fix: Convert everything to decimals at the start, or keep everything in % and square the % units consistently.
Applying the square-root rule when returns are autocorrelated or volatility is not constant.
The rule is memorized without its assumptions.
Fix: State the i.i.d. condition. If a question gives autocorrelation, expect the scaled volatility to differ from the simple √T result.
Using the wrong number of periods for T.
Students scale from weekly to annual using 12 or from monthly using 52.
Fix: Count periods per year in the data's own frequency: 252 daily, 52 weekly, 12 monthly, 4 quarterly.

Worked examples

Example 1

A fund had monthly returns over five months of 2%, 4%, −1%, 3% and 1%. Compute the sample standard deviation of monthly returns and annualize it.

Show the solution

Mean = (2 + 4 − 1 + 3 + 1) ÷ 5 = 9 ÷ 5 = 1.8%.
Deviations: 0.2, 2.2, −2.8, 1.2, −0.8.
Squared deviations: 0.04, 4.84, 7.84, 1.44, 0.64. Sum = 14.80.
Sample variance = 14.80 ÷ (5 − 1) = 3.70 (in %²).
Monthly volatility = √3.70 = 1.9235%.
Annualize: 1.9235 × √12 = 1.9235 × 3.4641 = 6.663%.

Answer: Monthly sample standard deviation ≈ 1.92%; annualized volatility ≈ 6.66%.

Example 2

A portfolio has an annual volatility of 20%. Assuming i.i.d. returns and 252 trading days, what is the approximate 10-day volatility?

Show the solution

Daily volatility = 20% ÷ √252 = 20% ÷ 15.8745 = 1.2599%.
Ten-day volatility = daily volatility × √10 = 1.2599% × 3.1623 = 3.984%.
Check directly: 20% × √(10 ÷ 252) = 20% × √0.039683 = 20% × 0.19921 = 3.984%.

Answer: About 3.98%.

Exam tips

Read whether the question says 'sample' or 'population'. This single word decides n or n − 1.
Check the trading-day count given in the question. Use 252 only when nothing else is stated.
Expect options that include the linear scaling error (× T) and the wrong divisor. Use them as a check on your own method.
For conceptual questions, remember the assumptions behind √T: i.i.d. returns, constant volatility, and log returns.
Keep extra decimals until the last step, because answer options are often close together.

Practice questions from Measuring Return, Volatility, and Correlation

Variance, Volatility and Annualization: frequently asked questions

Why is volatility scaled by the square root of time?

If returns are independent with the same variance, the variance of a T-period return is T times the one-period variance. Volatility is the square root of variance, so it scales by √T.

What is the difference between sample and population variance?

Population variance divides by N and uses the true mean. Sample variance divides by n − 1 because the mean is estimated from the data, which makes the estimator unbiased for variance.

How do I annualize daily volatility?

Multiply daily volatility by the square root of the number of trading days in a year, commonly √252. For monthly data use √12, and for weekly data use √52.

When does the square-root-of-time rule not work?

It fails when returns are autocorrelated or when volatility changes over time, as with volatility clustering. Positive autocorrelation makes the rule understate long-horizon risk.