FRM Exam Part I · Measuring and Monitoring Volatility

Volatility Basics and Return Measurement for FRM Part I

Updated 11 October 2026 · Fact-checked

Volatility is the standard deviation of an asset's returns over a chosen period. You measure returns as simple or log returns, compute the sample standard deviation, then scale it to another horizon by multiplying by the square root of time, assuming independent, identically distributed returns.

Understand Volatility Basics and Return Measurement

Volatility measures how widely returns spread around their average. In FRM it is the standard deviation of returns, not the variance. Variance is σ². Volatility is σ, in the same units as the return, so it is easier to read and compare.

Before you measure volatility you need a return. A simple return is the percentage change in price: R = (P₁ − P₀) ÷ P₀, plus any income. A log (continuously compounded) return is r = ln(P₁ ÷ P₀). For small moves the two are close. For large moves they differ, and log returns are slightly smaller than simple returns.

Log returns have a key advantage: they add across time. A two-day log return is the sum of the two daily log returns. Simple returns do not add; they compound. Simple returns have a different advantage: they add across assets in a portfolio, since a portfolio return is the weighted average of asset simple returns. Simple returns cannot fall below −100%, because a price cannot be negative. Log returns can take any real value, from −∞ to +∞, and they always imply a positive price, since P₁ = P₀·e^r.

To change the horizon, use the square root of time rule. If daily returns are independent and identically distributed with the same variance, variance grows in proportion to time, so volatility grows with √time. Daily volatility times √252 gives annual volatility. Market convention uses about 252 trading days, but always use the number the question gives.

The rule is an approximation. Real returns show stylized facts: fat tails (excess kurtosis), volatility clustering (large moves follow large moves, small follow small), and volatility that changes over time. Clustering means returns are not truly i.i.d. in their squared values, which is why models such as EWMA and GARCH exist.

Key formulas to remember

Simple return: R = (P₁ − P₀ + D) ÷ P₀; D is any income received in the period. Without income, R = P₁ ÷ P₀ − 1.
Log return: r = ln(P₁ ÷ P₀) = ln(1 + R); Also called continuously compounded return. Inverse: R = e^r − 1.
Multi-period log return: r(0,T) = r₁ + r₂ + … + r_T; Log returns add over time. Simple returns compound: 1 + R = (1 + R₁)(1 + R₂)…
Sample variance: s² = Σ(rᵢ − r̄)² ÷ (n − 1); Volatility is s = √s². Many risk-model questions assume a zero mean and divide by n instead; follow the question.
Square root of time rule: σ(T periods) = σ(1 period) × √T; Valid for i.i.d. returns. Variance scales with T, volatility with √T.
Annualizing daily volatility: σ(annual) = σ(daily) × √252; Use the trading-day count given. Monthly to annual uses √12; weekly uses √52.
Portfolio simple return: R_p = Σ wᵢRᵢ; Holds for simple returns across assets in one period, not for log returns.

How to solve Volatility Basics and Return Measurement questions

Use this order for any question on volatility and return measurement.

1Identify what is given: prices or returns, and whether they are simple or log returns.
2Convert if needed. Use r = ln(P₁ ÷ P₀) for log returns and R = P₁ ÷ P₀ − 1 for simple returns.
3If the question spans several periods, add log returns or multiply (1 + R) terms for simple returns.
4Compute the standard deviation. If given variance, take the square root. If given a return series, use n − 1 unless told to assume zero mean.
5Check the time unit of the volatility you have: daily, weekly, monthly or annual.
6Scale with √T. Multiply by √T to go to a longer horizon; divide to go to a shorter one. Scale variance by T, not √T.
7State the result in the right unit (percent per year, per day) and check it is reasonable.
8If the question asks about assumptions, say i.i.d. returns, and mention that clustering and fat tails weaken the rule.

Quickest way: Fast scaling and return check

When to use it: Use when the question gives a volatility in one unit and asks for another, or asks which return type to use.

Write the required factor first: √252 for daily to annual, √12 for monthly to annual, √10 for 1-day to 10-day.
Multiply or divide the volatility by that factor. Do not touch the mean unless asked.
For return type, remember: log returns add over time, simple returns add across assets.
For small returns, treat log and simple as nearly equal; for large moves, log return is smaller than simple return when positive.
Eliminate options that scale volatility by T instead of √T.

Common mistakes in Volatility Basics and Return Measurement

Scaling volatility by T instead of √T.
You remember that variance grows linearly with time and apply it to the wrong quantity.
Fix: Variance scales by T. Volatility scales by √T. Write which one you are scaling.
Adding simple returns across time to get a multi-period return.
Adding feels natural and works roughly for small returns.
Fix: Compound simple returns with (1 + R₁)(1 + R₂) − 1, or add log returns.
Averaging log returns across assets to get a portfolio return.
You carry the 'log returns add' rule to the wrong dimension.
Fix: Log returns add across time. Portfolio returns across assets use weighted simple returns.
Using 365 days to annualize daily volatility.
A year has 365 calendar days, so it seems right.
Fix: Markets trade about 252 days. Use 252 unless the question states another number.
Forgetting to take the square root of variance.
Questions give variance or squared returns and you stop early.
Fix: Volatility is the standard deviation. Always check whether your number is σ or σ².
Treating the square root of time rule as exact for real markets.
The rule is taught as a clean formula.
Fix: It needs i.i.d. returns. Volatility clustering, autocorrelation and fat tails make it approximate.

Worked examples

Example 1

A stock's daily return volatility is 1.5%. Assuming 252 trading days and i.i.d. returns, what is the annualized volatility?

Show the solution

Formula: σ(annual) = σ(daily) × √252.
√252 ≈ 15.8745.
1.5% × 15.8745 = 23.81%.

Answer: About 23.8% per year.

Example 2

A stock rises from ₹200 to ₹230 in one month. Calculate the simple and log returns. Then state the log return if the next month's log return is −4.00%.

Show the solution

Simple return: 230 ÷ 200 − 1 = 0.15, or 15.00%.
Log return: ln(230 ÷ 200) = ln(1.15) ≈ 0.13976, or 13.98%.
Two-month log return = 13.98% + (−4.00%) = 9.98%.
Check: the log return is below the simple return, as expected for a large positive move.

Answer: Simple return 15.00%, log return about 13.98%, two-month log return about 9.98%.

Exam tips

Read whether the question gives variance or volatility. Many wrong options come from missing the square root.
Keep the 252, 52 and 12 factors in mind, but use the day count the question states.
Know why each return type is used: log returns for time aggregation, simple returns for portfolio aggregation.
Questions on stylized facts expect you to name volatility clustering, fat tails and the failure of the i.i.d. assumption.
A financial calculator has ln and √ keys. Use them to avoid rounding errors on log returns.

Practice questions from Measuring and Monitoring Volatility

Volatility Basics and Return Measurement: frequently asked questions

What is volatility in FRM Part I?

Volatility is the standard deviation of an asset's returns over a period. It is usually quoted per year in percent. It is the square root of the variance.

How do I annualize daily volatility?

Multiply the daily volatility by √252, or by the trading-day count given in the question. This assumes i.i.d. daily returns. For monthly volatility, multiply by √12.

What is the difference between simple and log returns?

A simple return is the percentage price change. A log return is ln(P₁ ÷ P₀). Log returns add over time, while simple returns add across assets in a portfolio. For small moves they are nearly equal.

What is volatility clustering?

It is the tendency for large price moves to follow large moves and small moves to follow small ones. It shows that volatility changes over time. This is why constant-volatility assumptions and the square root of time rule are only approximations.