FRM Part I · FRM Exam Part I

Measuring Return, Volatility, and Correlation for FRM Part I

This chapter teaches you how to turn price data into return, risk and dependence numbers. You compute simple and log returns, estimate variance and volatility, annualize with the square-root-of-time rule, test normality with skewness and kurtosis, and measure covariance, correlation, beta and portfolio variance. Practise each formula with numbers.

What this chapter covers

This chapter is the statistical toolkit for market data. It starts with how you define a return (simple or log), then moves to how you measure the spread of returns (variance and volatility), and how you scale that spread across time. It also covers implied volatility, which is read from option prices rather than from history, and the VIX as a market gauge of it.

The second half looks at the shape of return distributions and at how two or more assets move together. You test whether returns look normal using skewness and kurtosis. You then measure co-movement with covariance, correlation and beta, learn where correlation misleads you, and finish by combining everything into the variance of a sum and of a portfolio.

The chapter feeds the rest of the paper. Value-at-risk, portfolio theory, CAPM, option pricing, hedge ratios and the models in Valuation and Risk Models all assume you can compute and interpret these numbers fast. Treat it as foundation material, not a standalone block.

Questions from this chapter are usually short and numerical, so they are among the most reliable marks on a 100-question, 4-hour paper if you know the formulas cold. The same ideas also sit inside questions in other topics, such as VaR, portfolio risk and hedging, so weak basics cost you marks in several places. Speed matters too: with about 2.4 minutes per question on average, you cannot afford to rederive formulas in the exam.

Measuring Return, Volatility, and Correlation: topics in the order to study them

1Simple and Log ReturnsEvery later calculation starts from a return, so you must know which type is used and how they convert.
2Variance, Volatility and AnnualizationOnce you have returns, you measure their dispersion and scale it to a yearly figure.
3Implied Volatility and VIXIt builds on volatility, contrasting the historical estimate with the market-implied one.
4Normality, Skewness and Kurtosis of ReturnsYou need volatility first to standardize returns and judge their shape against the normal.
5Covariance, Correlation and BetaThis moves from one asset to two, using the variance ideas you already know.
6Correlation Pitfalls and Dependence MeasuresYou can only see the limits of correlation after you know how it is calculated.
7Moments of Sums and Portfolio VarianceIt pulls variance, covariance and correlation together, so it comes last.

How to prepare Measuring Return, Volatility, and Correlation

Aim for fluency with formulas and a clear sense of what each number means. Work with small data sets by hand, then with your calculator.

Learn the core formulas on one page: simple return, log return, sample variance, annualization, covariance, correlation, beta and portfolio variance.
Work each formula with a tiny data set of four or five returns until you can finish it in under two minutes.
Learn your calculator's statistics mode so you can get mean, standard deviation and correlation quickly. Check whether it gives sample or population values.
For each concept, write one sentence on what it tells you and one on when it fails, such as correlation missing nonlinear dependence.
Do mixed practice questions that combine topics, such as annualizing a volatility and then using it in a portfolio variance.
Review every wrong answer and label the cause: formula, arithmetic, units or interpretation. Redo those questions after a few days.

Common mistakes in Measuring Return, Volatility, and Correlation

Reporting variance when the question asks for volatility, or the reverse.
Fix: Underline what the question asks for and take the square root last if it asks for volatility.
Annualizing variance by √T instead of volatility.
Fix: Variance scales by T and volatility scales by √T, assuming independent and identically distributed returns.
Using n instead of n − 1 for a sample, or the reverse.
Fix: Check whether the data is a sample or the whole population, and confirm your calculator's mode before computing.
Treating zero correlation as independence.
Fix: Remember that correlation measures linear dependence only, and a nonlinear relationship can have zero correlation.
Mixing up kurtosis and excess kurtosis.
Fix: Read the question carefully and subtract 3 when excess kurtosis is wanted.
Forgetting the covariance term, or its factor of 2, in portfolio variance.
Fix: Write the full expansion first, then fill in the weights, volatilities and correlation.

Last-day revision: Measuring Return, Volatility, and Correlation

Simple return = (P₁ + income − P₀) ÷ P₀; log return = ln(P₁ ÷ P₀).
Log returns add across time; simple returns compound by multiplication.
Sample variance divides the sum of squared deviations by n − 1.
Volatility is the standard deviation of returns, not the variance.
Under the square-root-of-time rule with independent returns, annual volatility = daily volatility × √(trading days).
Implied volatility is backed out of option prices; VIX measures expected 30-day S&P 500 volatility.
Normal distribution: skewness 0 and kurtosis 3, so excess kurtosis 0.
Fat tails mean kurtosis above 3; negative skew means a longer left tail.
Covariance = Σ(x − x̄)(y − ȳ) ÷ (n − 1) for a sample; correlation = covariance ÷ (σₓσᵧ).
Correlation lies between −1 and +1; beta = covariance(asset, market) ÷ variance(market).
Zero correlation does not imply independence; correlation captures only linear dependence.
Two-asset variance = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂.

Measuring Return, Volatility, and Correlation practice questions

Measuring Return, Volatility, and Correlation in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Measuring Return, Volatility, and Correlation: frequently asked questions

Do I need to memorize formulas for this chapter?

Yes. The formulas here are short and used often, and you will not have time to derive them in the exam. Memorize them and practise them with numbers until they feel automatic.

Should I use simple or log returns in the exam?

Use whichever the question specifies. Log returns are convenient for adding returns over time, while simple returns are needed for portfolio weights across assets at a single date.

How does this chapter link to value-at-risk?

VaR depends on volatility, correlation and the shape of the return distribution. If you can annualize or scale volatility and compute portfolio variance, parametric VaR calculations become much easier.

Is a financial calculator allowed, and how does it help?

GARP permits approved calculators, so check its current rules before exam day. Statistics mode helps you find means, standard deviations and correlations quickly, but you must know whether it gives sample or population values.