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

Covariance, Correlation and Beta for FRM Part I

Updated 11 October 2026 · Fact-checked

Covariance measures how two returns move together, in units that depend on the data. Correlation divides covariance by both standard deviations, giving a unit-free number from -1 to +1. Beta divides covariance with the market by market variance. Estimate each from data using sample formulas with n - 1.

Understand Covariance, Correlation and Beta

Covariance tells you whether two returns tend to be above or below their averages at the same time. If both are above average together, the product of deviations is positive. If one is high when the other is low, it is negative. Covariance is the average of these products.

Covariance has a weakness: its size depends on the units. A covariance of 0.0030 means little on its own. Correlation fixes this. You divide covariance by the product of the two standard deviations. The result always lies between -1 and +1. A value of +1 means a perfect positive linear relationship. A value of -1 means a perfect negative one. A value near 0 means little linear relationship.

In practice you estimate both from a sample of returns. The sample covariance sums the products of deviations from the sample means and divides by n - 1. The sample correlation uses the same n - 1 in the covariance and in the standard deviations, so the n - 1 terms cancel.

Beta links these ideas to the market. Beta of an asset equals its covariance with the market return divided by the variance of the market return. It also equals correlation × (asset volatility ÷ market volatility). Beta is the slope of a regression of asset returns on market returns. A beta of 1.2 means the asset moves about 1.2% for each 1% market move, on average.

Correlation measures only linear dependence. Zero correlation does not mean independence. Correlation also changes over time and tends to rise in crises, which matters for diversification.

Key formulas to remember

Population covariance
Cov(X, Y) = E[(X − μX)(Y − μY)] = E[XY] − E[X]E[Y]
Use the second form when you are given expected values of products.
Sample covariance
s(XY) = Σ (Xi − X̄)(Yi − Ȳ) ÷ (n − 1)
Divide by n − 1 for an unbiased estimate. Use n only if told the data is the whole population.
Correlation
ρ = Cov(X, Y) ÷ (σX × σY)
Always between −1 and +1. Has no units.
Covariance from correlation
Cov(X, Y) = ρ × σX × σY
Use it to rebuild covariance for portfolio variance.
Beta
β = Cov(Ri, RM) ÷ Var(RM) = ρ × σi ÷ σM
Slope of the regression of asset returns on market returns.
Variance of a sum
Var(X + Y) = σX² + σY² + 2Cov(X, Y)
For weights a and b: Var(aX + bY) = a²σX² + b²σY² + 2ab·Cov(X, Y).
Scaling property
Cov(aX, bY) = ab·Cov(X, Y); Corr is unchanged if a and b have the same sign
Correlation changes sign if exactly one of a, b is negative.

How to solve Covariance, Correlation and Beta questions

Use this method for any question on covariance, correlation or beta.

  1. 1Write down what you are given: covariance, correlation, standard deviations or variances, or raw return data.
  2. 2Identify what is asked. Decide if you need covariance, correlation, beta or a portfolio variance.
  3. 3If you have raw data, compute the sample means first.
  4. 4Compute the deviations from the means, multiply them in pairs and sum them.
  5. 5Divide by n − 1 for sample covariance. For sample variances, also divide by n − 1.
  6. 6Convert to correlation by dividing by the two standard deviations, or to beta by dividing by the variance of the market.
  7. 7Check that correlation is between −1 and +1 and that the sign of beta matches the sign of covariance.
  8. 8Watch for variance versus standard deviation. Square or take roots only when needed.

Quickest way: Shortcut with deviation sums

When to use it: Use when you are given a small set of paired returns and must find sample covariance, correlation or beta.

  1. Compute both means.
  2. Build three columns of deviations: X dev × Y dev, X dev squared, Y dev squared, and add each column.
  3. Correlation = Σ(XY dev) ÷ √(Σ X dev² × Σ Y dev²). The n − 1 cancels, so skip it.
  4. Beta of Y on X = Σ(XY dev) ÷ Σ(X dev²). Again n − 1 cancels.
  5. Only divide by n − 1 when the question asks for covariance or variance itself.

Common mistakes in Covariance, Correlation and Beta

  • Dividing by n instead of n − 1 for sample covariance

    Students recall the population formula and apply it to a sample.

    Fix: If the data is a sample of returns, use n − 1. Use n only when told it is the full population.

  • Treating zero correlation as independence

    Correlation is described as a measure of dependence.

    Fix: Correlation captures only linear dependence. Zero correlation does not imply independence, except in special cases such as jointly normal variables.

  • Mixing up variance and standard deviation in the correlation formula

    Questions give variances, and students divide by them directly.

    Fix: Take square roots first. ρ = Cov ÷ (σX × σY), not Cov ÷ (σX² × σY²).

  • Computing beta as covariance divided by the asset's variance

    Students divide by the wrong variance.

    Fix: Beta of asset i uses the market variance in the denominator: Cov(Ri, RM) ÷ Var(RM).

  • Forgetting the 2 × covariance term in portfolio variance

    Students add variances only, as if assets were uncorrelated.

    Fix: Always include 2ab·Cov(X, Y), or 2ab·ρ·σX·σY.

  • Thinking correlation changes when units change

    Covariance changes with scaling, so students assume correlation does too.

    Fix: Correlation is unit-free. Rescaling by positive constants leaves it unchanged.

Worked examples

Example 1

Monthly returns on asset X are 2%, 4%, 6% and on asset Y are 1%, 5%, 9%. Find the sample covariance and the sample correlation.

Show the solution
  1. Means: X̄ = (2 + 4 + 6) ÷ 3 = 4%. Ȳ = (1 + 5 + 9) ÷ 3 = 5%.
  2. X deviations: −2, 0, +2. Y deviations: −4, 0, +4.
  3. Products: (−2)(−4) = 8; 0; (2)(4) = 8. Sum = 16.
  4. Sample covariance = 16 ÷ (3 − 1) = 8 (in %², i.e. 0.0008).
  5. Σ X dev² = 4 + 0 + 4 = 8. Variance X = 8 ÷ 2 = 4, so σX = 2%.
  6. Σ Y dev² = 16 + 0 + 16 = 32. Variance Y = 32 ÷ 2 = 16, so σY = 4%.
  7. Correlation = 8 ÷ (2 × 4) = 1.0.

Answer: Sample covariance = 8 (%²), or 0.0008. Correlation = +1.0, a perfect positive linear relationship.

Example 2

A stock has volatility of 30% a year. The market has volatility of 20% a year. The correlation between them is 0.60. Find the stock's beta and the covariance between them.

Show the solution
  1. Covariance = ρ × σi × σM = 0.60 × 0.30 × 0.20 = 0.036.
  2. Market variance = 0.20² = 0.04.
  3. Beta = Cov ÷ Var(M) = 0.036 ÷ 0.04 = 0.90.
  4. Check: ρ × σi ÷ σM = 0.60 × 0.30 ÷ 0.20 = 0.90.

Answer: Covariance = 0.036 and beta = 0.90.

Exam tips

  • Questions often give correlation and volatilities and ask for covariance or beta. Memorise Cov = ρσXσY and β = ρσi ÷ σM.
  • In a data question, use the deviation-sum shortcut. The n − 1 cancels for correlation and beta, which saves time.
  • Check which variance a question gives: if it says variance, do not square it again.
  • Expect conceptual items on correlation limits: it is linear only, unstable over time and tends to rise in stress.
  • Portfolio variance questions need the covariance term. Plug in weights carefully and keep returns in decimals.

Practice questions from Measuring Return, Volatility, and Correlation

Covariance, Correlation and Beta in other exams

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

Covariance, Correlation and Beta: frequently asked questions

What is the difference between covariance and correlation?

Covariance shows the direction of co-movement but its size depends on the units of the data. Correlation is covariance divided by both standard deviations, so it is unit-free and lies between −1 and +1. Use correlation to compare strength across pairs of assets.

How do I calculate correlation between two assets?

Compute the sample covariance, then divide by the product of the two sample standard deviations. With raw data, you can use the sum of deviation products divided by the square root of the product of the sums of squared deviations. Both routes give the same answer.

Should I divide by n or n − 1 for sample covariance?

Divide by n − 1 when the returns are a sample, which is the usual case. This gives an unbiased estimate. Use n only when the question states the data is the whole population.

How is beta related to correlation?

Beta equals correlation times the ratio of asset volatility to market volatility. It is also covariance with the market divided by market variance. A highly volatile asset can have a high beta even with moderate correlation.