FRM Exam Part II · Empirical Properties of Correlation: How Do Correlations Behave in the Real World?

Correlation Basics and Why Correlations Matter for FRM Part 2

Q: What is the difference between correlation and covariance?

Covariance measures co-movement in the units of the two variables, so its size is hard to interpret. Correlation divides covariance by both standard deviations, giving a unit-free number between -1 and +1. Use correlation to compare relationships.

Q: How does correlation affect portfolio VaR?

Lower correlation means more offsetting between positions, so diversified VaR is lower. At a correlation of +1 the diversified VaR equals the sum of stand-alone VaRs. At lower values it is smaller.

Q: What is correlation risk?

It is the risk that correlations change in a way that hurts you, for example rising sharply in a crisis. Hedges and diversification then work less well than your models assumed.

Q: Why do real-world correlations differ from textbook assumptions?

Textbooks often treat correlation as a constant. In practice it changes over time, depends on market conditions, and is estimated with error from limited data.

Updated 11 October 2026 · Fact-checked

Correlation measures the strength and direction of linear co-movement between two variables, scaled from -1 to +1. Covariance is the unscaled version. Correlation sets how much diversification you get in portfolio variance and VaR. To solve questions, compute σ_p² with the covariance term, then take the square root and scale by the z-score.

Understand Correlation Basics and Why Correlations Matter

Covariance tells you whether two returns tend to move together. Positive means they move in the same direction. Negative means they move opposite. Its size depends on the units and on how volatile each asset is, so it is hard to compare across pairs.

Correlation fixes this by dividing covariance by the product of the two standard deviations. The result is always between -1 and +1. At +1 the two move in perfect lockstep (linear). At 0 there is no linear relationship. At -1 they move perfectly opposite. Correlation is a measure of linear dependence only. Zero correlation does not mean independence.

Correlation matters because portfolio risk is not the sum of individual risks. The lower the correlation, the more risk cancels out. With a correlation of +1, portfolio volatility is just the weighted average of volatilities. With anything lower, portfolio volatility is smaller. This is the diversification benefit. In VaR, the same logic applies: undiversified VaR is the sum of stand-alone VaRs, and diversified VaR is lower when correlation is below 1. Hedging also relies on correlation, because the hedge works only as well as the hedge instrument tracks the exposure.

In practice, correlations are not constant. They change over time, they tend to rise in market stress, and they differ across asset classes. Textbook models often assume a fixed correlation estimated from history. If correlation jumps in a crisis, the diversification you counted on shrinks and VaR is understated. This is correlation risk: the risk that correlations move against you.

Also remember that sample correlations are estimates. They carry estimation error and depend on the window and frequency used. A single number can hide a lot.

Key formulas to remember

Covariance: Cov(X, Y) = E[(X − μX)(Y − μY)]; Units are the product of the units of X and Y. Sign shows direction only.
Correlation: ρ(X, Y) = Cov(X, Y) ÷ (σX × σY); Always between -1 and +1. Needs non-zero standard deviations.
Covariance from correlation: Cov(X, Y) = ρ × σX × σY; Use this to move between the two in portfolio questions.
Two-asset portfolio variance: σp² = w1²σ1² + w2²σ2² + 2 w1 w2 ρ σ1 σ2; Portfolio volatility is the square root of this.
Two-asset diversified VaR: VaRp = √(VaR1² + VaR2² + 2ρ × VaR1 × VaR2); Holds for normally distributed returns with zero mean over the horizon. At ρ = 1 it equals VaR1 + VaR2.
Portfolio VaR (normal): VaR = z × σp × Portfolio value; z is about 1.645 at 95% and 2.326 at 99% one-tailed.

How to solve Correlation Basics and Why Correlations Matter questions

Use this method for any question on correlation, covariance or their effect on risk.

1Identify what is given: volatilities, weights, covariance or correlation, and whether the figures are daily or annual.
2If covariance is given, convert it to correlation with ρ = Cov ÷ (σ1σ2), or the reverse, as needed.
3Put everything on the same time horizon before combining. Scale volatility by the square root of time.
4Write the portfolio variance with the cross term 2w1w2ρσ1σ2, or combine stand-alone VaRs with the square-root formula.
5Take the square root, then multiply by the z-score and the portfolio value if VaR is asked.
6Check the bounds: portfolio risk must lie between the ρ = -1 and ρ = +1 cases.
7If the question is conceptual, state how correlation behaves in practice (unstable, rises in stress) and what that does to diversification.

Quickest way: Combine VaRs with the square-root rule

When to use it: When stand-alone VaRs for two positions and their correlation are given and you need diversified VaR.

Square each VaR.
Compute 2 × ρ × VaR1 × VaR2.
Add the three terms and take the square root.
Sanity check: the answer must be below VaR1 + VaR2 when ρ < 1 and above the smaller VaR unless ρ is very negative.
Eliminate options that equal the simple sum unless ρ = 1.

Common mistakes in Correlation Basics and Why Correlations Matter

Treating zero correlation as independence.
The two ideas feel the same.
Fix: Correlation captures only linear dependence. Nonlinear relationships can exist with zero correlation.
Adding stand-alone VaRs to get portfolio VaR.
It is the easy shortcut.
Fix: That is correct only at ρ = 1. Otherwise use the square-root combination.
Forgetting the factor of 2 on the cross term.
Students recall the formula loosely.
Fix: The covariance term appears twice in a two-asset variance, so write 2w1w2ρσ1σ2.
Mixing daily and annual figures.
Data in the question use different horizons.
Fix: Convert everything to one horizon first using √time scaling.
Assuming correlation is stable over time and across regimes.
Textbook models use one fixed number.
Fix: Remember that empirical correlations vary and tend to rise in stress, so diversification benefits shrink when you need them most.
Comparing covariances across pairs to judge strength of relationship.
Covariance looks like a strength measure.
Fix: Covariance depends on volatility. Use correlation to compare strength.

Worked examples

Example 1

A portfolio holds two positions with one-day 99% VaRs of $4 million and $3 million. Returns are jointly normal and the correlation is 0.5. What is the diversified portfolio VaR?

Show the solution

VaR1² = 16 and VaR2² = 9.
Cross term = 2 × 0.5 × 4 × 3 = 12.
Sum = 16 + 9 + 12 = 37.
VaRp = √37 ≈ 6.08.

Answer: About $6.08 million, below the undiversified sum of $7 million.

Example 2

Two assets have annual volatilities of 20% and 10%, and a covariance of 0.009. What is the correlation? What does it imply for a 50/50 portfolio's volatility relative to the average volatility of 15%?

Show the solution

ρ = 0.009 ÷ (0.20 × 0.10) = 0.009 ÷ 0.02 = 0.45.
Variance = 0.25×0.04 + 0.25×0.01 + 2×0.25×0.009.
= 0.0100 + 0.0025 + 0.0045 = 0.0170.
Volatility = √0.0170 ≈ 13.04%.
Compare with 15%: lower because ρ < 1.

Answer: Correlation is 0.45. Portfolio volatility is about 13.04%, below the 15% weighted average, showing diversification.

Exam tips

Expect applied questions where you compute diversified VaR and then explain what happens if correlation rises toward 1.
Know the direction of effects: higher correlation means higher portfolio risk and lower diversification benefit.
Read carefully for covariance versus correlation. Many traps swap one for the other.
For conceptual items, link correlation instability and stress behavior to understated VaR.
Check that horizons and units match before you calculate.

Practice questions from Empirical Properties of Correlation: How Do Correlations Behave in the Real World?

Correlation Basics and Why Correlations Matter: frequently asked questions

What is the difference between correlation and covariance?

Covariance measures co-movement in the units of the two variables, so its size is hard to interpret. Correlation divides covariance by both standard deviations, giving a unit-free number between -1 and +1. Use correlation to compare relationships.

How does correlation affect portfolio VaR?

Lower correlation means more offsetting between positions, so diversified VaR is lower. At a correlation of +1 the diversified VaR equals the sum of stand-alone VaRs. At lower values it is smaller.

What is correlation risk?

It is the risk that correlations change in a way that hurts you, for example rising sharply in a crisis. Hedges and diversification then work less well than your models assumed.

Why do real-world correlations differ from textbook assumptions?

Textbooks often treat correlation as a constant. In practice it changes over time, depends on market conditions, and is estimated with error from limited data.