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

Correlation Mean Reversion in Equity Markets

Updated 11 October 2026 · Fact-checked

Mean reversion means equity correlations drift back toward a long-run average after moving away from it. It is modeled with an autoregressive process: the change in correlation is a = −rate × (current correlation − long-run mean) plus noise. A positive reversion rate pulls the correlation back toward the mean each period.

Understand Correlation Behavior in Equities: Mean Reversion

Correlation between two equities is not fixed. It moves from day to day and month to month. The empirical finding is that it does not wander off forever. It tends to be pulled back toward a long-run average. This is mean reversion.

Think of a stretched spring. When correlation is far above its long-run level, the next move is more likely down. When it is far below, the next move is more likely up. The pull is stronger the further you are from the mean.

The standard model is a first-order autoregressive (AR) process. In discrete form, the change in correlation over one step is: ρ(t) − ρ(t−1) = a × (μ − ρ(t−1)) + ε(t). Here μ is the long-run mean, a is the reversion rate (between 0 and 1 for a stable process), and ε is a random error. Equivalent form: ρ(t) = a·μ + (1 − a)·ρ(t−1) + ε(t).

You estimate a by regressing the change in correlation on the lagged correlation level. If the regression is Δρ = α + β·ρ(t−1) + ε, then a = −β and μ = α ÷ a. A negative slope means mean reversion. A slope of zero means no reversion, so correlation behaves like a random walk.

Why it matters: if correlations revert, a very high or very low current correlation is a poor forecast of the far future. Risk models that freeze today's correlation can misstate diversification benefit over longer horizons. Keep in mind that reversion is a tendency, not a guarantee. Correlations can stay high for long periods in crises.

Key formulas to remember

Discrete mean-reverting correlation
ρ(t) − ρ(t−1) = a × (μ − ρ(t−1)) + ε(t)
a is the reversion rate per period, μ the long-run mean, ε a random error.
Expected next-period correlation
E[ρ(t)] = ρ(t−1) + a × (μ − ρ(t−1))
Drop the error term, since its expected value is zero.
Regression form
Δρ = α + β × ρ(t−1) + ε
Estimate by regression of correlation changes on the lagged level.
Recovering parameters
a = −β ; μ = α ÷ a
Mean reversion requires β < 0. Equivalent: μ = −α ÷ β.
Gap after n periods (no shocks)
(ρ(n) − μ) = (1 − a)^n × (ρ(0) − μ)
The distance from the mean shrinks by factor (1 − a) each period.

How to solve Correlation Behavior in Equities: Mean Reversion questions

Use this method for any question on equity correlation mean reversion, whether it asks for a forecast, a rate or an interpretation.

  1. 1Identify what is given: current correlation, long-run mean, reversion rate, or regression output.
  2. 2Check the time unit of the reversion rate (daily, monthly, annual) and match the horizon.
  3. 3If given regression output Δρ = α + β·ρ(t−1), convert: a = −β and μ = α ÷ a.
  4. 4Confirm mean reversion: a must be positive, so the slope β must be negative.
  5. 5For a one-period forecast, compute ρ(t−1) + a × (μ − ρ(t−1)).
  6. 6For several periods, apply (1 − a)^n to the gap from the mean, then add μ.
  7. 7Interpret: state whether correlation is above or below the mean and which way it is expected to move.
  8. 8Note that the forecast is an expected value and actual correlation includes random shocks.

Quickest way: Gap-shrink shortcut

When to use it: Use when you are given ρ, μ and a, and need an expected future correlation.

  1. Compute the gap: ρ − μ.
  2. Multiply the gap by (1 − a) for each period ahead.
  3. Add μ back to get the expected correlation.
  4. Sanity check: the answer must lie between ρ and μ when 0 < a < 1.

Common mistakes in Correlation Behavior in Equities: Mean Reversion

  • Treating the regression slope as the reversion rate with its sign unchanged.

    The slope β is negative for mean reversion, so students quote a negative rate.

    Fix: Use a = −β. A negative slope gives a positive reversion rate.

  • Computing the long-run mean as α alone.

    Students forget the intercept is a × μ, not μ.

    Fix: Use μ = α ÷ a, which equals −α ÷ β.

  • Mixing time units, such as a monthly rate over an annual horizon.

    The question states the rate and horizon in different units.

    Fix: Convert first, or apply the monthly factor (1 − a) for the correct number of months.

  • Subtracting a instead of multiplying the gap by (1 − a).

    Students confuse the size of the pull with a fixed step.

    Fix: The correction is a proportion of the gap, so the gap shrinks by (1 − a) each period.

  • Claiming correlation will return to the mean exactly or quickly.

    Reversion is read as a guarantee.

    Fix: State it as an expected tendency. Random shocks and crisis regimes can keep correlation away from the mean.

Worked examples

Example 1

The long-run mean equity correlation is 0.40. The current correlation is 0.70. The monthly reversion rate is a = 0.20. What is the expected correlation next month, and in three months?

Show the solution
  1. Gap now = 0.70 − 0.40 = 0.30.
  2. Next month: 0.70 + 0.20 × (0.40 − 0.70) = 0.70 − 0.06 = 0.64.
  3. Three months: gap = (1 − 0.20)^3 × 0.30 = 0.512 × 0.30 = 0.1536.
  4. Expected correlation = 0.40 + 0.1536 = 0.5536.

Answer: Expected correlation is 0.64 next month and about 0.554 in three months.

Example 2

A regression of monthly changes in average equity correlation on the previous month's level gives Δρ = 0.12 − 0.30 × ρ(t−1). Find the reversion rate and long-run mean, then the expected correlation next month if the current level is 0.20.

Show the solution
  1. Slope β = −0.30, so a = 0.30.
  2. Intercept α = 0.12, so μ = 0.12 ÷ 0.30 = 0.40.
  3. Expected change = 0.12 − 0.30 × 0.20 = 0.12 − 0.06 = 0.06.
  4. Expected next correlation = 0.20 + 0.06 = 0.26.

Answer: Reversion rate is 0.30 per month, long-run mean is 0.40, and the expected correlation next month is 0.26.

Exam tips

  • Questions often hand you regression output. Convert slope and intercept to a and μ before anything else.
  • A negative slope on the lagged level is the signature of mean reversion. Use it to eliminate options quickly.
  • Expected values must sit between the current level and the mean when 0 < a < 1. Use this to check your answer.
  • Watch for interpretation options that say reversion is certain or immediate. These are usually wrong.
  • Keep the time unit consistent between the rate and the horizon.

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

Correlation Behavior in Equities: Mean Reversion: frequently asked questions

What does mean reversion in correlation mean?

It means correlation tends to move back toward its long-run average after rising or falling away from it. High correlations tend to fall and low ones tend to rise. It is a statistical tendency, not a certainty.

How do I calculate the mean reversion rate of correlation?

Regress the change in correlation on the lagged correlation level. The reversion rate is the negative of the slope. The long-run mean is the intercept divided by the reversion rate.

What is the autoregressive model for correlation?

It is a model where this period's correlation depends on the previous period's. In the mean-reverting form, the change equals a × (μ − previous correlation) plus a random error. It is the standard way to capture reversion.

Does mean reversion mean correlations are stable in crises?

No. Correlations can stay elevated for long stretches in stressed markets. Mean reversion describes the average pull over time, so models should not assume correlations will quickly return to normal.