Use a standard regression to work out a mean reversion rate, and derive the matching autocorrelation.
The idea
Mean reversion in correlation has a speed a, a long-run level μ and a time step dt. Each period, the change in correlation is modelled as Change = a × (μ − C(t − 1)) × dt + noise. To estimate it, regress the change in correlation on the lagged level, which gives Fit = α + β × C(t − 1). The slope β equals minus a × dt, so with a unit time step the speed of reversion is Speed = −β. The long-run mean of the correlation series is the intercept divided by the speed, Mean = α ÷ a. The matching autocorrelation between successive observations is Autocorr = 1 − a × dt. For a unit time step this is 1 − a, which is also the slope if you regress the level on its own lag. The expected gap from the mean shrinks by a factor of 1 − a each period. Common errors are dropping the minus sign on the slope and reporting the intercept as the mean. Exam point: a slope near zero means slow reversion and autocorrelation near one.
Formulas
Mean reversion rate from a regression of the change in correlation on its lagged level
a = −β
- β
- the slope of the change in correlation regressed on its lagged level
Holds for a unit time step; in general the slope is −a × dt. Dropping the minus sign, or reading the intercept as the rate, are the usual errors.
Long-run mean of a mean-reverting correlation
μ = α ÷ a
- α
- the intercept of the regression of the change on the lagged level
- a
- the mean reversion rate, minus the slope
The intercept alone is not the mean: it must be divided by a. If the level is regressed on its lag instead, a is 1 minus that slope.
Autocorrelation implied by a mean reversion rate
autocorrelation = 1 − a
- a
- the mean reversion rate per period
With a unit time step. It equals the slope of the level regressed on its own lag, so a high autocorrelation means slow reversion.
A practice question
Which regression is used to estimate the mean reversion rate of a correlation series?
- Regress the correlation level C(t) on time t; the slope is the mean reversion rate.Why not: Fits a time trend, which captures drift rather than the pull back towards a long-run level.
- Regress the change in correlation on the change in equity volatility; the slope is the mean reversion rate.Why not: Mixes up mean reversion with the link between correlation and volatility, which is a different relationship.
- Regress C(t) - C(t-1) on C(t-1); with a unit time step, a is minus the slope. - correct
- Regress the lagged level C(t-1) on the change C(t) - C(t-1); the intercept is the mean reversion rate.Why not: Reverses dependent and independent variables and takes the intercept, which relates to the long-run mean, as the rate.
The model is C(t) - C(t-1) = a(mu - C(t-1)) dt + noise, i.e. change = alpha + beta C(t-1) with beta = -a dt. So a is minus the slope for dt = 1.
A harder one
A regression of the change in correlation on its lagged level gives: change = 0.0126 - 0.042 x C(t-1), with a unit time step. The implied mean reversion rate a and long-run mean correlation mu are:
- a = 0.0126 and mu = 0.30Why not: Takes the intercept as the speed of reversion; the speed is minus the slope.
- a = 0.042 and mu = 0.30 - correct
- a = 0.042 and mu = 0.0126Why not: Reports the intercept as the long-run mean instead of dividing it by a.
- a = -0.042 and mu = -0.30Why not: Forgets that a is minus the slope, so the signs of a and of mu = alpha / beta come out wrong.
a = -beta = 0.042 and mu = alpha / a = 0.0126 / 0.042 = 0.30. The matching autocorrelation is 1 - 0.042 = 0.958.
The hardest tier
An analyst regresses the level of an average correlation on its own lag and obtains C(t) = 0.021 + 0.93 x C(t-1), using a unit time step. The implied mean reversion rate a and long-run mean correlation mu are:
- a = 0.93 and mu = 0.30Why not: Reads the slope on the lag, which is the autocorrelation 1 - a, as the reversion rate itself.
- a = 0.07 and mu = 0.30 - correct
- a = 0.07 and mu = 0.021Why not: Reports the intercept as the long-run mean without dividing by a.
- a = 0.07 and mu = 0.023Why not: Divides the intercept by the slope 0.93 instead of by the reversion rate a = 0.07.
The level regression C(t) = a mu + (1 - a) C(t-1) gives 1 - a = 0.93, so a = 0.07, and mu = 0.021 / 0.07 = 0.30. The autocorrelation is 0.93.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
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