Aug 23

What is mean reversion in financial time series?

by David Harper, CFA, FRM, CIPM

Ashim made a good observation about mean reversion, based on the assigned Linda Allen reading.

The relevant FRM learning objective is: “Explain the implications of mean reversion in returns and return volatility, respectively have on VaR forecasts over long time horizons.”

The key idea refers to the application of the square root rule (variance scales directly with the multiple of time). The square root rule, while mathematically convenient, doesn’t really work in practice because it requires that normally distributed returns are independent and identically distributed (i.i.d.). What I mean is, we use it on the exam, but in practice, when applying the square root rule to scaling delta normal VaR/volatility, we should be sensitive to the likely error introduced.

Allen gives two scenarios that each illustrate “violations” in the use of the square root rule to scale volatility over time:

Mean reversion	Square root rule (SRR)
In returns	SRR will always overstate
In return volatility	If current volatility is less than (<) Long run mean (LRM) volatility,SRR will understate. If current volatility is greater than (>) Long run mean (LRM) volatility,SRR will overstate.

Several types of mean reversion

The real culprit is the ambiguous definition of mean reversion (here is a paper that reviews of no less than six definitions of mean reversion!).

For FRM purposes, three definitions of mean reversion are used:

Mean reversion in the asset dynamics. The price/return tends towards a long-run level; e.g., interest rate reverts to 5%, equity log return reverts to +8%
Mean reversion in variance. Variance reverts toward a long-run level; e.g., volatility reverts to a long-run average of 20%. We can also refer to this as negative autocorrelation, but it's a little trickier. Negative autocorrelation refers to the fact that a high variance is likely to be followed in time by a low variance. The reason it's tricky is due to short/long timeframes: the current volatility may be high relative to the long run mean, but it may be "sticky" or cluster in the short-term (positive autocorrelation) yet, in the longer term it may revert to the long run mean. So, there can be a mix of (short-term) positive and negative autocorrelation on the way being pulled toward the long run mean.
Autoregression in the time series. The current estimate (variance) is informed by (a function of) the previous value; e.g., in GARCH(1,1) and exponentially weighted moving average (EWMA), the variance is a function of the previous variance.

GARCH(1,1) as an example

GARCH(1,1) gives a good example of the differences. Here is a terrific paper (stochastic process toolkit); I was confused initially when the authors referred to GARCH(1,1) as non mean-reverting. They simply meant GARCH(1,1) does not mean revert in the asset dynamics.