en.wikipedia.org/wiki/Mean_reversion_(finance)
1 correction found
Quantitatively, it is the standard deviation of average annual returns that declines faster than the inverse of the holding period, implying that the process is not a random walk, but that periods of lower returns are then followed by compensating periods of higher returns, for example in seasonal businesses.
This sentence states the wrong scaling rule for return volatility. Standard deviation over longer horizons scales with the square root of time, so the benchmark is 1/√N, not 1/N.
Full reasoning
The quantitative claim here appears to omit "square root" and therefore states the wrong relationship.
For returns with constant expected returns, the standard finance/statistics result is that annualized standard deviation over an N-year holding period equals the one-year standard deviation divided by (\sqrt{N}) — the usual square-root-of-time rule. John Y. Campbell explains this explicitly in an NBER Reporter article discussing long-horizon stock risk: "the annualized standard deviation over a long holding period (N years) is the standard deviation over one year divided by the square root of N" and mean reversion would imply risk declines more rapidly than the square-root rule would imply.
That directly contradicts the Wikipedia sentence's wording that standard deviation declines faster than "the inverse of the holding period" (that is, faster than (1/N)). A (1/N) scaling is not the standard rule for standard deviation; it is consistent with a confusion with variance or a missing phrase such as "square root of".
So this is not just a stylistic issue: as written, the sentence gives the wrong mathematical benchmark for how return volatility scales with horizon.
2 sources
- Strategic Asset Allocation: Portfolio Choice for Long-Term Investors | NBER
"With constant expected returns, the annualized standard deviation over a long holding period (N years) is the standard deviation over one year divided by the square root of N... Under these circumstances, stock market risk declines more rapidly with the investment horizon than the square-root rule would imply."
- Standard Deviation | Morningstar
"Morningstar annualizes the monthly standard deviation by multiplying it by the square root of 12." This reflects the standard square-root-of-time scaling for standard deviation, not inverse-time scaling.