Lesson 10 of 25 · 6 min
Why a matrix estimated in a calm market misleads
The same nine holdings carried 15.41 percent volatility on one window's matrix and 23.45 percent on another's.
A matrix is an estimate from a window, and a window chosen for being long is usually a window chosen for being calm. Most months are ordinary months, so any sample large enough to estimate a matrix is dominated by them.
Nine equally weighted holdings, each with 26 percent annualised volatility. Estimated over a long calm window the average pairwise correlation comes out at 0.27. The bracket is one-ninth plus eight-ninths of 0.27 = 0.3511, so portfolio volatility is 26 times the square root of that, 15.41 percent, and the effective bets figure is 9 divided by 3.16 = 2.85.
Estimated over a drawdown window the same nine holdings show an average correlation of 0.79. The bracket becomes 0.8133, portfolio volatility becomes 23.45 percent, and effective bets fall to 9 divided by 7.32 = 1.23. Same holdings, same weights, same single-name volatilities, 8.04 points of portfolio volatility apart, and two thirds of the independent positions gone.
The error is asymmetric in a way that matters. The calm window has more observations, so by every ordinary standard it is the better estimate and it is the one that gets used; the drawdown window is short, noisy and easy to dismiss. So the figure carrying more confidence is systematically the figure that understates risk, by 8 points out of 23.
A calm-window matrix is not a wrong description of the market. It is a correct description of the market you are not worried about.
Key terms
- Estimation window
- The specific stretch of history a statistic was computed over. Two windows on the same holdings routinely give different matrices.
- Regime
- A stretch of time over which the market's behaviour is internally consistent and different from its behaviour in another stretch. Correlations are roughly stable within one and jump between them.
- Sample dominance
- The way a long window's result is set by whichever regime occupied most of it. Calm months outnumber stressed ones, so calm behaviour sets the estimate.
Worked example
Nine holdings, two matrices
Nine equally weighted holdings, each at 26% annualised volatility. Average pairwise correlation 0.27 on a long calm window and 0.79 on a drawdown window. Illustrative figures.
- Calm-window bracket
- 0.3511
- Calm-window portfolio volatility
- 15.41%
- Calm-window effective bets
- 2.85
- Drawdown-window bracket
- 0.8133
- Drawdown-window portfolio volatility
- 23.45%
- Drawdown-window effective bets
- 1.23
1/9 + (8/9 x 0.27).
26 x square root of 0.3511.
9 / 3.16.
1/9 + (8/9 x 0.79).
26 x square root of 0.8133.
9 / 7.32.
A position sized against 15.41 percent volatility behaves like 23.45 percent when the matrix is re-estimated, and the nine holdings never changed.
Check yourself
Sixteen equally weighted holdings at 22% volatility each show a 0.24 average correlation on a calm window and 0.71 on a stressed one. What are the two portfolio volatilities, and the two effective-bets figures?
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Educational only. Nothing in this lesson is investment, legal or tax advice, or a recommendation to enter into any transaction. Figures are illustrative and computed from the inputs stated in the lesson.