Downside correlation, measured on its own

The correlation that governs a collateral outcome is the one in the tail, and the tail is where the data is thinnest. Both halves of that sentence are the lesson, and the second half is the one usually left out.

Lesson 13's drawdown figure of 0.7599 came from ten months. Put an interval around it. For a correlation of 0.76 estimated on ten observations, the 95 percent interval runs from 0.25 to 0.94 — a width of 0.69, more than a third of the entire range a correlation can occupy. The same 0.76 on thirty observations narrows to 0.55 to 0.88, and on sixty to 0.63 to 0.85. Precision improves with the square root of the sample, and the tail never supplies a large sample, because a large sample of drawdown months would mean the drawdown never ended.

Carry the interval through lesson 6's division to see what it costs. For six holdings, effective bets are 6 divided by 1 plus 5 times the correlation. At 0.25 that is 6 divided by 2.25 = 2.67 bets. At the point estimate of 0.76 it is 1.25. At 0.94 it is 6 divided by 5.70 = 1.05. So the honest statement is that this portfolio behaves in a drawdown like somewhere between one and two and two thirds independent positions.

This is why a lender's schedule rounds rather than estimates. An institution that cannot pin a tail correlation closer than plus or minus a third has reason to prefer a rule needing no estimate at all: assign holdings to groups, treat each group as one position, and accept a crude answer that does not move when the data is re-sampled. Lesson 24 works out what that rounding amounts to.

The tail correlation is simultaneously the most important input and the least knowable one. Any method that appears to resolve that tension has hidden the uncertainty rather than removed it.