Why a lender's grouping rule is a crude correlation model

A schedule that collapses several holdings into one sector group looks like an administrative shortcut. It is a correlation matrix — specifically, the matrix with every entry rounded to 0 or 1. Within a group each pairwise entry becomes 1, so the group behaves as a single position. Across groups each entry becomes 0, so the groups are treated as independent.

Six equally weighted holdings, each at 30 percent volatility, with a true average pairwise correlation of 0.42. The bracket is one-sixth plus five-sixths of 0.42 = 0.5167, so portfolio volatility is 21.56 percent and effective bets are 6 divided by 3.1 = 1.94. Now apply a rule putting four of them in one group and two in another. The rounded matrix gives a variance factor of four-sixths squared plus two-sixths squared = 0.5556, a volatility of 22.36 percent, and effective bets of 1 divided by 0.5556 = 1.80.

Every grouping corresponds to an equivalent flat correlation, which is the clean way to see what the rule assumes. One group of six is equivalent to 1.00 and gives 1.00 bets. Five-and-one is equivalent to 0.6667 and gives 1.38. Four-and-two is equivalent to 0.4667 and gives 1.80. Three-and-three is equivalent to 0.40 and gives 2.00. Against a true 0.42 the first three are conservative and the last is not: the rounding errs the lender's way whenever the grouping is coarse enough that its equivalent correlation exceeds the truth.

What the rule buys is not accuracy but immunity. A rounded matrix does not move when the estimation window changes, so it cannot be fooled by the calm-window problem of lesson 10, the pooling artefact of lesson 11, or the ten-observation interval of lesson 16. It gives up the ability to be right in exchange for the inability to be wrong in a particular direction.

The limit of that immunity is worth stating precisely. If the true correlation rises to 0.80, effective bets fall to 1.20 and volatility rises to 27.39 percent, so the four-and-two grouping's 1.80 has become optimistic. Only coarser groupings hold up: one group of six reads 1.00 bets and stays conservative in every regime, at the cost of granting no diversification credit at all. Choosing how many groups is choosing where on that trade to sit, and Track 8 describes the discretion sitting alongside it.