Lesson 6 of 25 · 7 min

The effective number of independent bets

One division turns a count and a correlation into the number of separate positions a portfolio actually behaves like.

Volatility answers how much a portfolio moves. It does not answer how many separate things it is exposed to, and that second question has an answer just as short.

Track 7 lesson 10 computed this quantity as the reciprocal of a bracket. Written the other way round it is a single division, and this is the form the rest of this track uses without variation.

N = n / (1 + (n - 1) x r)

N: The effective number of independent bets: how many genuinely separate positions the portfolio behaves like. Rarely a whole number.
n: The number of equally weighted holdings actually held.
r: The average pairwise correlation among them, as a decimal from 0 to 1.

Eighteen holdings at an average correlation of 0.18: the denominator is 1 plus 17 times 0.18 = 4.06, so N is 18 divided by 4.06 = 4.43. The same eighteen holdings at 0.62: the denominator is 11.54 and N is 1.56. Four holdings at 0.45: the denominator is 2.35 and N is 1.70. That last comparison is the one worth sitting with — four holdings at 0.45 carried more independent risk, 1.70 bets against 1.56, than eighteen holdings at 0.62.

Two properties fix the formula's behaviour. At a correlation of zero the denominator is 1 and N equals n exactly, so the count is right only in the case that never occurs. At a correlation of 1 the denominator is n and N equals 1, so a perfectly correlated portfolio of any size is one position. Between those ends N moves far more with the correlation than with the count, which is lesson 7's subject.

The figure matters for collateral because it is what a lender's grouping rule is trying to estimate without a matrix. Track 7 lesson 11 listed the groupings a schedule uses — issuer, industry, look-through, factor — and each is a rule for deciding how many separate things are present. N is the number those rules are approximating.

Key terms

Effective number of independent bets: How many separate positions a portfolio behaves like once correlation is taken into account. At most the holdings count and at least one.
Denominator of the bets formula: 1 plus (n minus 1) times the average correlation. It measures how much of the portfolio's grid is shared rather than separate.
Equal weighting: The assumption that every holding is the same size. The bets formula is written for this case; unequal weights lower the figure further.

Worked example

The same count, and a smaller count that beats it

Three equally weighted portfolios, described only by their holdings count and average pairwise correlation. Illustrative figures.

Eighteen holdings at r = 0.18: 4.43 bets
Eighteen holdings at r = 0.62: 1.56 bets
Four holdings at r = 0.45: 1.70 bets
Which of the three carries the most independent risk: The first, by a factor of 2.8
Which of the three carries the least: The eighteen-holding portfolio at 0.62

The four-holding portfolio carried more independent risk than one of the eighteen-holding portfolios. A count cannot produce that ordering; this division does.

Check yourself

Twenty-three equally weighted holdings have an average pairwise correlation of 0.41. How many independent bets is that, and what would the figure be if the correlation were zero?

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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.