Lesson 7 of 25 · 6 min
Why forty holdings can behave like three
The ceiling on independent bets is one divided by the average correlation, and the ceiling does not move when the count does.
Forty equally weighted holdings at an average pairwise correlation of 0.32 behave like 2.97 independent bets. The reason is not the forty. It is the 0.32, and the arithmetic shows this by refusing to improve when the forty grows.
Apply lesson 6's division along a ladder of counts, holding the correlation at 0.32 throughout. Four holdings give 2.04 bets. Ten give 2.58. Twenty give 2.82. Forty give 2.97. One hundred give 3.06. Five hundred give 3.11. The sequence is climbing toward something and it is not climbing toward 500.
as n grows without limit, N approaches 1 / r
- n
- The number of equally weighted holdings.
- N
- The effective number of independent bets from lesson 6.
- r
- The average pairwise correlation, which sets the ceiling regardless of how large n becomes.
At a correlation of 0.32 the ceiling is 1 divided by 0.32 = 3.125 bets. Forty holdings already reach 2.97, which is 94.96 percent of it. Going from forty holdings to one hundred — more than doubling the statement — adds 0.09 of a bet. Past a point the count is not a weak measure of diversification but a measure of nothing, because the quantity it is supposed to track has stopped responding to it.
The volatility figures say the same thing from the other direction. Forty holdings at 29 percent each and a 0.32 average correlation carry 16.84 percent portfolio volatility. The same forty at a 0.05 average carry 7.88 percent — less than half the risk, on an identical count and identical single-name volatility. Nothing a lender can see on a holdings list distinguishes them.
Adding holdings buys diversification up to a ceiling set by the average correlation, and only lowering that correlation raises the ceiling.
Key terms
- Diversification ceiling
- The most independent bets a portfolio can reach at a given average correlation, equal to one divided by that correlation. Adding holdings cannot pass it.
- Diminishing returns to count
- The pattern by which each additional holding contributes less than the one before, with almost all of the available benefit present well before forty names.
Worked example
Climbing toward a ceiling of 3.125
Equally weighted holdings, average pairwise correlation held at 0.32 throughout. Only the count changes. Illustrative figures.
- Four holdings
- 2.04 bets
- Ten holdings
- 2.58 bets
- Forty holdings
- 2.97 bets
- One hundred holdings
- 3.06 bets
- The ceiling, at any count
- 3.125 bets
- Share of the ceiling already reached at forty
- 94.96%
4 / (1 + 3 x 0.32).
40 / 13.48.
Sixty more names bought 0.09 of a bet.
1 / 0.32.
Forty holdings captured 95 percent of everything a 0.32 average correlation permits. The rest is not available at any count.
Check yourself
Sixty equally weighted holdings have an average pairwise correlation of 0.25. How many independent bets is that, and how close is it to the ceiling?
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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.