Lesson 3 of 7
How many holdings is enough
Portfolio variance falls fast for the first few holdings and then flattens. See where the curve flattens, and what happens to it when correlations rise.
By Javier Sanz
Founder of ValueScreener. Built Ninety Nine, a securities services provider, and sold it to a tier-one global bank; ran operations and internationalization at Alpaca, the brokerage API.
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In one minute
- In an equally weighted portfolio of n holdings there are n variance terms and n times n minus 1 covariance terms, so covariance quickly becomes the whole story.
- The curve of portfolio volatility against the number of holdings flattens after a handful of names and never reaches zero.
- The floor is set by the average correlation, and that correlation rises in a decline, which moves the floor rather than the number of holdings.
Adding holdings buys less and less, and the reason is arithmetic rather than judgement: the terms that can be diversified away are outnumbered by the terms that cannot, and the ratio gets worse with every name.
Portfolio volatility as holdings are added, at two correlations
- Average correlation 0.2
- Average correlation 0.5
Show the numbers
| Average correlation 0.2 | Average correlation 0.5 | |
|---|---|---|
| 1 | 30.0% | 30.0% |
| 5 | 18.0% | 23.24% |
| 10 | 15.87% | 22.25% |
| 20 | 14.7% | 21.74% |
| 50 | 13.94% | 21.42% |
Counting the terms
A portfolio of 20 equally weighted holdings has 20 variance terms and 380 covariance terms. Each variance term is weighted by one over 400, so the whole block of them is one twentieth of the calculation. Everything else is co-movement.
That is the same two-asset formula extended to n equally weighted holdings with one average correlation. It says where the curve goes and where it stops.
Where it flattens
The number that actually moves the floor
The floor is the holding volatility multiplied by the square root of the average correlation. At 0.2 it is 13.42 percent. At 0.5 it is 21.21 percent.
So a portfolio of 20 names sits at 14.70 percent in calm conditions and 21.74 percent if average correlations rise to 0.5, which is what they do in a general decline. The holder did nothing. The diversification they thought they had was mostly the correlation assumption, and that assumption is the first thing a falling market takes away.
Two ways the curve lies about a real portfolio
The formula assumes each holding is the same size and each pair shares one average correlation. Both assumptions flatter the count.
Sizes drift. A portfolio of twenty names in which one has quadrupled is not a twenty-name portfolio for this purpose: if the largest holding is 30 percent of the account, the effective number of holdings is far below twenty and the curve above overstates the diversification by a wide margin.
Correlations cluster. Twenty companies drawn from three industries do not have one average correlation; they have high correlations inside each group and lower ones between groups. The portfolio then behaves closer to three holdings than to twenty, which is the arithmetic behind a rule most investors learn the hard way. Counting names is the easy measure. Counting distinct ways to be wrong is the one that matters.
What this does not settle
The arithmetic says where the volatility curve flattens. It does not say how many companies a person should own, because volatility is not the risk that matters and because the cost of the twenty-first holding is not statistical.
Each additional name is another set of filings to read, another thesis to keep current, and another chance to hold something you do not understand. A researched portfolio of twelve and a mechanical portfolio of fifty are different answers to different questions, and this formula is only about the second half of one of them.
Check yourself
In an equally weighted portfolio of 20 holdings, how many covariance terms are there?
- 20
- 190
- 380
Show the answer
Answer: C.
There are n times n minus 1 ordered pairs, so 20 times 19 is 380, against only 20 variance terms.At an average correlation of 0.2 and a holding volatility of 30 percent, what is the floor?
- 0 percent
- 13.42 percent
- 6.0 percent
Show the answer
Answer: B.
The floor is the holding volatility times the square root of the correlation: 30 times the square root of 0.2, which is 13.42 percent.What happens to a 20-holding portfolio if average correlation rises from 0.2 to 0.5?
- Nothing, the holdings are unchanged
- Volatility rises from 14.70 percent to 21.74 percent
- The floor disappears
Show the answer
Answer: B.
The correlation sets the floor, so a rise in correlation moves the whole curve up without anybody trading anything.Sources
- Computed for this lesson from the equally weighted portfolio variance formula shown above, with a holding volatility of 30 percent and average correlations of 0.2 and 0.5.
What this leaves out
- The 30 percent holding volatility and the two correlations are chosen to show the shape. They are not measured from any market.
- The formula assumes every holding has the same volatility and every pair the same correlation. Real portfolios have neither.
- It measures variation in returns, not the chance of permanent loss in any holding.
- Nothing here says how many holdings any reader should own, and none of it is advice.