Lesson 2 of 7
What two holdings do to risk
The two-asset variance formula, computed on a real pair of numbers, and the correlation at which adding a second holding stops reducing anything.
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
- A two-holding portfolio's variance is the two weighted variances plus twice the weighted covariance between them.
- The whole benefit of diversification lives in that third term, and it vanishes when the correlation reaches one.
- Volatility is not risk, and a lower number here is not automatically a better portfolio.
Two holdings, each with its own swings, combine into a portfolio whose swings are usually smaller than the average of theirs. How much smaller is decided by one term in one formula.
The same two holdings at four correlations
- Correlation 1.06.30%
- Correlation 0.81, as measured6.01%
- Correlation 04.60%
- Correlation minus 11.61%
The formula
With equal weights of one half, holding A at a standard deviation of 7.91 and holding B at 4.69, and the covariance of 30 computed in the previous lesson:
0.25 times 62.5, plus 0.25 times 22, plus 0.5 times 30, is 36.13. The square root is 6.01 percentage points.
Where the benefit comes from
The weighted average of the two standard deviations is 6.30. The portfolio's is 6.01. The difference, 0.29 points, is the entire benefit of holding both, and it came from the third term.
Reading the third term
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If the correlation is 1, the third term is as large as it can be and the portfolio's volatility equals the weighted average. Nothing was gained.
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As the correlation falls, that term shrinks and the portfolio's volatility falls below the average of its parts.
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At a correlation of 0 the term disappears, and the two holdings here give 4.60 instead of 6.30.
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At minus 1 the two swings cancel as far as the weights allow, leaving 1.61.
Why the measured case is so close to the worst case
The measured correlation of 0.81 bought 0.29 points out of a possible 4.69. Two holdings that move together strongly are, for this purpose, almost one holding.
That is the practical content of the formula, and it is the reason a portfolio of six companies in one industry is not diversified in any sense the arithmetic recognises. It is also why a reader who wants the benefit has to look for holdings whose fortunes rest on different things, which is a research question and not a statistical one.
Weights matter as much as correlation
The example uses equal weights because that isolates the correlation term. In an account the weights are whatever the prices have made them, and the formula is not symmetric in them.
Put 80 percent in holding A and 20 percent in holding B at the measured correlation, and the portfolio standard deviation is 7.11 rather than 6.01. The second holding is still there, still uncorrelated to the same degree, and it is doing much less work, because the term it appears in is multiplied by both weights. A holding at 2 percent of a portfolio changes almost nothing about the portfolio, whatever its correlation, which is worth remembering before adding a twenty-fifth name for diversification.
This is the arithmetic behind a rule that sounds like taste: a position too small to matter when it is right is also too small to diversify anything when it is wrong.
Check yourself
Which term in the formula carries the benefit of diversification?
- The first weighted variance
- The second weighted variance
- The covariance term
Show the answer
Answer: C.
The two variance terms do not depend on how the holdings relate. Only the covariance term falls as the holdings move together less.At the measured correlation of 0.81, how much did holding both reduce volatility?
- By more than half
- From 6.30 to 6.01, a saving of 0.29 percentage points
- Not at all
Show the answer
Answer: B.
Strongly correlated holdings behave almost like one holding, so most of the possible benefit was not available.What happens to the formula when the correlation is exactly 1?
- The variance becomes zero
- Portfolio volatility equals the weighted average of the two holdings' volatilities
- The formula cannot be computed
Show the answer
Answer: B.
At a correlation of 1 there is no reduction at all: the portfolio swings exactly as much as the weighted average of its parts.Sources
- Computed for this lesson from the invented five-year series in the previous lesson. The formula is standard and the arithmetic is shown in full so that it can be checked rather than cited.
What this leaves out
- Both holdings are invented, with round numbers chosen so the arithmetic is hand-checkable.
- The formula assumes fixed weights and a stable correlation. Neither holds in a real account, where weights drift with prices.
- Volatility is a measure of variation, not of the chance of permanent loss.
- Nothing here is advice about any security.