Lesson 5 of 25 · 6 min

Why a second holding at 0.9 changes almost nothing

A second position at 0.9 delivers eleven percent of the risk reduction the same position at 0.2 would have delivered.

Adding a holding always reduces volatility. The statement is true and nearly useless, because the size of the reduction spans almost the entire range between something and nothing.

Take one holding at 30 percent volatility and split it equally into two holdings, each also at 30 percent. Lesson 4's formula collapses to 30 times the square root of one half plus one half of the correlation. At a correlation of zero the pair carries 21.21 percent. At 0.20, 23.24 percent. At 0.50, 25.98 percent. At 0.90, 29.24 percent. At 1.00, 30.00 percent — the position you started with.

Read those as reductions rather than levels and the point sharpens. The second holding removed 6.76 points of volatility at a correlation of 0.20 and 0.76 points at 0.90. The 0.90 pair captured 11.23 percent of what the 0.20 pair captured, from an action that looks identical on a statement: one line became two, each half the size.

Track 7 lesson 2 separated a holding's move into a market-driven part and a company-specific part. Splitting into two names at 0.90 correlation leaves the market-driven part fully intact and cancels almost none of the company-specific part, because at 0.90 the two residuals are themselves moving together. The reduction is small because the thing being reduced was mostly shared to begin with.

The question a second holding raises is never whether it reduces risk. It is how much, and the answer ranges from a third of the volatility to three quarters of a point.

Key terms

Diversification benefit: The volatility points removed by holding several things instead of one. A quantity, computable from the correlations, not a property a portfolio either has or lacks.
Equal-weight pair: Two holdings at half the portfolio each. The simplest case in which a correlation's effect can be read directly off the arithmetic.

Worked example

Splitting one position in two, at four correlations

A single holding at 30% volatility is replaced by two equally weighted holdings, each also at 30% volatility. Illustrative figures.

One holding: 30.00%
Two at r = 0.20: 23.24%
Two at r = 0.50: 25.98%
Two at r = 0.90: 29.24%
Two at r = 1.00: 30.00%
Share of the 0.20 benefit delivered at 0.90: 11.23%

The same structural change — one position becoming two — produced 6.76 points of risk reduction in one case and 0.76 in another. The correlation was the whole difference.

Check yourself

You hold two equally weighted positions, each with 24% volatility, correlated 0.85. What is the pair's volatility, and what did the second holding buy?

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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.