Lesson 3 of 25 · 6 min
How correlation enters a portfolio's variance
Portfolio risk is a grid of pairs, and in a twenty-three holding portfolio 95.65 percent of the grid is pairs of different things.
Portfolio variance is not an average of its holdings' variances. It is a sum over every ordered pair of holdings, and in any portfolio worth calling a portfolio, most pairs are not a holding paired with itself.
portfolio variance = sum over all pairs i, j of (wi x wj x si x sj x rij)
- portfolio variance
- The square of portfolio volatility. Variances add across pairs; volatilities do not.
- i, j
- Labels for two holdings. Every combination is included, including each holding paired with itself.
- wi, wj
- The portfolio weights of those two holdings, as decimals that sum to one across the portfolio.
- si, sj
- The two holdings' own volatilities.
- rij
- The correlation between them. When i and j are the same holding this equals 1, and the term collapses to that holding's plain variance.
Count the terms and the subject rearranges itself. A portfolio of n holdings has n squared entries in its grid. Exactly n of them sit on the diagonal and are variances. The remaining n squared minus n are correlation terms. With seven holdings that is 49 entries, 7 variances and 42 correlation terms — 85.71 percent of the sum. With twenty-three holdings it is 529 entries, 23 and 506, so 95.65 percent. With forty it is 1,600 entries and 97.5 percent.
So the inputs a reader can look up — each holding's own volatility — are the small part of the answer, and they shrink as a share of it with every holding added. The large part is the part that requires a matrix nobody publishes. That is the structural reason diversification is hard to measure and easy to assume.
It also explains the shape of a lender's arithmetic. An advance rate assigned per holding is a diagonal input: it prices each position's own behaviour. The schedule's only off-diagonal input is usually a grouping rule, which assigns holdings to buckets. A lender pricing a twenty-three holding pledge with per-asset advance rates and one grouping rule is addressing 4.35 percent of the grid precisely and 95.65 percent of it by category. Lesson 24 computes what that approximation costs and which direction it errs in.
Key terms
- Portfolio variance
- The squared volatility of the whole portfolio. It is a sum over pairs of holdings, which is why correlation appears in it at all.
- Diagonal term
- A holding paired with itself. These are the plain variances, and there are only as many of them as there are holdings.
- Off-diagonal term
- A pair of two different holdings. Each one carries a correlation, and they outnumber the diagonal terms by n minus 1 to one.
Worked example
How much of the answer is correlation
Count the entries in the variance grid at three portfolio sizes.
- Seven holdings: entries, variances, correlation terms
- 49, 7, 42
- Correlation terms as a share of the grid
- 85.71%
- Twenty-three holdings
- 529, 23, 506
- Forty holdings
- 1,600, 40, 1,560
- Inputs a reader can obtain per holding
- The diagonal only
7 squared, then 49 - 7.
42 / 49.
95.65% correlation terms.
97.5% correlation terms.
The share of portfolio risk governed by correlation rather than by individual volatility rises with every holding added. At forty names it is 97.5 percent of the arithmetic.
Check yourself
A portfolio holds fifteen positions. How many entries does its variance grid contain, how many are correlation terms, and what share of the grid is that?
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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.