Lesson 14 of 25 · 7 min

Why the correlation input moves the answer more than the volatility input

Three years of data pins volatility to within a point. Nothing pins the number that supplies most of the variance.

Two reasons, and only the second is about estimation. The first is that in a diversified account almost all of the variance is correlation rather than volatility.

portfolio variance = σ² × (1 + (n − 1)ρ) ÷ n

σ: The volatility of each position, taken here to be the same across all of them.
n: The number of equally weighted positions in the account.
ρ: The common correlation between any pair of positions.

The share of variance carried by the correlation term is (n − 1)ρ ÷ (1 + (n − 1)ρ). For forty positions at a correlation of 0.25 that is 9.75 ÷ 10.75 = 90.7 percent. Nine tenths of the risk in a forty-name account is the assumption about how the names move together, and only one tenth is the assumption about each name individually.

Second, the two inputs are known to very different precisions. Three years of daily data pins a 26 percent volatility to within 1.34 points at two standard errors. Correlation has no comparable band, because it is not a stable quantity being estimated with noise — it is regime-dependent, and Track 4 lesson 15 established that the calm-market figure is not the stressed-market figure.

Put both together on the facility from the previous lesson. Forty positions at 26 percent each and a correlation of 0.25 give a portfolio volatility of 13.48 percent and a 9.04 percent chance of touching the trigger over three years. Moving volatility to the top of its statistical band, 27.34 percent, gives 11.06 percent. Moving correlation to 0.75 gives a portfolio volatility of 22.61 percent and a touch probability of 36.29 percent. The defensible range of one input moved the answer by two points; the regime range of the other moved it by twenty-seven.

A lender granting a better blended advance rate for a diversified account has priced the calm correlation. The effective number of independent positions in forty names is 3.72 at a correlation of 0.25 and 1.32 at 0.75 — the diversification credit and the diversification both leave at the same time.

Key terms

Common correlation: A single figure standing in for the correlation between every pair of holdings. A simplification that makes the arithmetic visible without changing its direction.
Effective number of positions: How many genuinely independent bets an account behaves like. Computed as n divided by one plus (n − 1) times the correlation.
Regime range: The span an input covers between calm and stressed conditions, as distinct from the statistical error in estimating it. For correlation the regime range is far wider.

Worked example

Forty names, two inputs, one answer

Forty equally weighted positions at 26 percent volatility each, applied to the facility from lesson 13. Illustrative.

Correlation 0.25: Portfolio volatility 13.48%
Volatility at the top of its two-standard-error band: 27.34%
Volatility raised to 31 percent: Portfolio volatility 16.07%
Correlation raised to 0.75: Portfolio volatility 22.61%
Share of variance from the correlation term: 90.7%
Effective independent positions: 3.72, then 1.32

Volatility across its honest uncertainty moved the answer 2.02 points. Correlation across its regime range moved it 27.25.

Check yourself

A forty-name account is modelled at 26 percent volatility per name and a 0.25 correlation. Which input would you most want to be right about, and why?

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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.