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Chapter 1 of 5 · 11 min

Two-asset portfolio volatility

Risk does not add. How much of two positions' risk survives the combination depends on one number: the correlation.

By the end of this chapter you can
  • Calculate the volatility of a two-asset portfolio
  • Measure the diversification benefit against the no-diversification case
  • Back out the correlation from a portfolio's realized volatility
  • Show what happens to portfolio risk as the correlation changes
1

The intuition

An ice-cream stand and an umbrella stand each have good and bad days, depending on the weather. Own both, and on a sunny day ice-cream sales make up for the umbrella slump; on a rainy day it is the other way round. Your total takings swing much less than either stand's. If both stands sold ice cream, owning two would just double the swings.

That is diversification, and correlation measures how much of it you get. At a correlation of 1 the two move in lockstep and the portfolio's volatility is the weighted average of theirs. Below 1, some of one position's bad days land on the other's good days, and the portfolio is calmer. Below zero, the two actively offset.

The key idea

Portfolio variance = wA²σA² + wB²σB² + 2 wA wB ρ σA σB. Portfolio volatility = √variance. With no diversification (ρ = 1), volatility = wA σA + wB σB. Diversification benefit = that weighted average − portfolio volatility.

2

Why it works

  • The conventions here: two assets whose weights (w) add to one, annual volatilities (σ) and one correlation (ρ). The worked steps show variances in %², so 30% volatility is 900 %².
  • Variances add, with a cross term. The cross term, 2 wA wB ρ σA σB, is the only place correlation enters, and it can be negative.
  • Volatility equals the weighted average only when ρ = 1. At any lower correlation it is below it.
  • At ρ = 0 the cross term disappears: volatility = √(wA²σA² + wB²σB²).
  • Correlations rise in a crash. A book that ran near its uncorrelated volatility all year can suddenly run near the weighted average, which is why stress tests use crisis correlations.
  • Back out the correlation from what the portfolio actually did: it shows how the two positions really behaved together, which may differ from what the risk system assumed.
60% in A at 30% volatility; 40% in B at 20%; correlation 0.2
A's term: (60% × 30%)²324 %²
B's term: (40% × 20%)²64 %²
Cross term: 2 × 60% × 40% × 0.2 × 30% × 20%57.6 %²
Volatility: √(324 + 64 + 57.6)21.1%
No diversification: 60% × 30% + 40% × 20%26.0%, so a 4.9-point benefit
Uncorrelated: √(324 + 64)19.7%

At a correlation of −1: √(324 + 64 − 288) = 10.0%. At 1: √(324 + 64 + 288) = 26.0%, exactly the weighted average.

3

The formulas

Portfolio variance = wA²σA² + wB²σB² + 2 wA wB ρ σA σB

Two own terms and a cross term.

Portfolio volatility = √variance

Back to a percentage.

No diversification = wA σA + wB σB

The ρ = 1 case.

Diversification benefit = weighted average − portfolio volatility

What the correlation saves.

Implied correlation = (σp² − wA²σA² − wB²σB²) ÷ (2 wA wB σA σB)

Run it backwards.

4

Worked example

Variance first: the two squared terms plus the cross term. Then take the square root.

Drawing the numbers…
5

See it move

Same two positions: A is the first company in the worked example, B the second. Change the weight in A, each one's volatility and the correlation between them.

Drawing the numbers…
Try this
  • Raise the correlation. Portfolio volatility rises, the benefit shrinks, and the weighted average does not move.
  • Set the correlation to 1. The benefit disappears: volatility equals the weighted average.
  • Take the correlation below zero. Portfolio volatility drops below the uncorrelated bar.
  • Raise A's volatility. The no-diversification bar rises.
6

Run it backwards

Same two positions, reversed: the portfolio's realized volatility is known. What correlation does it imply?

Drawing the numbers…

Square the portfolio volatility and take off the two own terms: what is left is the cross term. Divide it by 2 wA wB σA σB and only the correlation remains.

That is the correlation the two positions actually had over the period, which can be very different from the historical number the risk system used to size them.

7

Traps

Averaging the volatilities.
That is only right at a correlation of 1. Work in variances.
Forgetting the 2 in the cross term.
The cross term is 2 wA wB ρ σA σB.
Taking square roots term by term.
Add the variance terms first, then take one square root.
Stress-testing with calm-market correlations.
Correlations rise in a crash. Test with higher ones.
Treating negative correlation as free.
A negatively correlated position is often a hedge that costs return.
8

Say it in the interview

The interviewer asks

“Why is a portfolio less risky than the average of its parts?”

Say yours out loud first, then compare.
9

Check yourself

4 fresh questions, with new numbers. Answer each one correctly to finish the chapter. Get one wrong and you will see the full working, then you can try it again with new numbers.

Answers within 1% are marked right. Type the number; $, %, x and M are fine. First tries count toward Learned: the topic is Learned once every chapter is done and 75% of first tries were right.

0 of 4
Drawing your questions…
Remember
  • σp² = wA²σA² + wB²σB² + 2 wA wB ρ σA σB.
  • Only at ρ = 1 is volatility the weighted average.
  • Benefit = weighted average − portfolio volatility.
  • Correlations rise when markets fall.