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Statistics for finance · level 3

Two-asset portfolio risk

Risks do not add — variances do, plus a cross term that carries the correlation. That cross term is the whole story of diversification.

Theory: Portfolio Theory

Worked example: 60% at σ 20%, 40% at σ 10%, ρ = 0.3. Portfolio σ?

σ₁, σ₂20%, 10%
ρ0.3
variance188.8
σp13.74%

Step by step

  1. 1

    Variance, not σ, is what adds

    σp² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂

  2. 2

    Own terms

    144 + 16

  3. 3

    Cross term

    2 × 0.6 × 0.4 × 0.3 × 20 × 10 = 28.8

  4. 4

    Root the total

    √188.8 = 13.74%

  5. 5

    Diversification benefit

    16% weighted average − 13.74% actual = 2.26 pts saved

    With ρ < 1 the portfolio is always less risky than the weighted average of its parts.

60% at σ 20%, 40% at σ 10%, ρ = 0.3. Portfolio σ? = 13.74

The theory behind it

Intuition

Portfolio variance adds a covariance term, and it is that term — not the individual variances — that creates the diversification benefit.

Common pitfalls

  • ×Averaging the two standard deviations.
  • ×Dropping the factor 2 on the covariance term.

In the interview

Any 'should they diversify' question.