Chapter 14. Introduction to Modern Portfolio Theory

E(R) = Σ (Probabilityᵢ × Returnᵢ)

Excel: SUMPRODUCT(probabilities, returns)

Example →

Boom 0.3 × 15% + normal 0.5 × 10% + recession 0.2 × 2% = 9.9%.

Trap →

Check that the probabilities add up to 1 first.

σ² = Σ (Xᵢ − X̄)² ÷ (n − 1) · σ = √σ²

Excel: VAR.S(range) · STDEV.S(range)

Example →

Returns 10%, 14%, 8%, 12%: mean 11%, squared deviations sum to 20 → σ² = 20 ÷ 3 = 6.67, σ = 2.58%.

Trap →

Dividing by n instead of n − 1. Even 'all past data available to the adviser' is still a sample.

Cov(A,B) = ρ(A,B) × σA × σB

Example →

σA 5%, σB 8%, ρ 0.4 → 0.4 × 5 × 8 = 16 (%²).

Trap →

Covariance has no fixed range; −1 to +1 is the range of the correlation coefficient.

ρ(A,B) = Cov(A,B) ÷ (σA × σB) (range −1 to +1)

Example →

ρ = +1 → no diversification benefit; ρ = −1 → maximum benefit (risk can in theory fall to zero).

Trap →

Zero correlation doesn't eliminate risk; both variance terms remain.

E(Rp) = Σ wᵢ × Rᵢ

Excel: SUMPRODUCT(weights, returns)

Example →

A 0.2 × 9% + B 0.1 × 12% + C 0.3 × 15% + D 0.4 × 18% = 14.7%.

Trap →

Expected portfolio return is the weighted arithmetic mean, not a geometric average.

σp² = w₁²σ₁² + w₂²σ₂² + 2 w₁ w₂ ρ₁₂ σ₁ σ₂

Example →

Weights 0.5/0.5, σ₁ = σ₂ = 5%, ρ = 0.5 → 0.000625 × 3 = 0.001875 → σp = 4.33%.

Trap →

If covariance is given instead of ρ, don't multiply by σ₁σ₂ again. Portfolio SD is not the weighted average of SDs.

Covariance terms = (n² − n) ÷ 2 (plus n variance terms)

Example →

50 securities → (2,500 − 50) ÷ 2 = 1,225 covariances plus 50 variances.

Trap →

Not n² or n(n + 1) ÷ 2; the correct count is n(n − 1) ÷ 2.