How do I calculate portfolio risk for a two-asset portfolio? I keep messing up the formula.
I understand that portfolio risk isn't just the weighted average of individual risks because of correlation. But when I try to apply the formula, I always get confused about which terms go where. Can someone walk through the two-asset variance formula step by step?
You're right that portfolio risk depends on correlations, and that's exactly why the formula looks more complex than a simple weighted average. Let's break it down.
The two-asset portfolio variance formula:
σ²_p = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρ₁₂σ₁σ₂
Where:
- w₁, w₂ = weights of assets 1 and 2 (must sum to 1)
- σ₁, σ₂ = standard deviations of assets 1 and 2
- ρ₁₂ = correlation between the two assets
Step-by-step example:
Suppose you invest 55% in Orion Industrials (σ = 20%) and 45% in Summit Bonds (σ = 8%), with a correlation of 0.25.
| Component | Calculation | Value |
|---|---|---|
| w₁²σ₁² | (0.55)²(0.20)² | 0.0121 |
| w₂²σ₂² | (0.45)²(0.08)² | 0.001296 |
| 2w₁w₂ρ₁₂σ₁σ₂ | 2(0.55)(0.45)(0.25)(0.20)(0.08) | 0.00198 |
| Portfolio variance | Sum | 0.015376 |
| Portfolio std dev | √0.015376 | 12.4% |
Why this matters: The weighted average standard deviation would be 0.55(20%) + 0.45(8%) = 14.6%. But the actual portfolio risk is only 12.4% — a reduction of 2.2 percentage points purely from diversification.
Three critical scenarios to remember:
- ρ = +1: No diversification benefit. Portfolio std dev equals the weighted average.
- ρ = 0: Good diversification. Risk drops meaningfully below the weighted average.
- ρ = -1: Maximum diversification. You can theoretically reduce portfolio risk to zero with the right weights.
Common mistakes:
- Forgetting to square the weights AND the standard deviations in the first two terms
- Using covariance (Cov₁₂ = ρ₁₂σ₁σ₂) but then also multiplying by ρ again
- Forgetting the factor of 2 in the cross-term
Practice this calculation until it's automatic — CFA Level I frequently tests this in both item set and standalone format.
Check out our CFA Level I course for more portfolio management walkthroughs.
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