Two-Asset Portfolio Risk Decomposition
When you combine two risky assets into a portfolio, the portfolio's expected return is simply the weighted average of the individual expected returns. But portfolio risk is not a weighted average — it depends critically on how the two assets move together. With weights \(w_A\) and \(w_B = 1 - w_A\), the portfolio variance is
$$\sigma_p^2 \;=\; w_A^2\,\sigma_A^2 \;+\; w_B^2\,\sigma_B^2 \;+\; 2\,w_A\,w_B\,\rho_{AB}\,\sigma_A\,\sigma_B$$The third term, \(2\,w_A\,w_B\,\rho_{AB}\,\sigma_A\,\sigma_B\), is the covariance contribution. When \(\rho_{AB} < 1\) it is smaller than it would be under perfect co-movement, so total portfolio risk falls below what a simple weighted average would suggest. When \(\rho_{AB} \leq 0\) the term is zero or negative and actively reduces portfolio variance — the essence of diversification.
The visualization
The chart plots the portfolio's expected return against its standard deviation as the weight in Asset A varies from 0% to 100%. The highlighted point shows your currently selected portfolio. Adjusting the correlation slider changes the shape of the curve; adjusting the weight slider moves the highlighted point along it.
Asset A
Asset B
Portfolio
Portfolio Results
Variance: σp² = wA²σA² + wB²σB² + 2wAwBρABσAσB (not a weighted average)
Covariance: Cov(A, B) = ρAB·σA·σB
Key takeaways
- Expected return is a weighted average; risk is not. Combining assets with weights 60/40 gives a 60/40 weighted average return, but the portfolio standard deviation is generally less than the weighted average of the individual standard deviations.
- Correlation drives diversification. The lower the correlation, the more the frontier bows to the left, and the more risk is reduced at any given expected return.
- Negative correlation is the most powerful diversifier. When ρ < 0 the covariance term subtracts from variance, potentially reducing portfolio risk below either individual asset.
- The minimum-variance portfolio is not 0%/100% or 100%/0%. There is an interior weight that minimises risk, and that weight depends on both standard deviations and the correlation.
- Covariance, not variance, is added at the margin. When you add an asset to a portfolio, the incremental contribution to risk is determined by how it co-moves with the existing portfolio, not by its own variance.