Diversification and the Mean-Variance Frontier
The mean-variance framework, introduced by Harry Markowitz (1952), characterises each portfolio by two numbers: its expected return and its standard deviation. The mean-variance frontier traces out, for every level of expected return, the portfolio with the lowest achievable standard deviation. Investors prefer portfolios that lie on the upper portion of this frontier — the efficient frontier.
Holding multiple assets rather than a single asset can push the frontier to the left, meaning the same expected return is achievable with less risk. To see why, consider a two-asset portfolio with weights \(w_A\) and \(w_B = 1-w_A\). Its variance is
$$\sigma_p^2 \;=\; w_A^2\,\sigma_A^2 \;+\; w_B^2\,\sigma_B^2 \;+\; 2\,w_A\,w_B\,\rho_{AB}\,\sigma_A\,\sigma_B.$$When \(\rho_{AB} < 1\), the cross-term \(2\,w_A\,w_B\,\rho_{AB}\,\sigma_A\,\sigma_B\) is smaller than under perfect co-movement, so portfolio variance falls below the weighted average of the individual variances — the essence of diversification. The lower the correlation, the greater the reduction; when \(\rho_{AB} < 0\), the cross-term turns negative and actively offsets risk. But how much you gain from adding a third asset depends critically on how correlated it is with what you already hold. The interactive chart below makes this concrete.
The visualization
The portfolio universe contains three assets:
- Asset A — Domestic Stocks: expected return 9%, standard deviation 18%
- Asset B — Government Bonds: expected return 3%, standard deviation 6%; correlation with stocks: −0.20
- Asset C — A New Asset: expected return 7%, standard deviation 15%; its correlation with stocks (\(\rho_{AC}\)) is controlled by the slider
The light-grey dashed curve is the complete mean-variance frontier for assets A and B — including its lower, inefficient half — and you can see how it joins A and B along the full parabola. The solid grey curve highlights only the efficient portion of that frontier (from the minimum-variance portfolio upward, where higher return is achievable for the same risk). The solid orange curve is the efficient frontier achievable with all three assets, and the lightly-shaded region is the full feasible set of long-only three-asset portfolios. Drag the slider and observe how much — or how little — the orange efficient frontier expands relative to the grey baseline.
Key takeaways
- Diversification reduces risk. Even with two assets, combining them (when \(\rho_{AB} < 1\)) shifts the frontier to the left of both individual assets.
- Correlation determines the gain. The lower the correlation of the new asset with the existing portfolio, the farther the frontier moves leftward.
- Near-perfect correlation ≈ near-zero benefit. When \(\rho_{AC} \to 1\), asset C is essentially redundant — it duplicates what stocks already provide and does not expand the opportunity set.
- Negative correlation is especially valuable. When \(\rho_{AC} < 0\), the new asset hedges the existing portfolio, producing the largest frontier expansion and the lowest attainable risk.
Reference: Markowitz, H. (1952). Portfolio selection. The Journal of Finance, 7(1), 77–91.