Two-Asset Portfolio Variance and Minimum-Variance Weights (Worked Example)

Every Modern Portfolio Theory claim — "diversification reduces risk," "the efficient frontier," "you can sometimes do better than either asset alone" — comes out of two formulas: portfolio expected return and portfolio variance. Most explanations state the variance formula and stop. Below is a single worked example that carries one pair of assets through all three payoffs: the expected return and risk of an actual mix, why that risk sits below what you'd naively expect, and how to solve for the exact weights that minimize risk — including the special case where risk can be driven all the way to zero.

The two formulas

For a portfolio of two assets with weights w_A and w_B (where w_A + w_B = 1), expected returns E(R_A) and E(R_B), standard deviations σ_A and σ_B, and correlation coefficient ρ between the two assets' returns:

E(R_p) = w_A × E(R_A) + w_B × E(R_B)

σ_p² = w_A²σ_A² + w_B²σ_B² + 2 × w_A × w_B × ρ × σ_A × σ_B

Expected return is a simple weighted average — nothing surprising there. The variance formula is where diversification actually happens: that last term depends on ρ, and unless the two assets move in perfect lockstep (ρ = 1), the portfolio's risk comes out lower than a weighted average of the two assets' individual risks would suggest.

Worked example: a stock fund and a bond fund

Expected return: E(R_p) = 0.6(10%) + 0.4(6%) = 6% + 2.4% = 8.4%.

Variance: σ_p² = 0.6²(20²) + 0.4²(8²) + 2(0.6)(0.4)(0.20)(20)(8), which is 144 + 10.24 + 15.36 = 169.6.

Standard deviation: σ_p = √169.6 ≈ 13.02%.

Why 13.02% is the interesting number

If A and B moved in perfect lockstep (ρ = 1), the correlation term would simplify the whole formula down to a plain weighted average of the two standard deviations: 0.6(20%) + 0.4(8%) = 15.2%. That 15.2% is the "no diversification benefit" baseline — what you'd get if combining assets never helped. The actual portfolio risk, 13.02%, is more than 2 percentage points below that baseline, purely because the real correlation (0.20) is less than 1. Nothing about the expected return changed — it's still exactly the weighted average, 8.4% — but the risk didn't have to average the same way. That gap between 15.2% and 13.02% is the diversification benefit, and it's the entire reason Modern Portfolio Theory treats correlation, not just individual risk, as the variable that matters.

Finding the minimum-variance weights

60/40 was an arbitrary starting mix. For any given correlation, there's one specific weighting that minimizes portfolio variance. Minimizing the variance formula above with respect to w_A (subject to w_B = 1 - w_A) and setting the derivative to zero gives a direct formula for the minimum-variance weight in asset A:

w_A(min) = (σ_B² - ρ × σ_A × σ_B) ÷ (σ_A² + σ_B² - 2 × ρ × σ_A × σ_B)

Plugging in the same numbers (σ_A = 20, σ_B = 8, ρ = 0.20):

w_A(min) = (64 - 32) ÷ (400 + 64 - 64) = 32 ÷ 400 = 8%, so w_B(min) = 92%.

Running the variance formula again at these weights: 0.08²(400) + 0.92²(64) + 2(0.08)(0.92)(0.20)(20)(8) = 2.56 + 54.17 + 4.71 = 61.44, so σ_p = √61.44 ≈ 7.84%. That's the part that surprises people the first time they see it worked through: 7.84% is lower than Asset B's own standalone risk of 8%. Adding a small slice of the riskier asset (A, at 20% risk) to a bond-heavy portfolio can reduce total risk below what the "safe" asset carries on its own — and the expected return at this mix, 0.08(10%) + 0.92(6%) = 6.32%, is still slightly above Asset B's own 6%. Lower risk and higher return than the "safe" asset alone is only possible because ρ < 1.

The extreme case: ρ = −1 and a zero-risk portfolio

Push the correlation to its theoretical floor, perfect negative correlation, and the variance formula's cross term flips sign in a way that lets it collapse the whole expression into a perfect square: σ_p = |w_Aσ_A - w_Bσ_B|. Setting that equal to zero and solving (w_Aσ_A = w_Bσ_B, with w_B = 1 - w_A) gives:

w_A = σ_B ÷ (σ_A + σ_B) = 8 ÷ 28 ≈ 28.57%, so w_B ≈ 71.43%.

Check: 0.2857(20) = 5.71 and 0.7143(8) = 5.71 — equal, so the two risk contributions cancel exactly and σ_p = 0. This portfolio's expected return is 0.2857(10%) + 0.7143(6%) ≈ 7.14% — a completely risk-free result built from two individually risky assets, and it still beats the safe asset's own 6% return. In practice, real assets are never perfectly negatively correlated (ρ = -1 is a textbook limit, not an observed market condition), but this is exactly the case a two-asset quiz question is testing when it asks what's "theoretically possible" under perfect negative correlation: not just lower risk, but risk eliminated entirely.

Quick reference

This same mean-variance framework scales to any number of assets — real portfolio software optimizes across dozens of holdings using the matrix version of the same variance formula — but the two-asset case is where the mechanics are visible by hand, which is why it's the standard way the concept is taught and tested. To practice applying it, try PassDrill's portfolio theory practice questions, including the perfectly-negatively-correlated case worked through above.

This page is educational material to help you understand and practice portfolio math; it is not investment, financial, or professional advice, and no specific security or allocation is recommended.

Source: Harry M. Markowitz, "Portfolio Selection," The Journal of Finance, Vol. 7, No. 1 (1952) — the original mean-variance framework; CFA Institute curriculum, Quantitative Methods / Portfolio Risk and Return topic, for the standard two-asset expected-return and variance treatment.

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