Portfolio & Asset Pricingadvanced

The Efficient Frontier

Solving the mean-variance problem at every return target traces a curve in risk-return space called the minimum-variance frontier. Its leftmost point is the global minimum-variance portfolio, the single mix with the lowest possible risk among risky assets. The upper half of the curve is the efficient frontier, the set of portfolios offering the highest return for each level of risk. Portfolios below the frontier are dominated, since another portfolio gives more return for the same risk or less risk for the same return, so a rational investor holds only frontier portfolios.

Try it yourself

The efficient frontier

Sweep the weight w on asset A to trace the frontier of risk and return. A lower correlation ρ bows it toward the return axis, the diversification benefit. The minimum-variance portfolio is its leftmost point.

Current portfolio (w_A = 0.50)11.0% · σ 13.5%

MVP risk σ 11.3%MVP weight w* 0.83

Weight on A, w0.50

Correlation ρ0.20

E(R) of A8.0%

σ of A12.0%

E(R) of B14.0%

σ of B22.0%

Risk is below the weighted-average volatility whenever ρ < 1. The frontier bends most as ρ falls toward −1, where the right mix drives risk close to zero. The MVP holds w* = 0.83 in A.

Try this. Set ρ = +1 — the frontier collapses to a straight line, so there is no diversification benefit. Now set ρ = −1 — it bends sharply and the right mix can push risk almost to zero.

Why it matters

Plot every achievable portfolio and you get a bullet-shaped region. Its left tip is the calmest portfolio you can build, and the polished top edge of that bullet is the only part worth holding. Anything inside or along the bottom edge is dominated, because you could slide straight up to a portfolio with the same risk but more return. The frontier bends, rather than running straight, precisely because diversification keeps shaving risk as you blend assets, and the curvature is the visual signature of correlations below one.

Formulas

Global minimum-variance portfolio (two assets)

w_1^{\mathrm{GMV}} = \dfrac{\sigma_2^2 - \rho_{12}\,\sigma_1\sigma_2}{\sigma_1^2 + \sigma_2^2 - 2\,\rho_{12}\,\sigma_1\sigma_2}

The weight on asset 1 that minimises variance, found by setting

d\sigma_p^2/dw_1=0

. The remainder

w_2=1-w_1^{\mathrm{GMV}}

goes to asset 2.

Efficient set as an optimisation

\max_{\mathbf{w}}\; \mathbf{w}^{\mathsf{T}}\boldsymbol{\mu} \quad \text{s.t.} \quad \mathbf{w}^{\mathsf{T}}\boldsymbol{\Sigma}\,\mathbf{w}=\sigma_p^2, \;\; \mathbf{w}^{\mathsf{T}}\mathbf{1}=1

For each risk level

\sigma_p

, the efficient portfolio is the one with the greatest expected return. Sweeping

\sigma_p

upward traces the efficient frontier.

Worked examples

Scenario

Two assets have $\sigma_1=15\%$ , $\sigma_2=25\%$ , and correlation 0.3. Find the global minimum-variance weight on asset 1.

Solution

Substitute into the formula. The numerator equals $\sigma_2^2-\rho\,\sigma_1\sigma_2=0.0625-0.01125=0.05125$ and the denominator equals $\sigma_1^2+\sigma_2^2-2\rho\,\sigma_1\sigma_2=0.0225+0.0625-0.0225=0.0625$ . So $w_1^{\mathrm{GMV}}=0.05125/0.0625\approx 0.82$ , with about 18% in asset 2. The minimum-variance mix leans heavily toward the lower-risk asset.

Scenario

A portfolio earns 8% with 18% risk while another on the frontier earns 11% at the same 18% risk. Which is rational to hold?

Solution

The 8% portfolio is dominated. It carries identical risk yet a lower return, so any risk-averse investor strictly prefers the 11% frontier portfolio. Only portfolios on the efficient frontier survive this comparison, which is what makes the frontier the relevant menu.

Common mistakes

✗Every portfolio on the minimum-variance frontier is efficient. Only the upper half above the global minimum-variance point is efficient, since the lower half is dominated by portfolios directly above it.
✗The global minimum-variance portfolio is the best portfolio to hold. It has the least risk but a low return, so most investors prefer a higher point on the efficient frontier that suits their risk tolerance.
✗The efficient frontier is a straight line. With only risky assets the frontier is a curve, bending because diversification reduces risk as correlations below one combine.
✗A dominated portfolio can still be optimal for some risk-averse investor. By definition another portfolio offers more return at the same risk, so no risk-averse investor would ever choose the dominated one.

Revision bullets

•Sweeping the mean-variance solution over targets traces the minimum-variance frontier
•The global minimum-variance portfolio is its leftmost, lowest-risk point
•The efficient frontier is the upper half, most return for each risk level
•Portfolios below the frontier are dominated and never rational to hold
•The frontier curves because diversification keeps cutting risk

Quick check

Which portfolios make up the efficient frontier?

A portfolio is said to be dominated when

Connected topics

Diversification Portfolio Variance Mean-Variance Capital Allocation CAPM & SML

Sources

Markowitz (1952)
Markowitz, H. Portfolio Selection. Journal of Finance, 7(1), 77-91, 1952.
Introduces the efficient set and the dominance criterion that defines the frontier.
Brailsford, Heaney & Bilson (2015), Ch. 8
Brailsford, T., Heaney, R., & Bilson, C. Investments: Concepts and Applications. 5th ed. Cengage Learning Australia, 2015.
Derives the minimum-variance frontier, the global minimum-variance portfolio, and dominance.

How to cite this page

Dr. Phil's Quant Lab. (2026). The Efficient Frontier. Derivatives Atlas. https://phucnguyenvan.com/concept/im-efficient-frontier

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