On This Page
Key Takeaways
- The efficient frontier collects the portfolios that offer the highest expected return for each level of risk, so any portfolio below it is dominated.
- It is the upper part of the minimum-variance frontier, starting at the global minimum-variance portfolio and rising to the right.
- For positive weights, correlation below one reduces portfolio volatility relative to the weighted-volatility reference at perfect positive correlation.
- A capital allocation line combines a selected risky portfolio with a risk-free asset. Its slope is that portfolio’s Sharpe ratio; the optimal line uses the maximum-Sharpe mix.
- Define the asset universe, expected returns, covariance inputs and allocation constraints, then compare the efficient portfolios.
What is the efficient frontier?
The efficient frontier is the set of feasible portfolios for which expected return cannot be increased without increasing volatility, and volatility cannot be reduced without reducing expected return. It plots the best available risk-return trade-offs from a chosen set of assets, and any portfolio below the curve is dominated by one on it.
Harry Markowitz’s mean-variance framework turns an asset-selection question into an allocation question. Each point represents an entire set of weights.
Efficient frontier vs minimum-variance frontier
The minimum-variance frontier selects the lowest variance for every feasible expected-return target. Its global minimum is the lowest-volatility portfolio across those targets.
The efficient frontier is the nondominated branch extending from that minimum toward higher expected returns. Low-return points on the other branch are dominated by feasible portfolios with at least as much expected return and no more volatility.
Formula
Portfolio return E(Rp) = w1 × E(R1) + w2 × E(R2); Portfolio variance σ_p² = w1² × σ1² + w2² × σ2² + 2 × w1 × w2 × ρ12 × σ1 × σ2- w1 and w2 are the portfolio weights of the two assets and sum to one.
- E(R1) and E(R2) are the expected returns; σ1 and σ2 are the standard deviations of each asset.
- ρ12 is the return correlation. For positive weights in both assets, correlation below one lowers volatility relative to the same allocation at perfect positive correlation.
Efficient frontier example: a 50/
Asset A has an expected return of 6% and a standard deviation of 10%. Asset B has an expected return of 10% and a standard deviation of 18%. Their correlation is 0.2. A trader blends them 50/50.
- Portfolio return: 0.5 × 6% + 0.5 × 10% = 8%
- Portfolio variance: 0.25 × 100 + 0.25 × 324 + 2 × 0.25 × 0.2 × 10 × 18 = 124
- Portfolio risk: square root of 124 = 11.1%, below the 14% weighted average of the two standard deviations
The 50/50 allocation has 8% expected return and 11.14% volatility. Its perfect-correlation reference is 14% volatility. The global minimum for these inputs holds 81.8% A and 18.2% B, producing 6.7% expected return and 9.4% volatility. The efficient branch continues from that minimum toward higher expected returns.
A has 6% expected return and 10% volatility. B has 10% expected return and 18% volatility. Correlation is 0.20.
Equal weights produce 8% expected return. In percentage-point units, portfolio variance is 0.25 × 100 + 0.25 × 324 + 2 × 0.25 × 0.20 × 10 × 18 = 124.
Volatility is √124 = 11.14%. The same positive-weight allocation at perfect positive correlation has 14% volatility. That 2.86-point gap is the volatility diversification removes.
The global minimum holds 81.8% A and 18.2% B, with 9.4% volatility and 6.7% expected return. The chart follows the efficient branch from that mix to 100% B.
The capital allocation line and the maximum-Sharpe portfolio
Introduce a 2% risk-free rate. The maximum-Sharpe risky mix holds 60.6% A and 39.4% B. Its expected return is 7.6%, volatility is 10.2%, and Sharpe ratio is 0.55.
The line from 2% through that point has the same slope. Combining the risky mix with cash moves along the lending segment; borrowing extends the line when the selected financing rules allow it.
Every risky portfolio has a capital allocation line. The maximum-Sharpe portfolio selects the steepest available one for this positive-premium example.
Efficient frontier constraints and real allocations
Define long-only rules, holding limits and turnover constraints before solving. Compare portfolios using the same return horizon and risk measure. Read each candidate on expected return, volatility, concentration and scenario P&L before choosing it.
Correlation affects each pair through its actual weights. For two positive weights, lower correlation reduces the variance cross term. Across many assets, each pair’s correlation enters at its actual weights and volatilities, so the heaviest pairs move portfolio risk the most.
Limitations of the efficient frontier
- Stress the inputs before trusting the weights. Expected returns are forecasts, and the maximum-Sharpe weights shift with every change in them or in the covariances, so rerun the 60.6% / 39.4% split at nearby inputs before sizing it.
- Add a tail check. Volatility treats gains and losses alike, so measure drawdowns and fat-tail scenarios on the chosen mix before committing capital.
- Re-estimate on a schedule. Volatilities and correlations change, so rebuild the frontier as new data arrives and rebalance when the efficient mix moves.
- Write the rules into the solve. Long-only limits and holding caps change the curve and the maximum-Sharpe mix when they bind, and borrowing terms set how far the capital allocation line extends.
- Price the leverage at your real rate. The capital allocation line borrows and lends at the 2% risk-free rate, and a higher borrowing rate flattens the line beyond the risky mix.
Bottom line
The efficient frontier shows the best expected return available at each level of volatility from a chosen set of assets and rules. Use it to see what diversification buys and to compare allocations on one basis, and rerun the exact weights whenever the return and covariance inputs change.
Clear answers
Frequently Asked Questions
Sources and Further Reading
- CFA Institute: Building Optimized Portfolios with JPMorgan Forecasts
- CFA Institute: Portfolio Risk and Return, Part II
- CFA Institute: Portfolio Risk and Return, Part I
How entries are sourced, calculated and updated: editorial methodology.