Two Asset Efficient Frontier Calculator
Compare two asset return and risk assumptions, view the long only efficient frontier, and calculate minimum variance and maximum Sharpe mixes.
Earlier editorial review
- Reviewed
- 2026-09-22
- Methodology
- 1.1
Sources
How It Works
Enter expected annual returns, annualized volatilities, correlation, and a risk free rate. All rate inputs use annual percentage points, such as 8 for 8%, and no currency amounts are used. For each Asset A weight from 0% to 100%, the calculator assigns the remainder to Asset B and computes weighted return and portfolio variance. It draws 101 sampled blends, then calculates the long only minimum variance mix and finite maximum Sharpe candidate analytically.
The complete opportunity curve is not all efficient. When Asset A has the higher expected return, the highlighted efficient segment runs from the minimum variance mix toward 100% Asset A. If Asset B has the higher return, it runs in the other direction. A blend on the opposite branch has a lower expected return than an available blend with the same risk. Equal expected returns collapse the efficient branch to the minimum variance point because extra volatility buys no additional modeled return.
Use the chart to compare assumptions, not to forecast outcomes. Expected returns can strongly affect the tangency result. Recalculate with lower return differences, higher volatilities, and correlations closer to one. A range of scenarios is more informative than a precise weight based on uncertain estimates.
Record a base case and stress cases for return, volatility, and correlation. If the calculated weights change sharply across them, treat the apparent optimum as sensitive to the inputs.
Step by step
- Open the Two Asset Efficient Frontier and enter your current inputs.
- Review calculated outputs and summary tables.
- Adjust assumptions and compare scenarios side by side.
Worked example
Assume Asset A has an expected annual return of 8% and annual volatility of 10%, Asset B has an expected annual return of 12% and annual volatility of 20%, correlation is 0, and the risk free rate is 2%. The minimum variance weights are 80% in A and 20% in B. The modeled return is 8.8% and volatility is about 8.9443%.
For those same inputs, the maximum Sharpe weights are about 70.5882% in A and 29.4118% in B. The modeled return is about 9.1765%, volatility is about 9.1885%, and Sharpe ratio is about 0.7810. These are mathematical outputs from assumptions, not predicted results. The chart shows all 101 sampled mixes and highlights the upper efficient branch.
When to use this calculator
Use this calculator to compare two asset assumptions, see how correlation changes the modeled risk curve, or inspect the minimum variance and maximum Sharpe mixes. It is suited to transparent scenarios where each input can be varied.
Use a multi asset tool when you need to model more than two holdings, a risk parity tool when you do not want to optimize expected returns, or a drawdown tool to examine loss paths. Review assumptions carefully before making an allocation decision.
Common mistakes
Enter 10 for 10%, not 0.10. Keep expected returns, annualized volatility, and risk free rate in annual percentage points. Correlation is a coefficient between minus one and one, not a percentage.
Do not label the entire opportunity curve efficient, choose a tangency result without testing return estimates, or infer future drawdown from standard deviation. When both modeled excess returns are negative, the least unfavorable Sharpe result can still be negative and does not become an attractive investment recommendation.
The Formulas
rₚ = wₐ × rₐ + (1 − wₐ) × rᵇ
σₚ² = wₐ²σₐ² + (1−wₐ)²σᵇ² + 2ρσₐσᵇwₐ(1−wₐ)
wₘᵢₙ = (σᵇ² − ρσₐσᵇ) / (σₐ² + σᵇ² − 2ρσₐσᵇ)
Max Sharpe = w that maximizes (rₚ − rᶒ) / σₚ
Returns, volatility, and the risk free rate use the same annual percentage point units, so enter 8 for 8%. Weights are long only and range from 0% to 100%. A zero risk portfolio with positive excess return has no finite Sharpe maximum.
Limitations and assumptions
This is a two asset, long only, one period model. It treats entered annual returns, volatility, and correlation as internally consistent assumptions, though future values can differ and correlations can change. Variance does not distinguish upside from downside. The model excludes estimation error, tail behavior, drawdowns, taxes, fees, liquidity, rebalancing, borrowing spreads, and personal constraints. The chart is not a forecast, and the tangency result is not an investment recommendation.
Key terms
- What is the efficient frontier
- The efficient frontier shows the set of portfolios that offer the highest expected return for a given level of risk.
- What is the minimum variance portfolio
- The minimum variance portfolio is the combination of two assets that produces the lowest possible portfolio volatility.
- Tangency portfolio
- The tangency portfolio is the feasible long only mix that maximizes the Sharpe ratio, calculated as excess return per unit of volatility.
Compare alternatives
This tool compares two assets using entered annual return and volatility assumptions. Risk Parity Allocator uses volatility and correlation to model equal risk contributions without expected returns. Sharpe Sortino Ratio measures a historical return series rather than an expected return and risk pair.
Compare the frontier with Maximum Drawdown when the sequence of losses matters. A portfolio can have attractive expected return and variance assumptions yet experience an unacceptable historical or hypothetical peak to trough decline.
FAQ
What is the efficient frontier?
The opportunity set contains every portfolio available from the two assets under the weight constraint. The efficient frontier is only the upper branch of that set: from the minimum variance point toward the asset with the higher expected return. A point on the lower branch is inefficient because another available mix offers more expected return for the same volatility. This calculator draws the complete opportunity curve in a muted color and highlights the efficient branch so the distinction remains visible.
What is the minimum variance portfolio?
The minimum variance portfolio is the feasible mix with the lowest calculated standard deviation. Its analytic weight depends on both volatilities and their covariance. This implementation is long only, so an analytic weight below zero is clamped to zero and a weight above one is clamped to one. Minimum variance does not mean minimum drawdown, maximum return, or best personal portfolio. Those questions depend on return paths, liabilities, constraints, estimation error, and investor objectives that variance alone does not capture.
What is the tangency portfolio?
It is the long only mix with the highest modeled Sharpe ratio, calculated as expected return minus the entered risk free rate, divided by volatility. The result compares an interior analytical candidate with both pure asset endpoints, including when every excess return is negative. If a feasible zero risk portfolio has positive excess return, the modeled ratio is unbounded and no finite tangency point is shown. This mathematical case is not a borrowing or investment recommendation.
How does correlation affect the frontier?
Correlation ranges from minus one to one and combines with the two volatilities to determine portfolio variance. Lower correlation can reduce modeled volatility because the assets offset some variation. At minus one, a particular mix can have zero modeled risk; at plus one, the risk curve is a straight line. Real correlations change over time, so test a range instead of treating one estimate as permanent.
Which portfolio should I choose?
No single point is automatically right for every investor. Minimum variance answers a narrow volatility question, while tangency answers a narrow modeled Sharpe question. A suitable allocation must also reflect loss capacity, liquidity, time horizon, liabilities, taxes, trading cost, and confidence in the inputs. Compare conservative and optimistic return assumptions, then vary volatility and correlation. If small input changes cause a large weight change, treat the apparent optimum as fragile.
Related Tools
Measure realized risk adjusted performance with Sharpe Sortino Ratio, inspect loss paths with Maximum Drawdown, and expand to a covariance based multi asset allocation with Risk Parity Allocator. Each tool answers a different question from the same broad portfolio decision.