Irish Calculators
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Efficient Frontier Calculator for Long-Only Portfolios

Explore minimum-variance allocations at different expected returns using your own annual inputs. The frontier is long-only and does not establish which portfolio suits you.

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Your numbers

2–8 rows, one comma-separated row per asset. Decimal-return variance units: 20% volatility gives 0.04 on the diagonal. Same asset order throughout.
One percentage per asset, matching matrix order. These are assumptions, not forecasts.
Your figures stay on this device.

How the calculation works

The efficient frontier shows modelled trade-offs between expected return and variance across long-only portfolios. Unlike a single minimum-variance allocation, it solves several return targets and retains the upper branch where additional expected return requires the relevant risk trade-off.

For each target return, minimise wᵀΣw with sum(w)=1, μᵀw=target and w≥0.

  1. Locate the minimum-risk starting point

    The solver first minimises variance over non-negative weights that sum to one. When equal-risk solutions exist, it retains the one with the higher supplied expected return.

  2. Set feasible return targets

    Targets extend along the upper efficient branch towards the highest supplied asset return. Lower-return portfolios with no risk advantage are not presented as efficient choices.

  3. Solve each constrained allocation

    At each target, weights must meet both the total-weight and return constraints. The table and plot report their combined expected return, volatility and allocation.

Worked example: Check Minimum-variance expected return

Sample inputs demonstrate the method. They are not a forecast or product offer.

The first example has equally risky, uncorrelated assets with expected returns of 5% and 10%. The minimum-variance allocation splits equally, so its expected return is their weighted average of 7.5%.

Annual covariance matrix
0.04,0 0,0.04
Expected annual returns (%)
5,10
Frontier points
3

Expected result

Minimum-variance expected return
7.5%
Worked example: Check Minimum-variance expected return

Sample inputs demonstrate the method. They are not a forecast or product offer.

Annual covariance matrix
0.01,0 0,0.04
Expected annual returns (%)
5,10
Frontier points
3

Expected result

Minimum-variance expected return
6%

Understanding your result

Every point is conditional on the covariance matrix and expected returns. Small input changes can move weights substantially, especially when assets behave similarly.

The frontier does not include short selling, transaction costs, turnover limits or an optimised risk-free allocation. It is not a recommendation to select the highest-return point.

Questions you might have

What does the result include?

Build the long-only minimum-variance frontier from your expected annual returns and covariance matrix.

Which limits should I check?

2–8 assets; the upper efficient branch only. Covariances and expected returns must use the same annual basis.

Choosing the right inputs

Annual covariance matrix
Use a symmetric annual covariance matrix in decimal-return variance units, with two to eight assets.
Expected annual returns (%)
Enter one expected annual percentage for each asset in the same order as the matrix. These are assumptions, not downloaded forecasts.
Frontier points
This controls how many frontier points are requested. More points describe the curve more finely but do not improve the reliability of the underlying estimates.

Common mistakes to avoid

  • Do not mix monthly covariance with annual expected returns.
  • A mathematically feasible matrix is not evidence that historical relationships will persist.

Assumptions to check

  • 2–8 assets; the upper efficient branch only. Covariances and expected returns must use the same annual basis.
  • All calculations run on your device. Figures are scenarios based on your inputs, not investment recommendations.

Sources and method

Read our calculation and testing approach.