How to use the efficient frontier calculator
- List the assets. For each asset class or fund, enter an expected annual return and an annualized volatility. These are forward-looking assumptions: capital market assumptions published by asset managers, your own estimates, or long-run averages adjusted for today’s yields. The example uses four broad asset classes.
- Enter the correlation matrix, one row per line, in the same order as the assets. You can paste a full matrix, type only the lower triangle, or enter a single number to give every pair the same correlation. The matrix must be positive definite; the calculator tells you if it is not.
- Set the risk-free rate, typically the Treasury bill yield. It only affects the maximum Sharpe portfolio and the capital market line.
- Choose long-only or short selling allowed. Long-only keeps every weight between 0% and 100%, which is what most investors can actually hold.
The results show both special portfolios, the chart traces the frontier, and the tables list every asset’s weight and about 30 portfolios along the curve.
How the portfolios are calculated
With weights w, expected returns μ and covariance matrix Σ (built from the volatilities and correlations), a portfolio’s expected return and volatility are:
When short selling is allowed, all three answers have closed forms:
Maximum Sharpe: w = Σ−1(μ − rf1) ÷ 1ᵀΣ−1(μ − rf1)
Frontier: σ2(m) = (A m2 − 2B m + C) ÷ (AC − B2)1 is a vector of ones, rf the risk-free rate and m a target return. A = 1ᵀΣ−11, B = 1ᵀΣ−1μ, C = μᵀΣ−1μ. The tangency portfolio exists only when rf is below the minimum variance return.
Long-only weights have no formula. Each portfolio solves a quadratic program: minimize wᵀΣw subject to the weights summing to 1, a target return, and w ≥ 0. The calculator solves it exactly by active-set enumeration: for every possible set of held assets it solves the equality-constrained problem in closed form, discards sets that would need a negative weight, and keeps the lowest-variance survivor. Because the problem is strictly convex, that survivor is the global optimum. The maximum Sharpe portfolio is found the same way, after rescaling it into the equivalent problem of minimizing variance for one unit of excess return.
A worked example
Four asset classes: US stocks (7.5% expected return, 16% volatility), international stocks (8%, 18%), investment-grade bonds (4.5%, 6%) and commodities (5%, 20%). Stocks are correlated 0.8 with each other and only weakly with bonds; the risk-free rate is 4%, long-only.
The minimum variance portfolio holds 86.8% in bonds and small slices of everything else. Its volatility is 5.64%, below the 6% of bonds alone, because the other assets are barely correlated with bonds. It is expected to return 4.76%.
The maximum Sharpe portfolio holds 25.6% US stocks, 30.3% international stocks, 44.1% bonds and nothing in commodities. It is expected to return 6.33% with 9.64% volatility, a Sharpe ratio of 0.242 against 0.222 for the best single asset.
Commodities drop out of the tangency portfolio even though they help the minimum variance mix: with a 5% expected return they add too little excess return to pay for their 20% volatility.
Reading the chart
The blue curve is the efficient frontier: for each level of volatility, the highest expected return available from these assets. It starts at the minimum variance portfolio and rises to the highest-return asset (long-only) or beyond (with shorting). Portfolios below or to the right of the curve take more risk than necessary for their return.
The dashed line is the capital market line. It starts at the risk-free rate and touches the frontier at the maximum Sharpe portfolio. Every point on it is a mix of cash and that one portfolio, and every such mix has the same Sharpe ratio. This is Tobin’s separation result: an investor who can lend and borrow at the risk-free rate should hold the tangency portfolio and adjust risk with cash, rather than move along the curve. Points above the tangency assume borrowing at the risk-free rate, which few investors can do.
Long-only or short selling
| Setting | Max Sharpe ratio | Return | Volatility | Short positions |
|---|---|---|---|---|
| Long-only | 0.242 | 6.33% | 9.64% | none |
| Short selling allowed | 0.244 | 6.47% | 10.16% | 6.8% |
Removing the constraint lets the optimizer short commodities (-6.8%) to fund larger positions in the other three assets. The improvement in the Sharpe ratio is small, which is common: the long-only constraint often blocks positions that look attractive only because of noise in the inputs. The minimum variance portfolio is the same in both settings here because none of its weights wanted to go negative.
Why optimizers concentrate: estimation error
A mean-variance optimizer treats its inputs as exact. They are not. With realistic volatility, an asset’s average return measured over twenty years still has an error band of several percentage points, so the expected returns you type in are educated guesses. Studies in the early 1990s found that errors in expected returns do far more damage to optimized portfolios than errors in variances or covariances.
The optimizer’s job is to find differences between assets and exploit them. It cannot tell a real difference from an estimation error, so it piles into the assets whose returns were overestimated and whose risks were underestimated. Richard Michaud described mean-variance optimizers as “estimation-error maximizers” for this reason (1989). The example shows it clearly:
| Change to one input | US stocks | International stocks | Bonds | Max Sharpe |
|---|---|---|---|---|
| The example | 25.6% | 30.3% | 44.1% | 0.242 |
| US stocks expected 8.5% instead of 7.5% | 65.0% | 0.0% | 35.0% | 0.287 |
| International stocks expected 7% instead of 8% | 56.2% | 0.0% | 43.8% | 0.227 |
A one-point change in a single expected return, well inside any reasonable error band, removes international stocks from the portfolio entirely. Two highly correlated assets with slightly different expected returns are the classic trigger: the optimizer sees them as near substitutes and buys only the one that looks better.
Practitioners fight this in several ways:
- Constraints: long-only limits, maximum weights per asset, or bands around a benchmark.
- Better inputs: shrinking expected returns toward a common value, or deriving them from market equilibrium as the Black–Litterman model does, and shrinking the covariance matrix toward a structured target (Ledoit–Wolf).
- Resampling: Michaud’s resampled efficiency averages the optimal weights over many simulated input sets, which spreads weight across near-equivalent assets.
- Ignoring returns: minimum variance and risk parity portfolios avoid the least reliable input altogether.
Out-of-sample research has also found that simple equal weighting is surprisingly hard to beat once estimation error is counted. That is a reason to treat the optimizer’s answer as a starting point, not a verdict.
What moves the answer most
| Change | Max Sharpe weight in bonds | Max Sharpe ratio |
|---|---|---|
| The example (4% risk-free rate) | 44.1% | 0.242 |
| Risk-free rate 2% | 75.4% | 0.530 |
| Stock–bond correlations about 0.5 instead of about 0.1 | 0.0% | 0.232 |
A lower risk-free rate makes bonds’ excess return look larger, so the tangency moves down the frontier toward them. A higher stock–bond correlation removes the diversification benefit that earned bonds their place.
What this calculator does not include
- Uncertainty in the inputs. Results are only as good as the assumptions; the calculator does not resample or shrink them.
- A single period. Mean-variance analysis looks one period ahead and ignores rebalancing, compounding and changing correlations, which tend to rise in crises.
- Volatility as the only risk. Skewness, fat tails, liquidity and drawdowns are not considered.
- Costs, taxes and constraints such as borrowing costs on shorts, turnover limits or minimum position sizes.
- Leverage. With shorting allowed, weights still sum to 100%; the borrowing part of the capital market line is shown only for reference.