Enter each asset's expected return and volatility. Quantora runs Markowitz mean-variance optimization to find the minimum-variance and maximum-Sharpe portfolios, then plots the efficient frontier and the capital market line. Toggle short-selling on or off.
Assets
Asset
Exp. return %
Volatility %
How it works
For a set of assets with expected returns μ and a covariance matrix Σ, every portfolio plots somewhere on the risk-return plane. The efficient frontier is the upper edge - the best return for each level of risk. Two portfolios on it matter most: the minimum-variance portfolio (lowest possible risk) and the tangency portfolio (highest Sharpe ratio, where a line from the risk-free rate just touches the frontier). Mixing the tangency portfolio with cash traces the capital market line.
With short-selling (closed form):
A = 1' Σ⁻¹ 1 B = 1' Σ⁻¹ μ C = μ' Σ⁻¹ μ D = AC − B²
Min-variance: w = Σ⁻¹1 / A Tangency: w = Σ⁻¹(μ−rf) / 1'Σ⁻¹(μ−rf)
Long-only (no closed form): projected-gradient over the simplex {w ≥ 0, Σw = 1},
with a Monte-Carlo cloud of random long-only portfolios shown behind the frontier.
Unconstrained optimization can produce large short positions because it amplifies small estimated edges. Turning short-selling off restricts every weight to be non-negative - the realistic case for most mandates - which lowers the attainable Sharpe but yields investable weights. The faint dots are thousands of random long-only portfolios; the frontier is their upper envelope.
For analysis & education - not investment advice. Results depend entirely on your expected-return, volatility and correlation inputs, which are uncertain; mean-variance optima are highly sensitive to those estimates. Quantora is not a registered investment adviser or broker-dealer.