Finance & Insurance
Portfolio Optimization · Risk · Allocation
Since Markowitz’s 1952 landmark paper introduced mean-variance optimisation, mathematical programming has been the backbone of quantitative finance. Nobel-prize-winning frameworks — from CAPM to Black-Scholes — all rest on optimisation under uncertainty. Modern robust optimisation, pioneered by Ben-Tal and Nemirovski, replaces fragile point estimates with uncertainty sets, producing portfolios that perform well under the worst plausible realisation of asset returns.
Classical Markowitz mean-variance optimisation assumes the investor knows each asset’s expected return μ and covariance matrix Σ exactly. In practice, return estimates carry substantial estimation error — small perturbations in μ can produce wildly different allocations. Robust portfolio optimisation addresses this by defining an uncertainty set around the estimated returns and maximising the worst-case portfolio return within that set. The resulting allocations are more diversified and less sensitive to input noise, at the cost of a modest reduction in expected return under the nominal scenario.
Problem type: Quadratic programming / second-order cone programming. Allocate capital across n assets to minimise portfolio variance for a target return, or equivalently maximise return for a given risk budget. The robust variant replaces the point-estimate return constraint with a worst-case guarantee over an ellipsoidal uncertainty set.
s.t. wT μ ≥ rtarget // return target
wT 1 = 1 // fully invested
w ≥ 0 // no short selling
s.t. min{μ ∈ U} wT μ ≥ rtarget
U = {μ : ||Σ-½(μ - μ0)|| ≤ κ} // ellipsoidal uncertainty
wT 1 = 1, w ≥ 0
Portfolio Solver
- Markowitz, H. (1952). Portfolio Selection. The Journal of Finance, 7(1), 77-91. Published
- Ben-Tal, A. & Nemirovski, A. (1998). Robust Convex Optimization. Mathematics of Operations Research, 23(4), 769-805. Published
- Fabozzi, F.J., Kolm, P.N., Pachamanova, D.A. & Focardi, S.M. (2007). Robust Portfolio Optimization and Management. John Wiley & Sons. Published
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