Portfolio optimization chooses asset allocations to optimize an objective under specified constraints. In a one-period mean-variance optimization model, the return vector has mean and covariance matrix . Portfolio weights give expected return and return variance . A standard problem isThe constraints describe a fully invested portfolio, a target expected return and no short selling. Different assumptions produce different optimization problems. Harry Markowitz made the dependence on covariance and the tradeoff between mean and variance central to portfolio selection.[ref]
Portfolio Selection by Harry Markowitz. 1952, printed pages 80–82: weighted returns, covariance, budget and efficient portfolios
For two uncorrelated assets with expected returns and , each with standard deviation , a weight in the first gives variance . Its unconstrained minimum is at , yielding mean and standard deviation . A target of at least instead requires , making the constrained minimum . The same covariance matrix therefore gives a different solution when the objective's feasible set changes.
Estimated means and covariances are model inputs, rather than known future outcomes. Sampling uncertainty, transaction costs and additional allocation restrictions can materially change the resulting weights.
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Portfolio optimization is a quantitative method used in finance to allocate assets in a way that maximizes expected returns for a given level of risk, or alternatively, minimizes risk for a desired level of expected return. The goal is to create a well-balanced portfolio that achieves the best possible outcome based on the investor's risk tolerance, investment objectives, and constraints. Key concepts in portfolio optimization include: 1. **Risk and Return**: Investors seek to maximize returns while managing risk.