The objective typically maximizes factors such as expected return, and minimizes costs like financial risk, resulting in a multi-objective optimization problem.
Factors being considered may range from tangible (such as assets, liabilities, earnings or other fundamentals) to intangible (such as selective divestment).
[1][2] The model assumes that an investor aims to maximize a portfolio's expected return contingent on a prescribed amount of risk.
For realistic utility functions in the presence of many assets that can be held, this approach, while theoretically the most defensible, can be computationally intensive.
Harry Markowitz[6] developed the "critical line method", a general procedure for quadratic programming that can handle additional linear constraints and upper and lower bounds on holdings.
This is related to the topic of tracking error, by which stock proportions deviate over time from some benchmark in the absence of re-balancing.
(Alternatively, the model-implied weights are optimal in the sense of achieving the returns matching the manager's "views".)
Portfolio optimization assumes the investor may have some risk aversion and the stock prices may exhibit significant differences between their historical or forecast values and what is experienced.
In particular, financial crises are characterized by a significant increase in correlation of stock price movements which may seriously degrade the benefits of diversification.
Quantitative techniques that use Monte-Carlo simulation with the Gaussian copula and well-specified marginal distributions are effective.
Not accounting for these attributes can lead to severe estimation error in the correlations, variances and covariances that have negative biases (as much as 70% of the true values).
To minimize exposure to tail risk, forecasts of asset returns using Monte-Carlo simulation with vine copulas to allow for lower (left) tail dependence (e.g., Clayton, Rotated Gumbel) across large portfolios of assets are most suitable.
A suitable methodology that allows for the joint distribution to incorporate asymmetric dependence is the Clayton Canonical Vine Copula.