Portfolio optimization

Efficient portfolios

Modern portfolio theory, fathered by Harry Markowitz in the 1950s, assumes that an investor wants to maximize a portfolio's expected return contingent on any given amount of risk, with risk measured by the standard deviation of the portfolio's rate of return. For portfolios that meet this criterion, known as efficient portfolios, achieving a higher expected return requires taking on more risk, so investors are faced with a trade-off between risk and expected return. This risk-expected return relationship of efficient portfolios is graphically represented by a curve known as the efficient frontier. All efficient portfolios, each represented by a point on the efficient frontier, are well-diversified. For the specific formulas for efficient portfolios, see Portfolio separation in mean-variance analysis.

Implementation overview

The portfolio optimization problem is most generally specified as a constrained utility-maximization problem (see Constrained optimization). Although portfolio utility functions can take many forms, common formulations define it as the expected portfolio return (net of transaction and financing costs) minus a cost of risk. The latter component, the cost of risk, is defined as the portfolio risk multiplied by a risk aversion parameter (or unit price of risk). Practitioners often add additional constraints to improve diversification and further limit risk. Examples of such constraints are asset, sector, and region portfolio weight limits.

Methods of portfolio optimization

Different approaches to portfolio optimization measure risk differently. In addition to the traditional measure, standard deviation, or its square (variance), which are not robust risk measures, other measures include the Sortino ratio and the CVaR (Conditional Value at Risk).

Often, portfolio optimization takes place in two stages: optimizing weights of asset classes to hold, and optimizing weights of assets within the same asset class. An example of the former would be choosing the proportions placed in equities versus bonds, while an example of the latter would be choosing the proportions of the stock sub-portfolio placed in stocks X, Y, and Z. Equities and bonds have fundamentally different financial characteristics and have different systematic risk and hence can be viewed as separate asset classes; holding some of the portfolio in each class provides some diversification, and holding various specific assets within each class affords further diversification. By using such a two-step procedure one eliminates non-systematic risks both on the individual asset and the asset class level.

One approach to portfolio optimization is to specify a von Neumann-Morgenstern utility function defined over final portfolio wealth; the expected value of utility is to be maximized. To reflect a preference for higher rather than lower returns, this objective function is increasing in wealth, and to reflect risk aversion it is concave. 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.

Optimization constraints

Often portfolio optimization is done subject to constraints, which may be regulatory constraints, the lack of a liquid market, or any of many others. These constraints can lead to extreme weights being applied in the portfolio optimization process leading to portfolio weights that focus on a small sub-sample of assets within the portfolio. When the portfolio optimization process is subject to other constraints such as taxes, transaction costs, and management fees, the optimization process may result in an under-diversified portfolio.

Regulation and taxes

Investors may be forbidden by law to hold some assets. In some cases, unconstrained portfolio optimization would lead to short-selling of some assets. However short-selling can be forbidden. Sometimes it is impractical to hold an asset because the associated tax cost is too high. In such cases appropriate constraints must be imposed on the optimization process.

Transaction costs

Transaction costs are the costs of trading in order to change the portfolio weights. Since the optimal portfolio changes with time, there is an incentive to re-optimize frequently. However, too frequent trading would incur too-frequent transactions costs; so the optimal strategy is to find the frequency of re-optimization and trading that appropriately trades off the avoidance of transaction costs with the avoidance of sticking with an out-of-date set of portfolio proportions. 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.

Mathematical tools used in portfolio optimization

The complexity and scale of optimizing all but the simplest portfolio requires that the work be done by computer. Central to this optimization is the construction of the covariance matrix for the rates of return on the assets in the portfolio.

Techniques include:

Improving portfolio optimization

Investment is a forward looking activity, and thus the covariances of returns and risk levels must be forecast rather than observed. 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.

In a mean-variance optimization framework, accurate estimation of the variance-covariance matrix is paramount. Quantitative techniques that use Monte-Carlo simulation with the Gaussian copula and well-specified marginal distributions are effective. Allowing the modeling process to allow for empirical characteristics in stock returns such as auto-regression, asymmetric volatility, skewness, and kurtosis is important. Not accounting for these attributes lead to severe estimation error in the correlations, variances and covariances that have negative biases (as much as 70% of the true values). Other optimization strategies that focus on minimizing tail-risk (e.g., value at risk, conditional value at risk) in investment portfolios are popular amongst risk averse investors. 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.