Portfolio optimization
Portfolio optimization is the process of selecting an optimal
Modern portfolio theory
Optimization methods
The portfolio optimization problem is specified as a constrained utility-maximization problem. Common formulations of portfolio utility functions 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). For return distributions that are Gaussian, this is equivalent to maximizing a certain quantile of the return, where the corresponding probability is dictated by the risk aversion parameter. Practitioners often add additional constraints to improve diversification and further limit risk. Examples of such constraints are asset, sector, and region portfolio weight limits.
Specific approaches
Portfolio optimization often 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. For the specific formulas for efficient portfolios,[5] see Portfolio separation in mean-variance analysis.
One approach to portfolio optimization is to specify a
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. Moreover, in this context, the approach provides a method for determining the entire set of efficient portfolios. Its application here was later explicated by William Sharpe.[7]
Mathematical tools
The complexity and scale of optimizing portfolios over many assets means that the work is generally 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:
- Linear programming[8][9]
- Quadratic programming
- Nonlinear programming
- Mixed integer programming
- Meta-heuristic methods[10]
- Stochastic programming for multistage portfolio optimization[11]
- Copula based methods[12]
- Principal component-based methods
- Deterministic global optimization
- Genetic algorithm[13]
Optimization constraints
Portfolio optimization is usually done subject to constraints, such as regulatory constraints, or illiquidity. These constraints can lead 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.[14]
Regulation and taxes
Investors may be forbidden by law to hold some assets. In some cases, unconstrained portfolio optimization would lead to
Transaction costs
Transaction costs are the costs of trading 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.
Concentration risk
Concentration risk refers to the risk caused by holding an exposure to a single position or sector that is large enough to cause material losses to the overall portfolio when adverse events occur. If the portfolio is optimized without any constraints with regards to concentration risk, the optimal portfolio can be any risky-asset portfolio, and therefore there is nothing to prevent it from being a portfolio that invests solely in a single asset. Managing concentration risk should be part of a comprehensive risk management framework[15] and to achieve a reduction in such a risk it is possible to add constraints that force upper bound limits to the weight that can be attributed to any single component of the optimal portfolio.
Improving portfolio optimization
Correlations and risk evaluation
Different approaches to portfolio optimization measure risk differently. In addition to the traditional measure,
Investment is a forward-looking activity, and thus the covariances of returns 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.[16]
In a mean-variance optimization framework, accurate estimation of the
Other optimization strategies that focus on minimizing tail-risk (e.g.,
More recently, hedge fund managers have been applying "full-scale optimization" whereby any investor utility function can be used to optimize a portfolio.[20] It is purported that such a methodology is more practical and suitable for modern investors whose risk preferences involve reducing
Cooperation in portfolio optimization
A group of investors, instead of investing individually, may choose to invest their total capital into the joint portfolio, and then divide the (uncertain) investment profit in a way which suits best their utility/risk preferences. It turns out that, at least in the expected utility model,[22] and mean-deviation model,[23] each investor can usually get a share which he/she values strictly more than his/her optimal portfolio from the individual investment.
See also
- Outline of finance § Portfolio theory for related articles
- Asset allocation
- Chance-constrained portfolio selection
- Intertemporal portfolio choice
- Financial risk management § Investment management
- List of genetic algorithm applications § Finance and Economics
- Machine learning § Applications
- Marginal conditional stochastic dominance, a way of showing that a portfolio is not efficient
- Merton's portfolio problem
- Mutual fund separation theorem, giving a property of mean-variance efficient portfolios
- Portfolio theory, for the formulas
- Risk parity / Tail risk parity
- Stochastic portfolio theory
- Universal portfolio algorithm, giving the first online portfolio selection algorithm
References
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- ^ Merton, Robert. September 1972. "An analytic derivation of the efficient portfolio frontier," Journal of Financial and Quantitative Analysis 7, 1851–1872.
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- ^ The Critical Line Method in William Sharpe, Macro-Investment Analysis (online text)
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- ^ Sefiane, Slimane and Benbouziane, Mohamed (2012). Portfolio Selection Using Genetic Algorithm Archived 29 April 2016 at the Wayback Machine, Journal of Applied Finance & Banking, Vol. 2, No. 4 (2012): pp. 143–154.
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- ^ "Concentrate on Concentration Risk | FINRA.org". www.finra.org. 15 June 2022. Retrieved 16 March 2024.
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- ^ Grechuk, B., Molyboha, A., Zabarankin, M. (2013). "Cooperative games with general deviation measures", Mathematical Finance, 23(2), 339–365.
Bibliography
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