Portfolio optimization

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Portfolio optimization is the process of choosing the proportions of various assets to be held in a portfolio, in such a way as to make the portfolio better than any other according to some criterion. The criterion will combine, directly or indirectly, considerations of the expected value of the portfolio's rate of return as well as of the return's dispersion and possibly other measures of financial risk.

Efficient portfolios

Modern portfolio theory, fathered by Harry Markowitz[1][2] 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,[3] see Portfolio separation in mean-variance analysis.

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.

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:

Issues with 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.[6]

See also

References

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  2. Lua error in package.lua at line 80: module 'strict' not found. (reprinted by Yale University Press, 1970, ISBN 978-0-300-01372-6; 2nd ed. Basil Blackwell, 1991, ISBN 978-1-55786-108-5)
  3. 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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  6. Ragunathan V., Mitchell H., Modelling the Time-Varying Correlation Between National Stock Market Returns