Portfolio Optimization
Portfolio optimization - a practical approach
Andrzej Palczewski Institute of Applied Mathematics Warsaw University June 29, 2008
1
Introduction
The construction of the best combination of investment instruments (investment portfolio) is a principal goal of investment policy. This is an optimization problem: select the best portfolio from all admissible portfolios. To approach this problem we have to choose the selection criterion first. The seminal paper of Markowitz [8] opened a new era in portfolio optimization. The paper formulated the investment decision problem as a risk-return tradeoff. In its original formulation it was, in fact, a mean-variance optimization with the mean as a measure of return and the variance as a measure of risk. To solve this problem the distribution of random returns of risky assets must be known. In the standard Markowitz formulation returns of these risky assets are assumed to be distributed according to a multidimensional normal distribution N (µ, Σ), where µ is the vector of means and Σ is the covariance matrix. The solution of the optimization problem is then carried on under implicit assumption that we know both µ and Σ. In fact this is not true and the calculation of µ and Σ is an important part of the solution.
of market observations (so called stylized facts) shows that returns deviate from the i.i.d. assumptions. In addition, normal distribution seems to be a very coarse approximation of real returns (in a number of recent papers it is rather the tStudent distribution which fits better to reality). The error due to the fact that market returns are not normal and deviate form i.i.d. assumption is called model risk (or model error). Another source of errors in calculating µ and Σ stems from the finiteness of the sample. This kind of error (called estimation error or estimation risk) is particularly important in practical calculations where the sample is of a limited size. The effect of the estimation error to the
Andrzej Palczewski Institute of Applied Mathematics Warsaw University June 29, 2008
1
Introduction
The construction of the best combination of investment instruments (investment portfolio) is a principal goal of investment policy. This is an optimization problem: select the best portfolio from all admissible portfolios. To approach this problem we have to choose the selection criterion first. The seminal paper of Markowitz [8] opened a new era in portfolio optimization. The paper formulated the investment decision problem as a risk-return tradeoff. In its original formulation it was, in fact, a mean-variance optimization with the mean as a measure of return and the variance as a measure of risk. To solve this problem the distribution of random returns of risky assets must be known. In the standard Markowitz formulation returns of these risky assets are assumed to be distributed according to a multidimensional normal distribution N (µ, Σ), where µ is the vector of means and Σ is the covariance matrix. The solution of the optimization problem is then carried on under implicit assumption that we know both µ and Σ. In fact this is not true and the calculation of µ and Σ is an important part of the solution.
of market observations (so called stylized facts) shows that returns deviate from the i.i.d. assumptions. In addition, normal distribution seems to be a very coarse approximation of real returns (in a number of recent papers it is rather the tStudent distribution which fits better to reality). The error due to the fact that market returns are not normal and deviate form i.i.d. assumption is called model risk (or model error). Another source of errors in calculating µ and Σ stems from the finiteness of the sample. This kind of error (called estimation error or estimation risk) is particularly important in practical calculations where the sample is of a limited size. The effect of the estimation error to the
References: REVSTAT Stat. J., 5 (2007), 97–114. [1] Black, F., Litterman, R. – Global portfolio optimization, Financial Analysts J., 48 (1992), 28–43. [2] Chan, L., Karceski, J., Lakonishok, J. – On portfolio optimization: Forecasting covariances and choosing the risk model, Rev. Financial Stud., 12 (1999), 937–974. 4