Andrzej Palczewski Institute of Applied Mathematics Warsaw University June 29, 2008

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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 portfolio problem has been studied since 1980’s (see Merton [9], Jobson and Korkie [6], Michaud [10], Chopra and Ziemba [3]). Particularly the opinion of Michaud, who called portfolio optimizers – error maximizers, indicates practical diﬃculties in applying Markowitz method. All the above mentioned papers and a number of recent publications emphasize that the main source of errors in portfolio optimization is the inaccurate estimation of expected value µ of future returns. Merton claimed that to obtain a reasonable estimate of the mean we need about 100 years of monthly data. DeMiguel, Garlappi and Uppal [4] estimated that for a portfolio of 50 assets 600 months (50 years) of data is required.

Due to the paradigm of Markowitz µ and Σ should be the moments of the distribution of future returns from risky assets. The market provides only the information about historic (past) returns. This means that we have to predict the moments of future returns using past returns, In what follows we shall describe methods which can be justiﬁed only if the time series of market returns is a realization of an i.i.d. se- which use a reasonable set of market data and quence of random variables. In fact, a number produce ”good” optimal portfolios, i.e. portfo-

lios well diversiﬁed and with assets shares stable with respect to estimation errors. We restrict our analysis to the elliptic distributions which are fully characterized by their ﬁrst two moments (mean and covariance). The normal distribution and t-Student distribution are both examples of elliptic distributions. Hence the class is suﬃciently rich for practical purposes.

where q is the vector of portfolios’ returns and ε in the prediction error with the distribution N (0, Ω). Investor’s views play a role of an observation in Bayesian statistics. The posterior distribution of µ, given the predictions, is normal and has the mean given by the following formula: µeq + ΣP T (Ω/τ + P ΣP T )−1 (q − P µeq ). (1)

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This is the...