Multivariate Shrinkage for Optimal Portfolio Weights

Transcription

1 Multivariate Shrinkage for Optimal Portfolio Weights Vasyl Golosnoy a and Yarema Okhrin b,1 a Institute of Statistics and Econometrics, University of Kiel, Kiel, Germany b Department of Statistics, University of Frankfurt (Oder), Frankfurt (Oder), Germany Abstract The improvement of portfolio selection by means of multivariate shrinkage estimator for the optimal portfolio weights is a subject of this paper. The estimated classical Markowitz weights are shrunk to the vector of current portfolio weights, which is chosen as a shrinkage target. Assuming log asset returns to be Gaussian and i.i.d., the explicit solutions are derived for the optimal shrinkage intensity. The obtained theoretical results are compared with competing approaches within an extensive Monte Carlo simulation study and applied in an empirical example. JEL Classification: C13, C44, G11, G15. Keywords: portfolio optimization, shrinkage estimation, multivariate shrinkage. 1 Corresponding author. address: okhrin@uni-ffo.de, Phone:

2 1 Introduction The classical Markowitz s (1952) portfolio optimization, directly implemented in practice, often fails, despite its methodological appeal. The tendency to maximize the effects of errors in the input assumptions (cf. Michaud, 1989) seems to be the major problem of the classical analysis. Especially dramatic influence on the portfolio allocations is caused by estimation errors in the expected asset returns (cf. Merton, 1980, Best and Grauer, 1991). This paper is concerned with improving the results of Markowitz procedure by applying the shrinkage methodology for incorporating estimation uncertainty directly into investor s utility function. The portfolio selection can be improved by including some additional information into asset allocation procedure. Predictability of asset returns is captured by factor models (for overview see Ferson, 1995, Keim and Hawawini, 1995) by using exogenous variables for forecasting future outcomes. The exogenous variables in factor models is a source of conditional information, that should be included into portfolio optimization. The papers of Brennan, Schwartz and Lagnado (1997), Campbell and Viceira (1999), Brandt (1999), Barberis (2000), Ferson and Siegel (2001) etc. investigate portfolio problem for predictable returns. Another possibility to improve portfolio selection is to modify not the model of returns but the selection procedure itself by diminishing the undesirable effects of estimation uncertainty. Several approaches are developed for improving asset allocation procedure. Frost and Savarino (1988), Michaud (1989, 1998), Grauer and Shen (2000), Jagannathan and Ma (2003), Garlappi, Uppal and Wang (2003) suggest to constrain the portfolio weights in the portfolio optimization problem. Michaud (1998), Scherer (2002) use the resampling from the estimated distribution. They estimate the moments from the historical data, simulate paths of returns using these moments, calculate weights for each path and average over these weights. Klein and Bawa (1976), Jorion (1986), Frost and Savarino (1986), Kandel and Stambaugh (1996), Kan and Zhou (2004) advocate different types of Bayesian methods for portfolio selection. Jorion (1986) and Ledoit and Wolf (2003a,b) propose different types of shrinkage estimators by optimizing the trade-off between the classical estimation procedures and some shrinkage target. Here we introduce a multivariate shrinkage estimator for the optimal portfolio weights. Since the seminal paper of James and Stein (1961) the methodology of shrinkage estimation was further developed among others by Efron and Morris (1975), Berger (1982), Berger and Berliner (1984), George (1986a,b) and Longford (1999). The idea to improve the estimation procedure by shrinking has already been applied in portfolio analysis for estimation of the mean vector by Jorion (1986) and of the covariance matrix by Ledoit 2

3 and Wolf (2003a,b). However, these estimators are in some sense artificial, because there is no objective information what should be used as a shrinkage target. To the contrary, there is a natural shrinkage target for the optimal portfolio weights. This is the fractions of assets in the current portfolio composition. The main idea lies in using the shrinkage factors for linear weighting of the current portfolio weights, held by the investor, and the classical Markowitz estimated optimal portfolio weights, based on the estimated vector of means and covariance matrix. In case of large uncertainty about the estimated optimal weights the investor should stay close to his current holdings, if the uncertainty is reduced the investor should move in the direction of the estimated optimal portfolio weights. The usage of natural shrinkage for the optimal weights brings several advantages compared to the traditional estimation. First, the obtained shrinkage factors are optimal with respect to the investor s utility function to the contrary to the shrinkage estimators of Jorion (1986) and Ledoit and Wolf (2003a,b). The latter are optimal with respect to a quadratic loss criterion, that is not directly related to the investor s utility. The proposed shrinkage estimation procedure incorporates the variance of the classical estimated optimal portfolio weights. That means the estimation risk is also included into portfolio optimization. Additionally, because the current holdings are used as a shrinkage target, the shrinkage optimal weights impose smaller amount of wealth traded during portfolio revision compared to the Markowitz procedure, thus the proportional transaction costs are lower compared to the classical estimator. The explicit solutions for the optimal shrinkage factors for the portfolio problem with and without riskless asset are the main results of the paper. The optimal shrinkage factors are the function of the number of previous returns n, used for the estimation of the distribution parameters. If n reducing the variance of the estimators to zero, the optimal policy is to move far in the direction of the estimated optimal weights, for the small values of n the investor stays not far from the current vector of portfolio weights. The proposed shrinkage methodology is compared with competing approaches in terms of expected utility. The obtained solutions are analyzed both in a simulation study and in a study with empirical data. The rest of the paper is organized as follows. The multivariate shrinkage estimator for the optimal portfolio weights is introduced in Section 2. The explicit solutions for the optimal shrinkage intensity are given in Section 3. The competing approaches are compared analytically in Section 4 and analyzed in the simulation and the empirical study in Sections 5 and 6, respectively. Section 7 concludes the paper. 3

4 2 Shrinkage for the Optimal Weights The considered single-period investor aims to maximize the expected utility of the endperiod wealth. Here and further we assume a multivariate normal distribution for risky assets logarithmic returns. Thus the portfolio problem is equivalent to the expected utility maximization based on the portfolio return r p (cf. Huang and Litzenberger, 1988) max w E U( r p), (1) where w is a vector of selected portfolio weights of risky assets. The classical Markowitz (1952) optimal weights are the function of the unknown moments of returns, that are replaced by the sample vector of means and of the sample covariance matrix, i.e. ˆµ = n 1 t i=t n+1 x i and ˆΣ = (n 1) 1 t i=t n+1 (x i ˆµ)(x i ˆµ), where x i denotes the vector of historical asset returns at time point i. The solution of the portfolio problem is given by the estimated portfolio weights as a function of sample moments and the form of utility function. Unfortunately, the practical application of the Markowitz (1952) approach often leads to unsatisfactory results. Michaud (1989) provides an overview of reasons for this evidence. The main one is caused by estimation errors in the input parameters (cf. Merton, 1980, Best and Grauer, 1991). Here we propose the strategy improvement based on the multivariate shrinkage estimator for the optimal portfolio weight. The idea to use shrinkage methods in the portfolio theory is not a new one. It is initially suggested by Jorion (1986) with the aim to improve the estimator of the expected asset returns. The approach of Jorion (1986) is based on the observation of Stein (1955), that the maximum likelihood estimator for the expected return is inadmissible relative to a quadratic loss function. To the contrary, James-Stein shrinkage estimator (cf. James and Stein, 1961), constructed as a linear weighting of the sample estimator and some arbitrary chosen vector of constants, has lower risk than the sample mean and is admissible in terms of quadratic loss function. Ledoit and Wolf (2003a) extend the approach of Jorion (1986) by using a shrinkage methodology for improving the estimator of the covariance matrix. Their improved estimator is a weighted sum of the sample covariance matrix and of the covariance matrix from the single index model. While the sample covariance matrix is unbiased but has a lot of estimation error, the covariance matrix from the one factor model has comparatively little estimation error but is a biased one. The estimator of Ledoit and Wolf (2003a) is constructed in order to find an optimal balance between the estimation error and bias. 4

5 Both Jorion (1986) and Ledoit and Wolf (2003a) show that their estimator improves the portfolio performance in terms of expected utility. A natural extension of the shrinkage approach is to develop a multivariate shrinkage estimator directly to the optimal portfolio weights. We suggest a shrinkage estimator for the optimal portfolio weights of risky assets in the following form ŵ = û a + c (1 a), (2) where a is a vector of shrinkage factors, c is a vector of portfolio weights currently hold by the investor in his portfolio, û is a vector of optimal portfolio weights estimated from the classical Markowitz procedure, denotes element by element multiplication (Hadamard product), 1 is a vector of ones of corresponding dimension. The multivariate shrinkage can be seen as an extension of the single shrinkage approach (cf. George, 1986a, Longford, 1999), allowing individual trade-off between the true information and estimation risk for each single asset. The proposed shrinkage estimator for the portfolio weights can be justified in the following way. Before portfolio decision, the investor holds the vector of the current portfolio weights c. Then in some black box procedure based on the previous information he obtains the estimator for the vector of the optimal portfolio weights û, that can be seen as a realization of random variables. Now the investor faces the question to what extent he should trust the estimated optimal weights and move away from his current portfolio holdings. The answer to this question is the optimal shrinkage intensity a that maximizes the expected utility of the end-period portfolio return. The asset allocation problem (1) can now be written by max E U( r p ), (3) a where the portfolio weights ŵ are given by equation (2). The maximization problem (3) can be interpreted as follows. The classical Markowitz approach uses so-called two step procedure. First the distribution parameters are estimated, then the estimated parameters are substituted into portfolio optimization task as if they were not the estimators but the true values. Our approach extends the classical procedure one step further. The estimated optimal weights û are treated as a function of a realization of random vectors of returns and are substituted into maximization problem (3). Then the optimal shrinkage intensity a is determined. The investor maximizes the expected utility and takes into account the additional risk, caused by the estimation of the classical optimal weights û. Thus our approach includes the estimation risk into portfolio decision problem. On the one hand the introduction of individual shrinkage factors for each portfolio weight will obviously increase the expected utility. However, on the other hand this also increases 5

6 substantially the number of parameters in the model and leads to a rise in the general estimation error. The trade-off between the number of parameters and estimation precision is a long standing problem in statistical theory and the shrinkage estimation is not an exception. On the contrary to the case discussed above, another extreme is to assume that the only shrinkage factor is equal for all portfolio weights. This approach is advocated by Ledoit and Wolf (2003a,b) for the shrinkage estimation of covariance matrices. In this simplified framework the estimator of the portfolio weights in (2) reduces to and the expected utility maximization is given by ŵ s = a s û + (1 a s )c (4) max a s E U( r p ). (5) Since this special case is of importance in practical applications, we derive all results separately for a single and different shrinkage coefficients. 3 The Optimal Shrinkage Factors The investor chooses the optimal portfolio weights w by maximizing the expected utility function. Here we consider the investor with the quadratic utility function with the risk aversion coefficient γ > 0 given by EU = E( r p ) γ 2 V ar( r p). We assume in this paper that the log returns are i.i.d. with r t N k (µ, Σ) for future returns and x t N k (µ, Σ) for past returns, where N k (, ) stands for k-dimensional Gaussian distribution. Now we consider separately the cases with and without riskless asset in the portfolio. 3.1 Riskless asset is present The investor allocates his wealth among k risky and one riskless asset. The portfolio return is expressed by r p = w r + (1 w 1)r f, then the portfolio optimization problem is given by max w w (µ r f 1) γ 2 w Σw, (6) where r f is a riskless rate of return. The classical solution of this problem is represented by tangency portfolio weights u M with the unknown parameters usually replaced by their 6

7 sample counterparts, i.e. u M = γ 1 Σ 1 (µ r f 1) and û M = γ 1 ˆΣ 1 (ˆµ r f 1). Okhrin and Schmid (2003) derive the first two moments of the estimated optimal portfolio weights under normality assumption for log returns, E(û M ) and Cov(û M ) = Ω M. The formulas of Okhrin and Schmid (2003) are provided in Appendix A. Next we use these results to derive expressions and properties of the shrinkage factors. Taking into account the form of the utility function in (6), the optimization problem in (3) can be rewritten with respect to the weighting factors a in the following way max a M E(ŵ M(r r f 1)) γ 2 V ar(ŵ M(r r f 1)), (7) where ŵ M = (û M c) a M + c. The solution of this portfolio problem is given in the next Theorem. Theorem 1. Assume that the {x t } is a sequence of independent and normally distributed k-dimensional random vectors with the mean µ and the covariance matrix Σ. Then the solution of the maximization problem (7) is given by where a M = Q 1 M ((E(û M) c) Σ (u M c)), Q M = Ω M (Σ + (µ r f 1)(µ r f 1) ) + Σ (E(û M ) c)(e(û M ) c). The first term in the matrix Q M reflects the uncertainty in the two step estimation procedure. Small sample size leads to larger uncertainty about the estimated portfolio weights û M, i.e. the covariance matrix Ω M has a greater impact on the matrix Q M. This, consequently, implies smaller values of the optimal shrinkage coefficients and higher tendency to preserve current portfolio composition. The opposite situation is observed for large samples. In this case the investor is more certain about the model parameters and should tend to the optimal weights of Markowitz. These ideas are formalized in the next corollary, which provides results on the limiting behavior of the optimal shrinkage coefficients a M. Corollary 1. a) If the estimation risk is reduced to zero by setting n, the optimal shrinkage factors a M from Theorem 1 are given by lim n a M = 1. b) If the variance of the estimator of the optimal portfolio weights is infinite, i.e. n < k+5, the optimal shrinkage factors a M from Theorem 1 are given by a M = 0 for n < k

8 In the case of single shrinkage coefficient the optimization problem is given by max E(ŵ a M,s M,s(r r f 1)) γ 2 V ar(ŵ M,s(r r f 1)), (8) where ŵ M,s = a M,s (û M c) + c. The next corollary establishes the optimal shrinkage factors in this case. Corollary 2. Let x t follow the multivariate normal distribution with parameters µ and Σ. Then the solution of (8) is given by a M,s = (E(û M ) c) Σ(u M c) tr [ (Σ + (µ r f 1)(µ r f 1) )Ω M ] + (E(ûM ) c) Σ(E(û M ) c). The proof of the corollary follows along the lines of the proof of Theorem 1 by using the properties of Hadamard product and setting a M = a M,s 1. Similar statement as in Corollary 1 can also be proved in the case of a single shrinkage coefficient. When the optimal different shrinkage coefficients a M expected utility, it is not crucial if some elements of the vector a M from zero to one. coefficient. It appears that a M,s lead to the increase of investor s lie outside the interval However, this issue is important in the case of a single shrinkage portfolio is higher then on the current portfolio, i.e. 1 if the expected excess return on the tangency a M,s 1 if E(û M)(µ r f 1) c (µ r f 1) > 0. The proof of this statement follows along the lines of the proof of Corollary 5. Concerning the lower bound, note that the numerator is not exactly a quadratic form, because the estimator û is upward biased. Therefore we can always find such values of the parameters that the numerator of a M,s and consequently a M,s, are negative. James and Stein (1961) solve this problem by restricting the optimal shrinkage factor from below by zero, i.e. a JS = max{0, a }. In our case, the asymptotic unbiasedness of the estimator û makes the probability of a M,s the restriction a M,s < 0 negligible small for reasonable choices of n. Concerning 1, the condition provided above is sufficient, but not a necessary condition and in practice the inequality will be fulfilled for a wide range of parameters. 3.2 No Riskless Asset Here we analyze the portfolio problem without riskless asset. The portfolio return in this case is given by r p = w r. The asset allocation problem with k risky assets in the portfolio and quadratic utility is max w w µ γ 2 w Σw s.t. w 1 = 1. 8

9 The classical Markowitz solution of this problem is given by u Q = Σ Σ Rµ γ where R = Σ 1 Σ 1 11 Σ 1 1 Σ 1 1. Okhrin and Schmid (2003) derive the first two moments of the estimated weights û Q under normality assumption for log returns, given by E(û Q ) and Cov(û Q ) = Ω Q. The formulas are provided in Appendix A. Now we remark that in the absence of the riskless asset in the portfolio as well as in case when the riskless asset is present the vector of current weights c does not need to sum up to one. It may sum up to one in case of portfolio revision, but, for example, in the case of initial investment it is summed up to zero. We state the portfolio optimization problem with the restriction that the vector of current portfolio weights sums up to a number 1 υ = c 1 with υ [0; 1]. Then the maximization of the expected utility in case of portfolio revision can be expressed by max a Q E(ŵ Qr) γ 2 V ar(ŵ Qr), s.t. a Q(E(û Q ) c) = υ, (9) where ŵ Q = a Q (û Q c) + c. The solution of this problem is given in the following Theorem. Theorem 2. Assume that {x t } is a sequence of independent and normally distributed k- dimensional random vectors with the mean µ and the covariance matrix Σ. The solution of the stated above maximization problem (9) is given by a Q = Q 1 ( ) Q (E(û Q ) c) (µ γσc ξ1), γ where ξ = (E(û Q) c) Q 1 Q ((E(û Q) c) (µ γσc)) γυ (E(û Q ) c) Q 1 Q (E(û, Q) c) Q Q = Ω Q (Σ + µµ ) + Σ (E(û Q ) c)(e(û Q ) c). The covariance matrix Ω Q of the estimated portfolio weights plays here a similar role in reflecting the uncertainty in the estimated model parameters as in the case of tangency portfolio. Next we provide a corollary similar to Corollary 1 about the limiting behavior of the shrinkage factors a Q. Corollary 3. a) If the estimation risk is reduced to zero by using n, the optimal shrinkage factors a Q are given by lim n a Q = 1. 9

10 b) If the variance of the estimator of the optimal portfolio weights is infinite, i.e. n < k+4, the optimal shrinkage factors a Q are given by a Q = 0 for n < k + 4. The expected quadratic utility maximization that we solve in the case of single shrinkage factor is given by the following unconstrained maximization problem max a Q,s E(ŵ Q,sr) γ 2 V ar(ŵ Q,sr), (10) where ŵ Q,s = a Q,s (û Q c) + c. Note that the restriction ŵ Q,s 1 = 1 is fulfilled for all values of a Q,s if and only if û Q 1 = 1 and c 1 = 1. This implies that υ = 0. The next corollary establishes the solution of this optimization problem, the proof is similar to the proof of Corollary 2. Corollary 4. Let x t follow the multivariate normal distribution with the parameters µ and Σ. Then the solution of (10) is given by a Q,s = (E(û Q ) c) Σ(γ 1 Σ 1 µ c) tr [ (Σ + µµ )Ω Q ] + (E(ûQ ) c) Σ(E(û Q ) c). We can provide exact conditions, which guarantee that the single shrinkage coefficient takes the values between zero and one for the portfolio problem considered in this subsection. Corollary 5. It holds that a Q,s [0, 1] if the expected return on the quadratic utility optimal portfolio is higher then on the current portfolio and if the variance of the current portfolio is higher than the variance of the global minimum variance portfolio, i.e. a Q,s [0, 1] if the following two conditions hold (E(û Q ) c) µ > 0, c Σc 1 1 Σ 1 1 > 0. Similarly to the case of the problem with a riskless asset the condition provided here are sufficient, but not necessary conditions and the shrinkage coefficient belongs to the unit interval for large variety of other parameter constellations. As earlier the restrictions for the different shrinkage coefficients are not crucial. The shrinkage estimators obtained in this section are the functions of the unknown distribution parameters. By calculating the optimal shrinkage intensities the unknown true values are replaced by their estimated counterparts. Next we discuss the consequences of estimation of the shrinkage coefficients as well as the impact of different portfolio selection techniques on the investor s utility. 10

11 4 Comparison of Different Portfolio Rules Here we compare the portfolio performance of the shrinkage strategies with the performance of the classical Markowitz procedure. The performance is measured by the expected utility of portfolio return. The strategies to compare are (1) Markowitz allocation with known moments (µ, Σ); (2) Markowitz allocation with estimated moments (ˆµ, ˆΣ); (3a,b) single different shrinkage strategies with estimated moments but known optimal shrinkage intensities; (4a,b) single and different shrinkage strategies with estimated moments as well as estimated optimal shrinkage intensities. We show analytically that the expected utility of natural shrinkage strategies with estimated moments but known optimal shrinkage intensities are bounded above by the Markowitz allocation with full certainty and below by the Markowitz allocation with estimated moments. The estimation of the optimal shrinkage intensities leads to the reduction of the expected utility compared with the knowledge of the optimal shrinkage intensities. However, as it is shown later in the simulation study, the expected utility in case of estimated shrinkages is still bounded below by the expected utility from the Markowitz allocation with estimated moments. Relationship between expected utilities from the single and different shrinkage strategies in the case of estimated intensities needs further investigation. As earlier, we consider the portfolio allocation both in presence and absence of the riskless asset. 4.1 Riskless asset is present When the distribution of returns is known with certainty, the Markowitz expected utility is given by EU 1 = (µ r f1) Σ 1 (µ r f 1) 2γ + r f. We derive in Appendix C the Markowitz expected utility for the case of estimated moments of returns, given by EU 2 = n2 2nk 4n + 2k + 3 2γ(n k 2) 2 (µ r f 1) Σ 1 (µ r f 1) γ 2 tr ( Ω M (Σ + (µ r f 1)(µ r f 1) ) ) + r f with Ω M = Cov(û M ). The difference between the expected utilities with true and estimated parameters in Markowitz approach is given by 12 = EU 1 EU 2 = (k + 1)2 (µ r f 1) Σ 1 (µ r f 1) 2γ(n k 2) 2 + γ 2 tr ( Ω M (Σ + (µ r f 1)(µ r f 1) ) ). 11

12 Of course, 12 > 0 because Ω M and Σ are positive definite. If the natural shrinkage coefficients are known with certainty, the expected utility of different shrinkage strategy is given by EU 3b = E(ŵ M(r r f 1)) γ 2 V ar(ŵ M(r r f 1)) + r f with ŵ M = û M a M + (1 a M ) c. The expected utility from different shrinkage EU 3b EU 1, because EU 1 is a theoretically attainable maximum. The expected utility from single shrinkage EU 3a EU 3b because the single shrinkage approach can be seen as constrained optimization compared to the different shrinkage approach. On the other hand EU 3a EU 2, because EU 3a results from seeking for the maximum with respect to a M,s where EU 3a = EU 2 for a M,s = 1. Thus the expected utility of the shrinkage strategy with certain shrinkage coefficient is bounded by EU 1 EU 3b EU 3a EU 2. The comparison of the expected shrinkage utility with estimated shrinkage parameters is done in section Simulation Study. 4.2 Riskless asset is absent If the distribution parameters (µ, Σ) are known with certainty, the Markowitz expected utility is given by EU 1 = µ Rµ 2γ + µ Σ 1 1 γ/2. 1 Σ 1 1 We show in Appendix C, that the expected utility with uncertain parameters is given by EU 2 = (n 1)(n 2k 1) µ Rµ + µ Σ 1 1 γ/2 γ (n k 1) 2 2γ 1 Σ tr(ω QΣ), where Ω Q = Cov(û Q ). The difference between the expected utilities with true and estimated parameters in Markowitz approach is given by 12 = EU 1 EU 2 = k 2 µ Rµ + γ (n k 1) 2 2γ 2 tr(ω QΣ) 0, because Ω Q is positive semidefinite and Σ is positive definite. If the natural shrinkage coefficient is known with certainty, the expected utility of shrinkage strategy is given by EU 3b = E(ŵ Qr) γ 2 V ar(ŵ Qr) with ŵ Q = û Q a Q + (1 a Q ) c. Similar to the arguments provided in the previous subsection we can establish the following relationship EU 1 EU 3b EU 3a EU 2. 12

13 5 Simulation Study We clarify the obtained results within the simulation study. For the illustrative purposes we consider here the case of three risky and one riskless asset in the portfolio. Later, in the empirical study, we apply the methodology to the portfolio that consists of k = 10 risky assets. The study is designed as follows. The log risky asset returns are multivariate normal and i.i.d. and generated on the monthly basis, i.e. 12x t N(µ, Σ). The annualized distribution parameters of the model are given by.05 µ =.07, Σ = , the annualized riskless rate is r f =.02 and the risk aversion coefficient is taken from the set γ {10, 25}. If the distribution parameters of asset returns were certain, the corresponding optimal weights for the risky asset would be given by u M = (.12,.31,.32). We consider shrinkage strategies with k = 3 different shrinkage factors as well as with a single shrinkage factor. In case of no uncertainty about the true values of the distribution parameters the true optimal shrinkage factors (a M,s, a M ) as well as the optimal shrinkage weights of the risky asset w M are the functions of current portfolio holdings c and of the number of past returns n, used for the estimation. The shrinkage optimal weights w M, the shrinkage factors a M,s and a M, the classical optimal weights u M and the current weights c are plotted on the following Figures for different values of the risk aversion coefficient γ and for the distribution parameters, given above. Figures 1 about here. Because the true distribution parameters are unknown, they should be estimated. Using the sample estimators for the distribution parameters, we run a simulation study in order to compare the classical Markowitz portfolio strategy with the proposed shrinkage strategies. The simulation study is designed in the following way. A path of previous returns is generated every replication, the sample estimates for distribution parameters ˆµ and ˆΣ, optimal shrinkage factors (â M,s, â M ) and optimal portfolio weights û M, ŵ M are calculated from this path using n previous monthly returns. In the classical portfolio strategy the wealth is invested in proportions û M and 1 û M1 for risky and riskless assets, respectively. In the shrinkage portfolio strategy the wealth is invested in proportions ŵ M and 1 ŵ M 1. Following Ledoit and Wolf (2003a,b), the holding period of the portfolio investor is taken 13

14 to be one year (12 month). Then the expected utility is calculated for the classical and shrinkage strategies with and without knowledge of the exact distribution parameters. The study is done for the values of c 2 = u M,2, c 3 = u M,3, c 1 {.50,.49,...,.99, 1.0} and n {24, 60, 120}, the risk aversion coefficient is taken from the set γ {10, 25}. The expected utility is plotted on the following Figures for all considered portfolio strategies. The number of replications is chosen to be 10 5 for all cases. Figures 2 about here. The benchmark strategy with the knowledge of the true distribution parameters is clearly the best one and brings the investor the highest expected utility. The classical strategy with the estimated parameters is always the worst one. When the exact shrinkage intensities are known, the different shrinkage strategy dominates its single shrinkage counterpart. Concerning the case of unknown optimal shrinkage intensities the relationship between single and different shrinkage utilities changes depending on the distance between c 1 and u M,1. If c 1 is in the small neighborhood of u M,1 then the single shrinkage approach dominates, otherwise the different shrinkage approach is better. Next we apply the proposed methodology to the real financial data. 6 Empirical Results In this section we present the empirical evidence on the performance of shrinkage estimators defined in Section 3. The proposed estimators with single and different shrinkage factors are compared with the classical Markowitz (1952) sample estimator, with the Bayes-Stein estimator of Jorion (1986) as well as the estimator of Ledoit and Wolf (2003a,b) in terms of quadratic utility function. The investment strategy is constructed in the following way. First n previous monthly returns are used for the estimation of the distribution parameters ˆµ and ˆΣ. Then the classical Markowitz û Q, Jorion ĵr, Ledoit and Wolf ˆ lw as well as the single shrinkage ŵq,s and different shrinkage ŵ Q optimal weights are calculated based on the estimated parameters. The investor allocates his wealth accordingly to the estimated optimal weights for all compared strategies and holds the portfolio during the holding period. The length of the holding period s takes the values from 6 to 24 months. Thus, at the investment date t the in-sample period lasts from t n + 1 to t, the out-of-sample period goes from t + 1 to t + s. The quantities of interest are the out-of-sample mean and variance of the portfolio return, that are required for calculation of the expected utility. The expected utility is considered for the investor with the risk aversion coefficient γ {5, 10, 20, 35, 50, 100}. 14

15 The empirical study is conducted based on the MSCI monthly data set for the period Dec Apr 2004, totally 413 observations. The prices are taken for the last trading day of the month. The investor allocates his wealth among k = 10 developed financial markets, namely UK, Germany, France, the Netherlands, USA, Canada, Japan, Italy, Spain, Switzerland. The country MSCI indexes are denominated in US$, the dividends are cumulated. The MSCI World Index over the same period is chosen to be the proxy for the market index in the single factor model. The expected utilities are calculated for the different values of risk aversion coefficient γ, different holding periods s = {6, 7,..., 24} month with n {30, 31,..., 150} previous months, used for the estimation. The competing strategies are ranked from 5 (highest) to 1 (lowest) rank for each given s, γ and n. The ranks over different n {30, 31,..., 150} are averaged, the averaged ranks are plotted on Figure 3. Figure 3 about here. Based on the performed study we report the following empirical findings. The average ranking of alternative strategies strongly depends on the investor s risk aversion coefficient. For small γ < 10 the role of portfolio variance in the utility function is minor, the expected portfolio return dominates the investor s choice. The strategy of Jorion is clearly the best one for these values of γ. The different shrinkages strategy is the second best, the single shrinkage strategy is the third best. The strategy of Ledoit and Wolf dominates the approach of Markowitz for small investment horizon s, but starting approximately from 12 months is the worst one. The role of the portfolio variance increases with the increase of the investor s risk aversion. For γ {20, 35} the strategy of different shrinkages is the best one, the single shrinkage strategy is the second best. The strategy of Jorion occupies the third position, Ledoit and Wolf s strategy has position four for s < 12 months and position five otherwise. The investor with large risk aversion, i.e. γ > 50 is interested primarily in the minimization of the portfolio variance and neglects potential losses in the portfolio return. For such investors the different shrinkages strategy is again the best choice, the second best choice is the single shrinkage strategy. The strategy of Ledoit and Wolf has the third position approximately for s < 12 months, while for s > 12 the third position is occupied by the classical Markowitz strategy. The strategy of Jorion is almost everywhere the worst one for such choice of γ. Summarizing, we can conclude that the strategy with different shrinkages is the best one for almost for all γ s except the case of small risk aversion coefficients, while the single shrinkage strategy is the second best choice. The strategy of Jorion is preferable for small 15

16 γ, while the strategy of Ledoit performs well (third best) only for large γ s and investment horizon smaller than s < 1 year. 7 Summary The multivariate shrinkage for the optimal portfolio weights is developed in this paper. This development extends the univariate shrinkage estimators for the means (Jorion, 1986) and for the covariance matrix (Ledoit and Wolf, 2003a,b) proposed earlier in the context of portfolio selection. The idea lies in shrinking the Markowitz optimal weights for risky assets to the investor s current portfolio holdings. We propose two different estimators: one with single shrinkage coefficient and another one with different shrinkage coefficients. The shrinkage estimation uses the trade-off in terms of expected utility between the estimation uncertainty and the information about the true optimal weights that is contained in the classical estimator. If the information contained in the estimated Markowitz optimal weights is reliable, the investor should move more in the direction of the estimated weights. Alternatively, the investor moves not far from his current portfolio holdings if the information is not trusty enough. The main results of the paper are presented in Section 3. We develop analytically the explicit solutions for the optimal shrinkage intensity for both proposed estimators in case of k risky assets as well as in case of k risky and one riskless asset for the single period investor with the quadratic utility function. We also provide useful corollaries about the limiting behavior of the optimal shrinkage factors. The application of the shrinkage estimators for the portfolio weights shows their superiority over the classical Markowitz strategy in the simulation study. As a rule, the strategy with different shrinkage factors outperforms the single shrinkage factor strategy. The shrinkage strategies beat their Markowitz counterpart even for the estimated optimal shrinkage factors. We compare proposed shrinkage strategies in the empirical study with the classical Markowitz strategy, the Bayes-Stein strategy of Jorion (1986) and strategy of Ledoit and Wolf (2003a,b). The strategy with different shrinkage coefficients is the best one, while the single shrinkage strategy is the second best for all risk aversion coefficients γ except γ < 10. However, it should be noted that the shrinkage strategy requires input of additional information (vector of current portfolio holdings) compared to the competing approaches. 16

17 8 Appendix 8.1 Appendix A Riskless asset is present Okhrin and Schmid (2003) derive the first two moments of the optimal portfolio weights under normality assumption for log returns, when the usual sample estimators are used for the moments E(û M ) = γ 1 n 1 n k 2 Σ 1 (µ r f 1) [ n 2 Cov(û M ) = γ 2 z 1 n Σ 1 + (µ r f 1) Σ 1 (µ r f 1) Σ 1 n k + n k 2 Σ 1 (µ r f 1)(µ r f 1) Σ 1], with z 1 = (n 1) 2 /(n k 1)(n k 2)(n k 4) Riskless asset is absent Okhrin and Schmid (2003) derive the first two moments of the estimated weights û under normality assumption for log returns, given by E(û Q ) = Σ Σ n 1 n k 1 γ 1 Rµ. Cov(û Q ) = + 1 R (n k 1) 1 Σ q 1 γ 2 Rµµ R + q 2 γ 2 µ RµR 1 (n ) 1)2 (q γ q 2 (k 1) + n (n k 1) 2 R with q 1 = (n 1) 2 (n k + 1) (n k)(n k 1) 2 (n k 3), q 2 = (n 1) 2 (n k)(n k 1)(n k 3). 17

18 8.2 Appendix B Proof of Theorem 1 In order to proof the theorem it is necessary to solve the maximization problem, given by MEU = max a M E(ŵ M(r r f 1)) γ 2 V ar(ŵ M(r r f 1)), (11) where ŵ M = (û M c) a M + c and û M and r are independent by construction. Let Φ = Σ + (µ r f 1)(µ r f 1). Using properties of Hadamard product (cf. Zhang, 1999) and Theorem 3.2b.1 of Mathai and Provost (1992) it follows that E(ŵ M(r r f 1)) = a M((E(û M ) c) (µ r f 1)) + c (µ r f 1), E(ŵ M(r r f 1)) 2 = E ( ŵ MΦŵ M ) = E ( (a M û M ) Φ(a M û M ) + 2(a M û M ) Φ((1 a M ) c) + ((1 a M ) c) Φ((1 a M ) c) ) = a M(Φ E(û M û M))a M + 2a M(Φ E(û M 1 ))c 2a M(Φ E(û M c ))a M + a M(Φ E(cc ))a M 2a M(Φ E(c1 ))c + c Φc = a M[ Φ E(ûM û M) + 2Φ E(û M c ) + Φ (cc ) ] a M + 2a M[ Φ E(ûM 1 ) Φ (cc ) ] c + c Φc. Setting components together leads to the following optimization problem where MEU = max a M a M((E(û M ) c) (µ r f 1 γσc)) γ 2 a MQ M a M, Q M = Ω M Σ + Ω M ((µ r f 1)(µ r f 1) ) + ((E(û M ) c)(e(û M ) c) ) Σ. The first derivative of the objective function with respect to the vector a M is set to zero, then the solution of the maximization problem is given by ( ( )) µ a M = Q 1 rf 1 M (E(û M ) c) Σc. γ Note that Ω M Σ is positive definite, since Σ and Ω M are positive definite. Therefore Q M is positive definite as well, since the rest two terms in the expression for Q M are nonnegative definite. This implies that a M is a valid solution of the maximization problem. 18

19 8.2.2 Proof of Corollary 1 In order to prove the first part of the corollary, we take the limit ( ( )) µ lim n a M = lim Q 1 rf 1 n M (E(û M ) c) Σc. (12) γ Note that the estimator û M is asymptotically unbiased and consistent. This implies that lim n E(û M ) = u M and lim n Ω M = 0 k k and leads to the following limit of a M lim n a M = ( (u M c)(u M c) Σ ) 1 ( (um c) Σ(u M c) ). Let D denotes a diagonal matrix with the main diagonal equal to u M c. Then lim n a M = ( DΣD ) 1 DΣ(uM c) = D 1 Σ 1 D 1 DΣ(u M c) = D 1 (u M c) = 1 and this proves the first statement of the corollary. The second part of the corollary is proven with the observation, that all the elements of the matrix Q M are equal to infinity for n < k + 5. Then all the elements of the matrix Q 1 M are equal to zeros, thus a M = 0 for n < k + 5 follows immediately Proof of Theorem 2 The utility function is derived in this case similarly as for the tangency portfolio, however due to the constraint on the shrinkage coefficients the proof is based on solving the following Lagrange maximization problem max a a Q Q( (E(ûQ ) c) µ ) + c µ γ [ a 2 Q (Ω Q Σ + (µµ ) Ω Q +(E(û Q ) c)(e(û Q ) c) Σ) a Q + 2a ( Q (E(ûQ ) c) Σc ) + c Σc ] w.r.t. a Q(E(û Q ) c) = υ. (13) After setting the first derivatives with respect to the shrinkage factors a Q and Lagrangian coefficient λ to zero the system of two equation is solved. The resulting solution is given by a Q = Q 1 Q γ ( ( (E(û Q ) c) µ Σc (E(û Q) c) Q 1 Q ((E(û )) Q) c) (µ Σc)) γυ (E(û Q ) c) Q 1 Q (E(û 1 Q) c) where Q Q = Ω Q Σ + (µµ ) Ω Q + ((E(û Q ) c)(e(û Q ) c) ) Σ. The validity of the solution is motivated in a similar manner as in Theorem 1. 19

20 8.2.4 Proof of Corollary 5 Since the first term in the denominator of a Q,s is positive, we conclude that a Q,s (E(û Q) c) (µ γσc) γ(e(û Q ) c) Σ(E(û Q ) c). Substitution of the expression for E(û Q ) leads to a Q,s ( 1 Σ 1 1 Σ 1 1 γ 1 d 1 µ R c )(µ γσc) ( ( ) 1 γ Σ 1 1 Σ 1 1 γ 1 d 1 µ R c )Σ 1 Σ 1 1 Σ 1 1 γ 1 d 1 µ R c ( ) = 1 (1 + d 1 ) Σ 1 1 Σ 1 1 µ c + γ 1 d 1 µ 1 Rµ γ 1 Σ 1 1 γc Σc ( ) d 2 1 Σ Σ 1 1 µ c + γ 1 d 2 1µ 1 Rµ γ 1 Σ 1 1 γc Σc = I II To derive an economically sensible restriction on the portfolio characteristics we consider [ 1 Σ 1 µ ] I II = (1 d 1 ) 1 Σ γ 1 d 1 µ Rµ c µ = (1 d 1 ) [ E(u r) E(c r) ]. Since 1 d 1 < 0, then I II < 0 and consequently a s < 1, if E(u Q r) E(c r) > 0, i.e. if the expected return on the quadratic utility optimal portfolio is higher than on the current. Taking into account this assumption, c Σc 1/1 Σ 1 1 suffices to guarantee that a Q,s > 0, i.e. if the variance of the current portfolio is higher than the variance of the global minimum variance portfolio then a Q,s > 0. The asymptotic properties of a Q,s can be proved in the same way as for the different shrinkage coefficients. 8.3 Appendix C Derivation if the expected utility We consider here the situation when the riskless asset is present. The expected utility is given by EU 2 = E(û M ) (µ r f 1) γ 2 ( E(û M (r r f 1)(r r f 1) û M ) (E(û M ) (µ r f 1)) 2) + r f. Using the expression for expected optimal weights E(û M ) we consider E(û M(r r f 1)(r r f 1) û M ) = tr ( Ω M (Σ + (µ r f 1)(µ r f 1) ) ) + + z2 γ 2 ( (µ rf 1) Σ 1 (µ r f 1) + [(µ r f 1) Σ 1 (µ r f 1)] 2) 20

21 with z = (n 1)/(n k 2). Taking into account that E(û M ) (µ r f 1) = z γ (µ r f1) Σ 1 (µ r f 1), we obtain ) EU 2 = (z z2 (µ rf 1) Σ 1 (µ r f 1) γ 2 γ 2 tr (Ω M(Σ + (µ r f 1)(µ r f 1) )) + r f. The expected utility in case of no riskless asset is derived by analogy. References [1] Barberis, N., 2000, Investing for the long run when returns are predictable, Journal of Finance 55, [2] Berger, J., 1982, Selecting a minimax estimator of a multivariate normal mean, The Annals of Statistics 10, [3] Berger, J. and L.M. Berliner, 1984, Bayesian input in Stein-estimation and a new minimax empirical bayes estimator, Journal of Econometrics 25, [4] Best, M. and R. Grauer, 1991, On the sensitivity of mean-variance-efficient portfolios to changes in asset means: some analytical and computational results, Review of Financial Studies 4, [5] Brandt, M.W., 1999, Estimating portfolio and consumption choice: a conditional Euler equations approach, Journal of Finance 54, [6] Brennan, M.J., Schwartz, E.S. and R. Lagnado, 1997, Strategic asset allocation, Journal of Economic Dynamics and Control 21, [7] Campbell, J.Y. and L.M. Viceira, 1999, Consumption and portfolio decisions when expected returns are time varying, Quarterly Journal of Economics 114, [8] Efron, B. and C. Morris, 1975, Data analysis using Stein s estimator and its generalizations, Journal of the American Statistical Association 70, [9] Ferson, W., 1995, Theory and empirical testing of asset pricing models, in R. Jarrow, V. Maksimovic and W. Ziemba eds. Handbools in Operation Research and Management Science, Vol. 9 Finance, Amsterdam, Elsevier. [10] Ferson, W. and A. Siegel, 2001, The efficient use of conditional infermation in portfolios, Journal of Finance 56,

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23 [25] Ledoit, O. and M. Wolf, 2003a, Improved estimation of the covariance matrix of stock returns with an application to portfolio selection, Journal of Empirical Finance 10, [26] Ledoit, O. and M. Wolf, 2003b, A well-conditioned estimator for large-dimensional covariance matrices, Journal of Multivariate Analysis 88(2), [27] Longford, N.T., 1999, Multivariate shrinkage estimation of small area means and proposrtions, Journal of the Royal Statistical Society: Serie B 162, [28] Markowitz, H., 1952, Portfolio selection, Journal of Finance 7, [29] Mathai, A.M. and S.B. Provost, 1992, Quadratic forms in random variables, New York, Marcel Dekker. [30] Merton, R.C., 1980, On estimating the expected return on the market, Journal of Financial Economics 8, [31] Michaud, R.O., 1989, The Markowitz optimization enigma: is optimized optimal? Financial Analyst Journal 45(1), [32] Michaud, R.O., 1998, Efficient Asset Management, Boston, Massachusetts, Harvard Business School Press. [33] Okhrin, Y. and W. Schmid, 2005, Distributional properties of optimal portfolio weights. Forthcoming in Journal of Econometrics. [34] Scherer, B., 2002, Portfolio resampling: review and critique, Financial Analyst Journal 58, [35] Stein, C., 1955, Inadmissability of the usual estimator for the mean of a multivariate normal distribution. Proc. 3rd Berkeley Symp. Math. Statist. Prob. 1, [36] Zhang, F., 1999, Matrix Theory: Basic Results and Techniques, New York, Springer. 23

24 Figure 1: 24

25 Figure 2: 25

26 Figure 3: 26

27 Figure 1: The true optimal (black solid), true shrinkage (black dashed for single and black dashed-dotted for different shrinkage coefficients), current (dotted) portfolio weights of the first asset as well as optimal shrinkage coefficients (grey dashed for single and grey dasheddotted for different shrinkage coefficients) as a function of the current weight c 1. c 2 and c 3 are set equal to the true portfolio weights. Model parameters are given in Section 5. Left column corresponds to γ = 10, right column to γ = 25. n {24, 60, 120}. Figure 2: The expected utility for the true optimal (black solid), estimated optimal (grey solid), true shrinkage (black dashed for single and black dashed-dotted for different shrinkage coefficients), estimated shrinkage (grey dashed for single and grey dashed-dotted for different shrinkage coefficients), current (dotted) utilities as a function of the current weight c 1. c 2 and c 3 are set equal to the true portfolio weights. Model parameters are given in Section 5. Left column corresponds to γ = 10, right column to γ = 25. n {24, 60, 120}. Figure 3: The averaged over n [30, 150] ranks of expected utilities for competing strategies: (1) Markowitz strategy (solid black), (2) Jorion strategy (solid grey), (3) Ledoit and Wolf strategy (dotted), (4) single shrinkage strategy (dashed), (5) different shrinkage strategy (dashed-dotted) as a function investment horizon s (months), for the risk aversion coefficients γ {5, 10, 20, 35, 50, 100} (from left to right, from top to bottom). Design of the study is described in Section 6. 27