On the Markowitz mean variance analysis

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On the Markowitz mean-variance analysis of self-financing

portfolios

Zhidong Bai, Huixia Liu and Wing-Keung Wong

RMI Working Paper No. 09/02

Submitted: April 13, 2009

†

Abstract

This paper extends the work of Markowitz (1952), Korkie and Turtle (2002) and others by first proving that the traditional estimate for the optimal return of self-financing portfolios always over-estimates from its theoretic value. To circumvent the problem, we develop a bootstrap estimate for the optimal return of self-financing portfolios and prove that this estimate is consistent with its counterpart parameter. We further demonstrate the superiority of our proposed estimate over the traditional estimate by simulation.

Keywords: Optimal portfolio allocation, mean–variance optimization, self-financing portfolio, large random matrix, bootstrap method.

Zhidong Bai KLAS, Math & Stat

Northeast Normal University DSAP & RMI

National University of Singapore

Huixia Liu

National University of Singapore Department of Statistics

Wing-Keung Wong

Hong Kong Baptist University Department of Economics

Address for correspondence: Wing-Keung Wong, Department of Economics, Hong Kong Baptist University, Kowloon Tong, Hong Kong. Tel.: +852 3411 7542; Fax: +852 3411 5580; E-mail:awong@hkbu.edu.hk.

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DOI 10.3233/RDA-2008-0004 IOS Press

On the Markowitz mean–variance analysis of

self-financing portfolios

Zhidong Bai

_{, Huixia Liu}

_{and Wing-Keung Wong}

c,∗

a_{KLAS, Math & Stat, Northeast Normal University and DSAP & RMI, National University of Singapore,}

Singapore

b_{Department of Statistics, National University of Singapore, Singapore} c_{Department of Economics, Hong Kong Baptist University, Hong Kong}

Abstract.This paper extends the work of Markowitz (1952), Korkie and Turtle (2002) and others by first proving that the tradi-tional estimate for the optimal return of self-financing portfolios always over-estimates from its theoretic value. To circumvent the problem, we develop a bootstrap estimate for the optimal return of self-financing portfolios and prove that this estimate is consistent with its counterpart parameter. We further demonstrate the superiority of our proposed estimate over the traditional estimate by simulation.

Keywords: Optimal portfolio allocation, mean–variance optimization, self-financing portfolio, large random matrix, bootstrap method

1. Introduction

Markowitz [25,26] introduce a mean–variance (MV)1 portfolio optimization procedure in which in-vestors incorporate their preferences on risk and ex-pectation of return to seek the best allocation of wealth by selecting the portfolios that maximize anticipated profit subject to achieving a specified level of risk, or equivalently minimize variance subject to achiev-ing a specified level of expected gain, see, for exam-ple, Markowitz [25–27], Merton [29], Kroll et al. [20], Schwebach et al. [40], Meyer and Rose [31], Phengpis and Swanson [38] and Albuquerque [1].

This theoretical MV optimization procedure is ex-pected to be a powerful tool for portfolio optimizers to efficiently allocate their wealth to different investment alternatives to achieve the maximum expected profit as the conceptual framework of the classical MV portfo-lio optimization had first been set forth by Markowitz *_{Address for correspondence: Wing-Keung Wong, Department}

of Economics, Hong Kong Baptist University, Kowloon Tong, Hong Kong. Tel.: +852 3411 7542; Fax: +852 3411 5580; E-mail: awong@hkbu.edu.hk.

1_{We note that Leung and Wong [22] extend the MV theory by}

developing a multivariate Sharpe ratio statistic to test the hypothesis of the equality of multiple Sharpe ratios whereas Meyer [30], Levy [23], Wong [45], and Wong and Ma [46] establish some relationships between stochastic dominance and mean–variance rules.

more than half a century ago, and several procedures for computing the corresponding estimates (see, for example, [12,13,28,37,41,42,44]) have produced sub-stantial experimentation in the investment community for nearly four decades.

However, there have been persistent doubts about the performance of the estimates. Instead of imple-menting nonintuitive decisions dictated by portfolio optimizations, it has long been known anecdotally that a number of experienced investment profession-als simply disregard the results or abandon the en-tire approach, since many studies (see, for example, [11,32,43]) have found the MV-optimized portfolios to be unintuitive, thereby making their estimates do more harm than good. For example, Frankfurter et al. [14] find that the portfolio selected according to the Markowitz MV criterion is likely not as effective as an equally weighted portfolio, while Zellner and Chetty [48], Brown [10], Michaud [32] note that MV opti-mization is one of the outstanding puzzles in modern finance and that it has yet to meet with widespread ac-ceptance by the investment community, particularly as a practical tool for active equity investment manage-ment. He terms this puzzle the “Markowitz optimiza-tion enigma” and calls the MV optimizers “estimaoptimiza-tion- “estimation-error maximizers”.

To investigate the reasons why the MV optimiza-tion estimate is so far away from its theoretic

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36 Z. Bai et al. / Markowitz mean–variance analysis of self-financing portfolios

part, Jorion [16], Best and Grauer [8] and Britten-Jones [9] suggest the main difficulty concerns the extreme weights that often arise when constructing sample effi-cient portfolios that are extremely sensitive to changes in asset means. On the other hand, Laloux et al. [21] find that Markowitz’s portfolio optimization scheme which is based on a purely historical determination of the correlation matrix, is not adequate because its low-est eigenvalues dominating the smalllow-est risk portfolio are dominated by noise.

Many studies have tried to show that the difficulty can be alleviated by using different approaches. For example, Pafka and Kondor [35] impose some con-straints on the correlation matrix to capture the essence of the real correlation structure. Konno and Yamazaki [17] propose a mean–absolute deviation portfolio op-timization to overcome the difficulties associated with the classical Markowitz model but Simaan [43] finds that the estimation errors for both the mean–absolute deviation portfolio model and the classical Markowitz model are still very severe, especially in small samples. In this paper, we complement their theoretical work by starting off to first prove that when the number of assets is large, the traditional return estimate for the optimal self-financing portfolio obtained by plug-ging the sample mean and covariance matrix into its theoretic value is always over-estimated and, in re-turn, makes the self-financing MV optimization pro-cedure impractical. We call this return estimate “plug-in” return and call this phenomenon “over-prediction”. This inaccuracy makes the self-financing MV opti-mization procedure to be impractical. To circumvent this over-prediction problem, we invoke the bootstrap technique2to develop a new estimate that analytically corrects the over-prediction and reduces the error dras-tically. Furthermore, we theoretically prove that this bootstrap estimate is consistent to its counterpart para-meter and thus our approach makes it a possibility to implement the optimization procedure; thereby mak-ing it practically useful. Our simulation further con-firms the consistency of our proposed estimates; im-plying that the essence of the portfolio analysis prob-lem could be adequately captured by our proposed es-timates. Our simulation also shows that our proposed method improves the estimation accuracy so substan-tially that its relative efficiency could be as high as 205 times when compared with the traditional “plug-2_{See Scherer [39] for the review of bootstrap technique and refer}

to Kosowski et al. [19] for how to apply bootstrapping to portfolio management.

in” estimate for 350 number of assets with sample size of 500. Naturally, the relative efficiency will be much higher for bigger sample size and larger number of as-sets. The improvement of our proposed estimates are so big that there is a sound basis for believing our proposed estimate to be the best estimate to date for seeking the optimal return, yielding substantial bene-fits in terms of both profit maximization and risk re-duction.

The rest of the paper is organized as follows: Sec-tion 2 begins by providing a theoretical framework for the optimal self-financing portfolio and invoke the bootstrap technique to develop a new consistent esti-mate for the optimization procedure. In Section 3, we conduct simulation to confirm the consistency of our proposed estimates and show that our proposed method improves the estimation accuracy substantially. This is then followed up by Section 4 which summarizes our conclusions and shares our insights.

2. Theory

Suppose that there are p-branch of assets, S =

(s1,. . .,sp)T, whose returns are denoted by r =

(r1,. . .,rp)T with mean µ = (µ1,. . .,µp)T and

co-variance matrixΣ=(σij). In addition, we suppose that

an investor will invest capitalCon thep-branch of as-setsSsuch that s/he wants to allocate her/his investable wealth on the assets to obtain any of the following:

1. To maximize return subject to a given level of risk, or

2. To minimize her/his risk for a given level of ex-pected return.

Since the above two problems are equivalent, we only look for solution to the first problem in this pa-per. To obtain self-financing portfolio, we haveC=0. As her/his investment plan (or assets allocation) to be

c =(c1,. . .,cp)T, we havep_i₌₁ci = C = 0. Also,

her/his anticipated return,R, will then becTµwith risk cTΣ_c_{. In this paper, we further assume that short selling}

is allowed and hence any component ofccould be neg-ative. Thus, the above maximization problem can be re-formulated to the following optimization problem:

maxR=cTµ, subject to

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cT1=0 and cTΣ_cσ₀2,

where1 represents a vector of ones with length con-forming to the rules of matrix algebra andσ2

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risk level.3We callRsatisfying (1) theoptimal return and the solutioncto the maximization theoptimal allo-cation planfor the corresponding self-financing port-folio. The solution of (1) can be obtained in the follow-ing theorem.

Theorem 1. For the optimization problem shown in Eq.(1), the optimal return,R, and its corresponding investment plan,c, are, respectively,

R=σ0

µTΣ−1_µ

−(1

TΣ−₁ µ)2

1TΣ−1

1 (2)

and

c= σ0

µTΣ−1

µ−(1TΣ−1

µ)2_/₍₁TΣ−1 1) ×

Σ−1

µ−1

T_Σ−1_µ 1TΣ−₁

Σ−1 1

The proof of Theorem 1 is straightforward. The set of efficient feasible portfolios for all possible levels of portfolio risk forms the self-financing MV efficient frontier. For any given level of risk, Theorem 1 seems to provide us a unique optimal return with its corre-sponding self-financing MV-optimal investment plan or asset allocation to represent the best investment al-ternative given the selected assets, and thus it seems to provide a solution to the self-financing MV optimiza-tion procedure. Nonetheless, it is easy to expect the problem to be straightforward; however, this is not so as the estimation of the optimal self-financing return is a difficult task.

Suppose that {xjk} for j = 1,. . .,p and k =

1,. . .,n is a set of double array of independent

and identically distributed (iid) random variables with mean zero and varianceσ2_{. Let}_x

k =(x1k,. . .,xpk)T

andX =(x₁,. . .,xn). Then, the traditional return

esti-mate (we call itplug-in returnin this paper) for the op-timal self-financing portfolio is obtained by “plugging-in” the sample mean vectorXand the sample covari-ance matrixSinto the formulae in Theorem 1 such that

Rp= ˆcTpX, (3)

where

3_{We note that in this paper we study the optimal return. However,}

another direction of research is to study the optimal portfolio vari-ance; see, for example, Pafka and Kondor [34] and Papp et al. [36].

k=1

x_k n,

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S= 1

n₋1

k=1

(x_k₋x)(x_k₋x)T

and

cp= σ0

XT_S−₁_X

−(1T_S−₁_X₎₂_/₍₁_T

S−₁₁₎

S−1

X₋1

T_S−1_X 1T_S−1₁S

−1 1

Our simulation results shown in Fig. 1 show that this plug-in return is far from satisfaction, especially when sample size become large. From the figure, we find that the plug-in return is always larger than its theoretical value. In this paper, we prove this over-prediction phe-nomenon by the following theorem.

Theorem 2. Assume that X₁,. . .,Xn are normally

distributed with p_×1 mean vectorµand p×p co-variance matrixΣ. If the dimension-to-sample-size ra-tio indexp/n→y∈(0, 1), we have

µTΣ−1µ

n →a1,

1TΣ−1₁

n →a2,

1TΣ−1 µ

n →a3,

and ifa1a2−a23>0, then, we have

lim

n→∞ ˆ Rp

√_n→σ0

γ(a1a2−a2₃)

> lim

n→∞ R

√_n =_σ₀

a1a2−a2₃

whereγ=b

a x1dFy(x) = 1−1y >1,a=(1−

√_y₎2

andb=(1+√_y)2.

The proof of Theorem 2 can be obtained by apply-ing some fundamental limit theorems in the theory of large dimensional random matrix theory (see, for ex-ample, [2–7,15]). From this theorem, we know that the

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38 Z. Bai et al. / Markowitz mean–variance analysis of self-financing portfolios

Fig. 1. Empirical and theoretical optimal returns for different numbers of assets (solid line – the theoretic optimal return (R), dotted line – the plug in return (Rˆp)).

plug-in return,_Rˆ

p, is always bigger than the theoretical

optimal return,R=cTµ, defined in Eq. (1).

Employing the bootstrap technique, in this paper we further develop an efficient estimator for the optimal self-financing return to circumvent this over-prediction problem. We now describe the procedure to construct a parametric bootstrap estimate from the estimate of plug-in return,Rˆp, defined in Eq. (3) as follows: first,

we draw a resample χ∗ ₌

{X∗

1,. . .,X

∗

n} from the

p-variate ‘normal distribution’ with mean vector X

and covariance matrixS. Then, by invoking the self-financing optimization procedure again on the resam-ple χ∗

, we obtain the bootstrapped “plug-in” alloca-tion,ˆc∗

p, and the bootstrapped “plug-in” return, Rˆ

∗

such that

ˆ R∗

p= ˆc

∗_T

p X

∗

, (5)

whereX∗= 1

∗

i. Thereafter, we obtain the

boot-strap corrected return estimateRˆbas shown in the

fol-lowing theorem.

Theorem 3. Under the conditions in Theorem 2 and using the bootstrap correction procedure described above, the bootstrap corrected return estimate, Rˆb,

given by:

Rb= ˆRp+

√_γ(Rˆp−Rˆ∗p) (6)

possesses the following property: √_γ₍_R

−Rˆp)≃Rˆp−Rˆ∗p, (7)

whereγis defined in Theorem2,Ris the theoretic op-timal self-financing return obtained from Theorem1,

Rpis the plug-in return estimate defined in Eq.(3)and

obtained by using the original sampleχ, andRˆ∗

pis the

bootstrapped plug-in return estimate defined in Eq.(5) and obtained by using the bootstrapped sampleχ∗

, re-spectively.

The proof of Theorem 3 can be easily obtained by applying Theorem 2 and some fundamental limit the-orems in the theory of large dimensional random ma-trix theory (see, for example, [2,7,15]). We note that the relationA≃Bmeans thatA/B →1 in the limit-ing procedure. The above theorem has shown that the bootstrap corrected return estimate,Rˆb, is consistent to

the theoretic optimal self-financing return,R.

3. Simulation study

To illustrate the over-prediction problem, we, for simplicity sake, generate a p-branch of standardized asset returns from a multivariate normal distribution with mean µ = (µ1,. . .,µp)T and identity

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covari-ance matrix Σ = (Ijk) in which Ijk = 1 when

j =_kandIjk =0 otherwise. Given the level of risk,

the known population mean vector,µ, and the known population covariance matrix,Σ_{, we can compute the}

theoretic optimal allocation,c, and, thereafter, compute the theoretic optimal return,R, for the self-financing portfolios. These values will then be used to compare the performance of all the estimators being studied in our paper. Using this dataset, we apply the formula in Eq. (4) to compute the sample mean, X, and sample covariance,S, which, in turn, enable us to obtain the plug-in return,_Rˆ

p, and its corresponding plug-in

allo-cation,ˆcp, by substitutingXandSintoµandΣ respec-tively in the formula ofRˆpandˆcpas shown in Eq. (3).

To illustrate the over-prediction problem, we first plot the theoretic optimal self-financing returns,R, and the plug-in returns,Rˆp, for different values ofpwith the

same sample sizen=500 in Fig. 1.

From Fig. 1, we find the following: (1) the plug-in return_Rˆ

p defined in Eq. (3) is a good estimate to

the theoretic optimal returnRwhenpis small (30); (2) whenpis large (60), the difference between the theoretic optimal returnR and the plug-in return Rˆp

becomes dramatically large; (3) the larger the p, the greater the difference; and (4) whenpis large, both the plug-in return_Rˆ

pis always larger than the theoretic

op-timal return,R, computed by using the true mean and covariance matrix. These confirm the “Markowitz op-timization enigma” that the plug-in returnsRˆp should

not be used in practice.

In order to show the superiority of the performance of the bootstrap corrected optimal return estimate,_Rˆ

over that of plug-in returnRˆp, we define thebootstrap

corrected difference,dR

b (= ˆRb−R), for the return

es-timate,Rˆb, to be the difference between the bootstrap

corrected optimal return estimate, _Rˆ

b, and the

theo-retic optimal return,R, and theplug-in difference,dR p

(= ˆRp−R), for the return estimate,Rˆp, to be the

dif-ference between the plug-in return,Rˆp, and the

theo-retic optimal return,R, respectively. We then simulate 1000 times to computedR

x forx=pandb,n=500

andp=100, 200 and 300. The results are depicted in Fig. 2.

From Fig. 2, we find the desired property for our proposed estimate thatdR_b is much smaller thandRp in

absolute values for all the cases. This infers that the

Fig. 2. Comparison between the empirical and corrected portfolio allocations and returns (solid line – the absolute value ofdRp, dotted line – the

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40 Z. Bai et al. / Markowitz mean–variance analysis of self-financing portfolios

Table 1

MSE and relative efficiency comparison

p MSE(dRp) MSE(dR_b) RER_p_,_b

50 0.17 0.05 3.40

100 1.64 0.11 14.91 150 5.22 0.15 34.80 200 17.17 0.28 61.32 250 45.81 0.50 91.62 300 94.42 0.59 160.03 350 252.89 1.23 205.60

values of bootstrap corrected method are much more accurate in estimating the theoretic value than those obtained by using the plug-in procedure. Furthermore, as pincreases, the two lines on each level as shown in Fig. 2 further separated from each other; implying that the magnitude of improvement fromdRp todRb are

remarkable.

To further illustrate the superiority of our estimate over the traditional plug-in estimate, we present in Ta-ble 1 the mean square errors (MSEs) of the different estimates for differentpand present theirrelative effi-ciencies (RE)for returns such that

RER_p_,_b= MSE(d

R p)

MSE(dR b )

. (8)

Comparing the MSE ofdR_b with that ofdRp in Fig. 2

and Table 1,dR_b has been reduced dramatically from those ofdR

p, indicating that our proposed estimates are

superior. We find that the MSE of dR

b is only 0.05,

improving 3.4 times from that ofdRp when p = 50.

When the number of assets increases, the improvement become much more substantial. For example, when p=350, the MSE ofdR

b is only 1.23 but the MSE of

p is 252.89; improving 205.60 times from that ofdRp.

This is an remarkable improvement. We note that when bothnandpare bigger, the relative efficiency of our proposed estimate over the traditional plug-in estimate could be much larger.

4. Conclusions

Holding both theoretical and practical interest, the basic problem for MV analysis is to identify those combinations of assets that constitute attainable effi-cient portfolios. The purpose of this paper is to solve this problem by developing a new optimal return esti-mate to capture the essence of self-financing portfolio

selection. With this in mind, we go on to first theoret-ically prove that the estimated plug-in return obtained by plugging in the sample mean and sample covariance into the formulae of the self-financing optimal return is inadequate as it is always larger than its theoretical value when the number of assets is large. To recall, we call this problem “over-prediction”. To circumvent this problem, we develop the new estimator, the boot-strap corrected return, for the theoretic self-financing optimal return by employing the parametric bootstrap method.

Our simulation results confirm that the essence of the self-financing portfolio analysis problem could be adequately captured by our proposed bootstrap correc-tion method which improves the accuracy of the es-timation dramatically. As our approach is easy to op-erate and implement in practice, the whole efficient frontier of our estimates can be constructed analyti-cally. Thus, our proposed estimator facilitates the self-financing MV optimization procedure, making it im-plementable and practically useful.

We note that our model includes the situation in which one of the assets is a riskless asset so that in-vestors can risklessly lend and borrow at the same rate. In this situation, the separation theorem holds and thus our proposed return estimate is the optimal combination of the riskless asset and the optimal self-financing risky portfolio. We also note that the other as-sets listed in our model could be common stocks, pre-ferred shares, bonds and other types of assets so that the optimal return estimate proposed in our paper ac-tually represents the optimal self-financing return for the best combination of riskless rate, bonds, stocks and other assets.

For instance, the optimization problem can be for-mulated with short-sales restrictions, trading costs, liq-uidity constraints, turnover constraints and budget con-straints;4 see, for example, Lien and Wong [24] and Muthuraman and Kumar [33]. Each of these con-straints leads back to a different model for determining the shape, composition, and characteristics of the effi-cient frontier and, thereafter, makes MV optimization a more flexible tool. For example, Xia [47] investigates the problem with a non-negative wealth constraint in a semimartingale model. Another direction for further research is to adopt the continuous-time multiperiod 4_{We note that after imposing any of these additional constraints,}

there may not be any explicit solution. Even if an explicit solution can be found, it could be very complicated and the development of the theory could be very tedious. We will address this question in our further research.

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Markowitz’s problem. Further research could also in-clude conducting an extensive analysis to compare the performance of our estimators with other state-of-the-art estimators in the literature, for example, factor models or Bayesian shrinkage estimators.

Acknowledgements

The authors are grateful to Prof. Alain Bensous-san and Prof. Charles Tapiero for their substantive comments that have significantly improved this manu-script. The authors would like to thank participants at the Conference on Probability with Applications to Fi-nance and Insurance, a joint HKU–HKUST–CUHK– Fudan University meeting celebrating Professor Tze Leung Lais 60th Birthday; International Workshop on Scientific Computing: Models, Algorithms and Appli-cations; International Symposium on Financial Engi-neering and Risk Management 2007; and the 15th An-nual Global Finance Association and Conference for their valuable comments. The research is partially sup-ported by NSF grant No. 10571020 from China and the grant R-155-000-061-112 from Department of Sta-tistics and Applied Probability, National University of Singapore and the grant R-703-000-015-720 from Risk Management Institute, National University of Singa-pore.

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risk level.3We callRsatisfying (1) theoptimal return

and the solutioncto the maximization theoptimal allo-cation planfor the corresponding self-financing port-folio. The solution of (1) can be obtained in the follow-ing theorem.

Theorem 1. For the optimization problem shown in Eq.(1), the optimal return,R, and its corresponding investment plan,c, are, respectively,

R=σ0

µTΣ−1_µ

−(1 TΣ−₁

µ)2

1TΣ−1

1 (2)

and

c= σ0

µTΣ−1

µ−(1TΣ−1

µ)2_/₍₁TΣ−1 1) ×

Σ−1 µ−1

T_Σ−1_µ 1TΣ−₁

1 Σ−1

Suppose that {xjk} for j = 1,. . .,p and k =

1,. . .,n is a set of double array of independent and identically distributed (iid) random variables with mean zero and varianceσ2_{. Let}_x

k =(x1k,. . .,xpk)T

andX =(x₁,. . .,xn). Then, the traditional return

Rp= ˆcT_pX, (3)

where

3_{We note that in this paper we study the optimal return. However,} another direction of research is to study the optimal portfolio vari-ance; see, for example, Pafka and Kondor [34] and Papp et al. [36].

k=1

x_k n,

(4)

S= 1

n₋1

k=1

(x_k₋x)(x_k₋x)T

and ˆ

cp= σ0

XT_S−₁_X

−(1T_S−₁_X₎₂_/₍₁_T

S−₁₁₎

S−1 X₋1

T_S−1_X 1T_S−1₁S

−1 1

Theorem 2. Assume that X₁,. . .,Xn are normally distributed with p_×1 mean vectorµand p×p

co-variance matrixΣ. If the dimension-to-sample-size ra-tio indexp/n→y∈(0, 1), we have

µTΣ−1µ

n →a1, 1TΣ−1₁

n →a2, 1TΣ−1

n →a3,

and ifa1a2−a23>0, then, we have

lim

n→∞

√_n→σ0

γ(a1a2−a2₃)

> lim

n→∞ R

√_n =_σ₀

a1a2−a2₃

a2 ,

whereγ=b

a x1dFy(x) = 1−1y >1,a=(1−

√_y₎2

andb=(1+√_y)2.

(2)

Fig. 1. Empirical and theoretical optimal returns for different numbers of assets (solid line – the theoretic optimal return (R), dotted line – the plug in return (Rˆp)).

plug-in return,_Rpˆ _{, is always bigger than the theoretical} optimal return,R=cTµ, defined in Eq. (1).

{X∗

1,. . .,X

∗

n} from the p-variate ‘normal distribution’ with mean vector X

and covariance matrixS. Then, by invoking the self-financing optimization procedure again on the resam-ple χ∗

, we obtain the bootstrapped “plug-in” alloca-tion,ˆc∗

p, and the bootstrapped “plug-in” return, Rˆ ∗ p,

such that ˆ

R∗ p= ˆc

∗_T p X

∗

, (5)

whereX∗= 1

n 1X

∗

i. Thereafter, we obtain the boot-strap corrected return estimateRˆbas shown in the

fol-lowing theorem.

Theorem 3. Under the conditions in Theorem 2 and using the bootstrap correction procedure described above, the bootstrap corrected return estimate, Rˆb, given by:

Rb= ˆRp+

√_γ(Rpˆ −Rˆ∗

p) (6)

possesses the following property:

√_γ₍_R

−Rpˆ )_≃_Rpˆ ₋_Rˆ∗

p, (7)

whereγis defined in Theorem2,Ris the theoretic op-timal self-financing return obtained from Theorem1,

Rpis the plug-in return estimate defined in Eq.(3)and obtained by using the original sampleχ, and_Rˆ∗

pis the bootstrapped plug-in return estimate defined in Eq.(5)

and obtained by using the bootstrapped sampleχ∗ , re-spectively.

the theoretic optimal self-financing return,R.

3. Simulation study

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covari-ance matrix Σ = (Ijk) in which Ijk = 1 when j =_kandIjk =0 otherwise. Given the level of risk,

the known population mean vector,µ, and the known population covariance matrix,Σ_{, we can compute the}

To illustrate the over-prediction problem, we first plot the theoretic optimal self-financing returns,R, and the plug-in returns,Rpˆ , for different values ofpwith the same sample sizen=500 in Fig. 1.

From Fig. 1, we find the following: (1) the plug-in return_Rpˆ _{defined in Eq. (3) is a good estimate to} the theoretic optimal returnRwhenpis small (30); (2) whenpis large (60), the difference between the theoretic optimal returnR and the plug-in return Rpˆ

becomes dramatically large; (3) the larger the p, the greater the difference; and (4) whenpis large, both the plug-in return_Rpˆ _{is always larger than the theoretic} op-timal return,R, computed by using the true mean and covariance matrix. These confirm the “Markowitz op-timization enigma” that the plug-in returnsRˆp should

not be used in practice.

In order to show the superiority of the performance of the bootstrap corrected optimal return estimate,_Rˆ

over that of plug-in returnRpˆ , we define thebootstrap corrected difference,dR

b (= ˆRb−R), for the return

es-timate,Rˆb, to be the difference between the bootstrap

corrected optimal return estimate, _Rˆ

b, and the

theo-retic optimal return,R, and theplug-in difference,dR p

(= ˆRp₋R), for the return estimate,_Rpˆ _{, to be the} dif-ference between the plug-in return,Rpˆ , and the theo-retic optimal return,R, respectively. We then simulate 1000 times to computedR

x forx=pandb,n=500

andp=100, 200 and 300. The results are depicted in Fig. 2.

From Fig. 2, we find the desired property for our proposed estimate thatdR_b is much smaller thandRp in

absolute values for all the cases. This infers that the

Fig. 2. Comparison between the empirical and corrected portfolio allocations and returns (solid line – the absolute value ofdRp, dotted line – the absolute value ofdR_b).

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Table 1

MSE and relative efficiency comparison

p MSE(dRp) MSE(dR_b) RER_p_,_b

50 0.17 0.05 3.40

100 1.64 0.11 14.91

150 5.22 0.15 34.80

200 17.17 0.28 61.32

250 45.81 0.50 91.62

300 94.42 0.59 160.03

350 252.89 1.23 205.60

remarkable.

RER_p_,_b= MSE(d

R p)

MSE(dR b )

. (8)

Comparing the MSE ofdR_b with that ofdRp in Fig. 2

and Table 1,dR_b has been reduced dramatically from those ofdR

p, indicating that our proposed estimates are

superior. We find that the MSE of dR

b is only 0.05,

improving 3.4 times from that ofdRp when p = 50.

When the number of assets increases, the improvement become much more substantial. For example, when

p=350, the MSE ofdR

b is only 1.23 but the MSE of dR

p is 252.89; improving 205.60 times from that ofdRp.

This is an remarkable improvement. We note that when bothnandpare bigger, the relative efficiency of our proposed estimate over the traditional plug-in estimate could be much larger.

4. Conclusions

(5)

Acknowledgements

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