Conclusions On the Markowitz mean variance analysis

40 Z. Bai et al. Markowitz mean–variance analysis of self-financing portfolios Table 1 MSE and relative efficiency comparison p MSEd R p MSEd R b RE R 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 p increases, the two lines on each level as shown in Fig. 2 further separated from each other; implying that the magnitude of improvement from d R p to d R b 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 different p and present their relative effi- ciencies RE for returns such that RE R p,b = MSE d R p MSE d R b . 8 Comparing the MSE of d R b with that of d R p in Fig. 2 and Table 1, d R b has been reduced dramatically from those of d R p , indicating that our proposed estimates are superior. We find that the MSE of d R b is only 0.05, improving 3.4 times from that of d R p when p = 50. When the number of assets increases, the improvement become much more substantial. For example, when p = 350, the MSE of d R b is only 1.23 but the MSE of d R p is 252.89; improving 205.60 times from that of d R p . This is an remarkable improvement. We note that when both n and p are 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. Z. Bai et al. Markowitz mean–variance analysis of self-financing portfolios 41 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. 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