多約束條件下的資產組合分析及績效評價實證研究

劉　廣

（廣州大學經濟與統計學院，廣東廣州　510006）

多約束條件下的資產組合分析及績效評價實證研究

劉廣

（廣州大學經濟與統計學院，廣東廣州510006）

依據均值－方差規則，給出了同時存在證券數量變化、證券之間相互擾動、允許賣空機制約束條件下，資產組合有效前沿變動的統一分析框架和數理證明結果.基于證明結果，使用夏普比率作為績效評價工具，實證發現74只開放式基金的動態績效在考察期內并不顯著，且不同類別開放式基金的績效結果之間也不存在顯著差異.研究豐富了動態資產組合理論，并為評判機構投資者資產組合績效和資本市場建設效果提供了新視角和證據.

均值－方差規則；多約束條件；資產組合；績效評價

0　Introductions

Based on the mean-variance rule，MARKOWITZ［1］firstly formed the modern portfolio theory（MPT）by developing a standard analytical framework and offering its mathematical solutions for risk assets and riskless asset by two steps.Despite all this，there are still some unsolved issues we should give full consideration.

Firstly，what's the impact on the portfolio when some effective securities①Given a securities set I，if the effective frontier of Imoves when some very security joins（or removes），then the security is called effective security of I.Otherwise，it is called invalid security of I.Refer to HOU，et al［2］.are added to or removed？Initial public offering（IPO）and special transfer（PT）are themain reasons for securities change，and global asset allocation is another one.WU［3］studied on the global assetallocation strategy by using the data of 16 major stock markets，finding that compared with the domestic optimal portfolio，the global optimal portfolio had higher return and lower standard deviation.ZHANG，et al［4］achieved similar conclusions subsequently.

Secondly，what's the impact on the portfolio when there aremargin trading and short selling institutions？Our market mechanism has been improving by stock index futures，margin trading，and so on.The graded funds also provide sufficient samples for leveraged investment strategy.Those factors have already been proved their impact on risk-return characteristics of portfolio［5］.

Thirdly，what's the perturbation between securities impact on the portfolio？The correlations between securities are instable and time-varying if the market is not effective［6］.XUE，et al［7］proved that the perturbations would affect the value at risk（VaR）of portfolio.

In response to these issues，previous researches already gave corresponding mathematical proof and empirical test，which supplemented the portfolio theory and provided useful guidance for investment practice.However，there has not been developed a unified analytical framework taking into account the above factors synchronously yet.For this reason，this paper attempts to construct a unified model and provide its analytic solutions based on M-V rule.It expects to enrich the MPT theoretically，and provide a guide for global asset allocation and leverage investment practically.Furthermore，it also can be used to test the asset allocation efficiency by Sharpe ratio（SR）in horizon and vertical，and reveal our capital market construction achievements over the past decade.

This paper will be organized as follows.Literature review is delivered firstly.Then，a unified asset allocation model is constructed by two steps，and its solutions of optimal portfolio and efficient frontier are offered too.Thirdly，we investigate and demonstrate how the efficient frontier moves according to the abovemodel.Finally，the performance of74 open-end funds is evaluated，and some new evidence of institutional investors'asset allocation efficiency is acquired.

1　Literature reviews

Based on MPT，the constraints are gradually relaxed in subsequent researches.Those domestic and foreign researches can be roughly classified into three main categories.The first one involves the securities joining or quitting.The second one is short selling allowed or not allowed.The last one examines the consequence of perturbation or no perturbation between securities.Studies abroad have solved one or two of those problems.For example，SZEGO［8］provided a solution for short selling.Domestic studies also focus on those issues and can be summarized as below（Table 1）.

Table 1　Summary of portfolio change studies

Followed by Markowitz's study，SHARPE［24］simplified the allocation process，and provided SR for portfolio performance evaluation.SHARPE［25－26］studied the application of SR under short selling，so did some other foreign researches［27－28］.ZHANG，et al［29］evaluated the performance of 22 funds by SR，and found that their performance cannot surpass the average performance of market.YANG，et al［30］，YANG，et al［31］also proved that the SR ormodified SR was valid and applicable for closed-end funds.XU，et al［32］applied a modified SR to domestic hedge funds，finding that the risk-adjusted returns of hedge fundswere better than that of open-end funds，private equity funds or broker collection of financial products.LI，et al［33］conducted a natural experiment for liquor industry，finding that the daily SR was as high as 33％with short selling，which means stockswere grossly overvalued bymargin trading.

Both domestic and foreign studies always investigated whether the portfolio performance is significant statically，but have not yet made further inquiry in SR's dynamic change.In fact，though the performance of portfolio is better than that of benchmark，it can not reflect whether the performance itself becomes better and better，much less to reveal the achievement of our capitalmarket.

2　Hypotheses and modeling

2.1Hypotheses

Some necessary hypotheses will be given before modeling.Assuming that there are n kinds of risk assets and an individual riskless asset F，whose rate of returns are denoted by random riand fixed rfrespectively.Investor's initial wealth is fixed too，and will be allocated to the above n＋1 assets.is the expected rate of risk asset i.is the vector.σijis the covariance Cov（ri，rj）of risk asset i and j.wiis the allocation ratio of risk asset i.Wnis the vector（w1，w2，…，wn）T，i，j＝1，2，…，n.We also assume that themarket is perfect，risk assets are not linearly dependent.The covariance matrix Vn＝（σij）n×nis positive definite，and the risk assets'rate distribution is continuous.

2.2M odeling

Based on the M－V rule，if there is no riskless asset，the optimal risk portfolio P can be deduced by the following objective function and constraints.

3　Portfolio analyses w ithout risk less asset

The proof has been finished so far.Then，the following conclusions can be obtained.

Theorem 1 When k effective securities joining in the original securities set Sn，the efficient frontier curve will move to the left indefinitely.Furthermore，if Bn/An＜b/a，the curve willmove to the left upward.If Bn/An＞b/a，the curve willmove to the left downward.If Bn/An＝b/a，the curve will move to the left horizontally.

Proof.Referring to Eq.（3），the risks of the globalminimum variance portfolio of set Snand Sn＋kare 1/Anand 1/An＋k，the rate of returns are Bn/An

AnAn＋k＞0.Therefore，if Bn/An＜b/a，then Δr＞0，meaning the curve moves upward.If Bn/An＞b/a，thenΔr＜0，meaning the curvemoves downward.If Bn/An＝b/a，thenΔr＝0，meaning the curvemoves horizontally.Q.E.D.

Theorem 2 When k effective securities joining in the original securities set Sn，set Sn＋kwill completely contain set Sn.That is to say，Sn?Sn＋k.

Proof.Referring to Eq.（2），we can get the function ofˉrpby solving the equations formed by the efficient frontier of Snand Sn＋k.

Where A＝AnΔk＋Anχ－Δna，B＝BnΔk＋Bnχ －Δnb，C＝CnΔk＋Cnχ－Δnc.For Eq.（12），Δ＝4B2－4AC＝－4ΔnΔkΔn＋k＜0and A＞0，thus the efficient frontiers of two sets are separated，namely Sn?Sn＋k.Q.E.D.

Theorem 2 also suggests that when k effective securities joining in the original securities set Sn，the efficient frontier opening will become larger in

4　Portfolio analyses with riskless asset

Assuming that the original set Sncontains n risk assets and an individual riskless asset F now.Referring to Eq.（4），when k effective securities joining in set Sn，the optimal portfolio weights of Sn＋kand its efficient frontier are as follows.

Where Tn＋k＝－2Bn＋krf＋Cn＋kAn＋kBn＋kand Cn＋khave the samemeaning as Eq.（2）.

Theorem 3 When k effective securities joining in set Sn，the slope of efficient frontier line of Sn＋kwill become bigger in

Proof.Referring to Eq.（5），the slope difference of the efficient frontier lines between set Sn＋kand Snis

Theorem 3 suggests that investors ask for more expected revenue at the same level of risk.That is，risk premium increases，and SR of the optimal portfolio becomes bigger.

5　Empirical analyses on open-end funds

The proof above proves thatwhen the riskless return is fixed，the SR of the optimal portfolio will become bigger as the effective securities join.In other words，if the SR does not increase，it indicates indirectly that the efficiency of assets allocation is insufficient.In view of the domestic riskless return remaining ataround 3％stably in the past decade，we take a group of open-end funds as samples to examine their performance by two steps.Firstly，if the samples'SR's become bigger vertically，then their asset allocation efficiency is statistically significant，and vise versa.Secondly，if themean values of SR's between three type funds are statistically significantly different horizontally，then their asset allocation efficiency is statistically significantly different，and vise versa.

5.1Sam ples selecting and data collecting

We select74 stock funds thatare established before 2007 as samples，which contains 42 growth funds（including positive growth funds and steadygrowth funds），18 value funds（including investment value funds）and 14 index funds.To avoid missing data or abnormal fluctuations of the market，we use 78 weekly SR data from 2009 to June 30，2013.The data are from the CSMAR，and processed by Matlab7.0.

5.2Vertical trend analyses

The tendencies of SR's are examined by ADF test，and their directions are indicated by T value.Those test results are demonstrated in Table 2.

Table 2　Test results of SR's

The results above show that the SR's of65 funds are non-stable in all samples，where 35 funds have downward trend at10％significance level.The SR's of the other 9 funds are stable at the same significance level.It proves that those samples did not obtain better performance during the sampling period.Obviously，that is not an expected or satisfactory achievement.

5.3Horizontal com parison analyses

Table 3　Test results of SRmeans among three sample groups

The test results suggest that the null hypothesis can't be rejected at5％significance level.That is to say，the mean SR's of three group funds are equal，and the performance between different type funds has no significant difference either.

6　Conclusions

Institutional investors are themajor participators in our capital market.Their asset allocation behaviors and results are affected by external factors，and should be paid more theoretical and practical attention to.Based on the M-V rule，we first develop a unified analytical framework incorporating different external constraints synchronously.Then，assuming some effective securities joining，we derive the following results by logical deduction and mathematical proof.If riskless asset is not included，the efficient frontier curvewill move to the left，the opening of the curve will become larger，and the new efficient investment feasible setwill completely contain the original feasible set.If riskless asset F is included，the efficient frontier line will move around F to the left upward.The subsequent test results prove that the open-end funds'dynamic asset allocation performance evaluated by SR is not statistically significant vertically or horizontally.

The article is supposed to complete the asset selection and evaluation theory.The work provides a theoretical basis for institutional investors'positive management，and a new perspective to observe allocation performance.Taking into account the reality that the funds always have relative high degrees of holding stocks，and don't execute full diversified asset allocation strategy strictly，the results have important implications for institutional investors and market regulators.

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【責任編輯：孫向榮】

Em pirical research on dynam ic portfolio and performance evaluation w ith multi-constraints

LIU Guang

（School of Economics and Statistics，Guangzhou University，Guangzhou 510006，China）

Based on themean-variance rule（M-V rule），we try developing a unified analysis framework for the very condition that there are security quantity change，perturbations between securities and short-sellingmechanism at the same time，and prove how the efficient frontier changes.According to the above conclusions，the performance of74 open-end funds is evaluated by Sharpe Ratio.It's a pity that the total result is not significant，and the results between different type funds are not significant either.The study enriches the dynamic portfolio theory，and provides a new perspective and evidence for institution investors'asset allocation efficiency and regulators'capital market construction.

mean-variance rule；multi-constraints；dynamic portfolio；performance evaluation

F 830.59 Document code：A

date：2015－10－23；Revised date：2015－11－06

s：This work is supported by the Guangdong Province-sponsored Philosophy and Social Science Funding Program（GD14XYJ16）and the Guangdong Province Education Department-sponsored Platform Funding Program for Humanities and Social Sciences（2014WQNCX074）

F 830.59

A

1671-4229（2015）06-0009-08

Biography：LIU Guang（1980—），male，lecturer，Ph.D.E-mail：mrliuguang＠126.com