Portfolio Choice under the Mean-Variance Model with Parameter Uncertainty

HE Chao-lin (何朝林) ，XU Qian (許 倩)

School of Management Engineering，Anhui Polytechnic University，Wuhu 241000，China

Introduction

The conventional wealth allocation approach suggests choosing portfolio weights by maximizing the investor's expected utility.In the Markowitz［1］framework，the expected utility was maximized in the mean-variance procedure by finding the optimal trade-off between the expected portfolio return and risk associated with future investment outcomes.However，expected return，variance，and covariance are estimated with error，while classical mean-variance portfolio optimization ignores the estimation error.Consequently，the mean-variance portfolio formed using sample moments has extreme portfolio weight that fluctuates substantially over time and the performance of such a portfolio is often poor.So the research on how to effectively reduce the estimation error is very necessary during the process of mean-variance portfolio choice.

Knight and Berger［2-3］pointed that multi-prior approach could be used to deal with the problem of parameter uncertainty.Moreover，Ellsberg［4］proved that agents were not neutral to the ambiguity arising from having multi priors.Heath and Tversky［5］believed that with the aversion to uncertainty being particularly strong in the case that where people felt the pressure in assessing the relevant probabilities was low.Garleanu and Pedersen［6］gave the closed-form expression of multi-period optimal portfolio and showed that the optimal portfolio policy shrank toward the Markowitz portfolio at a fixed exchange rate in the absence of estimation error.Klein and Bawa［7］and Bawa et al.［8］examined the impact of estimation risk on portfolio allocation in a single-period model.Kandel and Stambaugh［9］allowed the estimation error in a multi-period model，but assumed that the investor only had a single-period horizon.Brennan and Xia［10-11］considered multi-period model where the investment opportunity set was a constant and the parameter was uncertain.Kan and Zhou［12］analytically characterized the utility loss of a mean-variance investor who suffered from parameter uncertainty.DeMiguel et al.［13］studied both portfolio computed from shrinkage estimators of the moments of asset returns as well as shrinkage portfolio obtained by shrinking the portfolio weights directly，carried out a comprehensive investigation of shrinkage estimators for asset allocation，and found that the shrinkage intensity played a significant role in the performance of the resulting optimal portfolio.Meng and He［14］examined the impact of parameter uncertainty and estimation risk on the dynamic portfolio choice of long-time investor.Yang and Chen［15］predicted the presence of estimation error on the perceived risk of investment portfolio and did an empirical analysis.Kourtis et al.［16］proposed a new estimation framework to enhance the portfolio performance，which directly applied the statistical methodology of shrinkage in the inverse covariance matrix.Tu and Zhou［17］considered the optimal trade-off between the sample mean-variance portfolio and the equally-weighted portfolio， while DeMiguel and Nogales［18］studied the empirical performance of a mixture of portfolio obtained as a combination of the sample minimumvariance portfolio and the equally-weighted portfolio.Liu et al.［19］developed a C-V model to minimize the variance of total expected return rate subject to chance constraint and to maximize the chance of achieving a prescribed return level subject to variance constraint，which was compared with meanvariance model.Teller［20］pointed the importance of estimation error from the aspect of portfolio risk management.Takano and Gotoh［21］studied a nonlinear control policy for multi-period portfolio and showed that their strategy not only reduced the estimation error，but also improved the portfolio performance.All these studies point that estimation error affects both the allocation of portfolio and its performance.

Given the difficulty in estimating moments of risky asset return，it is much more likely that an investor has multiple priors about the moments of risky asset return.In this paper，we introduce a set of constraint constants to measure the uncertainty degree of the expected return，which can be characterized as a set of confidence intervals around the estimated value of expected return.Then，we impose an additional constraint on the mean-variance portfolio optimization program that restricts the expected return to lie within a specified confidence interval of its estimated value，and permit the investor to minimize over the choice of expected return subject to the constraint.So，we build the max-min model of portfolio choice under the meanvariance model with parameter uncertainty and utilize the Lagrange method to obtain the closed-form solution of the multiprior portfolio，which is compared with the mean-variance portfolio and the minimum-variance portfolio; then， an empirical study is done based on the monthly returns for the period June 2011 to May 2014 of eight kinds of stocks in Shanghai Exchange 50 Index;at last，we analyze the results and obtain the conclusions.

1 Portfolio Choice Based on Mean-Variance Model

Assuming investor's initial wealth is unity，he may invest a fraction w in the N risky assets based on the mean-variance model.According to Markowitz［1］，the optimization problem of investor's portfolio choice is given by formula (1)，

where μ is the N-vector of the true expected returns，Σ is an N×N covariance matrix，1Nis an N-vector of ones，γ is the risk aversion coefficient.The solution to the above optimization problem is

A fundamental assumption of the above standard meanvariance portfolio optimum model is that the investor knows the true expected returns.In practice，however，the investor has to estimate the true expected returns.Denoting respectively the estimated value of true expected returns and covariance matrix byand，the actual portfolio weight associated with Eq.(2)is given as follows，

In reality，the expected returnand covariance matrix Σ∧are notoriously difficult to estimate.As a result，the portfolio weights obtained from Eq.(3)tend to consist of extreme position that swings dramatically over time.At the same time，the optimal portfolio often performs poorly.So，the uncertainty of expected return seriously affects the portfolio choice and its performance，we should deal with the estimation error caused by the uncertainty of expected return during the process of portfolio choice.Next，we apply the multi-prior approach to studying the problem of portfolio choice under the uncertainty about the expected return of risky asset based on the mean-variance model.

2 Portfolio Choice under the Expected Return with Uncertainty

2.1 The model of portfolio choice

Due to the existing of estimation error，the expected return can't be estimated with infinite precision，that is，= μ.Here，we introduce a set of constraint constants to the mean-variance optimization problem in formula (1)，which not only measure the different degree of uncertainty about the estimated expected return but also restrict the expected return to lie within a specified confidence interval of its estimated value.These constraint constants imply that the investor recognizes explicitly the possibility of estimation error，that is，the point estimate of the expected return is not the only possible value considered by the investor.According to Gilboa and Schmeidler［22］，the investor is allowed to minimize over the choice of expected return subject to the constraint constants.This minimization over expected return，μ，reflects the investor's uncertainty aversion.

Based on the mean-variance optimization problem in formula (1)，the multi-prior model of portfolio choice under the uncertainty about the expected return can take the following form

where f()· is a vector-valued function，and ∈is a constraint constant that reflects both the investor's uncertainty and his aversion to the uncertainty.In detail，as for the portfolio consisting of N risky asset，we suppose that the expected returns are estimated by their sample meanand the returns are drawn from a normal distribution， then the quantityfollows an F distribution with N and T-N degrees of freedom， where T is the number of observations in the sample for the risky asset.We letand ∈be a chosen quantity for the F distribution.Then the constraint in model (4)can be expressed as The constraint formula (5)can be explained as confidence intervals.In other words，the constraint corresponds to the probability statement- p for some appropriate significance level P.

As we know，whether the confidence interval or the significance level can be interpreted as a measure of the level of uncertainty associates with the parameters estimated.When it is combined with the max-min problem in formula (4)and is used in an investor's portfolio choice problem，it captures the investor's aversion to uncertainty.If the investor has high uncertainty aversion，he could use a ∈that corresponds to a 99% confidence interval.In other words，by picking the appropriate ∈，the investor can indicate his level of uncertainty about the estimated expected return of the portfolio as well as his level of uncertainty aversion.

2.2 Solution to the model

We apply the Lagrange method to solving the model(4)in two steps.According to Gilboa and Schmeidler［22］，the investor is allowed to minimize over the choice of expected return subject to the constraint constants.Based on formula (5)，we let εand first focus on the inner minimization in the model (4)，

The Lagrange function is

From the first order conditions with respect to μ in Eq.(7)，we obtain the optimal solution of μ，

Substituting Eq.(8)into Eq.(7)，we get

From the first-order conditions with respect to λ in Eq.(9)，we obtain

Substituting Eq.(10)into Eq.(9)，we solve the minimization problem in formula (6)，and replacing it in the model (4)，the original max-min problem in formula (4)is equivalent to the following maximization problem

The Lagrange function and its first order conditions with respect to w and λ are given by the following respectively

Let the variance of the portfolio be，that isFrom the first-order condition with respect to w in formula(12)，we obtain

From the first order condition with respect to λ in formula(12)，we obtain

Replacing Eq.(14)in Eq.(13)，we obtain the analytical solution of the model (4)

Replacing Eq.(15)in the equationwe obtain the following polynomial equation

On one hand，if ∈→0，then ε→0，the optimal portfolio w*in Eq.(17)converges to

which is similar with the portfolio choice based on the meanvariance model in Eq.(3);on the other hand，if ∈→∞，then ε→∞，the optimal portfolio w*in Eq.(17)converges to

which is similar with the portfolio choice based on the minimum-variance model in Kourtis［16］.

From Eqs.(17)-(19)，we can conclude that the optimal portfolio choice based on the multi-prior approach is a weighted average of the weights that based on the mean-variance model and the minimum-variance model，that is

where wMVis the portfolio choice based on the mean-variance model，wMINis the portfolio choice based on the minimumvariance model，and φ is the weighted coefficient and is given by

3 Empirical Study

Based on the optimal solution of multi-prior portfolio in Eq.(17)，we chose eight kinds of stocks in Shanghai Exchange 50 Index as the study sample and gave the weights，mean，variance，and performance of optimal portfolio under the investor's different degrees of risk-aversion，then，compared them with those based on the mean-variance model and the minimum-variance model.

3.1 Study sample selection

We select the monthly returns of eight stocks in Shanghai 50 index as the study sample，and their names and codes are:Shanghai Pudong Development Bank (60000)， CITIC Securities (60030)，China Shipping Development (60026)，Chinese Unicom (60050)，Shenergy (60642)，Shanghai Petrochemical (60688)，China Merchants Bank (60036)，and Tianjin Port (600717).Their returns are the continuous composite monthly returns from June 2011 to May 2014，so T =36，N = 8.Their return serial is ri，ri= ln(Pi+1/Pi)，i = 1，2，…，T，where Piis the closing price of the first trading day monthly about the selected eight stocks，which can be obtained from the data source at HuaTai Securities (The Second Professional Edition).

3.2 Empirical results

Based on the return serial of eight stocks，Table 1 gives the weights of optimal portfolio according to Eqs.(17)-(19)under different risk aversion coefficient (γ = 2，3，5).According to the weights in Table 1，Table 2 gives the mean and variance of optimal portfolio based on different method.At last，F(xiàn)ig.1 gives the performance of optimal portfolio by Sharpe ratio.The above calculation process is done by MATLAB software.

Table 1 The weights of optimal portfolio based on different methods

Table 2 The mean and variance of optimal portfolio based on different methods

3.3 Analysis of the results

Table 1 demonstrates that the portfolio based on multi-prior approach moves from the portfolio on account of mean-variance model to the portfolio which is based on the minimum-variance model with the increasing of the constraint constant ∈，specially，the case of ∈= 0 corresponds to the mean-variance portfolio while the case of ∈→∞corresponds to the minimumvariance portfolio.This can be explained by Eq.(20)，where the case of ∈= 0 corresponds to φ = 0 while the case of ∈→∞corresponds to φ = 1.Intuitively，when constraint constant ∈equals zero，the uncertainty-aversion investor thinks that the portfolio expected return is estimated with infinite precision，that is μ∧= μ，and puts a higher weight coefficient on the meanvariance portfolio.Conversely，the portfolio expected return is estimated extreme imprecisely， and the investor strongly recognizes the existing of estimation error and puts a higher weight coefficient on the minimum-variance portfolio.The above result is also demonstrated by Table 2 from the aspect of portfolio's mean and variance under different method.So，we conclude that the optimal portfolio choice based on the multiprior approach is a weighted average of the weights that based on the mean-variance model and the minimum-variance model，with the mean-variance portfolio shrinking toward the minimumvariance portfolio as the increasing of uncertainty about estimated expected return.

Table 1 also demonstrates that the optimal holding in a stock decreases with the increasing of the uncertainty about estimated expected return in that stock，and the decreasing speed is also smaller and smaller.As a result，the range of portfolio weight decreases with the increasing of the constraint constant∈，that is，the multi-prior portfolio weight is less unbalanced and varies much less over time compared with the meanvariance portfolio weight.Intuitively， because of the constrained minimization over expected return， when the constraint constant ∈is large for a stock，that is，the mean is estimated imprecisely，then the investor relies less on the estimated mean，and hence，reduces the weight invested in that stock.Conversely，when the constraint constant ∈is small for a stock，the minimization is constrained more tightly，and hence，the stock weight is closer to the standard weight given by the mean-variance model that ignores the estimation error of expected return，in the limit，when the constraint constant ∈is zero，the optimal weight come from the classic mean-variance model.So，the steady of multi-prior portfolio is strengthened compared with the mean-variance portfolio.

Figure 1 demonstrates that the Sharpe ratio curve firstly increases，and then decreases with the increasing of constraint constant ∈，that is，the Sharpe ratio of multi-prior portfolio is strongly greater than that of minimum-variance portfolio，and sometimes greater than that of mean-variance portfolio.In future，the varying trend of Sharpe ratio depends on risk aversion coefficient γ，that is，the smaller is the risk aversion coefficient，the more obvious is the varying trend.For the case of γ = 2，the Sharpe ratio arrives its maximization when the constraint constant is between 6 and 7;the case of γ = 3，the maximum Sharpe ratio corresponds to that the constraint constant is between 5 and 6;the case of γ = 5，the maximum Sharpe ratio corresponds to that the constraint constant is between 3 and 4.We can also find the above phenomena from the mean and variance of different portfolios in Table 2.Intuitively，estimation error on the expected return is also a kind of risk，the smaller is the risk aversion coefficient，the more is the fraction of estimation risk considered by the uncertaintyaversion investor，the bigger is the constraint constant，and this means that averting the uncertainty about expected return is equal to the increasing of investor's risk aversion degree.So the performance of multi-prior portfolio is strongly greater than that of minimum-variance portfolio，and sometimes greater than that of mean-variance portfolio.Combining with the analysis of the above two sections，the investor can improve the steady of multi-prior portfolio as well as its performance for some appropriate constraint constants ∈.

Fig.1 The performance of optimal portfolio based on multi-prior approach

4 Conclusions

The classical mean-variance portfolio optimization assumes that the expected return of risky asset is known with perfect precision.In practice，it is difficult to estimate the expected return precisely.Due to the existing of estimation error，the mean-variance portfolio weight has extreme value，fluctuates dramatically over time，and its performance is relatively poor.This paper utilizes the multi-prior approach to deal with the estimation error on the estimated expected return.The multiprior approach relies on imposing a set of constraint constants on the mean-variance portfolio optimization program， which restricts the expected return to lie within a specified confidence interval of its estimated value.This constraint recognizes the possibility of estimation error.So，in addition to the standard maximization of mean-variance objective function over the choice of weight，the investor also minimizes over the choice of expected return value subject to the constraint.Then，we utilize the Lagrange method to obtain the closed-form solution of multiprior portfolio，which is compared with the mean-variance portfolio and the minimum-variance portfolio;at last，an empirical study is done based on the monthly returns for the period June 2011 to May 2014 of eight kinds of stocks in Shanghai Exchange 50 Index.The results show，multi-prior portfolio is a weighted average of the mean-variance portfolio and the minimum-variance portfolio;the steady of multi-prior portfolio is strengthened compared with the mean-variance portfolio;the performance of multi-prior portfolio is greater than that of minimum-variance portfolio.The study demonstrates that the investor can improve the steady of multi-prior portfolio as well as its performance for some appropriate constraint constants.

Finally，market transaction friction，such as transaction cost，also plays an important role during the process of portfolio choice.In the research， market transaction friction is demonstrated as a function of model parameter，and is affected by the uncertainty of model parameter.This should be studied in the future.

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