Improving the Markowitz Model using the Notion of Entropy

Improving the Markowitz Model using the Notion of Entropy

Chao Cheng

U.U.D.M. Project Report 2006:11

Examensarbete i matematik, 20 poäng Handledare och examinator: Johan Tysk

December 2006

Department of Mathematics

Uppsala University

Abstract

The Mean-variance framework proposed by Markowitz is the most common model for portfolio selection problem. The most important concept in his theory is diversification.

Diversification means designing an investment portfolio that reduces exposure risk by combining a variety of investments. But actually, the portfolios’ weights are often extremely concentrated on few assets when using mean-variance framework; this is a contradiction to the notion of diversification. Entropy is a well accepted measure of diversity. In this thesis, we discuss an improved mean-variance model based on maximum entropy theory (MVME). Entropy can be viewed as a measure of disparity from the uniform probability distribution. This approach can be viewed as a direct shrinkage of portfolio weights. The estimation errors, stability of portfolio weights, portfolio performance and degree of diversification for both mean-variance and the MVME framework are tested. Compared with the mean-variance framework, the improved model leads to a well diversified portfolio.

Contents:

1. Introduction………...1

2. Background to the Mean-Variance framework………2

2.1 Introduction of Markowitz portfolio theory………..2

2.2 Assumptions of mean variance analysis………3

2.3 Parameters of Mean-Variance analysis……….4

2.4 Mean-Variance Model………...7

2.5 Efficient frontier………..9

3. An improved model based on Maximum Entropy Theory………...….11

3.1 Background………...………...11

3.2 Definition and Properties of Entropy……….14

3.3 Optimal portfolio diversification using Maximum Entropy Theory………18

4. The comparison test between two models………20

4.1 The choice of parameters………..20

4.2 Experiment for estimation errors………...20

4.2.1 The definition of estimation errors………...20

4.2.2 The comparison for estimation errors in different models………22

4.2.3 Estimation errors versus Sample size………24

4.3 Experiment on portfolio performance………...27

4.3.1 Efficient Frontier……….27

4.3.2 Sharp Ratio versus Sample size……….28

4.3.3 Portfolio returns………..30

4.4 Stability of Portfolio Weights………..32

4.5 Degree of diversification……….34

5. Conclusion………..37

Acknowledgements………...38

References……….39

Chapter 1. Introduction.

Modern portfolio theory (MPT) was first discovered and developed by Harry Markowitz in his paper "Portfolio Selection," [1] published in the1952. This article presents the method to construct a portfolio that could achieve a desired level of return while minimizing the investment risk. The mean-variance framework is the most widely used model in solving portfolio diversification problems. But it has one big weakness;

the portfolios’ weights are often extremely concentrated on few assets, which is a contradiction to the notion of diversification. In this thesis, an improved model based on maximum entropy theory is discussed and we also compare it with the classical mean-variance framework. This new approach could be viewed as a combination methodology of the mean-variance and the maximum entropy theory [2].

In Chapter 2, the classical mean-variance framework is presented comprehensively.

In Chapter 3, the conventional improvements of mean-variance framework are depicted; the properties of entropy and maximum entropy theory are introduced. At the end of this section, the improved model based on maximum entropy theory is proposed; the parameters in MVME model and unique solution are also discussed. In Chapter 4, the comparison between mean-variance and MVME framework will be made in four aspects:

• Estimation error.

• Portfolio performance, including the comparison on efficient frontier, Sharp ratio, actual return and final return.

• Stability of portfolio weights.

• Degree of diversification.

Finally, we present the conclusion in Chapter 5.

Chapter 2. Mean-Variance Framework.

2.1 Introduction of Markowitz Portfolio Theory.

Modern portfolio theory (MPT) is an attempt to find the balance relation of the risk-reward in the investment portfolios. MPT proposes the idea of diversification as a tool to optimize the portfolios.This theory was first discovered and developed by Harry Markowitz in the 1950’s. Markowitz showed the benefits of diversification, also known as “not putting all of your eggs in one basket” in this theory. In other words, investment is not only about picking stocks, but also about choosing the right combination of stocks. His theory emphasized the importance of risk, correlation and diversification on expected investment portfolio returns. His work changed the way that people invest.

Before Markowitz, people thought that there was one optimal portfolio which could offer the maximum expected return while minimizing risk. Markowitz clarified that it is impossible from the mathematic point of view. In the real world, the optimal portfolio selection is the problem about how much should be invested in each security to achieve a desired level of return while minimizing investment risk or getting the maximum expected return at a fixed risk level. Markowitz offered an answer by the Efficient Frontier. It is possible to construct a portfolio in the “efficient frontier” to offer the maximum return for any given level of risk. Based on the above concept, Markowitz developed the famous financial portfolio model Mean-Variance model (MV model), which was published in << Portfolio Selection >> in 1952. This model is the most common formulation of the portfolio selection problem. The mean-variance analysis provides the first quantitative treatment of the tradeoff between reward and risk. As we know, the two most important factors to be considered in Markowitz portfolio selection theory are reward and risk. A fundamental question is how to measure risk. In the MV model, reward is defined by expected return while the risk is defined by variance.

2.2 Assumptions of Mean-Variance Analysis.

The mean-variance analysis is based on the following assumptions [3]:

1). Investors are rational and behave in a manner as to maximize their utility with a given level of income or money.

2). Investors have free access to fair and correct information on the returns and the investment risk. Each investor could master the information sufficiently.

3). The markets are efficient and absorb the information quickly and perfectly.

4). All investors are risk-averse and try to minimize the risk and maximize return. It means that for some assets which offer the same return, the investors will prefer the lower risky one or for that level of risk an alternative portfolio which has higher expected returns exists.

5). Investors make decisions based on expected returns and variance or standard deviation of these returns. Investors will accept increased risk only if compensated by higher expected rewards. Conversely, an investor who wants to seek higher returns must accept more risk.

6). The returns of the investment security are random variables with a known multivariate normal distribution. With this assumption, portfolio efficiency is determined by simply compounding the expected returns and the standard deviations of their expected returns.

2.3 Parameters of Mean-Variance Analysis.

For building up the efficient set of portfolio, as laid down by Markowitz, we need to look into these important parameters [4]:

1). Rate of return.

The rate of return of the asset is defined by r, satisfying thatX_T = +(1 r X) ₀, where X₀ and X_T are the prices of the asset at purchase and selling

respectively. As an example, the rate of return from deposits in a bank account is the interest rate.

2). Expected return.

The rewards of an investment in an asset have some level of uncertainty. The value of X_T is unknown at time 0, which means the rates of returns are often not known in advance. We consider the rate of return as random variables. To characterize the asset we shall consider the expected rate of return. In the MV framework, we estimate the expected value

r r

μi for asset as follows: i

1

( ) 1

N

i i i i

t

r E r r

μ ⁻ N

=

= = =

∑

The estimated expected return is a useful way to describe the assets and gives us a general measurement of how large the return it is.

3). Variance and Standard deviation

To characterize the uncertainty of an asset, we usually use the variance or standard deviation of the historical returns. It quantifies how much the rate of return deviates from the expected rate of return. The variance is defined as the risk measurement in MV framework.

The estimated variance and standard deviation for asset is given by: i

2 2

1

(( ) ) 1 ( )

N

i it i it

t

E r r

N

2

σ μ μi

=

= − =

∑

4). Covariance between two assets.

In choosing an investment, one natural way to reduce the risk of losing value for an asset when a given event occurs is to find another asset with increasing value when this event occurs. So we should not only take into account the individual returns of assets but also consider the relationship of the returns among the assets. We use the covariance to exhibit the way asset returns move together or move inversely. The covariance between asset i and j is defined as follows,

1

(( )( )) 1 ( )( )

N

ij ji it i jt j it i jt j

t

E r r r r

σ σ μ μ N μ

=

= = − − =

∑

We note that σ_ij =σ_ji when i≠ j and σ_ii =σ_i² when i= j. If the return of asset and i j move in the same direction, we haveσ_ij >0, inversely, σ_ij<0. To describe the relation of possible assets, we define the covariance matrix as

follows:

n

11 12 1

1 2

n

n n nn

C

σ σ σ

σ σ σ

⎡ ⎤

⎢ ⎥

= ⎢ ⎥

⎢ ⎥

⎣ ⎦

L

M M M M

L

From the expression and the character of covariance, we know the covariance matrix C is symmetric and it also can be proved that matrix C is positive definite.

5). Investment weights.

Assume that the investor wants to select a portfolio from n possible assets, ω_i

is the proportion invested in asset i. So if all wealth is invested, we have

1

1

n i i

ω

=

∑

6).Expected rate of return of the portfolio.

The expected rate of return of the portfolio for assets will be expressed as follows,

n

1 1 1

( ) ( )

n n n

p i i i i

i i i

E r E r _{i i}

μ ω ω

= = =

=

∑

∑

∑

))2

)) μ

,

by using the linear property of expectations.

7).Variance of the portfolio.

The variance of the portfolio for n assets is given by

2 2

1

1 1

1, 1

1 1

( ) ( (

( ( ) (

(( )( ))

'

n

p p p i i i

i

n n

i i i j j j

i i

n

i j i i j j

i j

n n

i j ij

i j

E r E r

E r r

E r r

C

σ μ ω

ω μ ω μ

ω ω μ μ

ω ω σ ω ω

=

= =

= =

= =

= − = −

= − −

= − −

=

=

∑

∑ ∑

∑

∑∑

where ω'=(ω ω₁. ₂...ω_n) are the weight vector and C is the covariance matrix defined in (4).

2.4 Mean-Variance Model.

Assume that there are assets and for each asset it has an expected rate of return

n ( ,a a_{1 2}... )a_n

μi and a variance σ_ii. Defineμ'=(μ μ₁, ₂...μ_n) asthe expected rate of return vector, and C as the covariance matrix, where σ_ij is the covariance between asset i and j when i≠ j or the variance of asset wheni i= j.

The motivation of the Markowitz theory is to achieve a desired level of return while minimizing the investment risk or seek the maximum expected return at a fixed risk level. With this understanding, let ω'=(ω ω₁, ₂...ω_n) denotes the proportion invested

in each asset, e=(1,1,....1)∈Rⁿ, and then the mathematical formulation of the problems could be summarized to the following equations[5],

minω ²

1 1

'

n n

p i j ij

i j

σ ω ω σ ω ωC

= =

=

∑∑

Subject to '

' 1

fixed

e ω μ μ

ω

⎧ =

⎨ =

⎩ (1);

or

max

ω 1

'

n i i i

r μ ω ω μ

=

=

∑

Subject to (2).

' 2

' 1

pfixed

C e ω ω σ

ω

⎧ =

⎨ =

⎩

In equation (1), the objective function is the variance of the portfolio; the first constraint clarifies the desired level of returnμ_fixed . In equation (2) the objective function is the expected return of portfolio; the first constraint clarifies the desired portfolio risk level , and the second constraint in these two equations states that all weights add up to one, which means that all wealth should be invested. If short

2 pfixed

σ

selling is not allowed, we add the constraintsω_i ≥0. These two models are equivalent, it is more common to choose equation (1) since it is easier to solve.

To solve the constrained optimization problem (1), the most common method is Lagrange Multipliers. The Lagrangian for problem (1) is

1 2 1 2

( , , ) ' ( ' _fixed) ( ' 1);

Lω λ λ =ω ω λ ω μ μC − − −λ ω e−

The solution can be found in the following theorem [6].

Theorem 2.1

The optimization problem (1) has the unique primal-dual solution,

1

1 2 1 2

1 ( ) ; 2( ) / ; 2( )

2C e ^{fixed fixed} / .

ω= ⁻ λ μ λ+ λ = αμ −β δ λ = γ βμ− δ

Hereα =e C e' ⁻¹ ; β =e C' ⁻¹μ γ μ; = 'C⁻¹μ δ αγ β; = − ² .

Proof:

To find the critical point of the Lagrangian, we have to solve the first order equation,

1 2

1

2

2 0

' 0;

1 ' 0;

fixed

L C e

L

L e

ω ω λ μ λ

μ ω μ

λ λ ω

∇ = − − =

∂ = − =

∂

∂ = − =

∂

;

or the matrix form as follows

1

2

2 0

' 0 0 1

' 0 0 _fixed

C e

e

μ ω

λ

μ λ

⎛ ⎞

⎛ ⎞⎛ ⎞

⎜ ⎟

⎜ ⎟⎜− ⎟= ⎜

⎜ ⎟⎜ ⎟

⎜ ⎟⎜− ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠ ⎝μ ⎠^⎟

(3)

under the following conditions that covariance Matrix is positive definite and the expected rate of return

C

μ is not a multiple of . The left matrix of matrix (3) is a full rank matrix. So the optimization problem (1) has the unique primal-dual solution obtained from the strong convexity of the objective function and the full rank of constraints.

e

The optimal portfolio ω'=(ω ω₁.... _n) is obtained from the first row of the above

matrix, 1 ¹ _{1 2}

2C ( e)

ω= ⁻ λ μ λ+ . Substitutingωinto the last two rows of matrix, we obtain

( )

1 1

1 1

2

1 ' 2

' 1 2 1

2

fixed

fixed fixed

fixed

C e

e

αμ β

λ μ μ γ β μ

μ γ βμ

λ β α δ

− −

− −

⎛ ⎞ ⎛

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞

=⎜ ⎟ = = ⎜ ⎟

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎜⎝ ⎟⎠ ⎜⎝ ⎟ ⎜⎠ ⎝ ⎟⎠ −

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎞.

2.5 Efficient frontier.

Any solution of ω'=(ω ω₁.... _n) from the optimization problem (1) for a given μ_fixed is called a frontier portfolio. The set of such portfolios form a hyperbola in the σ μ− plane. This is illustrated below:

The hyperbola has a vertex which corresponds to the lowest risk portfolio. As we said before, investor will accept increased risk only if compensated by higher expected returns. So the upper half of hyperbola is called efficient frontier [7], while the bottom half is called inefficient frontier. It means that the optimal portfolio occurs on the upper half of the hyperbola. For any given value of standard deviation (risk), one would like to choose a portfolio that gives the highest expected rate of return from the efficient frontier.

One way to obtain the efficient frontier is to solve the following problem,

minω ω ω λω μ'C − '

Subject to ω'e 1= (4)

where λ is the trade-off parameter between returnμand risk. If short selling is not allowed, we could add constraintsω_i ≥0.

We also can use Lagrange Multipliers to solve optimization problem (4). The Lagrangian for problem (4) is

1 1

( , ) ' ' ( ' 1).

L ω λ =ω ω λω μ λ ωC − − e−

Theorem 2.2

Optimization problem (4) has the unique primal-dual solution:

1

1 1

1 2

( ) ;

2C e λβ

ω λμ λ λ

α

− −

= + = ,

where α =e C e' ⁻¹ ; β =e C' ⁻¹μ.

The proof is similar with the proof of Theorem 2.1.

Chapter 3. An Improved Model Based on Maximum Entropy .

3.1 Background.

As we mentioned before, the Markowitz mean-variance framework is the most common model for solving portfolio selection problems. The most important concept in his theory is diversification. Diversification means designing an investment portfolio that reduces exposure risk by combining a variety of investments. The goal of diversification is to reduce the risk in a portfolio. However, the portfolios’ weights obtained from mean-variance framework are often extremely concentrated on a few assets. This is a contradiction to the notion of diversification.

In practice, sample mean and covariance matrix are estimated from historical data.

There are lots of factors that influence the estimation, such as the sample size. If the sample size is too small, the sample mean and covariance could have large estimation errors. It is generally thought that the concentrated position problem is caused by the statistical errors when estimating the mean and covariance matrix.

Jobson and Korkie [8] showed that these statistical errors change the portfolio weights in such a way that often leads to that the portfolios’ weights are concentrated on some positions. And we also know that the mean-variance framework is extremely sensitive to input parameters. Small changes of the sample mean and covariance matrix will have a large effect on the optimal portfolios. So the precise estimation of sample mean and covariance matrix is the most important prerequisite for the mean-variance framework.

The method for reducing statistical errors in sample mean and covariance matrix has been widely researched. References showed that in order to reduce the statistical errors in mean-variance model, we should improve the estimation of the sample mean at least. Three different approaches may carry a good effect on estimation errors for the mean-variance model. Two of them are shrinkage estimators of sample means and the other is the bootstrap approach.

So called “shrinkage” estimator is intended to shrink the historical means to some grand mean. Consider R=( , ...., )r r_{1 2} r_T as a N T× matrix, where the rows are the time series of historical returns for each asset, the columns are the returns of different assets at a specific time. The first shrinkage estimator used to improve the sample means is called the James-Stein estimator [9].It is given by

I r_G

JS

+ −

−

= ω μ ω

μ (1 ) ,

where I is a vector of ones; μ'=(μ₁,...,μ_N) is the sample estimate, and

∑

= ^T

j ij

i r

T ₁

μ 1 ;

∑

=

− = ^N

i i

G N

r

1

1 μ is the grand mean. The shrinking factor is

))]

( )' (

/(

) 2 ( , 1

min[ N T r_G I C ¹ r_GI

− −

− −

−

×

−

= μ μ

ω

where C⁻¹is the sample covariance matrix.

The second shrinkage estimator used to improve sample means is called the Bayes-Stein estimator [9]. It is given by

I r_GBS

BS

+ −

−

= ω μ ω

μ (1 ) ,

where r_GBS I'C⁻¹ /(I'C⁻¹I),

− = μ ω α α= /( +T), and the shrinking factor is

)]

( )' /[(

) 2

(N r_GBS I C ¹ r_GBS I

− −

− −

− +

= μ μ

α .

The difference between these two shrinkage estimators is that they shrink the sample means to different targets. In the first case, the target is the arithmetic average of sample means, while the target is the mean of the MVP portfolio in sample in the second case. But we cannot say which shrinkage estimator is better in general. And we should note that the shrinking factor is biased since it is related to the sample.

The third method is the bootstrap approach [10].The bootstrap means using the resampling method to replace the actual data. The notion of bootstrap is to extract more information about the actual distribution of observed data by the generated bootstrap samples. Let the N T× matrixR=( , ....,r r_{1 2} r_T) be the return matrix, we resample the return matrix times, and at the end we will get the bootstrap returns matrix

T

*

R N T_× . Apply this resampled returns matrix to the MV framework, and then the corresponding optimal weightsω^* , return μ^* and standard deviation σ^* for R^*_{N T×} are obtained. By repeating this procedure B times, we will get B sets of bootstrapped optimal portfolios. The solution to the portfolio optimization problem for the return matrix R=( , ....,r r_{1 2} r_T) is the average solutions, which is obtained from the bootstrap procedure. The optimal weightsω , the return μ and standard deviation

σ for R_{N T×} are given by

* *

1 1 1

1 1 1

; ;

B B

b b b

B B B

*.

ω ω μ μ σ B σ

= =

=

∑

∑

∑

=

These three methods introduced above reduce statistical errors in the parameter estimations. Furthermore, they may improve the diversification for mean-variance framework. In the next section, we will introduce a different concept called entropy to improve the diversification. This method could also be understood as a form of shrinkage of portfolio weights [11].

3.2 Definition and Properties of Entropy.

The definition of entropy originally comes from thermodynamics. Now, the concept of entropy has already been developed in other fields of study, including information theory, statistical mechanics and psychodynamics. The concept in information entropy is occasionally called Shannon entropy [2]; Shannon first introduced his idea in his famous paper “A Mathematical Theory of Communication” [2] in 1948.

The concept of Shannon entropy describes how much information is included in an event. Consider that a n-states probability processX =( ,x x_{1 2},... )x_n , with probability

vector P=(p p₁, ₂,...,p_n) , where p_i ≥0 (i=1...n) ,

1

1

n i i

p

=

∑

function defined as the amount of information associated with the state

( _i) h p

X =xi,i=1...n. For each , we define by,

i

n H_n

1 2

1

( , ,..., ) ( )

n

n n i

i

H p p p p h p

=

=

∑

Hncan be thought as a measure of the average amount of information [12]. For simplicity, we write

1 2

1

{ ( , ,..., ); 0; 1}

n

n n i

i

P p p p p p

=

Δ = = ≥

∑

n

.

According to Shannon, the definition of information entropy should satisfy the following properties [13]:

1). For any n ,H_n(p p₁, ₂,...,p ) is a continuous function ofP=(p p₁, ₂,...,p_n)∈ Δ_n.

2). is maximized when the probability distribution is uniform. This means:

1 2

( , ,..., ) Hn p p p_n

1 2

1 1 1

( , ... ) ( , ... )

n n n

H p p p H

n n n

≤ ;

whereP=(p p₁, ₂,...,p_n)∈ Δ_n.

3). Events of probability zero do not contribute to the entropy, i.e.

1( 1, 2... , 0) ( 1, 2... )

n n n

H ₊ p p p =H p p p_n .

whereP=(p p₁, ₂,...,p_n)∈ Δ_n.

4). IfP=(p₁₁,p₁₂,...,p_1n,...,p_mn) _mn, and

1 n

i ij

j

q p

=

∈ Δ =

∑

1

11 12 1 1

1

( , ,..., ,..., ) ( ,..., ) ( ,..., )

m

i i

mn n mn m m i n

i i i

p pn

H p p p p H q q q H

q q

=

= +

∑

This property shows that total information is the sum of information gained from the so-called grouped information function and a weighted sum of the entropies conditioned on each group.

( ,...,1 ) Hm q q_m

These properties give us the precise expression of formula (5). Choosing ( _i) ln

h p = −K p_i

i i

, we arrive at

1 2

1

( , ... ) 1

n

n n

i

H p p p K p np

=

= −

∑

where P=(p p₁, ₂,...,p_n)∈ Δ_n, andK >0 is a constant.

The expression (6) is well-known as Shannon's entropy or measure of uncertainty [13].

From the expression of entropy, it easily shows that entropy is a non-negative and concave function. From property (2), we see that entropy has the maximum value when the probability distribution is uniform. So entropy can be viewed as a measure of the disparity of probability from the uniform distribution; the lower value of entropy the larger distance to uniform distribution. This property shows that, by maximizing entropy, it will give us a result which is closest to uniform distribution subject to the given constraints.

Based on the above idea, the optimal solution for reaching the maximum entropy [2]

could be obtained by solving the following optimization problem:

1

max ln 'ln

n

i i

i

ω ω ω

=

−

∑

Subject to '

' 1

fixed

e ω μ μ

ω

⎧ =

⎨ =

⎩ (7)

where μ'=(μ μ₁, ₂...μ_n),ω'=(ω ω₁, ₂...ω_n) and I =(1,1,....1)∈Rⁿ.

Equivalently to the approach for finding the minimum of negative entropy

1

min ln 'ln

n

i i

i

ω ω ω ω

=

∑

Subject to '

' 1

fixed

I ω μ μ

ω

⎧ =

⎨ =

⎩ (8)

where μ'=(μ μ₁, ₂...μ_n),ω'=(ω ω₁, ₂...ω_n) and I =(1,1,....1)∈Rⁿ

In order to find the solution of above problem, let us use the Lagrange multipliers technique. The Lagrangian for problem (8) is

1 2 1 2

( , , ) 'ln ( ' _fixed) ( ' 1);

Lω λ λ =ω ω λ ω μ μ− − −λ ω I−

We set the first derivatives of this function w.r.t.ω to zero, then we can obtain:

1 2

1 2

lnω+ −I λ μ λ− I = ⇔ =0 ω e^{λ μ λ+ I I−} ;

From the formula we see ω is naturally non-negative.

Differentiate the function L( ,ω λ λ₁, ₂) w.r.t. λ₁ and λ₂, then we get the following first order equations:

' 0

' 1 0;

fixed

I ω μ μ ; ω

− =

− =

Substituting ω into the last two equations we get,

1 2

1 2

' 0

' 1 0;

I I

fixed I I

e I e

λ μ λ λ μ λ

μ ^{+ −} μ

+ −

− = ;

− =

thus we have two equations and two unknown parameters λ₁ and λ₂.

The solution of ω is the closest to the uniform distribution, this property may have some nice effects on diversification.

3.3 Improve Portfolio Diversification Using Maximum Entropy Theory.

In this section, the proposed portfolio optimization approach could be viewed as one alternative of Mean-Variance approach. As we mentioned before, we want to improve the concentrated position of portfolio weights in the mean-variance framework by directly shrinking the portfolio weights. We have already seen in the last section that entropy is a well accepted measure for diversification. Due the nice property of entropy that the optimal solution obtained from maximizing entropy is closest to the uniform distribution, we want to add a shrink weights factor into mean-variance optimization model, hoping that it will lead to a well diversified portfolio.

The following new approach could be viewed as a combination of model (1) and (8).

The mean-variance model is sensitive to given data. On the other hand, the approach for finding maximum entropy is independent of given data. The use of entropy could be viewed as compensation to the risk part in MV model. It can thus decrease the reliance on data. This new approach not only uses given partial information obtained from the history sample efficiently, but also applies the entropy to adjust how much the portfolio is diversified. The improved MV model based on maximum entropy theory [14] (short to MVME model) is given as follows:

minω

1 1 1

ln ' 'ln

n n n

i j ij i i

i j i

ω ω σ α ω ω ω ω αωC ω

= = =

+ = +

∑∑ ∑

Subject to '

' 1

fixed

e ω μ μ

ω

⎧ =

⎨ =

⎩ (9)

where ω'=(ω ω₁, ₂...ω_n) denotes the proportion invested in each asset, μ'=(μ μ₁, ₂...μ_n) denotes the expected rate of return vector, C=(σ_ij)_nxn denotes the covariance matrix,

and μ_fixed is the specified target rate of return.

If short selling is not allowed, we could add another constraint,ω_i ≥0. Hereα≥0 is a parameter to adjust the effect of entropy. The larger the value ofαis, the stronger the effect of entropy. Whenα=0, we will come back to MV model. Here we should note that if the chosen value ofαis too large, the whole problem will resemble a minimum negative entropy problem, thus one has to pay great care to the selection of the parameterα.

Now we analyze the existence of the solution for MVME model (9). In the mean-variance framework, the covariance matrix is assumed to be symmetric and positive definite. Although the entropy is a concave function, the negative entropy is a convex function. So the combined problem is a nonlinear convex problem with linear constraints, and it has unique solution. This solution could be found by solving the Lagrange problem:

1 2 1 2

( , , ) ' 'ln ( ' _fixed) ( ' 1).

L ω λ λ =ω ω αωC + ω λ ω μ μ− − −λ ω e−

As the same with the MV model, a good way for computing the efficient frontier is to solve the following problem,

min

ω ω ω αω'C + 'lnω λμω−

.

s t

whereλ is the trade-off parameter between return μ and risk. If short selling is not allowed, we could limit for ω_i ≥0.

Chapter 4. The Comparison Tests Between Two Models.

4.1 The choice of the sample and parameters.

The following choices will be implemented in our comparison experiments:

• In this thesis, the number of assets in all tests isN =10.

• The samples used in calculation are simulated from a multivariate normal distribution using Matlab functionmvnrnd(μ_true,C_true,T ), where μ_true is the mean vector of the rate of return, is the covariance matrix, and is the number of observations.

Ctrue T

• For tests made to study the influence of parameter α in MVME-model, we chooseα =0.01, 0.05, 0.1.

• Short selling is not allowed, it meansω_i ≥0.

4.2 Experiment for estimation errors.

4.2.1 The definition of estimation errors.

The effect of using estimated sample mean and covariance instead of the true values when computing the portfolio optimization by MV and MVME models could be tested by the estimation errors of sample mean and covariance. First, we should clarify the following definitions:

• True efficient sets

Actually, the true efficient sets are unknown for investors, because the investor can not get the exactly future rate of returns. In this thesis, we use the true values as the seed to generate the observed data set. So the true efficient sets are calculated from the true values of μ_true and C_truewhich are mentioned above.

• Estimated efficient sets

The estimated efficient sets are the efficient frontier obtained from the simulated