Improving the Markowitz Modelusing 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.

1

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

2.2 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.

3

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 , satisfying thatr 0(1 )TX r X= + ,

where 0X and TX 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 TX 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

iμ for asset as follows: i

1

1( )N

i i i it

r E r rN

μ−

=

= = = t∑ 1.......t 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.

4

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

2 2

1

1(( ) ) ( )N

i it i itt

E r rN

2iσ μ μ

=

= − = −∑ and i isd σ=

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 and i j is defined as follows,

1

1(( )( )) ( )( )N

ij ji it i jt j it i jt jt

E r r r rN

σ σ μ μ μ=

= = − − = − −∑ μ .

We note that ij jiσ σ= when i j≠ and 2ii iσ σ= when i j= . If the return of

asset and i j move in the same direction, we have 0ijσ > , inversely, 0ijσ < .

To describe the relation of possible assets, we define the covariance matrix as follows:

n

11 12 1

1 2

n

n n nn

Cσ σ σ

σ σ σ

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

L

M M M M

L

From the expression and the character of covariance, we know the covariance

matrix is symmetric and it also can be proved that matrix is positive definite. C C

5

5). Investment weights.

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

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

1

1n

ii

ω=

=∑ . The situation that a weight iω is negative corresponds to a short selling

of the asset which means that the investor buys the asset and sells it to someone else, and uses the amount received to invest in other assets. When short selling is

not allowed, we require that 0iω ≥ .

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 ii i i

E r E r i iμ ω ω= = =

= = =∑ ∑ ∑μω

2))

))

μ

,

by using the linear property of expectations.

7).Variance of the portfolio.

The variance of the portfolio for assets is given by n

2 2

1

1 1

1, 1

1 1

( ) ( (

( ( ) (

(( )( ))

'

n

p p p i i ii

n n

i i i j j ji i

n

i j i i j ji j

n n

i j iji j

E r E r

E r r

E r r

C

σ μ ω

ω μ ω μ

ωω μ μ

ωω σ

ω ω

=

= =

= =

= =

= − = −

= − −

= − −

=

=

∑

∑ ∑

∑

∑∑

where 1 2' ( . ..... )nω ω ω ω= are the weight vector and C is the covariance matrix

defined in (4).

6

2.4 Mean-Variance Model.

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

rate of return

n 1 2( , ..... )na a a

iμ and a variance iiσ . Define 1 2' ( , ..... )nμ μ μ μ= as the expected rate

of return vector, and as the covariance matrix, where C ijσ is the covariance

between asset and i j when i j≠ or the variance of asset when . i 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 1 2' ( , ..... )nω ω ω ω= denotes the proportion invested

in each asset, (1,1,....1) ne R= ∈ , 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 iji j

Cσ ωω σ ω ω= =

= =∑∑

Subject to '

' 1fixed

eω μ μω=⎧

⎨ =⎩ (1);

or

max ω 1

'n

i ii

r μω ω μ=

= =∑

Su (2). bject to2'

' 1pfixedC

eω ω σ

ω⎧ =⎨

=⎩

In equation (1), the objective function is the variance of the portfolio; the first

constraint clarifies the desired level of return fixedμ . 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

2pfixedσ

7

selling is not allowed, we add the constraints 0iω ≥ . 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( , , ) ' ( ' ) ( ' 1);fixedL C eω λ λ ω ω λ ω μ μ λ ω= − − − −

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

Theorem 2.1

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

11 2 1 2

1 ( ); 2( ) / ; 2( )2 fixed fixedC e / .ω λ μ λ λ αμ β δ λ γ βμ δ−= + = − = −

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

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 eL

L e

ω ω λ μ λ

μ ω μλ

ωλ

∇ = − − =

∂= − =

∂∂

= − =∂

;

or the matrix form as follows

1

2

2 0' 0 0 1' 0 0 fixed

C ee

μ ωλ

μ λ

⎛ ⎞⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟− = ⎜⎜ ⎟⎜ ⎟

⎜ ⎟⎜ ⎟ ⎜ ⎟−⎝ ⎠⎝ ⎠ ⎝ ⎠μ⎟ (3)

8

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 1' ( .... )nω ω ω= is obtained from the first row of the above

matrix, 11 2

1 (2

C )eω λ μ λ−= + . Substitutingω into the last two rows of matrix, we obtain

( )1 1

1 1

2

'1 22' 1 12

fixedfixed fixed

fixed

C ee

αμ βλ μ μ γ β μμ

γ βμλ β α δ

− −−

−⎛ ⎞ ⎛⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ −⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠

⎞.

2.5 Efficient frontier.

Any solution of 1' ( .... )nω ω ω= from the optimization problem (1) for a given fixedμ is

called a frontier portfolio. The set of such portfolios form a hyperbola in the σ μ−

plane. This is illustrated below:

9

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

' 'Cω ω λω μ−

Subject to 'e 1ω = (4)

where λ is the trade-off parameter between returnμ and risk. If short selling is not

allowed, we could add constraints 0iω ≥ .

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

Lagrangian for problem (4) is

1 1( , ) ' ' ( ' 1).L C eω λ ω ω λω μ λ ω= − − −

Theorem 2.2

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

11 1

1 2( );2

C e λβω λμ λ λα

− −= + = ,

where 1 1' ; ' .e C e e Cα β μ− −= =

The proof is similar with the proof of Theorem 2.1.

10

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.

11

So called “shrinkage” estimator is intended to shrink the historical means to some

grand mean. Consider 1 2( , ...., )TR r r r= 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

IrGJS

−

+−= ωμωμ )1( ,

where I is a vector of ones; ),...,(' 1 Nμμμ = is the sample estimate, and

∑=

=T

jiji r

T 1

1μ ; ∑=

−

=N

iiG N

r1

1 μ is the grand mean. The shrinking factor is

))]()'(/()2(,1min[ 1 IrCIrTN GG

−−

−

−−×−= μμω

where is the sample covariance matrix. 1−C

The second shrinkage estimator used to improve sample means is called the

Bayes-Stein estimator [9]. It is given by

IrGBSBS

−

+−= ωμωμ )1( ,

where ,)'/(' 11 ICICIrGBS−−

−

= μ /( )Tω α α= + , and the shrinking factor is

)]()'/[()2( 1 IrCIrN GBSGBS

−−

−

−−+= μμα .

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.

12

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 matrixN T× 1 2( , ...., )TR r r r= be the return matrix, we

resample the return matrix times, and at the end we will get the bootstrap returns

matrix

T

*N TR × . Apply this resampled returns matrix to the MV framework, and then the

corresponding optimal weights *ω , return *μ and standard deviation *σ for *N TR ×

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 1 2( , ...., )TR r r r= is the average solutions, which is obtained from the

bootstrap procedure. The optimal weightsω , the return μ and standard deviation

σ for N TR × are given by

* *

1 1 1

1 1 1; ;B B

b b bB B B*.

B

ω ω μ μ σ 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].

13

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 -states probability processn 1 2( , ,... )nX x x x= , with probability

vector , where1 2( , ,..., )nP p p p= 0ip ≥ ( 1...i n)= , 1

1n

ii

p=

=∑ . Let be a

function defined as the amount of information associated with the

state

( )ih p

iX x= , . For each , we define by, 1...i = n

i

n nH

1 21

( , ,..., ) ( )n

n n ii

H p p p p h p=

= ∑ ; (5)

nH can be thought as a measure of the average amount of information [12]. For

simplicity, we write

1 21

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

n n ii

P p p p p p=

Δ = = ≥ =∑ i

n

.

According to Shannon, the definition of information entropy should satisfy the

following properties [13]:

1). For any , is a continuous function ofn 1 2( , ,..., )nH p p p 1 2( , ,..., )nP p p p= n∈Δ .

14

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

means:

1 2( , ,..., )nH p p pn

1 21 1 1( , ..... ) ( , ..... )n n nH p p p Hn n n

≤ ;

where . 1 2( , ,..., )nP p p p= n∈Δ

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

1 1 2 1 2( , ..... ,0) ( , ..... )n n nH p p p H p p p+ n= .

where . 1 2( , ,..., )nP p p p= n∈Δ

4). If , and11 12 1( , ,..., ,..., )n mnP p p p p= mn1

n

i ijj

q p=

=∈Δ ∑ , then

111 12 1 1

1( , ,..., ,..., ) ( ,..., ) ( ,..., )

mi i

mn n mn m m i ni i i

np pH p p p p H q q q Hq 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( ,..., )mH q qm

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

( ) lnih p K p= − i

i i

, we arrive at

1 21

( , ..... ) 1n

n ni

H p p p K p np=

= − ∑ ; (6)

where , and is a constant. 1 2( , ,..., )nP p p p= n∈Δ 0K >

15

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:

1max ln 'ln

n

i iiω ω ω

=

− = −∑ ω

Subject to'

' 1fixed

eω μ μω=⎧

⎨ =⎩ (7)

where 1 2' ( , ..... )nμ μ μ μ= , 1 2' ( , ..... )nω ω ω ω= and (1,1,....1) nI R= ∈ .

Equivalently to the approach for finding the minimum of negative entropy

1min ln 'ln

n

i iiω ω ω ω

=

=∑

Su bject to'

' 1fixed

Iω μ μω=⎧

⎨ =⎩ (8)

where 1 2' ( , ..... )nμ μ μ μ= , 1 2' ( , ..... )nω ω ω ω= and (1,1,....1) nI R= ∈

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 ( ' ) ( ' 1);fixedL Iω λ λ ω ω λ ω μ μ λ ω= − − − −

16

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

1 21 2ln 0 ;I II I eλ μ λω λ μ λ ω + −+ − − = ⇔ =

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

Differentiate the function 1 2( , , )L ω λ λ w.r.t. 1λ and 2λ , 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 Ifixed

I I

e

I e

λ μ λ

λ μ λ

μ μ+ −

+ −

;− =

− =

thus we have two equations and two unknown parameters 1λ and 2λ .

The solution of ω is the closest to the uniform distribution, this property may have

some nice effects on diversification.

17

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 ' ' lnn n n

i j ij i ii j i

Cωω σ α ω ω ω ω αω ω= = =

+ = +∑∑ ∑

Subject to '

' 1fixed

eω μ μω=⎧

⎨ =⎩ (9)

where 1 2' ( , ..... )nω ω ω ω= denotes the proportion invested in each asset,

1 2' ( , ..... )nμ μ μ μ= denotes the expected rate of return vector,

( )ij nxnC σ= denotes the covariance matrix,

and fixedμ is the specified target rate of return.

18

If short selling is not allowed, we could add another constraint, 0iω ≥ . Here 0α ≥ is a

parameter to adjust the effect of entropy. The larger the value ofα is, the stronger the

effect of entropy. When 0α = , we will come back to MV model. Here we should note

that if the chosen value ofα is 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 ( ' ) ( ' 1).fixedL C eω λ λ ω ω αω ω λ ω μ μ λ ω= + − − − −

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

solve the following problem,

minω

' ' lnCω ω αω ω λμω+ −

.s t 'e 1ω = (11)

whereλ is the trade-off parameter between return μ and risk. If short selling is not

allowed, we could limit for 0iω ≥ .

19

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 is 10N = .

• The samples used in calculation are simulated from a multivariate normal

distribution using Matlab function ( , ,true truemvnrnd C T )μ , where trueμ is the mean

vector of the rate of return, is the covariance matrix, and is the number

of observations.

trueC T

• For tests made to study the influence of parameter α in MVME-model, we

choose 0.01,0.05,0.1α = .

• Short selling is not allowed, it means 0iω ≥ .

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 trueμ and which are mentioned above. trueC

• Estimated efficient sets

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

sample. We also call them sample efficient sets.

20

• Actual efficient sets

Actual efficient sets are defined as follows: by calculating the estimated efficient

sets, each point on the estimated frontier corresponds to a set of portfolio weights.

These estimated weights are applied to the true parameters, i.e. the true expected

rate of return, and true covariance matrix. Actually, the actual efficient sets are the

investors really hold using the estimated weights invest. In short, the estimated

frontier is the analysis result based on the simulated data and estimated

parameter; but actual frontier is the real one based on the true data and

parameters.

One example of the effect of using estimated sample mean and covariance instead of

the true inputs when computing the portfolio optimization is graphed in Figure 2. The

estimated efficient frontier is located on the left of both the true and actual efficient

frontiers. It implies that the estimated frontier overestimates the expected return and

underestimates the risk of portfolios. From the figure, we see how large the difference

is between estimated frontier and actual frontier. Intuitively, the smaller distance

between those frontiers, the better performance of the portfolio model will be shown.

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.41.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

True Estimated

Actual

Mean

Standard Deviation

Figure 2. MVME=Model 0.1α = ,frontiers using 30 observations.

21

Investors obtain the estimated efficient frontier, but the performance of the portfolio is

given by the actual efficient frontier. We focus on the estimation errors between the

actual and true frontier. We can measure the estimation errors by root-mean- square

errors [9] (RMSE) to describe the difference between actual and true efficient frontier,

which is given by:

2

1

2

1

1( ) ( )

1( ) ( )

actual

actual

pi

truei

pi

truei

fp

fp

μ

σ

λ μ μ

λ σ σ

=

=

= −

= −

∑

∑ (12)

where trueμ and trueσ denote the target point on the true efficient frontier at the

specified value λ . Parameter and denote the actual mean rate of return

and the standard deviation which is calculated from the sets of simulated data at

specified value

actual

iμactual

iσ

thi

λ . Here is the repeating calculation time, chosen as 60 in our

experiments.

P

4.2.2 The comparison for estimation errors in different models.

The estimation errors between actual and true efficient portfolios of both MV model

and MVME model are presented in Table 1. The RMSE for mean return and standard

deviation on portfolios increases with λ . However the increasing speed tends to

decline as the increase ofλ . It implies that the difference between the actual and true

optimal solution obtained from those two optimization models will be larger for higher

return portfolio, but the difference does not increase without bound, it will tend to be

converge to some value.

22

λ

MV-model

fμ fσ

MVME-Model 0.01α =

fμ fσ

MVME-Model 0.05α =

fμ fσ

MVME-Model 0.1α =

fμ fσ

0 0.0715, 0.0463 0.0707, 0.0444 0.0678, 0.0387 0.0651, 0.0349 3 0.1851, 0.2082 0.1837, 0.2075 0.1785, 0.2054 0.1729, 0.2021 7 0.2617, 0.2612 0.2612, 0.2611 0.2577, 0.2593 0.2529, 0.2574 13 0.3020, 0.3161 0.3018, 0.3160 0.3010, 0.3153 0.2997, 0.3144 17 0.3194, 0.3527 0.3190, 0.3516 0.3177, 0.3478 0.3162, 0.3438

Table 1. RMSE in different models for N=10,T=30.

In the Figure 3 and Figure 4, they are clearly shown that the RMSE of both mean

return and standard deviation in MV model are higher than obtained from MVME

model, and as parameterα increases, the RMSE in MVME model tend to decrease.

The largerα implies the stronger effect of entropy on finding the optimal solution. This

result implies that the use of entropy makes a good effect on reducing the estimation

errors. We could argue that the use of entropy decrease the reliance on given data.

The effect to reduce the RMSE caused by entropy is stronger for smallerλ . This result

may be caused by the underlying constraints. Smaller value ofλ implies the smaller

target expected return. The smaller target expected return set in constraint offers a

more flexible space for the choice of optimal weight; moreover, this flexible space

makes entropy affect more on the optimal solution. Especially, for the global minimal

risk point when 0λ = , the only constraint in the mean-variance and the MVME

framework is . This offers the most flexible choices of portfolio weights.

Contrarily, for the larger value of

11

N

iiω

=

=∑

λ , the higher target expected return leads the optimal

solution to concentrate on fewer assets; it reduces the influence of entropy on the

optimization process.

23

RMSE of return

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 3 7 13 17

Lamda

RMSE

MV model

MVME-0.01

MVME-0.05

MVME-0.1

Figure 3. RMSE of rate of return in different models for N=10,T=30.

RMSE of Standard Dev

00.050.10.150.20.250.30.350.4

0 3 7 13 17

Lamda

RMSE

MV model

MVME-0.01

MVME-0.05

MVME-0.1

Figure 4. RMSE of rate of risk measure in different models for N=10,T=30.

4.2.3 Estimation errors versus Sample size.

In this section, we discuss the sensitivity of estimation errors relative to the sample

size. We generate random variables with normal distribution, by varying the sample

size from 30 to 550, and simulate 60 returns time series for each sample size, then

apply MV and MVME approach for 0.1α = to compute estimation errors. The

experiment results are shown in Figure 5 and Figure 6. In the left side, the sensitivity

of estimation errors of mean return to sample size changed under different target

expected return is shown; in the right side, the corresponding estimation errors of

standard deviation is described.

24

From the following figures, we can see the impact of larger sample size on the

reduction of the estimation error. Generally speaking, the higher sample size provides

a significant improvement of estimation errors for both the mean return and the

standard deviation. As we see from the figures, the estimation errors for both the

mean return and the standard deviation in MV and MVME-0.1 model tend to reach a

floor after 120 data points in the case of 0λ = ; the higher sample size after 120 does

not provide significant improvement. Compared with the case when 17λ = , the

estimation errors of mean return need at least 300 data points to reach the consistent

results. In order to obtain the more accurate results in this case, more data points

should be employed since the estimation errors for the standard deviation tend to

reach a floor after 450 data points. This result clearly shows that the lower the target

expected return portfolio, the less observations they need to reach the consistent

results. If you want a higher return portfolio, the consistent results should only be

obtained by employing more data. The result accords with the real life, for higher

return portfolio, the more risk you may have to face. Intuitively, the higher portfolio risk,

to avoid the estimation risk, the more observations are requested to describe the

sample distribution as precisely as possible for getting a good performance.

Comparing the RMSE between mean return and standard deviation, the RMSE of

mean return descends like a steep stair into a plane; but the RMSE of standard

deviation shows an obvious variation when the number of observations is smaller than

a given value, and especially the value ofλ is large.

25

RMSE VS Sample size

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

30 120 200 300 450 550

Sample size

RMS

E

Lamda=0

Lamda=3

Lamda=7

Lamda=13

Lamda=17

RMSE VS Sample size

00.050.10.150.20.250.30.350.4

30 120 200 300 450 550

Sample sizeRMS

E

Lamda=0

Lamda=3

Lamda=7

Lamda=13

Lamda=17

Figure 5. RMSE in MV model for N=10.

RMSE VS Sample size

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

30 120 200 300 450 550

Sample size

RMSE

Lamda=0

Lamda=3

Lamda=7

Lamda=13

Lamda=17

RMSE VS Sample size

00.050.10.150.20.250.30.350.4

30 120 200 300 450 550

Sample size

RMSE

Lamda=0

Lamda=3

Lamda=7

Lamda=13

Lamda=17

Figure 6. RMSE in MVME model for 0.1α = and N=10.

26

4.3 Experiment on the portfolio performance

4.3.1 Efficient Frontier

In order to analyze the relationship between risk and reward, we compare their

efficient frontiers depicted in the σ μ− plane.

The risk-reward graph is shown in Figure 7. The MVME efficient portfolios for 0.1α =

dominate mean-variance optimal portfolios even though it is not significant. For lower

return portfolios, the dominance is clearer. The slight dominance of MVME efficient

portfolios comes from the using of entropy. We argue that the well diversification may

be the reason for improving the efficient portfolios in the mean-variance coordinate.

0.8 1 1.2 1.4 1.6 1.8 21.6

1.8

2

2.2

2.4

2.6

2.8

3

+ MV model

o MVME-0.1 model

Mean

Standard dev

Figure 7. Efficient frontier in Mean-Standard dev

27

4.3.2 Sharp Ratio versus Sample size

A relative measure of portfolio performance is the Sharp ratio developed by William F.

Sharpe [15]. This performance measure is used by investors to compare the

performance of two portfolios in terms of risk-adjusted return. The Sharpe ratio is

calculated by subtracting the risk free rate from total portfolio return divided by the

standard deviation of the portfolio. The Sharpe ratio is shown below,

( )pp

p

fE R RS

σ−

= , (13)

where ( )pE R is the expected return of the portfolio, fR is the risk free rate of return,

and pσ is the standard deviation of the portfolio. The larger value of Sharp ratio, the

better performance of portfolio.

In this section, we use the Sharp ratio to compare the performance between the

mean-variance and the MVME framework under different sample sizes. We apply four

portfolio models, MV, MVME for 0.01α = ,0.05 and 0.1 to the sample with size from

12, 30, 60, 120, 240 to 300, then calculate their Sharp ratios to compare the

performance of the portfolios. The model with largest Sharp ratio has the best

performance in this test.

In this experiment, the total number of the model shows best performance for those

four models is 60 under the same sample size. However, different portfolio models

may provide the same optimal solutions. But if the Sharpe ratio shows the best

performance in these portfolios approach coincidentally, we think both of them have

the best performance. So the total number of best performance for these four models

should be equal to or bigger than 60 under the same sample size.

28

The results of this trial are shown in Figure 8 and Figure 9. In Figure 8, the times of the

best performance for those four models during 60 portfolios based on the portfolio

target is global minimize risk is shown respectively; in Figure 9, the corresponding

results based on the portfolio target is higher expected return is depicted.

From Figure 8, the performance of MVME framework when parameter 0.1α = is

generally better than the other three models under different sample size. Only when

sample size is 12, MV model has a better performance than MVME-0.1. We could

argue that the MV model has the dominance for very small sample size compared to

the MVME model. When the sample size increases, the MVME-0.1 model dominates

the MV model. From Figure 9, the performance of the MVME-0.1 framework has a

significant dominance to the MV model no matter the sample size is small or large.

We could argue that the MVME model has the dominance for the higher expected

portfolio return problem.

Sharp ratio VS Sample size

0

10

20

30

40

50

60

70

12 30 60 120 240 300

Sample size

Times

MV-model

MVME=0.01

MVME-0.05

MVME-0.1

Figure 8. Sharp Ratio test under different sample size

Sharp Ratio VS Sample size

0

10

20

30

40

50

60

70

12 30 60 120 240 300

Sample size

Times

MV-model

MVME-0.01

MVME-0.05

MVME-0.1

Figure 9. Sharp Ratio test under different sample size

29

4.3.3 Portfolio returns

• Actual returns versus Sample size

In this section, we use the actual return defined in 3.2.1 to compare the performance

between mean-variance and the MV ME framework. We apply four portfolio models,

MV, MVME for 0.01α = ,0.05 and 0.1, to the sample with size from 30 to 500, then

calculate their actual returns. We consider the model with the highest actual return as

the best performance in this test. In this experiment, we run the test for 200 times

under each sample size. The total number of the model shows the best performance

portfolios for those four models should be equal to or bigger than 200 under the same

sample size since different portfolio models may have the same optimal solutions.

Actual return VS Sample size

020406080100120140160

30 60 120 240 300 400 450 500

Sample size

Times

MV-model

MVME-0.01

MVME-0.05

MVME-0.1

Figure 10. Actual return test under different sample size

The results of this trial are shown in Figure 10. It clearly shows that for small sample

size, the performance of the MVME-0.1 framework has a significant dominance to the

MV model. But for large sample size, the results obtained from the MV model are

improved a lot. As the number of observations increases, the MVME-0.1 framework

tends to lose its significant dominance; we should not be surprised about this result.

When the number of observation rises, the sample distribution could be described

more precisely. The reduction of estimation errors for parameters improves the

performance of mean-variance framework.

30

• Final return

A simulated backtracking of the performance during five years is presented for the MV

and the MVME-0.1 model. The result is included in Figure 11. The figure represents

the most typical backtracking result during the tests. In this test, the number of history

observations for estimating the parameters is set by 60. The portfolio is the global

minimal risk portfolio. In Figure 11, the MVME-0.1 model performs slightly better than

the mean-variance model.

Figure 11. Performance for 10 assets during 60 months.

31

4.4 Stability of Portfolio Weights

As we mentioned before, the mean-variance framework is extremely sensitive to

parameters change, small changes of the sample mean and the covariance matrix will

take a large effect on the optimal portfolios weights. In this section, we will compare

the stability of portfolio weights between mean-variance and MVME-0.1 framework.

The experiment is described as follows: firstly we generate 50 sets of sample

return matrix

N T×

1 2( , ...., )TR r r r= from the multivariate normal distribution

( , ,true trueN C )Tμ generator, where is the time series of historical returns for

different assets, and is the number of different assets; secondly we estimate the

sample mean and the covariance matrix for each sample return matrix, ending up with

getting 50 sets of estimated sample mean and covariance matrix; at last, obtain the

optimal portfolio weights by applying the MV and the MVME model to the estimated

parameters.

T

N

Figure 12 and Figure 13 present how portfolio weights vary for different estimated

parameters. In Figure 12, the portfolio weights vary for different estimated parameters

in mean-variance framework is shown; the corresponding result obtained from the

MVME-0.1 framework is depicted in Figure 13. We see that both the mean-variance

and the MVME framework are sensitive to parameters change. Optimal portfolios

weights jump actively for each asset. We cannot see any difference from those two

figures, but intuitively, the MVME framework should affect the stability of portfolio

weights since maximizing entropy directly shrink the portfolio weights to the uniform

distribution despite of the sample size. For further analysis, we calculate the mean

and standard deviation of those portfolio weights for each asset to see if the stability of

portfolio weights is improved. The results are shown in Table 2. In the results, most of

standard deviations obtained from the MVME model are smaller than the one

obtained from the MV model; it implies that the stability of portfolio weights is

improved by using the MVME model. We believe that the stronger the effect of entropy,

the much more the improvement on stability of portfolio weights.

32

Stability of weights in MV model

00.050.1

0.150.2

0.250.3

0.350.4

1 2 3 4 5 6 7 8 9 10

Asset from 1 to 10

weight

Figure 12. Stability of Portfolio Weights for MV model.

Stability of weights in MVME-0.1 model

00.050.10.150.20.250.30.350.4

1 2 3 4 5 6 7 8 9 10

Asset from 1 to 10

weight

Figure 13. Stability of Portfolio Weights for MVME-0.1 model.

Asset MV-model

Mean Standard Dev MVME-Model 0.1α = Mean Standard Dev

1 0.0004, 0.0024 0.0092, 0.0077

2 0.1893, 0.0402 0.1689, 0.0334

3 0.1450, 0.0445 0.1265, 0.0329

4 0.1231, 0.0350 0.1272, 0.0289

5 0.0374, 0.0247 0.0488, 0.0200

6 0.2759, 0.0316 0.2581, 0.0295

7 0.0490, 0.0204 0.0521, 0.0173

8 0.0576, 0.0265 0.0676, 0.0219

9 0.0520, 0.0249 0.0692, 0.0182

10 0.0704, 0.0260 0.0725, 0.0208

Table 2. Mean and standard deviation of weights for MV and MVME-0.1 model.

33

4.5 Degree of diversification

In order to compare the degree of diversification between the mean-variance and the

MVME framework, we should quantify the measure of diversification first. It is very

common to use the deviation of a portfolio from the equal portfolio as the measure of

diversification [16]. The mathematical expression is given by

2 21

1 1

1( ) (N N

i m ii i m

DN

ω ω ω= =

= − = −∑ ∑ ) ; (14)

where iω is the weight of assets in the portfolio, 1

mmN

ω = is the weight that the

same asset would have in the equal portfolio. A lower value of indicates a higher

level of diversification.

1D

In this thesis, we consider using entropy to measure the portfolio diversification. The

notion of using this measure is similar to the former diversification measure .

Entropy is a measure of disparity from the uniform distribution, the lower value of

entropy the further deviation from the uniform. The measure of diversification is

defined below,

1D

2D

21

log 'log( );N

i ii

D ω ω ω ω=

= − = −∑ (15)

where iω is the weight of assets in the portfolio. Contrary to , a higher value of

indicates a higher level of diversification.

1D 2D

34

The results of 60 tests are graphed in Figure 14. In the figure, the values of entropy

obtained from three MVME frameworks are higher than the values calculated from the

mean-variance framework. As parameter α increases, the corresponding value of

entropy in MVME framework increases. This result implies that entropy has a good

effect on the improvement of diversification.

The more intuitive test results are shown in Figure 15 and Figure 16. In Figure 15, the

number of the zero-weight during 60 tests for each asset is shown. In the figure, for

asset 1, the number of the zero-weight is reduced by over 50% from the MV model to

the MVME-0.1 model, and asset 5 always be chosen when applying the MVME model

while the number of the zero-weight for this asset got from MV model is almost 0. It is

clearly shown from the figure that the number of the zero-weight obtained from all

three MVME models is less than the number got from the MV model. As parameter

α increases, the number of zero-weight in the MVME framework decreases. This

result accords with the conclusion obtained from the entropy analysis.

The test depicted in Figure 15 only shows the decrease of the number of the

zero-weight whether the portfolio weights deviated from zero is big or small. So we do

a further test to count how many zero-weight obtained from the MV model increases

over 1% by applying the MVME model. The results are shown by percent in Figure 16.

The proportion of portfolio weights increase over 1% from zero for each asset as

parameter α increases is shown in the figure. Especially for asset 5 and 8, both of

them get a very high proportion. The increase of the portfolio weights by applying the

MVME framework is significant to influence the investor’s decision. Hence, we think

the MVME model improves the portfolios diversification distinctly.

35

Entropy during 60 tests

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57

MVME-0.1

MVME-0.05

MVMV-0.01

MV model

Mean Entropy:

MVME-0.1: 1.7476

MVME-0.05: 1.7219

MVME-0.01: 1.6955

MV model: 1.5617

Figure 14. Degree of diversification measured by entropy.

Improvement of diversification

0

10

20

30

40

50

1 2 3 4 5 6 7 8 9 10

Asset from 1 to 10

Zero times during 60

tests

MV model

MVME-0.01

MVME-0.05

MVME-0.1

Figure 15. Zero weights for all models of each asset during 60 tests.

0

0.2

0.4

0.6

0.8

1

1.2

1 2 3 4 5 6 7 8 9 10

Asset from 1 to 10

Proportion

MVME-0.01

MVME-0.05

MVME-0.1

Figure 16. Proportion of portfolio weights increase over 1% in zero weights

for all MVME models of each asset during 60 tests.

36

5 Conclusion.

The present study has analyzed the effect of entropy on the portfolio selection

problem in the MVME framework. We conclude that the improved mean-variance

model using the Maximum Entropy has the following theoretic advantages:

(i). Comparing with the conventional method, we directly shrink the portfolio weights to

be uniformly distributed. It is a simplified procedure compared with the procedure

using the shrinkage estimator for sample mean and covariance matrix.

(ii) Isolating information from the sample mean and covariance matrix is crucial when

the number of observations is not large enough. Using entropy can decrease the

reliance on data. This new approach not only efficiently uses the information from the

observed data, but also applies the information obtained by maximizing entropy to

achieve a more diversified portfolio.

The test results show that compared with MV framework, the estimation errors

between the actual and the true efficient frontiers, as well as the stability of portfolio

weights are slightly improved by applying the MVME framework. In the test of portfolio

performance, the tests for efficient frontier and final return showed a slight dominance

of the MVME framework. The test for the Sharp ratio showed that the dominance of

the MVME framework is not influenced by the sample size. And the actual return test

showed that as sample size increases, the MVME framework tends to lose its

significant dominance. The test for the degree of diversification clearly shows that the

MVME framework improves the concentrated weights problem caused in the MV

framework significantly. In short, the MVME framework keeps the advantages of

mean-variance framework, improves the concentrated weights problem, and leads to

a well diversified portfolio.

37

Acknowledgements:

I would like to thank my supervisor Prof. Johan Tysk for his support and contribution

and my friend Lingwen Kong. Without them it would not have been possible to finish

this thesis on time.

38

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40