School of Education, Culture and CommunicationDivision of Applied Mathematics

BACHELOR THESIS IN MATHEMATICS / APPLIED MATHEMATICS

Markowitz vs Black–Litterman: A Comparison of Two PortfolioOptimisation Models

by

Kristel Eismann

Kandidatarbete i matematik / tillämpad matematik

DIVISION OF APPLIED MATHEMATICSMÄLARDALEN UNIVERSITY

SE-721 23 VÄSTERÅS, SWEDEN

School of Education, Culture and CommunicationDivision of Applied Mathematics

Bachelor thesis in mathematics / applied mathematics

Date:2018-05-31

Project name:Markowitz vs Black-Litterman: A Comparison of Two Portfolio Optimization Models

Author(s):Kristel Eismann

Version:1st June 2018

Supervisor(s):Lars PetterssonAnatoliy Malyarenko

Reviewer:Olha Bodnar

Examiner:Kimmo Eriksson

Comprising:15 ECTS credits

Abstract

Modern portfolio theory first gained its ground among researchers and academics, but hasbecome increasingly popular among practitioners. This paper examines the two popular port-folio optimization models, Markowitz mean-variance model and Black-Litterman formula andcompares their results on real data. In second chapter mean-variance model is derived step-by-step using Lagrange multipliers and matrices, whereas in third chapter Black-Littermanformula is proved by two different methods - by Maximum Likelihood method and Theil’smodel.

Two portfolio optimization models are used on real data, monthly data from November2007 to November 2017. In order to build the two models, Microsoft Excel is used. Swedish30-day Treasury Bill is taken as risk-free asset and SIXPRX as a benchmark. Detailed resultsare presented in Chapter 4. In Black-Litterman model two different views are implemented tosee if the model outperforms Markowitz mean-variance model. All in all there is a signific-ant difference in the outcomes, Black-Litterman portfolio performs better than mean-varianceportfolio.

Acknowledgement

I want thank my supervisors Senior Lecturer Lars Pettersson and Professor Anatoliy Mal-yarenko for their continuous support throughout the writing process. Their knowledge infinance and mathematics has been invaluable.

I would like to thank reviewer Olha Bodnar for her comments to improve this paper evenfurther. Also I want to acknowledge Mälardalen University and the program Analytical Fin-ance for having interesting and valuable courses that are one of a kind.

Last but not least, I want to express my gratitude to my family for unconditional supportduring my studies. I would have not achieved all this without them.

Kristel EismannMay 2018, Västerås

1

Contents

1 Introduction 51.1 Aim and Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Markowitz Mean-Variance Model 72.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Expected Return of Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Variance, Covariance, and Correlation . . . . . . . . . . . . . . . . . . . . . 82.5 Efficient Frontier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.6 Global Minimum Variance Portfolio . . . . . . . . . . . . . . . . . . . . . . 112.7 Orthogonal Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.8 Tangency Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.9 Capital Market Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.10 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Black-Litterman Model 153.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3 Proof of Black–Litterman Model . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3.1 Proof by Maximum Likelihood Method . . . . . . . . . . . . . . . . 173.3.2 Proof by Theil’s model . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.4 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Empirical Findings and Discussion 224.1 Markowitz Mean-Variance Portfolio . . . . . . . . . . . . . . . . . . . . . . 224.2 Black-Litterman Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.3 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5 Conclusion 295.1 Further Research Proposal . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.2 Objectives of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2

List of Figures

4.1 Efficient Frontier and Underlying Assets . . . . . . . . . . . . . . . . . . . . 224.2 Portfolio Weights of GMV and Optimal Portfolio . . . . . . . . . . . . . . . 234.3 Mean-Variance Portfolio Optimization . . . . . . . . . . . . . . . . . . . . . 244.4 Portfolio Weights of Black-Litterman Optimization . . . . . . . . . . . . . . 264.5 Different Portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3

List of Tables

4.1 Mean-Variance Portfolios (on Monthly Data) . . . . . . . . . . . . . . . . 234.2 Black-Litterman Portfolio (on Monthly Data) . . . . . . . . . . . . . . . . 274.3 Different Portfolios (on Monthly Data) . . . . . . . . . . . . . . . . . . . . 27

4

Chapter 1

Introduction

The ground for modern portfolio theory was laid when Harry Markowitz published hiswork Portfolio Selection in 1952. Markowitz’s paper tried to answer the important question:How to allocate wealth between different investment choices? [7] The main part of Markow-itz’s idea was to build and use mathematical models to diversify portfolios. He used probab-ility theory to quantify assets risk and return and proposed that they should viewed togetheras a risk-return trade-off. The idea is that portfolio’s risk depends on how correlated are theassets with one another in a portfolio. Furthermore, it will become a mean-variance optimiz-ation problem where investor should choose the portfolio that have the desired return, but thelowest level of risk. This area of finance has since then influenced modern thinking in bothinvestments and in the academics world.

Years it has had a big impact on research in financial industry, but also it has influencednew way of thinking about investments. As of 2013 there have been almost 20 000 articles inGoogle Scholar that refer to Markowitz’s original paper [7]. Yet it was not implemented byinvestment managers for a long time. The major problem is many pitfalls with the model. Forinstance, very sensitive nature of changes in inputs to output. Therefore, it was considered asopaque and unstable.

Since Markowitz model is considered as unintuitive, in early 1990s Fischer Black andRobert Litterman developed a new model which aimed to deal with mean-variance model’s pit-falls and shortcomings. Mean-variance model’s starting portfolio is the null portfolio, whereasBlack–Litterman model’s (B-L) initial point is the equilibrium portfolio. An investor then as-signs views, i.e. his/her opinion that one or many assets outperform the others. These differentviewpoints are are combined in calculation. Sometimes it is thought that B-L model is a totallynew model, yet it differs only in expected returns [10]. The first part of thesis gives an over-view and mathematical background of both models. The second part covers empirical findingswhere we compare the results from the two models. And finally, we draw a conclusion andsuggest further possible research topics.

5

1.1 Aim and PurposeThe aim of this paper is to describe the background of models for portfolio optimization anddiversification with important mathematical derivations and proofs. To make it more clearand perceivable, we analyze real data and compare the results from B-L and MV models.Furthermore, we want to know how much better performance has the B-L model compared toMarkowitz model since B-L model addresses the pitfalls of the latter model and therefore, isassumed to outperform mean-variance optimization approach.

1.2 MethodologyThe choice of possible range of equities that might be in portfolio is not easy. For the con-struction of Markowitz mean-variance and Black–Litterman model we use 29 equities thatform OMX Stockholm 30 Index. The reason for 29 equities and not all 30 is the lack of datafor Essity B stock (it has data available only from June 2017). The monthly data is taken over10 year period — from 30th of November 2007 to 30th of November 2017. One drawbackwith this period is that it also covers one year of the very turbulent times during the globalfinancial crisis. This means that the volatility could be higher than usual during that period oftime. The data is downloaded from Thomson Reuters EIKON platform. The 30-day SwedishTreasury Bill over 10 years is used as risk-free interest rate in the models. The reasoning be-hind it is that Treasury Bills are considered to be the closest to risk-free asset because financialworld does not have completely safe instrument. The benchmark is SIXPRX.

The data analyzing and portfolio constructions are done in Microsoft Excel.

1.3 Outline of the ThesisChapter 2 This chapter discusses the Markowitz mean-variance model with its assumptions

and limitations. Furthermore, the formulas for minimum variance, orthogonal, and tan-gency portfolios are derived.

Chapter 3 Gives the two different proofs of Black-Litterman master formula. Including shortdiscussion about the assumptions and limitations.

Chapter 4 In that chapter are showed and discussed the results of the model construction onreal data. The comparison and analysis is conducted.

Chapter 5 This chapter concludes and gives some further research topics.

Chapter 6 The last chapter presents the objectives of thesis and their fulfillment.

6

Chapter 2

Markowitz Mean-Variance Model

Harry Markowitz’s paper from 1952 established some key concepts that laid the ground toModern Portfolio Theory. He mathematically formulated the idea of diversifying the risk ofportfolio and discussed the components that possibly have an impact on return and volatility.The conceptual framework was based on key concept of investors risk appetite. In otherwords, an investor targets at all times at maximal return on his/her investment while wantingto minimize the risk. Although his work did not draw much attention at the time, in the 60’sacademics in finance started to pay attention. Nowadays his approach is used to constructa portfolio as well as measure its performance by many investors and portfolio managers.During the 65 years Markowitz model has been revised numerous ways [12]. This chapter iscovering first the assumptions of the model, followed by important parts of the model, efficientfrontier, global minimum variance and tangency portfolios and their mathematical derivations.We conclude the chapter with a brief discussion of the model’s limitations.

2.1 AssumptionsMarkowitz model is based on several assumptions which have been questioned time and timeagain [9]. First, investors are rational — it means that they like to maximize return whileminimizing risk. More risk is only accepted if there is a compensation by higher returns.Investors have limitless access to capital with risk-free rate. Furthermore, markets are held tobe efficient and there are no transaction costs or taxes.

2.2 SymbolsThe following notation is used in defining and deriving models:

w n-column vector with components w1,w2, . . . ,wn that denote weights of the assets in theportfolio where i = 1,2, . . . ,n.

R n-column vector of expected returns R1,R2, . . . ,Rn where i = 1,2, . . . ,n.

V n × n variance-covariance matrix with components σi j where i, j = 1,2, . . . ,n.

7

1 n-column vector of ones.

r the vector of expected excess returns.

> denotes the transpose of vector or matrix.

Rp portfolio’s return.

σ2p portfolio’s variance.

r f risk-free asset.

2.3 Expected Return of PortfolioMarkowitz model uses the expected return as a measure of central tendency [5]. The individualassets’ returns are observed as the historical performances. The expected return of a portfoliois the weighted average of the expected returns on the individual assets [3]:

Rp =N

∑i=1

wiRi = w>R. (2.1)

2.4 Variance, Covariance, and CorrelationAn investor prefers an asset with lower variance if two assets have the same expected return.The variance of portfolio (measure of dispersion) is the expected value of the squared devi-ations of the return on the portfolio from the mean return on the portfolio:

σ2p = E[(rp−Rp)

2] = E

(N

∑i=1

(wi(ri−Ri)

)2

=N

∑i=1

N

∑j=1

wiw jσi j = w>V w. (2.2)

Diversification cannot eliminate all risk, since there is unsystematic risk that can be re-duced significantly and systematic risk that is seen as market risk and therefore, cannot be getrid of.

Covariance expresses how assets move together [5]. This is an important aspect. It is usefulto standardize the covariance by dividing the covariance term with the standard deviations ofassets:

ρi j =σi j

σiσ j(2.3)

This is called correlation and is between -1 and +1. Negative correlation coefficient meansthat assets’ returns move in opposite directions and although perfectly negative correlationdoes not happen in real life due to the presence of systematic risk, it would be possible toconstruct a portfolio with zero risk. Positive correlation means that assets’ returns move insame direction and zero correlation means that they are independent given that they are jointlynormally distributed.

8

2.5 Efficient FrontierI derive the formulas by using matrices and Lagrange multipliers [3]. Efficient frontier is acurve or in some cases a line that represents all the portfolios of highest level of return withgiven level of risk. The points inside of the curve are inefficient, since an investor can havemore return with the same level of risk or vice versa. We would like to minimize risk given thetwo constraints - asset weights are summing up to one and second, portfolio earns expectedrate of return at a given level, i.e. the problem formulation for attainable portfolios is:

minimize w>V w

subject to w>1 = 1

Rp = w>R

To solve this, we set up the Lagrange function, L, with multipliers λ1 and λ2:

L = w>V w−λ1(w>R−Rp)−λ2(w>1−1). (2.4)

Next, we take the partial derivatives with respect to w, λ1, and λ2. The First Order Conditionsbecome:

δLδw

= 2V w−λ1R−λ21 = 0 ⇒ 2V w = λ1R+λ21, (2.5)

where 0 is the n-vector of zeros. From (2.5) we get the weights, w:

w =12

V−1(λ1R+λ21). (2.6)

The other two Lagrange equations become:

δLδλ1

= Rp−w>R = 0 ⇒ Rp = w>R,

δLδλ2

= 1−wT 1 = 0 ⇒ 1 = w>1.(2.7)

Writing the (λ1R+λ21) as matrix form in (2.6):

w =12

V−1(λ1R+λ21) =12

V−1 [R 1][λ1

λ2

]. (2.8)

Next, we want to solve for[

λ1λ2

]using the other two F. O. C rewritten as:

[R 1

]>w =[R 1

]> 12

V−1 [R 1][λ1

λ2

]=

[Rp1

]. (2.9)

Multiplying both side of Equation (2.8) by[R 1

]>, we get:[R 1

]>w =12[R 1

]>V−1 [R 1][λ1

λ2

]=

[Rp1

](2.10)

9

For convenience, let’s introduce a new symmetric matrix A that is called also informationmatrix:

A =[R 1

]>V−1 [R 1], (2.11)

where the entries are [a bb c

]=

[R>V−1R R>V−111>V−1R 1>V−11

]. (2.12)

Given that, we need to show that the matrix A is positive definite. If we have any y1,y2 whereat least one is nonzero, we see that

[R 1

][y1y2

]= [y1R+ y21]. (2.13)

This is a nonzero vector with n elements and by assumption the variables in R are not allequal. Given that, the A is positive definite, since[

y1 y2]

A[

y1y2

]=[y1 y2

][R1]>V−1 [R1

][y1y2

]= [y1R+ y21]>V−1[y1R+ y21]> > 0

(2.14)

by the positive definiteness of V−1. Next, we substitute the A in (2.11) and yield to the result:

12

A[

λ1λ2

]=

[Rp1

]. (2.15)

Since A is non-singular and there is an inverse, we can solve for multipliers:

12

[λ1λ2

]= A−1

[Rp1

]. (2.16)

From Equations (2.16) and (2.8) we obtain the n-vector of portfolio weights that minimizesthe portfolio variance for a given return:

w =12

V−1 [R 1][λ1

λ2

]=V−1 [R 1

]A−1

[Rp1

](2.17)

Given mean Rp, definitions of variance, derived previous solutions and matrix A, we canexpress the variance of minimum variance portfolio:

σp2 = w>V w =

[Rp 1

]A−1 [R 1

]>V−1VV−1 [R 1]

A−1[

Rp1

]=[Rp 1

]A−1

[Rp1

]=[Rp 1

] 1ac−b2

[c −b−b a

][Rp1

]=

a−2bRp + cR2p

(ac−b2).

(2.18)

10

2.6 Global Minimum Variance PortfolioIn the previous section, we discussed what the efficient frontier is and how to derive it. In thissection, we go through the derivation of Global Minimum Variance Portfolio (GMV), i.e. theportfolio that has the smallest amount of risk. This is the absolute minimum point on efficientfrontier. We denote the mean of GMV as RG and get it when we minimize equation (2.18)with respect to Rp which yields to:

RG =bc. (2.19)

By inserting this into the variance formula (2.18) we get the variance of GMV:

σ2p =

a−2bRG + cR2G

(ac−b2)=

a−2b(bc )+ c(b

c )2

(ac−b2)=

1c. (2.20)

For GMV weights, we insert RG into Equation (2.17) and yield:

wG =V−1 [R 1]

A−1[

RG1

]=

V−1 [R 1][ c −b−b a

][b/c1

](ac−b2)

=V−11

c. (2.21)

Note that the parameters for a,b and c are in information matrix A in Equation (2.12).

2.7 Orthogonal PortfolioBefore we can move on to tangency portfolio, we have to establish a concept of orthogonalportfolio. If the first portfolio has the mean Rp, then the orthogonal portfolio has mean Rhwith [3]:

Rh =a−bRp

b− cRp(2.22)

To set up Equation (2.22), we denote two arbitrary minimum variance portfolios, p and h, withweights wp given by Equation (2.17) and wh by:

wh =V−1 [R 1]

A−1[

Rh1

]. (2.23)

Two portfolios are orthogonal when their covariance is zero, this implies that

0 = w>h V wp =[Rh 1

]A−1

[Rp1

]. (2.24)

Rewriting (2.24) we get (2.22).

11

2.8 Tangency Portfolio"Risk-averse" investor is defined as an investor who prefers safer investments for riskier in-vestments with higher returns [5]. Most investors are not so risk-averse that they would chooseGMV, but would pick the tangency portfolio. Here we introduce the riskless asset r f , i.e. asecurity that has a certain future return. Although even this asset carries very small amount ofrisk, we assume the degree of risk for r f is so small that σ f = 0.

Let Ri where i = 1, 2, 3, . . . , n present the expected returns in gross terms. The expectedreturn in net terms is denoted by r and includes elements ri = Ri− r f where i = 1, 2, 3, . . . ,n. An investor is interested in growing assets in relation to net of future outflows. He/she isconcerned that his/her net worth may change, so here the term "net" is defined in terms ofexisting liabilities [5].

So the weight vector for risky assets is same as before, w and we denote the weight ofriskless asset as w f = 1−w>1. [3] The portfolio p mean excess return that an investor isinterested in is given by

Rp = w>R+(1−w>1)r f − r f = w>r (2.25)

and variance byσ

2p = w>V w (2.26)

We have at hand another optimisation problem

minimize w>V w = σ2p

subject to w>r = Rp.

Solving this problem as in Section 2.5 by using Lagrange multipliers:

L = w>V w−λ (Rp−w>r). (2.27)

By taking the partial derivatives with respect to w and λ , the First Order Conditions are:

δLδw

= 2V w−λr = 0 ⇒ 2V w = λr ⇒ w =λ

2V−1r, (2.28a)

δLδλ

= Rp−w>r = 0 ⇒ Rp = w>r (2.28b)

Next, we multiply both sides by r> in Equation (2.28a),

r>w = r>λ

2V−1r. (2.29)

Given the Equation (2.28b), we get

λ

2=

Rp

r>V−1r. (2.30)

12

Next we substitute Equation (2.30) into first order condition’s Equation (2.28)a and we get theweight-vector wT P of tangency portfolio,

wT P =Rp

r>V−1rV−1r. (2.31)

For obtaining the expected return of tangency portfolio, we multiply both sides of (2.31) by1> and use that the tangency portfolio consists only of risky assets, i.e. 1>wT P = 1

RT P =r>V−1r1>V−1r

. (2.32)

And finally, we have the variance of portfolio σ2T P where we plug in the obtained values,

σ2T P = w>V w

=

(Rp

r>V−1r

)2

r>V−1VV−1r

=R2

p

r>V−1r.

(2.33)

Next we find the Sharpe Ratio. It is given by formula SR =µp−r f

σpwhere µp is expected return

of a portfolio that contains only risky assets. We use the result in (2.33) to derive Sharpe RatioSR,

SR2T P =

(RT P

σT P

)2

=R2

pR2

pr>V−1r

= r>V−1r (2.34)

We take a square root of Equation (2.34) and have the Sharpe Ratio as SR =√

r>V−1r.

2.9 Capital Market LineThe Capital Market Line (CML) is a straight line that begins at the initial point that is equal torisk-free rate of return and touches one point on efficient frontier [5]. While tangency portfolioconsists only of risky assets, the portfolios on CML include the borrowing and lending of risk-free asset (except tangency portfolio). We have initial inputs to CML as r f = r0 and σ f = σ0.The line equation is mathematically presented as y = m+ kx where r lies on y-axis and σ onx-axis. Let k be the slope of CML where it is Sharpe Ratio by tangency portfolio. We havek = RT P−r0

σT P−σ0=

RT P−r fσT P−σ f

where σ f = 0, so the CML formula becomes:

CML = r f +RT P− r f

σT Pσ (2.35)

13

2.10 LimitationsIn theory mean-variance model is reasonable, but problems arise in practice. Different re-searchers over the years have discussed the limitations of Markowitz mean-variance model.

First, the estimates of risk and return are prone to estimation error. For instance, it putsmore weight on securities that have large estimated returns, small dispersion and negativecorrelation [11].

It is also argued that the substitution of the expected return with sample average does notwork well. Moreover, small changes in inputs (especially expected return) can lead to largechanges in output which makes the mean-variance model very unstable. Markowitz modeldoes not make a distinction if there is a possible uncertainty in estimated inputs of model [10].

14

Chapter 3

Black-Litterman Model

In section 2.10 we discussed the limitations of Markowitz mean-variance model. FischerBlack and Robert Litterman [1] tried to solve two problems of mean-variance model, namely,the difficulty of estimating expected returns and their extreme sensitivity of return assump-tions. Black and Litterman’s key is to combine mean-variance optimization and the capitalasset pricing model (CAPM) by Sharpe and Lintner. Their neutral starting point is marketequilibrium where investor views and the level of confidence are added, i.e views are combinedusing Bayesian mixed estimation techniques. This creates the B-L expected returns which areoptimized mean-variance way and finally, an optimal portfolio is constructed [10, 6].

3.1 AssumptionsThere are several assumptions for every model. The list of assumptions has many levels, forinstance, typical for quantitative models are normally distributed returns, no arbitrage, capitalmarkets are efficient etc.For portfolio models, only risk and return are used for making decisions, no transaction costsor taxes or risk-averse investor. Specifically, for B-L model first assumption is that investorshave views which create better portfolios, but they are not hundred percent certain about theviews. So, for every belief we have a level of confidence. Inefficiency in markets is assumed[10].

3.2 SymbolsThe following notation is used in Black–Litterman master model:

P the matrix of asset weights within each view, size k×N

q the vector of expected returns with investor views

µµµ the posterior expected return vector, size N×1

15

ΣΣΣ posterior covariance matrix containing variances and covariances of the model’s assets,size N×N

ΠΠΠ the column vector of expected returns for market estimates

τ the measure of uncertainty coefficient

q the vector of the returns including views, size k×1

Ω the covariance matrix with diagonal elements ω2j , size k× k

ΣΣΣ the known covariance matrix, size N×N

k the number of views in view matrix

3.3 Proof of Black–Litterman ModelIn this section, we prove the Black–Litterman model using first the Maximum Likelihoodmethod and second the proof by Theil’s model. First, let’s make an introduction to overallnotation and remarks about the variables that are used in the model and in proofs [10], [14].

There are d assets and m market observations r1, . . . , rm ∈Rd of normally distributed withmean value µµµ ∈ Rd and covariance matrix Σ.

Let P be the matrix of weights and q∈Rk the vector of expected returns of the k portfolioswhich contains the investor views where k ≤ d. The returns are calculated by the formula

q = PrI, (3.1)

where rI ∈ Rd is the vector of expected returns estimated by the investor.Furthermore, Ω is the k× k diagonal matrix with diagonal elements ω2

j = Var[q j]. ByEquation (3.1),

q j = Pr j, m+1≤ j ≤ m+n,

where n is the number for investor observations qm+1, . . . , qm+n.In the case where we only have market observations, the expected return for market estim-

ate is

rM :=1m

m

∑i=1

ri.

The random vector rM has normal distribution with mean µµµ and covariance matrix 1mΣ. The

standard notation:

ΠΠΠ = rM, q = qI =1n

m+n

∑j=m+1

q j,

where q is the investor’s estimate of the expected return with normal distribution N(Pµµµ, 1nΩ).

Theorem 1. Combining estimates from both investors and market, we yield to Black–Littermanmaster formula:

µµµ = [(τΣ)−1 +P>Ω−1P]−1[(τΣ)−1

ΠΠΠ+P>Ω−1q],

Σ = [(τΣ)−1 +P>Ω−1P]−1.

(3.2)

16

3.3.1 Proof by Maximum Likelihood MethodProof. [10] Let the probability density of the random vector ri be

f (ri,ννν) =1

(2π)d/2√

detΣexp(−1

2(ri−ννν)>Σ

−1(ri−ννν)

),

The probability density function of the random vector q j is

g(q j,ννν) =1

(2π)d/2√

detΩexp(−1

2(q j−Pννν)>Ω

−1(q j−Pννν)

).

By definition, the likelihood function is the product of both densities:

L(ννν) =m

∏i=1

f (ri,ννν)m+n

∏j=m+1

g(q j,ννν).

The maximum likelihood estimate, µµµ , is the value of ννν where the function L(ννν) attains itsmaximal value. It is difficult to take a derivative from this likelihood function, therefore, wefirst take the logarithm and get the logarithmic likelihood function

`(ννν) = lnL(ννν)

We obtain

`(ννν) =m

∑i=1

ln f (ri,ννν)+m+n

∑j=m+1

lng(q j,ννν)

= m ln1

(2π)d/2√

detΣ+n ln

1(2π)d/2

√detΩ

− 12

m

∑i=1

(ri−ννν)>Σ−1(ri−ννν)− 1

2

n+m

∑j=n+1

(q j−Pννν)>Ω−1(q j−Pννν).

Next we evaluate the gradient of `(ννν). For that, we let ek be the column vector with d com-ponents, where the kth component is equal to 1, and the remaining components are equal to 0.We get

(∇`(ννν))k =∂`(ννν)

∂νk

=−12

m

∑i=1

(−e>k Σ−1(ri−ννν)+(ri−ννν)>Σ

−1(−ek))

− 12

m+n

∑j=m+1

(−e>k P>Ω−1(q j−Pννν)+(q j−Pννν)>Ω

−1P(−ek))

= e>k

[Σ−1

(m

1m

m

∑i=1

ri−mννν

)+P>Ω

−1

(n

1n

m+n

∑j=m+1

q j−nPννν

)]= e>k

[mΣ−1(ΠΠΠ−ννν)+nP>Ω

−1(q−Pννν)],

17

By definition, the points where ∇`(ννν) = 0 are critical points of `(ννν). Then the equation is

mΣ−1(ΠΠΠ− µµµ)+nP>Ω

−1(q−Pµµµ) = 0.

which can be written as

τ−1

Σ−1(ΠΠΠ− µµµ)+P>Ω

−1(q−Pµµµ) = 0,

As we can see, we have introduced the variable τ by

τ =nm.

After moving the terms with µµµ to left and others to right, we get

[(τΣ)−1 +P>Ω−1P]µµµ = (τΣ)−1

ΠΠΠ+P>Ω−1q.

Next we multiply both hand sides by [(τΣ)−1 +P>Ω−1P]−1. We get the equation in (3.2). Aslast we prove that the critical point µµµ is a maximum. We calculate the Hessian of the function`(ννν), i.e. the matrix of its second partial derivatives. We yield

(∇2`(ννν))kl =∂ (∇`(ννν))k

∂νl

=∂

∂νle>k[mΣ−1(ΠΠΠ−ννν)+nP>Ω

−1(q−Pννν)]

= e>k[mΣ−1(−el)+nP>Ω

−1(−Pel)]

=−mΣ−1kl −n(P>Ω

−1P)kl.

Since both matrices Σ−1 and P>Ω−1P are positive-definite. We can see that the Hessian isnegative-definite, which means that the critical point µµµ is really a maximum. For calculatingΣ,

Σ := E[(µµµ−µµµ)(µµµ>−µµµ>)]

= E[µµµµµµ>]−E[µµµ]µµµ>−µµµE[µµµ>]+µµµµµµ

>.

We must evaluate E[µµµ] and E[µµµµµµ>],

E[µµµ] = [(τΣ)−1 +P>Ω−1P]−1[(τΣ)−1E[ΠΠΠ]+P>Ω

−1E[q]]

= [(τΣ)−1 +P>Ω−1P]−1[(τΣ)−1 +P>Ω

−1P]µµµ= µµµ,

This means that the estimates of the expected returns from both market and investor are un-biased.

Σ = E[µµµµµµ>]−µµµµµµ

>

For simplification we denote,A = (τΣ)−1 +P>Ω

−1P.

18

Since we haveµµµ> = [ΠΠΠ>(τΣ)−1 +q>Ω

−1P]A−1

andE[µµµµµµ

>] = A−1E[[(τΣ)−1ΠΠΠ+P>Ω

−1q][ΠΠΠ>(τΣ)−1 +q>Ω−1P]]A−1,

then expected value is

E[[(τΣ)−1ΠΠΠ+P>Ω

−1q][ΠΠΠ>(τΣ)−1 +q>Ω−1P]] = (τΣ)−1E[ΠΠΠΠΠΠ

>](τΣ)−1

+(τΣ)−1E[ΠΠΠq>]Ω−1P+P>Ω−1E[qΠΠΠ

>](τΣ)−1 +P>Ω−1E[qq>]Ω−1P.

We haveE[ΠΠΠΠΠΠ

>] = Σ+µµµµµµ>, E[ΠΠΠq>] = µµµµµµ

>P>,

E[qΠΠΠ>] = Pµµµµµµ

>, E[qq>] = Ω+Pµµµµµµ>P>.

Since the random vectors ΠΠΠ and q are independent,

E[[(τΣ)−1ΠΠΠ+P>Ω

−1q][ΠΠΠ>(τΣ)−1 +q>Ω−1P]] = (τΣ)−1(Σ+µµµµµµ

>)(τΣ)−1

+(τΣ)−1µµµµµµ>P>Ω

−1P+P>Ω−1Pµµµµµµ

>(τΣ)−1 +P>Ω−1(Ω+Pµµµµµµ

>P>)Ω−1P

= A+Aµµµµµµ>A.

It follows thatE[µµµµµµ

>] = A−1(A+Aµµµµµµ>A)A−1 = A−1 +µµµµµµ

>.

andΣ = A−1 +µµµµµµ

>−µµµµµµ> = A−1.

3.3.2 Proof by Theil’s modelProof. [14] We have

ΠΠΠ = Xµµµ +u, q = Pµµµ +v,

where X is the identity matrix, u is the normally distributed residual with mean 0 and covari-ance matrix Φ = τΣ, v is the normally distributed residual with mean 0 and covariance matrixΩ. The residuals are assumed to be independent. Notating it in matrix form,[

ΠΠΠ

q

]=

[XP

]µµµ +

[uv

], (3.3)

The covariance matrix of the random vector [u>v>]> is[Φ 00 Ω

].

The least square estimate of µµµ is

µµµ =

[[X P

] [Φ 00 Ω

]−1 [X>

P>

]]−1 [X> P>

][Φ 00 Ω

]−1[ΠΠΠ

q

].

19

In ordinary notation, this estimate gives

µµµ = [XΦ−1X>+PΩ

−1P>]−1[X>Φ−1

ΠΠΠ+P>Ω−1q]. (3.4)

Since X is the identity matrix and Φ = τΣ, we get the same equation as the first one in (3.2).Next, we prove the second equation in (3.2), we substitute (3.3) to (3.4). This gives

µµµ = [XΦ−1X>+PΩ

−1P>]−1[X>Φ−1(Xµµµ +u)+P>Ω

−1(Pµµµ +v)]

= [(τΣ)−1 +PΩ−1P>]−1[(τΣ)−1(µµµ +u)+P>Ω

−1(Pµµµ +v)]

= [(τΣ)−1 +PΩ−1P>]−1[(τΣ)−1

µµµ +(τΣ)−1u+P>Ω−1Pµµµ +P>Ω

−1v]

= [(τΣ)−1 +PΩ−1P>]−1[(τΣ)−1

µµµ +P>Ω−1Pµµµ]

+ [(τΣ)−1 +PΩ−1P>]−1[(τΣ)−1u+P>Ω

−1v]

= µµµ +[(τΣ)−1 +PΩ−1P>]−1[(τΣ)−1u+P>Ω

−1v].

It follows thatµµµ−µµµ = [(τΣ)−1 +PΩ

−1P>]−1[(τΣ)−1u+P>Ω−1v].

The covariance matrix of the left hand side is

Σ = E[(µµµ−µµµ)(µµµ−µµµ)>]

= [(τΣ)−1 +PΩ−1P>]−1E[[(τΣ)−1u+P>Ω

−1v][(τΣ)−1u+P>Ω−1v]>]

× [(τΣ)−1 +PΩ−1P>]−1

and the expected value inside the right hand side is

E[[(τΣ)−1u+P>Ω−1v][(τΣ)−1u+P>Ω

−1v]>] = (τΣ)−1E[uu>](τΣ)−1

+(τΣ)−1E[uv>]Ω−1P+P>Ω−1E[vu>]+P>Ω

−1E[vv>]Ω−1P.

The expected values of residuals are

E[uu>] = τΣ, E[uv>] = E[vu>] = 0, E[vv>] = Ω.

ThenE[[(τΣ)−1u+P>Ω

−1v][(τΣ)−1u+P>Ω−1v]>] = (τΣ)−1

τΣ(τΣ)−1

+P>Ω−1

ΩΩ−1P = (τΣ)−1 +P>Ω

−1P.

Σ = [(τΣ)−1 +PΩ−1P>]−1[(τΣ)−1 +PΩ

−1P>][(τΣ)−1 +PΩ−1P>]−1

= [(τΣ)−1 +PΩ−1P>]−1.

20

3.4 LimitationsBlack–Litterman model has obtained increasing popularity since its first publication. Blackand Litterman saw two strengths in their model - investor views can be easily added to theportfolio optimization process and B-L model does give more reasonable portfolios comparedto standard mean-variance optimization [2].

Over these years some researchers and managers have brought out some possible misusesand problems of the model. For instance, if we consider the active portfolio managementwhere the main objective is to maximize active alpha for the same level of active risk, then thedifference from the mean-variance portfolio efficiency (B-L is derived under this) may resultin unintentional trades which may cause losses [4].

As for any model, all the assumptions that the models have make it more sensitive, e.g.the assumption of independence of views. Moreover, since investor includes his/her views, itis not necessarily the best possible portfolio, but the best portfolio given the views.

21

Chapter 4

Empirical Findings and Discussion

In this chapter we present the main findings of the constructed portfolios and discuss theobtained results.

4.1 Markowitz Mean-Variance PortfolioThe mean-variance portfolio was constructed in Microsoft Excel using the matrix derivationsfrom Chapter 2. All calculations are done on monthly data and the results are presented alsoon monthly frequency. Figure 4.1 illustrates the efficient frontier and underlying assets. Wecan see that one asset (FING.ST) lies very much apart from the others.

Figure 4.1: Efficient Frontier and Underlying Assets

22

This is because of the very volatile nature of this stock. It had very small stock value (andreturn) in the beginning of our sample period, but the company experienced very large stockprice change starting from 2015. Therefore, much larger expected return, but also very highvolatility.

Global minimum variance, tangency, and portfolio with equal weights were calculatedusing the derivations from Chapter 2. The Table 4.1 presents the results of expected return andstandard deviation (risk) computed on monthly data over 10-year period.

Table 4.1: Mean-Variance Portfolios (on Monthly Data)

Portfolio Expected Return RiskGMV 0.3144% 3.0372%Equally weighted 0.8665% 5.2577%Tangency Portfolio 5.1581% 10.4141%

As we can observe, the GMV portfolio has very low expected return with low risk. Inequally weighted portfolio we do not have the short selling, i.e. all assets get the weight 1/Nwhich is in our analysis is 1/29. This portfolio does not have a good performance either. Thetangency portfolio gives us 5.1581% of return with 10.4141% of risk.

Figure 4.2: Portfolio Weights of GMV and Optimal Portfolio

23

When calculating the optimal portfolio, we take into account the risk-free asset which inour case is a 1-month Swedish Treasury bill, r f , with average rate of 0.8232%. The portfolioweights of GMV and equally weighted do not have the short selling constraint, but whencalculating the tangency portfolio, we used the Lintner short-selling definition 1.

The Figure 4.2 shows the clearly one flaw of mean-variance optimization, it recommendsto heavily short-sell some stocks and buy the others. One of the extreme short-selling sug-gestions is for TELIA.ST, followed by NDA.ST. ATCO.ST and SHBa.ST. get a strong buyrecommendation.

The Figure 4.3 presents all the assets individually on the return-standard deviation spacewith efficient frontier, capital market line (CML), and portfolios mentioned before.

We can see a quite steep efficient frontier, that means that the investor does not have totake on much risk to increase the expected return. On the other hand, if the efficient frontierwould be flatter, then an investor should take a lot of risk in order to have more return [8].

Figure 4.3: Mean-Variance Portfolio Optimization

1Short sales are a use of investor’s funds, but investor gets the riskless rate of the funds that are used in shortsales [5]

24

4.2 Black-Litterman PortfolioAs mentioned before, the main difference between Markowitz mean-variance and Black-Litterman portfolio construction is that the investor can express his/her views into the op-timized portfolio. We are going to apply two different views and calculate expected return andrisk for both portfolios.

Before we do that, we discuss some important parts in B-L portfolio model that we haveto use in computation part. The Black-Litterman formula includes the parameter τ that canbe confusing. There has been several discussions how to calibrate it. Charlotta Mankert inher paper [10] proposes that τ should be calibrated by using Maximum Likelihood Estimator(MLE). Then τ = 1

T is the biased MLE estimator and τ = 1T−N is the unbiased MLE estimator,

where T is sample size and N is the number of assets included in portfolio analysis (in thispaper it would be 29). In this work we use the τ = 1 as Satchell and Scowcroft suggestedin their paper [13]. We also need risk aversion coefficient. This is connected to tangencyportfolio and is calculated by δ = (RT P− r f )/σ2

T P and is δ = 3.997.The first views that are implied are rather strong and are as follows 2:

• ATCOa.ST will outperform TELIA.ST by 5%.

• SCABb.ST will outperform ABB.ST by 2%.

• SHBa.ST will outperform NDA.ST by 3%.

The second set of views are as follows:

• ALIVsdb.ST will outperform AZN.ST by 1%.

• FINGb.ST will outperform ERICb.ST by 1.5%.

• VOLVb.ST will outperform KINVb.ST by 2.5%.

In other words, let’s present these views in matrix form. The matrix P contains views givenabove,

P =

P1,1 . . . P1,N... . . . ...

Pk,1 . . . Pk,N

where k is number of different views, i.e. k = 3 and N is number of assets in portfolio con-struction, i.e. N = 29. The vector q is the vector of expected returns with investor views,

q =

q1...

qk

2Views are selected randomly.

25

In other words, vector q presents the percentages from the given views. The last part weneed for portfolio construction with the B-L model (3.2) in Chapter 3.3 is Ω.

Ω = P(τΣ)P> (4.1)

Since we have τ = 1 the (4.1) becomes,

Ω = PΣP> (4.2)

The Figure 4.4 presents the weights of Black-Litterman portfolios with different views.We can observe that the B-L portfolio 1 with first set of views has very large volume in thoseassets we chose into our view matrix. This is expected since we chose τ = 1 that means that wedid not apply uncertainty coefficient. Moreover, the randomly selected percentages are ratherlarger for being sure of one asset’s outperforming nature of other asset. When observing theweights of B-L portfolio 2, we can see that the larger percentage we have applied, the moreit differentiates from previous weights. The model is sensitive to inputs. When investor doesnot have views, then the equilibrium portfolio is selected.

Using these weights, we calculate the Black-Litterman portfolio’s risk and return that arepresented in Table 4.2. The stronger the investor views are, the more the portfolio’s outcomeis affected. The difference in expected returns is very large.

Figure 4.4: Portfolio Weights of Black-Litterman Optimization

26

Table 4.2: Black-Litterman Portfolio (on Monthly Data)

B-L Portfolio 1 B-L Portfolio 2Expected Return 9.0328% 1.6335%Risk 11.0652% 5.5374%

4.3 ComparisonTable 4.3 presents all the performances of different portfolios. We can observe very di-

verse results. Global minimum variance portfolio has very low return, only 0.3144%. SharpeRatio expresses the risk-return relationship. GMV portfolio has the lowest ratio from all theportfolios. Tangency portfolio has the highest Sharpe ratio out of Markowitz mean-varianceportfolio optimization. B-L portfolio 1 outperforms all other portfolios, but as mentioned be-fore the views incorporated into the modelling are rather too strong, especially consideringthat the uncertainty coefficient τ was equal to 1. B-L portfolio 2 is performing worse thantangency portfolio. One of the reasons could be that we used views on stocks that historicallywere not performing that well.

This once again highlights the sensitivity of the model. In Markowitz mean-variance modelthe small changes in historical returns can give very different portfolio composition and there-fore, changing the expected rate of return and risk.

Table 4.3: Different Portfolios (on Monthly Data)

Portfolio Expected Return Risk Sharpe RatioGMV 0.3144% 3.0372% 0.1035Equally Weighted 0.8665% 5.2577% 0.1648Tangency Portfolio 5.1581% 10.4141% 0.4953Black-Litterman 1 9.0328% 11.0652% 0.8163Black-Litterman 2 1.6335% 5.5374% 0.2950

27

On Figure 4.5 we can see that the Black-Litterman portfolio 1 is outside of the efficientfrontier. This means that we have reached the point which was unattainable before. By ac-cepting a bit more of risk, B-L portfolio 1 has much higher expected rate of return. However,the weights consists of heavy short-selling or buying recommendations. The Black-Littermanportfolio 2 is inside of efficient frontier. Although the portfolio performs better than under-lying assets or equally weighted portfolio, it is not efficient. Although B-L portfolio 1 canbe the most desirable out of others, an investor should be cautious because it is easy to beoverconfident.

Figure 4.5: Different Portfolios

28

Chapter 5

Conclusion

In this paper we derived and proved two popular portfolio optimization models and dis-cussed their application. The main difference in these two models is that an investor canexpress his/her views and these are taken into account.

In Chapter 2 we discussed thoroughly Markowitz mean-variance model. We derived globalminimum variance, orthogonal, and tangency portfolio using the Lagrange multipliers andmatrices. This laid the base and helped to apply the model in Microsoft Excel. The empiricalresults clearly demonstrated that while global minimum variance portfolio has very smallamount of risk, then tangency portfolio generates significantly higher expected return with alittle more risk.

In Chapter 3 we proved Black-Litterman master formula by two different methods, Max-imum Likelihood and Theil’s model. In the following chapter we applied real data on themodel and showcased that the portfolio outcomes can be very different depending on the in-puts. Global minimum variance portfolio had expectedly the least amount of risk and alsoless expected return compared to other portfolios. Two different portfolios created by Black-Litterman model had contrasting results. When we applied very strong views with uncertaintycoefficient τ = 1, the expected return was much higher than the one of tangency portfoliowhereas the risk was only slightly larger. The second B-L portfolio performed worse thantangency portfolio.

All in all the we can conclude that although both models are very useful tools for makingdecisions about portfolio construction, they are very sensitive to changes in inputs (historicalreturns or views) and can therefore have very different outcomes.

5.1 Further Research ProposalSince the Markowitz publish his paper in 1952, there have been several papers addressing

the flaws and limitations of mean-variance portfolio optimization model. There have beenimprovements on this model and investors could use the updated Markowitz 2.0 model. Itwould be interesting to make a comparison between BL and the latter model.

Moreover, it would be interesting to do a research on the different aspects of view vectorin B-L model. The investor’s behaviour affects a lot the views he/she may have. Therefore,

29

the behavioural side of finance is an interesting topic to combine with Black-Litterman model.

5.2 Objectives of the ThesisObjective 1 For Bachelor degree, student should demonstrate knowledge and understanding

in the major field of study, including knowledge of the field’s scientific basis, knowledgeof applicable methods in the field, specialization in some part of the field and orientationin current research questions.

Author has demonstrated knowledge and understanding in mathematics by applying andpresenting detailed mathematical methods and proofs. For instance, proof by Theil’smethod and Maximum Likelihood. Moreover, the step by step demonstration of Markow-itz mean-variance portfolio optimization is done using the matrices. Two different op-timization models are applied to real data using Microsoft Excel. Furthermore, thelimitations of Markowitz and Black-Litterman models are discussed.

Objective 2 For Bachelor degree, the student should demonstrate the ability to search, col-lect, evaluate and critically interpret relevant information in a problem formulation andto critically discuss phenomena, problem formulations and situations.

Author has read numerous academic articles on related topic as well as books and lecturenotes from different courses. This had helped to come up with study idea for this work.All the significant part from these sources are interpreted and referred in this thesis.Moreover, real data is used to highlight the theoretical parts.

Objective 3 For Bachelor degree, the student should demonstrate the ability to independentlyidentify, formulate and solve problems and to perform tasks within specified time frames.

Author has gotten the amount of help offered by the course outline. Otherwise thestudent has demonstrated the ability to identify, formulate, and solve problems withinspecified time frames.

Objective 4 For Bachelor degree, the student should demonstrate the ability to present or-ally and in writing and discuss information, problems and solutions in dialogue withdifferent groups.

This objective will be demonstrated on oral presentation that is planned on 1st of June2018.

Objective 5 For Bachelor degree, student should demonstrate ability in the major field ofstudy make judgments with respect to scientific, societal and ethical aspects.

Author has tried to demonstrate ability in applied mathematics to make judgments withrespect to different scientific, societal and ethical aspects.

30

Bibliography

[1] F. Black and R. Litterman. Global Portfolio Optimisation. Financial Analysts Journal,48(5):28–43, 1992.

[2] L.B. Chincarini and D. Kim. Uses and Misuses of the Black–Litterman Model in Port-folio Construction. Journal of Mathematical Finance, 3:153–164, 2012.

[3] G.M. Constantinides and A.G. Malliaris. Chapter 1. Portfolio theory. In Finance,volume 9 of Handbooks in Operations Research and Management Science, pages 1–30.Elsevier, 1995.

[4] Alexandre S. Da Silva, Wai Lee, and Bobby Pornrojnangkool. The Black–Littermanmodel for active portfolio management. The Journal of Portfolio Management, 35(2):61–70, 2009.

[5] E. J. Elton, M. J Gruber, S. J. Brown, and W. N. Goetzmann. Modern portfolio theoryand Investment Analysis. International Student Version. John Wiley and Sons, 2011.

[6] R. Jones, T. Lim, and P. J. Zangari. The Black-Litterman Model for Structured EquityPortfolios. Journal of Portfolio Management, pages 24–33, Winter 2011.

[7] P. N. Kolm, R. Tütüncü, and F. J. Fabozzi. 60 Years of Portfolio Optimization: Prac-tical Challenges and Current Trends. European Journal of Operational Research,234(2):356–371, 2014.

[8] Matt Maher and Harry White. Empirical Problems Using the Efficient Frontier to FindOptimal Weights in Asset Classes. Journal of Accounting and Finance, 11(4):47–62,2011.

[9] M. E. Mangram. A Simplified Perspective of the Markowitz Portfolio Theory. GlobalJournal of Business Research, 1(1), 2013.

[10] C. Mankert and M. J. Seiler. Mathematical Derivations and Practical Implications for theUse of the Black–Litterman Model. The Journal of Real Estate Portfolio Management,17(2):139–159, 2011.

[11] R. O Michaud. The Markowitz Optimization Enigma: Is Optimized Optimal? FinancialAnalysts Journal, 45(1):31–42, 1989.

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[12] M. Rubinstein. Portfolio Selection. Journal of Finance, 57:1041–1045, 2002.

[13] Stephen Satchell and Alan Scowcroft. A Demystification of the Black-Litterman Model:Managing Quantitative and Traditional Portfolio Construction. Journal of Asset Man-agement, 1(2):138–150, 2000.

[14] Jay Walters. The Black–Litterman model in detail. SSRN Electronic Journal, 2011.

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