the markowitz mean variance optimization model finance essay

8/20/2019 The Markowitz Mean Variance Optimization Model Finance Essay

1/34

The Markowitz Mean Variance Optimization Model Finance Essay

For assignment help please contact

at [email protected] or [email protected]

Abstract

This project examined the optimal allocation o stocks !ased on the Markowitz Mean

Variance Optimization Model. "t is mainly !ased on the comparison !etween two

samples o stock returns. The irst sample is reerred to as the ull data set# and

contains the returns o the stocks $E# %axter# &ow# 'aterpillar# (pple# and )rocter

and $am!le calculated o*er the period o i*e years. The second sample takes the +,-

most recent daily returns as the !ase data set. The o!jecti*e is to ind the optimal

allocation in a portolio o risky assets only and the 'omplete )ortolio that an

in*estor could choose in order to maximize his utility. %eore analyzing the results

rom the optimization process# we started !y analyzing the data# and *eriying that

the assumption o normality the model is *alid in !oth data sets. Then we descri!e

the results gi*en !y the dierent optimizations that we calculated and we compared

!oth samples showing the optimal allocation or risky assets only and the optimal

allocation that includes the risk ree asset. Then# we graph the eicient rontier along

with the capital allocation line# and we make some comments a!out the resultso!tained. Finally# we ormulate the recommendation to the in*estor !ased on the

results o!tained# and we point out some o the limitations o the model that could

explain some o the unrealistic results that we ound. The methodology o the

calculations# the ta!les and the charts used are all reerenced in the appendix o this

paper in order to illustrate the results o!tained.

Introduction

This project is !ased on the Markowitz Mean Variance Optimization Model or

deining the optimal weights o assets in a gi*en portolio !ased on *arious

in*estment constraints. The model generally seeks to maximize return or a gi*en

le*el o risk# or minimize risk or a gi*en le*el o return. Markowitz ormulated the

portolio construction pro!lem as a utility maximization pro!lem and used this to

de*elop a ramework or selecting a range o optimal portolios. "n his model#

Markowitz made se*eral assumptions on which this project is !ased. These


2/34

assumptions designed the portolio selection pro!lem to a mean*ariance portolio

optimization pro!lem and are as ollows/

(ll in*estors ha*e a single holding period during which they will maximize their

utility unction.

"n*estors don0t incur any transaction costs or taxes while trading the securities.

"n*estors ha*e a 1uadratic utility unction that they should maximize !ased on the

expected return# *ariance# and risk a*ersion.

The returns used should !e normally distri!uted.

"n*estors are assumed to !e risk a*erse where they preer to maximize the returnsgi*en a minimum le*el o risk

(ll the in*estors share the same economic *iew o the world# and they analyze the

securities in the same way.

I/ Analyzing Stock Returns and StandardDeviations

(ccording to Ta!le 2 and + in (ppendix +# the results show that the expected returns

in the ull data set o!tained were generally lower than the expected returns in the

sample o the +,- most recent returns. 3)lease reer to (ppendix 2 or the

methodology o calculations.4 The major explanation is that the ull data set co*ers

the +--56-7 inancial crisis in which the 8.9 stock market has dropped down !y

more than ,-:# which had an implication on the mean expected returns calculated

o*er the period rom +--; until +-22# and shows relati*ely lower returns. "n

comparison# the +,- most recent returns sample co*ers a period o one year starting

rom the +nd o March# +-2- until the # and is still acing a

market rally that has !een consistent until now# which aects the perormance o

indi*idual stocks at the exception o %axter# and )rocter ? $am!le# which

underperormed due to some undamental issues that are related to their !usiness

sectors.

egarding standard de*iation# we can see that the stocks in the ull data set present a

higher standard de*iation than in the second data set. This shows that during theperiod rom +--; to +-22# there has !een a high *olatility in the market# which is


3/34

illustrated !y the stocks demonstrating some extreme returns. This is mainly due the

shape o the reco*ery ater the inancial crisis# which displayed some signiicant

drops in the share prices and reco*ering !ack at a rapid rate o growth# ater March

+-->. Aowe*er# the *olatility has sta!ilized during the last year which means that

stocks are !ecoming less risky in comparison with the last three years.

II/ Analyzing the Data Sets

1) Testing for normality

%eore going through the process o selecting the optimal portolios# we irst examine

the returns computed and see i they match the assumptions o normality that ha*e

!een stated earlier or the model to !e *alid. "n order to do so# the =ar1ue%era testor normality is chosen# and pro*ides a ormal method in which 9kewness and

Burtosis are used to analyze the distri!ution o the returns. 3)lease reer to the

appendix or the description o the =ar1ue and %era test or normality4

Analysis of the results

The data in ta!les < and C rom (ppendix + show dierent results or the two sample

sizes. "n the Full data set# the =% test statistics computed are all higher than the >.+2

critical *alue with >>: conidence le*el# which means that the distri!ution o thereturns is not normal as we reject the null hypothesis. This is mainly explained !y the

act that the kurtosis o!ser*ed in all the stocks0 returns is positi*e and is higher than


4/34

Volatility clustering is mainly characterized !y the historical data o stock returns

showing periods o high *olatilities gi*ing some extreme returns# which are then

ollowed !y periods o relati*ely low *olatilities.

Analysis of the results (ccording to the *olatility charts in (ppendix +# we notice that all the stocks in the

complete data set showed e*idence o *olatility clustering with a certain similarity in

the clusters o!ser*ed. "n act# most o the *olatility clusters o!ser*ed can !e located

!etween the periods o late +--5 to the mid +-->. This is explained !y the high

*olatility o returns during the period o the inancial crisis# where most stocks

showed some extreme *ariations in the returns due to the in*estors trying to digest

the panic that was going on in the market# and then they reacted accordingly !yli1uidating their positions. This pattern lasted until the mid o +--> until some o the

conidence was restored and in*estors came !ack into the market again. "n

comparison with the sample o the +,- most recent returns and according to the

graphs# we cannot detect any signiicant e*idence o *olatility clustering or the

stocks o 'aterpillar and &OD# !ut we can clearly o!ser*e some e*idence regarding

(pple# )?$# $E# and %axter as the cluster can !e clearly identiia!le. This is mainly

interpreted !y the act that since the inormation arri*es in cluster# in*estors tend to

take more or less time to react relati*e to the kinds o inormation that they recei*e

with respect to the stocks.

III/ The $ortfolio Selection $rocess

The Main o!jecti*e o the portolio optimization process is to ind the optimal

allocation o assets in the two data sets gi*en two kinds o portolios/ one containing

risky assets only# and another one that would include a risk ree asset. For this# we

used the Excel 9ol*er to run three types o optimizations that result in three types o

portolios/ the Minimum Variance )ortolio# the Optimal isky )ortolio# and theOptimal 'omplete )ortolio.

1) Analyzing the results given by the %inimum "ariance $ortfolios

"n the Markowitz model# we ha*e assumed that in*estors seek to maximize the return

at a gi*en le*el o risk# or minimize the risk at a gi*en le*el o return. The minimum

*ariance portolio is the one that gi*es the optimal allocation o risky assets !y

minimizing the general risk o the portolio. "n order to do so# we used the Excel


5/34

9ol*er and we set the portolio *ariance as the target to minimize gi*en !oth the

constrained and unconstrained optimization criteria. The ormula or the portolio

*ariance is gi*en in the (ppendix and is deri*ed rom the correlation matrixes that

were computed or !oth data sets. The general parameters used in the Excel 9ol*er

are also explained or !oth the constrained and the unconstrained portolios.


(ccording to ta!le , and ; in (ppendix +# the minimum *ariance portolio or the ull

data gi*es two results !ased on the constrained optimization and the unconstrained

optimization. "n the ong Only )ortolio# the results suggest that the in*estor should

in*est in three stocks which are %axter# )rocter ? $am!le and# (pple with the

respecti*e weights o >:. This gi*es a portolio expectedreturn o ;.;-: with a standard de*iation o 27.;-:# 7C:# and C.2-:. The expected return o!tained is

+.-;:# or a standard de*iation o 2


6/34

!) The &'timal Risky $ortfolio

The optimal risky portolio is the one that maximizes the 9harpe atio and allows

identiying the optimal weights o the risky assets in the portolio. The ormula or

the 9harpe ratio is gi*en in the appendix in the Methodology section. The ratio seeksto know how much additional return an in*estor would recei*e or the additional risk

o holding the risky assets o*er a riskree asset. The higher is this ratio the !etter is

the perormance o the portolio with respect to risk and return.


(ccording to the ta!le 5 and 7 in (ppendix +# or the ull data set# the ong only

portolio suggests that the in*estor should !e ully in*ested in (pple stock# whichgi*es an expected return that is e1ui*alent to the stock0s expected return o 5:

and a standard de*iation also e1ui*alent o C-.+7:. %y comparison# the

unconstrained portolio suggests that the in*estors should go long the stocks %axter

3C7.55 :#4 'aterpillar 3>2.-+:#4 and (pple 3+22.CC:#4 and short the stocks $E 3

2;2.2;:#4 &ow 3


7/34

suggested o +C.;,:# and that is what one could consider as a realistic result since

the proportions to !e in*ested are shared !etween two stocks as it is the minimum

di*ersiication o risky assets that one could consider gi*en the constraint o

maximizing the 9harpe ratio.

() The &'timal #om'lete $ortfolio

The optimal complete portolio is the portolio that consists o the optimal risky

assets and the risk ree asset. The optimal complete portolio is determined !y

maximizing the in*estor0s utility unction which is deined in (ppendix 2. The result

o the optimization gi*es the proportions to !e in*ested in the risky assets and the

risk ree assets# which allows the in*estors to !e properly di*ersiied !ased on three

main components that are# expected return# *olatility# and risk a*ersion.


(ccording to ta!le > and 2- (ppendix +# or the ull data set# the ongOnly portolio

suggests that the optimal complete portolio is the one that pro*ides an expected

return o


8/34

Dhen comparing the two samples# we notice that the unconstrained portolios in

!oth data sets pro*ide some unrealistic results in terms o the portolio weights as

some o them e1ual or exceed 2--: o the portolio0s total weight. "n the second data

sets when we eliminate some o the constraints that allow us to ha*e more accurate

data# the result o!tained show *ery large *alues that are completely unrealistic and

pre*ent us rom drawing any meaningul conclusions. "n the ull data set# the ong

Only portolio although it suggests that the in*estor should !e mixed !etween risky

assets and the risk ree asset does not pro*ide proper di*ersiication as the portolio

o risky assets is concentrated on one stock which is (pple stock# howe*er# it seems

that it is the one that pro*ides the most realistic result.

) The *fficient +rontier and the #a'ital

Allocation ,ine&rawing the eicient rontier is the last step that allows the in*estors to *isualize the

optimal portolios computed pre*iously and choose the !est alternati*es that are

oered gi*en the results that are produced# and whether they are realistic or not. The

(ppendix + pro*ides C graphs or !oth data sets that show the eicient rontier gi*en

the constrained and the unconstrained optimized portolios. The capital allocation

line which is determined !y maximizing the 9harpe ratio is represented on each o

the graphs and shows the optimal risky portolios as well as the optimal completeportolios. The indierence cur*e drawn display the cur*e o e1ually preerred

portolios that will generate the same utility to the in*estor. "n the second data set# it

was not possi!le to draw the 'apital (llocation ine as the *alues o!tained were

inaccurate and extremely large. Moreo*er# the points showing the Optimal isky

)ortolio and the Optimal 'omplete )ortolio are those that were computed gi*en the

constraints that we ha*e set in (ppendix 2# or the purpose o o!taining more

accurate results. This shows one o the limitations o the Mean Variance optimization

model as it ails to pro*ide meaningul results when using a small num!er o

o!ser*ations in the sample that we used.

-) Recommendation to the Investor

The results o!tained rom each type o portolio are summarized in ta!les 22 and 2+.

For a matter o con*enience# we la!eled the portolios rom )ortolio 2 to 2+. "n the

recommendation to the in*estor we suggest to look irst at the portolios that gi*e

some realistic results# and then we can compare which ones pro*ide the !est results

!y using the 9harpe ratio as a measure o perormance. %ased on the resultso!tained# the portolios that ha*e !een chosen are )ortolio 2# +# C# 5 and 2-.


9/34

)ortolios 2 and 5 do not pro*ide a satisying perormance to the in*estor as the

returns are lower than the risk ree asset# and the risks o!ser*ed are too high. "n this

case the in*estor is !etter o in*esting all his money into the risk ree asset at


10/34

are more sophisticated# and thereore re1uire to use enhanced models such as the

%lack and itterman Model.

A''endi. 1 %ethodology Section

1 #om'uting Returns and Standard Deviations

The irst step is to process the historical prices into returns and determine the

indi*idual stock returns as well as the standard de*iations.

The daily stock returns rom the stocks0 closing prices are calculated using the oldest

historical prices as the !ase dates. The computation o the daily stock returns is done

using the ollowing ormula/

Daily stock return0 #losing 'rice of the currenttrading day 2 $revious day3s closing $rice) /$revious day3s closing $rice

De then compute the expected returns o the stocks in the portolio using the simple

arithmetic mean o returns ormula# which is as ollows/

AR 0 R1 4 R! 4 R( 4 567 4 R8) / 8The returns o!tained are then annualized !y multiplying the expected returns !y +,-

trading days on a*erage in a gi*en year.

"n order to calculate the risk o indi*idual stocks# we used the 9T&EV unction in

Excel !y selecting the daily returns calculated in the two data sets# and annualizing

the *alues o!tained !y multiplying !y the s1uare root o +,- trading days on a*erage

per year.

! The 9ar:ue and ;era Test for 8ormality

The =ar1ue%era test statistic or normality ollows a chis1uare distri!ution with +

degrees o reedom. 'oncerning the analysis o the stock returns in the two data sets#

" used the >>: conidence inter*al or which the critical *alue is >.+2 at + degrees o

reedom. " the test statistic is higher than the critical *alue# we reject the null

hypothesis that the monthly returns are normal# otherwise we accept it# and we can

assume the returns are normal.


11/34

9kewness gi*es the measure o asymmetry o the distri!ution# and is deined !y the

ollowing ormula/

Dhere T is the time period or the length o the data used# and

Burtosis is the measure o the latness o the distri!ution# and is deined as ollows/

This leads to the computation o the =ar1ue%era test statistic which is as ollows/

( "olatility #lustering

Volatility clustering is another important actor that can help to understand thepatterns o the *ariations among stock returns. To show e*idence o *olatility

clustering in the two data sets# we plotted the stock returns in a time series graph#

and analyzed the dierent patterns o the indi*idual stocks using the complete data

set irst# and then analyzing the most recent +,- stock returns to see i the patterns

display some periods o high *olatilities show some extreme returns# which are

ollowed !y periods o relati*ely low *olatilities.

%easurement of $ortfolio Risk and Return (ter computing the expected returns and risks o indi*idual stocks# we deine some

o the ormulas or computing the portolio risk and return that will !e used in the

optimization process.

Modeling the )ortolio isk

The portolio risk is generally reerred to as the portolio *ariance. "n our case o

many assets# the *ariance o the portolio takes the orm o a matrix and has theollowing general ormula/

Dhere w2 to wn is a matrix that reers to the weights o the assets 2 to H in the

portolio# and IJxy is the co*ariance !etween assets x and y.

"n practice it is more useul to use the correlation matrix to deri*e the co*ariance

matrix as it is generally unknown# and we ha*e already computed the standard

de*iations o the indi*idual stocks.

"n act# we can deri*e the co*ariance matrix using the ollowing ormula/


12/34


13/34

di*ersiication !etween the risky asset and the risk ree asset in !oth the constrained

and unconstrained portolios.

Shar'e Ratio

The sharpe ratio is a risk adjusted measure to e*aluate portolio perormance and is

!ased on the ollowing ormula/

9ource/ "n*estopedia.com

The investor3s utility function

The in*estor0s utility unction used in this project is a 1uadratic unction that is

represented as ollows/

Dhere is the utility *alue# and ( is the in*estor0s le*el o risk a*ersion. The num!er

-.--, is a scaling actor used !y con*ention to express the expected return and

standard de*iation as percentages.

The optimal complete portolio would !e the one that pro*ides the highest utility !y

maximizing the ormula using the Excel 9ol*er. The result gi*es the optimal

proportion to !e in*ested in the risky asset which is represented !y the ollowing

ormula/

The proportion in*ested in the risk ree asset is gi*en !y/ 2 G

A''endi. ! Tables and #harts

Asset Data +ull Data Set)

*.'ected ReturnStandard Deviation

*


14/34

,.C,:

C


15/34

+5.C:

:

;a.ter

2C.2-:

+C.C-:

#ater'illar

;C.;>:

+>.7,:

+,.C>:

Table ! Stock Return and Standard Deviations@sing the !-F %ost recent returns

Full &ata 9et

*

D&E


16/34

;AG

#AT

$H

AA$,

urtosis

>.+7

5.-C+C->

5.C5>>>-;<

,.-7->

5.;,>+;+

C.+;-CC;

Skeness-.C257


17/34

22,2.+;2

Table ( The 9ar:ue2;era Test for 8ormality@sing the +ull Data Set

+,- Most ecent eturns

*D&E

;AG

#AT

$H

AA$,

urtosis

+.C2C,>,

+.-,75,C

2.;-2C5>77C

+.,;+7C5

+.;2,+

Skeness


18/34

-.25-;52

-.2>

+.;2;C2

-.22+5+C5

-.+-;;,

-.2<

5.>555C7

Table The 9ar:ue2;era Test for 8ormality@sing !-F %ost recent returns

Full &ata 9etK

,ong #onstraint@nconstrained

$ortfolio *.'ected return0

;.;-:

5.C7:

$ortfolio "ariance 0


19/34


20/34

.7C:

2.C,:


+.+7:

2.,5:


21/34

$ortfolio SD 0

2,.22:

2+.,+:

Risk +ree Asset 0


22/34

+..57:

(()

+2.27:

5.7+:

Table J %inimum "ariance $ortfolio using the!-F %ost Recent Returns

Full &ata 9et

,ong #onstraint

@nconstrained

Shar'e Ratio 0

7;.5>:

22C.CC:

$ortfolio *.'ected Return0

5:

2-+.+C:


2;.+


23/34

C-.+>:

7;.5+:

Risk +ree Asset 0


24/34

-.--:

,5.,,:

(()

2--.--:

+22.CC:

Table K &'timal Risky $ortfolio using the +ullData Sets

+,- Most ecent eturns

,ong #onstraint

@nconstrained

Shar'e Ratio 0

+:

+>


25/34

Risk +ree Asset 0


26/34

Table L &'timal Risky $ortfolio using the !-F%ost Recent Returns

K Full &ata 9et

,ong #onstraint

@nconstrained

Investor @tility +unction to %a.imize

-.55>;>;

-.,,2+5>552

*.'ected Return on the #om'lete $ortfolio


27/34

7;.2;:

,+.55:

$ortfolio *.'ected Return05:

2-+.+>:


2;.+:

7;.5,:

Risk +ree Rate 0


28/34

-.--:

2;2.+C:

&OD

-.--:


29/34

Investor @tility +unction to %a.imize

-.+>


30/34


2


31/34

2--.--:

'(T

2.2;:

2--.--:

)?$

-.--:

;C.2


32/34

+C.;;:

$ortfolio ( &'timal #om'lete $ortfolio

+>.C:

$ortfolio J &'timal #om'lete $ortfolio


33/34

,ong2Short $ortfolio @nconstrained)

O

$ortfolio *.'ected Return

$ortfolio Standard Deviation

$ortfolio K %inimum "ariance $ortfolio

+.C+:

2+.,2:

$ortfolio L &'timal Risky $ortfolio

22-.


34/34

$ortfolio 11 &'timal Risky $ortfolio

2-+.+C:

7;.5+:

$ortfolio 1! &'timal #om'lete $ortfolio

,,.:

C,.57:

Table 11 Summary Table of the Results &btained

+rom the &'timizations

*fficient +rontiers +ull Data Set

*fficient +rontiers !-F %ost Recent Returns

"olatility of Returns +ull Data Set

"olatility of Returns !-F %ost Recent Returns

the markowitz mean variance optimization model finance essay

Documents