A Review of Merton’s Portfolio Problemtesi.eprints.luiss.it/17083/1/184031_CARBONE_ANGELO.pdf- 1 - Dipartimento di Economia e Finanza A Review of Merton’s Portfolio Problem Angelo

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DipartimentodiEconomiaeFinanza

AReviewofMerton’sPortfolioProblem

AngeloCarbone

Matricola184031

Relatore:Prof.MarcoPapi

AnnoAccademico2015/2016

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Index

CHAPTER1......................................................................................................31.1 INTRODUCTION ....................................................................................................................................31.2 ESTIMATING STOCK RETURNS ......................................................................................................51.2.1 STOCHASTIC PROCESSES .............................................................................................................51.2.2 PROPERTIES OF A STOCHASTIC PROCESS ...........................................................................61.2.3 MARKOV PROCESS ..........................................................................................................................81.2.4 BROWNIAN MOTION ....................................................................................................................101.2.4A WIENER PROCESS .....................................................................................................................111.2.4B GENERALIZED WIENER PROCESS.......................................................................................121.2.4C ITO PROCESS ................................................................................................................................141.2.5 STOCHASTIC PROCESSES FOR STOCK PRICES ................................................................151.2.5A HISTORY OF STOCHASTIC PROCESSES FOR STOCK PRICES .................................151.2.5B MODEL DERIVATION ................................................................................................................161.2.5C ITO’S LEMMA ..............................................................................................................................191.2.5D LOG-NORMAL PROPERTY .....................................................................................................201.3 TEST OF RETURN ESTIMATION ..................................................................................................221.3.1 DATA SET ..........................................................................................................................................221.3.2 KOLMOGOROV-SMIRNOV TEST .............................................................................................251.3.3 JARQUE-BERA TEST ....................................................................................................................271.3.4 APPLICATION OF THE J-B TEST .............................................................................................291.3.5 FAILURES OF THE LOG-NORMAL MODEL .........................................................................30

CHAPTER2...................................................................................................33

2.1 PORTFOLIO THEORY ......................................................................................................................332.2 MARKOWITZ PORTFOLIO .............................................................................................................332.3 PREFERENCES AND RISK AVERSIONS .....................................................................................382.4 MERTON PORTFOLIO PROBLEM ...............................................................................................432.4.1 POWER UTILITY ............................................................................................................................47

CHAPTER3...................................................................................................503.1 TEST OF PERFORMANCES .............................................................................................................503.2 RISK-FREE ASSETS ...........................................................................................................................503.3 RISK AVERSION COEFFICIENTS .................................................................................................523.4 PROCEDURE FOR THE MERTON PORTFOLIO ......................................................................533.5 APPLICATION OF MERTON PORTFOLIO THEORY .............................................................543.6 PROCEDURE FOR THE MARKOWITZ PORTFOLIO .............................................................563.7 APPLICATION OF MARKOWITZ PORTFOLIO .......................................................................573.8 ANALYSIS OF THE TWO RESULTS .............................................................................................593.9 CONCLUSION ......................................................................................................................................61

MATLABAPPENDIX...................................................................................63

REFERENCES................................................................................................70

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CHAPTER1

1.1INTRODUCTION

Investinginfinancialmarketsisadiffusedactivityperpetratedbydifferent

entities. Example of investors are not only individuals willing to gain a

return fromtheir savingbutalsobanks, investment fundsand insurance

companycanbequotedamongthenumberofinvestors.

Thoseentitiesareallnaturally interested ingainingasmuchaspossible

from their investment operations but at the same time they are also

concernedwiththeriskstheyhavetoface.Itiscommonsensetoconsider

investors as risk averse agents, that is to consider them as reluctant to

allocatetheirwealthonassetsbearingahighlevelofrisk.

A logical aim of an investor is as amatter of fact the allocation of their

wealth inaway thatmaximize theirreturnswhilealsonot trespassinga

risk level limit. It is possible to mathematically replicate this investor

behaviour through the theoryofstochasticcontroland themaximization

ofexpectedutility.

The objects taken into consideration by investors for potential financial

operations can generally be differentiated into two diverse categories:

risky assets, which include all the assets whose future returns are not

defined and are therefore uncertain, and risk-free assets, a group

containingthoseassetswhosereturnsarefixedandthereforebearnorisk.

Examples of risky asses could be stock, real estate, commodities,

derivatives and other several could come tomind. Examples of risk-free

assetsareinsteadbondsandt-bills.Basedonhisdegreeofriskaversion,

an investor could compose an investment portfolio as amixture of both

risky and risk-free assets in order to match the level of risk he is

comfortablewith. It is then natural to questionwhat allocation strategy

shouldbefollowedbysaidinvestorduringtheformulationofhisportfolio

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andwhatwillbetheresultthatwillmaximizehisutility.Suchmusthave

beenthequestionRobertC.Mertonposedhimselfwhilewritinghis1969

paper entitled “Lifetime portfolio selection under uncertainty: The

continuous-time case”. In this paper theNobel laureate formalizedwhat

hasbeenlaterregardedasthe“Merton’sportfolioproblem”.

The most basic version of the problem establishes a setting where an

investor has the limited choice of allocating his wealth between a risky

asset and a risk-free on. Given some additional assumptions, Merton

concluded that the best allocation strategy is to maintain a constant

fractionofthewealthintheriskyassetandconsequentiallytoholdfixed

the part of wealth invested in risk-free assets. This setting can later be

expanded and several risky assets can be incorporated in the model.

However, this variationdoesn’t affect the first conclusion as the optimal

allocation still accounts for a fixed proportion between the wealth

allocatedbetweenassetsbearingandnotbearingrisk.

Ifconsideredfromamorerealisticpointofview,theconclusionofMerton

when solving his portfolio problem is based on assumptions that don’t

holdwith the samestrength in the realworld.Forexample, the solution

assumesthatthedynamicsoftheriskyassetsfollowageometricBrownian

motionwhich implies normally distributed log returns. However, in the

case of real stock prices, this assumption is hardly ever held true. An

analysis of the distribution of real stock returns shows in fact that the

distributionstendtohaveheavierorfattertails,duetothefactthatprice

changes are normally higher than those a normal distribution would

forecast.

AnotherdrawbackofthesolutionproposedbyMertonisthefactthatsaid

conclusion is based on a continuous mathematical framework. The

investor should therefore rebalance his portfolio at the same rate of

change of the stock prices in order to rigorously follow the optimal

strategy. However modern financial markets are extremely liquid and

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pricechangeshappensalmostconstantly,aneventthatmakesimpossible

tostickstrictlytotheoptimalstrategy.Anadditionalfactorthatcould,and

should,betakenintoaccountisthefactthatportfoliorebalancinghappens

atthepriceofthetransactioncosts,makingsuchabehaviouranexpensive

one.

The aim of this thesis is to introduce the models necessary to the

application of a simple model of the Merton portfolio choice where

transactioncostsarenotconsidered.Theselectionwillbemadeusingthe

returnsof fourteenreal indexesoverthepastyears.Chapter1willcover

themodelsunderlyingthereturnsestimation,Chapter2willintroducethe

theorybehind thePortfolio selectionproblemwhileChapter3will focus

ontheapplicationofthemodelstotherealdata.

1.2ESTIMATINGSTOCKRETURNS

In order to apply portfolio theory to the previously shown data set it is

necessaryfirsttointroducetheprocessthroughwhichsaidreturnswillbe

estimated. Stochastic processes are generally regarded as the best

approachtosuchafeatureandarehereintroduced.

Firstitisnecessarytogiveadefinitionofastochasticprocess.

1.2.1STOCHASTICPROCESSES

A variablewhose value changes over the course of time in an uncertain

way is generally said to followa stochasticprocess. Stochasticprocesses

can be divided into discrete time processes and continuous time

processes.

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Inadiscretetimestochasticprocess,thevalueofthevariablecanassume

different value only in determined points in timewhile in a continuous

timestochasticprocessthosevaluecanchangeatanypoint.

Anotherpossibledivisionofstochasticprocessesmaybetheonebetween

continuousvariableprocessesanddiscretevariableones.Inacontinuous

variable stochastic process, the used variable can assume all the values

contained in a certain range, differently from the discrete variable

processeswheretheunderlyingvariablecantakeonlyadefinitenumber

ofvalues.

Amorerigorousdefinitionofastochasticprocessisherebyintroduced.

1.2.2PROPERTIESOFASTOCHASTICPROCESS

A stochastic process is a mathematical model for the occurrence of a

randomeventateachmomentafterthestartingtime.Therandomnessof

theoccurrence is representedby the introductionofameasurable space

(Ω, ℱ), denominated sample space, where probability measure can be

located.ThereforeastochasticprocessisacollectionofrandomvariableX,

with

𝑋 = 𝑋(; 0 ≤ 𝑡 < ∞ on Ω,ℱ ,(1.1)

which take values in a secondmeasurable space(𝑆, ℘), called the state

space. The index 𝑡 ∈ [0,∞) of the random variables Xt admits an

interpretationastime.

Givenafixedsamplepoint𝜔 ∈ Ω,thefunction

𝑡 ↦ 𝑋( 𝜔 ; 𝑡 ≥ 0(1.2)

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isthesamplepathoftheprocessXassociatedwithω.Thissamplepath,or

sampletrajectoryallowsforthecreationofmathematicalmodelwhichcan

beusedonarandomexperimentswhoseoutcomeareobservableovera

continuousperiodoftime.

Stochastic processes have several properties and those relevant to

successively discussed notions are hereby introduced, but it is first

necessarytodisclosethenotionsof𝜎-fieldsandfiltration.

• 𝜎-fields

A𝜎-fieldℱofsubsetsofXisacollectionℱofsubsetsofXsatisfying

thefollowingconditions:

a) ∅ ∈ ℱ

b) if𝐵 ∈ ℱthenitscomplementBcisalsoinℱ

c) if B1,B2,… is a countable collection of sets inℱthen their

unionis𝑈Z[\] 𝐵Z

• Filtration

Given that (𝐹(, 𝑡 ≥ 0) is a collection of 𝜎 -fields in the same

probability space(Ω, ℱ, 𝑃)withℱ( ⊆ ℱfor all𝑡 ≥ 0, said collection

iscalledafiltrationif

ℱc ⊆ ℱ(, ∀0 ≤ 𝑠 ≤ 𝑡(1.3)

In case the index t is discrete, the filtration(ℱZ, 𝑛 = 0,1, … )is a

sequenceof𝜎-fieldsonΩwithℱZ ⊆ ℱZk\forall𝑛 ≥ 0.

The reason for the inclusion of those notions in the study of

stochastic process is related to the temporal feature of the

stochasticprocessesanalysed.

Inordertoincludedifferenttimeperiodsintheprocessofchoiceit

is possible to equip the sample space(Ω, ℱ)with a filtration,

consisting in a nondecreasing family ℱ(; 𝑡 ≥ 0 of sub-𝜎-fields of

ℱ: ℱc ⊆ ℱ( ⊆ ℱfor0 ≤ 𝑠 < 𝑡 < ∞.Insuchsetting,italsoholdsthat

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ℱ] = 𝜎( .(mn ℱ().(1.4)

Havingintroducedthosetworelevantnotions,itisnowpossibleto

describesomerelevantpropertiesofastochasticprocess.

• Measurableprocesses

The stochastic process X is called measurable if, for every𝐴 ∈

ℬ ℝu ,the set 𝑡, 𝜔 ;𝑋((𝜔) ∈ 𝐴 belongs to the product𝜎-field

ℬ( 0,∞ )×ℱ;itmeansthatifthemappingissuchthat

𝑡, 𝜔 ↦ 𝑋( 𝜔 : ([0,∞)×Ω, ℬ( 0,∞ )×ℱ) → (ℝu, ℬ ℝu )(1.5)

thentheprocessismeasurable.

• Adaptedprocesses

AstochasticprocessXisdefinedasadaptedtothefiltration ℱ( ,if,

for each𝑡 ≥ 0, Xt is anℱ(-measurable random variable. Values of

𝑋((𝜔)canonlybedeterminedbytheinformationavailableattimet.

A stochastic process useful in the determination of the stocks returns is

theMarkovprocess.

1.2.3MARKOVPROCESS

AMarkovprocessisatypologyofstochasticprocesswheretheprediction

ofthefutureisbasedonlyonthevaluepresentlytakenbytheconsidered

variable.Thepasthistoryofthevariableandthepaththroughwhichthe

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present situation has been generated are therefore both considered as

irrelevant.

Stock prices are generally assumed to follow a Markov process. This

generalideaimpliesthattheonlyrelevantpieceofinformationregarding

a stock is the currentprice since thepast cannot affect the futureof the

stockitself.

Prediction for the futuremovementsof thepriceareuncertainandmust

therefore be expressed in terms of probability distributions. Those

distributions are at any time completely independent from the path

previouslyfollowedbythestock.

In order to have a mathematical definition of a Markov process, it is

necessarytotakeintoconsiderationapositiveintegerdandaprobability

measure on (ℝu, ℬ(ℝu) . An adapted, d-dimensional process 𝑋 =

𝑋(, ℱ(; 𝑡 ≥ 0 onsomeprobabilityspace(Ω, ℱ, 𝑃y)isthereforesaidtobea

Markov processwith intial distribution µ if the following conditions are

verified:

§ 𝑃y 𝑋n ∈ Γ = µ Γ , ∀Γ ∈ ℬ ℝu ;

§ 𝑃y 𝑋(kc ∈ Γ ℱc = 𝑃y 𝑋(kc ∈ Γ 𝑋c , 𝑓𝑜𝑟𝑠, 𝑡 ≥ 0𝑎𝑛𝑑Γ ∈ ℬ ℝu .

AparticulartypeofMarkovstochasticprocessinacontinuous-timesetting

is theWienerProcess. Inorder toexplainwhataWienerprocess is, it is

necessarytofirstintroducetheconceptofBrownianmotion,whichisitself

acaseofaMarkovprocess.

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1.2.4BROWNIANMOTION

Brownian movement is originally the name given to the irregular

movement of pollen suspended in water which was observed by the

botanistRobertBrownin1828.

Suchrandommotion,whichithasbeennowattributedtothebuffetingof

thepollenbywatermolecules,resultsinthedispersalofthepolleninthe

water. However, this originally botanic observation has been expanded

outsideitsoriginalfieldofstudy.

In mathematical terms, a one-dimensional Brownian motion is a

continuous and adapted process𝐵 = 𝐵(, ℱ(; 0 ≤ 𝑡 < ∞ , defined on a

probabilityspace(Ω, ℱ, 𝑃).Propertiesof thatprocessare that𝐵n = 0and

that the increment𝐵( − 𝐵c, given0 ≤ 𝑠 < 𝑡, is independent ofℱcand is

normallydistributedwithmeanzeroandvariance𝑡 − 𝑠.

AprocessBissaidtohavestationaryandindependentincrementsifsuch

a process is a Brownian motion and0 = 𝑡n < 𝑡\ < ⋯ < 𝑡Z < ∞. In that

case the increments 𝐵(� − 𝐵(�� [\

Zare independent and thedistribution

of𝐵(� − 𝐵(�� depends on tj and tj-1 only through the difference𝑡� − 𝑡��\,

thereforeitisnormalwithmeanequalzeroandvarianceequalto𝑡� − 𝑡��\.

Whilethe filtration ℱ( ispartof thedefinitionofBrownianmotion, it is

possible tohaveaBrownianmotionwhen the filtration isdifferent from

ℱ( .Specifically,ifnofiltrationisgivenforaprocess 𝐵(; 0 ≤ 𝑡 < ∞ butit

is known that𝐵( = 𝐵( − 𝐵nisnormalwithmeanzeroandvariance t and

that B has stationary, independent increments, it is possible to define

𝐵(, ℱ(�; 0 ≤ 𝑡 < ∞ asaBrownianmotion.

Anothercasecouldbethat ℱ( isalargerfiltration,meaningthatℱ(� ⊆ ℱ(

for𝑡 ≥ 0. Inthatcase, if𝐵( − 𝐵cis independentonℱcwhenever0 ≤ 𝑠 < 𝑡,

itispossibletoconsider 𝐵(, ℱ(; 0 ≤ 𝑡 < ∞ asaBrownianmotion.

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1.2.4aWIENERPROCESS

The Wiener Process, is another particular type of Markov Stochastic

process and a case of standard, one-dimensional Brownian motion. A

Wiener process has amean change of zero and a variance rate of 1 per

year.More formally,avariablez issaidto followaWienerprocess if the

followingtwopropertiesholdtrue:

• Property1

Thechange∆𝑥duringasmallperiodoftime∆𝑡is:

∆𝑧 = 𝜀 ∆𝑡

where𝜀hasastandardnormaldistribution𝜙(0,1).

• Property2

Thevaluesof∆𝑧foranytwodifferentshortintervalsoftime,∆𝑡,are

independent.

From the first property it is also possible to state that∆𝑧has a normal

distributionwiththefollowingcharacteristics:

o Meanof∆𝑧=0

o StandardDeviationof∆𝑧= ∆𝑡

o Varianceof∆𝑧=∆𝑡

ThesecondpropertyimpliesinsteadthatxfollowsaMarkovprocess.

Itisnowpossibletotakeintoconsiderationthechangeofvalueassumed

by zover a relatively longperiodof timeT.This changeof value conbe

definedas𝑧 𝑡 − 𝑧(0).

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Anotherway to interpret thisvariation isas thesumof thechanges inx

duringNsmalltimeintervalsoflength∆𝑡,whereN= �∆(.

Itisthereforepossibletoestablishthefollowingrelationship:

𝑧 𝑡 − 𝑧 0 = 𝜖��[\ ∆𝑡(1.6)

where the𝜖� (i=1,2,….N) are distributed𝜙(0,1). Those various𝜖� are

independent of each other from the second property of the Wiener

process. It is therefore possible to conclude that𝑧 𝑡 − 𝑧(0)is normally

distributedwith

o meanof𝑧 𝑡 − 𝑧(0)=0

o varianceof𝑧 𝑡 − 𝑧(0)=N∆𝑡=T

o standarddeviationof𝑧 𝑡 − 𝑧(0)= 𝑇.

1.2.4bGENERALIZEDWIENERPROCESS

The mean change per unit time for a stochastic process is generally

defined as the drift rate and the variance per time unit is known as the

variance rate. The basic Wiener process previously introduced, shortly

referredasdz,hasadriftrateofzeroandavariancerateof1.

Thedriftrateofzero indicatedthat theexpectedvalueofzatanyfuture

time is identical to its present value. The variance rate of one instead

suggeststhatthevarianceofthechangeinzinatimeintervalofduration

T equals T itself. A generalized Wiener process for a variable x can be

thereforedefinedintermsofdzas

𝑑𝑥 = 𝑎𝑑𝑡 + 𝑏𝑑𝑧(1.7)

whereaandbareconstants.

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Thisdefinitionismadeoftwoessentialcomponents: thefirst is theterm

“𝑎𝑑𝑡”whichimpliesthatxhasanexpecteddriftrateofaperunitoftime.

Thesecondterm“𝑏𝑑𝑧”canbeinsteadregardedasanadditionofnoiseand

variability to thepath followedbyx.Thequantityandmagnitudeof this

noiseisbtimesaWienerprocess.

Since a Wiener process has a standard deviation of 1, it follows that b

timesaWienerprocesshasastandarddeviationofb.Ifthissecondterm

wasn’ttobeadded,theequationwouldhavebeen𝑑𝑥 = 𝑎𝑑𝑡whichwould

have lead to𝑎 = u�u(. The integration of this relationship with respect to

timeyieldsthefollowingresult:𝑥 = 𝑥n + 𝑎𝑡,wherex0 indicatesthevalue

ofxattime0.

InaperiodoftimeT,thevariablexwouldthenincreasebyanamount𝑎𝑇,

a result which would leave out lots of possibilities which are instead

integratedbytheadditionofthesecondcomponent“𝑏𝑑𝑧”.

Asaresult,itispossibletostatethatinasmalltimeinterval∆𝑡,thechange

∆𝑥inthevalueofxis:

∆𝑥 = 𝑎∆𝑡 + 𝑏𝜀 ∆𝑡(1.8)

Asitwaspreviouslymentioned,𝜀hasanormalstandarddistribution,thus

itispossibletostatethatalso∆𝑥hasanormaldistributionwith:

o Meanof∆𝑥 = 𝑎∆𝑡

o Standarddeviationof∆𝑥 = 𝑏 ∆𝑡

o Varianceof∆𝑥 = 𝑏�∆𝑡

Withthesamereasoningitisalsopossibletodeterminethatthechangeof

valueinxinanytimeintervalTisnormallydistributedwith:

o Meanofchangein𝑥 = 𝑎𝑇

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o Standarddeviationofchangein𝑥 = 𝑏 𝑇

o Varianceofchangein𝑥 = 𝑏�𝑇.

Tosumup,thegeneralizedWienerprocesshasanexpecteddriftrateofa

andavariancerateofb2.

1.2.4cITOPROCESS

The Ito process is another type of stochastic process based on the

generalizedWienerprocess.ItisinfactageneralizedWienerinwhichthe

parameters𝑎and𝑏arefunctionsofthevalueoftheconsideredvariable𝑥

andofthetimevariable𝑡.AnItoprocessisalgebraicallyexpressedas:

𝑑𝑥(𝑡) = 𝑎(𝑥(𝑡))𝑑𝑡 + 𝑏(𝑥(𝑡))𝑑𝑧(𝑡)(1.9)

Bothe theexpecteddrift rateand thevariance rateofan Itoprocess can

change value over time. If the time interval taken into consideration is

small, that is in an interval between𝑡and𝑡 + ∆𝑡, then the variable value

changesfrom𝑥to𝑥 + ∆𝑥,where:

𝑥(𝑡 + ∆𝑡) − 𝑥(𝑡) ≈ 𝑎(𝑥(𝑡))∆𝑡 + 𝑏(𝑥(𝑡))𝜀(𝑡) ∆𝑡(1.10)

Asmallapproximationisassumedinthatrelationship.Theapproximation

consists in the fact that the mean and the standard deviation of𝑥,

conditional to𝑥(𝑡), remain constant, respectively equal to𝑎(𝑥(𝑡))∆𝑡and

𝑏(𝑥(𝑡)) ∆𝑡,duringthetimeintervalfrom𝑡to𝑡 + ∆𝑡.

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1.2.5STOCHASTICPROCESSESFORSTOCKPRICES

1.2.5aHISTORYOFSTOCHASTICPROCESSESFORSTOCKPRICES

The birth of the relationship between mathematics and financial model

mustbetrackedbacktoLouisBachelier’s1900dissertationonthetheory

of speculation in the Paris markets. That work gave life to both the

continuous timemathematicsof stochasticprocessesand thecontinuous

timeeconomicsofoptionpricing.

During his analysis of option pricing, Bachelier elaborated two different

derivationsof thepartialdifferentialequation for theprobabilitydensity

for the Wiener process. In one derivation, he worked out what is now

known as the Chapman-Kolmogorov convolution probability integral.

Bachelierexploited the ideasof theCentralLimitTheoremand, realizing

thatmarketnoiseshouldbewithoutmemory,hereasonedthatincrements

of stock prices should be independent and normally distributed.

Combining this reasoning with the Markov property, he was able to

connect Brownianmotionwith the heat equation in order tomodel the

market noise. Hewas also able to define other processes related to the

Brownianmotion,inparticularhecomputedthemaximumchangeduring

atimeintervalforaone-dimensionalBrownianmotion.AfterBachelier’s

crucialworks,ittookmorethan60yearsbeforeanewbreakthroughcould

happeninthefieldofstochasticprocessesforstockprices.

ItwasPaul Samuelson in 1965 that introduced theGeometricBrownian

Motion as a good model for stock price movements. In his research

Samuelson tried to improve Bachelier’s model and to fix its inability to

ensurepositivepricesforthestocks,acapacitythattheBrownianmotion

hadinstead.

Samuelsoncontributionsaremainlycontainedintwopapers.Inthose,he

gave,65yearsafterBachelierhadstated it,hiseconomicarguments that

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pricesmust fluctuate randomly. He postulated the idea that discounted

futurespricesfollowamartingaleandwentontoprovethatfuturesprices

changeswere uncorrelated across time.Moreover, his proposition could

be also extended to arbitrary functions of the sport price therefore

allowingforanapplicationonoptions.

Anotherturningpointwasthepublication1973iftheBlack-Scholesmodel

foroptionpricing.

ThetwoeconomistsFischerBlackandMyronScholesdeducedanequation

thatprovidedthefirststrictlyquantitativemodelforcalculatingtheprices

ofoptions.Thebasic insightunderlyingtheBlack-Scholesmodel isthata

dynamic portfolio trading strategy in the stock can replicate the returns

fromanoptiononthatstock.Thatactionisdefinedas“hedginganoption”

anditisthemostimportantideaunderlyingtheBlack-Scholesapproach.

Conceptualbreakthroughsinfinancetheoryinthe1980swerefewerand

lesscrucialthaninthe1960sand1970sbuttheresourcesemployedinthe

fieldweremorethanthatusedinthepreviousdecades.Thedevelopment

ofmorepowerful computingmachines and thegrowthof thenumberof

sophisticated mathematical models for financial practices happened

almostsimultaneously,allowingformoreeffectivemodelsthanthoseused

inthepast.

1.2.5bMODELDERIVATION

ThegeneralizedWienerprocessfailstocapturesomekeyaspectofstock

pricesduetoitsconstantexpecteddriftrateanditsconstantvariancerate.

Thebiggestshortfallistheincapacityofthemodeltocapturethefactthat

thepercentagereturnrequiredbyinvestorsfromastockisindependentof

thestock’sprice.

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Theassumptionof constantexpectdrift ratemust thereforebe removed

andreplacedbytheassumptionthattheexpectedreturn,representedby

theexpecteddriftdividedbythestockprice,isconstant.

Inthisassumption,ifSisusedtorepresentthestockpriceattimetand𝜇

isusedtorepresenttheexpectedrateofreturnonthestockexpressedin

decimal form, it is possible to state that the expected drift rate in S is

assumed tobe𝜇𝑆∆t and that represents the increase in Sduring a short

timeperiod∆t.

Assuming a constant stock price volatility of zero, the presented model

impliesthat:

∆𝑆 = 𝜇𝑆∆𝑡(1.11)

which,as∆t→ 0,becomes:

𝑑𝑆 = 𝜇𝑆𝑑𝑡(1.12)

oralso

u��= 𝜇𝑑𝑡(1.13)

Integratingthisrelationshipbetweentime0andtime𝑇itresultsthat:

𝑆� = 𝑆n𝑒y� (1.14)

Where S0 and ST are the stockprice at time0 and timeT.This equation

indicatesthatwhentheeffectofthevariancerateisabsent,thestockprice

growsatacontinuouslycompoundedrateofµ.

Itishoweverimpossibleinrealitytoexperienceanabsenceofvolatilityin

the growth of stock prices. A reasonable assumption that can be made

regardingvolatilityofstockpricesisthatthevariabilityofthepercentage

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returns in a short period of time∆𝑡is the same regardless of the stock

price.Theaimoftheassumptionistorepresentthefactthataninvestoris

indifferently uncertain about the future of stock independently from the

price of the stock itself. Moreover, the assumption suggests that the

standard deviation of the change in a short period of time∆𝑡should be

proportionaltothestockpriceandallowsforthecreationofthefollowing

model:

𝑑𝑆 = µ𝑆𝑑𝑡 + 𝜎𝑆𝑑𝑧(1.15)

or

u��= 𝜇𝑑𝑡 + 𝜎𝑑𝑧.(1.16)

Inthatequationthevariableµrepresentstheexpectrateofreturnofthe

stockandthevariable𝜎representsthevolatilityofthestockprices.

This model of stock price behaviours is mostly known as geometric

Brownianmotion.Itisalsopossibletoelaborateadiscretetimeversionof

themodel.Inthatversionithappensthat

∆��= µ∆𝑡 + 𝜎𝜀 ∆𝑡(1.17)

or

∆𝑆 = µ𝑆∆𝑡 + 𝜎𝑆𝜀 ∆𝑡(1.18)

The variable∆𝑆is the variation in the stock price𝑆during a small time

interval∆𝑡while𝜀isaparameterwithastandardnormaldistribution.The

parametersµand𝜎areagaintheexpectedrateofreturnperunitoftime

andthevolatilityofthestockprice.

-19-

Therightsideofthisequationisofparticularinterest.Thetermµ∆𝑡isthe

one representing the expected value of the return on a short amount of

time∆𝑡whiletheterm𝜎𝜀 ∆𝑡standsforthestochasticcomponentofthose

returns.Thevarianceofthatcomponent,andthereforeofthereturnsasa

whole, is𝜎�∆𝑡.In the end it is possible to state that∆��is normally

distributed with meanµ∆𝑡and standard deviation𝜎 ∆𝑡, or in another

notation:

∆��~𝜙(µ∆𝑡, 𝜎�∆𝑡)(1.19)

1.2.5cITO’SLEMMA

TheIto’slemma,namedafterthemathematicianK.Ito,isaninimportant

process in the understanding of the behaviour of functions of stochastic

variable.

Assumingthatavariable𝑥followstheItoprocess

𝑑𝑥 = 𝑎(𝑥(𝑡))𝑑𝑡 + 𝑏(𝑥(𝑡))𝑑𝑧(1.20)

where𝑑𝑧is a Wiener process and𝑎and𝑏are functions of𝑥and𝑡, it is

possibletodefinethedriftrateofthevariable𝑥as𝑎anditsvariancerate

asb2.Ito’slemmashowsthatafunction𝐺of𝑥and𝑡followstheprocess

𝑑𝐺 = ��𝑎 + ��

�(+ \

��

𝑏� 𝑑𝑡 + ��𝑏𝑑𝑧(1.21)

where𝑑𝑧indicatesthepreviouslymentionedItoprocess.𝐺alsofollowsan

Itoprocessandthedriftrateisdefinedby

��𝑎 + ��

�(+ \

��

𝑏�(1.22)

-20-

andthevariancerateis

��

�b2(1.23)

It is nowpossible to connect those resultswith the previous conclusion

that

𝑑𝑆 = µ𝑆𝑑𝑡 + 𝜎𝑆𝑑𝑧(1.15)

canbeconsideredavaluablemodelofstockpricemovementswithµand𝜎

constant.

FromIto’slemmaitfollowsthattheprocessundergonebyafunction𝐺of

𝑆and𝑡is

𝑑𝐺 = ��µ𝑆 + ��

�(+ \

��

𝜎�𝑆� 𝑑𝑡 + ��𝜎𝑆𝑑𝑧(1.24)

Itisrelevanttohighlighthowboth𝑆and𝐺areaffectedbytheunderlying

sourceofuncertainty𝑑𝑧.

1.2.5dLOG-NORMALPROPERTY

ItisnowpossibletouseIto’slemmatoderivetheprocessfollowedby𝑙𝑛𝑆,

which is thenatural logof thestockprice,when𝑆follows thepreviously

introducedequation

𝑑𝑆 = µ𝑆𝑑𝑡 + 𝜎𝑆𝑑𝑧.(1.15)

Firstitisnecessarytodefine𝑙𝑛𝑆as𝐺.Afterthatitispossibletostatethat:

-21-

��= \

�,�

��

= − \��, ��(= 0.(1.25)

Ito’s lemmacannowbeappliedand theprocess followedby𝐺results to

be:

𝑑𝐺 = µ − ¡�

�𝑑𝑡 + 𝜎𝑑𝑧.(1.26)

Since µ and𝜎are constant, the equation essentially states that the

equation𝐺 = 𝑙𝑛𝑆followsageneralizedWienerprocess.

Theconstantdriftrateisµ − ¡�

�andtheconstantvariancerateis𝜎�.

Therefore the variation in 𝑙𝑛𝑆 from time 0 to time 𝑇 is normally

distributedwithameanof µ − ¡�

�𝑇andavarianceof𝜎�𝑇.

Inamoreformalwayitmeansthat

𝑙𝑛𝑆� − 𝑙𝑛𝑆n~𝜙 µ − ¡�

�𝑇, 𝜎�𝑇 (1.27)

oralso

𝑙𝑛𝑆(~𝜙 𝑙𝑛𝑆n + µ − ¡�

�𝑇, 𝜎�𝑇 .(1.28)

Thisequationshowshow𝑙𝑛𝑆� isnormallydistributed.

A variable has a lognormal distribution if the natural logarithm of the

variable is normally distributed as it is in this case. Thismodel of stock

price behaviour implies that stock’s price at time T, given the current

price,islog-normallydistributed.

Thestandarddeviationofthelogarithmis𝜎 𝑇whichimpliesagrowthin

thestandarddeviationasthetimespanofinterestgrowslarger.

-22-

1.3TESTOFRETURNESTIMATION

Itisnowpossibletotestthesoundnessofthepreviousassumptions.Asa

matteroffact,thewholemodelreliesontheconditionthatstockpricesare

log-normally distributed and that therefore the returns are normally

distributed.Itispossibletoverifyifthoseconditionsholdalsoinrealityby

usingstatisticaltestsonaselecteddataset.

1.3.1DATASET

The data set taken into consideration for the application of the theories

presentediscomposedbythevariationsoffourteendifferentindexesover

thecourseofa16years’timespan.Thedataareanalysedfromthe1stof

January2000tothe4thofMarch2016.

The tablesbelowcontainasummaryof themostrelevantcharacteristics

ofthedailyreturnsofthosesamplesovertheconsideredtimelength.

INDEX SPXINDEXCCMP

Index

DAX

Index

SASEIDX

Index

NOFOBSERVATIONS 4065 4065 4110 4357

MEAN 0.01% 0.01% 0.02% 0.03%

STANDARD

DEVIATION1.27% 1.67% 1.55% 1.46%

KURTOSIS 11.08 8.67 7.37 12.77

SKEWNESS 0.01 0.22 0.12 -0.65

MEDIAN 0.05% 0.08% 0.07% 0.09%

MAXRETURN 11.58% 14.17% 11.40% 9.85%

MINRETURN -9.03% -9.67% -8.49% -9.81%

-23-

INDEXNKY

Index

CAC

Index

UKX

Index

FTSE MIB

Index

NOFOBSERVATIONS 3969 4134 4086 4102

MEAN 0.01% 0.05% 0.006% -0.007%

STANDARD

DEVIATION1.56% 1.50% 1.23% 1.56%

KURTOSIS 8.95 7.78 9.03 7.24

SKEWNESS -0.18 0.15 -0.003 0.05

MEDIAN 0.03% 0.03% 0.03% 0.04%

MAXRETURN 14.15% 11.18% 9.84% 11.49%

MINRETURN -11.41% -9.04% -8.85% -8.24%

INDEX INDEXCFIndex SHCOMPIndex SMIIndex

NOFOBSERVATIONS 4002 3909 4065

MEAN 0.08% 0.03% 0.01%

STANDARDDEVIATION 2.17% 1.66% 1.22%

KURTOSIS 18.90 7.29 9.80

SKEWNESS 0.30 -0.18 -0.002

MEDIAN 0.11% 0.06% 0.05%

MAXRETURN 28.69% 9.86% 11.39%

MINRETURN -18.66% -8.84% -8.67%

INDEX AS51Index SPTSXIndexMXBR

Index

NOFOBSERVATIONS 4091 4062 4215

MEAN 0.01% 0.01% 0.03%

STANDARDDEVIATION 1.02% 1.15% 2.21%

KURTOSIS 8.19 11.51 9.57

-24-

SKEWNESS -0.36 -0.45 0.07

MEDIAN 0.04% 0.06% 0.05%

MAXRETURN 5.79% 9.82% 18.08%

MINRETURN -8.34% -9.32% -16.74%

It is already possible to spot some of themost essential features of the

indexesthatwillbeusedfortheportfoliocomposition.

Themostglaringoneisthelowmeanofthedailychanges,alwayscloseto

zero, and thegeneral symmetryof thedata sets, representedby the low

skewnessvalue.

Moreover,alltheindexeshaveahighkurtosisvalue,apredictablefeature,

indicatingthatthetailsofthereturnsdistributionarefatterthanthoseofa

normaldistribution.

Themostvolatile indexesof thebunchare theRussian Index, the INDEX

CF,andtheBrazilian index,MXBR,whosedailyreturnvolatility ishigher

thanthe2%.

Still, the indexesseemto fit the logicalexpectationan investormayhave

whendealingwithassetsreplicatingan indexperformance, that isanon

excessivedailyvolatilityandexpectedreturns.

Havingintroducedthedataset,itisnowpossibletocomparehowthereal

returnswouldfarewhencomparedwithanormaldistribution.

Inordertoperformsuchacomparison,itisfirstnecessarytopresenttwo

useful statistical test for the normality of the returns. Those are the

Kolmogorov-SmirnovtestandJarque-Beratest.

-25-

1.3.2KOLMOGOROV-SMIRNOVTEST

TheKolmogorov-Smirnovtestisanonparametrictestthatcanbeusedto

compareasamplewithareferenceprobabilitydistributionortocompare

twosample.

Thetwosampletestmaybeusedtotestwhetherthetwounderlyingone-

dimensionalprobabilitydistributionsdifferwhiletheone-sampletestwill

indicate whether the sample has a distribution similar to a standard

normaloneornot..

• Kolmogorov-Smirnovone-sampletest

Assumingthat:

o x1,...,xmbeobservationsoni.i.d.r.vsX1,...,Xmwithac.d.f.F1,

Theaimistotestthenullhypothesis:

𝐻n ∶ 𝐹\(𝑥) = 𝐹�(𝑥),forallx(1.29)

where𝐹nisaknownc.d.f.

TheKolmogorov-Smirnovteststatistic𝐷Zisdefinedby

𝐷Z = 𝑠𝑢𝑝�∈§

𝐹 𝑥 − 𝐹n(𝑥) ,(1.30)

Inthatcase𝐹isanempiricalcumulativedistributionwhichcanbe

definedas

𝐹 𝑥 = #(�:�©ª�)Z

(1.31)

-26-

Itcanbenoticedthatthesupremumfromequation(1.30)canoccur

onlyatoneoftheobservedvalues𝑥� ortotheleftof𝑥� .

It is now possible to perform a Kolmogorov-Smirnov test and verify

whetherthelog-normalassumptionholdstruefortheanalysedindexesby

testing whether the returns appear to follow a standard normal

distributionornot.

The testwillbemadeatasignificance levelof0.05,or5%and itwillbe

performedusingtheMATLABsoftware.

Theresultsof thetestareherebyreported,where“REJECT”will indicate

the rejection of the null hypothesis of the sample following a standard

normaldistribution.

INDEX RESULTOFK-STEST(𝜶 = 𝟎. 𝟎𝟓)

SPXINDEX REJECT

CCMPIndex REJECT

DAXIndex REJECT

SASEIDXIndex REJECT

NKYIndex REJECT

CACIndex REJECT

UKXIndex REJECT

FTSEMIBIndex REJECT

INDEXCFIndex REJECT

SHCOMPIndex REJECT

SMIIndex REJECT

AS51Index REJECT

SPTSXIndex REJECT

MXBRIndex REJECT

MatlabcodeavailableintheMatlabAppendix.

-27-

As it can be seen the indexes’ returns distribution seems not to be

corresponding to a standardnormal, indicating that the assumption that

prices followa log-normaldistributionmaybean incorrectone. Inorder

toavoid thepossibilityofaKolmogorov-Smirnov test failure incorrectly

assessingthereturnsdistribution,it ispossibletointroduceandperform

anothernormalitytestastheJarque-Beratestis.

1.3.3JARQUE-BERATEST

The Jarque-Bera test is a statistical test capable of assessing whether a

sample data has a distribution approximately normal. The test tries to

determinethedistributionbymatchingthesampleskewnessandkurtosis

withthoseofanormallydistributedsample.

Thesampleskewnessisdefinedas:

𝑆 = \Z∙ (�©��)¯

°©±�

(¡�)¯�(1.32)

where

𝜎� = \Z

(𝑥� − 𝑥)�Z�[\ (1.33)

Skewness provides a measure of how symmetric the observations are

aroundthemean.Foradistributionskewedtotheright,theskewnesshas

a positive value while the opposite holds in case the observations are

skewedtotheleft.

Thesamplekurtosisisinsteaddefinedas:

-28-

𝐾 = \Z∙ (�©��)³

°©±�(¡�)�

(1.34)

Kurtosis gives a measure of the thickness in the tails of a probability

densityfunction.Foranormaldistributionthekurtosisis3.

Additionally,theExcessKurtosisisdefinedas:

𝐸𝐾 = 𝐾 − 3(1.35)

It follows that, for a normal distribution, the excess kurtosis is 0. A fat-

tailedor thick-taileddistributionhasavalue forkurtosis thatexceeds3.

Thatis,excesskurtosisispositive.Thisiscalledleptokurtosis.

ItisnowpossibletopresenttheJarque-Beratestfornormality.

Thetestissetbyformulatingthenullhypothesis:

H0:skewnessandexcesskurtosisarezero

againstthealternativehypothesis:

H1:non-normaldistribution.

TheJarque-Berateststatisticis:

𝐽𝐵 = 𝑛 ∙ ��

¶+ ·¸ �

�¹(1.36)

This test statistic canbe comparedwitha chi-squaredistributionwith2

degrees of freedom. The null hypothesis of normality is rejected if the

calculatedteststatisticexceedsacriticalvaluefromthe𝜒(�)� distributions.

Criticalvaluescanbechosenbystatingthepreferredsignificancelevelfor

the hypothesis test. The significance level, generally indicated by the

-29-

greek letter𝛼, represents theprobabilityof rejecting thenull hypothesis

whenitisindeedtrue.

1.3.4APPLICATIONOFTHEJ-BTESTItisnowpossibletoapplyaJarque-Beratestinordertoverifywhetherthe

returnsoftheindexesarenormallydistributed.Thesignificancelevelwill

again be of the five percent and the calculation will be performed in

MATLAB.

Theresultsareherebysummarizedbythefollowingtable:

INDEX RESULTOFJ-BTEST(𝜶 = 𝟎. 𝟎𝟓)

SPXINDEX REJECT

CCMPIndex REJECT

DAXIndex REJECT

SASEIDXIndex REJECT

NKYIndex REJECT

CACIndex REJECT

UKXIndex REJECT

FTSEMIBIndex REJECT

INDEXCFIndex REJECT

SHCOMPIndex REJECT

SMIIndex REJECT

AS51Index REJECT

SPTSXIndex REJECT

MXBRIndex REJECT

MatlabcodeavailableintheMatlabAppendix.

Againthestatisticaltestrefusesthehypothesisthatthedistributionofthe

indexesreturnsissimilartoanormaldistributionthereforeimplyingthat

thedistributionofthepricescannotbeconsideredalog-normalone.Itis

-30-

possible to highlight some of themain reasons behind the failure of the

log-normalmodel.

1.3.5FAILURESOFTHELOG-NORMALMODELThefailureofthelog-normalmodelincorrectlypredictingthedistribution

ofthestockreturnsisawidelyacceptednotionnowadays.Thereare,asa

matter of facts, a number of properties empirically experienced that are

hardlyreplicablebyastochasticprocess.Itispossibletolistsomeofsuch

properties inordertoprovideanoverviewof themaincharacteristicsof

returnsthatarehardtoincorporateinastochasticmodel.

• HeavyTails:thedistributionofreturnsgenerallydisplaysanheavy

tailwithpositiveexcesskurtosis.Suchabehaviourisnotcorrectly

reproducedbyanormaldistribution,moreoverthepreciseformof

thetailsishardtodetermineatall.

• Absence of autocorrelations: linear autocorrelations of assets

returnsaremostly insignificant,exceptionmadeforsmall intraday

timewindowswhereinsteadsomeformsofautocorrelationcanbe

observed.

• Gain/Loss asymmetry: drawdowns in stock prices and indexes

valuestendtobelargerthanupwardmovementsthereforecausing

asymmetricmovementsinthetwodirections.

• Volatilityclustering:volatilityclusteringisatermusedtoindicate

the fact that largechanges instockreturns tend tobe followedby

large changes, of either sign, while small changes tend to be

followed by small changes. This situation has also a quantitative

reflection.Whilecorrelationismostlyabsentamongreturns,thatis

not true for absolute returns. Absolute returns, defined as 𝑟((∆) ,

displayapositiveandsignificantautocorrelationfunction.

-31-

Themainresultofthoseeventsconsistsintheimpossibilityofreplicating

themovementofthestockpricesbyacontinuousmodel.Modelsassuming

thatpricesmoveinacontinuousmannerneglecttheabruptmovementsin

whichmostoftheriskisconcentrated.

-32-

ThepicturesrepresentthehistogramsofthereturnsandthepricesoftheindexSPX

plottedagainstanormaldistributionandalognormaldistributionrespectively.As

itcanbeseenthedistributionsdon’tseemtofitwellthedata.

-33-

CHAPTER2

2.1PORTFOLIOTHEORY

Portfoliotheorydealswiththeproblemofbuildingadesirableinvestment

outofacollectionofassets.Theobjectiveistoconstructaportfoliowhich

featuressatisfy thedemandof the financialagentwho isconsidering the

investmentitself.

The management of a portfolio is a fundamental aspect in modern

economics and finance. The first attempt to solving a portfolio problem

was the mean-variance approach introduced by H. Markowitz in a one-

perioddecisionmodel.Thesimplicityoftheapproach,causedbythestatic

nature of the problem it tackles, brings many drawbacks since the

investor’sjobislimitedtotheselectionoftheinitialportfolio.

As a matter of fact, after the initial selection procedure, the investor

becomeapassiveagentashecanonlywatchthepricesfluctuatewithout

any intervention possibility. It is however relevant to introduce the

Markowitzmodelbeforethediscussionofamoreadvancedintertemporal

model.

2.2MARKOWITZPORTFOLIO

Themean-varianceparadigmofMarkowitzisdefinitelythemostcommon

formulationoftheportfoliochoiceproblems.IttakesintoconsiderationN

riskyassetswitharandomreturnvectorRt+1andasingularriskfreeassets

withcertainreturnsRft.Theexcessreturns,indicatedasrt+1,aremadeby

the difference between Rt+1 and Rft and their conditional means and

-34-

covariancematrixareindicatedbyµtand∑trespectively.Inaddition,the

excessreturnsarealsoconsideredi.i.d.withconstantmoments.

It isnowfirstnecessarytotemporarilyeliminatetheriskfreeassetfrom

theproblemenvironment.That is, it isnecessary to supposean investor

whichcanonlyallocatehiswealthtotheNriskysecuritiesavailable.Inthe

absenceofarisk-freeasset , themean-varianceproblemis tochoosethe

vector of portfolio weights x, which indicate the investor’s relative

allocationsofwealthtothevariousriskyassets, inordertominimizethe

variance of the resulting portfolio return Rp,t+1 = x’Rt+1 while also

generating the predetermined goal of expected return Rft+µ. That is as

saying:

min�var[𝑅½,¾k\] = x’∑x(2.1)

Subjectto

𝐸 𝑅Â,(k\ = 𝑥Ã 𝑅Ä + µ = 𝑅Ä + µ and 𝑥� = 1��[\ (2.2)

Thefirstconstrainthastheroleofensuringthat theexpectreturnof the

portfolioisequaltothedesiredtargetwhilethesecondconstraintensures

that all wealth is invested in the risky assets. After setting up the

Lagrangian and solving the resulting first-order conditions, the optimal

portfolioweightsturnouttobe:

𝑥∗ = Λ\ + Λ�µ(2.3)

with

Λ\ =\Ç[𝐵 𝜄�\

. − 𝐴 µ�\. ]andΛ� =

\Ç[𝐶 µ) − 𝐴( 𝜄�\

.�\. ](2.4)

-35-

where𝜄denotes an appropriately sized vector of ones and where A

=𝜄′ µ�\. ,B=µ′ µ�\

. ,C=𝜄′ 𝜄�\. andD=BC-A2.

Thevarianceofthisportfolio,whichisthelowestvariancepossiblegiven

thefixedexpectedreturn,isequaltox*’∑x*.

The Markowitz paradigm has the virtue of highlighting two important

economic insights. The first is the effect of diversification, that is the

possibility of packaging imperfectly correlated assets into portfoliowith

betterexpectedreturn-riskcharacteristics.

The second one is the fact that once a portfolio is fully diversified, it is

possibletoachievehigherexpectedreturnsonlyatapriceoftheburdenof

ahigherrisk,thatisbyadoptingmoreextremeallocationsoftheportfolio

weights. These two relevant insights are also graphically visible.

The mean-variance frontier is here plotted as a hyperbola where every

point represent the minimized portfolio return volatility for a

predeterminedexpectedportfolioreturns.As itcanbeseen,portfoliosof

theassetsgeneratebetterrisk-returnperformancesanditisnecessaryto

movetotheright,thatistoincreasethevolatility,inordertoobtainbetter

returns.

-36-

It is now possible to reintroduce the risk-free asset in the model.

Introducingtherisk-freeassetsintroducesalsothepossibilityofunlimited

risk-freeborrowingandlendingattherisk-freerateRft.

Due to the addition of the risk-free asset, any portfolio on the mean-

variance frontier generated by the risky asset and represented by the

hyperbola can now be combined with the risk-free asset in order to

generateanewexpectedreturn-riskprofile.

Thenewlycreatedprofileliesonastraightlineoriginatingfromthepoint

indicated a risk-free portfolio, that is a portfolio fully composed of risk-

freeassets,andtangenttotheefficientfrontier.Theoptimalcombination

oftheriskyfrontierportfolioswithrisk-freeborrowingandlendingisthe

onemaximizingtheSharperatioofthewholeportfolio.TheSharperatiois

definedas:

·[ËÌ,ÍÎ�]c(u[ËÌ,ÍÎ�]

(2.5)

andcoincidewiththeslopeofthelinestartingfromtherisk-freeassetand

tangenttotheefficientfrontierofonlyriskyassets.

That line therefore represents theefficient frontierwhenborrowingand

lendingisallowedandiscomposedbycombinationsoftherisk-freeasset

andtheportfolioofriskyassettangenttothenewfrontier.

In the presence of a risky asset, the investor devoted a fraction x of his

wealthtotheriskyassetsandtheremainingwealth,indicatedby(1-𝜄′x),is

allocatedtotherisk-freeassets.

Theportfolioreturnisthereforeaweightedaverageofthereturnsofthe

risk-freeassetandofthetangencyportfolioofriskyassets.Inotherterms,

thereturnofthenewlycomposedportfoliois:

𝑅Â,(k\ = 𝑥Ã§ÍÎ� + 1 − 𝜄Ã𝑥 𝑅(Ä = 𝑥ÃËÍÎ� + 𝑅(

Ä(2.6)

-37-

and the mean-variance problem can be expressed in terms of excess

returns:

min�𝑣𝑎𝑟 𝑟Â = 𝑥Ã∑𝑥subjectto𝐸 𝑟Â = 𝑥Ãµ = µ(2.7)

Thesolutiontothisproblemisrepresentedby:

𝑥∗ = yyÃ y��.

Ð

× µ�\. (2.8)

where𝜆is a constant that scales proportionately all the elements in

µ�\. inordertoachievethepreferredportfolioriskpremiumµ.Fromthis

expression it is possible to find theweights of the tangencyportfolio by

noting that their summust be equal to one. Therefore, for the tangency

portfolio:

𝜆(ÒÓ =\

ÔÃ y��.andµ(ÒÓ =

yÃ y��.

�Õ y��.(2.9)

ItisnotdifficulttoidentifythereasonswhytheMarkowitzparadigmisan

appealing one. It captures the two essential aspects of portfolio choice,

diversificationandrisk-rewardtrade-off.

However,severalobjectionscouldbeposedtotheparadigm.Themainone

could be that the mean-variance problem is a myopic single-period

problem in which it is impossible to rebalance the portfolio during the

investmenthorizon.

-38-

2.3PREFERENCESANDRISKAVERSIONS

Inafinancialmarketwhereinvestorsarefacinguncertaintyit isrelevant

to include in the portfolio choice problem also the preferences of the

investor.VonNeumannandMorgensternhaveshownthatitispossibleto

represent the preferences of an individual who knows the probability

distributionoftherandomreturnsbyandexpectedutilitycriterion.More

precisely,denotingby!thepreferenceorderonthesetofrandomreturns,

it is possible to state that ! satisfies the Von Neumann-Morgenstern

criterion if there exists some increasing function U fro R into R, called

utilityfunction,suchthat:

𝑋\ ≻ 𝑋� ⟺ 𝐸 𝑈 𝑋\ > 𝐸[𝑈 𝑋� ](2.10)

The increasingpropertyof theutility function indicates that the investor

prefers more wealth to less wealth. The choice of the utility function

allows toapply thenotionsof riskaversionandriskpremiumstemming

fromthefundamentaluncertainty.

• Riskaversionandconcavityoftheutilityfunction

It is legitimate to consider an investor who dislikes risk. Therefore, in

respecttoarandomreturnX,hewillprefertoreceivewithcertaintythe

expectation E[X] of the investment return. This means that his utility

functionwillsatisfytheJensen’sinequality:

𝑈(𝐸 𝑋 ) ≥ 𝐸[𝑈 𝑋 ](2.11)

whichstandstrueonlyforconcavefunctions.Indeed,ifarandomreturnX

taking values x with probability𝜆 ∈ 0,1 and x’ with probability 1-𝜆is

chosen,itwillresultsthat:

-39-

𝑈 𝜆𝑥 + 1 − 𝜆 𝑥Ã ≥ 𝜆𝑈 𝑥 + 1 − 𝜆 𝑈(𝑥Ã)(2.12)

afactthatshowstheconcavityoftheutilityfunctionU.

• Degreeofriskaversionandriskpremium

Forarisk-averseagentwithaconcaveutilityfunctionU,theriskpremium

associatedtoarandomportfolioreturnXisdefinedasthepositiveamount

π=π(X) that the agentwould pay in order to receive a certain gain. It is

definedbytheequation:

𝑈 𝐸 𝑋 − 𝜋 = 𝐸[𝑈 𝑋 ](2.13)

Thequantityℰ 𝑋 = 𝐸 𝑋 − 𝜋iscalledthecertaintyequivalentofXandis

smallerthantheexpectationofX.

Denoting𝑋 = 𝐸[𝑋]andsupposingthattheportfolioreturnXislowlyrisky,

thefollowingapproximationcanbeobtained:

𝑈 𝑋 ≈ 𝑈 𝑋 + 𝑋 − 𝑋 𝑈Ã 𝑋 + \�(𝑋 − 𝑋)�𝑈′′(𝑋)(2.14)

andsobytakingexpectation:

𝐸 𝑈 𝑋 ≈ 𝑈 𝑋 + 𝑉𝑎𝑟(𝑋) ÜÃÃ(Ý)�(2.15)

-40-

itwillalsoresultthat

𝑈 𝑋 − 𝜋 ≈ 𝑈 𝑋 − 𝜋𝑈′(𝑋)(2.16)

whichgivestheapproximationfortheriskpremium

𝜋 ≈ − ÜÕÕ Ý�ÜÕ Ý

𝑉𝑎𝑟(𝑋)(2.17)

Hence,thecertaintyequivalentofXisgivenapproximatelyby

ℇ 𝑋 = 𝐸 𝑋 − \�𝛼 𝑋 𝑉𝑎𝑟(𝑋)(2.18)

whereα(x)isdefinedasthelocalabsoluteriskaversionatthereturnlevel

x:

𝛼 𝑥 = −ÜÃÃ(�)ÜÃ(�)

(2.19)

Thecoefficient𝛼isalsocalledtheArrow-Prattcoefficientofabsoluterisk

aversionofUatlevelx.

𝛼(𝑋)can therefore be considered as the factor by which an economic

agent with utility function U weights the risk and as the factor grows

largersomust theexpectationof returns inorder tocompensate for the

risk.Whenapplyingutilitytheory,itispossibletoconsiderthevarianceof

theportfolioasagoodindicatoroftheriskbeingfacedbytheinvestor.

-41-

Itisalsopossibletowritetherandomreturn𝑋as𝑋 = 𝑋(1 + 𝜀)where𝜀is

interpreted as the relative payoff of the return𝑋with respect to𝑋, and

definetherelativeriskpremium𝜌of𝑋as:

𝑈 𝑋 1 − 𝜌 = 𝐸 𝑈 𝑋 = 𝐸[𝑈 𝑋 1 + 𝜀 ](2.20)

Therelativeriskpremiumis interpretedastheproportionofreturnthat

the investor isreadytopayinordertoreceiveacertainpayoff.As ithas

happenedbefore,theriskpremium𝜌canbeapproximatedas:

𝜌 ≈ \�𝛾 𝑋 𝑉𝑎𝑟(𝜀)(2.21)

Where

𝛾 𝑥 = − �ÜÃÃ(�)ÜÃ(�)

(2.22)

Istherelativeriskaversionatlevelx.

• Commonutilityfunctions

o Constant Absolute Risk Aversion (CARA): 𝛼(𝑥) equals a

constant𝛼>0.

Since𝛼 𝑥 = −(lnUÃ)′(x),itfollowsthat𝑈 𝑥 = 𝑎 − 𝑏𝑒�â�.

Byusinganaffinetransformation,Ucanbenormalizedto

-42-

𝑈 𝑥 = 1 − 𝑒�â�(2.23)

o Constant Relative Risk Aversion (CRRA): 𝛾(𝑥) equals a

constant𝛾 ∈ 0,1 .

Duetoaffinetransformations,itispossibletoobtainthat

𝑈 𝑥 =ln 𝑥 ,forγ = 1

��ä

\�å,𝑓𝑜𝑟0 < 𝛾 < 1(2.24)

ExampleofaCARAutilityfunction

0 10 20 30 40 50 60 70 80 90 100x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

u(x)

-43-

ExampleofaCRRAutilityfunction

2.4MERTONPORTFOLIOPROBLEM

Another approach to selection of a portfoliowhich includes for the risk

aversion of the investor and is not subject to the static nature of the

MarkowitzparadigmwasintroducedbyRobertMertonin1969.

The scenario considered byMertonwas onewhere an investor had the

limitedchoiceofinvestinghiswealthinonlytwodifferentassets:arisky

asset and a risk-free asset. Given a limited time horizon, the goal of the

0 10 20 30 40 50 60 70 80 90 100x

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5u(x)

-44-

investor,who is risk averse,was tomaximize the expected utility of his

wealthattheendofthetimespantakenintoconsideration.

Merton’s goal was to determine how the investor should allocate and

reallocatehiswealth at each timepoint inorder to reach thepreviously

selectedgoal.

Inordertoexplainthesolutiontotheproblemitisnecessarytorecallthe

previously introduced dynamics concerning the price of the risky asset.

ThepriceoftheriskyassetwillthereforebedenotedasStattimet.

Theparametersµandσrepresentrespectivelythedriftandthevolatility

oftheriskyasset.Thepriceoftherisk-freeassetattimetisdenotedbyRt

andsatisfiesthefollowingdeterministicdifferentialequation:

𝑑𝑅( = 𝑟𝑅(𝑑t(2.25)

The parameter r stands for the risk-free continuously compounding

interestrate. Inthissetting, it iseasytoassumethat𝐸[𝑆(] > 𝐸[𝑅(]which

alsoindicatesthatµ > 𝑟.

Itisnownecessarytointroduceinthemodelthewealthoftheinvestorat

time𝑡, denoted by Vt. At each time point𝑡the investor must allocate a

fractionutofhiswealthintheriskyasset.

The remainingwealth 1-ut is invested in the risk-free asset. Thismeans

thatthevalueoftheriskyinvestmentattimetisutVtandthatthevalueof

therisk-freeinvestmentis(1 − 𝑢()𝑉( .Thestochasticdifferentialequation

ofthewealthorportfoliovalueistherefore:

𝑑𝑉( = 𝑑𝑢(𝑉( + 𝑑 1 − 𝑢( 𝑉( = µ𝑢(𝑉(𝑑𝑡 + 𝜎𝑢(𝑉(𝑑𝐵( + 𝑟 1 − 𝑢( 𝑉(𝑑𝑡 =

µ𝑢( + 𝑟 1 − 𝑢( 𝑉(𝑑𝑡 + 𝜎𝑢(𝑉(𝑑𝐵( .

-45-

Thegoalisnowtodecidetheoptimalallocationstrategyforutateachtime

point t in order to obtain the best possible outcome at some future

terminaltimeTfortheinvestor.

However,asithasbeenpreviouslydiscussed,aninvestorisnotconcerned

with wealth maximization per se but with utility maximization. It is

thereforepossibletointroduceanincreasingandconcaveutilityfunction

U(x)representingtheexpectedutilityofariskaverseinvestor.

Thegoaloftheproblemisnotanymoretomaximizetheexpectedportfolio

value but tomaximize the expected utility stemming from thewealth at

theterminaltimeT.

Ifatimehorizonrestrictedbyaninitialtimet0andterminaltimeTandan

initial portfolio value Vt0 are assumed, then it is possible to state the

maximizationproblemas:

𝐼 𝑡, 𝑥 = maxçÍ

𝐸[ 𝑈 𝑉� |𝑡n = 𝑡, 𝑉(n = 𝑥](2.26)

Thisconstitutesanoptimalcontrolproblem,wheretheallocationstrategy

utistheactualcontrolfunction.Defining

𝜙 𝑡, 𝑥 =𝜕𝐼(𝑡, 𝑥)𝜕𝑡 + µ𝑢( + 𝑟 1 − 𝑢(

𝜕𝐼 𝑡, 𝑥𝜕𝑥 +

12𝜎

�𝑢(�𝑥�𝜕�𝐼(𝑡, 𝑥)𝜕𝑥�

=𝜕𝐼(𝑡, 𝑥)𝜕𝑡 + 𝑟 + µ − 𝑟 𝑢(

𝜕𝐼 𝑡, 𝑥𝜕𝑥 +

12𝜎

�𝑢(�𝑥�𝜕�𝐼(𝑡, 𝑥)𝜕𝑥�

(2.27)

Theoptimalsolutionmustsatisfy

maxçÍ

𝜙 𝑡, 𝑥 = 0, 𝑡 ∈ [𝑡n, 𝑇],(2.28)

where𝜙indicates the cumulative distribution function of the standard

normaldistribution,

-46-

and

𝐼(𝑇, 𝑉�) = 𝑈(𝑉�).(2.29)

Thismaximizationproblem is a continuous-timeversionof theBellman-

Dreyfus fundamental equationofoptimality.This requirement alsogives

theoptimalsolutiontotheproblem.

Tofindasolutionthatiscompatiblewiththeutilityfunction𝑈(𝑥),andthat

thereforeisincreasingandconcave,itisrequiredthat

𝐼� =�ê((,�)��

>0and𝐼�� =��ê((,�)��

< 0.(2.30)

Also,afirstorderconditionforfindingamaximumis

µ − 𝑟 𝐼� + 𝜎�𝑢(𝑥𝐼�� = 0(2.31)

whichisequivalentto

𝑢( = − y�Ë êë¡��êëë

(2.32)

Ifthisequationissubstitutedintheequationfor𝜙(𝑡, 𝑥)ithappensthat:

𝐼( + 𝑥 𝑟 + µ − 𝑟 −µ − 𝑟 𝐼�𝜎�𝑥𝐼��

𝐼�𝑡 < 𝑇

+12𝜎

�(−µ − 𝑟 𝐼�𝜎�𝑥𝐼��

)�𝑥�𝐼�� = 0

𝐼 𝑡, 𝑥 = 𝑈 𝑥 ,𝑡 = 𝑇

↔ 𝐼( + 𝑟𝑥𝐼� −µ − 𝑟 �𝐼��

𝜎�𝐼��+12µ − 𝑟 �𝐼��

𝜎�𝐼��= 0,𝑡 < 𝑇

𝐼 𝑡, 𝑥 = 𝑈 𝑥 ,𝑡 = 𝑇

-47-

↔ 𝐼( + 𝑟𝑥𝐼� −µ − 𝑟 �𝐼��

2𝜎�𝐼��= 0,𝑡 < 𝑇

𝐼 𝑡, 𝑥 = 𝑈 𝑥 ,𝑡 = 𝑇

(2.34)

with

𝐼( =�ê((,�)�(

.(2.35)

2.4.1POWERUTILITY

InthesolutiontotheMerton’sportfolioproblemutilityfunctionhasbeen

left as an unknown functionU(x). It is now useful to identify a function

whichcouldbeusedinplaceofthegenericnotationU(x).

A classic solution to the problem is the use of the power function to

indicatetheutilityofwealthx.Inparticular,itcouldbesaidthat

𝑈 𝑥 = 𝑥å,0 < 𝛾 < 1(2.36)

The use of this utility function is coherent with assumption previously

made concerning the increasing and concave characteristics of the

function.Itispossibletoreferto𝛾astheriskaversionparameter.

Alowvalueoftheriskaversionparameterisassociatedwithhighaversion

toriskandviceversa.Inordertofindasolutiontotheproblemusingthe

newly introduced power utility function, it is necessary to insert the

equation𝐼 𝑡, 𝑥 = 𝑓(𝑡)𝑥åinthefinalresultoftheprevioussection.

Thatleadstothefollowingcalculations:

-48-

𝑓Ã 𝑡 𝑥å + 𝑟𝑥𝑓 𝑡 𝛾𝑥å�\ −µ − 𝑟 �𝑓� 𝑡 𝛾�𝑥� å�\

2𝜎�𝑓 𝑡 𝛾 𝛾 − 1 𝑥å�� = 0,𝑡 < 𝑇

𝑓 𝑡 𝑥å = 𝑥å,𝑡 = 𝑇

↔ −𝑓Ã 𝑡𝑓 𝑡 = 𝑟𝛾 +

µ − 𝑟 �𝛾2𝜎� 1 − 𝛾 ,𝑡 < 𝑇

𝑓 𝑡 = 1,𝑡 = 𝑇.

(2.37)

Solvingtheseequationswithrespectto𝑓(𝑡)yields

𝑓 𝑡 = exp 𝑟𝛾 + y�Ë �å�¡� \�å

𝑇 − 𝑡 (2.38)

Substitutingthissolutionintotheequation𝐼 𝑡, 𝑥 = 𝑓(𝑡)𝑥åitresultsthat

𝐼 𝑡, 𝑥 = exp 𝑟𝛾 + y�Ë �å�¡� \�å

𝑇 − 𝑡 𝑥å(2.39)

Finally,itispossibletofindtheoptimalcontrolut*bysolvingtheequation

𝑢( = − y�Ë êë¡��êëë

(2.40)

withrespecttothatlastequationfor𝐼(𝑡, 𝑥).

Theresultofthatoperationis

𝑢(∗ = −y�Ë íî½ Ëåk ï�ð �ä

�ñ� ��ä��( å�ä��

¡��ò�Â Ëåk ï�ð �ä�ñ� ��ä

��( å å�\ �ä��= y�Ë

¡�(\�å)(2.41)

-49-

which is a constant independent from the timevariable. It is possible to

concludethattheoptimalallocationstrategyistoholdaconstantfraction

u*of thewealth in theriskyasset,andhence,aconstant fraction1-u* in

therisk-freeasset.

Theratio

y�Ë

¡�(\�å)(2.42)

isalsoknowbythenameofMertonratio.

Thenumeratoroftheratio isthedifferencebetweentheriskyassetdrift

andtherisk-freerateofreturn.Inthecasethenumeratorisnegative,and

thereforerisbiggerthanµ,theinvestorwillallocateallhiswealthtothe

risk-free asset, as it is logical given that it would offer a higher return

withoutbearingarisk.

Inthecaseinsteadthatµisbiggerthanr,theinvestorwillinvestatleasta

fractionofhiswealthontheriskyasset.Thisfractionispartlydetermined

bythesizeofthedifferencebetweentheriskyassetdriftandtherisk-free

denominator.Thedenominator is theproductbetween thesquareof the

volatilityoftheriskyassetandoneminustheriskaversioncoefficient.

Aparticular featureof theMertonratio isgivenbythefact thattheratio

tends to decrease with an increase in volatility. This property of the

Mertonratioisanessentialonesinceitrepresentsthehigherreluctancein

investingintheriskyassetfromariskaverseagentinthecaseofahigher

volatility.Oneminustheriskaversionisinsteadascalingparameterthat

hastheroleofadjustingtheimpactofthevolatilityontheMertonratio.

Itcanbeseenhowalowvalueoftheparameter𝛾scalesuptheimpactof

volatility in relative terms and vice versa. That is another relevant

property since a low risk aversion parameter value must be associated

withhighriskaversion.

-50-

CHAPTER33.1 TESTOFPERFORMANCES

The aim of this third, and last, chapter is to apply the theoretical

frameworks introduced in the previous chapters and to test the

performances that they would have had if used in the real world. In

particular, this chapter will perform a confrontation between the

Markowitzmean-variancemodel and the portfolio solution provided by

Merton. The test will be performed on the same data set previously

introducedandwilldealspecificallywiththeSPXindex.SPXistheticker

bywhichtheStandards&Poor’s500isindicated.TheS&P500isbasedon

thecapitalizationof500largecompanieswhosestocksaretradedonthe

NYSEandNASDAQ.Allotherindexesquotedintheprevioussectioncould

have been used as well. The application will also show how the

compositionoftheportfoliointheMertonframeworkwillvaryastherisk

aversionoftheeconomicagentvaries.

Theproblemhererepresentedisofcourseabigsimplificationofthereal

world, since only one risky asset is taken into consideration. Moreover,

transactioncostswillbeleftoutofthemodel,anassumptionwhichcan’t

bemadeinrealityastransactioncostswillsignificantly limit theamount

ofrebalancingtheagentwilldoateachtimepoint.

It is now necessary to introduce the way in which the model has been

estimated.Thefirstimportantassumptiontomakeinordertorunthetwo

modelsistheexistenceofariskfreeasset.

3.2 RISK-FREEASSETS

Both theMarkowitz and theMerton portfolio problem solutions rely on

theexistenceofanassetdefinedasrisk-free.Thetermrisk-freemaymean

-51-

differentthingstodifferentsubjects.Theremaybeavarietyofobservable

ratesallofwhichmaybedescribedbysomeas“risk-free”.Themaingoal

in selecting an appropriate rate should be to pick a rate such that

valuationsofthemarketturnouttobeconsistent.Inthissense,therisk-

free rate may just stand for what could be otherwise described as the

“referencerate”.Agoodreferenceratemaythenbetheyieldavailableon

government debt. One justification for doing so is that the government

debt seems to fit more naturally the conventional meaning of the term

‘risk-free’,especiallyifthedebtisinthesamecurrencyoftheoneusedby

theagent.

It ishowevernoteworthy tounderlinehowdifferentmarketparticipants

may place different interpretations onwhat should be defined by “risk-

free” rate as the risks that they try to avoid may vary among them.

However, forthematterofthisapplication, itwillbesufficienttoholdto

common interpretationandassumethat therateavailableonshort-term

governmentbondsofthesamecountryoftheindexofreferenceisagood

approximationofarisk-freerate.

Anotherconsiderationconcerningrisk-freeratesmustbemadeinthelight

oftherecenteconomiccrisis.Since2008themarketitselfisasamatterof

fact questioning the level of risk behind government-backed securities.

During late 2008 the Bank of England started observe same anomalous

behaviour in the yield curves of Sterling-denominated UK government

debt.Itwasnoticedthatlong-termdebtwastradingatahigherratethan

overnightindexswapcontracts,aneventwhichindicatedthatthemarkets

wereattributinganon-zeromarketimpliedprobabilitytothegovernment

altering downwards previously ruling coupon rates on some of their

outstanding long-term debt. The same behaviour was spotted in other

Europeancountryandsignalledagrowingconcernaboutthelonger-term

creditworthiness of the governments of major developed countries. In

ordertoavoidsuchatrend,itisusefultoconsidershorttermdebtsinceit

-52-

hasbeenlessinfluencedbythismarketbehaviourduetoitslowerlevelof

expositiontofuturepoliticaldecisions.

SincetheindextakenintoaccountforthischapterwillbetheSPX,thatis

the S&P 500, an index based on themarket capitalizations of 500 large

companyhaving commonstock listedon theNYSEorNASDAQ, the risk-

free rate considered will be that of 13 weeks US government-backed t-

bills. Those are short term debit securities issued by the American

government.

Theratesthatwillbelaterusedaregatheredinthefollowingtable.

DATEOFISSUANCE 13WEEKSRATE

3rdJanuary2012 0.02

2ndJanuary2013 0.06

2ndJanuary2014 0.06

2ndJanuary2015 0.03

Source:U.S.DepartmentoftheTreasury

3.3 RISKAVERSIONCOEFFICIENTSThe Merton portfolio choice model, as it has been previously shown,

includes the utility of the economic agents into its framework. In

particular, in order to have the discretization of the problem shown in

equation (2.41) it is necessary to select a risk aversion coefficient. Risk

aversion is a commonly accepted notion in economics and is easily

recognizable in everyday life. Both investors and households would be

willing to give up at least a small fraction of their potential earnings in

ordertohaveaguaranteedincome.Intheaforementionedutilityfunction,

itcanbeeasilyseenhowriskaversionincreasesasthefactor𝛾decreases.

As a matter of fact, a value of𝛾close to one or equal to one would

therefore be an unpractical one as it would make the agent indifferent

betweencertainincomesandexpectedincomes.

-53-

Forthepurposeofthisapplication,differentvaluesof𝛾willbetakeninto

consideration. Those values have not been inferred from populations

studies, as it couldbepossible, since suchaneffort is beyond the scope.

However, they should be significant enough in understanding how

different relationships to risk by economic agents can influence their

approachtothemarket.

Theriskaversioncoefficientsusedaretherefore:

𝛾 ∈ 0.2; 0.3; 0.4; 0.5

3.4 PROCEDUREFORTHEMERTONPORTFOLIOHaving introducedtherisk freeratesandthevaluesofriskaversion it is

nowpossibletomoveontomorepracticalmatters.

Theapplicationof the twoportfolio theorieswillbedone ina simplified

way, as therewill be only one risky asset considered. The risky asset of

matter will be the SPX index, the index comprising the 500 American

companieswiththehighestcapitalization.

The Merton portfolio will be reallocated for different years and its

evolutionwill be reported in order to assesswhatwould have been the

finalreturnonahypotheticalinvestment.Giventhelengthofthedataset

introduced inChapterOne, the firstportfoliowill be createdon the first

availabledayoftheyear2012andthedatafromyear2000toyear2011

willbeusedasthebackgroundforthepurposeofestimatingthehistoric

expected returns and volatility of the market. As the portfolio will be

reallocated, the additional year of information will be merged into the

backgrounddataandusedforthehistoricestimations.

Thesameprocedurewillbedoneforthedifferentdegreesofriskaversion

selected so that in the end it will be possible to perform a comparison

withindifferenttimeperiodsanddifferenttypologyofinvestors.

-54-

The calculations will be performed on the MATLAB platform and the

commands used to complete the procedure will be reported in the

appendix.

3.5 APPLICATIONOFMERTONPORTFOLIOTHEORYTheMATLABcodesusedinthosecalculationareavailableintheappendix.

Forabetterclarity,only the findingsof theapplicationswillbereported

hereinthefollowingtables.

3.5.1 𝜸 = 𝟎. 𝟐

Ratioofportfolioonmarketassetfor

2012

36.73%

Returnofportfoliofor2012 4.73%


2013

59.38%



2014

100%



2015

100%


AnnualizedReturn 9.6%

3.5.2 𝜸 = 𝟎. 𝟑


2012

41.98%


Ratioofportfolioonmarketassetfor 67.87%

-55-

2013



2014

100%



2015

100%



3.5.3 𝜸 = 𝟎. 𝟒


2012

48.98%



2013

79.18%



2014

100%



2015

100%



3.5.4 𝜸 = 𝟎. 𝟓


2012

58.77%


Ratioofportfolioonmarketassetfor 95.01%

-56-

2013



2014

100%



2015

100%



3.6 PROCEDUREFORTHEMARKOWITZPORTFOLIO

Theprocedureused in theapplicationof theMarkowitzportfolio theory

willbeslightlydifferentfromtheoneusedintheMertonportfolio.Given

thedifferentnatureofthetwoportfolioproblem,theMarkowitzportfolio

won’tbe reallocatedover thecourseof timebutwill insteadbe fixedon

thefirsttradingdayof2012andhelduntiltheendof2015.Theriskfree

ratesusedherewillbethesameastheoneusedinprecedence.

However, another relevant difference must be highlighted. The main

investorcharacteristicbywhichtheMertonportfoliochoiceisinfluenced

is the investorriskaversion,a factorwhichaffects the finalchoiceof the

portfolio. In the Markowitz case the subjectivity of the investor is

incorporated through the selection of the preferred volatility rate or

return rate. As thematter of fact, it is impossible to determine an exact

portfolioallocationwithoutfirstfixingoneofthosetwoconstraints.

Asitwasdonebefore,thoseconstraintswillbearbitrarilyplacedinorder

toshowhowtheallocationoftheportfoliomayvaryastheriskaccepted

toburdenbytheinvestordecreasesorincreases.

Thevolatilityratesselectedfortheportfolioareherebysummarized:

-57-

AnnualPortfolioAcceptedVariance

5%

10%

15%

17.50%

3.7 APPLICATIONOFMARKOWITZPORTFOLIO

3.7.1 𝝈 = 𝟓%

Ratioinvestedonmarketasset 22.48%

Ratioinvestedonrisk-freeasset 77.52%

Portfolioreturnfor2012 2.92%




Annualizedportfolioreturn 3.12%

3.7.2 𝝈 = 𝟏𝟎%


Ratioinvestedonrisk-freeasset 55.03%%






-58-

3.7.3 𝝈 = 𝟏𝟓%








3.7.4 𝝈 = 𝟏𝟕. 𝟓%








Thedifferentallocationsoftheportfolioscreatedwiththetwomodelscan

now be summarized in the following graph. The percentage of the risky

asset isreportedonthey-axisandthepercentageofrisk-freeasset ison

thex-axis.Asitcanbeeasilyseenalltheportfoliosreliesonastraightline

astheyaresimplylinearcombinationsofthetwoassetstypology.

-59-

3.8 ANALYSISOFTHETWORESULTS

Itisnowpossibletohighlightssomeofthedifferencesthathavearisenin

theresultsofthetwomodels.

Themaindifferenceconsists inthe incapacityof theMarkowitzmodel in

adapting to awell performingmarket. TheMertonmodel as amatter of

fact partly “recognize” the way in which the market was strongly

outperforming the t-bills returnsandallocatesmoreandmoreresources

onthemarketasset.Itisimportanttoalsospecifythatinthisapplication

short-selling was not allowed. In 2014 and 2015 the Merton model

suggestedsomeamountofborrowingforalltheinvestorsandthatisthe

reason behind the fact that the entire portfolios were allocated on the

marketassets.

Anotherdifferences stems from thehighervariations in theallocationof

theportfoliostemmingfromchangesinthepreferenceforvolatilityinthe

Markowitzmodel thanthoserecognizable forchanges inriskaversion in

theMertonmodel.

Foryear2012,thedifferencebetweenthepercentageofriskyassetsinthe

leastaverseagentandthemostaverseagentisaroundtwentypercentfor

the Merton model. When the Markowitz model is instead taken into

account,thesamedifferencestandsatmorethan50%.

Somemoreclarificationsareindeednecessariesconcerningtheresultsof

the two application. In particular, it is striking how the performances of

theSPXindexintheanalysedyearsareincrediblyhigherthanthehistoric

recordsforthesameasset.Moreover,itisalsorelevanttounderlinehow

substantialinmagnitudeisthedifferencebetweenthereturnsontherisk

freeasset and the riskyasset.While those characteristics arenot crucial

fortheconsistencyofthisthesis,theywillbebrieflyintroduced.

Itiseasytoidentify2013astheyearwheretheperformancesoftheS&P

500 indexwerehighenough to increasesensibly theperformanceof the

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entireportfolios.Suchastrongeffectisalsotheproductofabehaviourof

theappliedmodelwhichcouldbeneverexperiencedinrealityandthatis

thelackofdiversification.

Theuseofanon-diversifiedportfolioobviouslyexposestheinvestor,and

consequentlythewholeportfolio,toanoutlyingbehaviourofthemarket.

Thisisessentiallyinthecaseinthepreviousapplication.

In2013theStandard&Poor’s500Indexposteditsbiggestannualadvance

since 1997 as an increase in consumer confidence and housing prices

bolsteredtheAmericaneconomy.

TheS&P500jumped30percentin2013andendedtheyearatanall-time

highpostinga173percentincreasefromits12-yearlowreachedin2009.

Thefactthatsuchalowwasinthehistoricdatausedtoestimatethestock

average return is certainly another factor that has had an effect on the

applicationsofthemodels.

Additionalremarkableeventswerethefactthatallthe10mainindustries

intheindexconcludedtheyearwithapositiveincreaseandatotalof460

stockswereupduring2013,aconditionthathadn’thappensince1990.

Thenumberofthosecompaniesincludedastonishingperformancesfrom

Netflix,up298percent,Micron,up243percentandBestBuy,up237.

Inthelightoftheseevents,itiseasiertounderstandthereasonbehindthe

unexpectedresultsofthetwoportfoliosintheanalysedtimespan.

Another condition which is strongly different from a typical setting for

thosekindsofportfolioproblemisconstitutedbytheextraordinarilylow

ratesofthet-bills,whicharehereusedasaproxyforarisk-freerate.

Onthe16thofDecemberof2008theFederalReservetookthedecisionof

cutting the fundsrate to thebandof0 to0.25%,concludingacutof500

basis points over the course of the precedent year. This move was a

reaction to the deepening recession that the American economy was

experiencing. The aim of such a move from the FED was to stimulate

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growthbyprompting individuals andbusinesses to increase the level of

investmentandspending.

Theeffectofsuchapolicycanbeeasilyseenalsointheapplicationofthe

twoportfolio theories reported before. Given the incredibly low interest

rates,theriskpremiumfromaninvestmentinmarketstocksisincredibly

higherandthatiswidelymanifestedintheportfolioallocationforMerton

theoryduringtheyears2014and2015.Inthoseyears,asamatteroffact,

theMerton ratio suggests to short-sell risk free assets and invest on the

S&P500index,aconditionnotallowedinthissetting.

3.9 CONCLUSION

It is now possible to briefly sum up the topics discussed before in this

work.

In the first chapter the main theoretical frameworks behind returns

estimationandpredictionhavebeenintroduced.Asithasbeenseen,such

models hardly adapt well when applied to real world assets. However,

thosemodelsarethepillarsuponwhichmodernmathematical finance is

foundedandarestillusefulifproperlyused.Themainassumptionwhich

doesn’ttranslatetotherealfinancialworldisthenormalityofthereturn

distribution. Due to a number of reasons already explained such an

assumption isnot consistentwithempiricaldataandaproofof thathas

beenprovidedbytheJarque-Beratest.

Thesecondchapter introduces theportfolioproblemtheories thatareat

thecoreofthisthesis.TheMertonportfolioprobleminparticularhasbeen

introduced and themathematical framework behind has been explained

anddescribed.Inordertohaveadiscreteformofthesamemodel,apower

utilityfunctionhasbeenadoptedandpluggedintothemodel.Duetothis

choice,anexplicitsolutiontotheproblemcanbefound.

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In the third and last chapter the two portfolio problem solutions

introduced in the second chapter, the Merton portfolio and Markowitz

portfoliomodels,havebeenappliedtorealworlddata intheformof the

S&P500indexandtheU.S.13-weekst-bills.Thetwoapplicationsshowed

how theportfolio composition could vary as the type of economic agent

changed.Thischangewas firstlyrepresentedbydifferentdegreesofrisk

aversion and then by different preferred portfolio volatility. The

performance of the various portfolios has then been reported and

analysedbothonayear-by-yearbasisandthenonanoverallpointofview.

Themainaimof this thesiswastoprovidea first lookatmoreadvanced

formofportfolio selection theory than thebasicmean-varianceportfolio

optimization problem. Many factors could have been added in order to

further advance the theoretical framework, in particular the addition of

transaction costs and stochastic volatility would have helped in the

processoffurtheraligningthetheoreticalworldwiththerealone.

Even though those elements were not part of the work, the topics

introducedhere should still represent a valid foundationandare a solid

foundationuponwhichfurtherdevelopmentscouldbemade.

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MATLABAPPENDIX

KOLOMOGOROV-SMIRNOVTEST

The codes here reported have been used to perform the Kolmogorov-

Smirnov test in Chapter One. The main command here used is “kstest”

whichisthebuilt-inMatlabcommandforsaidtest.

standSPX=(ReturnsSPX-nanmean(ReturnsSPX))/nanstd(ReturnsSPX) hKSspx=kstest(standSPX) standUKX=(ReturnsUKX-nanmean(ReturnsUKX))/nanstd(ReturnsUKX) hKSukx=kstest(standUKX) standAS51=(ReturnsAS51-nanmean(ReturnsAS51))/nanstd(ReturnsAS51) hKSas51=kstest(standAS51) standCAC=(ReturnsCAC-nanmean(ReturnsCAC))/nanstd(ReturnsCAC) hKScac=kstest(standCAC) standCCMP=(ReturnsCCMP-nanmean(ReturnsCCMP))/nanstd(ReturnsCCMP) hKSccmp=kstest(standCCMP) standDax=(ReturnsDax-nanmean(ReturnsDax))/nanstd(ReturnsDax) hKSdax=kstest(standDax) standFTSEMIB=(ReturnsFTSEMIB-nanmean(ReturnsFTSEMIB))/nanstd(ReturnsFTSEMIB) hKSftsemib=kstest(standFTSEMIB) standINDEXCF=(ReturnsINDEXCF-nanmean(ReturnsINDEXCF))/nanstd(ReturnsINDEXCF) hKSindexcf=kstest(standINDEXCF) standMXBR=(ReturnsMXBR-nanmean(ReturnsMXBR))/nanstd(ReturnsMXBR) hKSmxbr=kstest(standMXBR) standNKY=(ReturnsNKY-nanmean(ReturnsNKY))/nanstd(ReturnsNKY) hKSnky=kstest(standNKY) standSASEIDX=(ReturnsSASEIDX-nanmean(ReturnsSASEIDX))/nanstd(ReturnsSASEIDX) hKSsaseidx=kstest(standSASEIDX) standSHCOMP=(ReturnsSHCOMP-nanmean(ReturnsSHCOMP))/nanstd(ReturnsSHCOMP) hKSshcomp=kstest(standSHCOMP) standSMI=(ReturnsSMI-nanmean(ReturnsSMI))/nanstd(ReturnsSMI) hKSsmi=kstest(standSMI) standSPTSX=(ReturnsSPTSX-nanmean(ReturnsSPTSX))/nanstd(ReturnsSPTSX) hKSsptsx=kstest(standSPTSX) standUKX=(ReturnsUKX-nanmean(ReturnsUKX))/nanstd(ReturnsUKX) hKSukx=kstest(standUKX)

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JARQUE-BERATEST

ThecodesherereportedhavebeenusedtoperformtheJarque-Beratestin

ChapterOne.Themaincommandhereusedis“jbtest”whichisthebuilt-in

Matlabcommandforsaidtest.

hJBspx=jbtest(ReturnsAS51) hJBcac=jbtest(ReturnsCAC) hJBCCMP=jbtest(ReturnsCCMP) hJBDax=jbtest(ReturnsDax) hJBFTSEMIB=jbtest(ReturnsFTSEMIB) hJBINDEXCF=jbtest(ReturnsINDEXCF) hJBMXBR=jbtest(ReturnsMXBR) hJBNKY=jbtest(ReturnsNKY) hJBSASEIDX=jbtest(ReturnsSASEIDX) hJBSHCOMP=jbtest(ReturnsSHCOMP) hJBSMI=jbtest(ReturnsSMI) hJBSPTSX=jbtest(ReturnsSPTSX) hJBSPX=jbtest(ReturnsSPX) hJBUKX=jbtest(ReturnsUKX)

MERTONPORTFOLIOS

Thecodesherereportedarethoseusedtoestimatethecompositionofthe

portfolios under theMerton framework. First the historicalmoments of

theassethavebeenestimatedand then later theyhavebeen inserted in

theexplicitsolutionfoundinChapterTwo.Theprocesshasbeendonefour

times for each portfolio and the historical moments have been updated

everytimeinordertoincorporatetheadditionalinformationsprovidedby

thepreviousmarketyear.

SPX2011return=price2ret(SPX2011,[],'Periodic') MeanRet2011=((1+(nanmean(SPX2011return))).^252)-1 VAR2011=((1+var(SPX2011return)).^252)-1 RiskFree2012=((1.0002).^4)-1 MertonRatio201104=(MeanRet2011-RiskFree2012)./(VAR2011*(1-0.4)) MarketReturn2012=price2ret(SPXONLY2012,[],'Periodic') MeanRet2012ONLY=((1+(nanmean(MarketReturn2012))).^252)-1

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PortfolioReturn201204=(MertonRatio201104*MeanRet2012ONLY)+((1-MertonRatio201104)*RiskFree2012) SPX2012return=price2ret(SPX2012,[],'Periodic') MeanRet2012=((1+(nanmean(SPX2012return))).^252)-1 VAR2012=((1+var(SPX2012return)).^252)-1 RiskFree2013=((1.0006).^4)-1 MertonRatio201204=(MeanRet2012-RiskFree2013)./(VAR2012*(1-0.4)) MarketReturn2013=price2ret(SPXONLY2013,[],'Periodic') MeanRet2013ONLY=((1+(nanmean(MarketReturn2013))).^252)-1 PortfolioReturn201304=(MertonRatio201204*MeanRet2013ONLY)+((1-MertonRatio201204)*RiskFree2013) SPX2013return=price2ret(SPX2013,[],'Periodic') MeanRet2013=((1+(nanmean(SPX2013return))).^252)-1 VAR2013=((1+var(SPX2013return)).^252)-1 RiskFree2014=((1.0006).^4)-1 MertonRatio201304=(MeanRet2013-RiskFree2014)./(VAR2013*(1-0.4)) MarketReturn2014=price2ret(SPXONLY2014,[],'Periodic') MeanRet2014ONLY=((1+(nanmean(MarketReturn2014))).^252)-1 PortfolioReturn201404=(1*MeanRet2014ONLY)+((1-1)*RiskFree2014) SPX2014return=price2ret(SPX2014,[],'Periodic') MeanRet2014=((1+(nanmean(SPX2014return))).^252)-1 VAR2014=((1+var(SPX2014return)).^252)-1 RiskFree2015=((1.0003).^4)-1 MertonRatio201404=(MeanRet2014-RiskFree2015)./(VAR2014*(1-0.4)) MarketReturn2015=price2ret(SPXONLY2015,[],'Periodic') MeanRet2015ONLY=((1+(nanmean(MarketReturn2015))).^252)-1 PortfolioReturn201504=(1*MeanRet2015ONLY)+((1-1)*RiskFree2015) SPX2011return=price2ret(SPX2011,[],'Periodic') MeanRet2011=((1+(nanmean(SPX2011return))).^252)-1 VAR2011=((1+var(SPX2011return)).^252)-1 RiskFree2012=((1.0002).^4)-1 MertonRatio201105=(MeanRet2011-RiskFree2012)./(VAR2011*(1-0.5)) MarketReturn2012=price2ret(SPXONLY2012,[],'Periodic') MeanRet2012ONLY=((1+(nanmean(MarketReturn2012))).^252)-1 PortfolioReturn201205=(MertonRatio201105*MeanRet2012ONLY)+((1-MertonRatio201105)*RiskFree2012) SPX2012return=price2ret(SPX2012,[],'Periodic') MeanRet2012=((1+(nanmean(SPX2012return))).^252)-1 VAR2012=((1+var(SPX2012return)).^252)-1 RiskFree2013=((1.0006).^4)-1 MertonRatio201205=(MeanRet2012-RiskFree2013)./(VAR2012*(1-0.5)) MarketReturn2013=price2ret(SPXONLY2013,[],'Periodic') MeanRet2013ONLY=((1+(nanmean(MarketReturn2013))).^252)-1 PortfolioReturn201305=(MertonRatio201205*MeanRet2013ONLY)+((1-MertonRatio201205)*RiskFree2013) SPX2013return=price2ret(SPX2013,[],'Periodic') MeanRet2013=((1+(nanmean(SPX2013return))).^252)-1 VAR2013=((1+var(SPX2013return)).^252)-1 RiskFree2014=((1.0006).^4)-1 MertonRatio201305=(MeanRet2013-RiskFree2014)./(VAR2013*(1-0.5))

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MarketReturn2014=price2ret(SPXONLY2014,[],'Periodic') MeanRet2014ONLY=((1+(nanmean(MarketReturn2014))).^252)-1 PortfolioReturn201405=(1*MeanRet2014ONLY)+((1-1)*RiskFree2014) SPX2014return=price2ret(SPX2014,[],'Periodic') MeanRet2014=((1+(nanmean(SPX2014return))).^252)-1 VAR2014=((1+var(SPX2014return)).^252)-1 RiskFree2015=((1.0003).^4)-1 MertonRatio201405=(MeanRet2014-RiskFree2015)./(VAR2014*(1-0.5)) MarketReturn2015=price2ret(SPXONLY2015,[],'Periodic') MeanRet2015ONLY=((1+(nanmean(MarketReturn2015))).^252)-1 PortfolioReturn201505=(1*MeanRet2015ONLY)+((1-1)*RiskFree2015) SPX2011return=price2ret(SPX2011,[],'Periodic') MeanRet2011=((1+(nanmean(SPX2011return))).^252)-1 VAR2011=((1+var(SPX2011return)).^252)-1 RiskFree2012=((1.0002).^4)-1 MertonRatio201103=(MeanRet2011-RiskFree2012)./(VAR2011*(1-0.3)) MarketReturn2012=price2ret(SPXONLY2012,[],'Periodic') MeanRet2012ONLY=((1+(nanmean(MarketReturn2012))).^252)-1 PortfolioReturn201203=(MertonRatio201103*MeanRet2012ONLY)+((1-MertonRatio201103)*RiskFree2012) SPX2012return=price2ret(SPX2012,[],'Periodic') MeanRet2012=((1+(nanmean(SPX2012return))).^252)-1 VAR2012=((1+var(SPX2012return)).^252)-1 RiskFree2013=((1.0006).^4)-1 MertonRatio201203=(MeanRet2012-RiskFree2013)./(VAR2012*(1-0.3)) MarketReturn2013=price2ret(SPXONLY2013,[],'Periodic') MeanRet2013ONLY=((1+(nanmean(MarketReturn2013))).^252)-1 PortfolioReturn201303=(MertonRatio201203*MeanRet2013ONLY)+((1-MertonRatio201203)*RiskFree2013) SPX2013return=price2ret(SPX2013,[],'Periodic') MeanRet2013=((1+(nanmean(SPX2013return))).^252)-1 VAR2013=((1+var(SPX2013return)).^252)-1 RiskFree2014=((1.0006).^4)-1 MertonRatio201303=(MeanRet2013-RiskFree2014)./(VAR2013*(1-0.3)) MarketReturn2014=price2ret(SPXONLY2014,[],'Periodic') MeanRet2014ONLY=((1+(nanmean(MarketReturn2014))).^252)-1 PortfolioReturn201403=(1*MeanRet2014ONLY)+((1-1)*RiskFree2014) SPX2014return=price2ret(SPX2014,[],'Periodic') MeanRet2014=((1+(nanmean(SPX2014return))).^252)-1 VAR2014=((1+var(SPX2014return)).^252)-1 RiskFree2015=((1.0003).^4)-1 MertonRatio201403=(MeanRet2014-RiskFree2015)./(VAR2014*(1-0.3)) MarketReturn2015=price2ret(SPXONLY2015,[],'Periodic') MeanRet2015ONLY=((1+(nanmean(MarketReturn2015))).^252)-1 PortfolioReturn201503=(1*MeanRet2015ONLY)+((1-1)*RiskFree2015) SPX2011return=price2ret(SPX2011,[],'Periodic') MeanRet2011=((1+(nanmean(SPX2011return))).^252)-1 VAR2011=((1+var(SPX2011return)).^252)-1 RiskFree2012=((1.0002).^4)-1 MertonRatio201102=(MeanRet2011-RiskFree2012)./(VAR2011*(1-

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0.2)) MarketReturn2012=price2ret(SPXONLY2012,[],'Periodic') MeanRet2012ONLY=((1+(nanmean(MarketReturn2012))).^252)-1 PortfolioReturn201202=(MertonRatio201102*MeanRet2012ONLY)+((1-MertonRatio201102)*RiskFree2012) SPX2012return=price2ret(SPX2012,[],'Periodic') MeanRet2012=((1+(nanmean(SPX2012return))).^252)-1 VAR2012=((1+var(SPX2012return)).^252)-1 RiskFree2013=((1.0006).^4)-1 MertonRatio201202=(MeanRet2012-RiskFree2013)./(VAR2012*(1-0.2)) MarketReturn2013=price2ret(SPXONLY2013,[],'Periodic') MeanRet2013ONLY=((1+(nanmean(MarketReturn2013))).^252)-1 PortfolioReturn201302=(MertonRatio201202*MeanRet2013ONLY)+((1-MertonRatio201202)*RiskFree2013) SPX2013return=price2ret(SPX2013,[],'Periodic') MeanRet2013=((1+(nanmean(SPX2013return))).^252)-1 VAR2013=((1+var(SPX2013return)).^252)-1 RiskFree2014=((1.0006).^4)-1 MertonRatio201302=(MeanRet2013-RiskFree2014)./(VAR2013*(1-0.2)) MarketReturn2014=price2ret(SPXONLY2014,[],'Periodic') MeanRet2014ONLY=((1+(nanmean(MarketReturn2014))).^252)-1 PortfolioReturn201402=(1*MeanRet2014ONLY)+((1-1)*RiskFree2014) SPX2014return=price2ret(SPX2014,[],'Periodic') MeanRet2014=((1+(nanmean(SPX2014return))).^252)-1 VAR2014=((1+var(SPX2014return)).^252)-1 RiskFree2015=((1.0003).^4)-1 MertonRatio201402=(MeanRet2014-RiskFree2015)./(VAR2014*(1-0.2)) MarketReturn2015=price2ret(SPXONLY2015,[],'Periodic') MeanRet2015ONLY=((1+(nanmean(MarketReturn2015))).^252)-1 PortfolioReturn201502=(1*MeanRet2015ONLY)+((1-1)*RiskFree2015)

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MARKOWITZPORTFOLIOSThecodesherereportedarethoseusedtoestimatethecompositionofthe

portfolios under the Markowitz framework. The previously estimated

historical moments of the stock have been used and plugged into the

portfoliocompositionformula.Duetothestaticnatureoftheportfolio,the

initialcompositionhasbeenheldstillandasimulationofitsperformance

has been done by computing the performance which would have been

registered on the real market for the four years considered in this

application.

RiskFreeRatio5= (1-(0.05./(sqrt(VAR2011)))) RiskyRatio5=(1-RiskFreeRatio5) MarkPortReturn20125=(RiskFreeRatio5*RiskFree2012)+(RiskyRatio5*MeanRet2012ONLY) MarkPortReturn20135=(RiskFreeRatio5*RiskFree2013)+(RiskyRatio5*MeanRet2013ONLY) MarkPortReturn20145=(RiskFreeRatio5*RiskFree2014)+(RiskyRatio5*MeanRet2014ONLY) MarkPortReturn20155=(RiskFreeRatio5*RiskFree2015)+(RiskyRatio5*MeanRet2015ONLY) TotRetMark5=(1+MarkPortReturn20125)*(1+MarkPortReturn20135)*(1+MarkPortReturn20145)*(1+MarkPortReturn20155)-1 AnnualRetMark5=(1+TotRetMark5).^(1./4)-1 RiskFreeRatio10= (1-(0.1./(sqrt(VAR2011)))) RiskyRatio10=(1-RiskFreeRatio10) MarkPortReturn201210=(RiskFreeRatio10*RiskFree2012)+(RiskyRatio10*MeanRet2012ONLY) MarkPortReturn201310=(RiskFreeRatio10*RiskFree2013)+(RiskyRatio10*MeanRet2013ONLY) MarkPortReturn201410=(RiskFreeRatio10*RiskFree2014)+(RiskyRatio10*MeanRet2014ONLY) MarkPortReturn201510=(RiskFreeRatio10*RiskFree2015)+(RiskyRatio10*MeanRet2015ONLY) TotRetMark10=(1+MarkPortReturn201210)*(1+MarkPortReturn201310)*(1+MarkPortReturn201410)*(1+MarkPortReturn201510)-1 AnnualRetMark10=(1+TotRetMark10).^(1./4)-1 RiskFreeRatio15= (1-(0.15./(sqrt(VAR2011)))) RiskyRatio15=(1-RiskFreeRatio15) MarkPortReturn201215=(RiskFreeRatio15*RiskFree2012)+(RiskyRatio15*MeanRet2012ONLY) MarkPortReturn201315=(RiskFreeRatio15*RiskFree2013)+(RiskyRatio15*MeanRet2013ONLY) MarkPortReturn201415=(RiskFreeRatio15*RiskFree2014)+(RiskyRatio15*MeanRet2014ONLY)

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MarkPortReturn201515=(RiskFreeRatio15*RiskFree2015)+(RiskyRatio15*MeanRet2015ONLY) TotRetMark15=(1+MarkPortReturn201215)*(1+MarkPortReturn201315)*(1+MarkPortReturn201415)*(1+MarkPortReturn201515)-1 AnnualRetMark15=(1+TotRetMark15).^(1./4)-1 RiskFreeRatio175= (1-(0.175./(sqrt(VAR2011)))) RiskyRatio175=(1-RiskFreeRatio175) MarkPortReturn2012175=(RiskFreeRatio175*RiskFree2012)+(RiskyRatio175*MeanRet2012ONLY) MarkPortReturn2013175=(RiskFreeRatio175*RiskFree2013)+(RiskyRatio175*MeanRet2013ONLY) MarkPortReturn2014175=(RiskFreeRatio175*RiskFree2014)+(RiskyRatio175*MeanRet2014ONLY) MarkPortReturn2015175=(RiskFreeRatio175*RiskFree2015)+(RiskyRatio175*MeanRet2015ONLY) TotRetMark175=(1+MarkPortReturn2012175)*(1+MarkPortReturn2013175)*(1+MarkPortReturn2014175)*(1+MarkPortReturn2015175)-1 AnnualRetMark175=(1+TotRetMark175).^(1./4)-1 Portfoliofinalreturn02=((1+PortfolioReturn201102)*(1+PortfolioReturn201202)*(1+PortfolioReturn201302)*(1+PortfolioReturn201402))-1 Portfoliofinalreturn03=((1+PortfolioReturn201103)*(1+PortfolioReturn201203)*(1+PortfolioReturn201303)*(1+PortfolioReturn201403))-1 Portfoliofinalreturn04=((1+PortfolioReturn201104)*(1+PortfolioReturn201204)*(1+PortfolioReturn201304)*(1+PortfolioReturn201404))-1 Portfoliofinalreturn05=((1+PortfolioReturn201105)*(1+PortfolioReturn201205)*(1+PortfolioReturn201305)*(1+PortfolioReturn201405))-1 AnnualizedPortfolio02=((1+Portfoliofinalreturn02).^(1./4))-1 AnnualizedPortfolio03=((1+Portfoliofinalreturn03).^(1./4))-1 AnnualizedPortfolio04=((1+Portfoliofinalreturn04).^(1./4))-1 AnnualizedPortfolio05=((1+Portfoliofinalreturn05).^(1./4))-1 scatter(RFREERATIOS,RiskRATIOS)

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Documents

A Review of Merton’s Portfolio Problemtesi.eprints.luiss.it/17083/1/184031_CARBONE_ANGELO.pdf- 1 - Dipartimento di Economia e Finanza A Review of Merton’s Portfolio Problem Angelo