Question

Explain in detail the process of deriving the minimum variance portfolio MVP, and the process of estimating the expected return and risk in the relevant portfolio.

Answer 1

Portfolio ivariance iis ia imeasure iof ia iportfolio's ioverall irisk iand iis ithe iportfolio's istandard ideviation isquared.

· Portfolio ivariance itakes iinto iaccount ithe iweights iand ivariances iof ieach iasset iin ia iportfolio ias iwell ias itheir icovariances.

· A ilower icorrelation ibetween isecurities iin ia iportfolio iresults iin ia ilower iportfolio ivariance.

· Portfolio ivariance i(and istandard ideviation) idefine ithe irisk-axis iof ithe iefficient ifrontier iin imodern iportfolio itheory i(MPT).

Portfolio ivariance ilooks iat ithe icovariance ior icorrelation icoefficients ifor ithe isecurities iin ia iportfolio. iGenerally, ia ilower icorrelation ibetween isecurities iin ia iportfolio iresults iin ia ilower iportfolio ivariance.

Modern iportfolio itheory isays ithat iportfolio ivariance ican ibe ireduced iby ichoosing iasset iclasses iwith ia ilow ior inegative icorrelation, isuch ias istocks iand ibonds, iwhere ithe ivariance i(or istandard ideviation) iof ithe iportfolio iis ithe ix-axis iof ithe iefficient ifrontier

Formula iand iCalculation iof iPortfolio iVariance

The imost iimportant iquality iof iportfolio ivariance iis ithat iits ivalue iis ia iweighted icombination iof ithe iindividual ivariances iof ieach iof ithe iassets iadjusted iby itheir icovariances. iThis imeans ithat ithe ioverall iportfolio ivariance iis ilower ithan ia isimple iweighted iaverage iof ithe iindividual ivariances iof ithe istocks iin ithe iportfolio.

Portfolio ivariance i= iw12σ12 i+ iw22σ22 i+ i2w1w2Cov1,2

Where:

· w1 i= ithe iportfolio iweight iof ithe ifirst iasset

· w2 i= ithe iportfolio iweight iof ithe isecond iasset

· σ1= ithe istandard ideviation iof ithe ifirst iasset

· σ2 i= ithe istandard ideviation iof ithe isecond iasset

· Cov1,2 i= ithe icovariance iof ithe itwo iassets, iwhich ican ithus ibe iexpressed ias ip(1,2)σ1σ2, iwhere ip(1,2) iis ithe icorrelation icoefficient ibetween ithe itwo iassets

A iminimum ivariance iportfolio iis ia icollection iof isecurities ithat icombine ito iminimize ithe iprice ivolatility iof ithe ioverall iportfolio. iVolatility iis ia istatistical imeasure iof ia iparticular isecurity's iprice imovement i(ups iand idowns).

An iinvestment’s ivolatility iis iinterchangeable iin imeaning iwith i“market irisk”. iTherefore, ithe igreater ithe ivolatility iof ian iinvestment i(the iwider ithe iswings iup iand idown iin iprice), ithe ihigher ithe imarket irisk. iSo, iif iyou iwant ito iminimize irisk, iyou iwant ito iminimize ithe iups iand idowns.

To ibuild ia iminimum ivariance iportfolio, iyou ineed ito istick iwith ilow-volatility iinvestments ior ia icombination iof ivolatile iinvestments iwith ilow icorrelation ito ieach iother. Investments ithat ihave ilow icorrelation iare ithose ithat iperform idifferently icompared ito ithe iprevailing imarket iand ieconomic ienvironment. iThe istrategy iis ia igreat iexample iof idiversification.

i iThe iexpected ireturn iis ithe iprofit ior iloss ithat ian iinvestor ianticipates ion ian iinvestment ithat ihas iknown ihistorical irates iof ireturn i(RoR). iIt iis icalculated iby imultiplying ipotential ioutcomes iby ithe ichances iof ithem ioccurring iand ithen itotaling ithese iresults. iExpected ireturn icalculations iare ia ikey ipiece iof iboth ibusiness ioperations iand ifinancial itheory, iincluding iin ithe iwell-known imodels iof imodern iportfolio itheory i(MPT) ior ithe iblack-scholes ioptions ipricing imodel.

· The iexpected ireturn iis ithe iamount iof iprofit ior iloss ian iinvestor ican ianticipate ireceiving ion ian iinvestment.

· An iexpected ireturn iis icalculated iby imultiplying ipotential ioutcomes iby ithe iodds iof ithem ioccurring iand ithen itotaling ithese iresults.

· Essentially ia ilong-term iweighted iaverage iof ihistorical iresults, iexpected ireturns iare inot iguaranteed.

The iexpected ireturn iis ia itool iused ito idetermine iwhether ian iinvestment ihas ia ipositive ior inegative iaverage inet ioutcome. iThe isum iis icalculated ias ithe iexpected ivalue i(EV) iof ian iinvestment igiven iits ipotential ireturns iin idifferent iscenarios, ias iillustrated iby ithe ifollowing iformula:

Expected iReturn i= iSUM i(Returni ix iProbabilityi)

where: i"i" iindicates ieach iknown ireturn iand iits irespective iprobability iin ithe iseries

The iexpected ireturn iis iusually ibased ion ihistorical idata iand iis itherefore inot iguaranteed iinto ithe ifuture; ihowever, iit idoes ioften iset ireasonable iexpectations. iTherefore, ithe iexpected ireturn ifigure ican ibe ithough iof ias ia ilong-term iweighted iaverage iof ihistorical ireturns.