investors’ preferences Risk-adjusted performance of portfolio insurance and Finance Research Letters

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Finance Research Letters

journalhomepage:www.elsevier.com/locate/frl

Risk-adjusted performance of portfolio insurance and investors’ preferences

Dima Tawil

^∗

Rennes School of Business, 2 Rue Robert d’Arbrissel, 35065, Rennes, France

a rt i c l e i n f o

Article history:

Received 19 April 2017 Accepted 8 May 2017 Available online 10 May 2017 JEL classification:

G11 Keywords:

Option based portfolio insurance (OBPI) Constant proportion portfolio insurance (CPPI)

Risk-adjusted performance Risk preferences

a b s t ra c t

Thispaperdrawsaclearlinebetweeninvestors’riskpreferencesandtheirchoiceofeither OptionBasedPortfolioInsurance(OBPI)orConstantProportionPortfolioInsurance(CPPI).

Forthispurpose,OBPIandCPPIarecomparedusingpartial-moments-basedrisk-adjusted performancemeasures,whichareadequateforcomparingasymmetricreturndistributions and canbe easilytailoredtoreflect investors’preferences.The analysiscoversexpected utilityand prospectutility investors,amongothers, andthe resultsshow investors’risk preferencesinthegaindomainarethekeydeterminantsofthechoicebetweenOBPIand CPPI.

© 2017ElsevierInc.Allrightsreserved.

1. Introduction

Whetherinvestors’riskpreferencesdeterminetheirchoicebetweenOptionBasedPortfolioInsurance(OBPI)andConstant ProportionPortfolioInsurance(CPPI)remainsacontroversialquestionamongresearchers,who,inthequestforananswer, haveemployedawiderangeofcriteria.

Themostpopularcriterion,expectedutilitymaximization,¹doesnotgiveaclearrankingofOBPIandCPPI².Bernardand Kwak(2016)andPézierandScheller(2013) showedthat CPPIistheoptimalstrategyforhyperbolicabsoluterisk aversion (HARA)investors,whileElKarouietal.(2005)demonstratedthatOBPIisoptimalforconstantrelativeriskaversion(CRRA) investors.Furthermore,GrossmanandVila(1992)provedthatCPPIisoptimalintheframeworkoflong-termrisk-sensitive portfoliooptimization.Nevertheless,whentheguaranteeconstraintisexogenous,theoptimalstrategyhasbeenshownto beOBPI.³

Stochastic dominance (SD)criterion, which islinked to investors’risk preferences through theorder ofSD⁴, doesnot discriminate betweenOBPIandCPPI. Bertrand andPrigent(2005)documented nofirst-order SDbetweenOBPI andCPPI,

∗ Corresponding author.

E-mail address: dima.tawil@esc-rennes.com

1Expected utility theory assumes investors are risk averse at all wealth levels.

2The ranking produced by this criterion crucially depends on utility specification.

3These sutdies assumed the risky asset follows a geometric Brownian motion.

4The first-order SD corresponds to increasing utility, the second-order to a concave utility, meaning risk aversion and the third order to a convex utility, meaning risk seeking. See Levy (1992) for more details.

http://dx.doi.org/10.1016/j.frl.2017.05.004 1544-6123/© 2017 Elsevier Inc. All rights reserved.

andAnnaert etal.(2009)notednosecond-orderSD; however,Maalej andPrigent(2016) showedthatCPPI candominate OBPIatthethird-order,butonlywhentheimpliedvolatilityishighrelativelytotheempiricalvolatility.

Dierkesetal.(2010)andDichtlandDrobetz(2011)adoptedadifferentcomparativeapproachtoportfolioinsurance(PI) strategiesthatassumesinvestorsbehave accordingtoTverskyandKahneman’s(1992))cumulativeprospecttheory(CPT).⁵ However,although their resultsexplained thepopularityofPI strategiesversus uninsuredinvestment strategies,they did notdefineadominantPIstrategyforCPTinvestors.

Thispapercontributestothe previousliterature inthefollowingways.First, itprovidesasimple unifyingapproach to identifyingtheportfolioinsurancestrategy(OBPIorCPPI)thatcorrespondstothetypeofinvestorpreferenceswithoutusing formalpreferencefunctionstodescribetheinvestor’sbehavior.Itissufficientthattheinvestordefinesherriskpreferences, includingrisk aversion, riskseeking, orrisk neutrality ingeneralterms,andadapts her comparisoncriterion accordingly.

Second,thispaperconsidersdifferenttypesofpreferences,andcoversabroadersetofinvestorsthaninanypreviouswork onthissubject.

Partial-moments-basedrisk-adjustedperformancemeasuresand,inparticular,theFarinelliandTibiletti(2008)ratioare usedto compareOBPI andCPPI. Thisratiouses the upperpartialmoment (UPM) oforderpas ameasure ofreturnand thelowerpartialmoment(LPM)oforderqasameasureofdownsiderisk.Tworeasonsliebehindchoosingthisratio:first, itisaratiooftheexpectedupsidegain aboveacertain referencepointLrelativelytotherisk measured bytheexpected lossbelow L. It isthereforesuitable for describingboth the upsideparticipation andthe downsideprotectionprofiles of PI.Second,thisratiocanbeeasilyadaptedtosuitinvestors’riskpreferencesthroughtheordersofpartialmoments(pand q) aswell asthereferencepoint L.⁶ Awide rangeofpreferences,includingexpectedutilityandprospectutilityinvestors’

preferences,canbe modeled.Thismethodcanbequitenoteworthy,especiallyforindividualinvestorswhomaypreferthe simplicityandtheflexibilityitprovidesandwhoarethemainbuyersofPIinpractice(SeeFodoretal.(2013)).

Asa firststep,Ideriveanalytical solutionsfortheUPM andtheLPMofOBPI andCPPI underthegeometric Brownian motion(GBM).Then,Ianalyzetherelationbetweeninvestors’riskpreferences(describedbytheparametersp,q,andL)and theirchoicebetweenOBPIandCPPI.Ialsoanalyzethesensitivityoftheresultstomarketconditions,andusingrealmarket data,Iprovidearobustnesscheck.Theresultsdemonstratethatinvestorspreferencesinthegaindomain,ratherthaninthe lossdomain,playadeterminantroleindefiningtheirchoiceofaPIstrategy.

Thispaper’sresultsincludeageneralizationofBertrandandPrigent’s(2011))comparisonofOBPIandCPPI,whichthey performedusingtheOmegaratiointroducedbyKeatingandShadwick(2002)andKazemietal.(2004).TheOmegaratiois, infact,aspecialcaseoftheFarinelliandTibiletti(2008)ratio,where(p=q=1).⁷ Thissetofparameters impliesthat all deviationsfromthereferencepointareequallyweightedtotheiroriginalvalueandreflectsarisk-neutralbehavioringains andlosses.

Thereminderofthispaperisorganizedasfollows:inSection2,thepropertiesoftheFarinelli&Tibilettiratioandthose ofOBPI andCPPI are recalled, and analytical solutions for the LPMand the UPM of OBPI and CPPI under the GBM are derived.In Section 3,the impactofthe risk preferencesof investorsontheir choice ofa specific PI strategy isanalyzed, andasensitivityanalysisoftheresultstomarketconditionsisalsoprovided.Arobustnesscheckusingrealmarketdatais presentedinSection4,andSection5offersageneralconclusion.

2. Risk-adjustedperformancemeasurementofPIstrategies 2.1. TheFarinelliandTibiletti(2008)Ratio

Denotedhereafterby(^{p,q),theFarinelliandTibiletti(2008)ratioisdefinedasfollows:}

^p,q=

p

UPMp

(

^L

)

q

LPMq

(

^L

)

⁼

p

E[x−L]₊^p

q

E[L−x]^q₊. (1)

Wherexisaseriesofportfolioreturns,LisareferencepointthatseparatesthegainsaboveLfromthelossesbelowL. TheorderoftheUPMandtheLPMisp>0andq>0,respectively.Theratio^{p,qcanbeadaptedtofittherisk}preferences ofinvestorsthroughtwomainelements:thereferencepointLandtheordersofpartialmoments(pandq).

Theratiop,q is adecreasing functionw.r.t Lwhichis usually definedexogenuously.Unser(2000) demonstratedthat thelevelofLvariesaccordingtoinvestors’preferencesandincreaseswithriskaversion.⁸

Theparameterspandq,ontheotherhand,respectivelyreflectthedesiredrelevancegiventothemagnitudeofpositive andnegativedeviationsfromL.Thehigherthevalueofq(withq>1),thehighertheinvestor’sriskaversiontolargelosses andlowervaluesq<1reflectarisk-seekingbehaviorinthelossdomain.Similarly,p<1reflectsriskaversioninthegain domain,whilep>1reflectsrisk-seeking.Consequently,risk-averseexpectedutilityinvestorscanbemodeledbyq>1and

5CPT states that investors are risk averse in the gain domain, above a specific reference point and risk seeking in the loss domain below the reference point.

6Zakamouline (2014) established an explicit link between partial-moments-based performance measures and investor’s utility function.

7For p = 1 and q = 2 , 3 , we get the Kappa measures introduced by Kaplan and Knowles (2004) .

8For an investor buying PI, it is logical to think that the minimum value of L corresponds to the insured capital at maturity.

p<1.Prospecttheoryinvestors,whoarerisk-seekingbelowandrisk-averseaboveL,canbemodeledbyq<1andp<1.

ThetypeofinvestorsanalyzedinMarkowitz(1952),risk-aversebelowandrisk-seekingaboveL,⁹ canbemodeledbyq>1 andp>1,andrisk-seekinginvestorscanbemodeledbyq<1andp>1(Seee.g.Zakamouline(2011)).

2.2. theratiop,qfortheOBPIstrategy

2.2.1. TheOBPIstrategy

LelandandRubinstein(1976)introducedtheOBPIstrategy,whichconsistsofpurchasingjsharesofariskyassetScovered by j sharesofa European putoptionwritten onit. Theoption’s strikepriceK corresponds to theinsuredcapitalatthe maturitydateTwhateverthevalueofS_T.ThevalueoftheOBPIportfolioatmaturity,V_T^OBPI,isgivenby¹⁰:

V_T^OBPI=j

S_T+

(

^K−ST

)

⁺

=j

K+

(

^ST−K

)

⁺

. (2)

AndthevaluetheOBPIportfolioatanytimet≤T,V_t^OBPI,is:

V_t^OBPI=j

(

^St+P

(

^t,St,K

) )

^=j

Ke^−r(T−t)+C

(

^t,St,K

)

. (3)

WhereC(t,St,K)andP(t,St,K)arerespectivelythetimetBlack-ScholespricesofEuropeancallandputoptionswritten onSwitha maturityT anda strikepriceK.Eq.(3)showsforalldates t≤ T,thevalue oftheOBPIportfolio cannotfall belowthedeterministiclevel jKe^−r(T−t).

Ingeneral,investorsseektorecoverapercentage

α

^{oftheirinitialinvestmentV}0atmaturity.Inthiscase,thefollowing relationmustbeverified:

α

^V0=

α

^j

K.e^−rT+C

(

^0,S0,K

)

=jK. (4)

2.2.2. ComputationoftheLPMandtheUPMoftheOBPIstrategy

Proposition1. ThelowerpartialmomentofordernoftheOBPIstrategy,LPM_n^OBPI,isgivenby:

LPM^OBPI_n = n

i=0

n

i

(

⁻¹

)

^iLn−i∗

(

^jK

)

^iN

(

^−d0,K

)

⁺

β (

^jS0

)

ⁱ

N

−di,L/j

−N

(

^−di,K

)

. (5)

AndtheupperpartialmomentofordernoftheOBPIstrategy,UPM^OBPI_n ,isgivenby:

UPM^OBPI_n = n

i=0

n

i

(

⁻¹

)

^n−iLn−i

β (

^jS0

)

^iN

di,L/j

. (6)

Where

_n

i

= _i!(nⁿ_−i^!_)!,

β

=exp

i

μ

^T+¹2

i²−i

σ

^2T

andd_i,X=^ln(S0/X)+(^μ−σ2/2)^T+iσ2T

σ^√t . Proof. TheexpressionoftheLPM_n^OBPIcanbewrittenasfollows:

LPM^OBPI_n =E

L−V_T^OBPI

n +,

= n

i=0

n

i

(

⁻¹

)

^iLn−iE

V_T^OBPI

i

I

{

^VTOBPI≤L

}

^{. (7)}

Where

_n

i

=_i!(nⁿ₋^!_i)! and¹¹: E

V_T^OBPI

i

I

{

^VTOBPI≤L

}

^=E

(

^jK

)

^iI{^ST≤K}∗I_{jK≤L}

+E

(

^jST

)

^iI{^ST>K}∗I_{jST≤L}

,

=E

(

^jK

)

ⁱI_{ST≤K}+E

(

^jST

)

ⁱI_{ST≤L/j}−E

(

^jST

)

ⁱI_{ST≤K}. (8) Knowingthat,undertheGBM:

E

(

^ST

)

^iI{^ST≤X}=Sⁱ₀exp

i

μ

^T+1₂

i²−i

σ

^2T

N

(

^−di,X

)

^{, (9)}

Where:

d_i,X =ln

(

^S0/X

)

⁺

( μ

⁻

σ

^2/2

)

^T+i

σ

^T

σ

^{√T . (10)}

9Markowitz (1952) introduced this utility function to explain why people simultaneously buy insurance and lottery tickets.

10Due to the call-put parity, this strategy is equivalent to purchasing, at t = 0 , j shares of a European call option written on S with a strike price K and a maturity T and investing an amount jKe ^−rTin a risk-free asset.

11The indicator I _{jK≤L}is equal to 1 as L is always set to a higher value than the insured capital at maturity ( jK ).

TheexpressionoftheLPM^OBPI_n inEq.(7)canbewrittenasfollows:

LPM^OBPI_n = n

i=0

n

i

(

−1

)

^iLn−i∗

(

^jK

)

^iN

(

−d0,K

)

+

β (

^jS0

)

ⁱ

N

−di,L/j

−N

(

−di,K

)

. Where

β

=exp

i

μ

^T+¹2

i²−i

σ

^2T

.

Followingasimilarprocedure,wederivetheexpressionoftheUPM^OBPI_n asfollows¹²: UPM_n^OBPI=E

V_T^OBPI−L

n +,

= n

i=0

n

i

(

⁻¹

)

^n−iLn−i

β (

^jS0

)

^iN

di,L/j

. (11)

2.3.theratiop,qfortheCPPIstrategy

2.3.1. TheCPPIstrategy

BlackandJones (1987)introduced the CPPI strategy,¹³ whichconsists inmanagingadynamicportfolio ofa riskyand risk-freeassetsovertime.ToimplementtheCPPI,theinvestorselectsafloorbelowwhichshedoesnot wanttheportfolio valuetofallatmaturity.Thefloorisdefinedasapercentage

α

^{oftheinitialinvestmentV}0 andgrowsataconstantrater:

FT=

α

^{V0, (12)}

Ft=FTe^−r(T−t)=

α

^V0e^−r(T−t). (13)

Toachievethisguarantee,theinvestorallocatesanamountcalledtheexposuretoariskyasset,suchasamarketindex, andtheremainingamounttoarisk-freeassetthatevolvesaccordingtotheconstantrater.Theexposuremustbecontinu- ouslyadjustedtoaconstantmultiplemofthedifferencebetweentheportfoliovalueandfloor.Thisdifferenceiscalledthe cushionCt

Ct=max

V_t^CPPI−Ft,0

. (14)

Ifthe cushion becomesvery small,the exposure becomes very small too preventing the portfolio value fromfalling belowthefloor.Forallm>1,thepayoff functionoftheCPPIportfolioisaconvexfunctionw.r.tS_T.Thehigherthemultiple, themoreconvexthepayoff functionis.¹⁴

IftheunderlyingriskyassetfollowstheGBM,thetimetvalueofthecushionisgivenby:

Ct=C0exp

m

( μ

−r

)

+r−m²

σ

²

2 t+m

σ

^Wt

,

=C0exp

( μ

rt+

σ

rWt

)

. (15)

Where

μ

^{isaconstantdrift,}

σ

^{isthevolatilityoftheunderlyingriskyasset,Wt isthestandard Brownianmotion,}

μ

^r= m(

μ

−r)+r−^m22^σ2 and

σ

r=m

σ

.

2.3.2. ComputationoftheLPMandtheUPMoftheCPPIstrategy

Proposition2. ThelowerpartialmomentofordernoftheCPPIstrategy,LPM^CP_n ^PI,isgivenby:

LPM^CP_n^PI= n

i=0

n

i

(

−1

)

^ian−i

β

^C0ⁱN

(

−d_i,a

)

. (16)

AndtheupperpartialmomentofordernoftheCPPIstrategy,UPM^CP_n ^PI,isgivenby:

UPM^CP_n ^PI= n

i=0

n

i

(

⁻¹

)

^n−ian−i

β

^CN

(

^di,a

)

^{. (17)}

Where a=L−F_T,

β

=exp

i

μ

^rT+¹₂

i²−i

σ

r²T

andd_i,a= ^ln(C0/a_σ^)+μrT+iσrT

r√

T .

12 Detailed proofs of the expressions of UPM ^OBPI_n and UPM ^CP_n^PIare available at request.

13 See also Perold and Sharpe (1988) and Black and Perold (1992) .

14 Many extensions of the standard CPPI method are proposed in practice. See e.g. Lee et al. (2011) , Hamidi et al. (2014) and Ameur and Prigent (2014) .

Fig. 1. p,qas function of p and q for α= 0 . 9 and L = 102 .

Proof. TheexpressionoftheLPM^CP_n ^PIcanbewrittenasfollows:

LPM^CP_n^PI=E

L−V_T^CPPI

n +,

=E

(

^L−FT−CT

)

ⁿ+,

= n

i=0

n

i

(

⁻¹

)

^ian−iE

C_TⁱI_{CT≤a}

(18) Wherea=L−FT andE

C_TⁱI_{C_T≤a}

isgivenbyEq.(9). ThefinalformoftheLPM^CP_n^PIcanbewrittenasfollows:

LPM^CP_n^PI= n

i=0

n

i

(

⁻¹

)

^ian−i

β

^Ci0N

(

^−di,a

)

^{. (19)}

Where

β

=exp

i

μ

rT+¹2

i²−i

σ

r²T

.

Followingasimilarprocedure,wederivetheexpressionoftheUPM^CP_n ^PIasfollows:

UPM^CP_n^PI=E

V_T^CPPI−L

n

+ (20)

= n

i=0

n

i

(

⁻¹

)

^n−ian−i

β

^Ci0N

(

^di,a

)

^.

3. OBPIVersusCPPI

InordertocompareOBPIandCPPI,thetwostrategiesareassumedtohavethesameinitialportfoliovalue

V₀^OBPI=V₀^CPPI

and to providethe same guarantee (

α

^V0=jK=F_T) ^{atmaturity T. The second assumption implies that F}0=jKe^−rT, and thus,theinitialvalueofthecushionC₀ isequaltothecallpriceC(0,S₀,K).Theunderlyingriskyassetofbothstrategiesis supposedtoevolveaccordingtotheGBMwiththefollowingbaseparameters:T=1year,

μ

=8%,

σ

=20%,S0=100,and r=3%.Differentvaluesoftheinsuredpercentageatmaturityareconsidered,andCPPIstrategieswithdifferentlevelsofthe multiplearecompared.

3.1. Theimpactofinvestors’preferences

3.1.1. Theparameterspandq

Foraninsuredpercentage

α

=0.9andareferencepointL=102,aninvestorwithmoderategainpreferences(riskaver- sionin thegain domainwithp< 1) willprefer theOBPI tothe CPPI,regardless oftheinvestor’spreferences intheloss domain(q>1orq<1).Thesesetsofparameterscorrespondtoexpectedutilityandprospectutilityinvestors,respectively.

Increasingthevalueptop>1leadstoan inverseranking.CPPI strategiesstarttodominatetheOBPIstrategy whenpis around1.Forp>1andq>1,aninvestorwhoisriskseekingingainsandriskaverseinlosseswillprefertheCPPIstrategy.

However,noabsolutedominanceisobservedwhenp>1andq<1,whichisriskseekinginbothgainsandlosses.Inthis case,theOBPIdominatestheCPPIunlessthevalueofthemultiplemisrelativelysmall(m=4)(seeFig.1).

IncreasingthelevelofguaranteefavorsCPPIstrategies(seeFig.2).For

α

=1,CPPIstrategiesdominateOBPI foralmost allvaluesofpandq;thisresultalsoholdstrueforallvaluesofthemultiplem.Forthisnumericalexample,CPPIstrategies withalowervalueofmdominatethosewithhighervaluesofthisparameter.

3.1.2. ThereferencepointL

InvestorswithdifferentreferencepointsL(withL≥

α

^V0)areconsidered.ThevaluesofLareincorporatedwithselected setsoftheparameterspandqinordertoreflectinvestors’preferences.0.5,0.5 isusedforprospectutilityinvestors,1.5,0.5

forriskseekinginvestors,0.5,1.5forrisk-averseexpectedutilityinvestorsand1.5,1.5forMarkowitz (1952)investors.

Fig. 2. p,qas function of p and q for α= 1 and L = 102 .

Fig. 3. p,qas function of the reference point L for α= 0 . 9 .

Fig. 4. p,qas function of the reference point L for α= 1 .

Fig. 5. p,qas function of the volatility σfor α= 0 . 9 and L = 102 .

For

α

=0.9,increasingthe valueof L,which indicateshigherrisk aversion,favors theOBPI strategy,see Fig.3.Forall ratios^{p,q,thegapbetweenthecurvesofOBPIandCPPIdiminisheswhenLincreases.Neitherstrategydominatestheother} forallvaluesofLexceptfor1.5,1.5where theCPPIdominates.¹⁵Themultiplemplays adeterminant role in theranking of OBPIandCPPI.CPPIstrategieswithhighvaluesofmaredominatedbyOBPIforalmostallvaluesofL.Thisresultholdsfor 0.5,0.5,1.5,0.5 and0.5,1.5.CPPIstrategiesintersecteach otherathighvaluesofL(around103).Higher valuesofm are preferredwhenhighreferencepoints,relativelytotheinitialinvestment,areconsidered.Increasingtheinsuredpercentage to

α

=1favorstheCPPI strategies,whichstronglydominatetheOBPI,especially forsmallvaluesofL.However,the OBPI strategystilldominatesforhighvaluesofL,seeFig.4.

3.2.Theimpactofmarketconditions

Inordertoanalyzetheimpactofmarketconditionsontherelativerisk-adjustedperformanceofOBPIandCPPI,different levels ofthevolatility ofthe underlying(between 10% and40%) areconsidered. Fig. 5showsthat all ^{p,q ratios are de-} creasingfunctionsw.r.tthevolatility

σ

^.For

α

=0.9andL=102,neitherstrategydominatestheotherforallvolatilitylevels exceptfor1.5,1.5where the CPPIdominates.In all other cases,the OBPIdominatesCPPIstrategieswithhighvaluesofthe

15 This result is in line with the results of Bertrand and Prigent (2011) .

Fig. 6. p,qas function of the volatility σfor α= 1 and L = 102 . Table 1

Summery statistics of CAC40 returns for the period from 3/1/20 0 0 to 30/12/2016.

CAC40 statistics

Mean (Annualized) −1.12%

Standard deviation (Annualized) 23.48%

Skewness −0.034

Kurtosis 7.8033

ρ-value autocorrelation (Ljung-Box test) 0.0 0 0 ρ-value autocorrelation (Ljung-Box test squared returns) 0.0 0 0 ρ-value heteroscedasticity (Engle’s ARCH test) 0.0 0 0

multiple(m=6andm=9).Regardlessofthevalueofq,theOBPIstrategydominatesformoderatetohighvolatilitylevels (between20%and40%)whenp<1.Thisshowsthat,underallmarketconditions,thepreferencesinthegaindomain(risk aversionwhenp<1)haveamoresignificantimpactontherankingofOBPIandCPPIthanthoseinthelossdomain.

CPPIstrategiesdominatetheOBPIforalmost allvolatilityscenarioswhen

α

=1.However,whenp<1,theOBPIstarts todominatehighmultipleCPPIstrategiesatvolatilitylevels(

σ

≥20%),seeFig.6.

4. Robustnesscheck

ThemovingblockbootstrapmethodwasusedtosimulateaseriesofOBPIandCPPIportfolioreturnsusingFrenchmarket data.Thismethodpreservesthedependenceandtheautocorrelationfeaturesoftheoriginaldata.Tobeginthesimulations, arandomstartingdateisdrawnwithreplacement.TheperformanceofOBPIandCPPI issimulatedoverthe252daysthat followthestartingdate.Thisprocedureisthenrepeated10000timesandtheratios^{p,qarecomputedonthis10000OBPI} andCPPIyearlyreturns.

TheinitialinvestmentoftheOBPIandtheCPPIportfoliosisallocatedbetweenariskyasset(theCAC40index)andarisk- free assetthat evolvesaccordingtotheEuribor 12-monthrateprevailing atthefirst dayofthemanagement period.The pricedatausedfortheunderlyingriskyassetconsistofCAC40dailyclosingprices,realizedwithintheperiodfromJanuary 3,2000toDecember30,2016.Thisperiodisofparticularimportancebecauseitshowsdifferentmarketconditions,includ- ing thefinancial marketcrisis of2008.The statistics ofCAC40log-returns,presentedinTable1, showthe non-normality ofthesereturns.Forthisperiod,theCAC40hasanegativeannualmeanreturn(−1.12%),an annualizedstandard deviation (23.48%),asmallnegativeskewnessof(−0.034)andahighkurtosis(7.8033).TheresultsofLjung-Boxtestdetects signifi- cantserialcorrelationbetweenthereturnsoftheCAC40indexwithin thisperiod,andsignificantheteroscedasticityisalso detected.FortheOBPIstrategy,optionpricesarecalculatedaccordingtoBlack-Scholesoptionpricingformula.Thevolatility inputthat isusedtopricetheOBPI’scall optionscorrespondstotheannualizedstandard deviationcalculatedonthe252 dailyindexreturnspriortothestartingdateofthemanagementperiod.TheCPPIstrategyrebalancedonmonthlybasis.The initialfloorvalueiscomputedwiththeoneyearEuriborrateprevailingatthestartingdateofthemanagementperiod.The samerateisusedtocalculateoptionprices.

Simulationresults,presentedinTable2,showthattheinvestorswhoareriskaverseinthegaindomain,expectedutility (0.5,1.5)andprospectutility(0.5,0.5)investors,shallprefertheOBPIstrategy totheCPPI.Thisresultisparticularlytrue forhighervaluesofthereferencepoint Landlower insuredpercentages

α

^{.Thisresultholdsalsoforinvestorswho show}

arisk-seekingbehavior ingainsandlosses(1.5,0.5).However, investorswhoarerisk averseinlossesandriskseekingin gains(1.5,1.5) shall prefer the CPPI strategies especially for lower values ofLand a higher values of

α

^.

5. Conclusion

ThispaperanalyzedtherelationbetweentheriskpreferencesofinvestorsandtheirchoicesbetweenPIstrategies(OBPI andCPPI).Forthispurpose,theFarinelliandTibiletti(2008)ratio,whichcanbeeasily adjustedtofitthetypeofinvestor preferences, wasused asa risk-adjusted performance measure. The analysisproved that differentrisk preferencesimply differentrankingofOBPIandCPPI.Thisresultisparticularlytrueforlowlevelsofguarantee

α

^{.Inthiscase,where}

α

=0.9,

D.Tawil / FinanceResearchLetters24(2018)10–1817 The ratios p,qfor OBPI and CPPI strategies.

0.5,0.5 1.5,0.5 0.5,1.5 1.5,1.5

OBPI CPPI OBPI CPPI OBPI CPPI OBPI CPPI

α% m = 4 m = 6 m = 8 m = 4 m = 6 m = 8 m = 4 m = 6 m = 8 m = 4 m = 6 m = 8 L = 1%

90 0.94 0.68 0.50 0.53 2.27 2.32 1.06 2.10 0.32 0.32 0.28 0.31 0.66 0.93 1.00 1.06 92 0.98 0.74 0.49 0.46 2.36 2.49 2.08 2.08 0.32 0.34 0.28 0.28 0.68 0.99 1.02 1.11 94 1.04 0.82 0.50 0.41 2.48 2.77 2.18 2.08 0.34 0.37 0.29 0.26 0.70 1.08 1.07 1.15 96 1.14 0.99 0.56 0.40 2.65 3.29 2.44 2.17 0.36 0.42 0.31 0.26 0.74 1.24 1.17 1.21 98 1.24 1.39 0.72 0.46 2.85 4.57 3.13 2.55 0.39 0.55 0.38 0.29 0.79 1.65 1.44 1.40 100 1.35 3.49 1.40 0.80 3.09 11.2 6.09 4.55 0.41 1.15 0.66 0.46 0.84 3.52 2.68 2.41

L = 2%

90 0.75 0.44 0.38 0.42 1.90 1.66 1.63 1.73 0.38 0.24 0.24 0.26 0.59 0.74 0.84 0.81 92 0.77 0.44 0.35 0.35 1.95 1.67 1.57 1.65 0.38 0.24 0.22 0.23 0.60 0.75 0.85 0.83 94 0.80 0.44 0.33 0.29 2.01 1.69 1.53 1.56 0.39 0.24 0.21 0.21 0.61 0.77 0.83 0.83 96 0.86 0.42 0.31 0.26 2.09 1.69 1.50 1.47 0.30 0.24 0.21 0.19 0.63 0.78 0.83 0.80 98 0.90 0.39 0.29 0.23 2.18 1.67 1.49 1.40 0.31 0.23 0.20 0.18 0.66 0.82 0.84 0.79 100 0.93 0.38 0.27 0.22 2.27 1.62 1.45 1.37 0.32 0.25 0.20 0.18 0.68 0.88 0.90 0.93

L = 3%

90 0.60 0.30 0.29 0.34 1.59 1.20 1.30 1.43 0.24 0.19 0.23 0.23 0.53 0.60 0.71 0.79 92 0.60 0.28 0.26 0.28 1.60 1.14 1.20 1.32 0.24 0.18 0.20 0.20 0.53 0.58 0.69 0.79 94 0.61 0.25 0.23 0.23 1.61 1.05 1.1 1.19 0.25 0.17 0.17 0.18 0.53 0.56 0.65 0.76 96 0.63 0.21 0.20 0.18 1.64 0.91 0.98 1.04 0.25 0.16 0.16 0.15 0.54 0.53 0.61 0.70 98 0.63 0.17 0.16 0.16 1.64 0.71 0.81 0.86 0.25 0.14 0.14 0.14 0.54 0.46 0.54 0.62 100 0.61 0.13 0.13 0.13 1.63 0.48 0.58 0.65 0.25 0.12 0.12 0.12 0.54 0.35 0.44 0.51

L = 4%

90 0.48 0.21 0.24 0.28 1.32 0.88 1.05 1.19 0.21 0.16 0.17 0.20 0.47 0.49 0.61 0.69 92 0.47 0.19 0.20 0.23 1.30 0.78 0.94 1.08 0.21 0.15 0.16 0.18 0.47 0.47 0.58 0.68 94 0.46 0.16 0.17 0.18 1.28 0.67 0.81 0.94 0.21 0.13 0.14 0.15 0.46 0.45 0.53 0.63 96 0.45 0.14 0.15 0.15 1.26 0.53 0.67 0.77 0.21 0.12 0.12 0.13 0.45 0.36 0.46 0.56 98 0.43 0.12 0.13 0.13 1.21 0.36 0.50 0.58 0.20 0.11 0.12 0.12 0.44 0.28 0.38 0.46 100 0.39 0.10 0.11 0.11 1.13 0.20 0.30 0.38 0.19 0.10 0.11 0.11 0.43 0.18 0.26 0.32 The underlined numbers are the ones where OBPI dominates CPPI.

investors’preferences inthegain domainwere more relevantindescribing thechoice betweenOBPIandCPPI. OBPI was thepreferredstrategyforexpectedutilityandprospectutilityinvestorswho arerisk averseinthegaindomainand, CPPI providedahigherrisk-adjustedperformanceforinvestorswhoareriskseekingingainsandriskaverseinlosses;However, no absolutedominance between OBPI andCPPI is observed for risk seeking investors.Increasingthe insuredpercentage

α

^{wasalways infavorof theCPPI whichwasshowntobe preferredforall typesofinvestorswhen}

α

=1. However,the gapbetweentherisk-adjustedperformanceofOBPIandCPPIdiminishedwhenthereferencepointincreasedandwhenthe marketwasmorevolatile.

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