SSRN-id1971095

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Stocks,Bonds,Risk,andtheHoldingPeriod:

AnInternationalPerspective

JavierEstrada

IESEBusinessSchool,DepartmentofFinance,Av.Pearson21,08034Barcelona,Spain

Tel:+34932534200,Fax:+34932534343,Email:[email protected]

Abstract

Thetimediversificationcontroversy,oneofthemostcontentiousissuesinassetallocation,referstotherelationshipbetweenriskandtheholdingperiod.Oneoftheaspectsofthiscontroversy is

relatedtowhetherstocksbecomemoreorlessriskythanbondsastheholdingperiodlengthens.To

be sure, thisquestiondoesnothaveanunequivocalanswer.But thebulkof the comprehensive

evidenceanalyzed in thisarticle, spanningover19 countriesand110years, suggests that time

doesdiversifyrisk.Inotherwords,althoughnotallresultspointinexactlythesamedirection,theoverallpicture thatemerges is thatas theholdingperiod lengthens stocksdobecome lessrisky

thanbonds.Thisconclusionfollowsfromananalysisbasedontwowaysofassessingreturnsand

severalwaysofassessingrisk.

December,2011

1.Introduction

Risk is a slippery concept. Finance academics and practitioners have been wrestling

withitsquantificationeversinceMarkowitz(1952)defineditasthestandarddeviationofan

assets returns.Since then,manyothervariables havebeenproposed todefine it, andmany

morewillsurelybeproposedinthefuture.

Justasthornyastheissueofdefiningriskisthatofdetermininghowriskevolveswith

theholdingperiod.Doesthe(absolute)riskofanassetincreaseordecreasewiththeholding

period?Arestocksriskieroveroneyearorover30years?Therearenouniversallyaccepted

answerstothesequestions.

Athirdandrelatedcontroversialissueisthatofdetermininghowtherelativeriskoftwo

assetsevolveswiththeholdingperiod.Canoneassetberiskierthananotherintheshorttermbutlessriskyinthelongterm?Arestocksriskierthanbondsintheshorttermbutlessriskyin

thelongterm?Again,therearenouniversallyacceptedanswerstothesequestions.

Thisarticleismostlyaboutthethirdissue(relativeriskandtheholdingperiod),buton

thewaytoshedsomelightonit,thefirst(thedefinitionofrisk)andsecond(absoluteriskand

theholdingperiod)issuesarealsodiscussed.Moreprecisely,thisarticleultimatelyfocuseson

determininghowtherelativeriskofstocksandbondsevolveswiththeholdingperiod,andit

IwouldliketothankGabrielaGiannattasioandSergiCutillasprovidedvaluableresearchassistance.Theviewsexpressedbelowandanyerrorsthatmayremainareentirelymyown.

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doessobyassessingtheevidencefrom19countriesover110years.Thebulkofthisevidence

suggeststhattimedoesdiversifyrisk;thatis,astheholdingperiodlengthens,stocksgradually

becomelessriskythanbonds.

Unsurprisingly,theevidenceshowsthatintheshorttermstocksareriskierthanbonds;

thisisthecaseregardlessofthetypeofreturns(annualizedorcumulative)onwhichinvestors

focusand theway they assess risk (generally asuncertaintyormorenarrowlyasdownside

potential).Moreinterestingly,theevidencealsoshowsthatinthemediumtolongtermstocks

becomelessriskythanbonds;thisisclearlythecaseifinvestorsfocusonannualizedreturns,

andlargelythecaseiftheyfocusoncumulativereturns.

Therestofthearticleisorganizedasfollows.Section2introducestheissueatstakeby

defining time diversification, very briefly reviewing the relevant literature, and taking a

preliminarylookattheevidence.Section3discussestheevidencefrom19countriesover11decades by focusing on returns, uncertainty, downside potential, holding periods, expected

shortfalls, and risk premiums. Finally, section 4 provides an assessment. An appendixwith

tablesconcludesthearticle.

2.TheIssue

Time diversification is one of the issues most hotly debated in asset allocation and

portfoliomanagement.Thefactthatagoodpartofthisdebatedependsonhowriskisdefined

andhow investors perceive itdoesobviously not help the convergenceofdifferent pointsof

view.Thissectionfirstdefinesthescopeoftimediversification;thenverybrieflyreviewssome

of themain contributions on this subject; and finally illustrates the different aspects of this

controversywithevidencefromtwointernationallydiversifiedportfoliosofstocksandbonds.

2.1.TimeDiversification Investors assessrisk in differentways and have differentviewson howrisk evolves

withtheholdingperiod.Thelatteristypicallyreferredtoasthetimediversificationcontroversy,

whichencompassesthreerelatedbutdifferentissues.Inevitably,thesethreedifferentaspectsof

thesameconcepthaveaddedmuchconfusiontothedebateofanissuethatiscontentiousto

beginwith.

First, supporters of time diversification argue that the (absolute) risk of an asset,

particularly stocks, decreases with the holding period; critics argue the opposite. Second,

supportersoftimediversificationarguethatthelongeristheholdingperiod,theloweristhe

probabilitythatariskier(morevolatile)assetunderperformsalessrisky(lessvolatile)asset;

criticsdonotdisagreebutarguethatthisshortfallprobabilityprovidesanincompleteviewof

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risk.1Strictlyspeaking,the timediversificationcontroversyreferstothesetwo issues,namely,

howtheabsoluteriskofanassetevolveswiththeholdingperiod,andhowtherelativeriskof

twoassetsevolveswiththeholdingperiod.

However, athird issueisusuallybroughtintothe debate.A standardassetallocation

recommendation suggests that younger investors should have a higher proportion of their

portfolioallocatedtoriskierassetsthanolderinvestors;thatis,asinvestorsgetolderandtheir

holdingperiodshortens,theyshouldgraduallydecreasetheproportionof riskierassets(such

as stocks) and increase that of less risky assets (such as bonds) in their portfolio.2Both

supportersandcriticsoftimediversificationlargelyagreewiththisrecommendation,butthey

dosofordifferentreasons.

Importantly,notethatitispossiblefortwoinvestorstoagreeonhowagivenmeasureof

riskevolveswiththeholdingperiod,butnotnecessarilyonhowrisk

itselfdoesso.Thismaybesimplybecausethetwoinvestorsassessriskwithdifferentvariables.Toillustrate,asdiscussed

in more detail below, one may assess risk with the volatility of annualized returns, which

unequivocallydecreaseswiththeholdingperiod,andtheotherwiththevolatilityofcumulative

returns,whichunequivocallyincreaseswiththeholdingperiod.

Note,also,thatitis possible fortwoinvestors to agreeon the factthatthe shortfall

probabilitydecreaseswiththeholdingperiod,butnotnecessarilyonhowtherelativeriskofthe

two relevantassets evolveswith the holding period.Thismay bein part for the reason just

discussed (different investors may assess risk with different variables) but it may also be

becauseoneinvestorassessesriskonlywiththeshortfallprobability,andtheotherdoesitby

consideringboththeshortfallprobabilityandthemagnitudeofthepotentialshortfall.

Finally,notethatitispossiblefortwoinvestorstoagreeontheplausibilityofdecreasing

the exposure to riskier assets as their holding period shortens, but not necessarily on the

plausibilityoftimediversification.Thismaybebecausethereseemstobeabroadconsensuson

the fact thatwhenhumancapitalis considered(thatis,whenfuturewealthdoesnotdepend

exclusively on investment returns), then the standard asset allocation recommendation is

plausibleeveniftimedoesnotdiversifyrisk.Bodie,Merton,andSamuelson(1992)arguethat

younginvestorscanchoosetoworkharderiffacedwithpoorreturns(somethingoldinvestors

wouldfinditmoredifficulttodobecausetheirhumancapitalislargelydepleted)andtherefore

canaffordtotakeonmoreriskwhentheyareyoungthanwhentheyareold.

1Theshortfallprobabilityisingeneraldefinedastheprobabilityofnotmeetingachosenbenchmark.Indiscussionsoftimediversification,ittypicallyreferstotheprobabilitythatstocksunderperformbonds.2This recommendation is consistent with the often used and abused rule of thumb that an investor

splittinghisportfoliobetweenstocksandbondsshouldhaveanexposuretobonds( xB)roughlyequaltohisage,andanexposuretostocks(xS)roughlyequalto100minushisage;thatis,xBAge,xS100Age,andxS+xB=1.

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In short, then, time diversification refers to the relationship between risk and the

holding period. Supporters of time diversification believe 1) that the risk of an asset,

particularlystocks,decreaseswiththeholdingperiod;2)thatthelongeristheholdingperiod,

theloweristheprobabilitythatariskier(morevolatile)assetunderperformsalessrisky(less

volatile)asset;and3)thatinvestorsshouldgraduallydecreasetheirexposuretoriskierassets

astheirholdingperiodshortens.Criticsoftimediversificationdisagreewith1);agreewith2)

but find the argument incomplete as far as the relationship between relative risk and the

holdingperiod isconcerned;and agreewith3)but for reasonsunrelated tothe relationship

betweenriskandtheholdingperiod.

2.2.BriefOverviewoftheLiterature Supportersandcriticsoftimediversificationhaveusedawidevarietyofargumentstomake their case. Some arguments have focused on the properties of returns and utility

functions. Samuelson (1963)was thefirst to formallyarguethat aninvestorsexposuretoa

riskyassetshouldbeindependentfromtheholdingperiod.Hisargument,elaboratedfurtherin

Samuelson (1989,1990, 1994), holdsundervery specific conditions.3If these conditions are

accepted, thenSamuelsonsresultshave the forceofmathematical truth; thosethat disagree

withSamuelsondonotdisputehismathbuthisassumptions,whichheactuallydidhimself.

Infact,Samuelson(1994)admitsthatthereareatleastthreesettingsinwhichlonger

holdingperiodsdo call for ahigher exposuretoriskierassets. First,whenreturnsaremean

reverting (rather than IID) and investors are more risk averse than implied by a log utility

function.4Second,when investors have a subsistence (or minimum) levelof terminalwealth

theywish toattain. And finally,whenhumancapital playsa role in investing decisions(the

BodieMertonSamuelsonargumentalreadydiscussed).

Otherargumentshave focusedonoptionsandthecostofprovidinginsuranceagainst

thecontingencythatstockreturnsfallshortfrombondreturns;see,forexample,Bodie(1995),

Thorley (1995), Taylor and Brown (1996), Merrill and Thorley (1996), and Alles (2008).

AlthoughBodies(1995)influentialarticleopenedthislineofinquiry,hisconclusionthatthe

costofproviding insurance against a return shortfall increaseswith the holdingperiod (and

thereforethattimemagnifiestheriskofinvestinginstocks)haslargelybeendiscredited.

3Moreprecisely,Samuelsonarguesthatifaninvestoraimstomaximizehisexpectedutility;hisutilityfunction is given by the log of wealth; his future wealth depends exclusively on the results of hisinvestments; and returns are IID, thenhis optimal exposure toa riskyasset is independent from theholdingperiod.AslongasreturnsareIID,thisconclusionalsoholdsforanyutilityfunctionthatexhibitsconstantrelativeriskaversion.4FamaandFrench(1988)andPoterbaandSummers(1988)provideevidenceofmeanrevertingreturns

over long horizons. Although Brown, Goetzmann, and Ross (1995) argue that this may be due tosurvivorshipbiasinthedata,actuallyfoundtobenegligiblebyDimson,Marsh,andStaunton(2011),itisgenerallyacceptedthatoverthelongtermmarketsdotendtomeanrevert.

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Argumentsinfavorofandagainsttimediversificationhavealsobeenbasedonthejoint

considerationofthemeanandvarianceofreturnsthroughmeanvarianceoptimalallocations,

Sharpe ratios, or time diversification indices; see, for example, Levy and Gunthorpe (1993),

Hodges,Taylor,andYoder(1997),HanssonandPersson(2000),andFabozzi,Focardi,andKolm

(2006).Alternativewaysofassessingrisk,suchasfirstorderstochasticdominance(Butlerand

Domian,1991)andvalueatrisk(Anderson,Malone,andMarshall, 2009), havealsoplayed a

roleinthetimediversificationdebate.

Finally, severalbehavioral argumentshavebeenofferedtoexplainwhyinvestors and

advisorsoftenbelievethattimediversifiesrisk;see,forexample,OlsenandKhaki(1998)and

FisherandStatman(1999).KritzmanandRich(1998)provideagoodoverviewofmanyofthe

theoreticalargumentsinfavorofandagainsttimediversification,andalsoclarifythespecific

conditionsunderwhichitholds.5

2.3.AbsoluteRiskandtheHoldingPeriod Oneoftheissuessurroundingthetimediversificationcontroversyisthetypeofreturns

onwhichinvestorsfocus.Tobesure,thisnotamatterofrightorwrong;someinvestorsfindit

plausible to focus on annualized returns and some others on cumulative returns. However,

thesetwotypesofreturnsmayleadtoconflictingviewsabouttherelationshipbetweenriskand

theholdingperiod.

Exhibit1displaysinpanelAsomesummarystatisticsontherealreturnsoftheDimson

MarshStauntonindexfortheworldstockmarketoverthe19002009period;thereturnsare

annual,indollars,adjustedbyUSinflation,andaccountforcapitalgains/lossesanddividends.

Consider the figures inpanelB,which follow from the series ofannualized returns, for five

holdingperiodsbetween1yearand30years.6Toclarify,thesefiguresfollowfromcalculating

firsttheannualizedreturnforallpossible,say,30yearperiods,andthensummarystatisticsout

of such series of 30year annualized returns; the same is the case for all the other holding

periodsconsideredinthepanel.

5Theyestablishthattheoptimalexposuretoariskyassetisindependentfromtheholdingperiodif aninvestor hasa logutility function, regardless of the characteristics of returns;oran investor exhibitsconstant relativerisk aversion andreturnsareIID. They alsoestablish thatthe optimalexposure toariskyassetdecreasesastheholdingperiodshortensifaninvestorexhibitsconstantrelativeriskaversion,

ismoreriskaversethanimpliedbyalogutilityfunction,andreturnsaremeanreverting.6Thetermsannualizedreturn,meanannualcompoundreturn,andgeometricmeanannualreturnarealldifferentnamesforthesameconceptandthereforeusedinterchangeablythroughoutthearticle.

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Exhibit1:TheWorldMarketThis exhibit shows information for the DimsonMarshStaunton (DMS) indexof the world stockmarket over the19002009 period. Panel A shows, for the series of annual returns, the sample size (T), arithmetic (AM) andgeometric(GM)meanreturn,standarddeviation(SD),andsemideviationfora0%benchmark(SSD).PanelBshows,fortheseriesofannualizedreturns,thestandarddeviation,lowestandhighestreturns,andspreadbetweenthem(Spread=HighestLowest). Panel C shows, for the series of cumulative returns, the arithmetic mean, standarddeviation,ratioofthelattertotheformer,lowestandhighestreturns,andspreadbetweenthem.PanelDshowstheshortfallprobability(SP),annualizedshortfallmagnitude(ASM),cumulativeshortfallmagnitude(CSM),annualizedexpectedshortfall(AES),andcumulativeexpectedshortfall(CES),allasdefinedinthetext,withrespecttotheDMSindexoftheworldbondmarket.Returnsarereal(adjustedbyUSinflation),indollars,andaccountforbothcapitalgains/lossesandcashflows(dividendsandcoupons).AllfiguresbutTandSD/AMin%.

PanelA T AM GM SD SSD

110 6.9 5.4 17.7 9.4

PanelB 1Year 5Years 10Years 20Years 30Years

SD 17.7 8.1 5.3 3.1 1.7 Lowest 40.4 13.3 6.5 0.6 2.1 Highest 70.1 22.5 18.2 13.5 9.6 Spread 110.5 35.8 24.6 14.2 7.5

PanelC 1Year 5Years 10Years 20Years 30Years

AM 6.9 40.9 94.7 263.0 521.9 SD 17.7 54.8 101.5 222.9 278.8 SD/AM 2.5 1.3 1.1 0.8 0.5 Lowest 40.4 51.1 48.8 12.0 87.3 Highest 70.1 175.7 430.9 1163.4 1473.4 Spread 110.5 226.8 479.7 1175.4 1386.1

PanelD 1Year 5Years 10Years 20Years 30Years

SP 33.6 26.4 23.8 7.7 2.5 ASM 12.8 5.3 2.0 1.4 0.4 CSM 12.8 29.5 29.6 81.4 50.8 AES 4.3 1.4 0.5 0.1 0.0 CES 4.3 7.8 7.0 6.3 1.3

Notethatboththestandarddeviationofannualizedreturnsandthespreadbetweenthe

highestandlowestannualizedreturnsdecreaseastheholdingperiodlengthens.Theintuitionis

clear:Thereturnoveranyshortholdingperiodcanbeextraordinarilyhighorlow,butasthe

holding period lengthens, extreme sustained returns become more and more unlikely. An

investorcangainover70%orloseover40%inanygivenyear(aspanelBshowsthatitdid

happen),butitwouldbefarmoreunlikely(andpanelBshowsthatitactuallyneverhappened)

foraninvestortolosethatmuchperyearoverfive,ten,ormoreyears.Someinvestorsconclude

outofthiskindofevidencethatriskdecreaseswiththeholdingperiod;or,inotherwords,that

timediversifiesrisk.

ConsidernowthefiguresinpanelC,whichfollowfromtheseriesofcumulative(ortotal)

returns,forfiveholdingperiodsbetween1yearand30years.Toclarify,thesefiguresfollow

fromcalculatingfirstthecumulativereturn(definedasthatbetweenthebeginningandtheend

ofeachholdingperiod)forallpossible,say,30yearperiods,andthensummarystatisticsoutof

suchseriesof30yearcumulativereturns;thesameisthecaseforalltheotherholdingperiods

consideredinthepanel.

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Notethatboththestandarddeviationofcumulativereturnsandthespreadbetweenthe

highestandlowestcumulativereturnsincreaseastheholdingperiodlengthens.Theintuitionis

againclear:Giventhecompoundingofcapitalandthetypicalupwardtrendofstockmarkets,

cumulativereturnstendtoincreasewiththelengthoftheholdingperiod(dailyreturnstendto

besmallerthanmonthlyreturns,whichtendtobesmallerthanannualreturns,whichtendtobe

smallerthanfiveyearreturns,),andsodomeanreturns,thedispersionaroundthosemean

returns,andthespreadbetweenthehighestandlowestreturns,aspanelCclearlyshows.Some

investorsconcludeoutofthiskindofevidencethatriskincreaseswiththeholdingperiod;or,in

otherwords,thattimemagnifiesrisk.

Investors that assess risk on the basis of annualized returns focus on the fact that

although inshortholdingperiods thestockmarket ishighly unpredictable, the longer is the

holdingperiod,themorelikelyitbecomesthatannualizedreturnswillreverttotheirlongtermmean.Asmentionedbefore,thereturninanygivenyearmaybeextraordinarilyhighorlow,but

as theholdingperiod lengthens, thereturnperyear(thatis,theannualizedreturnovereach

holdingperiod)becomeslessandlessextraordinaryandconvergestothelongtermgeometric

meanreturn.Thisconvergenceastheholdingperiodlengthensisviewedbysomeinvestorsasa

decreaseinrisk.

Investorsthatassessriskonthebasisofcumulativereturns,ontheotherhand,focuson

the levelofwealth accumulatedat the endofa holdingperiod.To illustrate,consider a$100

investment in the worldstockmarket. Notice that thespread in annualized returns shrinks

considerablywhencomparing,say,1yearto30yearholdingperiods(from110.5%to7.5%,as

panelB shows).Butalsonoticethat thespread intheterminalvalueoftheinvestmentisjust

$110.5 (=$170.1$59.6) over 1 year, grows to $479.7 ($530.9$51.2) over 10 years, and to

$1,386.1(=$1,573.4$187.3)over30years,aspanelCimplies.Thisincreaseinthedispersionof

terminalwealthastheholdingperiodlengthensisviewedbysomeinvestorsasanincreasein

risk.

Again,thereisnorightorwrongapproach;itisonlyamatterofperspective.Inthesame

waythatsomeinvestorsassessriskwithvolatility,otherswithdownsidevolatility,andothers

withfactors,someinvestorsassesstherelationshipbetweenriskandtheholdingperiodwith

annualizedreturnsandotherswithcumulativereturns.Obviously,thefocusoneithertypeof

returnsdoesnotchangetheabsoluteriskofanasset,therelativeriskoftwoassets,orhow

absoluteandrelativeriskevolvewiththeholdingperiod;itonlyhasanimpactonaninvestors

perceptionofrisk.AsplausiblyarguedbyMcEnally(1985),riskisultimatelyintheeyesofthe

beholder.

Thatbeingsaid,twothingsareworthhighlighting.First,notethatcomparingcumulative

returnsoverholdingperiodsofdifferentlengthmaybeakintocomparingapplesandoranges.

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Inordertomakeanapplestoapplescomparison,thereturnsofholdingperiodsofdifferent

lengthneedtobestandardized,andthatispreciselywhatannualizingreturnsdoes.Afocuson

cumulativereturnsmakesitarguablewhethera20%returnoverthirtyyearsisbetterorworse

thanan18%returnoverfiveyears.Butitisunambiguousthatthereturnperyearhasbeen

muchhigherinthesecondcase(3.4%)thaninthefirstcase(0.6%).

Arguingthatcomparingcumulativereturnsoverdifferentholdingperiodsmakeslittle

sense,Fabozzi,Focardi,andKolm(2006)proposetostandardizeriskbydividingthechosen

measureofriskoveragivenholdingperiodbytheexpectedreturnoverthatholdingperiod.

Theyalsoarguethattimediversifiesriskifsuchratiobetweenriskandreturndecreaseswith

theholdingperiod.ThethirdrowofpanelCofExhibit1showstheratiobetweenvolatilityand

meanreturn(thesecondrowdividedbythefirst),bothbasedoncumulativereturns.Asthese

figuresclearlyshow, risk

per

unit

of

return(orreturnadjustedrisk),clearlydecreaseswiththeholdingperiod,thussuggestingthattimediversifiesrisk.

Second,panelCshowsthatthelowestcumulativereturndecreaseswhengoingfroma

1year(40.4%)toa5year(51.1%)holdingperiod,butthenitsteadilyincreasesfromthat

point on. This implies that although the spread between the highest and lowest cumulative

returnssteadilyincreaseswiththeholdingperiod,andsodoestheuncertaintyaboutterminal

wealth,theworstcasescenariodoesnotgetworse; itactuallygetsbetter.Inotherwords,most

of the increase in the spread is due to an increase in upside potential. Unlike the gamble

consideredbySamuelson(1963),inwhichpotentiallossesmountasthenumberoftimesthe

gamble is played increases, lengthening the holding period in the stock market typically

decreases the downside potential by shrinking, and eventually reversing the sign of, the

potentiallosses.

2.4.RelativeRiskandtheHoldingPeriod Theissuesdiscussedintheprevioussectionconcerntherelationshipbetweentherisk

of an individual asset (absolute risk) and the holding period. This section focuses on the

relationshipbetweentherelative riskof twoassetsandtheholdingperiod,which isthemain

issuediscussedinthisarticle.

Supportersoftimediversificationoftenhighlightthattheshortfallprobabilitydecreases

astheholdingperiodlengthens(seeSiegel,2008);thatis,thelongeristheholdingperiod,the

loweristheprobabilitythatariskier(morevolatile)asset,suchasstocks,underperformsaless

risky(lessvolatile)asset,suchasbonds.PanelDofExhibit1showsinitsfirstlinetheshortfall

probability (SP), calculated as the proportion of holding periods in which the world stock

marketunderperformedtheworldbondmarket,overfiveholdingperiodsbetweenone1year

and30years.Asthesefiguresshow,althoughstocksunderperformedbondsinoveronethirdof

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all1yearperiods,theydidsoinlessthanonefourthofall10yearperiods,andinjust2.5%of

all30yearperiods.Inotherwords,theevidencedoesshowthattheshortfallprobabilityclearly

decreasesastheholdingperiodlengthens.

Criticsoftimediversificationareundeterredbythisfact.Theyarguethattheprobability

of underperformance is important but so is themagnitude of the underperformance. Put

differently, they argue that focusing on the shortfall probability and ignoring the shortfall

magnitude results in a misleading assessment of relative risk. So, how does the shortfall

magnitude evolve with the holding period? Here again the use of annualized or cumulative

returnsleadstoconflictingresults.

ThesecondlineofpanelDshowstheannualizedshortfallmagnitude(ASM),definedas

theaveragedifferencebetweentheannualizedreturnofthebondmarketandthatofthestock

marketovertheholdingperiodsinwhichthelatterunderperformedtheformer.Thethirdlineof the same panel shows the cumulative shortfall magnitude (CSM), defined as the average

differencebetweenthecumulativereturnofthebondmarketandthatofthestockmarketover

theholdingperiodsinwhichthelatterunderperformedtheformer.Asthesetwolinesshow,the

ASMclearlydecreases,andtheCSMlargelyincreases(peakingat20years),withtheholding

period,thusyieldingconflictingresults.

Thatbeingsaid,theshortfallprobabilityandtheshortfallmagnitudecouldandshould

beconsideredjointly.ThefourthlineofpanelDshowstheannualizedexpectedshortfall(AES),

defined as the product between the shortfall probability and the annualized shortfall

magnitude;thatis,AES=SPASM.Thefifthlineshowsthecumulativeexpectedshortfall(CES),

defined as the product between the shortfall probability and the cumulative shortfall

magnitude;thatis,CES=SPCSM.NotethattheAESandtheCESquantifyexpected losses,not

expected returns.More precisely, they account for the probability that stocks underperform

bondsandthemagnitudeoftheshortfall,butnotforprobabilitythatstocksoutperformbonds

and themagnitude oftheoutperformance. Byway ofanalogy, flipping acoinwithpayoffsof

+30%and10%wouldhaveanexpectedlossof5%(=0.100.5)butanexpectedreturnof10%

(=0.100.5+0.30.5).Inshort,theAESandtheCESisolatethedownsidebutdonotaccountfor

theupside.

AspanelDshows,theAESsteadilydecreaseswiththeholdingperiod;theCES,inturn,

peaksat5yearsandthenalsosteadilydecreaseswith theholdingperiod.Thissuggeststhat

regardlessofwhetherinvestorsfocusonannualizedorcumulativereturns,jointlyconsidering

theshortfallprobabilityandtheshortfallmagnitudeleadstotheconclusionthatforinvestment

horizonslongerthanfiveyearsstocksbecomelessriskythanbonds,atleastinthesensethat

theirexpectedshortfalldecreaseswiththeholdingperiod.

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3.Evidence

Theresultsdiscussedsofar,basedontwointernationallydiversifiedportfoliosofstocks

andbonds,havehighlightedtheissuesatstake,introducedsomerelevantdefinitions,andset

thestageforamorethoroughanalysisatthecountrylevel.Therelationshipbetweenrelative

riskandtheholdingperiodcanonlybeevaluatedinameaningfulwaywithacomprehensive

sample,andtheDimsonMarshStaunton (DMS) dataset,whichcovers 19 countriesover 110

years,isidealforthispurpose.Exhibit2summarizessomecharacteristicsoftheseriesofannual

realreturnsofstocksandbondsoverthe19002009periodforallthecountriesinthesample.

Returnsareinlocalcurrency,adjustedbylocalinflation,andaccountforcapitalgains/losses

andcashflows(dividendsorcoupons).7

Exhibit2:SummaryStatisticsThisexhibitshows,fortheseriesofannualreturns,thearithmetic(AM)andgeometric(GM)meanreturn,standarddeviation (SD), and semideviation for a 0%benchmark (SSD) for all the stock (S)and bond (B)markets intheDimsonMarshStaunton(DMS)datasetoverthe19002009period.Returnsarereal(adjustedbylocalinflation),inlocalcurrency,andaccountforbothcapitalgains/lossesandcashflows(dividendsorcoupons).Allfiguresin%.

AM GM SD SSD Country S B S B S B S B

Australia 9.1 2.3 7.5 1.4 18.2 13.2 9.3 7.7 Belgium 5.2 0.6 2.5 0.1 23.6 12.0 12.6 8.3 Canada 7.2 2.5 5.8 2.0 17.2 10.4 8.5 5.5 Denmark 6.7 3.6 4.9 3.0 20.7 11.6 8.9 5.1 Finland 9.1 1.0 5.1 0.3 30.3 13.7 14.1 11.1 France 5.7 0.7 3.1 0.2 23.5 13.0 12.6 9.7

Germany 8.1 0.7 3.0 2.0 32.2 15.5 15.1 12.6 Ireland 6.5 2.1 3.8 1.1 23.1 14.6 12.2 7.9 Italy 6.2 0.4 2.1 1.6 29.0 14.1 15.8 11.9 Japan 8.6 1.5 3.8 1.2 29.8 20.1 15.5 15.0 Netherlands 7.1 1.8 4.9 1.4 21.8 9.4 10.4 5.2 NewZealand 7.6 2.4 5.9 2.0 19.7 9.0 9.2 4.9 Norway 7.2 2.4 4.1 1.7 27.4 12.2 11.9 7.0 S.Africa 9.5 2.2 7.2 1.7 22.5 10.4 9.2 5.9 Spain 6.0 2.0 3.8 1.4 22.1 11.7 11.1 7.0 Sweden 8.6 3.2 6.2 2.5 22.8 12.4 10.9 6.1 Switzerland 6.1 2.5 4.3 2.1 19.8 9.3 10.3 4.3 UK 7.2 2.2 5.3 1.3 20.0 13.6 9.9 7.2 USA 8.2 2.4 6.2 1.9 20.3 10.1 10.6 5.3

Average

7.4

1.9

4.7

0.9

23.4

12.4

11.5

7.8

3.1.Returns TheDMSdatasetprovidesavery broadperspective on theperformance of stock and

bondsmarketsoverthelast11decades.ThecolumnslabeledGMinExhibit2show,perhaps

unsurprisingly,thattheannualizedrealreturnofstockshasbeenbothpositiveandhigherthan

thatofbondsineverycountry in the sample. Inotherwords, inthe longterm,not onlydid

stocksneverfailtobeatinflationbutalsoprovidedbetterprotectionagainstitthanbonds.On

7Jorion(2003)alsoconsidersalargesampleofcountries,bothdevelopedandemerging,buthissampleperiodisshorterthanthatinthisarticle.

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averageacross the 19countries inthe sample, stocks provided investorswith anannualized

realreturnof4.7%,3.8percentagepointshigherthanthatofbonds(0.9%).

Somewhatmoresurprisingmaybethefactthatinsixofthe19countriesinthesample

(Belgium,Finland,France,German,Italy,andJapan)bondsdelivereda negativeannualizedreal

return;hence,inthesecountries,inthelongterm,bondsdidnotkeepupwithinflation.Thus,

although ineverycountry forwhich longtermdataexiststocks beat inflation and increased

purchasing power, in almost one thirdof those countries bonds failed tobeat inflation and

actuallydecreasedpurchasingpower.Andyetmostinvestorsviewstocksasriskierthanbonds.

Why?

3.2.Uncertainty Asalreadymentioned,riskisaslipperyconceptthatcanbemeasuredinmanydifferentways,andvolatilityisperhapsthevariablemostwidelyusedtoassessit.Thecolumnslabeled

SD in Exhibit 2 and the columns labeled 1 Year in panel A of Exhibit 3 show, perhaps

unsurprisingly,thatthevolatilityofstockshasbeenhigherthanthatofbondsineverycountry.

Onaverageacrossthe19countriesinthesample,thestandarddeviationofannualreturnswas

23.4%inthecaseofstocks,almosttwiceashighasthatofbonds(12.4%).

Thepicturedoesnotchangesubstantiallyifriskismeasuredwiththespreadbetween

thehighestandlowestannualreturnsoverthewhole19002009periodinstead.Thecolumns

labeled 1Yearin panelB ofExhibit3 showthat thespreadofstocks ishigher thanthat of

bondsineverycountry.Onaverageacrossthe19countriesinthesample,theannualspreadof

stockswas154.0%,substantiallyhigherthanthatofbonds(87.7%).8

Inshort,aslongasinvestorsviewriskasshorttermuncertainty,thereislittlequestion

thatstocksareriskierthanbonds.Bothvolatilityandspreadsunequivocallyindicatethatinthe

short term stock returns are more unpredictable than bond returns. But, clearly, not all

investorsassessriskthisway.

8ToclarifythecalculationofspreadsconsidertheUSmarket.The94.5%spreadofstocksresultsfromthe

differencebetweenthehighest(56.5%,in1933)andlowest(38.0%,in1931)annualreturnoverthewhole19002009period.Similarly,the54.5%spreadofbondsresultsfromthedifferencebetweenthehighest(35.1%,in1982)andlowest(19.4%,in1918)annualreturnoverthe19002009period.

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Exhibit3:UncertaintyAnnualizedReturnsThisexhibitshowsthevolatility(panelA)andspread(panelB,asdefinedinExhibit1)ofstocks(S)andbonds(B)overfivedifferentholdingperiods,basedonannualizedreturns.ThedataisdescribedinExhibit2.Allfiguresin%.

1Year 5Years 10Years 20Years 30Years PanelA S B S B S B S B S B

Australia 18.2 13.2 7.5 7.2 4.6 6.0 2.9 4.4 2.3 2.8

Belgium 23.6 12.0 10.4 7.4 7.0 6.4 4.6 4.7 3.1 3.4 Canada 17.2 10.4 7.3 6.0 4.1 4.9 2.6 3.6 1.7 2.1 Denmark 20.7 11.6 6.9 5.8 4.0 4.9 2.6 3.7 1.8 2.7 Finland 30.3 13.7 14.0 10.0 8.8 8.1 5.2 5.1 3.2 2.8 France 23.5 13.0 11.3 9.6 7.4 8.4 4.5 6.4 2.9 5.3 Germany 32.2 15.5 14.3 12.9 9.9 10.8 5.9 7.5 4.3 4.8 Ireland 23.1 14.6 9.4 7.4 6.6 6.0 4.1 4.3 2.6 2.7 Italy 29.0 14.1 12.2 11.2 8.4 9.6 4.2 6.5 2.2 4.6 Japan 29.8 20.1 16.2 13.8 11.6 12.3 6.9 9.7 5.0 7.5 Netherlands 21.8 9.4 9.8 5.3 6.5 4.4 4.3 3.5 2.8 2.6 NewZealand 19.7 9.0 7.6 5.8 4.0 4.8 2.0 3.5 1.6 2.1 Norway 27.4 12.2 9.0 6.9 5.6 5.7 3.8 3.7 2.5 2.2

S.Africa 22.5 10.4 8.8 5.5 5.3 4.5 3.1 3.1 1.7 2.0 Spain 22.1 11.7 11.7 5.8 7.4 4.4 4.6 3.1 2.3 2.1 Sweden 22.8 12.4 9.9 6.6 6.3 5.1 4.4 3.7 3.2 2.6 Switzerland 19.8 9.3 9.1 5.0 5.8 3.5 3.7 2.0 2.1 1.1 UK 20.0 13.6 8.0 7.0 5.3 5.5 3.1 3.9 1.8 2.6 USA 20.3 10.1 8.2 5.1 5.2 4.1 3.2 3.1 1.7 2.0 Average 23.4 12.4 10.1 7.6 6.5 6.3 4.0 4.5 2.6 3.1

1Year 5Years 10Years 20Years 30Years PanelB S B S B S B S B S B

Australia 94.0 88.8 43.1 37.7 22.8 22.5 12.3 14.7 8.0 10.3 Belgium 166.6 71.2 53.0 32.9 27.4 26.1 21.2 17.5 13.4 11.7 Canada 89.0 67.6 36.5 29.7 19.2 19.9 10.6 14.4 6.7 8.3 Denmark 157.0 68.3 37.8 32.1 19.2 21.7 11.9 13.0 7.7 9.1

Finland 222.5 99.7 81.6 51.8 42.5 31.4 23.4 17.4 14.1 10.6 France 108.7 79.4 55.7 47.5 34.4 33.5 19.3 21.9 12.5 15.6 Germany 245.4 157.5 91.5 62.2 57.0 35.7 32.2 21.4 20.6 14.3 Ireland 133.8 95.3 42.9 35.9 26.8 24.5 18.5 15.9 11.6 11.1 Italy 193.5 92.9 54.6 59.5 35.1 39.2 16.7 23.1 10.8 15.2 Japan 206.6 147.3 98.9 80.0 62.0 53.8 30.8 30.1 22.9 20.5 Netherlands 152.0 50.9 42.2 27.8 24.4 19.0 20.0 11.2 10.9 8.8 NewZealand 160.0 57.8 52.6 28.9 24.8 19.3 8.4 12.9 7.1 8.5 Norway 220.5 110.2 57.6 41.3 26.1 29.7 15.5 15.2 13.3 9.3 S.Africa 155.1 69.6 49.2 26.2 23.5 18.2 14.5 12.0 7.0 6.9 Spain 142.7 83.5 63.0 24.3 35.3 18.0 20.6 12.9 12.0 9.2 Sweden 133.3 104.8 52.2 35.9 32.2 22.5 22.5 15.0 13.0 8.7 Switzerland 97.2 77.5 45.0 33.8 29.3 21.1 14.6 11.8 8.1 5.3 UK 153.7 89.6 40.5 34.1 23.2 24.5 15.2 14.7 8.4 10.6 USA 94.5 54.5 38.6 28.2 20.8 16.6 11.8 11.7 7.7 8.1 Average 154.0 87.7 54.6 39.5 30.8 26.2 17.9 16.1 11.4 10.6

3.3.DownsidePotential Arguably,investorsdonotdislikevolatilityoruncertainty;rather,theydislikedownside

volatilityandnegativesurprises,particularlywhenthesearelarge.ThecolumnslabeledSSDin

Exhibit 2 and the columns labeled 1 Year in panelA of Exhibit 4 show that the downside

volatilityofstockshasbeenhigherthanthatofbondsineverycountry.Onaverageacrossthe

19 countries in the sample, the annual semideviationwith respect to a 0%benchmark was

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11.5% for stocks and 7.8% for bonds.9Importantly, this measure of risk does not simply

measuredeparturesfromthemeanreturn;itmeasuresdownsidedeparturesfromanychosen

benchmark.Putdifferently,thesemideviationsinExhibits2and4measurevolatilitybelow0%,

orthevolatilityofnegative(real)returns.

Exhibit4:DownsidePotentialAnnualizedReturnsThis exhibit shows the semideviation for a 0%benchmark(panel A)and the lowestreturn over the 19002009period(panelB)ofstocks( S)andbonds(B)overfivedifferentholdingperiods,basedonannualizedreturns.ThedataisdescribedinExhibit2.Allfiguresin%.


Australia 9.3 7.7 2.8 4.0 0.9 3.2 0.0 1.8 0.0 1.4 Belgium 12.6 8.3 6.0 5.6 3.3 5.0 1.5 3.5 0.5 2.3 Canada 8.5 5.5 2.2 3.2 0.4 2.1 0.0 1.0 0.0 0.4 Denmark 8.9 5.1 2.0 2.5 0.3 1.5 0.0 0.5 0.0 0.2 Finland 14.1 11.1 6.9 8.6 3.8 6.8 0.8 4.1 0.0 2.3

France 12.6 9.7 6.0 7.9 3.3 6.9 1.3 5.0 0.3 4.0 Germany 15.1 12.6 7.9 12.0 4.7 10.0 2.7 7.1 1.5 4.6 Ireland 12.2 7.9 3.8 4.3 2.1 3.2 0.6 1.9 0.0 1.3 Italy 15.8 11.9 7.5 10.1 4.6 8.8 1.7 6.4 0.5 5.3 Japan 15.5 15.0 10.5 12.2 7.3 11.1 3.4 9.1 1.8 7.4 Netherlands 10.4 5.2 3.5 2.7 1.5 2.1 0.4 1.5 0.0 1.3 NewZealand 9.2 4.9 2.8 2.8 0.6 2.2 0.0 1.2 0.0 0.8 Norway 11.9 7.0 4.8 4.2 2.4 2.9 0.5 1.4 0.2 0.8 S.Africa 9.2 5.9 1.5 2.6 0.6 2.0 0.0 1.3 0.0 0.9 Spain 11.1 7.0 5.9 3.5 3.9 2.4 1.3 1.6 0.2 1.3 Sweden 10.9 6.1 3.8 3.1 2.1 1.7 0.5 1.0 0.0 0.8 Switzerland 10.3 4.3 4.3 2.7 2.2 1.5 0.7 0.5 0.0 0.0 UK 9.9 7.2 3.2 4.2 1.1 3.0 0.2 1.6 0.0 1.1 USA 10.6 5.3 2.8 2.5 0.9 1.5 0.0 0.8 0.0 0.6 Average 11.5 7.8 4.6 5.2 2.4 4.1 0.8 2.7 0.3 1.9 1Year 5Years 10Years 20Years 30Years PanelB S B S B S B S B S B

Australia 42.5 26.6 19.4 14.8 5.6 8.4 1.6 4.0 2.9 4.0 Belgium 57.1 30.6 23.6 19.8 11.3 16.0 7.5 9.5 3.1 5.4 Canada 33.8 25.9 10.1 14.2 2.5 8.8 0.9 4.7 3.0 1.4 Denmark 49.2 18.2 11.8 12.6 2.0 7.9 0.5 2.3 2.3 0.7 Finland 60.8 69.5 31.8 36.6 14.6 20.8 3.5 9.9 0.1 5.6 France 42.7 43.5 25.8 32.5 15.2 22.2 5.7 12.0 1.6 7.5 Germany 90.8 95.0 41.8 45.5 19.3 25.1 10.2 14.2 6.4 9.2 Ireland 65.4 34.1 14.2 16.3 9.4 11.4 4.2 5.9 1.0 4.1 Italy 72.9 64.3 28.5 45.3 13.8 28.2 6.0 15.3 2.2 10.0 Japan 85.5 77.5 52.3 58.5 30.9 39.6 12.7 22.6 8.6 14.7 Netherlands 50.4 18.1 12.1 11.3 5.5 7.2 2.7 3.7 0.2 3.4 NewZealand 54.7 23.7 19.2 10.9 3.7 7.1 1.8 4.0 2.1 2.5 Norway 53.6 48.0 23.4 23.2 10.4 13.2 2.3 5.4 1.2 2.4 S.Africa 52.2 32.6 8.4 10.7 5.1 6.5 0.1 3.9 4.0 2.1 Spain 43.3 30.2 26.8 11.6 16.8 7.6 5.3 4.9 0.8 3.5 Sweden 43.6 36.7 20.2 13.9 9.2 6.9 3.5 3.4 0.1 2.3 Switzerland 37.8 21.4 20.7 15.0 11.1 8.4 3.8 3.1 0.3 0.4 UK 57.1 30.7 18.0 14.6 5.6 11.2 1.7 5.8 2.4 4.1 USA 38.0 19.4 11.4 10.4 4.0 5.4 0.9 3.1 2.8 2.0 Average 54.3 39.3 22.1 22.0 10.3 13.8 3.3 7.3 0.2 4.4

9

ThesemideviationwithrespecttoabenchmarkB(B)isgivenbyB={(1/T)tMin(RtB)2

}1/2

,whereRdenotesreturns,Tthenumberofobservations,andtindexestime.Throughoutthisarticlethebenchmarkusedis0%.Foranintroductiontothesemideviation,seeEstrada(2006).

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Besidesbeingconcernedaboutdownsidevolatility,whichisproperly capturedby the

semideviation, investors often are also concerned about worstcase scenarios. The columns

labeled1YearinpanelBofExhibit4showthelowestannualreturnoverthewhole19002009

periodforallthestockandbondmarketsinthesample.Withonlythreeexceptions(Finland,

France,andGermany),theworstannualreturnforstockswaslowerthanthatforbonds.Across

allthemarketsinthesample,theworstannualreturnforstocksandbondsaveraged54.3%

and39.3%,adifferenceof15percentagepoints.Thus,withminorexceptions,intheworstof

timesstockspunishedinvestorswithhigherlossesthandidbonds.

In short, if investors view risk not as uncertainty in general but more narrowly as

downsidepotential,itstillremainsthecasethatintheshorttermstocksareriskierthanbonds.

Boththesemideviationandworstcasescenariosindicatethatstocksaremorelikelytodeliver

unpleasantsurprisestoinvestorsthanbonds.Thus,aslongasinvestorsfocusontheshortterm,andregardlessofwhethertheyviewriskasuncertainty(measuredbyvolatilityorspreads)or

downside potential (measured by the semideviation or worstcase scenarios), the evidence

clearlysuggeststhatstocksareriskierthanbonds.

3.4.TheHoldingPeriod Thatbeingsaid,itisobviousthatnotallinvestorsfocusontheshortterm;thosesaving

fortheirchildrenscollegetuition,theirdreamhouse,orretirement,amongmanyothers,have

muchlongerinvestmenthorizons.Itisnecessarytoexplore,then,howthelengthoftheholding

periodaffectstherelativeriskofstocksandbonds.

PanelAinExhibit3showsthatfora10yearholdingperiod,theannualizedvolatilityof

stocks(6.5%)was,onaverage,almostthesameasthatofbonds(6.3%).For20/30yearholding

periods,theannualizedvolatilityofstockswas,inmostcountriesandonaverage,actuallylower

thanthatofbonds.PanelBshowsthatfor20/30yearholdingperiods,theannualizedspreadof

stockswas,onaverage, just slightlyhigherthan thatofbonds. Thus, the figures inExhibit3

essentiallyshowthat,althoughstocksareunquestionablyriskierthanbondsintheshortterm,

astheholdingperiodlengthens,theuncertaintyabouttheexpectedreturnfromstocksdeclines

muchmorerapidlythantheuncertaintyabouttheexpectedreturnfrombonds.

Afocusaway from uncertainty in general andmorenarrowly on downsidepotential

actuallystrengthenstheideathatastheholdingperiodlengthens,stocksgraduallybecomeless

riskythanbonds.PanelAinExhibit4 showsthatfora 5yearholdingperiod, the annualized

semideviationofstocks(4.6%)was,onaverage,lowerthanthatofbonds(5.2%).Furthermore,

for10/20/30yearholdingperiods,thesemideviationofstockswaslowerthanthatofbonds

notonlyonaveragebutalsoinmostcountries.Inotherwords,astheholdingperiodlengthens,

bondsexposeinvestorstohigherdownsidepotentialthandostocks.

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Importantly,panelBinExhibit4showsthatforholdingperiodsasshortasfiveyears,

the lowest annualized return of stocks over the whole 19002009 period (22.1%) was on

averagevirtuallyidenticaltothatofbonds(22.0%);inroughlyhalfthecountries,infact,the

lowest5yearannualizedreturnofstockswashigherthanthatofbonds.Therestofthecolumns

inthispanelshowthatfor10/20/30yearholdingperiods,inmostcountriesandonaverage,

thelowestannualizedreturnofstocksoverthewhole19002009periodwashigherthanthatof

bonds.Infact,althoughfora20yearholdingperiodthelowestannualizedreturnofbondswas

negative in all countries, that of stocks was positive in six countries; for a 30year holding

period, although the lowest annualized return of bonds was negative in all countries but

Switzerland,thatofstockswaspositivein11countries.

Thus, the evidenceshows thatintheshort term stockshave highervolatility, higher

spreads,higherdownsidepotential,anddelivermorepainfullossesthandobonds.However,for holding periods longer than ten years, the opposite is largely the case; that is, stocks

graduallybecomelessriskythanbonds.TheseconclusionsfollowfromExhibits3and4,bothof

whicharebasedonannualizedreturns.However,asalreadydiscussed,afocusoncumulative

returnsmayormaynotleadtothesameconclusions.ExhibitsA1andA2intheappendix,both

basedoncumulativereturns,explorethisissue.

ExhibitA1showsinpanelAvolatilityandinpanelBspreadsforstocksandbondsover

differentholdingperiods.Clearly,astheholdingperiodlengthens,stocksbecomeincreasingly

morevolatilethanbondsandthespreadofstocksgraduallyincreasesrelativetothatofbonds.

Perhapsunsurprisingly,thesefindingsbased oncumulativereturnsseem tocontradict those

basedonannualizedreturnsbysuggestingthat,astheholdingperiodlengthens,stocksbecome

riskier than bonds. However,Exhibit A2, also in the appendixand also basedoncumulative

returns,putsthemessagefromExhibitA1intoperspective.

Panel A in Exhibit A2 shows that for holding periods as short as five years, the

semideviationofstocks is lowerthan thatofbonds, bothonaverageand inoverhalf of the

countriesinthesample.Furthermore,forholdingperiodstenyearsorlonger,thesemideviation

ofstocksislowerthanthatofbondsnotonlyonaveragebutalsoinalmosteverycountry.The

importantimplicationofthisevidenceisthatalthoughstocksbecomegraduallymorevolatile

thanbondsastheholdingperiodslengthens,mostoftheincreaseinvolatilityisontheupside.

Infact,panelB,whichshowsthelowestcumulativereturnoverthe19002009period

fordifferentholdingperiodsstrengthensthisconclusion.Foraholdingperiodoffiveyears,the

lowest return delivered by stocks was onaverage just slightly lower than that delivered by

bonds. For holding periods ten years or longer, the lowest return delivered by stocks was

actuallyhigherthanthatdeliveredbybonds,bothonaverageandinalmosteverycountry.Fora

30yearholdingperiod,thelowestreturndeliveredbybondswasnegativeineverycountrybut

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Switzerland,butthatdeliveredbystockswaspositivebothonaverageandinmorethanhalfof

thecountriesinthesample.

These results suggest that if investors focus on cumulative returns, volatility and

spreadstakeninisolationmayleadthemtobelievethatastheholdingperiodlengthens,stocks

becomeriskierthanbonds.However,amorethoroughassessmentthatdistinguishesbetween

uncertaintyanddownsidepotentialshouldleadthemtoreconsider.Theevidencesuggeststhat

mostof the increase in the uncertaintyofstocks relative tobonds isactuallyan increase in

upsidepotential;thatis,itisuncertaintyabouthowmuchmorestockswilldeliverthanbonds.

Andthatcanhardlybecalledrisk.

3.5.TheExpectedShortfall As already discussed, when assessing the relationship between relative risk and theholding period, both the shortfall probabilityand the shortfallmagnitude combined into the

expectedshortfallcanandshouldplayaroleintheevaluation.Tothatpurpose,Exhibit5shows

the shortfall probability (SP), the annualized shortfall magnitude (ASM), and the annualized

expectedshortfall(AES)forallthemarketsinthesample,allasdefinedinsection2.4.

Panel A shows that with only minor exceptions, the SP steadily decreases with the

holdingperiod.Onaverageacrossallcountries,stockmarketsunderperformedbondmarketsin

40%ofall1yearperiods,inlessthan25%ofall10yearperiods,andinlessthan10%ofall30

yearperiods.Insixmarkets,infact,stocksneverunderperformedbondsover30years.These

figures clearly suggest that although in the short term stocks are far from guaranteed to

outperformbonds,inthelongtermtheyareverylikelytodoso.

PanelBshowsthatbothonaverageandin everycountrytheASMsteadilydecreases

withtheholdingperiod.PanelC,whichcombinesthefiguresfrompanelsAandB,showsthat

bothonaverageandineverycountrytheAESalsosteadilydecreaseswiththeholdingperiod.In

otherwords, as the investment horizon lengthens, stocks gradually become less risky than

bondsinthesensethattheannualizedexpectedshortfallsteadilydecreaseswiththeinvestment

horizon.

A focus on cumulative returns, summarized in Exhibit A3 in the appendix, tells a

somewhat(butnottotally)differentstory.PanelAshowsthesameshortfallprobabilitiesshown

inpanelAofExhibit5.PanelBshowsthatthecumulativeshortfallmagnitude(CSM)increases

with the holding period inmost countries, although inmany countries itpeaksat20years.

PanelC,whichcombines the figures frompanels A and B, shows that onaverage across all

countries,aswellasinsevencountries,thecumulativeexpectedshortfall(CES)peaksatthe20

yearholdingperiod.InsixcountriestheCESpeaksataninvestmenthorizonof10years,infour

countriesat5years,andintwocountriesat30years.Thus,in13ofthe19countriesinthe

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sampletheCESpeaksatholdingperiodsbetween10and20years,afterwhichstocksbecome

lessriskythanbondsasmeasuredbytheircumulativeexpectedshortfall.Inshort,focusingon

cumulative(ratherthanonannualized)returnsdoesnotreallyreversetheconclusionthat,as

the holding period lengthens, stocks become less risky than bonds; itmerely increases the

lengthoftheholdingperiodafterwhichthishappens.

Exhibit5:ExpectedShortfallsAnnualizedReturnsThis exhibit shows shortfall probabilities (panel A), annualized shortfall magnitudes (panel B), and annualizedexpectedshortfalls(panelC),allasdefinedinthetext,over1year(1Y),5year(5Y),10year(10Y),20year(20Y),and30year(30Y)holdingperiods.ThedataisdescribedinExhibit2.Allfiguresin%.

PanelA 1Y 5Y 10Y 20Y 30Y 1Y 5Y 10Y 20Y 30Y

Australia 33.6 19.8 12.9 2.2 0.0 Netherlands 40.0 27.4 25.7 12.1 21.0 Belgium 41.8 33.0 24.8 12.1 1.2 NewZealand 34.5 18.9 12.9 11.0 0.0 Canada 40.0 29.2 22.8 17.6 4.9 Norway 47.3 34.9 25.7 17.6 8.6 Denmark 42.7 41.5 27.7 20.9 14.8 S.Africa 38.2 23.6 12.9 3.3 0.0

Finland 41.8 28.3 8.9 1.1 0.0 Spain 47.3 39.6 30.7 19.8 6.2 France 44.5 31.1 31.7 22.0 18.5 Sweden 39.1 23.6 23.8 27.5 24.7 Germany 40.0 28.3 28.7 19.8 1.2 Switzerland 37.3 31.1 27.7 27.5 22.2 Ireland 37.3 28.3 13.9 12.1 4.9 UK 33.6 18.9 18.8 13.2 0.0 Italy 40.0 36.8 35.6 23.1 13.6 USA 38.2 26.4 16.8 3.3 0.0 Japan 41.8 32.1 27.7 22.0 13.6 Average 40.0 29.1 22.6 15.2 8.2

PanelB 1Y 5Y 10Y 20Y 30Y 1Y 5Y 10Y 20Y 30Y

Australia 13.8 4.7 1.9 1.1 N/A Netherlands 14.2 8.4 4.9 4.3 1.2 Belgium 13.8 6.6 4.0 1.7 0.6 NewZealand 11.8 7.3 5.3 2.7 N/A Canada 13.0 5.3 2.7 1.4 0.7 Norway 13.8 5.8 3.2 1.8 1.0 Denmark 12.3 3.6 2.3 1.4 0.8 S.Africa 12.6 3.8 2.6 0.1 N/A Finland 15.1 6.5 2.5 0.9 N/A Spain 12.8 5.5 3.7 1.6 0.5 France 13.5 6.6 3.6 1.9 0.8 Sweden 16.1 9.6 5.7 3.4 2.5 Germany 15.6 7.6 4.7 1.9 0.1 Switzerland 13.6 6.1 3.3 1.6 1.1 Ireland 13.4 4.2 3.7 1.0 0.5 UK 11.3 5.5 2.2 0.7 N/A Italy 16.4 7.4 3.6 2.1 1.2 USA 15.8 6.3 3.1 1.3 N/A Japan 16.8 8.2 5.7 3.5 1.9 Average 14.0 6.3 3.6 1.8 1.0

PanelC 1Y 5Y 10Y 20Y 30Y 1Y 5Y 10Y 20Y 30Y

Australia 4.6 0.9 0.2 0.0 0.0 Netherlands 5.7 2.3 1.3 0.5 0.2 Belgium 5.8 2.2 1.0 0.2 0.0 NewZealand 4.1 1.4 0.7 0.3 0.0 Canada 5.2 1.5 0.6 0.2 0.0 Norway 6.5 2.0 0.8 0.3 0.1 Denmark 5.3 1.5 0.6 0.3 0.1 S.Africa 4.8 0.9 0.3 0.0 0.0 Finland 6.3 1.8 0.2 0.0 0.0 Spain 6.1 2.2 1.1 0.3 0.0 France 6.0 2.0 1.1 0.4 0.1 Sweden 6.3 2.3 1.3 0.9 0.6 Germany 6.3 2.1 1.3 0.4 0.0 Switzerland 5.1 1.9 0.9 0.5 0.2

Ireland 5.0 1.2 0.5 0.1 0.0 UK 3.8 1.0 0.4 0.1 0.0 Italy 6.5 2.7 1.3 0.5 0.2 USA 6.0 1.7 0.5 0.0 0.0 Japan 7.0 2.6 1.6 0.8 0.3 Average 5.6 1.8 0.8 0.3 0.1

3.6.TheRiskPremium Theexpected shortfallcombines theshortfall probabilityandthe shortfallmagnitude,

thusfocusingonlyontheexpectedlossofstocksrelativetobonds.Theriskpremium,inturn,

accounts for both the expectedloss and the expectedgain, the latter definedas the product

between the probability that stocks outperform bonds and the average magnitude of the

outperformance.Putdifferently,theexpectedshortfallisgivenbySPSMandtheriskpremium

by SPSM+(1SP)OM, where SM and OM denote the shortfall magnitude and the

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outperformancemagnitude(andSP,asbefore,theshortfallprobability). 10Exhibit6showsrisk

premiums calculated this way for five holding periods between 1 and 30 years, based on

annualized returns; Exhibit A4 in the appendix shows risk premiums based on cumulative

returns.

Exhibit6:RiskPremiumsAnnualizedReturnsThisexhibitshowsriskpremiums,asdefinedinthetext,over1year(1Y),5year(5Y),10year(10Y),20year(20Y),and30year(30Y)holdingperiods,basedonannualizedreturns.ThedataisdescribedinExhibit2.Allfiguresin%.

Country 1Y 5Y 10Y 20Y 30Y Country 1Y 5Y 10Y 20Y 30Y

Australia 6.9 6.1 5.9 5.9 6.0 Netherlands 5.3 4.1 4.1 4.1 4.1 Belgium 4.5 3.4 3.2 3.2 2.9 NewZealand 5.3 4.3 4.0 4.0 4.4 Canada 4.7 3.9 3.9 4.0 4.3 Norway 4.8 2.8 2.4 2.1 2.2 Denmark 3.1 2.2 2.0 1.8 2.0 S.Africa 7.2 5.7 5.7 6.2 6.6 Finland 8.1 6.2 6.0 5.8 5.8 Spain 4.0 2.9 2.6 2.6 2.4 France 5.0 3.7 3.6 3.7 3.7 Sweden 5.4 4.1 3.7 3.6 4.0 Germany 7.4 5.2 5.1 5.4 5.5 Switzerland 3.6 2.7 2.5 2.4 2.5

Ireland 4.3 3.4 3.6 3.7 3.9 UK 5.0 4.2 4.2 4.6 4.7 Italy 6.5 4.2 4.2 4.6 5.1 USA 5.8 4.5 4.5 4.8 5.0 Japan 7.0 5.6 5.1 5.5 6.4 Average 5.5 4.2 4.0 4.1 4.3

Resultsvarybycountry,insomecasessubstantially(compare,forexample,Australiaor

FinlandtoSpainorSwitzerland),butonaverageacrossallcountriesstockscanbeexpectedto

outperformbondsby5.5%over1yearholdingperiods,by4%ayearover10years,andby

4.3%ayearover30years.Needlessto say, theseareunconditionalexpectations;obviously,

valuationmeasuresshouldconditionandplayacriticalroleinananyforecast.Inotherwords,

the expected relative performance of stocks and bonds over any given holding period is

criticallydependentonwhethereachasset isovervalued,undervalued,orproperly valuedat

themomenttheforecastismade.

3.7.AWordOnAbsoluteRisk Beforeconcluding,consideroncemoretheissueofabsoluterisk;thatis,therelationship

betweentheriskofanindividualassetandtheholdingperiod.Becausetheassetconsideredin

thissectionisstocks,focusonthecolumnslabeledSofalltheexhibitsmentioned.

Exhibits3and4showthatineverycountrythevolatility,spreadbetweenthehighest

andlowestreturns,andsemideviationwithrespectto0%allsteadilydecreasewiththeholding

period. At the same time, the lowest return steadily increases with the holding period,

eventuallyturningpositiveinmanycountriesatinvestmenthorizonsof20or30years.These

resultssuggestthatalthoughstocksmaysubjectinvestorstolargeuncertaintyanddownside

potentialintheshortterm,bothgraduallydecreasewiththeholdingperiod.Inotherwords,the

10Both SM and OM can be expressed in terms of annualized or cumulative returns. SP, in turn, isobviouslyindependentfromthewayreturnsareexpressed.

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longertheinvestmenthorizonthemorelikelyarestockstodelivertheirlongtermmeanannual

compoundreturn.

Furthermore,thelefthalfofExhibitA5(labeledAnnualizedReturns)intheappendix

shows the ratio between volatility and arithmetic mean return for the five holding periods

considered. As these figures clearly show, risk per unit of return (or returnadjusted risk)

steadilydecreaseswiththeholdingperiodineverycountryinthesample.Themessagefrom

Exhibits3and4andthelefthalfofExhibitA5,then,isclear:Timedoesdiversifytheriskof

investingininternationalstockmarkets.

Thisconclusion follows from a focuson annualized returns;would it be altered by a

focus on cumulative returns instead? Exhibit A1 in the appendix shows that, with few

exceptions, volatility andspreads increasewith theholdingperiod,whichsuggests that time

magnifiestheuncertaintyofinvestinginstocks,thuscontradictingthepreviousconclusion. However,panelAofExhibitA2intheappendixshowsthat,onaverageacrosscountries,

downside potential asmeasured by the semideviation peaksat a 5year holding period and

decreases from that point on. In fact, with only three exceptions, in every country the

semideviationover20/30yearholdingperiodsislowerthanorequaltothesemideviationover

a5yearholdingperiod.Atthesametime,panelBofExhibitA2showsthat,withfewexceptions,

thelowestreturn(thelowerendofthespreads)peaksatthe5yearholdingperiodandthen

increasesfromthatpointon.Finally,therighthalfofExhibitA5(labeledCumulativeReturns)

in the appendixshows that inmost cases risk per unit of return clearly decreases with the

holdingperiod.

Afocusoncumulativereturns,then,suggeststhatalthoughuncertaintyasmeasuredby

volatilityandspreadsclearlyincreaseswiththeholdingperiod,thisincreaseislargelydrivenby

an increase in the probability and magnitude of beneficial outcomes. In other words, the

increaseinuncertaintyisnotdrivenbyanincreaseinpotentiallydetrimentaloutcomesbutbya

combinationofadecreasingdownsidepotentialandanincreasingupsidepotential.

Theseresultsprovideabroadinternationalperspectiveontherelationshipbetweenthe

riskofinvestinginstocksandtheholdingperiod.Theywillprobablynotsettleacontroversy

thathasbeenragingonfordecades,butthebigpictureisnothardtosee.Compoundingandthe

longtermupwardtrend ofstockmarketshaveanaturalandstraightforward implication for

investors:Thelongeristheholdingperiod,thehigheristheuncertaintyabouthowlargethe

accumulatedwealthwillbeattheendofthatperiod.Butitisessentialtonoticethatmostofthis

increasinguncertaintyisupsiderisk.Worstcasescenariostypicallyimprove(notworsen),and

bestcasescenariostypicallyimprovedramatically,withthelengthoftheholdingperiod.

Inshort,then,swingsinaccumulatedwealththattendtoincreasewiththelengthofthe

investmenthorizonareanaturalconsequenceofcompoundingandupwardtrendingmarkets.

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But,ultimately,thelongeristheholdingperiodthemorelikelyareinvestorstobeexposedto

positive(ratherthannegative)surprises,andthemoretheuncertaintyisabouthowhighthe

upside(ratherthanhowlowthedownside)willbe.

4.Assessment

Ifdiversificationistheprimemethodforimprovingthetradeoffbetweenriskand

return,thendiversificationovertimeisjustasimportantasdiversificationacross

assetgroupsatanygivenmoment.Bernstein(1976).

Thedefinitionof risk andthe evolutionof absolute andrelative riskwith theholding

periodareissuesthatfinanceacademicsandpractitionershavebeenhotlydebatingforseveral

decades.Thisarticleaimstocontributetothisdebateandhopefullyhelptoclarifysomeofthe

contentiousissuesbyanalyzingtheevidencefromacomprehensivedatasetthatspansover19countriesand110years.Althoughboththedefinitionofriskandtheevolutionofabsoluterisk

with the holding period are discussed, the ultimate issue addressed in this article is the

evolutionoftherelativeriskofstocksandbondswiththeholdingperiod.

As faras this last issueis concerned,theevidence shows, unsurprisingly, that in the

short term stocks are riskier than bonds; this is the case regardless of the type of returns

(annualizedorcumulative)onwhichinvestorsfocusandthewaytheyassessrisk(generallyas

uncertainty or more narrowly as downside potential). The evidence also shows that in the

mediumtolongtermstocksbecomelessriskythanbonds;thisisclearlythecaseifinvestors

focusonannualizedreturns,andlargelythecaseiftheyfocusoncumulativereturns.

Wheninvestorsfocusonannualizedreturns,alltheriskmeasuresconsidered(volatility,

spreads,semideviation,worstcase scenarios, andexpectedshortfall) unambiguously suggest

that as the holding period lengthens, stocks steadily become less risky than bonds. When

investorsfocusoncumulativereturnsinstead,volatilityandspreadstakeninisolationmaylead

investors tobelieve that as the holding period lengthens, stocks become riskier than bonds.

However,a more thoroughassessment thatdistinguishesbetweenuncertainty anddownside

potentialshouldleadthemtoreconsider.Theevidencesuggeststhatmostoftheincreaseinthe

uncertaintyofstocksrelativetobondsisactuallyanincreasein upsidepotential;thatis,itis

uncertaintyabouthowmuchmorestockswilldeliverthanbonds.

Althoughforallthewrongreasonssomeinvestorsthinkofthestockmarketasacasino,

thereisanimportantsimilaritybetweenthem.Theownerofacasinoisnotguaranteedtomake

aprofitin any givenperiod.Herunsanoperationwith favorable oddsbut nocertainties; in

someperiodshewilldowellandin someothersbadly,andarunofverybadluckmayeven

bankrupthisbusiness.Butanunlikelycatastrophenotwithstanding,thepatientcasinoownerisverylikelytocomeoutahead;afterall,herunsabusinesswithapositiveexpectedvalueand

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timeisonhisside.Thesituationisverysimilarinthestockmarket;therearenocertainties,and

nothingpreventsaninvestorfrombeinghitbyarunofbadluckwithdramaticconsequences.

Butmuchlikethecasinoowner,investorsinthestockmarketareplayingwithaloadedcoin;

theoddsareintheirfavorandthereforetimeisontheirside.

Theanalogycanbeextendedtotherelationshipbetweenthestockandbondmarkets.A

casinoownerrunsseveralgames,all ofthemwithfavorable,butnot thesameodds. Inshort

periods, a game with less favorable odds may generate more profit than one with more

favorableodds.Butinthelongterm,itisalmostcertainthatthemoreprofitablegameswillbe

thosewithbetteroddsforthecasinoowner.Thesituationissimilarinfinancialmarkets.Inthe

short term, the stock market may underperform the bond market; in fact, it may vastly

underperformandtheshorttermmaybemuchlongerthanmanyinvestorswouldlike.Butthe

longerisaninvestorsholdingperiod,themorelikelyheistofindthatstocksoutperformbonds,andthattheydosobyanincreasingmagnitude.Timeisonthesideofthepatientinvestorthat

canbearthehighershorttermvolatilityanddownsidepotentialofstocks;time,infact,loads

thecointhatwillcompensatetheseinvestorswithahigherfuturepayoff.

As argued by Reichensteinand Dorsett (1995),just as therisk ofan assetcannotbe

assessed independently from the portfolio to which it belongs, nor it cannot be assessed

independentlyfromtheholdingperiod.Anassetmaybeveryriskyinisolationandmuchless

riskywithinaportfolio.Similarly,anassetmaybeveryriskyintheshorttermandmuchless

riskyinthelongterm;oritmayberiskierthananotherintheshorttermandlessriskyinthe

longterm.

Allinall,thecomprehensiveevidencediscussedinthisarticlesuggeststhattimedoes

diversifyrisk;thatis,astheholdingperiodlengthensstocksgraduallybecomelessrisky,bothin

absolute terms and relative to bonds. To be sure, not all investors will agree with this

assessment.Butthosewhothinkthattimemagnifiestheriskofinvestinginstocksshouldnotice

that cumulative returns over different holding periods are not really comparable; that

increasinguncertaintyisundesirableunlessitmostlycapturesincreasingupsidepotential;and

thatthedownsidepotentialofstocksclearlydecreaseswiththeholdingperiod.Allthatbeing

said,bothbeautyandriskareandwillalwaysbeintheeyesofthebeholder.

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AppendixExhibitA1:UncertaintyCumulativeReturnsThisexhibitshowsthevolatility(panelA)andspread(panelB,asdefinedinExhibit1)ofstocks(S)andbonds(B)overfivedifferentholdingperiods,basedoncumulativereturns.ThedataisdescribedinExhibit2.Allfiguresin%.


Australia 18.2 13.2 48.0 42.3 85.9 80.7 253.2 161.3 595.9 143.6Belgium 23.6 12.0 62.1 35.7 103.6 63.2 275.7 124.5 386.7 167.2Canada 17.2 10.4 48.8 33.5 75.8 65.1 172.4 146.0 322.5 150.1Denmark 20.7 11.6 47.1 34.3 71.5 74.4 183.7 187.7 326.1 296.3Finland 30.3 13.7 115.5 39.5 195.9 71.3 621.2 115.3 922.7 108.4France 23.5 13.0 74.4 40.0 121.3 75.1 277.1 179.8 410.2 305.4Germany 32.2 15.5 113.5 42.2 372.2 72.2 634.9 121.7 678.8 148.9Ireland 23.1 14.6 65.6 42.0 115.8 79.7 313.8 163.7 558.5 167.5Italy 29.0 14.1 70.3 40.6 137.3 71.3 186.8 123.1 187.3 106.2Japan 29.8 20.1 104.7 46.4 246.7 81.5 544.1 125.5 1071.7 165.8Netherlands 21.8 9.4 69.8 29.8 129.3 55.5 446.3 106.4 562.6 124.2NewZealand 19.7 9.0 54.0 33.7 79.9 66.5 119.4 132.5 273.1 113.4Norway 27.4 12.2 57.8 37.0 79.1 80.1 224.6 139.6 443.7 137.8S.Africa 22.5 10.4 74.6 31.4 114.5 60.0 287.9 102.8 467.6 87.7Spain 22.1 11.7 82.7 30.7 110.6 54.0 330.3 101.6 347.9 91.4Sweden 22.8 12.4 69.6 39.6 124.3 77.2 491.2 164.1 834.0 167.3Switzerland 19.8 9.3 57.8 27.8 96.9 47.1 193.4 78.2 245.5 73.2UK 20.0 13.6 49.6 38.7 96.6 70.0 245.6 138.1 338.2 159.3USA 20.3 10.1 55.3 28.6 97.7 54.0 250.2 113.5 386.5 112.8Average 23.4 12.4 69.5 36.5 129.2 68.4 318.5 132.9 492.6 148.8


Australia 94.0 88.8 255.6 235.7 430.8 333.0 1214.6 711.4 2005.9 585.8

Belgium 166.6 71.2 337.6 152.3 415.3 245.7 1277.4 445.2 1886.9 599.5Canada 89.0 67.6 264.7 159.5 391.5 245.5 763.1 589.6 1334.3 679.8Denmark 157.0 68.3 264.2 192.3 404.4 319.0 932.2 701.2 1564.41058.7Finland 222.5 99.7 740.8 192.9 1153.0 263.2 3768.5 412.0 4955.0 415.2France 108.7 79.4 347.5 187.0 556.5 283.4 1244.4 662.7 2182.01046.1Germany 245.4 157.5 745.7 211.3 2436.3 266.6 5362.3 401.9 5328.2 439.8Ireland 133.8 95.3 306.4 203.4 458.1 314.4 1392.1 642.4 3427.4 730.2Italy 193.5 92.9 300.6 189.3 668.5 280.8 733.5 444.3 1116.6 461.4Japan 206.6 147.3 674.0 263.2 1501.1 375.9 2805.9 421.8 5527.6 539.6Netherlands 152.0 50.9 319.8 159.6 510.8 256.4 2388.0 373.0 2266.5 450.7NewZealand 160.0 57.8 387.8 172.4 609.8 270.5 552.8 512.8 1222.1 528.9Norway 220.5 110.2 408.9 203.0 398.6 435.3 1116.8 617.5 3051.3 682.2S.Africa 155.1 69.6 489.5 149.1 482.7 250.5 1420.7 427.3 1944.8 355.8Spain 142.7 83.5 447.8 127.5 528.1 224.2 1680.3 431.7 2303.8 489.6

Sweden 133.3 104.8 368.6 223.6 755.7 380.3 3169.1 843.1 3903.6 591.0Switzerland 97.2 77.5 265.4 192.6 499.4 290.2 741.4 480.3 992.5 409.3UK 153.7 89.6 238.8 198.7 448.8 318.3 1194.2 522.1 1991.4 631.6USA 94.5 54.5 277.9 168.6 407.3 231.8 962.9 467.8 1793.4 543.0Average 154.0 87.7 391.7 188.5 687.2 293.9 1722.1 532.0 2568.3 591.5

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ExhibitA2:DownsidePotentialCumulativeReturnsThis exhibit shows the semideviation for a 0%benchmark(panel A)and the lowestreturn over the 19002009period(panelB)ofstocks( S)andbonds(B)overfivedifferentholdingperiods,basedoncumulativereturns.ThedataisdescribedinExhibit2.Allfiguresin%.

1Year 5Years 10Years 20Years 30Years

PanelA S B S B S B S B S B Australia 9.3 7.7 11.0 16.5 7.7 24.2 0.0 27.6 0.0 27.9 Belgium 12.6 8.3 21.9 21.5 23.9 31.4 20.0 38.9 12.8 42.6 Canada 8.5 5.5 9.6 13.1 3.6 16.7 0.0 15.4 0.0 10.3 Denmark 8.9 5.1 8.8 10.7 3.2 12.1 0.0 8.8 0.0 5.1 Finland 14.1 11.1 23.4 26.1 24.7 36.3 12.6 45.3 0.5 43.6 France 12.6 9.7 21.8 25.5 22.2 36.2 17.9 48.3 7.7 58.0 Germany 15.1 12.6 25.3 30.5 27.9 42.8 31.2 57.2 27.4 62.9 Ireland 12.2 7.9 15.6 17.3 15.9 24.1 8.9 28.3 0.0 27.4 Italy 15.8 11.9 25.5 27.9 30.4 39.4 23.1 51.6 11.1 60.3 Japan 15.5 15.0 25.5 27.2 30.5 35.4 33.4 46.9 24.8 60.1 Netherlands 10.4 5.2 14.6 12.0 12.4 17.3 6.6 24.0 0.0 26.5

NewZealand 9.2 4.9 10.7 12.0 5.2 17.1 0.0 17.8 0.0 18.0 Norway 11.9 7.0 16.9 15.3 17.3 20.5 8.3 21.4 4.0 17.8 S.Africa 9.2 5.9 6.6 11.5 5.2 17.0 0.0 19.9 0.0 21.2 Spain 11.1 7.0 20.1 15.0 23.7 18.9 18.6 22.8 4.3 27.0 Sweden 10.9 6.1 14.6 13.1 15.1 14.4 8.3 16.3 0.0 18.7 Switzerland 10.3 4.3 16.0 10.9 15.8 11.4 11.0 7.5 0.0 0.0 UK 9.9 7.2 13.0 16.7 9.0 22.4 3.0 24.2 0.0 23.1 USA 10.6 5.3 11.8 10.8 8.1 12.8 0.0 13.9 0.0 15.4 Average 11.5 7.8 16.5 17.6 15.9 23.7 10.7 28.2 4.9 29.8


Australia 42.5 26.6 65.9 55.0 44.0 58.5 38.0 56.2 133.771.0 Belgium 57.1 30.6 73.9 66.8 69.7 82.4 79.0 86.6 60.681.2

Canada 33.8 25.9 41.2 53.5 22.2 60.4 20.1 62.1 140.134.7 Denmark 49.2 18.2 46.5 49.0 18.5 56.1 11.3 36.8 100.718.5 Finland 60.8 69.5 85.2 89.8 79.3 90.3 50.6 87.7 4.482.3 France 42.7 43.5 77.6 86.0 80.9 91.9 69.3 92.2 37.690.2 Germany 90.8 95.0 93.3 95.2 88.3 94.5 88.4 95.3 86.394.5 Ireland 65.4 34.1 53.6 58.9 62.8 70.0 57.8 70.4 35.971.2 Italy 72.9 64.3 81.3 95.1 77.3 96.4 70.8 96.4 49.495.7 Japan 85.5 77.5 97.5 98.8 97.5 99.4 93.3 99.4 93.399.1 Netherlands 50.4 18.1 47.6 45.1 43.0 52.7 41.7 53.4 6.364.4 NewZealand 54.7 23.7 65.5 44.0 31.2 51.9 42.7 55.6 85.053.3 Norway 53.6 48.0 73.7 73.4 66.5 75.7 37.7 66.7 30.251.8 S.Africa 52.2 32.6 35.5 43.2 40.6 49.2 1.9 55.2 220.647.3 Spain 43.3 30.2 79.0 46.2 84.1 54.5 66.3 63.1 22.566.1 Sweden 43.6 36.7 67.7 52.6 61.9 50.9 51.0 50.4 2.050.6 Switzerland 37.8 21.4 68.7 55.5 69.2 58.4 53.7 46.7 8.3 11.3 UK 57.1 30.7 62.9 54.5 43.8 69.5 28.5 69.9 106.671.2 USA 38.0 19.4 45.4 42.4 33.3 42.3 19.1 46.3 129.445.4 Average 54.3 39.3 66.4 63.4 58.6 68.7 34.5 67.9 30.862.0

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ExhibitA3:ExpectedShortfallsCumulativeReturnsThis exhibit shows shortfall probabilities (panel A), cumulative shortfallmagnitudes (panel B), and cumulativeexpectedshortfalls(panelC),allasdefinedinthetext,over1year(1Y),5year(5Y),10year(10Y),20year(20Y),and30year(30Y)holdingperiods.ThedataisdescribedinExhibit2.Allfiguresin%.

PanelA 1Y 5Y 10Y 20Y 30Y 1Y 5Y 10Y 20Y 30Y

Australia 33.6 19.8 12.9 2.2 0.0 Netherlands 40.0 27.4 25.7 12.1 21.0 Belgium 41.8 33.0 24.8 12.1 1.2 NewZealand 34.5 18.9 12.9 11.0 0.0 Canada 40.0 29.2 22.8 17.6 4.9 Norway 47.3 34.9 25.7 17.6 8.6 Denmark 42.7 41.5 27.7 20.9 14.8 S.Africa 38.2 23.6 12.9 3.3 0.0 Finland 41.8 28.3 8.9 1.1 0.0 Spain 47.3 39.6 30.7 19.8 6.2 France 44.5 31.1 31.7 22.0 18.5 Sweden 39.1 23.6 23.8 27.5 24.7 Germany 40.0 28.3 28.7 19.8 1.2 Switzerland 37.3 31.1 27.7 27.5 22.2 Ireland 37.3 28.3 13.9 12.1 4.9 UK 33.6 18.9 18.8 13.2 0.0 Italy 40.0 36.8 35.6 23.1 13.6 USA 38.2 26.4 16.8 3.3 0.0 Japan 41.8 32.1 27.7 22.0 13.6 Average 40.0 29.1 22.6 15.2 8.2

PanelB 1Y 5Y 10Y 20Y 30Y 1Y 5Y 10Y 20Y 30Y

Australia 13.8 29.9 41.2 79.3 N/A Netherlands 14.2 43.4 65.7146.7 68.6

Belgium 13.8 31.3 44.5 34.4 14.2 NewZealand 11.8 41.4 95.2177.9 N/A Canada 13.0 30.2 48.3 99.0115.8 Norway 13.8 30.7 52.6 69.7112.9 Denmark 12.3 20.6 41.6101.2114.7 S.Africa 12.6 22.7 45.3 7.0 N/A Finland 15.1 31.7 28.9 54.4 N/A Spain 12.8 24.1 28.0 33.9 19.5 France 13.5 34.5 41.9 53.7 82.5 Sweden 16.1 54.0 79.6145.3185.5 Germany 15.6 38.6 64.0 76.6 14.4 Switzerland 13.6 30.5 35.7 56.0 59.5 Ireland 13.4 24.5 50.8 51.5 68.1 UK 11.3 31.8 37.7 41.2 N/A Italy 16.4 36.8 39.3 65.1 41.8 USA 15.8 32.4 41.8 56.3 N/A Japan 16.8 44.4 75.2135.3166.6 Average 14.0 33.3 50.4 78.1 81.9

PanelC 1Y 5Y 10Y 20Y 30Y 1Y 5Y 10Y 20Y 30Y


ExhibitA4:RiskPremiumsCumulativeReturnsThisexhibitshowsriskpremiums,asdefinedinthetext,over1year(1Y),5year(5Y),10year(10Y),20year(20Y),

and30year(30Y)holdingperiods,basedoncumulativereturns.ThedataisdescribedinExhibit2.Allfiguresin%.

Country 1Y 5Y 10Y 20Y 30Y Country 1Y 5Y 10Y 20Y 30Y


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ExhibitA5:ReturnAdjustedRiskStocksThisexhibitshowsriskperunitofreturn,definedastheratioofvolatilitytoarithmeticmeanreturn,over1year(1Y),5year(5Y),10year(10Y),20year(20Y),and30year(30Y)holdingperiods,basedonbothannualizedandcumulativereturns.ThedataisdescribedinExhibit2.

AnnualizedReturns CumulativeReturns

PanelA 1Y 5Y 10Y 20Y 30Y 1Y 5Y 10Y 20Y 30Y Australia 1.99 0.94 0.61 0.39 0.31 1.99 0.91 0.70 0.68 0.69 Belgium 4.58 2.90 2.25 1.62 1.20 4.58 1.99 1.61 1.71 1.61 Canada 2.39 1.18 0.68 0.43 0.27 2.39 1.19 0.83 0.66 0.57 Denmark 3.08 1.28 0.78 0.53 0.37 3.08 1.33 0.95 0.95 0.87 Finland 3.32 2.21 1.53 1.04 0.61 3.32 1.89 1.49 1.78 1.49 France 4.13 2.91 2.14 1.46 1.02 4.13 2.09 1.63 1.64 1.68 Germany 3.98 3.43 2.80 1.79 1.20 3.98 2.39 2.92 2.46 1.58 Ireland 3.57 1.95 1.34 0.82 0.54 3.57 1.75 1.27 1.24 1.20 Italy 4.71 4.02 3.34 2.04 1.05 4.71 2.19 1.96 1.71 1.45 Japan 3.48 2.91 2.59 1.75 1.25 3.48 1.77 1.78 1.69 1.65 Netherlands 3.07 1.73 1.19 0.80 0.53 3.07 1.60 1.26 1.45 1.04

NewZealand 2.58 1.19 0.66 0.34 0.27 2.58 1.24 0.86 0.52 0.54 Norway 3.82 1.93 1.37 1.05 0.70 3.82 1.67 1.18 1.37 1.53 S.Africa 2.38 1.16 0.73 0.43 0.23 2.38 1.37 0.92 0.76 0.53 Spain 3.67 2.73 1.98 1.42 0.83 3.67 2.09 1.45 1.77 1.77 Sweden 2.65 1.46 1.01 0.75 0.54 2.65 1.37 1.12 1.44 1.14 Switzerland 3.24 1.89 1.29 0.83 0.47 3.24 1.60 1.25 0.99 0.72 UK 2.78 1.37 0.92 0.51 0.29 2.78 1.24 1.01 0.87 0.61 USA 2.47 1.26 0.81 0.49 0.27 2.47 1.22 0.91 0.78 0.60 Average 3.26 2.02 1.47 0.97 0.63 3.26 1.63 1.32 1.29 1.12

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