.lU

IJ -l

and )rate tSSA

)f 10 nslve

LeorningAboutReturnond Risk from the HistoricolRecord

)00on nagers , sothe r .3'75, ) repeat of ..54o,

CASUALoBSERVATIoN AND formal research both suggestthat investmentrisk is as impor-

risk that investorsactually anticipated,from historical data. (There is an old saying that

tant to investors as expected return. While

forecasting the future is even more difficult than forecasting the past.) In this chapter,

we have theories about the relationshipbetween risk and expected return that would prevailin rationalcapital markets,there is no

we present the essentialtools for estimating expected returns and risk from the historical

theory about the levelsof risk we should find in the marketplace.We can at best estimate

record for future investments.

record and consider the implicationsof this

the levelof risk likelyto confront investorsby analyzinghistoricalexperience.

We begin by discussinginterest rates and investments in safe assets and examine the This situation is to be expected because historyof risk-freeinvestmentsin the U.Sover prices of investment assets fluctuate in the last 80 years. Moving to risky assets,we responseto news about the fortunes of begin with scenario analysisof risky investcorporations,as well as to macroeconomic ments and the data necessaryto conduct it. developments that affect interest rates. With this in mind, we develop statisticaltools Thereis no theory about the frequency and needed to make inferencesfrom historical importanceof such events; hence we cannot time series of portfolio returns. We present cletermine a "natural" level of risk. a global view of the history of returns over Compoundingthis difficulty is the fact that 100 years from stocks and bonds in various neitherexpected returns nor risk are directly countries and anàlyze the historical record observable.We observe only realized rates of five broad asset-classportfolios. We end of returnafter the fact. Hence,to make forecastsabout future expected returns and risk, we first must learn how to ,,forecast,,their

the chapter with discussionsof implications of the historicalrecord for future investments

Pastvalues, that is, the expected returns and

used in the industry.

and a variety of risk measures commonly

Practic€ PARI ll PortfolioTheoryand

ffi

sting in a relativelYlong-term rnitting anYfundsto long-term ^s ^--lipd mâcÎoparts of applied macrocDs. : - one ^-^ of ^r the the most notoriously difficult that factors is J*" fundameniat T::::rï;#;, Forecastinginterestrates unq good a have do i',înetheless' we """";;;. rates: interest of the level ;;;;;" households' from savers'primarily in plant' 1. The supply of fooA'

f"';;u;fr"* 2. rhe demand

uusjnesll::ml;*minvestments

, n:U**ln::::m$"'#;:ffii;i;;;;."' Bank' FedeialReserve

on these forces Before we elaborate interestrates' ,"Jt o* nominal

ofthe bvac'ions modined

1. -:-+^iàotrâtes rates' we Ineedto distinguish and resultant interest

t

(S't) r-R-' by the loss of purchasing

of interestis the In words,the real rate t"r"utng ftom inflation' between o"** 'ïn ru"t, theàxactrelationship

bY nterestrateis given

6.2)

l+r=1'-

1+i

i

CHAPTLR 5 Learning About Returnand Riskfromthe HistoricalRecord -1 I

I .1 J

,o le oe )a dil ;fm )rm lrothat

This is becausethe growthfactor of your purchasingpower,1 * i; equalsthe growth factorof your money,1 * R, dividedby thenewpricelevel,thatis, I * i timesits valuein the previousperiod.The exactrelationshipcanbe rearrangedto

R-t r=1+t

(s.3)

which shows that the approximation rule overstatesthe real rate by the factor 1 * r.

If thenominalinterestrateon a 1-yearCD is 87o,andyou expectinflationto be 5Vooverthe comingyear,thenusing the approximationformula,you expectthe real rate of interest

the guish

:einga rtive to rgpowaverage rmPtion ' the last hced bY ,dsit can rrchasing :ate,after Yith a net .nominal 'owth raE

Before the decision to invest, you should realize that conventionalcertificatesof deposit offer a guaranteednominal rate of interest. Thus you can only infer the expectedreal rate on theseinvestmentsby subtracting your expectationofthe rate ofinflation. It is alwayspossible to calculatethe real rate after the fact. The inflation rate is published by theBureauof Labor Statistics(BLs). The future real rate, however,is unknown, and one hæto rely on expectations.In other words, becausefuture inflation is risky, the real rate of retumis risky even when the nominal rate is risk-free.

TheEquilibrium RealRateof Interest Ttneebasic factors-supply, demand, and government actions-determine the real interest rate.The nominal interest rate, which is the rate we actually observe,is the real rate plus tlrcexpectedrate of inflation. So a fourth factor affecting the interest rate is the expected rateof inflation.

Althoughthere are many different interest rates economywide(as many as there are types of securities), theseratestendto movetogether,so economistsfrequentlytalk asif a singlerepresentative rate.we .* or" this abstractionto gain someinsights I:1.-,""." ntotherealrateof interestif we considerthe supplyanddemand "u*"-, for funds. Figure5.1 showsa downward-sloping demani curve and an upward-slopingsupply ve'on thehorizontalaxis,we measurethe quantityof funds,andon the verticalaxis, measure the real mte of interest.

'Thcsupplycurveslopesup from left to right because thehighertherealinterestrate,the t|! supplyof householdsavings.Theassumption is thatat higherrealinterestrates f::l ttds will chooseto postponesome current consumption and set asideor invest more otsposableincome for future use.l

ivenbY

curveslopesdown from left to right becausethelower thereal interestrate, -ottuld re.businesses will want to investin physicalcapital.Assumingthatbusinesses rank by theexpectedreal return on investÀd.upitui, firms will undertakemoreprojects r therealinterestrateon thefundsneededio financethoseprojects. rlsconsiderable disagreementamong experts on the extent to which household saving does increasein reso anincrease in theieal interest rate.

125

and Pradice PAETll Portfolioïheory

lnterest Rate

governmeNos -v"^the governrtlÇ'r .:::lï^

i"" mg i"**a

Equilibrium

and shiftsthe

i"i*a curveto ttrefeht' theequilibricauses ;fi nse

o{the"ouiltÆ F I G U R E5.1 Determination

to um real interestrate

a risethrough a forecastthatindicatesT"TÏ:;i$:i;;ii';t;ch to pointB' rhat is' "*p"":1,T::,","'*L"ï"lt'ffÈd;'"nntt'*" increases "itt"** *.*:-1T*-,^nens*\ bonowing t""-ï:*: m?n":u iî".p*tJr""uty *ffff -',-^-^-*ora"rerminantj';;;; prol tundam"i].i sav il;prontabilitv)orinthe .oora ffi iffiJriil;ild;"ntaldeterminf.:ïl;iJff *-ï;,rlough *::T:ffffilîo.-* ,- _+ anc fiscaland t Êonar golernm *9 bv save well to as househoids afflcted of "'i:i:;flJà:ii'i""îJ*"*p":"d.ï:1i:H;::X;ilve'o*''*"o'nscarand De P:,:Xi*it"*i: can rate real physicalcapital'the ilil;;tt

;;';"":y:::,T"n'n' ill""îG'"'""'*:'#iSrTii;:iïi'ffi ::i#ilffiffi"'ilDoricv'whichwill'ntoi;Ji:i"""'*:*:".ffi

l"'ïit;'fl

monetaryPolicies'

H::ÏÏ:î"ÏÏ"i:i'Ï::#':'i"':::#irr,*,rv, rate. q"ï::::',ï:i:i::i:'Ëffi;; * *" "o*Hf"|ii::Ïi ,",",,ii" *nation *":'':;;;; investors rateincreases' Because ;t theinfla-t]on "oil;;;î;i' ffit ilildution t*' rate *"'1" higherrate power-we rhis higher rhis i*"r*"*. investments' in ttreirpurcry11sn"Ïl-"Ï:,1:iii.iH;;;;ir increase ffiIlJJ #;purchasine investmenl-^ u'vv'---* investment' an demancnrqt"'::1111lijïi",ffi Dvdr willdemÏl,Ttff:ï#itJ;"** wilt H";;lîvbv offered investors inu""o" ç^,-nne witt re$rn^o'ereo with expectedreal t1::"1*,: re-for-one ,o maintatn to ought rate :i:::#rJïffi;-the i, n""".tury ----^r ihâr the nomrnal

-'f;iiiti{îîi::i;*"âthuttt'"**"-"'ii""#;;;;l':.'*n"",*:ff ^; rrvingFisher(1e30)-iii{Ëiii}ïHf.Ji""triilîï',iiË '"':It l:^,if,n;;ï"j"]ffi;",iod, ;ryn:::i;Ï,Ïii""", innation penoo',h"nwecanstate eæe-ae! the;;p#o in À; thecoming increases ,,"ÏJ"i:',

lin*"u; ;1tttinnutiolS *11iÏ,n'""'' as rormallv ni"t'"' equation ï:ï:"ïi;i

(s.4)

R=r*E(j)

-

reas hat if real ratesare Thisr r inflationrates' Its æemixed; alth-o:Clt r ratesseemto PreûcÎ ri"-i"t"""t inflationn ttte [frcultto determine

";;;;"i-l,l1"j,!:"ff:iffiJ ""ry"d'"TY^i;:ïi#i-;;, changes vltw

rvEv-_-

ast on nominally risk-free required real rate bet relationship the i-ur oo,t r .*r 4' we discuss -orecastsfor long-tefl r ser rates incorPorate

"Fisherhypothesisthat at-su is that the real fate

:Hli','H-ïii:l .lÏ Ïi,L"u-#' t" F* -;^-^ i.terest !âd tsofl âIoûc, ruu-'--

CHAPTLR5

LearningAbout Returnand Riskfrom the HistoricalRecord

on bonds of different maturity may diverge. In addition, we will see that prices of longer-termbonds are morevolatile than thoseof short-term bonds. This implies that expected returns on longer-term bonds maY include a risk premium, so that the expectedreal rate offeredby bondsof varying maturity also may vary.

he he ES,

the :ral IESE

rves rto and

127

(a) Supposethe real interestrate is 3% per year and the exoectedinflationrate is 8%. What is the nominalinterestrate? (b) Supposethe expectedinflationrate risesto 107o, but the real rate is unchanged.What happensto the nominalinterestrate?

Taxesand the RealRateof Interest

r exrease bud-

Taxliabilitiesarebasedon nominalincome andthe tax ratedeterminedby the investor'stax "bracket creep" (when nominal income grows bracket.Congressrecognizedthe resultant dueto inflation and pushestaxpayersinto higher brackets)and mandatedindex-linkedtax bracketsin the Tax Reform Act of 1986. Index-linkedtax bracketsdo not provide relief from the effect of inflation on the taxation of savings,however.Given a tax rate (r) and a nominal interestrate (R), the after-tax interestrate is R(1 - r). The real after-taxrate is approximatelythe after-taxnominal rate minusthe inflation rate:

:EASES

)ITow.ts the right, uilibnto flse ;filITI0flt

through )penslty y) of iniscaland

minal rate increases' higher rate rr-onewith the cunent -r^tê

sto'" ilÊ Cê0

(s.4)

;"#îil':ïi;

;:fiïîiii"

;niï:Iil il,,'l:'f,

R(l - ô - i:(r

+,Xl - t)- i:

r(1- t)- it

(5.5)

Thusthe after-taxreal rate ofreturn falls as the inflation raterises.Investorssufferan inflationpenaltyequalto the tax rate times the inflationrate.If, for example,you arein a307o taxbracketand your investmentsyield 127o,while inflation runs at the rate of 87o,then yourbefore-taxreal rate is approximately47o, and you should, in an inflation-protected taxsystem,net after taxesa real return of 47o(l - .3) : 2.8Vo.BuLthe tax code doesnot recognize that the f,rst 8Voof your return is no more than compensationfor inflation-not realincome-and henceyour after-taxreturn is reducedby 87o x .3 :2.4Vo, so that your after-tax reai interestrate, at .4Vo,is almostwiped out.

lnflationand Rates The FederalReserveBank of St. Louishas informationavailableon interestrates and economicconditions.A publicationcalled MonetaryTrendscontainsgraphs andtableswith informationabout currentconditionsin the capitalmarkets.Go to theWebsitewww.stls.frb.organd clickon EconomicResearch on the menu at the top of the page. Findthe most recentissueof MonetaryTrendsin the RecentData Publications sectionand answerthesequestions. 1. Whatis the professionals'consensus forecastfor inflationfor the next 2 years? (Usethe FederalReserveBankof Philadelphia lineon the graphto answerthis.) i 2' Whatdo consumers expectto happento inflationover the next2 years?(Use the University of Michiganline on the graphto answerthis.) '' 3' Haverealinterestratesincreased, decreased,or remainedthe sameou"r.ih" ;, tast2years? ;'4' Whathashappenedto short-termnominalinterestratesoverthe last 2years? t:, Whataboutionq-termnominalinterestrates? i5' !o* do recent n and long-terminterestratescomparewith thoseof Ë:r theothercount 1 6 ' W n aat r et h e m a v a i l a b l lee v e l so f 3 - m o n t ha n d 1 O - y e ayri e t d so n ', rreasury securit .-

E - l n v e s t m en t s

and Practice PARTll Portfolio Theory

T HOLDING

urvvsrr'^î-r""*iti"r safeinvestment'-l:::*Tlî a ù4e a seeks whosoeKs wno matuninvestor aninvestor ffi::H;i#Consider Consider an several different with ::"ii

i"t.;-itpp*e we observe.:"to-"olpjl-li.?i.i; chanrer14.arebondsthataresoldat a

H::;"î:lî:,i:;"':iilH:"".Ë.fii*-::"1*:*ï,ï::*i:J##*:;l .i;#t'^1t:i:.:5j'"f; ii:"1ï,î"""J',i*T:';iiïiï:ffi 5iï:ïî'*:3Ë ï,ïiî,i:":":""i,.îï"ilfl :iï,"#.'TJi,ii:iii#i"ïffi #?--i;*iffi

ff

;;:::1'::f "tu:-ij: *îî;iliitffi bondwith $IUUpariilli"liàiffi;:îf :1$:$',îJf'I;:i"re'urn life of thebond. overthe ::jÏ "i; ,;u"r*n invesl the of value the in increase ;r"-p"r""otuge

(s.6)

100

r,(T):6-, 1, Equation5.6 providesthe

horizon of 1 year' risk-freeratefor an investment

Supposepricesofzero-couponTreasurieswith.$lO0facevalueandvariousmaturitiesare or "u"tt t"" asfollows. We find th" t"*i;;;

lloo/P(nl 1

Horizon, T Half-year 1 year 25 years

$97.36

iôs.iz izà.eo

1 O O l 9 7 ' 3 61- = ' 0 2 7 1

I ='0580 1oote5'521 =3'2e18 1oo/23.30'

Risk-free Return for Given Horizon

rt(.5):2'71"h rdl) = 5'800/" rt(25)= 329'18o/"

t" EiT*il horizons ronger Notsurprisingry, 3#ïËïil"ff:rtHiJ:ffi ::tff investmenl:lon returns the ,t *tO *" .orop*" .^. â .ômïnon oeriod.We typicatty ex' il;;î"o,;oonpoioq.wetvpicalrvex*:î:1"Ë::."'gï,H"ffi ïffiffi (EAR), defrnedasthe percentage u, an effective "oooJ."tu r"*, oress "t"*; all investrnent a l-yeæ horizon' over i" Àta. invested EAR equalsthe total retur the For a l-year investment' For i vato" ot a $1 inve^stment' (1 + EAR), i, tr," t"JïJ For e tro"p"t-periodreturnfor atulI vear' l;;t;ilnt"oo fr return the2'7170half-year Example 5-2,we""*oJ"tâ = 1'0549'j =(l'0271)2 I een obtaina terminuruau"'oïr Forinvestmentslongerthanayear,theconventionistot tft" samevalue as the actual rate that *ootO "o*poinJ i" i in Example5.2 growsby investmentin tte zs-yearlond 4.2918,soits EAR is :4'2918 (1 + EAR)25 = -1 + EAR 4'2gl8rt2s 1'0600

web' for exampleatYahoo! arewidely availableon the 2yields on Treasurybills andbondsof variousmaturities ftom the FederalReserve' 'ith maturitiesof up.o firr-.", f,ASll Uoney, or tlirectly

aren*-" which issuesr-bils, l*" u.r.rreasury *:î::r::'.,,ï:r"JJIJ:IiffH:'Jîtnî'îtiil"*u" crearczero-coup* claimsrcme institutions r'vTr] n"*"""" Ànancial

;d silins

:::iË;.""*r;;;;,r, byo*t",*"nii-n"r'lti"",'.1 lllilHl i".r'. "J* royears -:if:ï|.'S H"nJlTT separatet' value face of p"y*"nt frna -â "5"n* n"t*"L

CHAPTER5

LearningAbout Retumand Riskfrom the HistoricalRecord

ln general,we can relate EAR to the total return, r7(T),over a holding period of length Z by using the following equation:

I

(s.7)

I + EAR: [t + ,y(T)Jt,,

d

We can illustate with an example.

ln-

Total

rriata

For the 6-monthTreasuryin Example5.2,T = Yz,andIIT : 2. Therefore, 1+ EAR : (1.027D2: 1.0549and EAR: 5.497o

lufiury 1ras

(s.6)

in Example5.2, T = 25.Therefore, Forthe 21-yearTreasury :1.060 and EAR:6.0Vo 1+EAR :4.291812s

AnnualPercentageRates Rateson short-terminvestments(by convention,T 1 1 year)often are annualizedusing simpleratherthancompoundinterest.Thesearecalledannualpercentagerates,or APRs. to a monthlyrate suchas that chargedon a credit Forexample,the APR corresponding cardis calculatedby multiplying the monthlyrate by 12. More generally,if thereare z periodsin a year,andtheper-periodrateis dZ), thentheAPR : n x rlT). compounding you canfind the trueper-periodratefrom theAPR asrlD :7 X APR. Conversely, Usingthis procedure,the APR of the 6-monthbond in Example5.2 (which had a notethatfor short-terminvestrateof 23 l7o) is 2 x 2.7| : 5.427o.To generalize, 6-month periodsin a year.Therefore,therelationof length4 therearen : 1/Zcompounding ments period,theEAR, andtheAPR is shipamongthecompounding

tes ale

:11 + r/D)ttr: t + EAR: [1 + r1(7))n ns. How uiresthat ically ex' ercentage )ss retuftl' t less than

onthbillin periods tt = 5'4910' , the

rarnPle' r a factor

âfiiPle

at

atunties

[1 + Z x APRltr

(5.8)

Equivalently,

APR:

(1+EAR)I-1

WeuseEquation5.8 to find the APR correspondingto an EAR of 5.87owith various commoncompoundingperiods, and, conversely, the values of EAR implied by an APR of 5.8%.Theresutts;p;earin Table 5.1.

Compounding evidentfrom Table 5.1 (and Equation 5.8) that the difference betweenAPR and EAR vswith thefrequencyof compounding. This raisesthe question: how far will thesetwo qNergeasthe compounding frequency continuesto grow? Put differently, what is the of.[l + T x Repjrr as T gets ever smaller? As Z approacheszero, we effectively r continuouscompounding (CC), and the relation of EAR to the annual percentcenotedby r". for the continuously compoundedcase,is given by the exponential

of

,swittt ling clatrns

1 + EAR : exp (r"") : e,cc I is approximat ely2.71g2g.

(s.e)

129

150

PARTll Portfolio Theory

and Practice

----aooT'JiFtrlD

-EIFZIa1IîII'F: r = o'osa -n,t -An ** - tr,_rtmj:i]I 1 year 6 months 1 quarter 1 month 1 week 1 daY Continuous TABLE

5'1

Annual Percentage

.05800 1.0000 .0580 .05718 o.5o0o .0286 .0s678 o.25OO .0142 .05651 0.0833 .0047 .05641 0.0192 .0011 .05638 0.0027 .0002 r . " = l n ( 1+ E A R ) = ' 0 5 6 3 8

.0580 .0290 .0145 .0048 .0011 .0002

*ii,,

= o'oss * ot*n",ff .05800 .05884 .05927 .05957 .05968 .05971

A\R=exP(rJ-1='OY:

eXce.l "nnu31 121ss(EAR) rate (APR)and effective

at leasevishus www'mhhe'com/bkm

5'9 for r"" asfollows: rate' we solveEquation annual effective the To frnd r." ftom = ln (1 + EAR) r"" and exponential inverse-of the tun1tion, ::ql:] ^B-:tthe logarithm andEXP0' respectivelv' wheretn (.) is thena$ral t;n*""r, *i *" "u'"oiNtl togarithmicfunctions;';ilffi

ffi

to provlde ârl equrvo"'^" APR is necessary

----

While continuous comPoul

4"interest A bankoffersvouty:i':"j|:::Ir^Il"

look agarn * _arjÏË*.." for ample, schedules reladonsnlP the There, ùl t.,. 3 years:(a) a nominalrateR' an9

rate,r the 01',:Pol9o9^l1ilfri account asavinss :i"i'L,5srycom-real ffi#i'u[ i,r-R- iwasonllan

t"rT:liiiiijtlt;;*x*ïr-'#ïli'"ii; = e'*1 (1 + EAR)Î + EAR = ercc,then +Ihis follows from equation5'9 lf 1

CHAPTLR 5 Learning AboutReturnandRiskfromthe Historical Record asdemonstrated approximation, by equation5.3.But if we expressall ratesascontinuously thenequation5.1 is exact,5thatis, r..(real): r",(nominal)- i,,. compounded,

il and vely.

i.840\s ng.But provide Lhigher

In this chapter we will often work with a history that begins in 1926, and it is fair to ask why.The reasonis simply that January l,1926, is the starting date of the most widely availableaccuratereturn database. Table 5.2 summarizesthe history of short-term interest rates in the U.S., the inflation rate,and the resultant real rate. You can find the entire 80-year history of the annual rates of theseserieson the text's Web site, www.mhhe.com/bkm (link to the Online Learning Centerfor Chapter 5). The annual rates on T-bills are computed from rolling over 12 onemonthbills during each year. The real rate is computed from the annual T-bill rate and the percentchangein the CPI according to Equation 5.2. Table5.2 shows the averages,standarddeviations, and the first-order serial correlations for the full 80-year history as well as for various subperiods.The first-order serial correlation measuresthe relationship between the interest rate in one year with the rate in the precedingyear.Ifthis correlation is positive, then a high rate tends to be followed by another highrate,whereasif it is negative,a high rate tends to be followed by a low rate. Thediscussionof equilibrium real ratesof interestin Section 5.1 suggeststhat we should startwith the seriesof real rates. The averagereal rate for the full 8O-yearperiod, .72V0,is quitedifferentfrom the averageover the most recent 40 years, 1966-2005, which is l.25Vo. Weseethat the real rate has been steadily rising, reaching a level of2.28Vo for the generation of 1981-2005. The standarddeviation of the real rate over the whole period, 3.9iVo, wasdrivenby much higher variability in the early years.The real rate was far more stable in theperiodof 1981-2005,with a srandarddeviationof only 2.35V0. We can attribute a good part of these trends to policies of the Federal ReserveBoard. Sincethe early 1980s,the Fed has adopted a policy of maintaining a low rate of inflation anda stablereal rate. Some believe that the higher level of real rates in recent years may alsobe attributableto increasedproductivity of capital, particularly investmentsin informationtechnologywhen applied to a better educatedlabor force.

:e,work' risk.For r..(l), is ion to the g disc rother

ation5'

tween R, and

vasonlY

Ulonquiteclosely since the 1950s. Indeed, the correlation between the T-bill rate and the _ I +r(nominai) I + inflation

b0 +'(rearll = r^(l rrAsueU) r , ( m a l ;= , , ( n o m i n a l ) _i

: mtr+,tno.inal)l- ln(t+inflation)

t5l

o ro o.

I

N

o.

ËEË5ëÈËiË

lt|

ro or I G'

o\

SBEl*:i3=

3:33$3+3Ë 3 5 È * 4 3 3 5 33 s J I

o ;j v'=E

X HË o Ë t

i:Èn3Ë:RË

ui u e .F 6 a

ËRË3n3:Ë5 )

|r!o 6 o g , îl 5

Ë i B * 3 È n : ,Ëà r N A 6 E F

!

o ' F c . = O E

# = o . :

o

$ È E ç H Ë ! R bô5.; çr

N

É 0 , d !

€ F o r -

.9c

Ii5:$ËÈ]3

irt G

C

o - Eo

9= è>

ËE E É Èe riE EË

u

Ë ut

,

- c

I

g

U€

,i = "i+ 1T j È

132

: ) . :

r e

CHAPTE.R5LeamingAboutReturnandRiskfromtheHistorica|Record inflation rate is .41 for the full 80-Year history, .69 for the later 40 Years,and ' 72fot the most recent generation,1981-2005' Figure 5'3 shows the progressionof the nominal andreal value of $1 invested in T-bills at the beginning of 1926' accumulated to 2005. The Progresston of the value of a $ 1 investmentis called a wealth index.The wealth index in a currentYearis obtained bY compounding the Portfolio value from the end of thepreviousYearbY 1 * 4 the gross rate of return in the current Year. Deviations of the curve of the nominal wealth index in Figure 5.3 from a smooth exponentialline are due to variation over time in the rate of return. The lines in Figure5.3, which grow quite smoothly, clearlY demonstrate that shortterminterestrate risk (real as well as nominal) is small even for long-term horizons. It certainly is less risky by an order of magnitude than investmentsin stocks,as we will soonsee. One important lesson

155

15 't0

8

s

6

o

Æ

o

1946'

4 I

-5

'1966

1956

'1976

1986

1996

-10 -15

F I G U R E5.2

Interestand inflationrates, 1926-2005

12

-

10

!

6

;

Nominalwealth index Realwealth index

8 6

19Ë

r9s

19€

1955 rt6

1975

19E5 1S5

m5

(!

o

B

4 2 t'tt

-

icço

1e9s

FlGuRE5.3Nominalandrealwea|thindexes,1966-2005(insertfigure, 1925-2005)

fromthishistoryis the efinflationratewas3'02%o fætofinflation*h"n "o*poondedoverlong periods.Theaverage between 1926and2005, ind 4.29VobetweenigOO*O 2005.Theseratesmaynot seemimpressive, butaresufficientto reducetheterminalvalueof $1 investedin 1966from a nomirnalvalueof $10.08in 2005to a real (constantpurchasingpower)valueof only $1'63.

Retums arcconsidering investinsin a stockindexfund.The fund currentlysellsfor $100per :' Withaninve-stment hùzon of 1 vear,the realizedrate of return on your investment

154

Theory PARTll Portfolio

and Practice

willdependon(a)thepricepershareatyear,sendand(b)thecashdividendsyouwill

t:,tr]gï:dï-]j',ïi"ii,Ëii:#tr" co[ecroverrheveæ. end year,s theprice Suppose P* :li:jt#; i'n" nou'n''perior t:T:i *ooJ to$4'Therealized is definedas holdingPeioA is 1 Year)'

priceorastræ;t enaine Pl*;ni*:rice HpR=

In our casewe have HPR=

+ cash0

(s'10)

g4 = .r4, or !4vo s110- g100+ ï1ô6period. he endof thehording income 'gnot"t reinvesffnent r-gPeriod'The Percentreturn gatns ni Yi"tOPfotthecaPital

StandardDeviation ExpectedReturnand

(s'11) E(r)=)r(s)r(s)

ApplyingthisformulatothedatainSpreadsheet5.l,wefindthattheexpectedlateolre., ls nlrn on the index tund x ?167o)]: l4lo + (5 x I4Vo)+ [0'20 34lo) X =(0'30 E(r)

thatthis"T :T :"-î;1;ff3"""iY r ltrow Spreadsheet .s 1 calculatesT"ti',rt"ït"uabilitv*ïilîf'f;;;ffi of eac tunction'which first

J,i:

;"*ô;èi Here'the numberPau Products' il;;;e rateof reûrn '"'"T;^;;;ard or deviationot tne yte 11 firn : in variance'which -ouar; ,oot of the the higher vo ;p"cted return'The ir";;;"

agevalueof these"-:î:*"1ÏH:-r*T;:, SYmbolicallY'

ffi:Jffi ^ïî

oi"o*"*'

oz =)r(s)[r(s)-E(r)12

, (5'

CHAPTLR5

ar he

Ratesof returnexoressedas decimals PurchasePrice= Cash Year-end Dividends Price '129.50 4.50 0.3 4.00 110.00

r0) Expectedvalue(mean) Standarddeviationof HPR Riskoremium

riod. )ome eturn gains

LeamingAbout Returnand Riskfrom th€ HistoricalRecord

0.34 0.14 -0.16 111, ,E9:E11) E 9 : E 1 '=1 ) = 0.14 0.14 SUMPRODUCT(Bg:B1 = 0.1732 F9:F11)^.5 SUMPRODUCT(89:811, = G9:G11) SUMPRODUCT(89:811,

S P R E A D S H E5E. Î1

eXce|

Distributionof HPRon the stock indexfund

lÎïïiil:ii,lrrn-

Therefore,in our example, 1 year ify our lle scechæacr Ieturn' reighted scenario we may

- t+12+ .5(14- t+12+ 0.2(-16- rq2: 300, o2: 0.30(34 and o:",Æ00:I7.3270 Clearly,what would trouble potential investorsin the index fund is the downside risk of of a34Vorateof return.Thestandarddeviaa-lîVorateof return,nottheupsidepotential tion of the rate of return does not distinguish between these two; it treats both simply as deviationsfrom the mean.As long as the probability distribution is more or less symmetric aboutthe mean,o is an adequatemeasureof risk. In the special casewhere we can assume thatthe probability distribution is normal-represented by the well-known bell-shaped curve-E(r) and o are perfectly adequateto characterizethe distribution.

(s.11)

Excess Returnsand RiskPremiums :ate of re'

the SUM' ' Pairs,and rio andthe f,ned as 1 dev be the ion

Getting backto theexample,how much,if anything,shouldyou investin theindexfund? Fint,youmustaskhow much of an expectedrewardis offeredfor the risk involvedin investing moneyin stocks. Wemeasure the rewardasthe differencebetweentheexpectedIIPR on the index stock frrndandtherisk-free rate, that is, the rate you can earnby leavingmoneyin risk-free tgætssuchasT:bills, moneymarketfunds,or the bank.We call this differencethe risk prcmiumon commonstocks.If the risk-freeratein the exampleis 6Voperyear,andthe clp€cted indexfund returnis l4To,thenthe risk premiumon stocksis 870per year.The ûfier.erce in anyparticularperiodherweenthe.actunl.rate of returnon a risky assetand : h risk-free .* i! ""ff"A .ri""r, ."to*. Therefore,the risk premiumis the expectedexreturn,andthe standarddeviation ofthe excessreturn is an appropriatemeasureofits {SeeSpreadsheet 5.1 for thesecalculations.)

edegree to which investorsarewilling to commit fundsto stocksdependson risk ton'Financialanalystsgenerally assumeinvestorsarerisk aversein the sensethat, IiSk oremi,,*

Premiumwere zero, people would not be willing to invest any money in stocks.

0.08

155

Theory PAff ll Portfolio

15Ô

and Practice

ffitrffi;:riîffi,ru:Ï*'ffi [îilirri,s,:?';itiî*fri["Fs:xï.r:"'"'Ë l\i:u';$ilu:l*i.kiir*i*";*s î-î;*iiïÏïrri"i;"*"'î i:Ë:$Ï:ff ii"tau*i in+arêSt rare) r..--

lnalySis:

Low

ntfr'Æ

I

devlauu"' standd9 ;tandaro\rerrs---

'#::":lîÏ::

lidti i

, \

iiil;" arenobserva 5'tt'_i'-'-r."r,, inEquation

,

; '.

I Sr .. -^-^,in_,,Sg rr-. lilzalv ..scenario."

.^ri. Averagc ÀvefâgG lgtic

:r;iï,*ir;"*i"nx* '' is t

(5.1

tj.

Lr? 1l

.. :

l

of ratesof = ariÙrmetcavefage

return

Ratum

t,

',

CHAPTLR5

be in )rs :ks 'in

c

A

B

Period

rmpilcrily ASSumeo Prohrabilitu- 1/5

D

E

F

gross nrn = 1+HPR

weafin lndex*

t

,' 3 4

10 11

12 13 14 15

,9S Of

htsto ecord nakes -atistirefirst

Squarecl HPR (decimall

De-viâtiôn

-u,t r uv 0 . 0 19 t 0.058( 4.2210 0.0701 0.2869 0.1088 0.007i 0 000f 2005 0.0491 = 0.0210 Arithmeticaverage AVEMGE(C5:C9) 0.0210 C5:C9)= ExpectedHPR SUMPRODUCT(85:89, = 0-1774 D5:D9)^.5 Standarddeviation SUMPRODUCT(85:89, = 0.1983 STDEV(C5:C9) - 1= GEOMEAN(E5:E9) Geometricaveraoereiurn 'Thevalueof $1 investedat the beqinninqof the sampleperiod(1/1/2001)'

zuul 2002 2003 2004

5 6 7 a

ano be1 re) get ndo

.z .2 .2 .2 .2

2 S P R E A D S H E5E. T forthe5&P500 of HPR Timeseries

EXCEI mili:::Lr-.

I.

an equalprobability of 1/5, or .2. Column B in Spreadsheet5.2 therefore uses .2 as probabilities,and Column C shows the annual HPRs. Applying Equation 5.13 (using Excel's SUMPRODUCT function) to the time series in Spreadsheet5.2 demonstratesthat adding up the products of probability times HPR amounts to taking the arithmetic average of the HPRs(comparecells C10 and Cl1). :i' I associ'.,{ cludebY proPosed1È' '.* meseries nts of the {PRs.We se returns re$rn and .d

Example5.6 illustratesthe logic for the wide useof the arithmeticaveragein investthetrueunderlyingprobabilments. If thetime seriesof historicalreturnsfairly represents ity distribution, thenthe arithmeticaveragereturnfrom a historicalperiodprovidesa good forecast of theinvestment's expectedHPR.

ïhc Geometric(Time-Weighted) AverageReturn rile saw that the arithmetic average provides an unbiased estimate of the expected rute of. rctum.But what does the time seriestell us about the actual performance of the portfolio overthe full sample period? Column F in Spreadsheet5.2 shows the wealth index from investing$l in an S&P 500 index fund at the beginning of 2001. The value of the wealth indexat theend of 2005, $l .0275 , is the terminal value of the $ 1 investment,which implies t S'yearholding-periodreturn (HPR) of 2.75Vo. . An intuitive measureof performance over the sample period is the (fixed) annual HPR thotwouldcompoundover the period to the same terminal value as obtained from the seof actualreturns in the time series.Denote this rate by g so that

Terminalvalue : (l * r,) x (1 + rùx.'. x (1 + r) : 1.0275 (l * 8)" = Terminalvalue: 1.0275 (cellF9 in Spreadsheet (s.14) 5.2) = - l:1.02751/5- 1 : .0054: .54Vo (cellE14) 8 Terminal valuer/n

Jo rs for the

'Us"ruadof s and

137

LearningAbout Returnand Riskfrom the HistoricalRecord

I * g is the geometricaverageof the grossreturns(1 + r) from the time series canbecomputedwith Excel'sGEOMEANfunction)andg is the annualHPR that rcplicate thefinal valueof our investment. tuonersof investments also call g the time-weighted(asopposedto dollar-weighted) retum,to emphasize that each past return receivesan equal weight in the process

0.8811 0.7794 1.2869 1.1088 1.0491

0.8811 0.686r 0.883( 0.9792 1.027!

lheck: L0054^5= 0.0054 1.0275

158

and Practice PARTII PortfolioTheory

jilïïïîïA:îï:Jff ::i:l',iii oraveraging.ffi :-.:"1,îJJ changes- t:1":-i:^*:;.;;"r tund is largeproo the

sisnificant

ilË;;;**o'uta,n"oouriîiff iXïrYtli#JÏi,.ï;o;m"" whe. than ratesobtained "t p"Àtrt*ce

evaluation

r

, :':ffiJ;

*lffijii""'"s" tiltiril; (2.107o)' averase effectot P asvmmetric âSlfIUuvurw

of arithmetic average

"i

ratr negative

:ïÏ"f"":"ffi#

'J"/'Ytvo'

otreluL1f1.i::îilhffîJ"ffi"ï'JïÏ:"'ffii inrates theswings rherarger rù' uvrr'----r urùurvçÉ---' IIIaL avefages' -^ârma "nme ffom a noflnar

medcandgeometric**3î::':Y::i'"'.:ii"":::-"';J;;;"f'"Tili medC and geOmetfiC of the annual re$rnr that is' ot: the distribution' oeriod and the average equalshalfthe variance îh" diff"r"n"e exactly _z - rrvz u = Arithmetic average average Geometric

(5.15)

(Awarning:touseEquation5.15,youmustexpressreturnsasdecimals,notpercentages.)

Deviation Varianceand Standard

valueof squæeddeviations Væiance= sxpected - E(r)i2 oz = !ds) [r(s) as we estirnatevariance n observations' with data Usinghistorical -' -= !* rr(s)-Fl2 ol rar.l

6.16

About Returnand Riskfromthe HistoricalRecord 5 Learning CHAPTÉR

res. rfits pter

5.2. ColumnD showsthe squaredeviationsfrom the Takeanotherlook at Spreadsheet andiell D12 givesthe standarddeviationasthe squareroot of thesum Jtt r"ti. average, the squareddeviations(.1'174). Jifrooo"o of the (equal)probabilitiestimes

ffi netic n the rf the

u =(h)a 1i t,t,r- r,' : *V.,rr(s)- Fl2

erage u had re end which .enec- '

r 2003 rateto r value 2-yeat hanthe

re arith, SamPIe ibution,

(s.ls) entages.)

o=

frÉ"u,-','

deviation is .1983, which is a bit Cell Dl3 showsthat the unbiasedestimateof the standard higherthan the .1774value obtainedin CellD12'

(Sharpe)Ratio the Reward-to-Variability

with the risk of that expectedercessreturn, and hence it's best to measure commensurate riskby the standarddeviation of excess,not total, returns. Th! importanceof the trade-off between reward (the risk premium) and risk (as measuredby standarddeviation or SD) suggeststhat we measurethe attaction of an investment portfolioby the ratio of its risk premium to the SD of its excessreturns.

Sharperatio(for porfolios) : m the at' : estimate 'etunr,the

(s.17)

Risk oremium ,,, SD "f ;"."*"t

(s.18)

Thisreward-to-volatility measure(frrstproposedby William Sharpeandhencecalledthe Sharpe of investmentmanagers' ratio)is widelyusedto evaluatetheperformance

lual lableE(r),

'Ïhke

anotherlook at Spreadsheet5.1. The scenario analysis for the proposed investment l'h thestockindex fund resulted in a risk premium of 8Vo,and standarddeviation of excess

of 17.32Vo. This implies a Sharperatio of .46, a valuethat is pretty muchin line pastperformance of stockindex funds.We elaborateon this importantmeasurem chaptersand show that while it is an adequatemeasureof the risk-return trade-off

r59

140

Theory PAff II Portfolio

and Practice

of subject (the pomorios ilJ;'ri;iixitil:i ,'il;ïÏîli"i; t fordiversined shares of stocK

applied to diversified

such as individual assets

portfolios'

ii l.'i;: lT'Il,,n, ::::.î:::::::J:: petYeat' tL"lÏ:#"r#, ratewas60/o ic) compute,lt risk-free (Q Compute the stanoi

..

theSharq

(O) aotp""

- ^-. .-ina rhe the ris'

--

:''l''"+"nu**"**'"'

W'

rhebel'shap"eÏ::tïi;Hfiî;i:"1îîï$iliîHtii "t heiehtsandweights

ï;;;;;_ult

of multip

a, ationsimplicit in

iffï*:h;*nr;*;:ËrïËf Tï'r"i*fxn jï;:ï""'"ffi

i,#':ilîlBii:i"iï;i;''orT*:1"^:rydistributedæoun

:t"'fJJHffi,1nU,qry.,;çii;rff omesa*er2da comes at the end

ot any period' Here

is an eve

Two good daYs'Profit

= $200

=0 Two bad daYs'Profit

CHAPTER5

the HistoricalRecord LearningAbout Returnand Riskfrom

l 'f

FIGURE

al al,

-shapednormal distribution' with midrange teastlikelY'6 of uitft *"* of lOToand standarddeviation

rfrt llS,

)ut-

ttil::ff*.anagementisfarmoretractablewhenratesof returncanbewellapproximated probability distributionis slnnmetric,thatis, the bythenormaldistribution.FJJ thenormal of the "quut a thai of a negativedeviation of anypositivedeviationabovethe meanis. returns risk as the standarddeviation of ume magnitude.Absent symmetry,measlnnq to a uniquefamily of distributions Second,tn" ""r."f distributionbelongs is inadequate. When assetswith normally as "stable,"becauseof the following iroperty: characterized return also is normally pà.roto distributed retumsare mixed to constructa porrtblio, trte (mean ontV ir greatly simplified ylren distributed. fnirO, sceniîL"f'rit rylparameters

the probabilities of future scenanos' 5.rnd SD) needto be estimated to obtain justify its use in fit the normal curve to , $i How closely *ust altuul return distributions realt-not be a perfect description of $invcstment management?Clearly, the normal "u*" canplo' one bad

rbilitY both

rple' r the -^ ic 3Ie r"

number oution

,r ç^autPlg, i1çtui1l KËïiilffiË:;#i**

rçLuuù

distribution É*o* beressthan-100%,whichthenormal

I not rule out. But this does not mean that u issuearisesin many other contexts. For er ,icallyevaluatedby comparing it to a normal ising, becausea normal distribution admit werebasedon the Lirtoricalfootnote,earlydescriptionsof the normal distributionin the eighteenthcentury

ro'*" "*!ny1îX; B ora"binomial drawn t ..i lik"th"on"wehave

see ln unaPtçr îiiJî:i: as we wlll :1î*",9.îîi:: ,pæsentationis used in practice to price many option conEacts'

Filili#

,iîi,n"l,iiiî*oruon

sotorvwwicu.ect'/matbÂsep/ quickly"ppro*mrÀtrrenormal,

t4l

1L2

and Practice PARTII PôrtfotioTheory

,."*ffi##i:6",,-.

.in Excel ffi,l]u;l'ds'@4çj9*!

ofre1u1T * :f:;tÎ;ili51 themonthlv suppose late deviationor ovc of rlo andstandæd

ili";rïî rir::djitk;ï Ë:"$lf r*"àH

;iii;;;

answerthis questton:irrr,;ilU". functionis gi'

t!ry-Ti1.irrï;;;;ant cording

rokngwtheprobab:

so d:i1liïis670' iî:î'ï:',I}"ll iâ,i:;'i*a'a

iÏiff ii11i:J;"""J'"'"*:îË:ff ::f#;'::T'i,i be thesame:NORMsurs

r r