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Risk and Rates of Return
(Ch. 10 & 11)Stand-Alone Risk
Portfolio Risk
Risk and ReturnQuick overview of CAPM/SML
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A First Look at Risk and Return
•Standard & Poor’s 500: 90 U.S. stocks uto !9"# and "00 after t$at. Leaders in t$eirindustries and a%on& t$e lar&est 'r%s tradedon U.S. Markets.
• Sma sto!ks( Securities traded on t$e )*S+wit$ %arket caitali,ations in t$e otto%!0.
• "ord Portfoio( nternational stocks fro%all t$e worlds %a1or stock %arkets in )ort$A%erica2 +uroe2 and Asia.
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A First Look at Risk and Return(!ont’d)
• Cor#orate $onds( Lon&-ter%2 AAA-rated U.S. cororate onds wit$
%aturities of aro3i%atel4 50 4ears.
• %reasur $is: An invest%ent in
t$ree-%ont$ 6reasur4 ills.
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'aue of 100 n*ested at the +nd of1,-5
Source: Chicago Center for Research in Security Prices (CRSP) for U.S. stocks and
CPI, Global inance !ata for the "orld Inde#, $reasury bills and cor%orate bonds.
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$ottom Line
• Sma sto!ks had the hi/hest on/term returns w$ile 6-7ills $ad t$elowest lon&-ter% returns.
• Sma sto!ks had the ar/est2u!tuations in #ri!e2 w$ile 6-7ills$ad t$e lowest.
So ke %akea3a
4i/her risk reuires a hi/her return.
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n*estment Returns
6$e rate of return on an invest%ent cane calculated as follows(
8or instance2 if !2000 is invested and
!2!00 is returned after one 4ear2 t$erate of return for t$is invest%ent is(
:!2!00 ; !2000
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"hat is in*estment risk• 6wo t4es of invest%ent risk
– Stand-alone risk
– Portfolio risk
• nvest%ent risk is related to t$e %robability of earnin& a low ornegative actual return.
• %he /reater the !han!e of o3er
than e6#e!ted or ne/ati*ereturns the riskier thein*estment.
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Probability Distributions• A listin& of all ossile outco%es2 and t$e roailit4 o
eac$ occurrence.• Can e s$own &ra$icall4.
+3ected Rate of Return
Rate of
Return :
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Probability !istribution of Returns for a sto &n illustration
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?ow does it translate &ra$icall4@
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+%irical istriution of Annual Returns for U.S. Lar&e Stocks :SBP"00
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Avera&e Annual Returns for U.S. S%all Stocks2 Lar&eStocks :SBP "00
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A word on calculating portfolio return and portfolio risk
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Volatility is more tricky to compute as you shall take
into account dependence
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15
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nvest%ent Alternatives in a c$an&in& world
+cono%4 Pro. A $ C 8
Recession 0.! "." -7.08 5#.0 .0 -!#.0
7elow av& 0.5 "." 7.08 !E.0 -!F.0 -E.0
Avera&e 0.F "." 15.08 0.0 E.0 !0.0
Aove av& 0.5 "." 90.08 -!!.0 F!.0 5".0
7oo% 0.! "." 5.08 -5!.0 5.0 ED.0
6reasur4@
" diGerent asset class2 " ortfolios
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nvest%ent Alternatives in a c$an&in& world
+cono%4 Pro. $ C 8
Recession 0.! -7.08 5#.0 .0 -!#.0
7elow av& 0.5 7.08 !E.0 -!F.0 -E.0
Avera&e 0.F 15.08 0.0 E.0 !0.0
Aove av& 0.5 90.08 -!!.0 F!.0 5".0
7oo% 0.! 5.08 -5!.0 5.0 ED.0
6reasur4@
" diGerent asset class2 " ortfolios
!5.F
:F"
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Ca!uatin/ Standard ;e*iation for $
∑=
−=σ
σ==σ
=σ
)
!i
i
5
5
P
"ei/hted
A 02! -5# !52F -02E9F 02!""5 020!""57 025 -# !52F -02!9F 020E# 0200#"5
C 02F !" !52F 0205 02000 020005F
025 E0 !52F 02!# 020E09 020050
+ 02! F" !52F 02E5 02!0E 020!00
E%?%AL 020F0!!
E "ransform #ariance into standard de#iation =$ (,411!&(,5! = 2,%
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Co%arin& Standardeviations
Pro.A
7
0 "." 9.D !5.F Rate of Return :<
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Comments on Standard ;e*iationas a @easure of Risk
• Standard deviation :Ji< %easures total2 orstand-alone2 risk.
• 6$e lar&er Ji is2 t$e lower t$e roailit4t$at actual returns will e closer toe3ected returns.
• Lar&er Ji is associated wit$ a wider
roailit4 distriution of returns.
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Co%arin& Risk and Return
S+C A
S+C $S+C CS+C ;@AR+
%
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CoeKcient of Iariation :CI<
A standardi,ed %easure of
disersion aout t$e e3ectedvalue2 t$at s$ows the risk #er unitof return.
rHreturn+3ected
deviationStandard
CI
σ==
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Risk Rankin/s
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llustratin& t$e CI as a Measure ofRelative Risk
Je = Jf 2 ut e is riskier because of a larger %robability
of losses. n ot$er words2 the same amount of risk(as measured
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nvestor Attitude towards Risk
Risk a*ersion ; assu%es investors dislike
risk and reuire $i&$er rates of return toencoura&e t$e% to $old riskier securities.
Risk #remium :w$ic$ serves as
co%ensation for investors to $old riskiersecurities< - the diEeren!e
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Portfolio Construction( Risk and Return
• Assu%e a t'ostock %ortfolio is created
wit$ "02000 invested euall4 in 5ortfolios.
• Portfoio’s e6#e!ted return =
wei&$ted avera&e of t$e returns of t$eortfolios co%onent assets.
•Standard deviation is a little %oretrick4 as we saw earlier..
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Calculatin& Portfolio +3ected Return
.#:!.0
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An Alternative Met$od for eter%inin&Portfolio #%ected Return
+cono%4 Pro. 7 C Port.Port.Recession 0.! -5#.0 5#.0 0.00.0
7elow
av&
0.5 -#.0 !E.0 E.0E.0
Avera&e 0.F !".0 0.0 #."#."
Aoveav&
0.5 E0.0 -!!.0 9."9."
7oo% 0.! F".0 -5!.0 !5.0!5.0
.#:!5.0
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Calculatin& Portfolio Standardeviation and CI
0."!.#
E.F CI
E.F
.#
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Co%%ents on Portfolio RiskMeasures• J = E.F is %uc$ lower t$an t$e Ji of
eit$er stock :J?6 = 50.0O JColl. = !E.5
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eneral Co%%ents aout Risk
J ≈ E" for an avera&e stock.
Most stocks are ositivel4 :t$ou&$not erfectl4< correlated wit$ t$e%arket :i.e.2 etween 0 and !
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Returns istriution for 6wo Perfectl4)e&ativel4 Correlated Stocks : = -!.0<
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Correlation
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Partial Correlation2 = 0.E"
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Creatin& a Portfolio( 7e&innin& wit$ neStock and Addin/ Random See!tedSto!ks to Portfolio
D# de!reases as sto!ks added2ecause t$e4 would not be %erfectlycorrelated wit$ t$e e3istin& ortfolio.
+3ected return of t$e ortfolio wouldre%ain relativel4 constant.
+ventuall4 t$e diversi'cation ene'ts of
addin& %ore stocks dissiates :afteraout !0 stocks
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llustratin& iversi'cation +Gects of aStock Portfolio
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7reakin& own Sources of Risk
Stand-alone risk = Market risk iversi'ale risk
• @arket risk ; ortion of a securit4sstand-alone risk t$at cannot e eli%inatedt$rou&$ diversi'cation. Measured 4eta.
• ;i*ersiGa
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8ailure to iversif4
• f an investor c$ooses to $old a one-stock
ortfolio :doesnt diversif4
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Caital Asset Pricin& Model:CAPM<
Model linkin& risk and reuired returns. CAPMsu&&ests t$at t$ere is a Securit4 Market Line:SML
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4
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7eta
Measures t$e %arket risk of a stock2and s$ows a stocks volatilit4relative to t$e %arket.
Indicates ho' risky a stock is if thestock is held in a 'elldi-ersied
%ortfolio.
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Co%%ents on 7eta
• f eta = !.02 t$e securit4 is 1ust as risk4as t$e %arket ortfolio.
• f eta T !.02 t$e securit4 is riskier t$an
t$e %arket ortfolio.• f eta !.02 t$e securit4 is less risk4
t$an t$e %arket ortfolio.
Most stocks $ave etas in t$e ran&e of 0."to !.".
$ f i
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Can t$e eta of a securit4 ene&ative@
• *es2 if t$e correlation etween Stock iand t$e %arket is ne&ative :i.e.2 i2% 0
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Calculatin& 7etas
• Vell-diversi'ed investors are ri%aril4concerned wit$( how a stock is expected tomove relative to the market in the future.
• Anal4sts are forced to rel4 on $istorical data.
A typical approach to estimate beta is torun a regression of the security’s pastreturns against the past returns of themarket .
• 6$e slo%e of t$e re&ression line is de'ned ast$e beta coe/cient for t$e securit4.
ll i $ C l l i f
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llustratin& t$e Calculation of7eta
.
.
.
ri
W
rM-" 0 " !0 !" 50
50
!"
!0
"
-"
-!0
Re&ression line(
ri = -5."9 !.FF rMN N
*ear rM ri
! !"!D
5 -" -!0
E !5 !
+etas with espect to the
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+etas with espect to the
-. 5 for /ndi#idua
toc0s (based on month)
data for 24–2!
6$ S it M k t Li :SML
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6$e Securit4 Market Line :SML
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V$at is t$e market risk#remium@
• ASV( Additional return over t$e risk-freerate needed to co%ensate investors forassu%in& an a-erage a%ount of risk.
• ts si,e deends on( :i< t$e erceivedrisk of t$e stock %arketO and :ii<investors de&ree of risk aversion.
• Iaries fro% 4ear to 4ear2 ut %ostesti%ates su&&est t$at it ran&esetween F and D er 4ear.
C l l ti R i d R t f
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Calculatin& Reuired Rates ofReturn
+ t d R i d R t
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+3ected vs. Reuired Returns
ll t ti t$ S it M k t
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llustratin& t$e Securit4 MarketLine
..Coll.
.?6
6-ills
.USR
SML
rM = !0."
rR8 = "."
-! 0 ! 5
.
SML( ri = "." :".0
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5
An +3a%le(
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An +3a%le(+uall4-Vei&$ted 6wo-Stock Portfolio
• Create a ortfolio wit$ "0 invested in?6 and "0 invested in Collections.
• %he
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Calculatin& Portfolio ReuiredReturns
• 6$e reuired return of a ortfolio is t$ewei&$ted avera&e of eac$ of t$e stocksreuired returns :alread4 seen efore
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8actors 6$at C$an&e t$e SML
• f investors raise inXatione3ectations 4 E2 w$at would$aen to t$e SML@
SML!
ri :<
SML5
0 0." !.0 !."
!E."!0."
D." "."
Y = E
Risk2 i
8actors 6$at C$an&e t$e SML
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8actors 6$at C$an&e t$e SML
• f in*estors’ risk a*ersionin!reased2 causin& t$e %arket riskre%iu% to increase 4 E2 w$at
would $aen to t$e SML@SML!
ri :<
SML5
0 0." !.0 !."
!E."!0."
"."
YRPM = E
Risk2 i
Ierif4in& t$e CAPM +%iricall4
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Ierif4in& t$e CAPM +%iricall4
• 6$e CAPM $as not een veri'edco%letel4.
• Statistical tests $ave role%s t$at
%ake veri'cation al%ost i%ossile.• So%e ar&ue t$at t$ere are
additional risk factors2 ot$er t$an
t$e %arket risk re%iu%2 t$at %uste considered.
Aout t$e CAPM
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Aout t$e CAPM
• nvestors see% to e concerned wit$ ot$%arket risk and total risk. 6$us2 t$e SML%a4 not roduce a correct esti%ate of r i.
ri = rR8 :rM ; rR8