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PORTFOLIO CONCEPTS
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MEAN –VARIANCE ANALYSIS
Mean–variance analysis is the fundamental implementation of modern portfoliotheory, and describes the optimal allocation of assets between risky and risk-free
assets when the investor knows the expected return and standard deviation of those
assets.
Assumptions necessary for mean–ariance efficiency ana!ysis"
#$ A!! inestors are ris% aerse& t'ey prefer !ess ris% to more for t'e same !ee!of e(pecte) return$
*$ E(pecte) returns for a!! assets are %no+n$
,$ T'e ariances an) coariances of a!! asset returns are %no+n$
-$ Inestors nee) %no+ on!y t'e e(pecte) returns. ariances. an) coariances
of returns to )etermine optima! portfo!ios$ T'ey can i/nore s%e+ness.%urtosis. an) ot'er attri0utes of a )istri0ution$
1$ T'ere are no transaction costs or ta(es$
*
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EFFICIENT PORTFOLIOS
Efficient portfolios (assets) offer the hihest level of return for a iven levelof risk as measured by standard deviation in modern portfolio theory.
2 3ecause inestors are ris%4aerse. 0y assumption. t'ey +i!! c'oose to a!!ocate
t'eir assets to portfo!ios t'at 'ae t'e 'i/'est possi0!e !ee! of e(pecte) return
for a /ien !ee! of ris%$
2 T'ese portfo!ios are %no+n as efficient portfo!ios$
4 5e can use optimi6ation tec'ni7ues to )etermine t'e necessary +ei/'ts to
minimi6e t'e portfo!io stan)ar) )eiation for a specifie) set of e(pecte)
returns. stan)ar) )eiations. an) corre!ations for t'e assets comprisin/ t'e
portfo!io$
,
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PORTFOLIO E8PECTE9 RET:RN AN9 RIS;
2 5e can ca!cu!ate t'e e(pecte) return an) ariance of a t+o asset portfo!io as"
2 5e can ca!cu!ate t'e e(pecte) return an) ariance of a t'ree asset portfo!io as"
2 Stan)ar) )eiation is. of course. t'e positie s7uare root of ariance in 0ot'
cases$
2
-
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PORTFOLIO E8PECTE9 RET:RN AN9 RIS;
!ocus "n# $alculations
2 You are e(aminin/ t'ree internationa! in)ices$ 5'at is t'e e(pecte) return an)
stan)ar) )eiation of a portfo!io compose) of 1erman e7uities?
2 T'e E @r is ,$B1D=.
an) t'e stan)ar) )eiation
is #
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TE EFFICIENT FRONTIER
*he efficient frontier is a plot of the set of expected returns and standarddeviations for all efficient portfolios (assets) above the lobal minimum-
variance portfolio.
2 T'e minimum4ariance frontier
@solid reen line is t'e set of
a!! portfo!ios t'at represent t'e!o+est !ee! of ris% t'at can 0e
ac'iee) for eac' possi0!e !ee!
of return$
4 T'e portfo!io +it' t'e !o+est
ariance of a!! t'e portfo!ios.
+it' t'e !o+est !ee! of ris%
t'at can 0e ac'iee). is
%no+n as t'e lobal
minimum-variance portfolio$
B
Efficient !rontier
Stan)ar) 9eiation
E @r
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TE EFFICIENT FRONTIER
%ortfolios on the efficient frontier provide the hihest possible level ofreturn for a iven level of risk.
2 3ecause portfo!ios on t'e
efficient frontier use ris%
efficient!y to /enerate returns.
inestors can restrict t'eirse!ection process to portfo!ios
!yin/ on t'e frontier$
4 T'is approac' simp!ifies t'e
ris%y4asset se!ection process
an) re)uces se!ection cost$4 T'e !i/'t /reen portfo!ios in
t'e fi/ure are inefficient
portfo!ios$
Efficient !rontier
Stan)ar) 9eiation
E @r
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9IVERSIFICATION AN9 CORRELATION
*he trade-off between portfolio risk as measured by standard deviation andportfolio expected return is affected by asset returns, variances, and
correlations.
2 Reca!! t'e e(pecte) return an) ariance
of a t+o4asset portfo!io$
2 A!! t'e terms in t'e ariance ca!cu!ationare strict!y positie. e(cept t'e !ast
term. +'ic' inc!u)es t'e corre!ation.
+'ic' ran/es from perfect ne/atie @ –#"
0!ue to perfect positie @G#" purp!e
+it' 6ero corre!ation in 0et+een @
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FIN9IN> TE MINIM:M4VARIANCE FRONTIER
+e can use an optimier, such as the 'olver in Excel, to solve for theweihts in the minimum-variance portfolios and thus the minimum-variance
frontier.
2 Reca!! t'at t'e set of +ei/'ts in any portfo!io must
sum to # an). if t'ere are no s'ort sa!es. must a!!
0e positie$2 T'e e(pecte) return an) ariance for a /ien set of
+ei/'ts are
2 For eery return. z . 0et+een z min an) z max .
+e so!e for t'e set of +ei/'ts t'atminimi6es t'e portfo!io ariance su0Hect to
E @r p z.
4 If +e )o so iteratie!y. +e 0e/in at z min an) iterate
0y a fi(e) amount of E @r p unti! +e reac' z max .
( ) ( )∑=
=n
i
ii p r E wr E 1
( ) ,1 1
Varσ σ ρn n
p i j i j i ji j
r w w= =
=
∑∑
∑=
=n
i
iw1
1
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EJ:AL45EI>TE9 PORTFOLIOS
2 T'e e(pecte) return to an e7ua!!y +ei/'te) portfo!io is Hust t'e sum of t'e
e(pecte) returns to t'e assets )ii)e) 0y t'e num0er of assets$
2 It can 0e s'o+n t'at t'e ariance of an e7ua!!y +ei/'te) portfo!io is"
+'ere n is t'e num0er of assets in t'e portfo!io. is t'e aera/e ariance oft'ose assets. an) is t'e aera/e coariance of t'e assets$
2 Consi)er a #
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TE CAPITAL ALLOCATION LINE
*he capital allocation line ($) describes the optimal expected return andstandard deviation combinations available from combinin risky assets
with a risk-free asset.
2 T'is is a !ine ori/inatin/ at t'e e(pecte) return–stan)ar) )eiation coor)inates
of t'e ris%4free asset an) !yin/ tan/ent to t'e efficient frontier$
4 T'e s!ope of t'is !ine is %no+n as t'e S'arpe ratio. an) it represents t'e 0estpossi0!e ris%–return tra)e4off 0y construction$
4 As can 0e seen from t'e e7uation for t'e CAL"
4 T'e intercept is t'e ris%–return coor)inate for t'e ris%4free asset or KR F .
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TE CAPITAL ALLOCATION LINE
#*
CAL
Efficient Frontier
Stan)ar) 9eiation
E @r
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TE CAPITAL ALLOCATION LINE
!ocus "n# $alculations
Consi)er an inestor facin/ a ,= ris%4free rate +it' access to a tan/ency
portfo!io +it' a #*= return an) an #D= stan)ar) )eiation$
4 If t'e inestor re7uires a #
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TE CAPITAL MAR;ET LINE
+hen all investors share identical expectations about the expectedreturns, variances, and covariances of assets, the $ becomes the $M.
T'e capita! mar%et !ine @CML represents t'e case in +'ic' a!! inestors 'ae t'e
same e(pectations an). t'erefore. 'o!) t'e same ris%y portfo!io as t'e tan/ency
portfo!io$
4 In e7ui!i0rium. t'is +i!! 0e a!! ris%y assets in t'eir mar%et a!ue +ei/'ts&
'ence. a!! inestors +i!! 'o!) t'e mar%et portfo!io as part of t'eir portfo!io$
4 T'e s!ope of t'e CML is %no+n as t'e mar%et price of ris% an) is t'e S'arpe
ratio for t'e mar%et portfo!io$
2
#-
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CAPITAL ASSET PRICIN> MO9EL
*he capital asset pricin model, or $%M, describes the expected return toany asset as a linear function of its /beta.0
2 T'e CAPM proposes t'at a!! security e(pecte) returns can 0e 0ro%en )o+n into t+o
components"
4 A ris%4free component @in re)$
4 A component receie) for 0earin/ mar%et ris% @in 0!ue$
4 T'is component is t'e amount of ris%. βi . times t'e price of ris%. E @R M – R F $
4 βi is a measure of t'e assets sensitiity to mar%et moements @mar%et ris%$
4 βi # is t'e 0eta for t'e mar%et. or βM $
4 βi # is /reater t'an t'e 0eta for t'e mar%et an) +e +ou!) e(pect returns in
e(cess of mar%et returns$
4 βi # is !ess t'an t'e 0eta for t'e mar%et an) +e +ou!) e(pect returns !o+er
t'an mar%et returns$
4 βi < is 6ero mar%et ris% @ris% free an) +e +ou!) e(pect t'e ris%4free return$
4 E @R M – R F is %no+n as t'e mar%et ris% premium$
#1
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CAPM ASS:MPTIONS
#$ Inestors nee) on!y %no+ t'e e(pecte) returns. t'e ariances. an) t'e
coariances of returns to )etermine +'ic' portfo!ios are optima! for t'em$
4 T'is assumption appears t'rou/'out a!! of mean–ariance t'eory$
*$ Inestors 'ae i)entica! ie+s a0out ris%y assets mean returns. ariances of
returns. an) corre!ations$,$ Inestors can 0uy an) se!! assets in any 7uantity +it'out affectin/ price. an)
a!! assets are mar%eta0!e @can 0e tra)e)$
-$ Inestors can 0orro+ an) !en) at t'e ris%4free rate +it'out !imit. an) t'ey can
se!! s'ort any asset in any 7uantity$
1$ Inestors pay no ta(es on returns an) pay no transaction costs on tra)es$
#B
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TE SEC:RITY MAR;ET LINE
*he raphical depiction of the $%M is often known as the security marketline, or 'M.
#
E @r m
r f
β
E @r
βm#
SML
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MAR;O5IT 9ECISION R:LE
*he Markowit decision rule provides several principles by which investorscan determine how to allocate their assets.
2 5'en c'oosin/ to a!!ocate a!! of your money to Asset A or Asset 3. c'oose A
+'en
4 T'e mean return on A is /reater t'an or e7ua! to t'at of 3. 0ut A 'as a
sma!!er stan)ar) )eiation t'an 3. or
4 T'e mean return of A is strict!y !ar/er t'an t'at of 3. an) A an) 3 'ae t'e
same stan)ar) )eiation$
4 5'en eit'er of t'ese is t'e case. +e say t'at A Qmean–variance )ominates
3$
2 If +e can 0orro+ an) !en) at t'e ris%4free rate. t'en
4 T'e portfo!io +it' t'e 'i/'er S'arpe ratio mean–ariance )ominates t'e
asset +it' t'e !o+er S'arpe ratio an) s'ou!) 0e c'osen$
#D
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A99IN> AN ASSET CLASS
+e will add a new asset class to our existin portfolio when makin thataddition provides a hiher 'harpe ratio for the resultin portfolio.
2 In or)er to )etermine +'et'er +e +i!! 'ae a 'i/'er S'arpe ratio. +e nee)
4 T'e S'arpe ratio of t'e ne+ asset c!ass&
4 T'e S'arpe ratio of t'e e(istin/ portfo!io. p& an)4 T'e corre!ation 0et+een t'e ne+ inestments returns an) t'ose of our
e(istin/ portfo!io$
4 If t'is con)ition is true. our ris%–return re!ations'ip is improe) 0y a))in/ t'e
ne+ asset c!ass$
2
#
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LIMITATIONS OF MEAN–VARIANCE ANALYSIS
1istorical estimates of model parameters involve two potential problems# (2) thelare number of estimates needed, and (3) the 4uality of such estimates.
2 T'e num0er of parameters nee)e) for mean–ariance efficiency ana!ysis is n3 53
6 7n 53$
4 For een a sma!! set of assets. t'is num0er is ery !ar/e$
4 For e(amp!e. if +e 'ae *D assets. t'at is -,- parameters t'at must 0e use) in
t'e optimi6ation process$
2 T'e 7ua!ity of t'e estimates t'emse!es is /enera!!y !o+$
4 T'e estimate of mean return 'as a !ar/e ariance. an) sma!! c'an/es can
)ramatica!!y effect mean–ariance estimation outcomes$
4 T'e estimate of t'e ariance 'as a sma!!er re!atie ariance. 0ut is a!so
measure) +it' error$
4 T'e estimates of t'e coariance a!so 'ae a !ar/e amount of measurement
error$
*<
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TE MAR;ET MO9EL
*he market model can be estimated via linear reression and is often usedto estimate unad8usted firm betas.
2 T'e estimate) re/ression e7uation for t'e mar%et mo)e! is"
2 From t'is. +e can ca!cu!ate t'e e(pecte) return. ariance. an) stan)ar) )eiation of
any stoc% as"
2 :sin/ t'e mar%et mo)e! to )etermine t'e necessary mean–ariance parameters
re)uces t'e set of parameter estimates to 7n 6 3$
2
*#
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TE MAR;ET MO9EL
!ocus "n# $alculations
2 5e are e(aminin/ t+o in)ustry in)ices. one of +'ic' 'as a beta of 2.9: an) a
residual standard deviation of 2;.
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3ETA" TO A9:ST OR NOT A9:ST
1istorical betas may not be as useful for predictin future behaviorbecause we know that betas chane over time.
2 5e can mo)e! 0eta itse!f from past a!ues of 0eta$
4 3eta can 0e mo)e!e) as an AR@# process as in C'apter #
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A9:STE9 3ETA
!ocus "n# $alculations
2 :se t'e 0eta a)Hustment mo)e!"
2 5'at is t'e a)Huste) 0eta for a firm +'ose una)Huste) 0eta is #$D?
2
*-
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INSTA3ILITY IN TE EFFICIENT FRONTIER
+hen small chanes in the input values lead to lare chanes in theefficient frontier, it is called /instability in the efficient frontier.0
2 Insta0i!ity arises 0ecause +e use parameter estimates as inputs rat'er t'an t'e
true un)er!yin/ parameter a!ues$
4 If t'e )ifferences in parameters are sma!! @statistica!!y or economica!!y
insi/nificant. t'e optimi6ation process +i!! !i%e!y oerfit t'e mo)e!$
4 Lar/e ne/atie +ei/'ts in t'e a0sence of s'ort4se!!in/ restrictions may 0e
in)icatie of t'is pro0!em$
4 T'e mo)e! may in)icate fre7uent re0a!ancin/ in response to on!y sma!!
aria0!e c'an/es$
2 Insta0i!ity may a!so arise across time 0ecause of true c'an/es in t'e un)er!yin/
parameters or 0ecause of t'e same estimation pro0!em as a!rea)y note)$
*1
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M:LTIFACTOR MO9ELS
Models of asset returns that use more than one underlyin source of risk,known as a factor, are known as multifactor models.
2 Features of mu!tifactor mo)e!s"
4 T'e un)er!yin/ sources of ris% are %no+n as systematic factors an) referre) to
as price) ris%s$
4 Mu!tifactor mo)e!s e(p!ain asset returns 0etter t'an t'e mar%et mo)e!$
4 Mu!tifactor mo)e!s proi)e a more )etai!e) ana!ysis of ris% t'an sin/!e4factor
mo)e!s$
2 Cate/ories of mu!tifactor mo)e!s"
#$ Macroeconomic
*he factors are surprises in macroeconomic variables.*$ Fun)amenta! *he factors are attributes of stocks or companies.
,$ Statistica! *he factors are determined statistically and are often the
return on differin portfolios.
*B
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MACROECONOMIC FACTOR MO9ELS
macroeconomic factor surprise is the component of the factor@s returnthat was unexpected.
2 T'e surprise is /enera!!y measure) as t'e )ifference 0et+een t'e rea!i6e)
a!ue an) t'e pre)icte) a!ue prior to rea!i6ation$
2 A k -factor macroeconomic mo)e! is e(presse) as"
+'ere Qa is t'e e(pecte) return to t'e asset. t'e Qb terms are factor
sensitiities. an) t'e QF terms are t'e surprises in t'e macroeconomicfactors$
2
*
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MACROECONOMIC FACTOR MO9ELS
!ocus "n# $alculations
2 Suppose I 0e!iee a mu!tifactor asset pricin/ mo)e! is a correct )escription of
t'e ris%–return re!ations'ip for e7uity returns$ T'e mo)e! ta%es t'e fo!!o+in/
form"
R i = ai G bi,f Fore( G bi,d 9efau!t + bi,sSi6e + εi
2 I p!an on 0uyin/ t+o stoc%s +it' t'e fo!!o+in/ factor sensitiities"
2 5'at is t'e e(pecte) return to a portfo!io of *1= Stoc% 8 an) 1= Stoc% Y?
r i =
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AR3ITRA>E PRICIN> TEORY
*he %*, as it is known, describes the expected return to an asset as alinear function of the risk of the asset with respect to a set of factors.
2 T'e APT is an e7ui!i0rium mo)e!
+'ere t'e βs represent factor sensitiities an) t'e λs represent ris% premiums$2 T'e APT re!ies on t'ree assumptions"
#$ A factor mo)e! )escri0es asset returns$
*$ T'ere are many assets. so inestors can form +e!!4)iersifie) portfo!ios t'at
e!iminate asset4specific ris%$
,$ No ar0itra/e opportunities e(ist amon/ +e!!4)iersifie) portfo!ios$
2 In contrast to mu!tifactor mo)e!s. t'e APT mo)e!s t'e e(pecte) return in
e7ui!i0rium @t'e first term of t'e e7uation. in essence restrictin/ t'e first term
in t'e /enera! mu!tifactor e(pression to t'e APT a!ue for t'at term$
2
*
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AR3ITRA>E PRICIN> TEORY
!ocus "n# $alculations
2 You are consi)erin/ purc'asin/ s'ares in C!ee!an) Corp$. an) you 0e!iee t'e
APT +it' t'ree price) ris% factors is an accurate )escription of t'e e(pecte)
return to C!ee!an) Corp$ T'e first ris% factor. Macro. 'as a ris% premium of ,=
an) C!ee!an) Corp$ 'as a β for t'is ris% factor of #$#$ T'e secon) ris% factor.
Term. 'as a ris% premium of *= an) C!ee!an) 'as a β of
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TE APT AN9 AR3ITRA>E
!ocus "n# $alculations
2 Consi)er t'e fo!!o+in/ stoc% returns an) factor sensitiities for a sin/!e factor
APT$
2 Can +e com0ine 8 an) Y to ac'iee an ar0itra/e possi0i!ity +it' ?
4 5'at +ei/'ts create a portfo!io +it' e7ua! sensitiities so t'at t'e sensitiity
of t'e portfo!io t'e sensitiity of ?
4 Is t'e e(pecte) return to t'is portfo!io t'e same as t'e e(pecte) return to ?
,#
'tock Expected &eturn 'ensitivity
8
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TE APT AN9 AR3ITRA>E
!ocus "n# $alculations
4 5'at +ei/'ts create a portfo!io +it' e7ua! sensitiities so t'at t'e sensitiity
of t'e portfo!io t'e sensitiity of ?
4 Is t'e e(pecte) return to t'is portfo!io t'e same as t'e e(pecte) return to ?
4 Ao. *herefore, if we o short B, we can use the proceeds to o lon C
and D in weihts 3
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F:N9AMENTAL FACTOR MO9ELS
Gn contrast to macroeconomic models, fundamental models use expectedreturns (instead of surprises) as factors.
2 3ecause t'e e(pecte) returns no !on/er 'ae an e(pecte) a!ue of 6ero. as )o t'e
surprises in macroeconomic factor mo)e!s. t'e intercept. ai , is no loner an expected
return but the intercept term from a reression.
2 *he bi terms are typically factor sensitivities that have been standardied by the
sensitivity across all stocks.
4T'is is )one 0y su0tractin/ t'e aera/e sensitiity across a!! stoc%s an) t'en )ii)in/ t'eresu!t 0y t'e stan)ar) )eiation of t'e attri0ute across a!! stoc%s$
4 9oin/ t'is ena0!es us to interpret a!! factor sensitiities as unit!ess an) 0y comparison
+it' t'e Qtypica! stoc%$
4 A factor sensitiity of
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TE INFORMATION RATIO
"ften denoted G&, the information ratio takes a form similar to the 'harperatio.
2 T'e information ratio can 0e use) to capture t'e mean actie return
per unit of actie ris%$
4 T'e 'istorica! IR is
+'ere t'e su0scripts p an) B in)icate t'e portfo!io 0ein/ ea!uate)
an) t'e 0enc'mar%. respectie!y$
4 T'e information ratio is t'e )ifference in mean return for t'e
portfo!io an) t'e 0enc'mar% )ii)e) 0y t'e stan)ar) )eiation oft'e )ifference in return for t'e portfo!io an) t'e 0enc'mar%$
2 T'is can 0e use) to set /ui)e!ines for t'e amount 0y +'ic' t'e
portfo!io performance can )eiate from its 0enc'mar% @trac%in/ ris%$
2
,-
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ASSESSIN> ACTIVE RET:RN
!ocus "n# $alculations2 Returnin/ to our preious e(amp!e. consi)er a firm t'at uses t'e fo!!o+in/
asset pricin/ mo)e! to )etermine e(pecte) return"
2 T'is is an empirica! mo)e! suc' t'at t'e factor sensitiities use) are
)etermine) ia re/ression an) are not stan)ar)i6e)$
2 If t'e portfo!io factor sensitiities. 0enc'mar% sensitiities. an) factor returns
are as fo!!o+s. 'o+ +ou!) you )ecompose t'e sources of actie return for t'e
portfo!io?
2
,1
!actor 'ensitivity
!actor %ortfolio Henchmark ifference !actor&eturn
Macro #$# #
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ASSESSIN> ACTIVE RET:RN
!ocus "n# $alculations2 If t'e mana/er ac'iee) ,$-= actie return from asset se!ection. t'e actie
return sources are t'en"
2 T'is is an actie asset se!ection mana/er. as seen 0y t'e !ar/e proportion of
return attri0uta0!e to asset se!ection$ T'e mana/er a!so 'a) a positie
contri0ution from a macro ti!t. 0ut )i) poor!y +it' t'e inf!ation an) term ti!ts$
,B
&eturn $omponents bsolute$ontribution
of *otal ctive
Macro
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ASSESSIN> ACTIVE RIS;
!ocus "n# $alculations
2 Reca!! t'at Actie ris% s7uare) Actie factor ris% G Actie specific ris%
2 Consi)er t'e fo!!o+in/ portfo!ios an) t'eir ris% ca!cu!ations"
,
ctive !actor
%ortfolio Gndustry&iskGndex
*otal!actor
ctive'pecific
ctive &isk'4uared
A #*$*1 #$#1 *$- #$B -
3 #$*1 #,$1 #1 #< *1
C #$*1 #$1 #D$1 B$*1 *1
9
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ASSESSIN> ACTIVE RIS;
!ocus "n# $alculations
2 Portfo!io 9 is effectie!y a passie portfo!io +it' !itt!e or no trac%in/ ris% @!ast
co!umn$
2 Portfo!io A /ets most of its actie ris% from an in)ustry component fo!!o+e) 0y a
stoc%4specific component. t'en a ris%4in)e( component$
2 Portfo!ios 3 an) C 'ae simi!ar !ee!s of trac%in/ error. 0ut C 'as more from ris%
factor se!ection an) 3 from a stoc%4specific component$
,D
ctive !actor ( of total active)
%ortfolio Gndustry&iskGndex *otal !actor
ctive'pecific ctive &isk
A *1= ,1= B
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ACTIVE RIS;" M:LTIFACTOR MO9ELS
!ocus "n# $alculations
2 Factor mar/ina! contri0ution to actie ris% s7uare) is
2 Reca!! our t'ree4factor mo)e! +it' actie factor e(posures of
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ACTIVE RIS;" M:LTIFACTOR MO9ELS
!ocus "n# $alculations
2 5'at are t'e factor mar/ina! contri0utions to actie ris% s7uare) if tota! actie
ris% s7uare) is #1D$D?
2 E($ ca!cu!ation" NumFMCAR@Term –
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TRAC;IN> RIS;
!ocus "n# $alculations
-#
2 Consi)er t'at a mutua! fun)
an) its re!eant 0enc'mar%
'ae t'e returns an) trac%in/
error s'o+n in t'e ta0!e$
2 T'e c!ient is a foun)ation t'at
+ants to earn an actie return
a0oe t'e cost of mana/in/ its
account an) %eep trac%in/ ris%
0e!o+ 1=$ 5e current!y
receie #$1= for mana/in/ t'e
account$
2 Ea!uate t'e performance of
t'e fun). ca!cu!ate t'e IR. an)
interpret it$
ate
Gndex
&eturn
!und
&eturn
*rackin
Error
an *
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TRAC;IN> RIS;
!ocus "n# $alculations
-*
2 Ea!uate t'e performance of t'e fun)$
4 T'e fun) is current!y earnin/ s!i/'t!y in e(cess of its 0enc'mar%. 0ut it is
current!y not meetin/ its actie return o0Hectie 0ecause its aera/e trac%in/
error is 0e!o+ current mana/ement fees @#$# #$1$
4 It is a!so not meetin/ its trac%in/ ris% o0Hectie 0ecause t'e trac%in/ ris%
ca!cu!ate) as t'e actie ris% of $1= in t'e prior e(amp!e is /reater t'an 1=$
2 T'e information ratio for t'is portfo!io is
*he client is earnin 2
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FORMIN> A TRAC;IN> PORTFOLIO
!ocus "n# $alculations
2 Trac%in/ portfo!ios are portfo!ios +it' factor sensitiities t'at matc' t'ose of t'e
0enc'mar% portfo!io$
2 5e can formu!ate t'e +ei/'ts for a trac%in/ portfo!io of n factors as !on/ as +e
'ae n + # +e!!4)iersifie) portfo!ios$
2 Consi)er t'e fo!!o+in/ t'ree +e!!4)iersifie) portfo!ios an) tar/et 0enc'mar%
+ei/'ts"
-,
!actors
%ortfolio Husiness$ycle
*erm
#
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FORMIN> A TRAC;IN> PORTFOLIO
!ocus "n# $alculations
2 5'at are t'e +ei/'ts for a trac%in/ portfo!io t'at 'as t'e 0enc'mar%s
sensitiities? So!e t'is set of e7uations"
+'ic' /ies +ei/'ts of w2 F ?.;22:, w3 F –?.?993, and w7 F ?.=:=
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MAR;ET RIS; AN9 NONMAR;ET RIS; PREMI:MS
Inestors can earn su0stantia! premiums from e(posure to ris%s
unre!ate) to mar%et ris% +'en 'is or 'er factor ris% e(posures to ot'er
sources of income an) 'is or 'er ris% aersion )iffers from t'e aera/e
inestor$
4 In suc' cases. ti!ts a+ay from in)e(e) inestments may 0e optima!$4 For e(amp!e. 'uman capita! ris% increases t'e factor sensitiity of an
inestor +'o re!ies on earne) emp!oyment income to recession ris%$
Suc' an inestor +i!! 0i) up t'e price of countercyc!ica! stoc%s an)
se!! )o+n t'e price of countercyc!ica! stoc%s. causin/ a recession ris%
premium to e(ist for procyc!ica! stoc%s$
-1
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S:MMARY
2 Portfo!io mana/ement 'as a 'ost of 7uantitatie tec'ni7ues t'at are use) to
4 Se!ect assets
4 Assess e(pecte) returns an) ris%s
4 Trac% performance
2 Mean–ariance efficient ana!ysis forms t'e foun)ation of mo)ern portfo!iot'eory an) )escri0es 'o+ inestors +i!! c'oose 0et+een ris%y assets an) 'o+
t'ey +i!! +ei/'t a portfo!io of ris%y an) ris%4free assets$
2 Asset pricin/ mo)e!s /enera!!y )escri0e t'e e(pecte) return to assets
@portfo!ios as a function of t'e types an) !ee!s of ris% t'ey 0ear an) t'e
re+ar)s )ue for 0earin/ eac' type of ris%$