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Quantitative Chapter11

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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%$