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# Quantitative Chapter11

Aug 07, 2018

## Documents

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PORTFOLIO CONCEPTS

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MEAN –VARIANCE ANALYSIS

Mean–variance analysis is the fundamental implementation of modern portfolio theory, 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!! inestors are ris% aerse& t'ey prefer !ess ris% to more for t'e same !ee!of e(pecte) return\$

*\$ E(pecte) returns for a!! assets are %no+n\$

,\$ T'e ariances an) coariances of a!! asset returns are %no+n\$

-\$ Inestors nee) %no+ on!y t'e e(pecte) returns. ariances. an) coariances

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 hihest level of return for a iven level of risk as measured by standard deviation in modern portfolio theory.

2 3ecause inestors are ris%4aerse. 0y assumption. t'ey +i!! c'oose to a!!ocate

t'eir assets to portfo!ios t'at 'ae t'e 'i/'est possi0!e !ee! of e(pecte) return

for a /ien !ee! 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) )eiation for a specifie) set of e(pecte)

returns. stan)ar) )eiations. 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) )eiation is. of course. t'e positie 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) )eiation of a portfo!io compose) of 1erman e7uities?

2 T'e E @r  is ,\$B1D=.

an) t'e stan)ar) )eiation

is #

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TE EFFICIENT FRONTIER

*he efficient frontier is a plot of the set of expected returns and standard deviations for all efficient portfolios (assets) above the lobal minimum-

variance portfolio.

2 T'e minimum4ariance frontier

@solid reen line is t'e set of

a!! portfo!ios t'at represent t'e !o+est !ee! of ris% t'at can 0e

ac'iee) for eac' possi0!e !ee!

of return\$

4 T'e portfo!io +it' t'e !o+est

ariance of a!! t'e portfo!ios.

+it' t'e !o+est !ee! of ris%

t'at can 0e ac'iee). is

%no+n as t'e lobal

minimum-variance portfolio\$

B

Efficient !rontier

Stan)ar) 9eiation

E @r 

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TE EFFICIENT FRONTIER

%ortfolios on the efficient frontier provide the hihest possible level of return for a iven level of risk.

2 3ecause portfo!ios on t'e

efficient frontier use ris%

efficient!y to /enerate returns.

inestors can restrict t'eir se!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) 9eiation

E @r 

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9IVERSIFICATION AN9 CORRELATION

*he trade-off between portfolio risk as measured by standard deviation and portfolio 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!ation are strict!y positie. e(cept t'e !ast

term. +'ic' inc!u)es t'e corre!ation.

+'ic' ran/es from perfect ne/atie @ –#"

0!ue to perfect positie @G#" purp!e

+it' 6ero corre!ation in 0et+een @

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FIN9IN> TE MINIM:M4VARIANCE FRONTIER

+e can use an optimier, such as the 'olver in Excel, to solve for the weihts 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 positie\$ 2 T'e e(pecte) return an) ariance for a /ien set of

+ei/'ts are

2 For eery return. z . 0et+een z min an) z max .

+e so!e for t'e set of +ei/'ts t'at minimi6es t'e portfo!io ariance su0Hect to

E @r  p  z.

4 If +e )o so iteratie!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 ρ n n

p i j i j i j i j

r w w = =

=

∑∑

∑ =

= n

i

iw 1

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 )ii)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 aera/e ariance of t'ose assets. an) is t'e aera/e coariance of t'e assets\$

2 Consi)er a #

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TE CAPITAL ALLOCATION LINE

*he capital allocation line (\$) describes the optimal expected return and standard 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) )eiation 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 0est possi0!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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TE CAPITAL ALLOCATION LINE

#*

CAL

Efficient Frontier

Stan)ar) 9eiation

E @r 

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TE CAPITAL ALLOCATION LINE

!ocus "n# \$alculations

Consi)er an inestor facin/ a ,= ris%4free rate +it' access to a tan/ency

portfo!io +it' a #*= return an) an #D= stan)ar) )eiation\$

4 If t'e inestor re7uires a #

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TE CAPITAL MAR;ET LINE

+hen all investors share identical expectations about the expected returns, variances, and covariances of assets, the \$ becomes the \$M.

T'e capita! mar%et !ine @CML represents t'e case in +'ic' a!! inestors 'ae 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!! inestors +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 to any 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 receie) 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 assets sensitiity to mar%et moements @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

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