bkm 10e chap008 sm final

8/12/2019 BKM 10e Chap008 SM Final

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CHAPTER 8: INDEX MODELS


PROBLEM SETS

1. The advantage of the inde !ode"# $o!%a&ed to the Ma&'o(it) %&o$ed*&e# i+ theva+t", &ed*$ed n*!-e& of e+ti!ate+ &e*i&ed. In addition# the "a&ge n*!-e& ofe+ti!ate+ &e*i&ed fo& the Ma&'o(it) %&o$ed*&e $an &e+*"t in "a&ge agg&egatee+ti!ation e&&o&+ (hen i!%"e!enting the %&o$ed*&e. The di+advantage of the inde!ode" a&i+e+ f&o! the !ode"/+ a++*!%tion that &et*&n &e+id*a"+ a&e *n$o&&e"ated. Thi+a++*!%tion (i"" -e in$o&&e$t if the inde *+ed o!it+ a +ignifi$ant &i+' fa$to&.

0. The t&adeoff entai"ed in de%a&ting f&o! %*&e indeing in favo& of an a$tive",!anaged %o&tfo"io i+ -et(een the %&o-a-i"it, 2o& the %o++i-i"it,3 of +*%e&io& %e&fo&!an$e again+t the $e&taint, of additiona" !anage!ent fee+.

4. The an+(e& to thi+ *e+tion $an -e +een f&o! the fo&!*"a+ fo& w5 2e*ation 8.053and w6 2e*ation 8.013. Othe& thing+ he"d e*a"# w5 i+ +!a""e& the g&eate& the&e+id*a" va&ian$e of a $andidate a++et fo& in$"*+ion in the %o&tfo"io. 7*&the&# (e +eethat &ega&d"e++ of -eta# (hen w5 de$&ea+e+# +o doe+ w6. The&efo&e# othe& thing+e*a"# the g&eate& the &e+id*a" va&ian$e of an a++et# the +!a""e& it+ %o+ition in theo%ti!a" &i+', %o&tfo"io. That i+# in$&ea+ed fi&!+%e$ifi$ &i+' &ed*$e+ the etent to(hi$h an a$tive inve+to& (i"" -e (i""ing to de%a&t f&o! an indeed %o&tfo"io.

. The tota" &i+' %&e!i*! e*a"+: α 9 2β Ma&'et &i+' %&e!i*!3. ;e $a"" a"%ha anon!a&'et &et*&n %&e!i*! -e$a*+e it i+ the %o&tion of the &et*&n %&e!i*! that i+inde%endent of !a&'et %e&fo&!an$e.

The Sha&%e &atio indi$ate+ that a highe& a"%ha !a'e+ a +e$*&it, !o&e de+i&a-"e.A"%ha# the n*!e&ato& of the Sha&%e &atio# i+ a fied n*!-e& that i+ not affe$ted -, the+tanda&d deviation of &et*&n+# the deno!inato& of the Sha&%e &atio. Hen$e# anin$&ea+e in a"%ha in$&ea+e+ the Sha&%e &atio. Sin$e the %o&tfo"io a"%ha i+ the %o&tfo"io(eighted ave&age of the +e$*&itie+/ a"%ha+# then# ho"ding a"" othe& %a&a!ete&+ fied#an in$&ea+e in a +e$*&it,/+ a"%ha &e+*"t+ in an in$&ea+e in the %o&tfo"io Sha&%e &atio.

81




<. a. To o%ti!i)e thi+ %o&tfo"io one (o*"d need:

n = >5 e+ti!ate+ of !ean+

n = >5 e+ti!ate+ of va&ian$e+

??5#10

0

=

− nn

e+ti!ate+ of $ova&ian$e+

The&efo&e# in tota": 8@5#10

40

=+ nn

e+ti!ate+

-. In a +ing"e inde !ode": r i − r f = i 9 B i 2r M – r f 3 9 e i

E*iva"ent",# *+ing e$e++ &et*&n+: R i = i 9 B i R M 9 e i

The va&ian$e of the &ate of &et*&n $an -e de$o!%o+ed into the $o!%onent+:

2"3 The va&ian$e d*e to the $o!!on !a&'et fa$to&:00

M i σ β

203 The va&ian$e d*e to fi&! +%e$ifi$ *nanti$i%ated event+: 32 0ie

In thi+ !ode": BB3#2Cov ji ji r r =

The n*!-e& of %a&a!ete& e+ti!ate+ i+:

n = >5 e+ti!ate+ of the !ean E 2r i3

n = >5 e+ti!ate+ of the +en+itivit, $oeffi$ient B i

n = >5 e+ti!ate+ of the fi&!+%e$ifi$ va&ian$e 02ei 3

1 e+ti!ate of the !a&'et !ean E 2r M 3

1 e+ti!ate of the !a&'et va&ian$e 0 M σ

The&efo&e# in tota"# 180 e+ti!ate+.

The +ing"e inde !ode" &ed*$e+ the tota" n*!-e& of &e*i&ed e+ti!ate+ f&o!1#8@5 to 180. In gene&a"# the n*!-e& of %a&a!ete& e+ti!ate+ i+ &ed*$ed f&o!:

3042to0

40

+

+n

nn

>. a. The +tanda&d deviation of ea$h individ*a" +to$' i+ given -,:0D1000

3E2BF i M ii eσ σ +=Sin$e B A = 5.8# B B = 1.0# 2e A 3 = 45G# 2e B 3 = 5G# and M = 00G# (e get:

A = 25.80 000 9 450 310 = 4.?8G

B = 21.00 000 9 50 310 = ?.@4G

80




-. The e%e$ted &ate of &et*&n on a %o&tfo"io i+ the (eighted ave&age of thee%e$ted &et*&n+ of the individ*a" +e$*&itie+:

E2& P 3 = (A E2& A3 9 ( E2& 3 9 (f & f

E2& P 3 = 25.45 14G3 9 25.< 18G3 9 25.0< 8G3 = 1G

The -eta of a %o&tfo"io i+ +i!i"a&", a (eighted ave&age of the -eta+ of theindivid*a" +e$*&itie+:

BP = (A BA 9 ( B 9 (f B f

BP = 25.45 5.83 9 25.< 1.03 9 25.0< 5.53 = 5.?8

The va&ian$e of thi+ %o&tfo"io i+:

32CBC 0000

P M P P e+= σ

(he&e 00 B M P i+ the +,+te!ati$ $o!%onent and 320

P eσ i+ the non+,+te!ati$

$o!%onent. Sin$e the &e+id*a"+ 2ei 3 a&e *n$o&&e"ated# the non+,+te!ati$va&ian$e i+:

0 0 0 0 0 0 02 3 2 3 2 3 2 3 P A A B B f f e w e w e w eσ σ σ σ = × + × + ×

= 25.450 450 3 9 25.<0 50 3 9 25.0<0 53 = 5<

(he&e 02e A 3 and 02e B 3 a&e the fi&!+%e$ifi$ 2non+,+te!ati$3 va&ian$e+ ofSto$'+ A and # and 02e f 3# the non+,+te!ati$ va&ian$e of T-i""+# i+ )e&o. The&e+id*a" +tanda&d deviation of the %o&tfo"io i+ th*+:

2e P3 = 25<310 = 05.10G

The tota" va&ian$e of the %o&tfo"io i+ then:

?.>@@5<300?8.52C 000=+×= P

The tota" +tanda&d deviation i+ 0>.<G.

?. a. The t(o fig*&e+ de%i$t the +to$'+/ +e$*&it, $ha&a$te&i+ti$ "ine+ 2SCL3. Sto$' Aha+ highe& fi&!+%e$ifi$ &i+' -e$a*+e the deviation+ of the o-+e&vation+ f&o!the SCL a&e "a&ge& fo& Sto$' A than fo& Sto$' . Deviation+ a&e !ea+*&ed -,the ve&ti$a" di+tan$e of ea$h o-+e&vation f&o! the SCL.

-. eta i+ the +"o%e of the SCL# (hi$h i+ the !ea+*&e of +,+te!ati$ &i+'. The SCLfo& Sto$' i+ +tee%e& hen$e Sto$' /+ +,+te!ati$ &i+' i+ g&eate&.

84




$. The R2 2o& +*a&ed $o&&e"ation $oeffi$ient3 of the SCL i+ the &atio of thee%"ained va&ian$e of the +to$'/+ &et*&n to tota" va&ian$e# and the tota" va&ian$ei+ the +*! of the e%"ained va&ian$e %"*+ the *ne%"ained va&ian$e 2the +to$'/+&e+id*a" va&ian$e3:

32CCB

CB000

00

0

i M i

M i

e R +=

Sin$e the e%"ained va&ian$e fo& Sto$' i+ g&eate& than fo& Sto$' A 2the

e%"ained va&ian$e i+ 00B M B # (hi$h i+ g&eate& +in$e it+ -eta i+ highe&3# and it+

&e+id*a" va&ian$e0 2 3 Beσ i+ +!a""e&# it+ R0 i+ highe& than Sto$' A/+.

d. A"%ha i+ the inte&$e%t of the SCL (ith the e%e$ted &et*&n ai+. Sto$' A ha+ a+!a"" %o+itive a"%ha (he&ea+ Sto$' ha+ a negative a"%ha hen$e# Sto$' A/+a"%ha i+ "a&ge&.

e. The $o&&e"ation $oeffi$ient i+ +i!%", the +*a&e &oot of R0# +o Sto$' /+

$o&&e"ation (ith the !a&'et i+ highe&.

8. a. 7i&!+%e$ifi$ &i+' i+ !ea+*&ed -, the &e+id*a" +tanda&d deviation. Th*+# +to$' Aha+ !o&e fi&!+%e$ifi$ &i+': 15.4G J @.1G

-. Ma&'et &i+' i+ !ea+*&ed -, -eta# the +"o%e $oeffi$ient of the &eg&e++ion. A ha+ a"a&ge& -eta $oeffi$ient: 1.0 J 5.8

$. R0 !ea+*&e+ the f&a$tion of tota" va&ian$e of &et*&n e%"ained -, the !a&'et&et*&n. A/+ R0 i+ "a&ge& than /+: 5.<?> J 5.4>

d. Re(&iting the SCL e*ation in te&!+ of tota" &et*&n 2r 3 &athe& than e$e++&et*&n 2 R3:

2 3

21 3

A f M f

A f M

r r r r

r r r

α β

α β β

− = + × − ⇒

= + × − + ×

The inte&$e%t i+ no( e*a" to:

21 3 1G 21 1.03 f f r r α β + × − = + × −

Sin$e & f = >G# the inte&$e%t (o*"d -e: 1G >G21 1.03 1G 1.0G 5.0G+ − = − = −

8




@. The +tanda&d deviation of ea$h +to$' $an -e de&ived f&o! the fo""o(inge*ation fo& R0:

==0

00

0

C

CB

i

M i

i R

The&efo&e:

G45.41C

@8505.5

05?.5CBC

00

0

000

=

=×

==

A

A

M A

A

R

7o& +to$' :

G08.>@

855#10.5

050.1

000

=

=×

=

B

B

15. The +,+te!ati$ &i+' fo& A i+:

0 0 0 05.?5 05 1@> A M β σ × = × =

The fi&!+%e$ifi$ &i+' of A 2the &e+id*a" va&ian$e3 i+ the diffe&en$e -et(eenA/+ tota" &i+' and it+ +,+te!ati$ &i+':

@85 K 1@> = ?8

The +,+te!ati$ &i+' fo& i+:

0 0 0 01.05 05 <?> B M β σ × = × =

/+ fi&!+%e$ifi$ &i+' 2&e+id*a" va&ian$e3 i+:

#855 K <?> = #00

11. The $ova&ian$e -et(een the &et*&n+ of A and i+ 2+in$e the &e+id*a"+ a&e a++*!ed to -e *n$o&&e"ated3:

44>5505.1?5.5BB32Cov 0 =××== M B A B A ,r r

The $o&&e"ation $oeffi$ient -et(een the &et*&n+ of A and i+:

1<<.508.>@45.41

44>

3#2CovL =×== B A

B A AB r r

8<




10. Note that the $o&&e"ation i+ the +*a&e &oot of R0: 0L R=

10

#

10

#

2 3 5.05 41.45 05 085

2 3 5.10 >@.08 05 85

A M A M

B M B M

Cov r r

Cov r r

ρσ σ

ρσ σ

= = × × =

= = × × =

14. 7o& %o&tfo"io P (e $an $o!%*te:

P = F25.>0 @853 9 25.0 8553 9 20 5. 5.> 44>310 = F1080.5810 = 4<.81G

B P = 25.> 5.?3 9 25. 1.03 = 5.@5

@<[email protected] 00000 =×−=−= M P P P e

Cov2r P ,r M 3 = B P 0 M =5.@5 55=4>5

Thi+ +a!e &e+*"t $an a"+o -e attained *+ing the $ova&ian$e+ of the individ*a" +to$'+(ith the !a&'et:

Cov2r P ,r M 3 = Cov25.>r A 9 5.r B# r M 3 = 5.> Cov2r A# r M 3 9 5. Cov2r B#r M 3

= 25.> 0853 9 25. 853 = 4>5

1. Note that the va&ian$e of T-i""+ i+ )e&o# and the $ova&ian$e of T-i""+ (ith an, a++eti+ )e&o. The&efo&e# fo& %o&tfo"io :

[ ]

[ ] G<<.0134>54.5<.5023554.52358.080#1<.52

3#2Cov0

0D100

0D10000

=×××+×+×=

×××++= M P M P M M P P Q

r r wwww

25.< 5.@53 25.4 13 25.05 53 5.?<Q P P M M

w wβ β β = + = × + × + × =

<0.04@355?<.52<0.>B32 00000 =×−=−= M QQQe

45555?<.5B3#2Cov 0 =×== M Q M Q r r

1<. a. eta oo'+ ad*+t+ -eta -, ta'ing the +a!%"e e+ti!ate of -eta and ave&aging it(ith 1.5# *+ing the (eight+ of 04 and 14# a+ fo""o(+:

ad*+ted -eta = F2043 1.0 9 F2143 1.5 = 1.1>

-. If ,o* *+e ,o*& $*&&ent e+ti!ate of -eta to -e Bt K1

= 1.0# then

Bt = 5.4 9 25.? 1.03 = 1.1>8

8>




1>. 7o& Sto$' A:

F 2 3 .11 F.5> 5.8 2.10 .5>3 5.0G A A f A M f r r r r α β = − + × − = − + × − =

7o& +to$' :

F 2 3 .1 F.5> 1.< 2.10 .5>3 1G B B f B M f r r r r α β = − + × − = − + × − = −

Sto$' A (o*"d -e a good addition to a (e""dive&+ified %o&tfo"io. A +ho&t %o+ition inSto$' !a, -e de+i&a-"e.

1?. a.

A"%ha 23 E%e$ted e$e++ &et*&ni = & i K Fr f 9 Bi 2r M K r f 3 E 2r i 3 K r f

A = 05G K F8G 9 1.4 21>G K 8G3 = 1.>G 05G K 8G = 10G = 18G K F8G 9 1.8 21>G K 8G3 = K .G 18G K 8G = 15GC = 1?G K F8G 9 5.? 21>G K 8G3 = 4.G 1?G K 8G = @G

D = 10G K F8G 9 1.5 21>G K 8G3 = K .5G 10G K 8G = GSto$'+ A and C have %o+itive a"%ha+# (he&ea+ +to$'+ and D havenegative a"%ha+.

The &e+id*a" va&ian$e+ a&e:

σ02e A3 = <80 = 4#4>

σ02e B3 = ?10 = <#51

σ02eC 3 = >50 = 4#>55

σ02e D3 = <<0 = 4#50<

8?




-. To $on+t&*$t the o%ti!a" &i+', %o&tfo"io# (e fi&+t dete&!ine the o%ti!a" a$tive %o&tfo"io. +ing the T&e,no&"a$' te$hni*e# (e $on+t&*$t the a$tive %o&tfo"io:

5.555?> K5.>10

K5.5558?4 1.10><C 5.555@ K1.0181

D K5.551400 1.?5<8

Tota" K5.555??< 1.5555

e *n$on$e&ned (ith the negative (eight+ of the %o+itive +to$'+the enti&ea$tive %o+ition (i"" -e negative# &et*&ning eve&,thing to good o&de&.

;ith the+e (eight+# the fo&e$a+t fo& the a$tive %o&tfo"io i+:

= FK5.>10 1.> 9 F1.10>< 2K .3 K F1.0181 4. 9 F1.?5<8 2K .53

= K1>.@5GB = FK5.>10 1.4 9 F1.10>< 1.8 K F1.0181 5.?5 9 F1.?5<8 1 = 0.58

The high -eta 2highe& than an, individ*a" -eta3 &e+*"t+ f&o! the +ho&t %o+ition+in the &e"ative", "o( -eta +to$'+ and the "ong %o+ition+ in the &e"ative", high -eta +to$'+.

σ02e3 = F2K5.>103044> 9 F1.10><0<51 9 F2K1.0181304>55 9 F1.?5<80450<

= 01#85@.>

σ 2e3 = 1?.>8G

The "eve&ed %o+ition in F(ith high σ02e3 ove&$o!e+ the dive&+ifi$ation

effe$t and &e+*"t+ in a high &e+id*a" +tanda&d deviation. The o%ti!a" &i+',

%o&tfo"io ha+ a %&o%o&tion w6 in the a$tive %o&tfo"io# $o!%*ted a+ fo""o(+:

0

5 0 0

2 3 .1>@5 01#85@.>5.5<10

F 2 3 .58 04 M f M

ew

E r r

α σ

σ

−= = = −

−

The negative %o+ition i+ *+tified fo& the &ea+on +tated ea&"ie&.

The ad*+t!ent fo& -eta i+:

58>.535<10.53258.0121

5<10.5

3B1216

5

5 −=−−+

−=

−+=

w

ww

Sin$e w6 i+ negative# the &e+*"t i+ a %o+itive %o+ition in +to$'+ (ith %o+itivea"%ha+ and a negative %o+ition in +to$'+ (ith negative a"%ha+. The %o+ition inthe inde %o&tfo"io i+:

1 K 2K5.58>3 = 1.58>

88




$. To $a"$*"ate the Sha&%e &atio fo& the o%ti!a" &i+', %o&tfo"io# (e $o!%*te theinfo&!ation &atio fo& the a$tive %o&tfo"io and Sha&%e/+ !ea+*&e fo& the !a&'et %o&tfo"io. The info&!ation &atio fo& the a$tive %o&tfo"io i+ $o!%*ted a+ fo""o(+:

A =2 3e

α

σ = K1>.@51?.>8 = K5.11

A0 = 5.5141

Hen$e# the +*a&e of the Sha&%e &atio 2S3 of the o%ti!i)ed &i+', %o&tfo"io i+:

141.55141.504

8 0

000 =+

=+= AS S M

S = 5.4>>0Co!%a&e thi+ to the !a&'et/+ Sha&%e &atio:

S M = 804 = 5.4?8 A diffe&en$e of: 5.518

The on", !ode&ate i!%&ove!ent in %e&fo&!an$e &e+*"t+ f&o! on", a +!a""

%o+ition ta'en in the a$tive %o&tfo"io A -e$a*+e of it+ "a&ge &e+id*a" va&ian$e.

d. To $a"$*"ate the !a'e*% of the $o!%"ete %o&tfo"io# fi&+t $o!%*te the -eta# the!ean e$e++ &et*&n# and the va&ian$e of the o%ti!a" &i+', %o&tfo"io:

B P = w M 9 2w A B A 3 = 1.58> 9 F2K5.58>3 × 0.58 = 5.@<

E 2 R P 3 = P 9 B P E 2 R M 3 = F2K5.58>3 × 2K1>.@5G3 9 25.@< 8G3 = 8.0G

( @.<08>.85@#01358>.52304@<.5232CCBC 000000=×−+×=+=

P M P P e

G55.04C = P

Sin$e A = 0.8# the o%ti!a" %o+ition in thi+ %o&tfo"io i+:

<>8<.5@.<088.051.5

0.8=

××= y

In $ont&a+t# (ith a %a++ive +t&ateg,:

<51.5048.051.5

80 =

××= y A diffe&en$e of: 5.508

The fina" %o+ition+ a&e 2M !a, in$"*de +o!e of +to$'+ A th&o*gh D3:

i""+ 1 K 5.<>8< = 4.1<G

M 5.<>8< × ".58> = <@.>1

A 5.<>8< × 2K5.58>3 × 2K5.>103 1.?5 5.<>8< × 2K5.58>3 × 1.10>< = K 4.11

C 5.<>8< × 2K5.58>3 × 2K1.01813 4.4?

D 5.<>8< × 2K5.58>3 × 1.?5<8 = K .?1

2+*-e$t to &o*nding e&&o&3 155.55

8@




18. a.If a !anage& i+ not a""o(ed to +e"" +ho&t# he (i"" not in$"*de +to$'+ (ith negativea"%ha+ in hi+ %o&tfo"io# +o he (i"" $on+ide& on", A and C:

Α σ02e3

A 1.> 4#4> 5.555?> 5.44<0C 4. 4#>55 5.555@ 5.>>8

5.55105 1.5555

The fo&e$a+t fo& the a$tive %o&tfo"io i+:

= 25.44<0 1.>3 9 25.>>8 4.3 = 0.85G

B = 25.44<0 1.43 9 25.>>8 5.?3 = 5.@5

σ02e3 = 25.44<00 4#4>3 9 25.>>80 4#>553 = 1#@>@.54

2e3 = .4?G

The (eight in the a$tive %o&tfo"io i+:

[email protected]

54.@>@#1D85.0

D32

32DA00

0

5 === M M R E

ew

Ad*+ting fo& -eta:

[email protected]@[email protected]

[email protected]

(3121

(6(

5

5 =×−+

=β−+

=

The info&!ation &atio of the a$tive %o&tfo"io i+:

0.85 5.5>412 3 .4?

Ae

α σ

= = =

Hen$e# the +*a&e of the Sha&%e &atio i+:

0

0 085.5>41 5.10<5

04S

= + = ÷

The&efo&e: S = 5.4<4<

The !a&'et/+ Sha&%e &atio i+: SM = 5.4?8

;hen +ho&t +a"e+ a&e a""o(ed 2P&o-"e! 1?3# the !anage&/+ Sha&%e &atio i+

highe& 25.4>>03. The &ed*$tion in the Sha&%e &atio i+ the $o+t of the +ho&t +a"e&e+t&i$tion.

The $ha&a$te&i+ti$+ of the o%ti!a" &i+', %o&tfo"io a&e:

815




0 0 0 0 0 0

21 5.5@413 25.5@41 5.@3 5.@@

2 3 2 3 25.5@41 0.8G3 25.@@ 8G3 8.18G

2 3 25.@@ 043 25.5@41 1@>@.543 <4<.<

04.1G

P M A A

P P P M

P P M P

P

w w

E R E R

e

β β

α β

σ β σ σ

σ

= + × = − + × =

= + × = × + × =

= × + = × + × =

=

;ith A = 0.8# the o%ti!a" %o+ition in thi+ %o&tfo"io i+:

<<<.5<.<4<8.051.5

18.8, =

××=

The fina" %o+ition+ in ea$h a++et a&e:

i""+ 1 K 5.<<< = <.<G

M 5.<<< × 21 − 5.5@413 = @.?

A 5.<<< × 5.5@41 × 5.44<0 = 1.?5

C 5.<<< × 5.5@41 × 5.>>8 = 4.48

155.55

b. The mean and variance of the optimized complete portfolios in theunconstrained and short-sales constrained cases, and for the passive strategy

are:

E 2 RC 3 0

CC

n$on+t&ained 5.<>8< 8.0G = .?@ 5.<>8<0 <08.@ = 1?5.@<

Con+t&ained 5.<<< 8.18G = .> 5.<<<0 <4<.< = 1<@.4>

Pa++ive 5.<51 8.55G = .40 5.<510 <[email protected] = 1<.41

The *ti"it, "eve"+ -e"o( a&e $o!%*ted *+ing the fo&!*"a:055<.532 C C Ar E −

n$on+t&ained 8G 9 .?@G K 25.55< 0.8 1?5.@<3 = 15.5G

Con+t&ained 8G 9 .>G K 25.55< 0.8 1<@.4>3 = 15.04G

Pa++ive 8G 9 .40G K 25.55< 0.8 1<.413 = 15.1>G

811




1@. A"" a"%ha+ a&e &ed*$ed to 5.4 ti!e+ thei& va"*e+ in the o&igina" $a+e. The&efo&e# the

&e"ative (eight+ of ea$h +e$*&it, in the a$tive %o&tfo"io a&e *n$hanged# -*t the a"%ha

of the a$tive %o&tfo"io i+ on", 5.4 ti!e+ it+ %&evio*+ va"*e: 5.4 −1>.@5G = −<.5?GThe inve+to& (i"" ta'e a +!a""e& %o+ition in the a$tive %o&tfo"io. The o%ti!a" &i+',

%o&tfo"io ha+ a %&o%o&tion w6 in the a$tive %o&tfo"io a+ fo""o(+:

0

5 0 0

2 3 5.5<5? 01#85@.>5.51<4?

2 3 5.58 04 M f M

ew

E r r

α σ

σ

−= = = −−

The negative %o+ition i+ *+tified fo& the &ea+on given ea&"ie&.

The ad*+t!ent fo& -eta i+:

51<1.53E51<4?.52358.01F21

51<4?.5

3B1216

5

5 −=−×−+

−=

−+=

w

ww

Sin$e w6 i+ negative# the &e+*"t i+ a %o+itive %o+ition in +to$'+ (ith %o+itive a"%ha+and a negative %o+ition in +to$'+ (ith negative a"%ha+. The %o+ition in the inde

%o&tfo"io i+: 1 K 2K5.51<13 = 1.51<1To $a"$*"ate the Sha&%e &atio fo& the o%ti!a" &i+', %o&tfo"io (e $o!%*te the info&!ation&atio fo& the a$tive %o&tfo"io and the Sha&%e &atio fo& the !a&'et %o&tfo"io. Theinfo&!ation &atio of the active portfolio i+ 5.4 ti!e+ it+ %&evio*+ va"*e:

A =<.5?

2 3 1?.>8e

α

σ

−= = K5.544 and A0 =5.55118

Hen$e# the +*a&e of the Sha&%e &atio of the optii!ed ri"#y portfolio i+:

S 0 = S 0 M 9 A0 = 28G04G30 9 5.55118 = 5.1000

S = 5.4@<

Co!%a&e thi+ to the !a&'et/+ Sha&%e &atio: S M =8G04G

= 5.4?8

The diffe&en$e i+: 5.551?

Note that the &ed*$tion of the fo&e$a+t a"%ha+ -, a fa$to& of 5.4 &ed*$ed the +*a&edinfo&!ation &atio and the i!%&ove!ent in the +*a&ed Sha&%e &atio -, a fa$to& of:

5.40 = 5.5@

05. If ea$h of the a"%ha fo&e$a+t+ i+ do*-"ed# then the a"%ha of the a$tive %o&tfo"io (i""a"+o do*-"e. Othe& thing+ e*a"# the info&!ation &atio 2IR3 of the a$tive %o&tfo"io a"+o

do*-"e+. The +*a&e of the Sha&%e &atio fo& the o%ti!i)ed %o&tfo"io 2S +*a&e3 e*a"+the +*a&e of the Sha&%e &atio fo& the !a&'et inde 2SM +*a&e3 %"*+ the +*a&e ofthe info&!ation &atio. Sin$e the info&!ation &atio ha+ do*-"ed# it+ +*a&e *ad&*%"e+.The&efo&e: S +*a&e = SM +*a&e 9 2 $R3Co!%a&ed to the %&evio*+ S +*a&e# the diffe&en$e i+: 4 $R No( ,o* $an e!-a&' on the $a"$*"ation+ to ve&if, thi+ &e+*"t.

810




CFA PROBLEMS

1. The &eg&e++ion &e+*"t+ %&ovide *antitative !ea+*&e+ of &et*&n and &i+' -a+ed on!onth", &et*&n+ ove& the five,ea& %e&iod.

B fo& AC (a+ 5.>5# $on+ide&a-", "e++ than the ave&age +to$'/+ B of 1.5. Thi+indi$ate+ that# (hen the SQP <55 &o+e o& fe"" -, 1 %e&$entage %oint# AC/+ &et*&n onave&age &o+e o& fe"" -, on", 5.>5 %e&$entage %oint. The&efo&e# AC/+ +,+te!ati$ &i+'2o& !a&'et &i+'3 (a+ "o( &e"ative to the t,%i$a" va"*e fo& +to$'+. AC/+ a"%ha 2theinte&$e%t of the &eg&e++ion3 (a+ K 4.0G# indi$ating that (hen the !a&'et &et*&n (a+5G# the ave&age &et*&n on AC (a+ K4.0G. AC/+ *n+,+te!ati$ &i+' 2o& &e+id*a"&i+'3# a+ !ea+*&ed -, 2e3# (a+ 14.50G. 7o& AC# R0 (a+ 5.4<# indi$ating $"o+ene++of fit to the "inea& &eg&e++ion g&eate& than the va"*e fo& a t,%i$a" +to$'.

B fo& X (a+ +o!e(hat highe&# at 5.@?# indi$ating X/+ &et*&n %atte&n (a+ ve&,+i!i"a& to the B fo& the !a&'et inde. The&efo&e# X +to$' had ave&age +,+te!ati$&i+' fo& the %e&iod ea!ined. A"%ha fo& X (a+ %o+itive and *ite "a&ge# indi$ating

a &et*&n of ?.4G# on ave&age# fo& X inde%endent of !a&'et &et*&n. Re+id*a" &i+'(a+ 01.<G# ha"f again a+ !*$h a+ AC/+# indi$ating a (ide& +$atte& of o-+e&vation+a&o*nd the &eg&e++ion "ine fo& X. Co&&e+%onding",# the fit of the &eg&e++ion !ode"(a+ $on+ide&a-", "e++ than that of AC# $on+i+tent (ith an R0 of on", 5.1?.

The effe$t+ of in$"*ding one o& the othe& of the+e +to$'+ in a dive&+ified %o&tfo"io!a, -e *ite diffe&ent. If it $an -e a++*!ed that -oth +to$'+/ -eta+ (i"" &e!ain +ta-"eove& ti!e# then the&e i+ a "a&ge diffe&en$e in +,+te!ati$ &i+' "eve". The -eta+ o-tainedf&o! the t(o -&o'e&age ho*+e+ !a, he"% the ana",+t d&a( infe&en$e+ fo& the f*t*&e.The th&ee e+ti!ate+ of AC/+ B a&e +i!i"a&# &ega&d"e++ of the +a!%"e %e&iod of the*nde&",ing data. The &ange of the+e e+ti!ate+ i+ 5.>5 to 5.?1# (e"" -e"o( the !a&'etave&age B of 1.5. The th&ee e+ti!ate+ of X/+ B va&, +ignifi$ant", a!ong the th&ee

+o*&$e+# &anging a+ high a+ 1.< fo& the (ee'", data ove& the !o+t &e$ent t(o ,ea&+.One $o*"d infe& that X/+ B fo& the f*t*&e !ight -e (e"" a-ove 1.5# !eaning it!ight have +o!e(hat g&eate& +,+te!ati$ &i+' than (a+ i!%"ied -, the !onth",&eg&e++ion fo& the five,ea& %e&iod.

The+e +to$'+ a%%ea& to have +ignifi$ant", diffe&ent +,+te!ati$ &i+' $ha&a$te&i+ti$+. Ifthe+e +to$'+ a&e added to a dive&+ified %o&tfo"io# X (i"" add !o&e to tota" vo"ati"it,.

814




0. The R0 of the &eg&e++ion i+: 5.?50 = 5.@

The&efo&e# <1G of tota" va&ian$e i+ *ne%"ained -, the !a&'et thi+ i+non+,+te!ati$ &i+'.

4. @ = 4 9 β 211 − 43 ⇒ β = 5.?<

. d.

<. -.

81

bkm 10e chap008 sm final

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