bkm 10e chap007 sm final

8/12/2019 BKM 10e Chap007 SM Final

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CHAPTER 7: OPTIMAL RISKY PORTFOLIOS


PROBLEM SETS

1. (a) and (e). Short-ter rate! and "a#or $!!%e! are &a'tor! that are 'oon to a""&$r! and there&ore %!t #e 'on!$dered a! aret r$! &a'tor!. The rea$n$n three&a'tor! are %n$*%e to th$! 'or+orat$on and are not a +art o& aret r$!.

,. (a) and ('). A&ter rea" e!tate $! added to the +ort&o"$o there are &o%r a!!et '"a!!e! $nthe +ort&o"$o: !to'! #ond! 'a!hand rea" e!tate. Port&o"$o ar$an'e no/ $n'"%de! aar$an'e ter &or rea" e!tate ret%rn! and a 'oar$an'e ter &or rea" e!tate ret%rn!/$th ret%rn! &or ea'h o& the other three a!!et '"a!!e!. There&ore +ort&o"$o r$! $!a&&e'ted #0 the ar$an'e (or !tandard de$at$on) o& rea" e!tate ret%rn! and the'orre"at$on #et/een rea" e!tate ret%rn! and ret%rn! &or ea'h o& the other a!!et '"a!!e!.

(ote that the 'orre"at$on #et/een rea" e!tate ret%rn! and ret%rn! &or 'a!h $! o!t"$e"0 2ero.)

3. (a) An!/er (a) $! a"$d #e'a%!e $t +ro$de! the de&$n$t$on o& the $n$% ar$an'e+ort&o"$o.

4. The +araeter! o& the o++ort%n$t0 !et are:

E(rS) 5 ,6E(rB) 5 1, 8S5 36 8B5 19 5 6.16

Fro the !tandard de$at$on! and the 'orre"at$on 'oe&&$'$ent /e enerate the

'oar$an'e atr$; ond! Sto'!

>ond! ,,9 49

Sto'! 49 ?66

The $n$%-ar$an'e +ort&o"$o $! 'o+%ted a! &o""o/!:

wM$n(S) 5 173?.6)49,(,,9?66

49,,9

)(Co,

)(Co,,

,

=

+

=

+

BSBS

BSB

,rr

,rr

wM$n(B) 5 1 6.173? 5 6.@,1

The $n$% ar$an'e +ort&o"$o ean and !tandard de$at$on are:

E(rM$n) 5 (6.173? B .,6) (6.@,1 B .1,) 5 .133? 5 13.3?

8M$n

5 ,D1,,,, )=-(Co., are +er&e't"0 neat$e"0 'orre"ated a r$!-&ree +ort&o"$o'an #e 'reated and the rate o& ret%rn &or th$! +ort&o"$o $n e*%$"$#r$% /$"" #e ther$!-&ree rate. To &$nd the +ro+ort$on! o& th$! +ort&o"$o = !et the !tandardde$at$on e*%a" to 2ero. J$th +er&e't neat$e 'orre"at$on the +ort&o"$o !tandardde$at$on $!:

8P5 A#!o"%te a"%e


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13. Fa"!e. I& the #orro/$n and "end$n rate! are not $dent$'a" then de+end$n on theta!te! o& the $nd$$d%a"! (that $! the !ha+e o& the$r $nd$&&eren'e '%re!) #orro/er!and "ender! 'o%"d hae d$&&erent o+t$a" r$!0 +ort&o"$o!.

14. Fa"!e. The +ort&o"$o !tandard de$at$on e*%a"! the /e$hted aerae o& the'o+onent-a!!et !tandard de$at$on! only$n the !+e'$a" 'a!e that a"" a!!et! are+er&e't"0 +o!$t$e"0 'orre"ated. Other/$!e a! the &or%"a &or +ort&o"$o !tandardde$at$on !ho/! the +ort&o"$o !tandard de$at$on $! lessthan the /e$hted aeraeo& the 'o+onent-a!!et !tandard de$at$on!. The +ort&o"$o variance$! a /e$htedsumo& the e"eent! $n the 'oar$an'e atr$; /$th the +rod%'t! o& the +ort&o"$o+ro+ort$on! a! /e$ht!.

19. The +ro#a#$"$t0 d$!tr$#%t$on $!:

Pro#a#$"$t0 Rate o& Ret%rn

6.7 166

6.3 G96

Mean 5


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( ) 466 and ( ) ( ) 6.9I J I J Min MinCov r r w I w J = = = =

Th$! $nt%$t$e re!%"t $! an $+"$'at$on o& a +ro+ert0 o& an0 e&&$'$ent &ront$er nae"0that the 'oar$an'e! o& the "o#a" $n$% ar$an'e +ort&o"$o /$th a"" other a!!et! onthe &ront$er are $dent$'a" and e*%a" to $t! o/n ar$an'e. (Other/$!e add$t$ona"

d$er!$&$'at$on /o%"d &%rther red%'e the ar$an'e.) In th$! 'a!e the !tandardde$at$on o& (I ) red%'e! to:

1D,( )


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1?36! -1.,9 [email protected]

1?46! ?.11 -1.,9

1?96! 1?.41 ?.11

1?6! 7.@4 1?.41

1?76! 9.?6 7.@4

1?@6! 17.6 9.?61??6! 1@.,6 17.6

ote that ea'h 'orre"at$on $! #a!ed on on"0 !een o#!erat$on! !o /e 'annot arr$eat an0 !tat$!t$'a""0 !$n$&$'ant 'on'"%!$on!. Loo$n at the re!%"t! ho/eer $t a++ear!that /$th the e;'e+t$on o& "are-'o+an0 !to'! there $! +er!$!tent !er$a"'orre"at$on. (Th$! 'on'"%!$on 'hane! /hen /e t%rn to rea" rate! $n the ne;t+ro#"e.)

,1. The ta#"e &or rea" rate! (%!$n the a++ro;$at$on o& !%#tra't$n a de'ade! aerae$n&"at$on &ro the de'ade! aerae no$na" ret%rn) $!:

Rea" Rate!

Sa""Co+an0

Sto'!

LareCo+an0

Sto'!

Lon-Teroernent

>ond!

Intered-Teroernent

>ond!Trea!%r0

>$""!

1?,6! -,.7, 1?.3 4.?@ 4.77 4.9

1?36! ?.3, 6.7? .4 9.?9 ,.34

1?46! 19.,7 3.79 -1.77 -3. -4.??

1?96! 1.7? 17.1? -1.?7 -1.11 -6.39

1?6! 11.,6 9.3, -1.3@ 6.@? 1.37

1?76! 1.3? -1.4 -6.73 -1.,9 -1.67

1?@6! 7.3 1,.96 .46 .?1 3.?6

1??6! 16.?1 19.,7 9.7 4.@1 ,.6?

Ser$a"Corre"at$on

6.,? -6.,7 6.3@ 6.11 6.66

Jh$"e the !er$a" 'orre"at$on $n de'ade nominalret%rn! !ee! to #e +o!$t$e $ta++ear! that rea" rate! are !er$a""0 %n'orre"ated. The de'ade t$e !er$e! (a"tho%haa$n too !hort &or an0 de&$n$t$e 'on'"%!$on!) !%e!t that rea" rate! o& ret%rn are$nde+endent &ro de'ade to de'ade.

,,. The 3-year risk premium for the S&P portfolio is

, the 3-year risk premium for the hedge fund

portfolio is S&P 3-year standard deviation is 0

. The hedge fund 3-year standard deviation is 0

. S&P Sharpe ratio is 15.76/34.64 = 0.4550, and the

hedge fund Sharpe ratio is 33.10/60.62 = 0.5460.

23. With a = 0, the optimal asset allocation is

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,

.

With these weights,E

The resulting Sharpe ratio is 22.81/32.10 = 0.7108. Greta has a risk aversion of A=3,Therefore, she will invest

y

of her wealth in this risky portfolio. The resulting investment composition will be S&P:0.7138 59.32 = 43.78% and Hedge: 0.7138 40.68 = 30.02%. The remaining 26% will

be invested in the risk-free asset.

24. With = 0.3, the annual covariance is .

25. S&P 3-year standard deviation is . The hedge fund 3-

year standard deviation is . Therefore, the 3-year

covariance is 0 .

STOPPED

26. With a =.3, the optimal asset allocation is

,

.

With these weights,

E

. The resulting Sharpe ratio is 23.49/37.55 = 0.6256. Notice that the higher covarianceresults in a poorer Sharpe ratio.Greta will invest

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y

of her wealth in this risky portfolio. The resulting investment composition will be S&P:

0.5554 55.45 =30.79% and hedge: 0.5554 44.55= 24.74%. The remaining 44.46%

will be invested in the risk-free asset.

CFA PROBLEMS

1. a. Re!tr$'t$n the +ort&o"$o to ,6 !to'! rather than 46 to 96 !to'! /$"" $n'rea!ethe r$! o& the +ort&o"$o #%t $t $! +o!!$#"e that the $n'rea!e $n r$! /$"" #e $n$a".S%++o!e that &or $n!tan'e the 96 !to'! $n a %n$er!e hae the !ae !tandard

de$at$on () and the 'orre"at$on! #et/een ea'h +a$r are $dent$'a" /$th 'orre"at$on

'oe&&$'$ent . Then the 'oar$an'e #et/een ea'h +a$r o& !to'! /o%"d #e 8 , andthe ar$an'e o& an e*%a""0 /e$hted +ort&o"$o /o%"d #e:

,,,:8

18

18

n

n

nP

+=

The e&&e't o& the red%'t$on $n non the !e'ond ter on the r$ht-hand !$de

/o%"d #e re"at$e"0 !a"" (!$n'e 4?D96 $! '"o!e to 1?D,6 and 8 ,$! !a""er than

8,) #%t the deno$nator o& the &$r!t ter /o%"d #e ,6 $n!tead o& 96. For

e;a+"e $& 8 5 49 and 5 6., then the !tandard de$at$on /$th 96 !to'!

/o%"d #e ,6.?1 and /o%"d r$!e to ,,.69 /hen on"0 ,6 !to'! are he"d.

S%'h an $n'rea!e $ht #e a''e+ta#"e $& the e;+e'ted ret%rn $! $n'rea!ed

!%&&$'$ent"0.

#. Henne!!0 'o%"d 'onta$n the $n'rea!e $n r$! #0 a$n !%re that he a$nta$n!

rea!ona#"e d$er!$&$'at$on aon the ,6 !to'! that rea$n $n h$! +ort&o"$o. Th$!

enta$"! a$nta$n$n a "o/ 'orre"at$on aon the rea$n$n !to'!. For e;a+"e

$n +art (a) /$th 5 6., the $n'rea!e $n +ort&o"$o r$! /a! $n$a". A! a

+ra't$'a" atter th$! ean! that Henne!!0 /o%"d hae to !+read h$! +ort&o"$o

aon an0 $nd%!tr$e!Q 'on'entrat$n on %!t a &e/ $nd%!tr$e! /o%"d re!%"t $n

h$her 'orre"at$on! aon the $n'"%ded !to'!.

,. R$! red%'t$on #ene&$t! &ro d$er!$&$'at$on are not a "$near &%n't$on o& the n%#er o&$!!%e! $n the +ort&o"$o. Rather the $n'reenta" #ene&$t! &ro add$t$ona" d$er!$&$'at$onare o!t $+ortant /hen 0o% are "ea!t d$er!$&$ed. Re!tr$'t$n Henne!!0 to 16$n!tead o& ,6 $!!%e! /o%"d $n'rea!e the r$! o& h$! +ort&o"$o #0 a reater ao%nt than/o%"d a red%'t$on $n the !$2e o& the +ort&o"$o &ro 36 to ,6 !to'!. In o%r e;a+"ere!tr$'t$n the n%#er o& !to'! to 16 /$"" $n'rea!e the !tandard de$at$on to,3.@1. The 1.7 $n'rea!e $n !tandard de$at$on re!%"t$n &ro $$n %+ 16 o& ,6

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!to'! $! reater than the 1.14 $n'rea!e that re!%"t! &ro $$n %+ 36 o& 96 !to'!.

3. The +o$nt $! /e"" taen #e'a%!e the 'o$ttee !ho%"d #e 'on'erned /$th theo"at$"$t0 o& the ent$re +ort&o"$o. S$n'e Henne!!0! +ort&o"$o $! on"0 one o& !$; /e""-

d$er!$&$ed +ort&o"$o! and $! !a""er than the aerae the 'on'entrat$on $n &e/er$!!%e! $ht hae a $n$a" e&&e't on the d$er!$&$'at$on o& the tota" &%nd. Hen'e%n"ea!h$n Henne!!0 to do !to' +$'$n a0 #e adantaeo%!.

4. d. Port&o"$o Y 'annot #e e&&$'$ent #e'a%!e $t $! do$nated #0 another +ort&o"$o.For e;a+"e Port&o"$o ha! #oth h$her e;+e'ted ret%rn and "o/er !tandardde$at$on.

9. '.

. d.

7. #.

@. a.

?. '.

16. S$n'e /e do not hae an0 $n&orat$on a#o%t e;+e'ted ret%rn! /e &o'%! e;'"%!$e"0

on red%'$n ar$a#$"$t0. Sto'! A and C hae e*%a" !tandard de$at$on! #%t the'orre"at$on o& Sto' > /$th Sto' C (6.16) $! "e!! than that o& Sto' A /$th Sto' >(6.?6). There&ore a +ort&o"$o 'o+r$!ed 'o+o!ed o& Sto'! > and C /$"" hae "o/ertota" r$! than a +ort&o"$o 'o+r$!ed 'o+o!ed o& Sto'! A and >.

11. F%nd re+re!ent! the !$n"e best add$t$on to 'o+"eent Ste+hen!on! '%rrent+ort&o"$o $en h$! !e"e't$on 'r$ter$a. F%nd ! e;+e'ted ret%rn (14.6 +er'ent) ha! the+otent$a" to $n'rea!e the +ort&o"$o! ret%rn !oe/hat. F%nd ! re"at$e"0 "o/'orre"at$on /$th h$! '%rrent +ort&o"$o (6.9) $nd$'ate! that F%nd /$"" +ro$dereater d$er!$&$'at$on #ene&$t! than an0 o& the other a"ternat$e! e;'e+t F%nd >. The

re!%"t o& add$n F%nd !ho%"d #e a +ort&o"$o /$th a++ro;$ate"0 the !ae e;+e'tedret%rn and !oe/hat "o/er o"at$"$t0 'o+ared to the or$$na" +ort&o"$o.

The other three &%nd! hae !hort'o$n! $n ter! o& e;+e'ted ret%rn enhan'eentor o"at$"$t0 red%'t$on thro%h d$er!$&$'at$on. F%nd A o&&er! the +otent$a" &or$n'rea!$n the +ort&o"$o! ret%rn#%t $! too h$h"0 'orre"ated to +ro$de !%#!tant$a"o"at$"$t0 red%'t$on #ene&$t! thro%h d$er!$&$'at$on. F%nd > +ro$de! !%#!tant$a"o"at$"$t0 red%'t$on thro%h d$er!$&$'at$on #ene&$t!#%t $! e;+e'ted to enerate a

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ret%rn /e"" #e"o/ the '%rrent +ort&o"$o! ret%rn. F%nd C ha! the reate!t +otent$a" to$n'rea!e the +ort&o"$o! ret%rn#%t $! too h$h"0 'orre"ated /$th the '%rrent +ort&o"$oto +ro$de !%#!tant$a" o"at$"$t0 red%'t$on #ene&$t! thro%h d$er!$&$'at$on.

1,. a. S%#!'r$+t OP re&er! to the or$$na" +ort&o"$o A>C to the ne/ !to' and Pto the ne/ +ort&o"$o.

$. E(rP) 5 w!PE(r!P) wABCE(rABC) 5 (6.? 6.7) (6.1 1.,9) 5 6.7,@

$$. Co 5 !PABC5 6.46 ,.37 ,.?9 5 ,.7? ,.@6

$$i" P5


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Rane o& ret%rn!, /h$'h 'on!$der! the h$he!t and "o/e!t e;+e'ted ret%rn! $nthe &%t%re +er$od /$th a "arer rane #e$n a !$n o& reater ar$a#$"$t0 andthere&ore o& reater r$!.

Se$ar$an'e 'an #e %!ed to ea!%re e;+e'ted de$at$on! o& ret%rn! #e"o/ the

ean or !oe other #en'har !%'h a! 2ero.E$ther o& the!e ea!%re! /o%"d +otent$a""0 #e !%+er$or to ar$an'e &or ra'e.Rane o& ret%rn! /o%"d he"+ to h$h"$ht the &%"" !+e'tr% o& r$! !he $!a!!%$n e!+e'$a""0 the do/n!$de +ort$on o& the rane a#o%t /h$'h !he $! !o'on'erned. Se$ar$an'e /o%"d a"!o #e e&&e't$e #e'a%!e $t $+"$'$t"0 a!!%e!that the $ne!tor /ant! to $n$$2e the "$e"$hood o& ret%rn! &a""$n #e"o/ !oetaret rateQ $n ra'e! 'a!e the taret rate /o%"d #e !et at 2ero (to +rote'taa$n!t neat$e ret%rn!).

13. a. S0!teat$' r$! re&er! to &"%'t%at$on! $n a!!et +r$'e! 'a%!ed #0 a'roe'ono$'&a'tor! that are 'oon to a"" r$!0 a!!et!Q hen'e !0!teat$' r$! $! o&ten

re&erred to a! aret r$!. E;a+"e! o& !0!teat$' r$! &a'tor! $n'"%de the#%!$ne!! '0'"e $n&"at$on onetar0 +o"$'0&$!'a" +o"$'0 and te'hno"o$'a"'hane!.

F$r-!+e'$&$' r$! re&er! to &"%'t%at$on! $n a!!et +r$'e! 'a%!ed #0 &a'tor! thatare $nde+endent o& the aret !%'h a! $nd%!tr0 'hara'ter$!t$'! or &$r'hara'ter$!t$'!. E;a+"e! o& &$r-!+e'$&$' r$! &a'tor! $n'"%de "$t$at$on +atent!anaeent o+erat$n 'a!h &"o/ 'hane! and &$nan'$a" "eerae.

#. Tr%d0 !ho%"d e;+"a$n to the '"$ent that +$'$n on"0 the to+ &$e #e!t $dea!/o%"d o!t "$e"0 re!%"t $n the '"$ent ho"d$n a %'h ore r$!0 +ort&o"$o. Thetota" r$! o& a +ort&o"$o or +ort&o"$o ar$an'e $! the 'o#$nat$on o& !0!teat$'

r$! and &$r-!+e'$&$' r$!.The !0!teat$' 'o+onent de+end! on the !en!$t$$t0 o& the $nd$$d%a" a!!et! toaret oeent! a! ea!%red #0 #eta. A!!%$n the +ort&o"$o $! /e""d$er!$&$ed the n%#er o& a!!et! /$"" not a&&e't the !0!teat$' r$! 'o+onento& +ort&o"$o ar$an'e. The +ort&o"$o #eta de+end! on the $nd$$d%a" !e'%r$t0#eta! and the +ort&o"$o /e$ht! o& tho!e !e'%r$t$e!.

On the other hand the 'o+onent! o& &$r-!+e'$&$' r$! (!oet$e! 'a""ednonsystematic ris#) are not +er&e't"0 +o!$t$e"0 'orre"ated /$th ea'h other anda! ore a!!et! are added to the +ort&o"$o tho!e add$t$ona" a!!et! tend to red%'e+ort&o"$o r$!. Hen'e $n'rea!$n the n%#er o& !e'%r$t$e! $n a +ort&o"$o red%'e!&$r-!+e'$&$' r$!. For e;a+"e a +atent e;+$rat$on &or one 'o+an0 /o%"d not

a&&e't the other !e'%r$t$e! $n the +ort&o"$o. An $n'rea!e $n o$" +r$'e! $! "$e"0 to'a%!e a dro+ $n the +r$'e o& an a$r"$ne !to' #%t /$"" "$e"0 re!%"t $n an $n'rea!e$n the +r$'e o& an ener0 !to'. A! the n%#er o& rando"0 !e"e'ted !e'%r$t$e!$n'rea!e! the tota" r$! (ar$an'e) o& the +ort&o"$o a++roa'he! $t! !0!teat$'ar$an'e.

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bkm 10e chap007 sm final

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