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Solutions for Chapter 18: Questions and Problems

CHAPTER 18

EVALUATION OF PORTFOLIO PERFORMANCE

Answers to Questions

1. The two major factors would be: (1) attempt to derie ris!"adjusted returns that e#ceed a

naie bu$"and"hold polic$ and (%) completel$ diersif$ " i.e.& eliminate all uns$stematicris! from the portfolio. ' portfolio manaer can do one or both of two thins to derie

superior ris!"adjusted returns. The first is to hae superior timing reardin mar!et

c$cles and adjust $our portfolio accordinl$. 'lternatiel$& one can consistentl$ see!t

un"er#$ue" sto!%s& 's lon as $ou do not ma!e major mista!es with the rest of the

portfolio& these actions should result in superior ris!"adjusted returns.

%. Tre$nor (1*+) diided a fund,s e#cess return (return less ris!"free rate) b$ its beta. -or a

fund not completel$ diersified& Tre$nor,s T/ alue will understate ris! and oerstate performance. Sharpe (1**) diided a fund,s e#cess return b$ its standard deiation.

Sharpe,s S/ alue will produce ealuations er$ similar to Tre$nor,s for funds that arewell diersified. 0ensen (1*8) measures performance as the difference between a fund,s

actual and reuired returns. Since the latter return is based on the C'P2 and a fund,s

beta& 0ensen ma!es the same implicit assumptions as Tre$nor " namel$& that funds arecompletel$ diersified. The information ratio (34) measures a portfolio,s aerae return

in e#cess of that of a benchmar!& diided b$ the standard deiation of this e#cess return.

5i!e Sharpe& it can be used when the fund is not necessaril$ well"diersified.

6. -or portfolios with 4 % alues noticeabl$ less than 1.7& it would ma!e sense to compute

both measures. ifferences in the ran!ins enerated b$ the two measures would suestless"than"complete diersification b$ some funds " specificall$& those that were ran!edhiher b$ Tre$nor than b$ Sharpe.

9. 0ensen,s alpha (α) is found from the euation 4 jt 4-4 t ; α j < β j=4 mt 4-4 t> <e jt. The a j

indicates whether a manaer has superior (α j ? 7) or inferior (α j @ 7) abilit$ in mar!et

timin or stoc! selection& or both. 's suested aboe& 0ensen defines superior (inferior)

performance as a positie (neatie) difference between a manaer,s actual return and his

C'P2"based reuired return. -or poorl$ diersified funds& 0ensen,s ran!ins would moreclosel$ resemble Tre$nor,s. -or well"diersified funds& 0ensen,s ran!ins would follow

those of both Tre$nor and Sharpe. A$ replacin the C'P2 with the 'PT& differences

between funds, actual and reuired returns (or alphas/) could proide fresh ealuationsof funds.

168

Cop$riht B %717 b$ elson Dducation 5td.

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+. The 3nformation 4atio (34) is calculated b$ diidin the aerae return on the portfolio

less a benchmar! return b$ the standard deiation of the e#cess return. The 34 can beiewed as a benefit"cost ratio in that the standard deiation of return can be iewed as a

cost associated in the sense that it measures the uns$stematic ris! ta!en on b$ actie

manaement. Thus 34 is a cost"benefit ratio that assesses the ualit$ of the inestor,sinformation deflated b$ uns$stematic ris! enerated b$ the inestment process.

*. The difference b$ which a manaer,s oerall actual return beats hisEher oerall

benchmar! return is termed the total alue"added return and decomposes into anallocation effect and a selection effect. The former effect measures differences in weihts

assined b$ the actual and benchmar! portfolios to stoc!s& bonds and cash times the

respectie differences between mar!et"specific benchmar! returns and the oerall benchmar! return. The latter effect focuses on the mar!et"specific actual returns less the

correspondin mar!et"specific benchmar! returns times the weihts assined to each

mar!et b$ the actual portfolio. Ff course& the foreoin anal$sis implicitl$ assumes that

the actual and benchmar! mar!et"specific portfolios (e..& stoc!s) are ris!"euialent. 3f this is not true the anal$sis would not be alid.

G. Hhen measurin the performance of an euit$ portfolio manaer& oerall returns can berelated to a common total ris! or s$stematic ris!. -actors influencin the returns achieed

b$ the bond portfolio manaer are more comple#. 3n order to ealuate performance based

on a common ris! measure (i.e.& mar!et inde#)& four components must be considered thatdifferentiate the indiidual portfolio from the mar!et inde#. These components include:

(1) a polic$ effect& (%) a rate anticipation effect& (6) an anal$sis effect& and (9) a tradin

effect. ecision ariables inoled include the impact of duration decisions& anticipationof sectorEualit$ factors& and the impact of indiidual bond selection.

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CHAPTER 18

Answers to Pro'ems

1(a).

+7.1.79

.7*

.79

.7G.162ar!et

1.*G.7*

.17

.7*

.7G.1GS

1.77.76

.76

.76

.7G.17S

1.67.17

.16

.17

.7G.%7S

*7.1.7+.78

7.7+.7G.1+S

S

4

Q

P

==−

=

==−

=

==−

=

==−

=

==−=

1(b).

7*77.1.77

.7*

1.77

.7G.162ar!et

.771.17

.17

1.17

.7G.1GT

.7+77.*7

.76

.*7

.7G.17T

.78*G1.+7

.16

1.+7

.7G.%7T

.78771.77

.78

1.77

.7G.1+T

S

4

Q

P

==−

=

==−

=

==−

=

==−

=

==−

=

Sharpe Tre$nor

P % 6

Q 9 %

4 + +

S 1 1

2ar!et 6 9

1(c). 3t is apparent from the ran!ins aboe that Portfolio Q was poorl$ diersified since

Tre$nor ran!ed it I% and Sharpe ran!ed it I9. Ftherwise& the ran!ins are similar.

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%(a). Portfolio 2F enjo$ed the hihest deree of diersification since it had the hihest 4 %

(9.8J). The statistical loic behind this conclusion comes from the C'P2 which sa$sthat all full$ diersified portfolios should be priced alon the securit$ mar!et line. 4 % is a

measure of how well assets conform to the securit$ mar!et line& so 4 % is also a measure

of diersification.

%(b). ote the mean returns are net of the ris!"free rate. oin the calculations we obtain:

Fun" Tre(nor )*$rpe +ensen

'AC 7.G+(9) 7.8+G(9) 7.1%(9)

D- 7.G1+(+) 7.*1(+) "7.7+6(+)

KL3 1.+G9(1) 1.1G(1) 7.9*6(1)

0M5 1.%*%(%) 7.1+(6) 7.6++(%)2F 1.169(6) 1.777(%) 7.%*(6)

%(c).

Fun" t,$p*$-

'AC 1.G9++(6)D- "7.%G8(+)

KL3 %.96*8(1)0M5 1.*16*(9)

2F %.1196(%)

Fnl$ KL3 and 2F hae sinificantl$ positie alphas at a +J leel of confidence.

6(a). (3nformation ratio) 34 j ; α jEσu where σu ; standard error of the reression

34 ' ; .7+8E.+66 ; 7.1788

34 A ; .11+E+.889 ; 7.71+

34 C ; .%+7E%.1*+ ; 7.11++

6(b). 'nnualiNed 34 ; (T)1E%(34)

'nnualiNed 34 ' ; (+%)1E%(7.1788) ; 7.G89*

'nnualiNed 34 A ; (%*)1E%(7.71+) ; 7.79

'nnualiNed 34 C ; (1%)1E%(7.11++) ; 7.9771

6(c). The hiher the ratio& the better. Aased upon the answers to part a& 2anaer C would be

rated the hihest followed b$ 2anaers ' and A& respectiel$. Loweer& once the alues

are annualiNed& the ran!in chane. Specificall$& based upon the annualiNed 34& 2aner ' is rated the hihest& followed b$ C and A. (3n both cases& 2anaer A is rated last).

Aased upon the Krinold"Mahn standard for ood/ performance (7.+77 or reater)& onl$

2anaer ' meets that test.

9(a). Ferall performance (-und 1) ; %*.97J " *.%7J ; %7.%7J

Ferall performance (-und %) ; 16.%%J " *.%7J ; G.7%J

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9(b). D(4 i) ; *.%7 < β(1+.G1 *.%7)

; *.%7 < β (.+1)

Total return (-und 1) ; *.%7 < (1.6+1)(.+1) ; *.%7 < 1%.8+ ; 1.7+Jwhere 1%.8+J is the reuired return for ris!

Total return (-und %) ; *.%7 < (7.7+)(.+1) ; *.%7 < 8.*1 ; 19.81Jwhere 8.*1J is the reuired return for ris!

9(c)(i). Selectiit$1 ; %7.%J " 1%.8+J ; G.6+J

Selectiit$% ; G.7%J " 8.*1J ; "1.+J

9(c)(ii).4atio of total ris! 1 ; σ1Eσm ; %7.*GE16.%+ ; 1.+*

4atio of total ris! % ; σ%Eσm ; 19.%7E16.%+ ; 1.7G

4 1 ; *.%7 < 1.+* (.+1) ; *.%7 < 19.86+* ; %1.79J4 % ; *.%7 < 1.7G (.+1) ; *.%7 < 17.1G+G ; 1*.68J

iersification1 ; %1.79J 1.7+J ; 1.J

iersification% ; 1*.68J 19.81J ; 1.+GJ

9(c)(iii). et Selectiit$ ; Selectiit$ iersification

et Selectiit$1 ; G.6+J " 1.J ; +.6*J

et Selectiit$% ; "1.+J " 1.+GJ ; "6.1*J

9(d). Den accountin for the added cost of incomplete diersification& -und 1,s performancewas aboe the mar!et line (best performance)& while -und % fall below the line.

+.

a. Oear

2r

4eturn 2r O 4eturn

1 "1.+ "*.+% "1.+ "6.+

6 "1.+ "1.+

9 "1.7 6.+

+ 7.7 9.+* 9.+ *.+

G *.+ G.+

8 8.+ 8.+ 16.+ 1%.+

17 1G.+ 16.+

'erae 9.+ 9.+Std e *.7 *.*6

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Semi"de 7.*+ 9.%7

Semi"deiation considers onl$ the returns that are below the aerae.

b. Sharpe ratio: (aerae return minus ris!"free rate) E standard deiation

2r : 7.96+

2r O: 7.9+% Aest performer

*.*(a)(i). .*("+) < .6("6.+) < .1(7.6) ; "9.7%J

*(a)(ii). .+("9) < .%("%.+) < .6(7.6) ; "%.91J

*(a)(iii). .6("+) < .9("6.+) < .6(7.6) ; "%.81J

2anaer ' outperformed the benchmar! fund b$ 1*1 basis points while 2anaer A beat

the benchmar! fund b$ 1%1 basis points.

*(b)(i). =.+("9 < +) < .%("%.+ < 6.+) < .6(.6 ".6)> ; 7.G7J

*(b)(ii). =(.6 " .*) ("+ < 9.7%) < (.9 " .6) ("6.+ < 9.7%) < (.6 ".1)(.6 < 9.7%)> ; 1.%1J

2anaer ' added alue throuh her selection s!ills (G7 of 1*1 basis points) and her

allocation s!ills (G1 of 1*1 basis points). 2anaer A added alue totall$ throuh his

allocation s!ills (1%1 of 1%1 basis points).

G (a). ollar"Heihted 4eturn

2anaer 5:

+77&777 ; "1%&777E(1<r) " G&+77E(1<r)%" 16&+77E(1<r)6 " *&+77E(1<r)9" 17&777E(1<r)+<

*%+&777E(1<r)+

Solin for r& the internal rate of return or H44 is %.G+J

2anaer 2:

G77&777 ; 6+&777E(1<r) < 6+&777E(1<r)%<6+&777E(1<r)6<6+&777E(1<r)9<6+&777E(1<r)+ <

*%+&777E(1<r)+

Solin for r& the internal rate of return or H44 is %.8J.

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G(b). Time"weihted return

2anaer 5:

Periods LP4

1 =(+%G&777 +77&777) 1%&777>E+77&777 ; .76 % =(+67&777 +%G&777) G&+77>E+%G&777 ; ".778+ 6 =(+++&777 +67&777) 16&+77>E+67&777 ; .7%1G

9 =(+87&777 +++&777) *&+77>E+++&777 ; .7666

+ =(*%+&777 +87&777) 17&777>E+87&777 ; .7*76

TH44 ; =(1 < .76)(1 " .778+)(1 < .7%1G)(1 < .7666)(1 < .7*76)> 1E+ " 1

; (1.196) 1E+ 1; 1.7%G1% 1 ; .7%G1% ; %.G1J

2anaer 2:

Periods LP4

1 =(*%&777 G77&777) < 6+&777>EG77&777 ; .768+G % =(**6&777 *%&777) < 6+&777>E*%&777 ; .778*G

6 =(*%1&777 **6&777) < 6+&777>E**6&777 ; ".717+*

9 =(*1%&777 *%1&777) < 6+&777>E*%1&777 ; .7918G

+ =(*%+&777 *1%&777) < 6+&777>E*1%&777 ; .7G89

TH44 ; =(1 < .768+G)(1 < .778*G)(1 " .717+*)(1 < .7918G)(1 < .7G89)>1E+ " 1

; (1.1*9*) 1E+ 1; 1.7679 1 ; .7679 ; 6.79J

D (1 H)(Contribution)

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