Table Of ContentEmilio Barucci, Claudio Fontana
Financial Markets Theory
Equilibrium, Efficiency and Information
Second Edition
Solutions Manual
Springer
2
ExercisesofChapter2
SolutionofExercise2.2:
(i):ItsufficestoapplyDel’Hoˆpital’stheorem:
1 (cid:16) b−1 (cid:17)
lim (a+bx) b −1
b→1b−1
(cid:18) (cid:19)
b−1 1 x(b−1)
=lim(a+bx) b log(a+bx)+
b→1 b2 b(a+bx)
=log(a+x).
(ii):Itsufficestonotethat:
lim 1 (cid:18)1+bx(cid:19)b−b1 =−lim(cid:16)1+ x (cid:17)−n=−e−ax.
b→0b−1 a n→∞ an
(iii):ItsufficestoapplyDel’Hoˆpital’stheorem:
1−eax
lim =−limxeax=−x.
a→0 a a→0
SolutionofExercise2.4:
Duetocondition(2.4),theriskpremiumρ (x˜)solvesthefollowingequation:
u
µ−ρ (x˜)−b(cid:0)µ−ρ (x˜)(cid:1)2=u(cid:0)E[x˜]−ρ (x˜)(cid:1)=E(cid:2)u(x˜)(cid:3)=µ−b(cid:0)σ2+µ2(cid:1).
u u u
2 2
Theaboveconditionamountstothefollowingsecondorderequationforρ (x˜):
u
b b
− ρ (x˜)2+(µb−1)ρ (x˜)+ σ2=0.
u u
2 2
Undertheconditionthatµ <1/b,thepositiverootisgivenby
(cid:114)
(cid:16) (cid:17)2
1− 1+ b σ2
µb−1
ρ (x˜)= .
u
b/(µb−1)
SolutionofExercise2.5:
Duetocondition(2.4),theriskpremiumρ (x˜)solvesthefollowingequation:
u
−1exp(cid:0)−a(cid:0)µ−ρ (x˜)(cid:1)(cid:1)=u(cid:0)E[x˜]−ρ (x˜)(cid:1)=E(cid:2)u(x˜)(cid:3)=−1exp(cid:16)−a(cid:0)µ−aσ2(cid:1)(cid:17)
u u
a a 2
3
and,therefore,
ρ (x˜)=aσ2/2.
u
SolutionofExercise2.6:
Recall that, if x˜ is distributed as a log-normal random variable, meaning that
logx˜∼N (µ,σ2),thenx˜1−a isalsodistributedasalog-normal,since(1−a)logx˜
isdistributedasN (cid:0)(1−a)µ,(1−a)2σ2(cid:1).Moreover
1
logE[x˜]=µ+ σ2.
2
Duetocondition(2.4),theriskpremiumρ (x˜)solvesthefollowingequation:
u
(cid:0)E[x˜]−ρ (x˜)(cid:1)1−a=E(cid:2)x˜1−a(cid:3).
u
Equivalently,passingtologarithmsandusingtheabovepropertiesofthelog-normal
distribution,
log(cid:0)E[x˜]−ρ (x˜)(cid:1)=µ+1−aσ2,
u
2
sothat
ρu(x˜)=E[x˜](cid:0)1−e−a2σ2(cid:1).
Inanapproximateform,itholdsthatρ (x˜)≈aE[x˜]σ2/2.
u
SolutionofExercise2.11:
Theindifferencebetweengamblesc)andd)impliesthat
1 1
20.25γ = 16γ+ 25γ,
2 2
which is verified for γ =1/2. For this value, the expected utility of gamble a) is
250001/2≈158.11,whileforgambleb)wehave
0.2×320001/2+0.7×100001/2+0.1×10001/2≈108.94.
Therefore,since108.94<158.11,theagentwillprefergamblea).
ExercisesofChapter3
SolutionofExercise3.2:
(i):Theoptimalportfoliocanbeexplicitlycomputedbyrelyingonformula(3.10):
4
(1−bw r )E[r˜−r ] (1−0.006×100×1.1)(1.3−1.1)
w∗= 0 f f = ≈7.359.
b(cid:0)σ2(r˜)+E[r˜−r ]2(cid:1) 0.006×(1.5+0.22)
f
(ii):Fromformula(3.10)fortheoptimaldemandoftheriskyassetw∗,wegetthat
w∗<0ifandonlyif1−b×100×1.1<0,i.e.,b>1/110.
(iii):Byapplyingformula(3.10)withr(cid:48) =1.2,wecancomputethatthenewopti-
f
malvaluebecomesw∗≈3.09.ObservethatthisresultiscoherentwithProposition
3.5,sincethequadraticutilityfunctionisIARAandthedemandoftheriskyassetis
lessthantheinitialwealth.
(iv):Formula(3.10)givesusavalueofw∗≈0.22fortheoptimalportfolioincor-
respondenceoftheinitialwealthw(cid:48) =150.Observethatthisresultiscoherentwith
0
Proposition 3.3, since the quadratic utility function is IARA and, hence, the risky
assetdemandisdecreasingwithrespecttoinitialwealth.
(v): Formula (3.10) gives us a value of w∗ ≈3.75 for the optimal portfolio in cor-
respondenceofanexpectedreturnofE[r˜(cid:48)]=1.2.Again,observethatthisresultis
coherentwithProposition3.5.
SolutionofExercise3.6:
(i):Thevarianceoftheportfolio(w,1−w)canbeeasilycomputedas
σ2(r˜)=w2σ2+(1−w)2σ2+2w(1−w)σ σ ρ,
1 2 1 2
sothat
σ2(r˜)≤w2σ2+(1−w)2σ2+2w(1−w)σ σ =(cid:0)wσ +(1−w)σ (cid:1)2≤max{σ2;σ2},
1 2 1 2 1 2 1 2
wherewehaveusedtheassumptionthat0≤w≤1.
(ii):Ifρ=1,thenσ2(r˜)=(cid:0)wσ +(1−w)σ (cid:1)2,sothatthevarianceoftheportfolio
1 2
canbereducedtozerobylettingw=−σ /(σ −σ ).
2 1 2
(iii):Ifρ =−1,thenσ2(r˜)=(cid:0)wσ −(1−w)σ (cid:1)2,sothatthevarianceoftheport-
1 2
foliocanbereducedtozerobylettingw=σ /(σ +σ ).
2 1 2
SolutionofExercise3.8:
(i): The problem consists in maximizing the expected utility function, so that, de-
notingbywthedemandoftheriskyasset,
(cid:18) (cid:113) (cid:113) (cid:19)
max π w r +w(u−r )+(1−π) w r +w(d−r ) .
0 f f 0 f f
w∈R
Differentiating with respect to w at equating the derivative to zero, we obtain the
followingoptimalitycondition:
π(u−r ) (1−π)(d−r )
f f
=− .
(cid:112) (cid:112)
w r +w∗(u−r ) w r +w∗(d−r )
0 f f 0 f f
Solvingforw∗weget
5
π2(u−r )2−(1−π)2(d−r )2
w∗=−w r f f .
0 f π2(u−r )2(d−r )−(1−π)2(d−r )2(u−r )
f f f f
(ii):Theproblemconsistsinmaximizingtheexpectedutilityfunction,sothat,de-
notingbywthedemandoftheriskyasset,
(cid:16) (cid:0) (cid:1) (cid:0) (cid:1)(cid:17)
max πlog w r +w(u−r ) +(1−π)log w r +w(d−r ) .
0 f f 0 f f
w∈R
Differentiatingwithrespecttow,equatingthederivativetozeroandsolvingforw∗,
weget
(cid:0) (cid:1)
r − uπ+d(1−π)
w∗=w r f .
0 f
(u−r )(d−r )
f f
(iii):Theproblemconsistsinmaximizingtheexpectedutilityfunction,sothat,de-
notingbywthedemandoftheriskyasset,
(cid:16) (cid:0) (cid:1)γ (cid:0) (cid:1)γ(cid:17)
max π w r +w(u−r ) +(1−π) w r +w(d−r ) .
0 f f 0 f f
w∈R
Differentiating with respect to w at equating the derivative to zero, we obtain the
followingoptimalitycondition:
π(u−r )(cid:0)w r +w∗(u−r )(cid:1)γ−1=−(1−π)(cid:0)d−r )(w r +w∗(d−r )(cid:1)γ−1.
f 0 f f f 0 f f
Lettingα :=1/(γ−1)andsolvingforw∗weget
(cid:0) (cid:1)α (cid:0) (cid:1)α
−(1−π)(d−r ) − π(u−r )
w∗=w r f f .
0 f (cid:0) (cid:1)α (cid:0) (cid:1)α
(u−r ) π(u−r ) −(d−r ) −(1−π)(d−r )
f f f f
SolutionofExercise3.9:
Notefirstthatthevariance-covariancematrixV oftherandomreturnsanditsinverse
V−1takethefollowingform:
V =(cid:20) 4 6ρ(cid:21) and V−1=(cid:34) 4(1−1ρ2) −6(1−ρρ2)(cid:35).
6ρ 9 − ρ 1
6(1−ρ2) 9(1−ρ2)
(i):Byformula(3.11),theoptimalportfoliocanbeexpressedas
1
w∗= V−1(e−r 1),
f
a
andthus,inthecontextofthepresentexercise,
6
1 ρ
w∗= 0.1− 0.2,
1 4a(1−ρ2) 6a(1−ρ2)
ρ 1
w∗=− 0.1+ 0.2.
2 6a(1−ρ2) 9a(1−ρ2)
Fromtheaboveexpressionitcanbecheckedthatw∗,w∗>0ifandonlyifρ<3/4.
1 2
Notealsothatw∗>0foreveryρ ∈(−1,1).
2
(ii):Frompart(i),itfollowsthatforρ=0.5theoptimalportfolioisdiversified.Sup-
posethenthatρ =0.5.AsshowninSection3.1,theoptimalportfolioisdiversified
ifandonlyif
ρσ σ
10.2<0.1< 1 0.2.
σ ρσ
2 2
Ascanbeeasilychecked,theaboveinequalitiesholdifandonlyif
(cid:26) (cid:27)
σ 2σ
ρ ≤min 2 ; 1 .
2σ σ
1 2
Hence,giventhevaluesofthevariancesandpreservingtheinequalityσ ≤2σ ,if
2 1
thecorrelationcoefficientρ increases,σ mustincreaseand/orσ mustdecreasein
2 1
orderfortheoptimalportfoliotoremaindiversified.
(iii):Forρ =0.5,theoptimalportfolioisgivenbyw∗≈ 1(0.011,0.019)(cid:62),sothat
a
the agent invests an amount greater than one in the risky assets if a≤0.03. For
ρ=−0.5,theoptimalportfolioisgivenbyw∗≈ 1(0.056,0.041)(cid:62),sothattheagent
a
investsanamountgreaterthanoneintheriskyassetsifa≤0.097.
SolutionofExercise3.16:
Letusdenotebye=(e ,e )(cid:62) theexpectedreturnsofassets1and2and,similarly,
1 2
by σ2 and σ2 the variance of r˜ and r˜ , respectively, and by ρ their correlation
1 2 1 2
coefficient.Then,
1 (cid:18) σ2 −ρσ σ (cid:19)
V−1= 2 1 2 ,
σ12σ22(1−ρ2) −ρσ1σ2 σ21
sothat
σ2e +σ2e −ρσ σ (e +e )
A=1(cid:62)V−1e= 2 1 1 2 1 2 1 2 ,
σ2σ2(1−ρ2)
1 2
σ2e2+σ2e2−2ρσ σ e e
B=e(cid:62)V−1e= 2 1 1 2 1 2 1 2,
σ2σ2(1−ρ2)
1 2
σ2+σ2−2ρσ σ
C=1(cid:62)V−11= 1 2 1 2.
σ2σ2(1−ρ2)
1 2
and D=BC−A2. The claim then follows by replacing the above expression into
thegeneralformula(3.16)giveninTheorem3.12.
7
SolutionofExercise3.21:
(i):Inthecaseoftworiskyassetsonly,theexpectedreturnofaportfolio(w,1−w)is
completelydeterminedbytheproportionofwealthinvestedinthetworiskyassets,
meaningthat
E[r˜ ]=wE[r˜ ]+(1−w)E[r˜ ]=w×1.2+(1−w)×1.3.
w 1 2
Hence, in order to yield a given expected return µ ∈R, the portfolio w has to be
equaltow=(µ−1.3)/(1.2−1.3).Thecorrespondingvarianceisgivenby
σ2(r˜ )=w2σ2(r˜ )+(1−w)2σ2(r˜ )+2w(1−w)ρσ(r˜ )σ(r˜ )
w 1 2 1 2
(cid:18) µ−1.3 (cid:19)2 (cid:18) µ−1.3 (cid:19)2 µ−1.3 (cid:18) µ−1.3 (cid:19)
=4 +9 1− +2ρ 1− 6
1.2−1.3 1.2−1.3 1.2−1.3 1.2−1.3
(cid:18) µ−1.3 (cid:19)2 µ−1.3
=(13−12ρ) +(12ρ−18) +9,
1.2−1.3 1.2−1.3
thusconfirmingthat,inthevariance-expectedreturnplane,theportfoliofrontieris
representedbyaparabola.
(ii):Inthepresentsetting,itholdsthat
V−1=(cid:32) 4(1−1ρ2) −6(1−ρρ2)(cid:33)=(cid:18) 1/3 −1/9(cid:19).
− ρ 1 −1/9 4/27
6(1−ρ2) 9(1−ρ2)
RecallfromSection3.2thatσ2(r˜ )=1/C,where
wMVP
(cid:18) (cid:19)(cid:18) (cid:19)
1/3 −1/9 1
C=1(cid:62)V−11=(1,1) =7/27,
−1/9 4/27 1
sothatσ2(r˜ )=27/7.Moreover:
wMVP
V−11 27(cid:18) 1/3 −1/9(cid:19)(cid:18)1(cid:19) (cid:18)6/7(cid:19)
wMVP= = = .
C 7 −1/9 4/27 1 1/7
(iii):RecallfromProposition3.15that
e−r 1
we=V−1 f .
1(cid:62)V−1(e−r 1)
f
Thedenominatorcanbeexplicitlycomputedas
(cid:18) (cid:19)(cid:18) (cid:19)
1/3 −1/9 0.1
1(cid:62)V−1(e−r 1)=(1,1) ≈0.02963.
f −1/9 4/27 0.2
Therefore:
(cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)
1 1/3 −1/9 0.1 0.375
we≈ = .
0.02963 −1/9 4/27 0.2 0.625
8
LetusalsocomputetheexpectedreturnE[r˜we]ofthetangentportfolio:
E[r˜we]≈0.375×1.2+0.625×1.3=1.2625.
(iv): In Exercise 3.9, we have shown that the optimal portfolio for the exponential
utilityisgivenbyw∗= 1(0.011,0.019)(cid:62).Hence,thevalueofaforwhichtheop-
a
timalportfoliofortheexponentialutilitycoincideswiththetangentportfoliowe is
givenbya≈0.0296.
(v):Withoutariskfreeasset,thedesiredportfolioissimplyobtainedbyfindingthe
valuew∈Rthatsatisfiesthefollowingequation:
1.25=w×1.2+(1−w)×1.3.
Thisimpliesthat,inthecaseoftworiskyassetsonly,theequallyweightedportfolio
(0.5,0.5) achieves the given expected return. In the case of two risky assets and a
riskfreeasset,sinceanyfrontierportfoliocanbewrittenasthelinearcombination
of the risk free asset and the tangent portfolio we, it suffices to determine the pro-
portionofwealthα investedinthetangentportfolio.Theoptimalα isobtainedas
thesolutiontothefollowingequation:
1.25=αE[r˜we]+(1−α)rf ≈α×1.2625+(1−α)×1.1,
whichyieldsα =0.923.Hence,theoptimalportfolioforthegivenexpectedreturn
µ =1.25isgivenby(0.923×0.375,0.923×0.625)(cid:62),while1−0.923isinvested
intheriskfreeasset.
(vi):Letuscomputetheexpectedreturnandthevarianceofthereturnofthegiven
portfoliow(cid:48):
E[r˜w(cid:48)]=0.6×1.2+0.5×1.3−0.1×1.1=1.26,
σ2(r˜w(cid:48))=0.62×4+0.52×9+2×0.6×0.5×3=5.49.
Letusnowcomputethefrontierportfoliowhichminimizesthevarianceforanex-
pected return equal to µ =1.26. As done in part (v), it suffices to determine the
optimal proportion of wealth invested in the tangent portfolio we, solving the fol-
lowingequation:
1.26=αE[r˜we]+(1−α)rf ≈α×1.2625+(1−α)×1.1,
yieldingassolutionα ≈0.985.Hence,thevarianceofthecorrespondingportfolio
w(cid:48)(cid:48):=αwecanbecomputedasfollows,forρ =0.5:
(cid:18) (cid:19)(cid:18) (cid:19)
σ2(r˜w(cid:48)(cid:48))=α2σ2(r˜we)≈0.9842×(cid:0)0.3750.625(cid:1) 4339 00..367255 ≈5.31.
Sinceσ2(r˜w(cid:48)(cid:48))=5.31<5.49=σ2(r˜w(cid:48)),whileE[r˜w(cid:48)(cid:48)]=E[r˜w(cid:48)],wehavethusshown
thattheportfoliow(cid:48)cannotbelongtotheportfoliofrontier.
9
SolutionofExercise3.22:
(i): The expected return of a portfolio w investing in the two risky assets only is
givenby
E[r˜ ]=wE[r˜ ]+(1−w)E[r˜ ],
w 1 2
sothat,inordertoachieveagivenexpectedreturn µ ∈R,theoptimalportfoliow
mustbegivenbyw=(µ−E[r˜ ])/(E[r˜ ]−E[r˜ ]),withacorrespondingvariance
2 1 2
(cid:18) µ−E[r˜ ] (cid:19)2 (cid:18) µ−E[r˜ ] (cid:19)2
σ2(r˜ )=2 2 +4 1− 2
w E[r˜ ]−E[r˜ ] E[r˜ ]−E[r˜ ]
1 2 1 2
√ µ−E[r˜ ] (cid:18) µ−E[r˜ ] (cid:19)
+4 2ρ 2 1− 2 .
E[r˜ ]−E[r˜ ] E[r˜ ]−E[r˜ ]
1 2 1 2
(ii):TheminimumvarianceportfoliowMVPcanbecomputedbyrelyingonformula
(3.26), which gives wMVP =(0.82,0.18)(cid:62), if ρ =0.5, and wMVP =(0.61,0.39)(cid:62),
inthecaseρ =−0.5.
(iii): The tangent portfolio we can be computed by relying on Proposition 3.15,
which gives we =(−1/3,4/3)(cid:62), if ρ =0.5, and we =(0.51,0.49)(cid:62), in the case
ρ =−0.5.
(iv):Fromthegeneraltheory,weknowthattheagentwillchooseanoptimalport-
foliow∗ belongingtotheefficientportfoliofrontier.Hence,duetothepropertiesof
theportfoliofrontier,theoptimalportfoliow∗canalwaysberepresentedbyalinear
combinationofthetangentportfolioweandtheriskfreeasset.Hence,wecandeter-
minetheoptimalportfoliow∗byapplyingformula(3.10)withrespecttothecouple
tangent portfolio - risk free asset. Therefore, for a fixed initial wealth w =1, the
0
optimalinvestmentα∗inthetangentportfolioweisgivenby
α∗= (1−brf)E[r˜we−rf] .
b(cid:0)σ2(r˜we)+E[r˜we−rf](cid:1)2
Clearly,theagentwillinvestmorethanoneunitofwealthinthetangentportfoliowe
ifandonlyifα∗>1.Intermsoftherisk-aversionparameterb,thelattercondition
holdsifandonlyif
b(cid:0)(σ2(r˜we)+E[r˜we]−rf)2−(E[r˜we]−rf)rf(cid:1)<E[r˜we]−rf.
SolutionofExercise3.23:
(i):Thevarianceofthereturnr˜ associatedtoaportfolio(w,1−w)isgivenbythe
w
generalformula
σ2(r˜ )=w2σ2+(1−w)2σ2+2w(1−w)ρσ σ .
w 1 2 1 2
Inordertoshowthatthereexistsw∈(0,1)suchthatσ(r˜ )<σ ,itsufficestoshow
w 1
that the first derivative of the function w(cid:55)→σ2(r˜ ) evaluated at 1 is positive. The
w
firstderivativeisgivenby
10
dσ2(r˜ )
w =2wσ2−2(1−w)σ2+2(1−2w)ρσ σ ,
dw 1 2 1 2
and,whenevaluatedat1,itisequalto2σ2−2ρσ σ andispositiveifρ <σ /σ ,
1 1 2 1 2
while,whenevaluatedat0,gives−2σ2+2ρσ σ .Sincethelatterquantityisneg-
2 1 2
ativeifρ<σ /σ ,thisimpliestheexistenceofw∈(0,1)suchthatσ(r˜ )<σ (as
1 2 w 1
canbeverifiedbyminimizingthefunctionw(cid:55)→σ2(r˜ ))andthat,forallw∈/[0,1],
w
itholdsthatσ(r˜ )≥σ .
w 1
(ii): This follows from the general formula for the variance σ2(r˜ ), using the as-
w
sumptionthatσ2≤σ2.
1 2
(iii):Thisfollowsbyargumentsanalogoustothoseusedtoprovepart(i)oftheex-
ercise.
(iv):Foranyw∈R,itholdsthat
σ2(r˜ )=σ2(cid:0)w2+(1−w)2+2ρw(1−w)(cid:1).
w
Thevalueofwwhichminimizesσ2(r˜ )canbefoundbysettingequaltozerothe
w
derivativeofthefunctionw(cid:55)→σ2(r˜ ),yielding
w
dσ2(r˜w) =2σ2(cid:0)w−(1−w)+ρ(1−2w)(cid:1)=2σ2(2w−1)(1−ρ),
dw
whichimmediatelyimpliesthatwMVP=(1/2,1/2)(cid:62),providedthatρ(cid:54)=1.Notethat
ifρ =1thenthevarianceofaportfolioisconstantandalwaysequaltoσ2.
SolutionofExercise3.27:
Theagentmustsolvethefollowingoptimizationproblem:
(cid:0) (cid:1)
max πu(w −D−wp+w)+(1−π)u(w −wp)
0 0
w∈R+
(cid:18) (cid:16) b (cid:17)
= max π w −D−wp+w)− (w −D−wp+w)2
0 0
w∈R+ 2
(cid:16) b (cid:17)(cid:19)
+(1−π) (w −wp)− (w −wp)2 .
0 0
2
Thefirstorderconditioncanbewrittenas
π(1−p)−πb(w −D−w∗p+w∗)(1−p)=(1−π)p−(1−π)pb(w −w∗p),
0 0
whichgivesthefollowingoptimalvalue
π−p−bπ(w −D)(1−p)+b(1−π)w p
w∗= 0 0 .
bπ(1−p)2+b(1−π)p2
If p=π,thenitiseasytocheckthattheaboveexpressionreducestow∗=D.On
theotherhand,ifp>π,thenonecanverifyfromtheaboveexpressionthatw∗<D,
thusconfirmingtheresultofProposition3.23.
