Elton, Gruber, Brown, and Goetzmann. Modern Portfolio Theory and Investment Analysis, 7th Edition. Solutions to Text Problems: Chapter 16

Transcription

1 lton, Gruer, rown, and Goetzmann Modern Portfolo Theory and Investment nalyss, 7th dton Solutons to Text Prolems: hapter 6 hapter 6: Prolem From the text we know that three ponts determne a plane. The PT equaton for a plane s: λ + λ + λ 0 ssumng that the three portfolos gven n the prolem are n equlrum (on the plane), then ther expected returns are determned y: λ + λ (a) 0 + λ λ + λ () 0 + λ 0. λ λ (c) 0 + λ 0. 5 The aove set of lnear equatons can e solved smultaneously for the three unknown values of λ0, λ and λ. There are many ways to solve a set of smultaneous lnear equatons. One method s shown elow. Sutract equaton (a) from equaton ():.4 λ 0. λ (d) Sutract equaton (a) from equaton (c): 0 λ λ (e) Sutract equaton (e) from equaton (d):.4 0.7λ or λ Susttute λ nto equaton (d):.4 λ 0.6 or λ Susttute λ and λ nto equaton (a): λ or λ 0 0 Thus, the equaton of the equlrum PT plane s: lton, Gruer, rown, and Goetzmann 6- Modern Portfolo Theory and Investment nalyss, 7th dton Solutons To Text Prolems: hapter 6

2 hapter 6: Prolem ccordng to the equlrum PT plane derved n Prolem, any securty wth and 0 should have an equlrum expected return of %: % ssumng the derved equlrum PT plane holds, snce portfolo has and 0 wth an expected return of 0%, the portfolo s not n equlrum and an artrage opportunty exsts. The frst step s to use portfolos n equlrum to create a replcatng equlrum nvestment portfolo, call t portfolo, that has the same factor loadngs (rsk) as portfolo. Usng the equlrum portfolos, and n Prolem and recallng that an nvestment portfolo s weghts sum to and that a portfolo s factor loadngs are weghted averages of the ndvdual factor loadngs we have: ( X X ) X + X + ( X X ) X + ( X X ) 0.5X + 0.X 0.5( X X ) 0 X + Smplfyng the aove two equatons, we have: X or X Snce X, X 0 and X 0.7X + X X X. Snce portfolo was constructed from equlrum portfolos, portfolo s also on the equlrum plane. We have seen aove that any securty wth portfolo s factor loadngs has an equlrum expected return of %, and that s the expected return of portfolo : X % So now we have two portfolos wth exactly the same rsk: the target portfolo and the equlrum replcatng portfolo. Snce they have the same rsk (factor loadngs), we can create an artrage portfolo, comnng the two portfolos y gong long n one and shortng the other. Ths wll create a self-fnancng (zero net nvestment) portfolo wth zero rsk: an artrage portfolo. lton, Gruer, rown, and Goetzmann 6- Modern Portfolo Theory and Investment nalyss, 7th dton Solutons To Text Prolems: hapter 6

3 In equlrum, an artrage portfolo has an expected return of zero, ut snce portfolo s not n equlrum, nether s the artrage portfolo contanng and, and an artrage proft may e made. We need to short sell ether portfolo or and go long n the other. The queston s: whch portfolo do we short and whch do we go long n? Snce oth portfolos have the same rsk and snce portfolo has a hgher expected return than portfolo, we want to go long n and short ; n other words, we want X and X. Ths gves us: X X 0 (zero net nvestment) ut snce portfolo conssts of a weghted average of portfolos, and, X s the same thng as X, X 0 and X, so we have: X X (zero net nvestment) X X (zero factor rsk) X X (zero factor rsk) X X % (postve artrage return) s artrageurs explot the opportunty y short sellng portfolo, the prce of portfolo wll drop, therey pushng portfolo s expected return up untl t reaches ts equlrum level of %, at whch pont the expected return on the artrage portfolo wll equal 0. There s no reason to expect any prce effects on portfolos, and, snce the artrage wth portfolo can e accomplshed usng other assets on the equlrum PT plane. lton, Gruer, rown, and Goetzmann 6- Modern Portfolo Theory and Investment nalyss, 7th dton Solutons To Text Prolems: hapter 6

4 hapter 6: Prolem From the text we know that three ponts determne a plane. The PT equaton for a plane s: λ + λ + λ 0 ssumng that the three portfolos gven n the prolem are n equlrum (on the plane), then ther expected returns are determned y: λ + λ + λ (a) 0 λ + λ () 0 +.5λ 7 λ λ (c) λ Solvng for the three unknowns n the same way as n Prolem, we otan the followng soluton to the aove set of smultaneous lnear equatons: λ 8; λ 6 ; λ ; 0 Thus, the equaton of the equlrum PT plane s: hapter 6: Prolem 4 ccordng to the equlrum PT plane derved n Prolem, any securty wth and 0 should have an equlrum expected return of %: % ssumng the derved equlrum PT plane holds, snce portfolo has and 0 wth an expected return of 5%, the portfolo s not n equlrum and an artrage opportunty exsts. The frst step s to use portfolos n equlrum to create a replcatng equlrum nvestment portfolo, call t portfolo, that has the same factor loadngs (rsk) as portfolo. Usng the equlrum portfolos, and n Prolem and recallng that an nvestment portfolo s weghts sum to and that a portfolo s factor loadngs are weghted averages of the ndvdual factor loadngs we have: ( X X ) X +.5X + 0.5( X X ) X + ( X X ) X ( X X ) 0 X + lton, Gruer, rown, and Goetzmann 6-4 Modern Portfolo Theory and Investment nalyss, 7th dton Solutons To Text Prolems: hapter 6

5 Smplfyng the aove two equatons, we have: X 4 X 5X + Solvng the aove two smultaneous equatons we have: X, X and X X X. Snce portfolo was constructed from equlrum portfolos, portfolo s also on the equlrum plane. We have seen aove that any securty wth portfolo s factor loadngs has an equlrum expected return of 4%, and that s the expected return of portfolo : X % So now we have two portfolos wth exactly the same rsk: the target portfolo and the equlrum replcatng portfolo. Snce they have the same rsk (factor loadngs), we can create an artrage portfolo, comnng the two portfolos y gong long n one and shortng the other. Ths wll create a self-fnancng (zero net nvestment) portfolo wth zero rsk: an artrage portfolo. In equlrum, an artrage portfolo has an expected return of zero, ut snce portfolo s not n equlrum, nether s the artrage portfolo contanng and, and an artrage proft may e made. We need to short sell ether portfolo or and go long n the other. The queston s: whch portfolo do we short and whch do we go long n? Snce oth portfolos have the same rsk and snce portfolo has a hgher expected return than portfolo, we want to go long n and short ; n other words, we want X and X. Ths gves us: X X 0 (zero net nvestment) lton, Gruer, rown, and Goetzmann 6-5 Modern Portfolo Theory and Investment nalyss, 7th dton Solutons To Text Prolems: hapter 6

6 ut snce portfolo conssts of a weghted average of portfolos, and, X s the same thng as X, X and X, so we have: X X + 0 (zero net nvestment) X X (zero factor rsk) X X (zero factor rsk) X X % (postve artrage return) s artrageurs explot the opportunty y uyng portfolo, the prce of portfolo wll rse, therey pushng portfolo s expected return down untl t reaches ts equlrum level of 4%, at whch pont the expected return on the artrage portfolo wll equal 0. There s no reason to expect any prce effects on portfolos, and, snce the artrage wth portfolo can e accomplshed usng other assets on the equlrum PT plane. hapter 6: Prolem 5 The general K-factor PT equaton for expected return s: K + λ 0 λk k k where λ0 s the return on the rskless asset, f t exsts. lton, Gruer, rown, and Goetzmann 6-6 Modern Portfolo Theory and Investment nalyss, 7th dton Solutons To Text Prolems: hapter 6

7 Gven the data n the prolem and n Tale 6. n the text, along wth a rskless rate of 8%, the Sharpe mult-factor model for the expected return on a stock n the constructon ndustry s: % The last numer,.59, enters ecause the stock s a constructon stock. hapter 6: Prolem 6. From the text we know that, for a -factor PT model to e consstent wth the λ m β. Gven that ( m ) 4 and usng results from standard PM, j ( F ) λj Prolem, we have: 4β λ or β λ 0. 5 ; 4β λ or β λ From the text we know that β β λ + β. So we have: λ F β β β ssumng all three portfolos n Prolem are n equlrum, then we can use any one of them to fnd the rsk-free rate. For example, usng portfolo gves: f + ( m F ) β or F ( m F ) β Gven that %, 0. 5 β and ( m ) 4% F, we have: F % lton, Gruer, rown, and Goetzmann 6-7 Modern Portfolo Theory and Investment nalyss, 7th dton Solutons To Text Prolems: hapter 6