Elton, Gruber, Brown, and Goetzmann. Modern Portfolio Theory and Investment Analysis, 7th Edition. Solutions to Text Problems: Chapter 9

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1 Elton, Gruber, Brown, and Goetzmann Modern Portfolo Theory and Investment Analyss, 7th Edton Solutons to Text Problems: Chapter 9 Chapter 9: Problem In the table below, gven that the rskless rate equals 5%, the securtes are ranked n descendng order by ther excess return over beta. R R R R R R β R R β β β e e e e C Securty Rank ( ) β ( ) The numbers n the column above labeled C were obtaned by recallng from the text that, f the Sharpe sngle-ndex model holds: Thus, gven that m 0: C C m + m ( ) β e β e etc. C Wth no short sales, we only nclude those securtes for whch Elton, Gruber, Brown, and Goetzmann 9- Modern Portfolo Theory and Investment Analyss, 7th Edton Solutons To Text Problems: Chapter 9 β > C. Thus, only securtes and 6 (the hghest and second hghest ranked securtes n the above table) are n the optmal (tangent) portfolo. We could have stopped our R R calculatons after the frst tme we found a ranked securty for whch < C, β (n ths case the thrd hghest ranked securty, securty ), but we dd not so that we could demonstrate that as well. β < C for all of the remanng lower ranked securtes

2 Snce securty 6 (the second hghest ranked securty, where ) s the last ranked securty n descendng order for whch β > C, we set C * C and solve for the optmum portfolo s weghts usng the followng formulas: β R R e β C * Ths gves us: 30 ( ) ( ) Snce for securty and for securty 6, the optmum (tangent) portfolo when short sales are not allowed conssts of 47.5% nvested n securty and 5.5% nvested n securty 6. Chapter 9: Problem Ths problem uses the same nput data as Problem. When short sales are allowed, all securtes are ncluded and C * s equal to the value of C for the lowest ranked securty. Referrng back to the table gven n the answer to Problem, we see that the lowest ranked securty s securty 3, where 6. Therefore, we have C * C Elton, Gruber, Brown, and Goetzmann 9- Modern Portfolo Theory and Investment Analyss, 7th Edton Solutons To Text Problems: Chapter 9

3 To solve for the optmum portfolo s weghts, we use the followng formulas: and or β R R e β C 6 (for the standard defnton of short sales) 6 (for the Lntner defnton of short sales) * So we have: 30 ( ) ( ) ( ) ( ) ( ) ( ) Elton, Gruber, Brown, and Goetzmann 9-3 Modern Portfolo Theory and Investment Analyss, 7th Edton Solutons To Text Problems: Chapter 9

4 Ths gves us the followng weghts (by rank order) for the optmum portfolos under ether the standard defnton of short sales or the Lntner defnton of short sales: Standard Defnton Lntner Defnton Securty ( ) Securty 6 ( ) Securty ( 3) Securty 5 ( 4) Securty 4 ( 5) Securty 3 ( 6) Chapter 9: Problem 3 Wth short sales allowed but no rskless lendng or borrowng, the optmum portfolo depends on the nvestor s utlty functon and wll be found at a pont along the upper half of the mnmum-varance fronter of rsky assets, whch s the effcent fronter when rskless lendng and borrowng do not exst. As s descrbed n the text, the entre effcent fronter of rsky assets can be delneated wth varous combnatons of any two effcent portfolos on the fronter. One such effcent portfolo was found n Problem. By smply solvng Problem usng a dfferent value for R, another portfolo on the effcent fronter can be found and then the entre effcent fronter can be traced usng combnatons of those two effcent portfolos. Elton, Gruber, Brown, and Goetzmann 9-4 Modern Portfolo Theory and Investment Analyss, 7th Edton Solutons To Text Problems: Chapter 9

5 Chapter 9: Problem 4 In the table below, gven that the rskless rate equals 5%, the securtes are ranked n descendng order by ther excess return over standard devaton. ( ) Securty Rank R R R R R R ρ ρ + ρ C The numbers n the column above labeled C were obtaned by recallng from the text that, f the constant-correlaton model holds: C ( ) ρ ρ + ρ Thus, gven that ρ for all pars of securtes: C etc. C Wth no short sales, we only nclude those securtes for whch > C. Thus, only securtes,, 5 and 6 (the four hghest ranked securtes n the above table) are n the optmal (tangent) portfolo. We could have stopped our calculatons R R after the frst tme we found a ranked securty for whch < C, (n ths case the ffth hghest ranked securty, securty 4), but we dd not so that we could demonstrate that well. < C for all of the remanng lower ranked securtes as Elton, Gruber, Brown, and Goetzmann 9-5 Modern Portfolo Theory and Investment Analyss, 7th Edton Solutons To Text Problems: Chapter 9

6 Snce securty 6 (the fourth hghest ranked securty, where 4) s the last ranked securty n descendng order for whch for the optmum portfolo s weghts usng the followng formulas: > C, we set C * C and solve R R C ( ρ ) * Ths gves us: 4 ( )( ) ( 0.78) ( )( ) ( 0.78) ( )( ) ( 0.78) ( )( ) ( ) Snce for securty, for securty, 3 for securty 5 and 4 for securty 6, the optmum (tangent) portfolo when short sales are not allowed conssts of 3.75% nvested n securty, 5.8% % nvested n securty, 47.49% % nvested n securty 5 and.95% nvested n securty 6. Elton, Gruber, Brown, and Goetzmann 9-6 Modern Portfolo Theory and Investment Analyss, 7th Edton Solutons To Text Problems: Chapter 9

7 Chapter 9: Problem 5 Ths problem uses the same nput data as Problem 4. When short sales are allowed, all securtes are ncluded and C * s equal to the value of C for the lowest ranked securty. Referrng back to the table gven n the answer to Problem 4, we see that the lowest ranked securty s securty 7, where 7. Therefore, we have C * C To solve for the optmum portfolo s weghts, we use the followng formulas: and or R R C ( ρ ) 7 (for the standard defnton of short sales) 7 (for the Lntner defnton of short sales) * So we have: ( )( ) ( 0.75) ( )( ) ( ) 0367 ( )( ) ( ) 00 3 ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) Elton, Gruber, Brown, and Goetzmann 9-7 Modern Portfolo Theory and Investment Analyss, 7th Edton Solutons To Text Problems: Chapter 9

8 Ths gves us the followng weghts (by rank order) for the optmum portfolos under ether the standard defnton of short sales or the Lntner defnton of short sales: Standard Defnton Lntner Defnton Securty ( ) Securty ( ) Securty 5 ( 3) Securty 6 ( 4) Securty 4 ( 5) Securty 3 ( 6) Securty 7 ( 7) Chapter 9: Problem 6 Wth short sales allowed but no rskless lendng or borrowng, the optmum portfolo depends on the nvestor s utlty functon and wll be found at a pont along the upper half of the mnmum-varance fronter of rsky assets, whch s the effcent fronter when rskless lendng and borrowng do not exst. As s descrbed n the text, the entre effcent fronter of rsky assets can be delneated wth varous combnatons of any two effcent portfolos on the fronter. One such effcent portfolo was found n Problem 5. By smply solvng Problem 5 usng a dfferent value for R, another portfolo on the effcent fronter can be found and then the entre effcent fronter can be traced usng combnatons of those two effcent portfolos. Elton, Gruber, Brown, and Goetzmann 9-8 Modern Portfolo Theory and Investment Analyss, 7th Edton Solutons To Text Problems: Chapter 9