Elton, Gruber, Brown, and Goetzmann. Modern Portfolio Theory and Investment Analysis, 7th Edition. Solutions to Text Problems: Chapter 14

Transcription

1 Elton, Gruer, Brown, nd Goetznn odern Portfolo Theory nd Investent Anlyss, 7th Edton Solutons to Text Proles: hpter 14 hpter 14: Prole 1 Gven the zero-et securty rket lne n ths prole, the return on the zeroet portfolo equls.4 (4%), the ntercept of the lne, nd the excess return of the rket ove the zero-et portfolo s return (lso clled the rket rsk preu ) equls.1 (1%), the slope of the lne. The return on the rket portfolo ust therefore e , or 14%. hpter 14: Prole hs the se role n the zero-et odel s does n the stndrd odel. So, referrng ck to the nswer to Prole 5 n hpter 13, sply replce wth to otn: P 1 Y r cov ( ) ( Y Y ) Y r P vr ( Y ) 1. where r ( ) hpter 14: Prole 3 As s shown n the text, the post-tx for of the AP s equlru prcng equton s: ( τ ( δ )) β τ ( ) δ errngng the ove equton to solte δ we hve: 1 ( ) β τδ ( ) τ ( δ ) τ oprng the ove generl equton to the specfc one gven n the prole,. 5 we see tht ( 1 τ ). 5, or, nd thtτ. 4. Therefore: 1 τ ( ).5 ( 1.4).658 ( 6.58% ) Elton, Gruer, Brown, nd Goetznn 14-1 odern Portfolo Theory nd Investent Anlyss, 7th Edton Solutons To Text Proles: hpter 14

2 hpter 14: Prole 4 Snce we re gven nd only one, nd snce >, ths stuton s where there s rskless lendng t nd no rskless orrowng. The effcent fronter wll therefore e ry n expected return-stndrd devton spce tngent to the nu-vrnce curve of rsky ssets nd ntersectng the expected return xs t the rskless rte of 3% plus tht prt of the nu-vrnce curve of rsky ssets to the rght of the tngency pont. Ths s depcted n the grph elow, where the effcent fronter extends long the ry fro to the tngent portfolo L, then to the rght of L long the curve through the rket portfolo nd out towrd nfnty (ssung unlted short sles). Note tht, unless ll nvestors n the econoy choose to lend or nvest solely n portfolo L, the rket portfolo wll lwys e on the nu-vrnce curve to the rght of portfolo L. Snce oth nd re on the nu-vrnce curve, the entre nuvrnce curve of rsky ssets cn e trced out y usng contons (portfolos) of nd. Lettng e the nvestent weght for the rket portfolo, the expected return on ny conton portfolo P of nd s: P ( ) 1 (1) ecognzng tht nd re uncorrelted, the stndrd devton of ny conton portfolo P of nd s: σ σ ( ) σ P 1 () Elton, Gruer, Brown, nd Goetznn 14- odern Portfolo Theory nd Investent Anlyss, 7th Edton Solutons To Text Proles: hpter 14

3 Susttutng the gven vlues for nd nto equton (1) gves: P ( ) (3) Susttutng the gven vlues for σ nd σ nto equton () gves: σ P ( 1 ) (4) Usng equtons (3) nd (4) nd vryng (the frcton nvested n the rket portfolo ) gves vrous coordntes for the nu-vrnce curve; soe of the re gven elow: P σ P The zero-et for of the securty rket lne descres equlru et rsk nd expected return reltonshp for ll securtes nd portfolos (ncludng portfolo L) except those conton portfolos coposed of the rskless sset nd tngent portfolo L long the ry - L n the ove grph: 5 1β ( ) The equlru et rsk nd expected return reltonshp for ny conton portfolo coposed of the rskless sset nd tngent portfolo L long the ry - L n the ove grph s descred y the followng lne: β ( L ) β L β Elton, Gruer, Brown, nd Goetznn 14-3 odern Portfolo Theory nd Investent Anlyss, 7th Edton Solutons To Text Proles: hpter 14

4 onng the two lnes yelds the followng grph: hpter 14: Prole 5 If the post-tx for of the equlru prcng odel holds, then: ( ) ( δ ) τ ) β ( δ )τ If the stndrd AP odel holds, then: ( ) β Assue tht the post-tx odel holds nsted of the stndrd odel, nd δ. or stock wth ( δ ) τ >, the nsttuton tht uses the post-tx odel would correctly eleve tht the stock hs hgher expected return thn the stock s return expected y the nsttuton usng the stndrd odel. Slrly, for stock wth ( δ ) τ <, the nsttuton tht uses the post-tx odel would correctly eleve the stock hs lower expected return thn the stock s return expected y the nsttuton usng the stndrd odel. Elton, Gruer, Brown, nd Goetznn 14-4 odern Portfolo Theory nd Investent Anlyss, 7th Edton Solutons To Text Proles: hpter 14

5 Now consder specfc exple usng the followng dt for stocks A nd B, the rket portfolo nd the rskless sset: β 1. ; δ 8% ; β 1. ; δ % ; 14% ; δ 4% ; 4% ; τ. 5 A A B B If the post-tx odel holds, then the nsttuton usng tht odel would correctly eleve tht the equlru expected returns for the two stocks re: A 4 (( 14 4) ( 4 4).5) 1. ( 8 4) %.5 B 4 (( 14 4) ( 4 4).5) 1. ( 4) %.5 The nsttuton usng the stndrd odel would ncorrectly eleve tht the stocks equlru expected returns re: A B 4 ( 14 4) % 4 ( 14 4) % The nsttuton usng the post-tx odel would tend to uy stock A nd sell stock B short. Of course, resdul rsk puts lt to the ount of unlncng the nsttuton would do. But y soe unlncng, the nsttuton erns n excess return. The nsttuton usng the stndrd odel would e ndfferent etween the two stocks. However, y uyng stock B, the nsttuton loses excess return. Elton, Gruer, Brown, nd Goetznn 14-5 odern Portfolo Theory nd Investent Anlyss, 7th Edton Solutons To Text Proles: hpter 14

6 hpter 14: Prole 6 Usng oss s APT odel, we cn crete n rtrge portfolo s follows: AB 1 (1) () AB AB (3) AB AB Snce the ove portfolo hs zero net nvestent nd zero rsk wth respect to the gven two-fctor odel, y the force of rtrge ts expected return ust lso e zero: AB AB (4) ro theore of lner lger, snce the ove orthogonlty condtons (1), () AB nd (3) wth respect to the result n orthogonlty condton (4) wth respect to AB the, cn e expressed s lner conton of 1, nd : 1 1 (5) We cn crete zero-rsk nvestent portfolo s follows: 1 Susttutng the ove equtons nto equton (5) gves: 1 Elton, Gruer, Brown, nd Goetznn 14-6 odern Portfolo Theory nd Investent Anlyss, 7th Edton Solutons To Text Proles: hpter 14

7 Elton, Gruer, Brown, nd Goetznn 14-7 odern Portfolo Theory nd Investent Anlyss, 7th Edton Solutons To Text Proles: hpter 14 We cn crete strctly rket-rsk nvestent portfolo s follows: 1 1 Susttutng the ove equtons nto equton (5) gves: 1 1 or 1 We cn crete strctly nterest rte-rsk nvestent portfolo s follows: 1 1 Susttutng the ove equtons nto equton (5) gves: 1 or Susttutng the derved vlues for, 1 nd nto equton (5), we hve: ( ) ( )

8 hpter 14: Prole 7 In the grph elow, the effcent fronter wth rskless lendng ut no rskless orrowng s the ry extendng fro to the tngent portfolo L nd then long the nu-vrnce curve through the rket portfolo nd out towrd nfnty (ssung unlted short sles). All nvestors who wsh to lend wll hold tngent portfolo L n soe conton wth the rskless sset, snce no other portfolo offers hgher slope. urtherore, unless ll nvestors lend or nvest solely n portfolo L, the rket portfolo wll e long the nu-vrnce curve to the rght of portfolo L, snce the rket portfolo s welth-weghted verge of ll the effcent rsky-sset portfolos held y nvestors, nd no rtonl nvestor would hold rsky-sset portfolo long the curve to the left of L. The expected return on zero-et sset s the ntercept of lne tngent to the rket portfolo, nd the zero-et portfolo on the nu-vrnce fronter ust e elow the glol nu vrnce portfolo of rsky ssets y the geoetry of the grph. urtherore, y the geoetry of the grph, snce the rskfree lendng rte s the ntercept of the lne tngent to portfolo L, nd snce L s to the left of on the nu-vrnce curve, the rsk-free lendng rte ust e elow the expected return on zero-et sset. Elton, Gruer, Brown, nd Goetznn 14-8 odern Portfolo Theory nd Investent Anlyss, 7th Edton Solutons To Text Proles: hpter 14

9 The zero-et securty rket lne s the lne n the grph elow extend fro the expected return on zero-et sset through the rket portfolo nd out towrd nfnty (ssung unlted short sles). The expected return-et reltonshps of ll rsky securtes rsky-sset portfolos (ncludng the rket portfolo nd portfolo L) re descred y tht lne. The other lne fro the rsk-free lendng rte to portfolo L only descres the expected return-et reltonshps of conton portfolos of the rsk-free sset nd portfolo L; those conton portfolos re not descred y the zero-et securty rket lne. hpter 14: Prole 8 Assue the se stuton s n Prole 5. The nvestor who eleves n the stndrd (pre-tx) AP expects return of 14% on ether securty. You expect return efore txes of 15% on stock A nd 13% on stock B. If your tx fctor ws elow the ggregte tx fctor (τ lower thn.5) then you should uy stock B fro the other nvestor nd sell tht nvestor stock A. The fct tht ths wll led to hgher fter-tx csh flows for you s strghtforwrd. Elton, Gruer, Brown, nd Goetznn 14-9 odern Portfolo Theory nd Investent Anlyss, 7th Edton Solutons To Text Proles: hpter 14

10 hpter 14: Prole 9 Ths prole cn e nswered drectly y usng the equton developed for nonrketle ssets. The equton lso holds for deleted ssets, wth the suscrpt H now stndng for those ssets tht were left out: P cov H PH ( ) P σ cov H P H ( ) cov( ) The effect of levng out onds depends on two fctors: 1.) Whether or not the returns on the ggregte of ll onds re negtvely or postvely correlted wth the returns on the ggregte of ll stocks;.) The correlton etween the returns on prtculr stock nd the returns on the ggregte of ll onds. ro the ove equton, f returns on stocks nd onds re generlly postvely correlted (s eprcl evdence shows), then the denontor n the second ter of the equton wll tend to lower the expected return on ny stock. If the return on prtculr stock s negtvely correlted wth onds, tht wll further lower the stock s expected return. However, f the stock s postvely correlted wth onds, ths wll offset the effect of postve correlton etween ll stocks nd onds nd y ctully result n hgher expected return for the stock. Elton, Gruer, Brown, nd Goetznn 14-1 odern Portfolo Theory nd Investent Anlyss, 7th Edton Solutons To Text Proles: hpter 14