4.3 DJIA. From French Franc to Euro (7-2)

Transcription

1 4.3 DJIA The Dow Jones ndex frst ht 1,000 onts on November , 80 years after ts launchng. But t took only 14 years for the ndex to reach,000 onts (January, 8, Has the market grown too fast over the second erod? Is ths attern ndcatve of a bubble n rces? Soluton In the frst sub-erod, t takes 80 years for the ndex to go from 1 to 1,000. It the second sub-erod, t takes (14+/1 years to go from 1,000 to,000. In fact, the startng values are not the same. Use a sem-log grah to analyze the value of the sloe. Ln(1000/80->8.63% and Ln(/14,17->4.89%. The ncrease s smaller over the nd sub-erod. 1 Lex Dao sute < < > Menu From French Franc to Euro (7- As of January 3, 1999, rces for stocks traded on the Pars Bourse are exressed n euros. Pror to ths date, rces were exressed n French Francs (FF. Devse a soluton that allows to make re- and ost- January 3, 1999 rces homogeneous. All data are converted n euros On January 3, 1999, a common rce factor of 0.15 (1/ ales to each stock. Data are made homogeneous by multlyng all rces and dvdends ror to ths date by 0.15 (convertng FF to Lex Dao sute < < > Menu OST 1

2 Prce aths: arthmetc and geometrc mean (7-4 The exected return on stock s equal to 10%. On every1- year erod, the rce ether goes u by 30% or down by 10% wth ½ and ½ robablty. 1. Plot the trajectores over successve years. Comute the average return usng arthmetc mean 3. Comute the average return usng geometrc mean 4. What s the exected return? 5. What s your nterretaton for the results you obtan? 3 Lex Dao sute < < > Menu 1. Prce aths Prce aths: arthmetc and geometrc mean (soluton 1 to 4 1,3 1,69 Fnal wealth 1,69 1,69 ½ -130% 1,17 1,17 1,17 ½ ,17% 0,81 0,81 0,81 ½ -1-0,9 10% & 3. Returns are equal to +30% and 10%. Arthmetc ¼ x 1,69 0,45 mean s equal to +10% and geometrc mean s equal to +8,17% ½ x 1,117 0, (1+8.17%^ 1.17 whereas (1+10%^1.1 ¼ x 0,81 0,05 5. Geometrc mean gves the mode and the medan of Sum 1,100 exected returns whereas the arthmetc mean accounts for all ossble aths. E(R (1,1-1/ Lex Dao sute < < >

3 Funds (7-6 Fund A Fund B 1 16% 30% 10% -10% 3 14% 8% 4 % 17% 5 4% -% Ths table reorts the annual returns of two funds over the ast 5 years. 1. Comute the annualzed rate of return over the 5 years. What s the best estmate of next year's return? 3. Dscuss the robustness of ths estmate (assumtons, lmtatons. 5 Lex Dao sute < < > Funds (7-6, soluton Are these returns dscrete or contnuous returns? We assume hereafter that the table reorts dscrete returns Assumng the ntal value of the funds s 100 (wth no loss of generalty, Fund A end-of-erod value s equal to and Fund B end-of-erod value s The geometrc mean of returns s found to be 9.06% for Fund A and 11.4% for Fund B. The resectve arthmetc means are found to be 9.% and 1.6%. The geometrc mean gves the correct answer..geometrc means 3.Does hstory reeat tself? In addton the length of the estmaton erod s very short. 6 Lex Dao sute < < > 3

4 10% loan (7-7 Suose you borrow 10,000 wth an annual nterest rate of 10%. How much wll you have to reay n 1 year? Soluton Assumng 10% s a contnuous return and nterests are contnuously comounded, the amount that has to be read n 1 year s equal to: e 11051,7 And the dscrete nterest rate s thus equal to %. Assumng there s no comoundng of nterets (let's have a dream!, the amount that has to be read s equal to (but bankers are not that stud!. 7 Lex Dao sute < < > Menu Comounded returns (7-7 (Perre Mnut and the urchase of the Manhattan Island In Setember 166, Perre Mnut, the governor of the West Inda Comany, bought the 31-squared mle Island of Manhattan from the local Indans for 60 gulders (about $4 (Joron et Goetzman, What are the $4 worth today assumng the followng annual comound rates: 3%, 5%, 6%, and 7%, Dscuss the results 8 Lex Dao sute < < > Menu 4

5 P Manhattan (soluton P P R n, t e c R c 377,00,00 e Rc 377 Ln P, t ( ( Ln P,00 Ln 4 + Rc 377 Rc 377 P, 00 4 e 0% 4 1% % % % % % % % % Ths s an nfernal machne but not necessarly the deal of the century because these numbers rely on the assumton that one can nvest at a 6% nterest rate over 377 years and that the nterest themselves can be re-nvested at a 6% nterest rate. 9 Lex Dao sute < < > Menu The contrbuton of stock returns to ortfolo return (7-8 Portfolo contans n stocks. x denotes the ercentage of the ortfolo that s nvested n stock (Σx 1. E [ R ] n 1 Queston : What s the contrbuton of stock to ortfolo exected return? x R 10 Lex Dao sute < < > Menu 5

6 E Contrbuton of a stock to the exected return of a ortfolo [ R ] n 1 x R The contrbuton of stock s comuted by dfferentatng the above exreson wth resect to x [ ] E R x R Not surrsngly, ths s equal to stock exected return (detaled roof 11 Lex Dao sute < < > Contrbuton of a stock to the exected return of a ortfolo (detaled roof n x jr j j 1 x { x R + x R + L+ x R + L+ x R } 1 x1r x 1 1 xr + x x xr + L+ x n xnr + x n n R 1 Lex Dao sute < < > 6

7 Oxygène (7-10 On June 1, 198, stockholders from Oxygène et Acétylène de l Extrême Orent were awarded 1 free share (1/5 rato. The rce factor for ths oeraton s.833. The comany then made a catal ncrease n cash wth rghts (1/9 rato wth an ssung rce set to FRF00. The subscrton erod started on November 15, 198. The assocated rce factor s.911. The newly-ssued stocks were assmlated on January 0, On June 1, 198, stockholders were ad a FRF60 net dvdend. The table below reorts the rces as well as other nformaton over the erod March 198 March Comuted the adjusted rces and the adjusted dvdend. Comuted the adjusted return over the erod March 198 March Draw a grah of raw and adjusted rces. 4. Comute adjusted and suer-adjusted (.e. accountng for dvdends ayment rces 13 Lex Dao sute < < > Menu OST Oxygène, rces, 7-10 Date Coeffcent Cours coté Dvdende 31 mars , avrl , ma , jun , jun 198 0,833 1, jun , jullet , novembre , novembre 198 0,911 1, janver , mars , Lex Dao sute < < > Menu OST 7

8 R 7-10 Oxygène, soluton Date Coeffcent Cours Dvdende rodut Coté corrgé net corrgé Π[1+(D/P] P* 31 mars 8 1 1, ,00 0 0,00 1, ,00 30 avr 8 1 1, ,00 0 0,00 1, ,00 8 ma 8 1 1, ,00 0 0,00 1, ,00 18 jun 8 1 1, ,00 0 0,00 1, ,00 1 jun 8 0,833 0, , ,03 1, ,68 30 jun 8 1 0, ,73 0 0,00 1, ,47 30 jul 8 1 0, ,54 0 0,00 1, ,8 14 nov 8 1 0, ,69 0 0,00 1, ,6 15 nov 8 0,911 0, ,08 0 0,00 1, ,45 0 janv , ,38 0 0,00 1, ,96 30 mars , ,81 0 0,00 1, ,34 1. Adjusted rces and dvdends. Yearly rate of return wth the tyes of adjustments (left ustream / rght dowstream (60 0,911 (1508 0,911 0, ,911 0, ,05% R ,911 0,833 0, Lex Dao sute < < > Menu OST 76,05% Ustream or dowstream adjustement 7-10 Adjusted rce n blue (left dowstream / rght ustream. Same attern but the values on the y axs are dfferent. If ustream adjustement s used, recent rce values are left unchanged and are thus equal to current market rces The rate of return s nsenstve to the way the adjustment s erformed. 16 Lex Dao sute < < > Menu OST 8

9 Varance of returns and varance of rces (8-3 How can you deduce the varance of returns from the varance of rces? 17 Lex Dao sute < < > Menu From varance of rces to varance of returns (Soluton 8-3 ( R% E ( R ( E R% ~ ( R P P E 0 E P 0 ~ ( P [ ] ~ ( R 1 E ( P E( P ~ P 0 P 0 P 0 ( R% E ~ ( R E ~ ( P ( % P 0 P P E P P0 + P 0 E P0 0 ~ ( R 1 ~ ( P P 0 18 Lex Dao sute < < > 9

10 Prce aths and volatlty (8-4 The exected return on a stock s equal to 10%. Every year, the rce ether goes u by 30% or down by 10% wth ½ / ½ robablty. See questons 1 to 5 (returns seres 6. Comute volatlty 19 Lex Dao sute < < > Menu Prce aths and volatlty (soluton 8-4 See questons 1 to 5 (returns seres 6. The varance s equal to 1 ( R ( 30% 10% + ( 10% 10% 4% ( R 4 % 0% 1 And volatlty s equal to: 0 Lex Dao sute < < > 10

11 Varance and squared returns 8-9 Is t ossble to estmate volatlty usng squared returns (more recsely the exectaton of squared returns? 1.Exress the varance usng the exectaton of squared returns. What are the assumtons for the aroxmaton to be accurate? 3.Accordng to you, what s the mact of the tme-nterval over whch returns are comuted? 1 Lex Dao sute < < > Menu Varance and squared returns (1 & 1. The exresson for the varance s: [( ] ( [ ( ] ( ( ~ ~ ~ ~ X E X E X E X E X 1 n [ ] ( R R, t ( R n 1 t 1 n 1 n. When mean s close to 0, the varance can be exressed usng squared returns Lex Dao sute < < > > sute 11

12 Varance and squared returns (3 US stocks 1973_1994 Tme horzon N Mean Mean / 1 year 1, ,100 15,40 0,708 1 quarter 0,500,775 7,70 0, month 0,8333 0,95 4,45 0,081 1 week 0,019 0,13,13 0, day 0,0040 0,044 0,97 0, hour 0,0005 0,006 0,34 0,0161 Joron, 1997, 83, gve detals about the results he obtans usng ortfolos of NYSElsted stocks 5 tradng days n a year and 8 hours n a tradng day From 1973 to 1994, one can comute annual returns (N 1 year. The eaverage return s 11.1% and the standard devaton s foud to be 15.40%. It s ossble to comute *4 quarterly retunrs (N1/4. The average return s.775% wth corresondng std devaton 7.70% The rato of the mean over the standard devaton decreases wth the tme nterval 3 Lex Dao sute < < > < Correlaton and deendence (10-4 Let X and Y be two random varables such that YX². 1 Assume X takes on the followng values: -11%, - 9.5%, -3%, 3%, 9.5% et 11%; Comute the correlaton coeffcent between X and Y. Assume the correlaton coeffcent between two varables s 0. Are the two varables ndeendent? 4 Lex Dao sute < < > Menu 1

13 Correlaton and deendence (soluton 10-4 X Y XY -11% 0,011-0, ,50% 0, , % 0,0009-0, % 0,0009 0, ,50% 0, , % 0,011 0, E(X 0% E(Y 0,00734 E(XY -3,614E-0 Covarance -1,9E-0 1,4% 1,% 1,0% y x + 3E-16x R 1 E(X0 et E(XY-3,610-0 YX² 0,8% 0,6% 0,4% 0,% y -3E-18x + 0,0073 R E-33 0,0% -15% -10% -5% 0% 5% 10% 15% The correlaton coeffcent s found to be zero although the varables are tghtly lnked Ths comes from the fact that the (lnear correlaton coeffcent only catures lnear deendences 5 Lex Dao sute < < > Margnal rsk (10-8 Let be a ortfolo that comrses n rsky assets, ncludng one asset denoted k r~k k k, ortfolo varance of returns stock k random return stock k varance of returns covarance between the returns on stock k and ortfolo Assume we ncrease the roorton of stock k n ortfolo by nvestng m euros n stock k er euro nvested n ortfolo. The nvestment s fnanced at the rsk-free rate (denoted r f. 1. What s the exresson for the return of the new ortfolo (denoted '.. What s the exresson of the varance of the new ortfolo? 3. Based on revous result, what s the ncdence of a margnal change n the amount nvested n stock k (e when m s small Source: Grnblatt and Ttman (1997, 13 6 Lex Dao sute < < > Menu 13

14 Margnal rsk (soluton 10-8 The exresson of returns on ortfolo ' s found to be: r% ' r% + m ( r% k rf The exresson for the varance of ortfolo s: + m + m ' k, k When m s small, ths yelds : ' k, m The contrbuton to the varance of the ortfolo s catured by the dfferental: ( m + ' m k k, Varaton n rsk s commensurate wth the covarance between stock k and ortfolo. 7 Lex Dao sute < < > Margnal rsk (detal 10-8 ( r ' ( r% + m ( r% k rf ( r ' ( r% + m r% k m rf ( r ' ( r + m rk % % Snce Rf s a constant The varance of the sum of random varables wth a0, bm, see relatonsh (6. ( (% ( % r r + m r + m % % ' k r ; rk Proertes of varance oerator, roof #. 16 wth am ( (% (% r r + m r + m % % ' k r ; r k Smlfyng yelds r % ; r % k becomes k, + m + m ' k, k 8 Lex sute < < > 14

15 Mnmum Varance Portfolo (MVP, The data are gven n the followng tables 1. Comute the exectaton and the varance of the followng ortfolos : Portfolo A (% B(% Probab lty Stock A Stock B 0. 18% 0% 0. 5% -3% 0. 1% 15% 0. 4% 1% 0. 6% 1%. Fnd the weghts that yeld the MVP 3. Comute the covarance between ortfolos 3 and 5 4. Comute the covarance between ortfolo 3 and MVP 5. Comute the covarance between ortfolo 5 and MVP 6. What s the ntuton for the results n questons 4 and 5 7. Comare results between questons, 4 and 5 9 Lex Dao sute < < > Menu MVP (1., soluton ProbabTtre A Ttre B.A.B (A-E(A² (B-E(B² (A-E(A(B-E 0. 18% 0% % -3% % 15% % 1% % 1% Characterstcs of the stocks E( R Var( R ( R Cov r(a,b Average and varance of returns : n a 1-a E( VAR( SD( 1 1,5-0,3 0,1 0, , ,09 0,008 0, ,75 0,5 0,08 0, , ,5 0,5 0,07 0, , ,5 0,75 0,06 0, , ,05 0, , ,5 1,5 0,04 0, , Lex Dao sute < < > 15

16 . Comoston of the MVP MVP (., soluton ~ ~ ~ ~ ~ ~ ~ ( R a ( X + ( 1 a ( Y + a ( 1 a ( X ; Y ( X ( Y ρ ~ ( R a ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ( ( ( ( ; ( ( ( ; ( ( a X Y + a Y + ρ X Y X Y 4aρ X Y X Y 0 ~ ~ ~ ~ ~ ( Y ρ( X ; Y ( X ( Y ~ ~ ~ ~ ~ ~ ( X + ( Y ( X ; Y ( X ( Y a ρ Numercal results : Excel : a 0, Mathematca : 31 Lex Dao sute < < > MVP (3. to 7., soluton Covarance between 3 and 5 0, 008 0, , 5 Cov ( P3 ; P5 0,75 0,5 0, , ,75 4. Covarance between 3 and MVP 0, 008 0, , 6404 Cov ( P3 ; P PVM 0,75 0,5 0, , , Covarance between 5 and MVP 0, 008 0, ,6404 Cov ( P5 ; P PVM 0,5 0,75 0, , ,3596 COV(3, COV(3,PVM COV(5,PVM ²(PVMCOV(PVM,PVM Lex Dao sute < < > 16

17 Volatlty after a merger At the begnnng of 1996, Chemcal Bank merged wth Chase Manhattan and Westnghouse Electrc acqured CBS. The followng table gves the 1 returns for the 4 frms durng 1994 CBS Chase Chemcal BanWestnghouse Jan Feb Mar Ar May June July Aug Set Oct Nov Dec Cat $3,387.0 $6, $8, $4, Grnblatt, Ttman (1997, Market values are reorted on the last lne (n mllon dollars Predct the varances of Chase and Westnghouse after ther merger s consummated? 33 Lex Dao sute < < > Menu Volatlty after a merger CBS Chase Chemcal BanWestnghouse Jan Feb Mar Ar May June July Aug Set Oct Nov Dec Cat $ $ $ $ Chemcal Bank wth Chase CBS wth Westnghouse electrc 1. Comute stock varances. Comute covarances 3. Comute weghts after merger 4. Comute varance after merger 34 Lex Dao sute < < > 17