te Finance (4th Edition), July 2017.

Transcription

1 Appndx Chaptr. Tchncal Background Gnral Mathmatcal and Statstcal Background Fndng a bas: 3 2 = 9 3 = 9 1 /2 x a = b x = b 1/a A powr of 1 / 2 s also quvalnt to th squar root opraton. Fndng an xponnt: 3 2 = 9 2 = ln 9 ln 3 a x = b x = ln b nc to Ivo W rat Fnanc ton), J on), July orat ch, Corporat Fnanc ton), July 20 4th Edton), July orat Fna. orat Fnanc ton), July I nanc ton), July orat Fnanc ( uly orat Fnanc ton), July Ivo W rporat Fnanc ton), July orat Fnanc (4th E ln a (Instad of th natural log ln, you could us any othr log, too.) Summaton notaton: N f = f 1 + f f N Ths should b rad as th sum ovr all from 1 to N. Thr ar N trms n ths sum. s not a ral varabl: It s smply a dummy countr to abbrvat th notaton. Whn 1 and N ar omttd, t usually mans ovr all possbl. 621 Wlch, C 7. Ivo c (4th E rporat Fnanc ton), July Ivo Wlch July orat Fnanc (4t t Fnanc ton), July orat F anc ton), Jul

2 622 Appndx Chaptr. Tchncal Background Summaton ruls: N a f + b For xampl, = [a f 1 + b] + [a f 2 + b] + + [a f N + b] N = a f + N b nc to Ivo W rat Fnanc ton), J on), July orat ch, Corporat Fnanc ton), July 20 4th Edton), July orat Fna. orat Fnanc ton), July I nanc ton), July orat Fnanc ( uly orat Fnanc ton), July Ivo W rporat Fnanc ton), July orat Fnanc (4th E Wlch, C = [ ] + [ ] + [ ] = = 166 = = = = 166 Lnar functons: A functon s calld a lnar functon f and only f a + b x = a + b x = a + b x, whr a and b ar constants. Hr s an llustraton. Th (wghtd) avrag s a lnar functon. For xampl, start wth (5, 10, 15) as a data srs. Th avrag s 10. Pck a = 2 and b = 3. For avragng to b a lnar functon, t must b that Avrag Data = Avrag Data Lt s try ths th lft-hand sd (LHS) would bcom th avrag of (17, 32, 47), whch s 32. Th rght-hand sd (RHS) would bcom = 32. It works: Avragng ndd bhavs lk a lnar functon. In contrast, th squar root s not a lnar functon, bcaus Th LHS s 5, th RHS s 7. Lnar functons ar vry mportant n fnancal conomcs: Smlar to avragng, xpctd valus ar lnar functons. Ths s what has prmttd us to ntrchang xpctatons and lnar functons: E a + b X = a + b E X Ths wll b xpoundd n th nxt scton. Th rat of rturn on a portfolo s also a lnar functon of th nvstmnt wghts. For xampl, a portfolo rat of rturn may b r x = 20% r x +80% r y, whr r x s th rat of rturn on th componnt nto whch you nvstd $20. For r x to b a lnar functon, w nd r x = r x Substtut n a + b r x = r a + b x 7. Ivo c (4th E rporat Fnanc ton), July Ivo Wlch July orat Fnanc (4t t Fnanc ton), July orat F anc ton), Jul

3 Gnral Mathmatcal and Statstcal Background (20% r x + 80% r y ) = 20% (2 + 3 r x ) + 80% (2 + 3 r y ) Both sds smplfy to % r x + 240% r y, so our statmnt s tru and a portfolo rturn s ndd a lnar functon. Howvr, not all functons ar lnar. Th varanc s not a lnar functon, bcaus You wll confrm ths n th nxt scton. Var a + b X a + b Var X Q A.1. If (1 + x) 10 = (1 + 50%) = 1.5, what s x? Q A.2. If (1 + 10%) x = (1 + 50%) = 1.5, what s x? N N Q A.3. Ar x and x s th sam? s=1 b Q A.4. In f x, y, what ar th varabls? x=a Q A.5. Wrt out and comput (3 + 5 x). Is x a varabl or just a placholdr to x=1 wrt th xprsson mor convnntly? Q A.6. Wrt out and comput 3 +5 y. Compar th rsult to th prvous nc to Ivo W rat Fnanc ton), J on), July orat ch, Corporat Fnanc ton), July 20 4th Edton), July orat Fna. orat Fnanc ton), July I nanc ton), July orat Fnanc ( uly orat Fnanc ton), July Ivo W rporat Fnanc ton), July orat Fnanc (4th E y=1 y=1 xprsson. Q A.7. Is ( ) th sam as? Wlch, C 7. Ivo c (4th E rporat Fnanc ton), July Ivo Wlch July orat Fnanc (4t t Fnanc ton), July orat F anc ton), Jul

4 624 Appndx Chaptr. Tchncal Background Laws of Probablty, Portfolos, and Expctatons Lt s go ovr th algbra of probablts and portfolos, whch you had to us n th nvstmnts chaptrs. It s prsntd n a mor mathmatcal fashon than t was n th chaptrs, whch you may fnd asr or hardr, dpndng on your background. If you hav a statstcs background, ralz that our book s notaton s smplfd, bcaus w do not plac tlds ovr random varabls. Sngl Random Varabls Th law of xpctatons for sngl random varabls ar as follows: An xpctaton s dfnd as E X = N Prob [X = X()] nc to Ivo W rat Fnanc ton), J on), July orat ch, Corporat Fnanc ton), July 20 4th Edton), July orat Fna. orat Fnanc ton), July I nanc ton), July orat Fnanc ( uly orat Fnanc ton), July Ivo W rporat Fnanc ton), July orat Fnanc (4th E Wlch, C It s bascally a probablty-wghtd avrag. Th xpctd valu of a lnar transformaton (a and b ar known constants): E a X + b = a E X + b (A.1) To s ths, consdr a far con that can b thr 1 or 2. Say a = 4 and b = 10. In ths cas, th LHS s E a X+b = E 4 X+10 = 0.5 (4 1+10)+0.5 (4 2+10) = = 16. Th RHS s 4 ( ) + 10 = 16. Ths all workd bcaus xpctaton s a lnar oprator. (It s a fancy way of sayng that t s a summaton, whch allows you to rgroup th summaton trms of th lnar combnaton a X + b nsd th xpctaton, whch s also a probablty-wghtd lnar combnaton.) A lttl mor gnrally, you could rnam X as f X, so E a f x + b = a E f x + b Howvr, you cannot always pull xpctatons n, so E f x s not always f(e X ). For xampl, f f x = x 2, t s th cas that E X X E X E X To s ths, rconsdr th far 1 or 2 con. Th LHS s E X 2 = 0.5 (1 1) (2 2) = 2.5, but th RHS s [E X ] 2 = ( ) 2 = (1.5 2 ) = Dfnton of varanc: Var X = E X E X 2 It s somtms asr to rwrt ths formula as Var X = E X 2 [E X ] 2. Lt m show you that ths works. For our far 1 or 2 con xampl, th varanc accordng to th man formula s 0.5 (1 1.5) (2 1.5) 2 = For th scond formula, w just computd E X 2 = 2.5 and [E X ] 2 = Subtractng ths trms ylds th sam Dfnton of a standard dvaton: 7. Ivo c (4th E rporat Fnanc ton), July Ivo Wlch July orat Fnanc (4t t Fnanc ton), July orat F anc ton), Jul

5 Laws of Probablty, Portfolos, and Expctatons 625 Sdv X = Var X Th varanc of a lnar combnaton (whr a and b ar known constants): Var a X + b = a 2 Var X (A.2) For our far 1 or 2 con xampl, wth a = 4 and b = 10, th LHS s 0.5 [( ) 16] [( ) 16] 2 = 0.5 [ 2] [2] 2 = 4. Th RHS s = 4. Hr s an xtndd llustraton. A con, whos outcom w call X, has 4 and 8 wrttn on ts two sds. Ths two outcoms can b wrttn as 4, whr s thr 1 or 2. Thrfor, th xpctd valu of X s E X = Prob X = (4 ) (4 ) = Prob X = 4 (4) + Prob X = 8 (8) = 50% % 8 = 6 Var X = Prob X = (4 ) [(4 ) 6] 2 = Prob X = 4 (4 6) 2 + Prob X = 8 (8 6) 2 = 50% % 4 = 4 Th standard dvaton s th squar root of th varanc, hr 2. E X 2 s, of cours, not th sam as [E X ] 2 = [3] 2 = 9, bcaus nc to Ivo W rat Fnanc ton), J on), July orat ch, Corporat Fnanc ton), July 20 4th Edton), July orat Fna. orat Fnanc ton), July I nanc ton), July orat Fnanc ( uly orat Fnanc ton), July Ivo W rporat Fnanc ton), July orat Fnanc (4th E E X 2 = Prob X = (2 ) (2 ) 2 = Prob X = 2 (2 2 ) + Prob X = 4 (4 2 ) = 50% % 16 = 10 Now work wth a lnar transformaton of th X, say, Z = $2.5 X + $10. Ths s a fundamntal opraton n fnanc, bcaus th rats of rturn on portfolos ar such lnar transformatons. For xampl, f you own 25% n A and 75% n B, you wll arn 0.25 r A r B. Thus, Prob Con X Z 1/ 2 Hads 4 $20 1/ 2 Tals 8 $30 You want to convnc yourslf that th xpctd valu of Z, dfnd as $2.5 X + $10, s $2.5 E X + $10 = $25. Frst, comput by hand th xpctd valu th long way from Z, Wlch, C 7. Ivo c (4th E rporat Fnanc ton), July Ivo Wlch July orat Fnanc (4t t Fnanc ton), July orat F anc ton), Jul

6 626 Appndx Chaptr. Tchncal Background E Z = Prob X = (4 ),.., sam as Z = $2.5 X + $10 (Z ) = Prob X = 4,.., sam as Z = $20 ($20) + Prob X = 8,.., sam as Z = $30 ($30) = 50% $ % $30 = $25 Unlk th man (th xpctd valu), th varanc s not a lnar functon. Th varanc of Z = $2.5 X + $10 s not $2.5 Var X + $10 = $ $10 = $20. Instad, Var Z = Var a X + b = a 2 Var X = ($2.5) 2 Var X = $$ = $$25. You can confrm ths workng wth Z drctly: Var Z = Prob X = (4 ) (Z ) E Z 2 = Prob X = 4,.., sam as Z = $20 ($20 $25) 2 + Prob X = 8,.., sam as Z = $30 ($30 $25) 2 = 50% ($5) % ($5) 2 = $$25 Th standard dvaton of Z s thrfor $$25 = $5. Lt us quckly confrm Formula A.1 for Z = $2.5 X + $10: $25 = E $2.5 X + $10 = $2.5 E X + $10 = $ $10 = $25 E Z = E a X + b = a E X + b nc to Ivo W rat Fnanc ton), J on), July orat ch, Corporat Fnanc ton), July 20 4th Edton), July orat Fna. orat Fnanc ton), July I nanc ton), July orat Fnanc ( uly orat Fnanc ton), July Ivo W rporat Fnanc ton), July orat Fnanc (4th E Lt us also quckly confrm Formula A.2: $$25 = Var $2.5 X + $10 = $2.5 2 Var X = $$ = $$25 Var Z = Var a X + b = a 2 Var X Q A.8. What s th xpctd valu and standard dvaton of a bt B that pays off th numbr of ponts on a far d, squard? For xampl, f th d lands on 3, you rcv $9. Q A.9. Assum that you hav to pay $30, but you rcv twc th outcom of th bt B from Quston A.8. Ths s a nw bt, calld C. That s, your payoff s C = $ B. What s th xpctd payoff and rsk of your poston? (Suggston: Mak your lf asy by workng wth your answrs from Quston A.8.) Wlch, C 7. Ivo c (4th E rporat Fnanc ton), July Ivo Wlch July orat Fnanc (4t t Fnanc ton), July orat F anc ton), Jul

7 Laws of Probablty, Portfolos, and Expctatons 627 Portfolos A portfolo s a st of wghts n possbl nvstmnt assts. That s, for a st of assts, ach known nvstmnt s usually dnotd as w. Th rat of rturn on a portfolo s r P = w r whr r s th scurty rturn on scurty. Portfolo rturns ar th wghtd sum of multpl random varabls. Portfolo rturn xpctatons: E w r = w E r Although th wghts ar fxd and known constants, thy cannot b pulld out of th summaton, bcaus thy ar ndxd by (ach could b dffrnt from th othrs). Portfolo rturn rsknss: Var w r = = N N w w j Cov r, r j j=1 N N w w j Cov r, r j j=1 Of cours, for ntuton, on would oftn comput th standard dvaton by takng th squar-root of th varanc. Hr s an llustraton. A con toss outcom s a random varabl, T, and t wll rturn thr $2 (hads) or $4 (tals). You hav to pay $2 to rcv ths bt. Ths looks lk a good bt: Th man rat of rturn on ach con toss, E r T, s 50%. Th varanc on ach con toss s nc to Ivo W rat Fnanc ton), J on), July orat ch, Corporat Fnanc ton), July 20 4th Edton), July orat Fna. orat Fnanc ton), July I nanc ton), July orat Fnanc ( uly orat Fnanc ton), July Ivo W rporat Fnanc ton), July orat Fnanc (4th E Var r T = 1/ 2 (0% 50%) / 2 (100% 50%) 2 = 2, 500%% = 0.25 Thrfor, th standard dvaton of ach con toss s 2, 500%% = 50%. Now, bt on two ndpndnt such con toss outcoms. Say you nvst $10 on th frst bt (w 1 = $10) and $20 on th scond bt (w 2 = $20). Your portfolo s {w 1, w 2 } = {$10, $20}. You can also comput your portfolo s nvstmnt wghts nstad of ts absolut nvstmnts. w 1 = $10 $ and w 2 = (1 w 1 ) = $20 $ Your ovrall portfolo rat of rturn s r = w r W can now us th formulas to comput your xpctd rat of rturn (E r ) and rsk (Sdv r ). To comput your xpctd rat of rturn, us Wlch, C 7. Ivo c (4th E rporat Fnanc ton), July Ivo Wlch July orat Fnanc (4t t Fnanc ton), July orat F anc ton), Jul

8 628 Appndx Chaptr. Tchncal Background E r = w E r = w 1 E r 1 + w2 E r 2 = 1 / 3 (50%) + 2 / 3 (50%) = 50% (Rcall that an xpctaton s a lnar oprator, that s, a summaton. A portfolo s a summaton, too. Bcaus both ar ultmatly nothng but summatons, you can rgroup trms, whch mans that th abov formula works.) To comput your varanc, us Var r = nc to Ivo W rat Fnanc ton), J on), July orat ch, Corporat Fnanc ton), July 20 4th Edton), July orat Fna. orat Fnanc ton), July I nanc ton), July orat Fnanc ( uly orat Fnanc ton), July Ivo W rporat Fnanc ton), July orat Fnanc (4th E Wlch, C j=1 w w j Cov r, r j = w 1 w 1 Cov r 1, r 1 + w1 w 2 Cov r 1, r 2 + w2 w 1 Cov r 2, r 1 + w2 w 2 C = w 2 1 Cov r 1, r w1 w 2 Cov r 1, r 2 + w 2 2 Cov r 2, r 2 = w 2 1 Var r w1 w 2 Cov r 1, r 2 + w 2 2 Var r 2 = ( 1 / 3 ) 2 Var r w1 w ( 2 / 3 ) 2 Var r 2 = ( 1 / 9 ) Var r 1 + ( 4/ 9 ) Var r 2 = ( 1 / 9 ) ( 4 / 9 ) Th standard dvaton s thrfor %. Ths s lowr than th 50% that a sngl con toss would provd you wth. Q A.10. Rpat th xampl, but assum that you nvst $15 nto ach con toss rathr than $10 and $20, rspctvly. Would you xpct th rsk to b hghr or lowr? (Hnt: What happns f you choos a portfolo that nvsts mor and mor nto just on of th two bts?) Cumulatv Normal Dstrbuton Tabl Exhbt 1 allows you to dtrmn th probablty that an outcom X wll b lss than a prspcfd valu x, whn standardzd nto th scor z, f X (and thus z follow a normal dstrbuton). For xampl, f th man s 15 and th standard dvaton s 5, an outcom of X = 10 s 1 standard dvaton blow th man. Ths standardzd scor can b obtand by computng z(x) = [x E x ]/Sdv x = (x 15)/5 = (10 15)/5 = ( 1). Ths tabl thn ndcats that th probablty that th outcom of X (.., drawn from ths dstrbuton wth man 15 and standard dvaton 5) wll b lss than 10 (.., lss than ts scor of z = 1) s 15.87%. Exhbt 2 shows what th tabl rprsnts. Exhbt 2(a) shows th classcal bll curv. Rcall that at z = 1, th tabl gvs (z = 1) = 15.87%. Ths 15.87% s th shadd ara undr th curv up to and ncludng z = 1. Exhbt 2(b) just plots th valus n th tabl tslf, that s, th ara undr th graph to th lft of ach valu from Exhbt 2(a). If you vr nd to approxmat th cumulatv normal dstrbuton n a spradsht, you can us th bult-n functon normsdst. 7. Ivo c (4th E rporat Fnanc ton), July Ivo Wlch July orat Fnanc (4t t Fnanc ton), July orat F anc ton), Jul

9 Cumulatv Normal Dstrbuton Tabl 629 z z z z z z z z z z z z Exhbt 1: Cumulatv Normal Dstrbuton Tabl. Normal scor (z) vrsus standardzd normal cumulatv dstrbuton probablty z nc to Ivo W rat Fnanc ton), J on), July orat ch, Corporat Fnanc ton), July 20 4th Edton), July orat Fna. orat Fnanc ton), July I nanc ton), July orat Fnanc ( uly orat Fnanc ton), July Ivo W rporat Fnanc ton), July orat Fnanc (4th E n(z) Ara =15.87% Probablty z Cumulatv Probablty Wlch, C N(z) Exhbt 2: Th Normal Dstrbuton. N( 1)=15.87% z 7. Ivo c (4th E rporat Fnanc ton), July Ivo Wlch July orat Fnanc (4t t Fnanc ton), July orat F anc ton), Jul

10 630 Appndx Chaptr. Tchncal Background Law of xpctaton, 624. Normal dstrbuton, 628. Q A.1 x 4.138%. Chck: ( %) Q A.2 x Chck: Q A.3 Ys! and s ar not varabls, but notaton! Q A.4 x s not a varabl, but smply a notaton shortcut. Wrttn out, th xprsson s f(a, y) + f(a + 1, y) f(b 1, y) + f(b, y), whch maks t clar that a, y, and b ar th varabls. Q A.5 Th xprsson s (3 + 5 x) = ( ) + ( ) + ( ) x=1 = = 39 x s not a varabl. It s smply a countr dummy usd for wrtng convnnc. It s not a part of th xprsson tslf. Q A.6 Th xprsson s y = ( ) + 5 ( ) = 39 y=1 y=1 Th rsult s th sam. Ths s an xampl of why a + b x = Kywords Answrs nc to Ivo W rat Fnanc ton), J on), July orat ch, Corporat Fnanc ton), July 20 4th Edton), July orat Fna. orat Fnanc ton), July I nanc ton), July orat Fnanc ( uly orat Fnanc ton), July Ivo W rporat Fnanc ton), July orat Fnanc (4th E a + b x. Q A.7 No. Th two xprssons ar ( ) = = 14 = ( ) ( ) = 36 Th two ar not th sam! Thus, b carful not to try to pull out multplyng s! You can only pull out constants, not countrs. Incdntally, ths s also why E X 2 E X 2, as statd n th nxt scton. Q A.8 Th xpctd valu s Th varanc s E B = ( 1 / 6 ) $1 + ( 1 / 6 ) $4 + ( 1 / 6 ) $9 + ( 1 / 6 ) $16 + ( 1 / 6 ) $25 + ( 1 / 6 ) $36 $15.17 Var B = ( 1 / 6 ) ($1 $15.17) 2 + ( 1 / 6 ) ($4 $15.17) 2 + ( 1 / 6 ) ($9 $15.17) 2 + ( 1 / 6 ) ($16 $15.17) 2 + ( 1 / 6 ) ($25 $15.17) 2 + ( 1 / 6 ) ($36 $15.17) 2 $$ Th standard dvaton s thrfor Sdv B = Var B $$ $12.21 Q A.9 You xpct to rcv E C = $ E B $ $15.17 $0.34 Var C = 2 2 Var B 4 $$ = $$ Sdv C = Var C $24.42 Q A.10 Your nvstmnt wghts ar now w 1 = w 2 = 0.5. Th man rat of rturn rmans th sam 50%. Th varanc of th rat of rturn s computd smlarly to th xampl n th txt: Var r = ( 1 / 2 ) ( 1 / 2 ) = Thrfor, th rsk (standard dvaton) s 35.35%. Ths s lowr than t was whn you put mor wght on on of th con tosss. Ths maks sns: As you put mor and mor nto on of th two con tosss, you los th bnft of dvrsfcaton! Wlch, C 7. Ivo c (4th E rporat Fnanc ton), July Ivo Wlch July orat Fnanc (4t t Fnanc ton), July orat F anc ton), Jul