Investment Science Chapter 3

Transcription

1 Ivestmet Scece Chapte 3 D. James. Tztzous <[email protected]> 3. se P wth 7/.58%, P $5,, a 7 84, to obta $ Obseve that sce the et peset value of X s P, the cash flow steam ave at by cyclg X s equvalet to oe obtae by ecevg paymet of P evey peos sce k,...,. Let /. The P P k. k Solvg explctly fo the geometc sees, we have that P Deotg the aual woth by, we must have P. P, so that solvg fo P as a fucto of P a substtutg the esult to the equato fo, we ave at P.

2 Ivestmet Scece Chapte 4 Solutos to Suggeste Poblems D. James. Tztzous <[email protected]> 4. Oe fowa ate f, s s % 4. Spot pate se. Hece, fo example,. ll values ae [ sk k ] /k f,k s [ ].6 6 /5 f,k 6.3%.5 f, f,3 f,4 f,5 f,

3 4.3 Costucto of a zeo se a combato of the two bos: let x be the umbe of 9% bos, a y teh umbe of 7% bos. Select x a y to satsfy 9x 7y, x y. The fst equato makes the et coupo zeo. The seco makes the face value equal to. These equatos gve x 3.5, a y 4.5, espectvely. The pce s P Istataeous ates a e st t e st t e ft,t t t f t,t st t st t t t b t lm t t sttst t tt c We have [stt] t st s t l xt tt, stt s tt, [stt]. Hece, l xt l x stt, a fally that xt xe stt. Ths s ageemet wth the vaace popety of expectato yamcs. Ivestg cotuously gve the same esult as vestg a bo that matues at tme t. 4.6 Dscout coveso

4 The scout factos ae fou by successve multplcato. Fo example,,,, The complete set s.95,.893,.77,.77, o taxes Let t be the tax ate, x be the umbe of bo puchase, c be the coupo of bo, p be the pce of bo. To ceate a zeo coupo bo, we eque, fst, that the afte tax coupos match. Hece whch euces to x tc x tc, x c x c. Next, we eque that the afte tax fal cash flows match. Hece p x p x p. sg ths last elato the equatofo fal cash flow, we f Combg these equatos, we f that fte pluggg the gve values, we f that x x. p c p c p c c. p Real zeos We assume that wth coupo bos thee s a captal gas tax at matuty. We eplcate the zeocoupo bo s afte-tax cash flows usg bos a. Let x be the amout of bo eque 3

5 a,,,3,4,5, NP V b Yea Cash Flow 4 Dscout PV Pue uato P λ P λ λ P λ P λ λ x k s k /m k k x k s k /meλ/m k, k k x k s k m /meλ/m k s k /me λ/m, k k x k s k /m k, m k k x k k m sk /m k k x k s k /m k D. Ths D exactly coespos to the ogal efto of uato as a cash flow weghte aveage of the tmes of cash paymets. No mofcato facto s eee eve though we ae wokg scete tme. 4.4 otgage vso a P k k k k, 6 k,

6 Soluto of HW3 Poblem 6. X outlay ths case t s equal to the epost. X amout eceve, equal to the etue epost plus the poft fom shotg. Thus, X X X X The total etu, R X/X X X X/X Thus, R X X X Poblem 6. Let a a b be the outcomes of two e olls. The Zab. y epeece, we kow: [ ] [ ] [ ] ab a b 6 va[ Z] ab [ ab] a b [ a] [ b] a b [ a] [ b] Poblem 6.3 sg the aswe of the Poblem 6.4: a α 9 / 3 b α α 9 / / % c α α.4 Poblem 6.4 Let α equal the pecet of vestmet stock, the the pecet stock s -α. Now, poblem becomes mze

7 , 4 / 4 / * * α α α α α α α α α α α α α α α α ω ω ω ω ω ω j j j : take evatves espect to α Thus pecetage of asset -α The mea ate of etu of ths potfolo s * * α α ]/ [ ] / [ ] / [ Poblem 6.5 oey veste fo cocet yea fom ow mllo Reveue expecte uless t as 3 mllo Chaces of a5% Ra Isuace $.5 pe ut oey obtae fom suace f t as $ pe ut Let o. of uts of suace bought u a.now total moey veste, X moey veste o. of uts of suace bought *^6.5*u

8 Reveue obtae X 3*^6 *u Now sce thee ae 5% chaces of a we have total eveue:- X 3*^ 6u *.5.5*^6.5u Total Retu X/X.5*3,,.5*u/,,.5u Now ate of etu ate of etu X/X-.5*^6.5u - *^6.5*u/ *^6.5*u 5,/,,.5*u b. To get the mmum vaace we wll have to buy all the 3 mllo uts whch wll gve us a vaace of. We have to mze Va Hece u 3*^ 6 uts. Va Theefoe ^6.5-/ *^6.5*3*^6.5/.5. % Poblem 6.6 The effcet set at mmum-vaace pot assets

9 Sce assets ae ucoelate, The, µ µ µ ω µ ω ω µ µ ω ω ω µ ω ω L L Va Theefoe, Va 4 4 ω ω Va Poblem 6.7 Covaace matx V xpecte ates of etu

10 a mum vaace potfolo cose that w w 3 by symmety akowtz fomulato: jw j w w w w 3 j λ µ. Fo solvg the system of equatos. we omt the last omalzg costat. The,,, w, w, w that satsfes we omalze the obtae soluto { v v v } to the soluto { } the costat. System wth 6 vaables a 6 equatos: 3 w w w 3 λ µ y usg calculato: w w w 3 λ µ b λ, µ System wth 3 vaables a 3 equatos: v v v y usg calculato: v v v 3 w w w

11 c Gve that thee s a sky-fee pat, thee s a sgle fu of sky assets such that ay effcet potfolo ca be costucte as a combato of the fu a the sk-fee stumet. y usg equato 6., we get a system of 3 equatos wth 3 vaables. whee v k f k. f, k,.., v v v v v v 3 w w w Poblem 6.8 a Va α ST.. α j j j L α α α µ α L α j j µ α j L α µ fo each So, j µ α j j α fo each

12 b Va α ST.. α α ~ ~ j j j L α α α µ α λ α L α j j µ λ α j L α µ ~ L α λ fo each So, j µ α j j λ α α ~ fo each Poblem 6.9 eetg Wheel Cose a geeal bettg wheel wth segmets. The payoff fo a $ bet o a segmet s. Suppose you bet a amout / o segmet fo each. Show that the amout you w s epeet of the outcome of the wheel. What s the sk-fee ate of etu fo the wheel? pply ths to the wheel xample 6.7 Sol:

13 Data: segmets fom segmet to segmet mout eceve fom segmet fo each $ veste Facto of segmet veste / W a xpecte etu fo segmet, R / $ mout to w f the outcome equal to segmet : xpecte etu fo segmet tmes the facto veste segmet R * W R * * $ Fo ay outcome o the wheel the amout we wll w wll be $. Rsk-fee ate of etu fo the wheel: Fo ay outcome of the wheel, the amout that we wll w wll be $., X $ amout eceve The amout veste wll be X R X X $ R Poblem 6. Deve Soluto λw. k k f ta, j w jww j f /

14 ta, /, /, j j j j j j j j kj f j j j f k k w w w w w w w w w, k, Sce, we have:, > j j j w w, j j kj f j j j f k w w w w, f k j j kj j j j f w w w w lets eote j j j f w w w, λ f k k w λ

15 Poblem ssume that the expecte ate of etu o the maket potfolo s 3% a the ate of etu o T-bllsthe sk-fee ate s 7%.The staa evato of the maket s 3%.ssume that the maket potfolo s effcet a What s the equato of the captal maket le? fomulato expecte ate of etu [ [ sk-fee ate expecte ate of etu fom maket-sk fee ate]/ staa evato of maket ] * staa evato expecte ate of etu fom maket.3 sk-fee ate.7 staa evato of maket.3 expecte ate of etu /.3 *staa evato.7.5 staa evato b f expecte etu of 39% s ese, what s the staa evato of ths posto? expecte etu.39 staa evato posto.64 If you have $, to vest, how shoul you allocate t to acheve the above posto? xpecte etu s.39 we ca get x.7 -x.3 x - sk fee asset - maket potfolo c If you vest $3 the sk-fee asset a $7 the maket potfolo, how much moey shoul you expect to have at the e of the yea? 3.77*.38 Poblem 7. small wol

16 Cose a wol whch thee ae oly two sky assets, a, a a sk fee asset F. Te two sky assets ae equal supply the maket; that s, ½. The followg fomato s kow: F a F a geeal expesso fo,, Sce the maket has oly two sky assets a, the the expecte ate of etu a the vaace of the maket epe solely o the expecte ate of etu a the vaace of the assets a. - - Sce a ae equal amouts the maket -.5 [cov, ]/ cov, cov, - - [ - ]/[ - - ] [ ]/ [ ]/ [cov, ]/ cov, cov, - - [ - ]/[ - - ] [ ]/ [ ]/ bccog to CP, what ae the umecal values of a f - f - f f [ [ ]/ ] - f f [.4./.5.4..][.8-.].. - f f [ [ ]/ ] - f f

17 [../.5.4..][.8-.]..6 Poblem ous o etus Cose a uvese of just thee secutes. They have expecte ates of etu of %, %, a %, espectvely. Two potfolos ae kow to le o the mmum-vaace set. They ae efe by the potfolo weghts w [.6,.,.], v [.8, -.,.4]. It s also kow that the maket potfolo s effcet. a Gve ths fomato, what ae the mmum a maxmum possble values fo the expecte ate of etu o the maket potfolo? b Now suppose you ae tol that w epesets the mmum-vaace potfolo. Does ths chage you aswes to pat a? a aket potfolo ca t cota egatve amout of secuty.! #"$&% *, -#./ -#. -#.%-#. - #34 *,! -#. %-#. %-#. -#. 4 - # -#.5 *,6 -#. -#. 4 -#. 4 %-#. - # #.5 3#3 xpecte ate of etu o the maket potfolo: : * 9 :;< > :? * 9 : "! "$!% *,!-#@ -#./ -#. -#.-%-#. - *,! -#@ -#. %-#. %-#.- -#.- *,6!-#@ -#. -#. -#.- %-# #.- -#.- 4 Fom.8.4 a.53 3, we kow

18 b If gve potfolo s the mmum-vaace potfolo, ate of etu of potfolo s the mmum ate of etu of maket potfolo. Rate of etu of potfolo Theefoe, the expecte ate of etu of maket potfolo became Poblem 7.4 Quck CP evato Deve the CP fomula fo Chapte 6. [Ht: Note that w cov,.] pply 6.9 both to asset k a to the maket tself. Soluto k k k f by usg quato 6.9 quato 6-9 fom the textbook λw k,,. k k f We apply the above equato both to asset k a to maket. So we get [ w w ] f λ a λ [ w w ] f Fom the ht: [ w w ] cov, a [ w w ] cov, Substtutg above equatos we get: λ f

19 λ f lmatg λ fom the above two equatos a solvg, we get: f λ. Theefoe, f f. Fom the textbook, β. Substtutg, we obta β Whch s the eque captal asset pcg CP fomula. f f Poblem 7.5 β / j j x cov x j j j cotbutes accog to ts weght j j j / Theefoe β x x Poblem 7.6 Poblem 7.6 Smplela I Smplela, thee ae oly two sky stocks, a, whose etals ae lste below: - Numbe of shaes outstag: o : o : 5 - Pce pe shae o : $.5 o : $. - xpecte ate of etu o : 5% o : % - Staa evato of etu o : 5% o : 9% Futhemoe, the coelato coeffcet betwee the etus of stocks a s ab/3. Thee s also a sk fee asset, a Smplela satsfes the CP exactly.

20 a. xpecte ate of etu of the maket potfolo aket Captalzato - Stock : *.55 - Stock : 5*.3 - Total: 45 We ca euce the espectve weghts the maket potfolo: - Stock : /3 5/45 - Stock : /3 So the expecte ate of etu of the maket potfolo s: m.5*/3.*/3 m3% b. Staa evato Vam a * a b * b * a* b* ab* a* b Vam/3 *.5 /3 *.9 */3*/3*/3*.5*.9 Vam.8 m.9 c. eta of stock acova,m/ m we kow that: Hece: So the eta of stock s: a.5/.9 a.96 m/3*a/3*b Cova,m Cova, /3*a/3*b Cova,m /3*Cova,a/3*Cova,b Cova,m /3* a /3* ab* a* b Cova,m /3*.5 /3*/3*.5*.9 Cova,m.5. Rsk Fee sset Smplela Relato 7. gves us: a-f a*m- f So: f a- a*m / - a f.5-.96*.3 / -.96 f 6.5% Poblem 7.7

21 Zeo-beta assets Let w be the potfolo weghts of sky assets coespog the mmum-vaace pot the feasble ego. Let w be ay othe potfolo o the effcet fote. Defe a to be the coespog etus. a Thee s a fomula of the fom ². F. [Ht: Cose the potfolos - w w, a cose small vaatos of the vaace of such potfolos ea. Let w o w α α α α a w a w α α α α α. α be two potfolos o the the effcet fote. α α α 4α 4 α th potfolo ca be costucte base b Coespog to the potfolo w thee s a potfolo w z o the mmumvaace set that has zeo beta wth espect to w ; that s,,z. Ths potfolo ca be expesse as w z - w w. F the pope value of.

22 4 4 z * * ecause 4 : Set the : the VP, s w Sce - : that t ca be coclue the thee potfolos, y lea combato of a w the the weghte combato of zeo, s If pot potfolo. the mmum vaace be Let w α α α α β β β β β β β β β α α β β α β β β β α α α α z z z z z z z z z z z z z z z z z z z z z z z z z α c Show the elato of the thee potfolos o a agam that clues the feasble ego.

23 w z w wz z If thee s o sk-fee asset, t ca be show that othe assets ca be pce accog to the fomula z β z whee the subscpt eotes the maket potfolo a z s the expecte ate of etu o the potfolo that has zeo beta wth the maket potfolo. Suppose that the expecte etus o the maket a the zeo-beta potfolo ae 5% a 9% espectvely. Suppose that a stock has a coelato coeffcet wth the maket of.5. ssume also that the staa evato of the etus of the maket a stock ae 5% a 5% espectvely. F the expecte etu of stock. z 5% 9% 5% 5% ρ.5 z β.9. %.5*.5* z

24 Poblem 7.8 lecto Wzas, Ic. has a ew ea fo poucg TV sets, a t s plag to ete the evelopmet stage. Oce the pouct s evelope whch wll be at the e of yea, the compay expects to sell ts ew pocess fo a pce p, wth expecte value p $ 4. Howeve, ths sale pce wll epe o the maket of TV sets at the tme. y examg the stck hstoes of vaous TV compaes, t s eteme that the fal sales pce p s coelate wth the maket etu us [ p p ] $ Μ. To evelop the pocess, WI must vest a eseach a evelopmet poject. The cost c of the poject wll be kow shotly afte th poject s begu. The cuet estmate s that the cost wll be ethe c$ o c$6, a each of these s equally lkely. Ths ucetaty s ucoelate wth the fal pce a s also ucoelate wth the maket. ssume that the sk fee ate s f 9% a the expecte etu of the maket 33%. a What s the expecte ate of etu of ths poject? b What s the beta of ths poject? Ht: p p [ p p ] c c Is ths a acceptable poject base o a CP cteo? I patcula what s the excess ate of etu o - above the pecte by the CP? Soluto: p c p p. Due to the fact that p,c ae epeet we c c c p.565 * 4. c a have 35 b [ ] p c p c p p c [ p p ].5.5 The: β. 5 c ase o the CP the expecte etu s: c c Μ c c

25 f β β The expecte excess ate of etu s:.7 f f f Thus we coclue that, base o the CP moel, the poject s ot acceptable sce t gves smalle etu ate tha CP. Nevetheless, the ffeece s oly. theefoe the fal ecso shoul ot be base oly o the CP cteo. f Poblem 7.9 Fomulato: Gav s poblem Pove to Gav Joes that the esults he obtae egs. 7.5 a 7.7 wee ot accets. Specfcally, fo a fu wth etu f - m show that both CP moels gve the pce of $ woth of fu assets as $. We have to pove that the fomulas, cetaty equvalet fom of the CP & the CP as a pcg fomula both wll gve the same esults fo pcg a asset P by appopately scoutg the fal etu Q. we have to o ths fo the case metoe fo Gav Joes egs. 7.5 & 7.7 of the book,.e. fo a asset combato of two secutes wth weghts a -. The etu fo a asset mxtue Q wth above weghts s gve by please ote that Q o wth a ba o top s epesete as Q o.e., bolface Q P f - lso fo the covaace we ca wte, cov Q, cov P f -,. P- fom cetaty equvalet fom of the CP, we have, P [/ f ] * [ Q- {cov Q, * - f / } Substtutg a fom above, P [/ f ]*[ P f - {P- * - f / }] [/ f ]* P [ f - { - * - f }]

26 expag a cacelg out commo tems, P [/ f ]* f P. hece the poof.

27 Poblem 8. smple potfolo Someoe who beleves that the collecto of all stocks satsfes a sgle-facto moel wth the maket potfolo sevg as the facto gves you fomato o thee stocks whch makes up a potfolo. I ato, you kow that the maket potfolo has a expecte ate of etu of % a a staa evato of 8%. The sk-fee ate s 5%. a What s the potfolo s expecte ate of etu? b ssumg the facto moel s accuate, what s the staa evato of ths ate of etu? aket Rate of etu Staa evato % 8% Rsk fee ate 5% Stock eta Staa evato of aom eo tem Weght pofolo. 7.% %.8.3% 5% C.% 3% a The equato fo a sgle-facto moel fo stock etus s: β f m f So, solvg we have: R.* % R.8* % R3.* % Sce R potfolo R potfolo.7*..6*.5.* % b Staa Devatos: w potfolo We kow that: 3 b f e b w b Solvg, we have:.*..5*.8.3*^ e w e

28 .8^ % e.^*.7^.5^*.3^.3^*.^.3% Potfolos staa evato 84.6%*3.4%.3%^ % Poblem 8. Two stocks ae beleve to satsfy the two-facto moel a f f a 3f 4f I ato thee s a sk-fee asset wth a ate of etu o %. It s kow that -ba 5% a -ba %. What ae the values of,, a fo ths moel? Sce thee s a sk-fee asset wth ate-of-etu of %:. sg the elatoshp: f a b j f j the -ba b j j Yels: Poblem 8.3 Suppose thee ae aom vaables x, x, x a let V be the coespog covaace matx. ege vecto of V s a vecto v v, v, v such that Vv λv fo some λ calle a egevalue of V. The aom vaable v x v x v x s a pcple compoet. The fst pcple compoet s the oe coespog to the lagest egevalue of V, the seco to the seco lagest, a so foth. goo caate fo the facto a oe-facto moel of asset etus, s the fst pcple compoet extacte fom the etus themselves: that s, by usg the pcple egevecto of the covaace matx of the etu. F the fst pcple compoet fo the ata of example 8.. Does ths facto whe omalze esemble the etu o the maket potfolo? [Note: Fo ths pat, you ee a egevecto calculato as avalable most matx opeato packages.]

29 Fom the example 8. we have the followg ata Yea Stock Stock Stock 3 Stock 4 aket Rskless veage Va Cov β α e-va Fom the above fst we calculate the aveages, vaace a fally the covaace fo the above ata I the covaace we have to ve the value that s obtae fom the excel solve by 9* to ajust the bas. sg the above Covaace values the Covaace matx s costucte Fom the excel we get t as Covaace matx Fom the above matx we have to calculate the ege values

30 To f the ege values we solve usg xcel the followg equato: et V λ. Ths equato has seveal oots. We ae teeste the lagest ege value. We get the lagest ege value 3.6 The coespog ege vecto s fou to be V [ ] Sce ths s the omalze fom we have the ege vecto as V Poblem 8.4 Let, fo,,,, be epeet samples of a etu of mea µ a vaace. Defe the estmates. ˆ ˆ s µ µ Show that s. Soluto. the expectato by the leaty of ˆ ˆ ˆ ˆ ˆ the expectato by popetes of ˆ ˆ µ µ µ µ µ µ µ s Sce Y Y V Y, the

31 s µ µ µ µ. q.e.. Poblem 8.5 a Show that ˆ s epeet of a ˆ ˆ ˆ va ˆ va * va ˆ. sce va a ˆ va * Fom, va * It shows that ˆ s epeet of b Show that how ˆ epes o va ˆ va * ˆ * va ˆ... ssume that the etu ae omally stbute,

32 4 4 va * Theefoe, 4 va ˆ The, ˆ oe ata oes ot help to estmate the mea moe pecsely but t mpoves the estmato of the volatlty. Poblem 8.6 eco of aual pecetage ates of etu of the stock S s show the followg table. Reco of Rates of Retu: oth Pecet ateof etu oth Pecet ateof etu a stmate the athmetc mea ate of etu, expesse pecet pe yea. b stmate the athmetc staa evato of these etus, aga as pecet pe yea. c stmate the accuacy of the estmates fou pats a a b. How o you thk the aswes to woul chage s you ha yeas of weekly ata stea of mothly ata? See execse 5. Peo moth umbe of peos 4 ˆ

33 a ˆ ˆ ˆ y ˆ.% b s 4 ˆ s y % c The accuacy of the estmatos s eteme by takg the followg staa evatos: ccuacy of the mea estmato ˆ % whee sce we ae ealg wth yealy staa evatos ccuacy of the vaace estmato stev s whee 4 a s the mothly. mothly stev s yealy * stev s mothly.6.6% % Sce we ae quag the staa evato wth uts as pecet, the we ee to multply by to get the aswe as a pecet a ot pecet squae

34 xecse 8-5 poves that the accuacy of the mea estmato s epeet of the umbe of peos,. s a esult, the accuacy of the mea estmato woul ot chage by usg yeas of weekly ata stea of yeas of mothly ata. The accuacy of the staa evato estmato howeve, s epeet o the umbe of peos. The accuacy woul cease wth moe peos to estmate the staa evato. Poblem 8.7 Gav Joes fgue out a cleve way to get 4 samples of mothly etus just ove oe yea stea of oly samples; he takes ovelappg samples; that s, the fst sample coves Ja. to Feb. a the seco sample coves Ja. 5 to Feb. 5, a so foth. He fgues that eo hs estmate of, the mea mothly etu, wll be euces by ths metho. alyze Gav s ea. How oes the vaace of hs estmate compae wth that of the usual metho of usg oovelappg mothly etus? Soluto Hea othly ate of etu Half mothly ate of etu / 4 / / / 4 / / / y y y y V V ρ ρ Fom the ht: ρ ρ Cov * ] [ ] [ ] [ ] [ * ] [!, ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ Fom the ht: ρ ae ucoelate- epeet. / / V ρ ρ ρ Fom the fomula above

35 / / 4 / 4 / 3* / * / / / / * / / * / ] [ ] [ ] [ ] [ ] [ ] [ ] [ ρ ρ ρ ρ ρ ρ ρ Fom the page 5 a the ht: 4 4 ] / *4 [4 4 4] 3 cov... 3 cov, cov 4... [ V V V V V Hea V T he esult s equal to page 5 8. Poblem 8.8 We use the geeal moel wth ppe whee P s a mx matx, s a -mesoal vecto, a P a e ae m-mesoal vectos. The vecto p s a set of obsevato values a e s a vecto of eos havg mea. The eo vecto has covaace matx Q. The the estmate of s vtpvqptpvqp whee v s the vese of a matx a t s the taspose of a matx. a If thee s a sgle asset a just oe measuemet of the fom pe, we must show that p. Soluto: Suppose that pe, the P a Q ae scalas wth tpp. Thus vpvqppvqpqvqpp.

36 b Suppose we have two ucoelate measuemets wth values p a p havg vaaces a espectvely. We ee to show that Soluto: Hee we have [// ^ / ^]p/ ^ p/ ^. Pt ptp p a Q s the matx wth etes Q ^, Q, Q, Q ^ wth the vese of Q, vq, gve by the etes Puttg these to the above fomula we get vq/ ^ vq vq vq/ ^ [// ^ / ^]p/ ^ p/ ^. c We cose example 8.5. Thee ae measuemets of the fom pe pe 3p3e3 4p4e4 f *f f *f 3f 3*f 4f 4*f whee the e s ae ucoelate a whee cove,f.5 ^. ssumg the s ae kow exactly, f the best estmates of the s. Soluto: Fst we ote that the vese of a x matx s gve wth etes a, b, c,. Iv[/a-bc]

37 Whee s the matx wth etes, -b, -c, a a. Fo pat c we use the above fomula state at the begg wth We also use the fomula Pt. ef -f whee e a ae the expecte values of the etu fo secuty usg the equlbum moel a the maket espectvely, f s the sk fee ate, a e / fo the vaace of the eo the measuemet of e. We have as well that cove, f.5 ^ whee the eos e fo the measuemets h ae ucoelate. Fo stock we have h5. a e3.5 as compute example 8.5. So pt a Q s the matx wth etes Q3.^, QQ.53.^, a Q.4^. Isetg these to the above fomula we get Note that the fst ety fo Q s / 3.7%. Fo stock we have h4.34% a e3.99% usg. a we also compute pt a Q s the matx wth etes Q3.75^, QQ.53.75^, a Q.74^ whee.74. /. Isetg these to the above fomula we get 4.%.

38 Fo stock 3 we have h.9% a e7.3% usg 3.4 a we also compute pt a Q s the matx wth etes Q4.7^, QQ.54.7^, a Q3.76^ whee /. Isetg these to the above fomula we get 4.5%. Fo stock 4 we have h5.9% a e.7% usg 3.68 a we also compute pt5.9.7 a Q s the matx wth etes Q.63^, QQ.5.63^, a Q.86^ whee /. Isetg these to the above fomula we get.9%.

39 Poblem vesto has a utlty fucto x x^/4 fo salay. He has a ew job offe whch pays $8, wth a bous. The bous wll be $, $,, $,, $3,, $4,, $5,, o $6,, each wth equal pobablty. What s the cetaty equvalet of ths job offe? Cetaty quvalet x x^/4 Salay $8, wth bous ous $, $,, $,, $3,, $4,, $5,, $6, p of each /7 F the cetaty equvalet Salay $8,, $9,, $,, $,, $,, $3,, $4, x /7 $6.8 $7.3 $7.78 $8. $8.6 $8.99 $9.34 x /7$7.77 x $8.54 Nee to f value C such that C $8.5 Cx^/4 $8.5 C $8.5^4 C $8,6.4 The value C above s the Cetaty quvalet.

40 Poblem Suppose a vesto has expoetal utlty fucto x -e -ax a a tal wealth level of W. The vesto s face wth a oppotuty to vest a amout w W a obta a aom payoff x. Show that hs evaluato of ths cemetal vestmet s epeet of W. To evaluate the cemetal vestmet, we wll compae the vestmet vesus ot makg the vestmet. [ W w x ] mae, a [ W ] s the expecte value of the utlty fucto f the vestmet s s the expecte value of the utlty fucto of wealth f the vestmet s ot mae. [ W w x ] > [ W ] a W w x aw [ e ] > [ e ] aw a xw aw [ e e ] > [ e ] aw a xw aw [ e ] [ e ] > [ e ] a xw [ ] < e The tal wealth, W, s o loge pat of the equato; oly the vestmet w a the payoff x ae. Ths shows that the evaluato s epeet of W.

41 Poblem 3 3 Suppose x s utlty fucto wth ow-patt sk aveso coeffcet ax Let Vx C bx. What s the sk aveso coeffcet of V? ax - x/ x V x b x V x b x av b x/b x av x/ x Theefoe the sk aveso coeffcet -ax

42 Poblem 4 The ow-patt elatve sk aveso coeffcet s x x * x / x Show that the sk aveso coeffcets ae costat fo x l x a x * x x / x x - / x So, x x * - / x / / x - / x / / x - costat x * x - x 3 * x - -* x - So, x - costat

43 Poblem 5 youg woma uses the fst poceue escbe Secto 9.4 to euce he utlty fucto x ove the age <x<. She uses the omalzato,. To check he esult, she epeats the whole poceue ove the age <x<, whee < < <. The esult s a utlty fucto Vx, wth V, V. If the esults ae cosstet, a V shoul be equvalet; that s, Vxaxb fo some a> a b. F a. Gve:, fom to V, fom to Vxaxb a> F: a a b Soluto: b b b b b b b b b a b a V b a V a a a a a b b a V b a V

44 Poblem 6 The HR fo hypebolc absolute sk aveso class of utlty fuctos s efe by γ γ ax x b, b>. γ γ Show how the paametesγ, a a b ca b chose to obta the followg specal cases o a equvalet fom. a Lea o sk eutal: x x Let λ so we have axx the a, b. b Quaatc: x x cx Let λ so we wll have.5ax^.5b^abx/-b So ab c xpoetal: x e λ - ax lm b x γ the b a Powe: x cx ax x lm x x γ γ γ ax x b γ γ e Logathmc: x l x γ x cx the a b γ γ γ ax x b γ γ γ * γ λ ax b γ γ γ γ γ γ γ γ γ γx γ x γ γ ax b γ γ γ x x γ γ, a, b the x l x γ γ x ax b γ γ γ γ γ x The ow-patt sk aveso coeffcet s,

45 a b x b x a a b ax a b ax a x x x a b ax a x b ax a x γ γ γ γ γ γ γ γ γ γ

46 Poblem 7 The vetue captalst vetue captalst wth a utlty fucto xsqtx cae out the poceue of xample 9.3. F a aalytcal expesso fo C as a fucto of e, a fo e as a fucto of C. Do the values Table 9. of the example agee wth these expessos? Vetue captalst xsqtx Lottey outcomes, ethe $ o $9 pecevg $ vaes Fo pethe two outcomes.5, [x]$5. Cetaty equvalet, C$4 Ce p e C alytcal expesso fo C as a fucto of e s fou: Fo ths poblem, we ca solve C fo C. Fom ths, we wll have a equato wth a ukow, vaable pobablty p. Sce we kow the pobabltes the table, we ca calculate the expecte values tems of pobablty a vce vesa. Substtutg pobabltes as a fucto of expecte values to the cetaty equvalet equato yels the ese cetaty equvalet as a fucto of expecte values. To f the seco pat of the poblem, we smply solve the equato fom the fst pat fo expecte values to get expecte value as a fucto of cetaty equvalet. Fally, we compae these two equatos to the table by substtutg values fom the table fo C a e to eteme that the equatos ema tue statemets. If they wee ot tue, the table a equatos woul ot agee. CSqtCe[x] Fo a ueteme pobablty value fo the $ lottey outcome, Cp*- p*9 Fom the table ep*-p*9 ep9-9*p9-8*p Sce we ae lookg fo a expesso fo C as a fucto of e, we ca solve the last equato fo p a substtute ths to the equato fo C to f the aswe. e9-8*p e-9-8*p

47 p9-e/8 a equato fo C Cp*-p*9SqtC C[p*-p*9]^ Sqt 9Sqt93 C[p*-p*3]^[p3*-p]^p3-3*p^3-*p^4*p^-*p9 Substtutg fo p C4*[9-e/8]^-*[9-e/8]9 4*[9-e^]/64-*9-e/89 4/64*[e^-8*e8]-8*e/89 /6*e^-8*e8-6/64*e/644/6 /6*e^-8/6*e8/6-6/64*e/644/6 /6*e^6/6*e9/6/6*e^6*e9/6*e3^ So, Ce[e3^]/6 alytcal expesso fo e as a fucto of C s fou: CSqtCe[x] Cp*-p*9SqtC Sqt 9Sqt93 p9-e/8 CSqtC9-e/8[-9-e/8]*39-e/83-[3*9-e]/8 9-e/83-7/83*e/89-e4-73*e/8 *e6/8e3/4 SqtCe3/4 4*SqtC-3e So, ec4*sqtc-3 Do the values of Table 9. xample 9.3 agee wth these expessos? Substtutg values fom table to these equatos C6.6[6.63^]/65.76, the same value as the table e5.764*sqt , the same value as the table Fom these obsevatos, the values the table agee wth these equatos.

48 Poblem 8 Thee s a useful appoxmato to cetaty equvalet that s easy to eve. secooe expaso ea x x gves ] [ x Va x x x X x x X x x x x O the othe ha, f we let c oate the cetaty equvalet a assume t s close to x, we ca use the fst-oe expaso X C X X C sg these appoxmatos, show that X X C X C s the efto " ] " [ " ] [ X X Va X X C X X X Va X X X C X Va X X C X C

49 Poblem 9 vesto wth ut wealth maxmzes the expecte value of the utlty fucto x ax bx / a obtas a mea-vaace effcet potfolo. fe of hs wth wealth W a the same utlty fucto oes the same calculato, but gets a ffeet potfolo etu. Howeve, chagg b to b oes yel the same esult. What s the value of b? I geeal; [x] [ax /bx ] a[x] /b[x ] a[x] /bva[x] [x] I ths stuato, f the aom payoff of the potfolo of the vesto wth ut wealth s R, t woul maxmze: [R] a[r] /bva[r] [R] Smlaly, f the vesto wth wealth W puchases the same potfolo, the payoff wll be WR a R shoul maxmze: [x ] a[rw] /b va[rw] [RW] aw[r] /b W va[r] W [R] W[a[R] /b Wva[R] [R] ] If b b b/w s substtute the fal equato fo the seco vesto, the same R wll solve the expecte value of the utlty fucto as the R usg ut wealth.

50 Poblem Suppose a vesto has utlty fucto. Thee ae sky assets wth ate of etu,,,, a oe sk-fee asset wth ate of etu f. The vesto has tal wealth W. Suppose that the optmal potfolo fo ths vesto has aom payoff x*. Show that [ x* - f ] fo,,,.. Fom 9.4 p. 43 we kow that [ x* ] P. If thee s a sk-fee asset wth ate of etu R, the R a P. Thus, [ x*] R > [ x*] f. If thee s a asset wth total etu R, the R a P. Thus, [ x* R ] > [ x* ] > [ x* ] [ x*] f [ x* ] - [ x* f ] [ x* - x* f ] [ x* - f ]

51 Poblem W W W thus W W W W W W λ λ λ λ λ The pce of moey back guaatee vestmet P $,5

52 Poblem Fomulato: The followg s a geeal esult fom matx theoy: Let be mx matx. Suppose that the equato x p ca acheve o p except p. The thee s a vecto y > wth T y. se ths esult to show that f thee s o abtage, thee ae postve state pces; that s, pove the postve state pce theoem Secto 9.9. [Ht: If thee ae S states a N secutes, let be a appopate S xn matx] Soluto: Let costuct a matx... S p... S p S 3 p N N... SN p N. Hee j s ve of the secuty j the state. Let take vecto 3... N as vecto of weghts of the secutes the potfolo.... S p... S p S3 p N N... SN p N 3... * Hee D s ve state a P N s pce of the potfolo. Now lets assume that P N o P N < a thee s some k s.t. D k >. I ths case we have abtage because wth o egatve pce thee s a possblty to get ve oe of the states. I oe to avo the abtage we ee to coclue that f P N the fo all D. It meas that system * ca acheve wth T. ccog to the algeba we have that y > s.t. N D D D 3... P N T y... S p y... S p T y y3 ** N N... SN pn y S

53 ecause y S we ca ve all y by y S a efe them as state pce. xpesso ** wll looks lke: ψ... S p ψ... S p T ψ ψ 3 whee ψ > N N... SN pn y solvg ths we have fo each state that p S j ψ j whch meas that fo each state we costucte a postve state pce ψ s.t. p S j ψ j. j j

54 Poblem 3 * Fom the above execse, we have [ x ] the4 quaatc case, we have x-cx. We eote by the k, whee s the sk fee ate. I R the etu of the WR, potfolo, a usg the fact that tal captal s W we get k [ ] equvaletly [ cwr R R ], so R R cw [ R R R R] cw [ cov R, R R R R ], so, R R cwr R R cw [ cov R, R ], a equvaletly R R γcov R, R, whee γ CWR. If we apply ths elato to the potfolo, we obta Cov R, R R R γ Cov R, R γva R, so R R R R β R Va R a so, the poblem s solve. R,

55 Poblem 9.4. t the tack t the hose ace oe Satuay afteoo Gav Joes stues the acg fom a coclues that the hose No btage has a 5% chace to w a s poste at 4 to os. Fo evey olla Gav bets, he eceves $5 f the hose ws a othg f t loses. He ca ethe bet o ths hose o keep moey hs pocket Gav eces that he has a squae-oot utlty fo moey. a What facto of hs moey shoul Gav bet o No btage? b What s the mple wg payoff of a $ bet agast No btage? Soluto: a If we eote by α facto of hs moey m G. J. shoul bet we ee 3 max [ ] m 4 αm α m 4 4 So we ee the st evatve to be equal to zeo, that s, m 3 m α m 4 αm 8 α m 7 Solvg ths equato we obta α b Imple wg payoff of a $ bet agast No btage s 5.5 4

56 Poblem 5 Geeal sk-eutal pcg We ca tasfom the log-optmal pcg fomula to a sk-eutal pcg equato. Fom the log-optmal pcg equato we have P R * Whee R* s the etu o the log-optmal potfolo. We ca the efe a ew expectato opeato by Rx x. R * Ths ca be egae as the expectato of a atfcal pobablty. Note that the usual ules of expectato hol. Namely: a If x s ceta, the x x. Ths s because * R R. ax by a x b y. b Fo ay aom vaables x a y, thee hols c Fo ay oegatve aom vaable x, thee hols x. sg ths ew expectato opeato, wth the mple atfcal pobabltes, show that the pce of ay secuty s P. R Ths s sk eutal pcg. Fom the ules b, we kow * R R ccog to the efto of the opeato, fo ay vaable x, Rx x, assume R * /R as a vaable, we ca get R * R * * R R R Fom the log-optmal pcg equato we have: P 3 R * So, P y3 R * R * R y R * R * y R