Ivestmet Scece Chapte 3 D. James. Tztzous <jmt@ams.jhu.eu> 3. se P wth 7/.58%, P $5,, a 7 84, to obta $377.3. 3. 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.
Ivestmet Scece Chapte 4 Solutos to Suggeste Poblems D. James. Tztzous <jmt@ams.jhu.eu> 4. Oe fowa ate f, s s.69.63 7.5% 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,6 5.6 5.9 6.7 6.5 6.3
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 3.5. 4.5 93. 65.9. 4.5 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
The scout factos ae fou by successve multplcato. Fo example,,,,.95.94.893. The complete set s.95,.893,.77,.77,.646. 4.7 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 37.64. 4.8 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
a,,,3,4,5,6.954.98.849.798.747.7 NP V 9.497 b Yea 34 5 6 Cash Flow 4 Dscout.954.9469.946.9399.936 938 PV 9.497 5.97 44.34 36.453 8.44 9.38. 4. 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,
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: [ ] 3 4 5 6 [ ] [ ] 3.5.5 ab a b 6 va[ Z] ab [ ab] a b [ a] [ b] 4 4 9 6 5 36 a b [ a] [ b] 3.5 79.97 6 Poblem 6.3 sg the aswe of the Poblem 6.4: a α 9 / 3 b α α 9 / 3.5 9 / 3.3 3.4% 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
, 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
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
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.4.8 3.8
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 -.4 - -.8 - -.8 -.4.8.8 - - y usg calculato: w w w 3 λ µ.5.5.6 b λ, µ System wth 3 vaables a 3 equatos: v v v 3.4.8.8 y usg calculato: v v v 3 w w w 3...3.67.333.5.733 3
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 3..6.6 v v v 3 w w w 3...5.5 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
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:
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 /
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
Poblem 7. 7. 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.7.3-.7/.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.39 39 39 x.7 -x.3 x - sk fee asset - maket potfolo c If you vest $3 the sk-fee asset a $7 the maket potfolo, how much moey shoul you expect to have at the e of the yea? 3.77*.38 Poblem 7. small wol
Cose a wol whch thee ae oly two sky assets, a, a a sk fee asset F. Te two sky assets ae equal supply the maket; that s, ½. The followg fomato s kow: F..4...8 a F a geeal expesso fo,, Sce the maket has oly two sky assets a, the the expecte ate of etu a the vaace of the maket epe solely o the expecte ate of etu a the vaace of the assets a. - - Sce a ae equal amouts the maket -.5 [cov, ]/ cov, cov, - - [ - ]/[ - - ] [ ]/ [ ]/ [cov, ]/ cov, cov, - - [ - ]/[ - - ] [ ]/ [ ]/ bccog to CP, what ae the umecal values of a f - f - f f [ [ ]/ ] - f f [.4./.5.4..][.8-.].. - f f [ [ ]/ ] - f f
[../.5.4..][.8-.]..6 Poblem 7.3 7.3 ous o etus Cose a uvese of just thee secutes. They have expecte ates of etu of %, %, a %, espectvely. Two potfolos ae kow to le o the mmum-vaace set. They ae efe by the potfolo weghts w [.6,.,.], v [.8, -.,.4]. It s also kow that the maket potfolo s effcet. a Gve ths fomato, what ae the mmum a maxmum possble values fo the expecte ate of etu o the maket potfolo? b Now suppose you ae tol that w epesets the mmum-vaace potfolo. Does ths chage you aswes to pat a? a aket potfolo ca t cota egatve amout of secuty.! #"$&% *, -#./ -#. -#.%-#. - #34 *,! -#. %-#. %-#. -#. 4 - # -#.5 *,6 -#. -#. 4 -#. 4 %-#. - #3 78 9 -#.5 3#3 xpecte ate of etu o the maket potfolo: : * 9 :;< > :? * 9 : "! "$!% 4 4 4 *,!-#@ -#./ -#. -#.-%-#. - *,! -#@ -#. %-#. %-#.- -#.- *,6!-#@ -#. -#. -#.- %-#. - 7 -#.- -#.- 4 Fom.8.4 a.53 3, we kow. 3 3.6.
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.6.4... Theefoe, the expecte ate of etu of maket potfolo became. 3 3.6. 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
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.
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
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.
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.
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*.5.375 z.375.5.9.5
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
f β β.5.33.9.9. 36 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 }]
expag a cacelg out commo tems, P [/ f ]* f P. hece the poof.
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.*.-.5.5.7.7% R.8*.-.5.5.6.6% R3.*.-.5.5..% Sce R potfolo R potfolo.7*..6*.5.*.3.44.44% b Staa Devatos: w potfolo We kow that: 3 b f e b w b Solvg, we have:.*..5*.8.3*^.846 3 e w e
.8^.34 3.4% e.^*.7^.5^*.3^.3^*.^.3% Potfolos staa evato 84.6%*3.4%.3%^.5 6.66% 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:.5... 3 4.. 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.]
Fom the example 8. we have the followg ata Yea Stock Stock Stock 3 Stock 4 aket Rskless.9 9.59 3.7 7.4 3 6. 8.37 5.5 9.47 7.5 7.54 6.7 3 3.64 3.53-6.58..7 6.4 4 4.37 7.67 5.8.6 9.34 5.8 5 3.4.74 6.4 9.84 9.8 5.9 6 -.45 -.56-5.5.5-4.39 5. 7. 5.46 7.8.4 8.9 4.9 8 9.8 6.9 8.8 6..78 5.5 9 7.63 9.73 3.5.93 3.34 6. 5.7 5.9 6.94 3.49 5.3 5.84 veage 5. 4.34.9 5.9 3.83 5.854 Va 9.6 7.3 6. 68.5 7. Cov 65.9 73.6.78 48.99 7. β.9..4.68 α.95.34-6. 3.8 e-va 3.54 3.9.37 34.99 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 3 4 9.6 5.89 79. 4.8 5.89 7.3 5.38 3.99 3 79. 5.38 6. 56.54 4 4.8 3.99 56.54 68.5 Covaace matx Fom the above matx we have to calculate the ege values
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 [.7.63.36.53] Sce ths s the omalze fom we have the ege vecto as V.7.63.363.534 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
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,
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. 3 4..5 4 4.5 3 4. 5 -.5 4 -.7 6. 5 -. 7 -.7 6 3.5 8 3.7 7-3. 9 3. 8 4. -.4 9.7.7..9 -.4 3 -.9 3. 4. 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. 4.9338-3.5.5 3 4..4 4 -.7 3.69 5 -. 9 6 3.5 6.5 7-3. 6.8
a 8 4. 9.6 9.7.49..8 -.4.56 3. 4.84 3 4..4 4 4.5.5 5 -.5.5 6.. 7 -.7 7.9 8 3.7 7.9 9 3. 4.84 -.4.56.7.89.9 3.6 3 -.9 8.4 4.. ˆ. 65.4 4 ˆ ˆ y ˆ.% b s 4 ˆ 7.934 s y 9.89545% c The accuacy of the estmatos s eteme by takg the followg staa evatos: ccuacy of the mea estmato ˆ 6.5687% 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% 5.447.5447% 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
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
/ / 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... [ 4 4 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.
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]
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 pt5. 3.5 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 pt4.34 3.99 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.%.
Fo stock 3 we have h.9% a e7.3% usg 3.4 a we also compute pt.9 7.3 a Q s the matx wth etes Q4.7^, QQ.54.7^, a Q3.76^ whee 3.76.4 /. 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.86.68 /. Isetg these to the above fomula we get.9%.
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.
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.
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
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
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
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,
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
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...3.4.5.6.7.8.9 e 9 8. 7.4 6.6 5.8 5 4. 3.4.6.8 C 9 7.84 6.76 5.76 4.84 4 3.4.56.96.44 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
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*sqt5.76-36.6, the same value as the table Fom these obsevatos, the values the table agee wth these equatos.
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
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.
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 ]
Poblem 5.6 5. 5...6..8 3 6. 3.8..3..4 3 6. 3.8..36..48 3 6. 3.36..4 3 6. 3.9 4 3 4 3 4 4 3 4 3 4 4 3 4 4 3 4 3 W W W thus W W W W W W λ λ λ λ λ The pce of moey back guaatee vestmet P $,5
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... 3 3 S 3 p 3............ 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... 3 3 S3 p 3............ 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
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
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,
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.346 5 b Imple wg payoff of a $ bet agast No btage s 5.5 4
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