Comparing Sharpe and Tint Surplus Optimization to the Capital Budgeting Approach with Multiple Investments in the Froot and Stein Framework.

Transcription

1 Comparng Sharpe and Tn Surplus Opmzaon o he Capal Budgeng pproach wh Mulple Invesmens n he Froo and Sen Framework Harald Bogner Frs Draf: Sepember 9 h 015 Ths Draf: Ocober 1 h 015 bsrac Below s shown ha full relave surplus opmzaon followng Sharpe and Tn 1 and Sharpe 00 leads o he same porfolo as a smplfyng varaon of he capal budgeng approach wh mulple nvesmens ha forms a par of he rsk managemen, capal budgeng and capal srucure polcy framework of Froo and Sen. 3 The followng frs descrbes he Froo and Sen-based approach, and subsequenly an alernave formulaon of he arge funcon for a porfolo reurn mean-varance opmzer (used e.g. by Sharpe ), whch f modfed o accoun for nal unchangeable exposures s smlar o he arge funcon used by Sharpe and Tn and Sharpe (00). Movng o opmzaon wh respec o relave surplus s hen a smple rescalng exercse. The ex concludes wh a bref dscusson of he remanng purely ermnologcal dfferences beween he descrbed conceps. Inal porfolo n nvesor holds a porfolo ncludng an nal posve or negave poson n a rskless asse c wh a rsk-free rae of reurn labelled r, 5 and (posve or negave) exposures o rsky asses, whch have known curren values bu are assumed o be unradeable,.e. hese exposures canno be changed. 6 The 1 ables New pproach, Journal of Porfolo Managemen (16, ), wner 1990, p. 5-10, Wllam F. Sharpe and awrence G. Tn. Budgeng and Monorng Penson Fund Rsk, Fnancal nalyss Journal, Sepember/Ocober 00, Rsk Managemen, Capal Budgeng, and Capal Srucure Polcy for Fnancal Insuons: n Inegraed pproach, The Journal of Fnancal Economcs, 1998, no. 47, 55-8, enneh. Froo and Jeremy C. Sen. s n earler poss n hs blog, he smplfcaons conss of usng reurns before subracng he premums for prced rsk, and usng an exogenously gven uly funcon nsead of a concave payoff funcon (for deals on hese dfferences see he Ocober 014 pos n hs blog (hp:// )). 4 Wllam F. Sharpe, Capal sse Prces Wh nd Whou Negave Holdngs, Nobel ecure, December 7, Noe ha Sharpe and Tn do no explcly menon he exsence of a rskless asse (or lendng and borrowng opporuny a he same rsk-free rae). Sharpe 00 uses cash as rskless asse. bsence of a rskless asse and dfferen lendng and borrowng raes wll be dscussed laer n hs blog. 6 n exsng long exposure may be a poson n he asse drecly, or a poson n a forward on hs asse (oher dervaves wll be dscussed laer n hs blog). For an exsng shor poson n a rsky asse may or may no be a correspondng long poson of equal sze n he rskless asse.e. he shor poson n he rsky asse could resul from an earler shor sale or from a shor poson n a forward conrac on he rsky asse.

2 aggregae noonal exposure 7 of he nal unradeable sochasc asse and lably posons n he porfolo amouns o. 8 The oal nal wealh or capal of he nvesor s herefore: c The porfolo of exsng exposures s assumed o generae a fuure payoff P, so ha he fuure wealh W a he end of consdered perod, f no new exposures were enered, would be: W cr P The porfolo reurn could hence be wren as: W 1 wcr wr wh he reurn of he exsng porfolo of rsky exposures: P r, and weghs of he rsky exposures and he rskless asse beng w and w c c respecvely. 9 New exposures The nvesor has he opporuny o ake on addonal, new exposures o n asses, here ndexed wh, where each asse provdes a sochasc reurn r. 10 Ths may be done va a drec purchase or (shor) sale, 7 Noe ha shor posons here are added wh a negave sgn as opposed o Sharpe and Tn and Sharpe 00, where lables have a posve sgn and are subraced from he porfolo value. 8 The symbol here s chosen o avod havng o nroduce addonal symbols n a laer secon ha says close o he Sharpe and Tn noaon however should be sressed here s no he sum of lably values only, bu he aggregae of he noonal values of nal, unchangeable posve and negave exposures o rsky asses. 9 I may be ha s zero. Then w r would have o be replaced wh P. 10 The reurns of all rsky asses are assumed o be mulvarae normal. Noe ha n Froo and Sen he new exposures are o unradeable asses, as he opmal rsk managemen polcy derved earler n he model mples ha he nsuon consdered n he model wll no ake any radable rsks,.e. n fuure perods he nsuon canno change hese exposures. However, n

3 n whch case he poson n he rskless asse wll be affeced, as well as va enerng long or shor (zero ne-worh) posons va forwards, whch wll provde a payoff equal o he noonal poson value a mes he underlyng asse s excess reurn (reurn mnus he rsk-free rae). 11 The (effecve) wegh of he asse (underlyng he new exposure) n he oal porfolo s: a w Changng he effecve wegh va a cash ransacon or enerng a forward changes he effecve wegh of he aggregae poson n he rskless asse accordngly for every new exposure o a rsky asse wh sze a, here wll be a poson wh sze -a n he rskless asse added o he porfolo. The resulng wegh of he effecve aggregae poson n he rskless asse s can be expressed as: rp wc w r wr w r Correspondngly, f several new posons are esablshed: r p wc w r wr w r, or: I. r w r w r w r r p c nd he fuure wealh s hence: II. W 1 r w r w r w r r p c Maxmzng expeced uly of (fuure) wealh wc w, and hence he porfolo reurn The nvesor has a concave uly funcon U of fuure wealh W, and wans o fnd he wegh of every asse ha maxmzes her expeced uly EU(W) a he end of he consdered perod when all payoffs are realzed,.e. asses sold and forwards and lables seled. necessary condon for a maxmum of EU(W) s ha he frs dervave wh respec o w equals zero: III. E U w 0 one-perod expeced uly maxmzaon fuure reallocaon s no consdered, so ha he resul here would be he same f he new exposures were assumed o be unchangeable n he fuure. 11 s a forward could hence be replcaed wh a combnaon of a posve or negave poson n a (posve for a long and negave for a shor poson) and a poson wh he same absolue value bu oppose sgn n he rskless asse, n he followng he erm effecve wegh s used, o emphasze ha exposures creaed by forwards may be ncluded. 3

4 pplyng he aw of he Unconscous Sascan, whch saes ha he expeced value of he dervave of a funcon equals he dervave of he expeced value of hs funcon, III can be wren as: 1 U E w 0 Usng he chan rule: U E W W 0 w For more compac noaon, wre: U U W W, and: W Ww w So ha he condon for a local expeced uly maxmum can be expressed as: IV. E U W W w 0 By applyng an equaly ha can be derved from he defnon of he covarance: 13 U, W EU W EU EW cov W w W w W w, IV. can be expressed as: V. U W covu, W EU EW 0 E W w W w W w. The Rubnsen-Sen lemma saes, ha for normally dsrbued w : 14 U, W EU covw, W cov W w WW w pplyng hs lemma o V. gves: VI. E U W EU covw, W EU EW 0 W w WW w W w 1 For reference see foonoe 4 of he pos led on Froo and Sen revsed n hs blog. 13 For he dervaon see for example: hps://en.wkpeda.org/wk/lgebrac_formula_for_he_varance#generalzaon_o_covarance. Ths and all followng onlne sources are as of Sepember 6 h The lemma s more commonly known as Sen s lemma. However, n he form used here, he lemma s shown on p.41 n: The Valuaon of Unceran Income Sreams and he Prcng of Opons uhor(s): Mark Rubnsen Source: The Bell Journal of Economcs, Vol. 7, No. (auumn, 1976), pp

5 The frs dervave of wealh (equaon II) wh respec o he wegh of asse s: 15 VII. W r r w Wrng for he expecaon of he excess reurn VIII. EW w r r, he expeced value of VII s: From II and VII he covarance of wealh and s frs dervave wh respec o he wegh of s: IX. covw, Ww covr, rp p Rewrng VI wh VIII and IX: EUWW p EUW 0 Solvng for derved): gves he opmaly condon (from whch furher below he opmal weghs wll be Wh X. G p EU G WW EUW Ths condon was frs derved by Mark Rubnsen. 16 Noe ha X s he condon for a local opmum, however he concavy of he uly funcon ensures ha he porfolo meeng X s he global expeced uly maxmum. 17 The rsk averson parameer G, aka Rubnsen s measure of absolue rsk averson 18, may or may no be consan (or vary self wh changng porfolo composon 19 ), dependng on he 15 Ww s he change of he fuure porfolo value, f he wegh of asse s ncreased margnally expressed per measuremen un of wegh whch s 1 (.e. 100%). 16 See equaon 6 page 614 n: Comparave Sacs nalyss of Rsk Premums, Mark E. Rubnsen, The Journal of Busness, Vol. 46, No. 4 (Oc., 1973), pp For he slgh dfferences o equaon X. see he pos Treynor-Black n hs blog. Noe ha Rubnsen n he above source derves he equaon wh a Taylor seres expanson, whle n a laer source, n ggregaon Theorem for Secures Markes, Journal of Fnancal Economcs, Ocober 1974, he apparenly uses he Rubnsen- Sen emma o derve he equaon (here equaon 16, apparenly derved from equaon 1). lso on page 614 n Comparave Sacs, Rubnsen pons ou ha he equaon holds for a quadrac uly funcon as defned by Mossn as well. The same equaon can also be used by he quadrac suggesed by Markowz and evy, whch s n a relavely wde range a very good approxmaon of oher uly funcons. Furher deals, proofs and references wll be provded laer n hs blog. 17 proof wll be gven laer n hs blog. 18 The name was gven o hs measure by and Zemba see he pos on he -Zemba approxmaon n hs blog. 19 See n hs conex he dscusson on nvesmens nfluencng he value of G n Froo and Sen, p.67. Furher, as dscussed n he pos Treynor-Black, f he porfolo s effcen, he rsk averson parameer s equal o he rao of expeced reurn o 5

6 uly funcon. In case of negave exponenal uly s a consan, 0 and equal o he rrow-pra measure. 1 Wrng now for he aggregae of new exposures and he nal poson n he rskless asse: a c c The wegh of hs aggregae n he oal porfolo s: w wc w wc nd he payoff of he aggregae s: P a r r cr Dvdng P by gves he reurn of he aggregae: XI. P r a r r cr Wh w and r defned lke hs, equaon I can be wren as: rp wr wr So ha: XII. p w w wh he covarances: covr, r and covr, r X can hen be wren as: 3 XIII. Gw w or:. varance whch depends on weghs, varances and covarances. See also foonoe 5 n Sharpe 1990, where Sharpe hence refers o he rsk olerance parameer as beng he rsk olerance parameer n case of opmal holdngs. 0 s Sharpe poned ou n Sharpe proof of hs, followng Sargen, can be found a: hp:// 1 Furher deals o be provded laer n hs blog. Recall ha for every new exposure a o a rsky asse, here wll be a poson - a n he rskless asse added o he porfolo and ncluded n he sum n hs equaon. Noe ha s no he sum of he values of new asses only, bu he sum of all new posve and negave exposures o rsky asses see also foonoe 8 above. 3 If =0 he covarance of r wh he payoff P dvded by s o be used here nsead. 6

7 XIV. G G Comparng XIV and X shows ha porfolo opmzaon akng nal unradeable exposures no accoun s equvalen o he porfolo opmzaon of an nvesor whou nal exposures and capal, f G s subraced from he expeced reurn of every asse, as poned ou by Froo and Sen. 4 Ths may be parcularly helpful n cases lke hose wh negave exponenal uly funcon, where he rsk averson measure s a consan or n applcaons where can be assumed o be approxmaely consan, e.g. n sages of an nvesmen process durng whch relavely small varaons n porfolo composon are o be analyzed. 5 In general hough, G may vary sgnfcanly wh he porfolo allocaon, so ha he erms o be subraced from expeced reurns are no known before he porfolo opmzaon s compleed. 6 Solvng for opmal exposure szes: From XI: j a j j So ha XIII can be wren as: G G j Rearrangng: a j j XV. G G j a j j For every asse, here wll be one equaon lke XV. In marx noaon, wh μ beng he excess reurn vecor, beng he reurn covarance marx, a he vecor of new exposures o rsky asses, and vecor of covarances of new asses reurns wh he exsng porfolo reurn, hs se of n equaons can be expressed as: C a 4 See Froo and Sen p See e.g. Sharpe s web se: hp://web.sanford.edu/~wfsharpe/ma/rr/ma_rr.hm#ndfference and hp://web.sanford.edu/~wfsharpe/ma/rr/ma_rr.hm#negave 6 See Froo and Sen p.67, where dependences of nvesmen decsons due o her mpac on G are dscussed. To fnd he opmal porfolo n cases wh G varyng sgnfcanly wh porfolo composon, one could follow he approach by Bro and fnd he varance-mnmzng hedge for he exsng exposures, and hen add uns of he porfolo of new asses ha has he maxmum excess reurn per un of porfolo sandard devaon. The opmal allocaon o ha porfolo would be a he pon where he slope of an ndfference curve n a mean-sandard devaon dagram equals ha rao, see page 77 In Tobn, qudy Preference as Behavor Towards Rsk, Revew of Economc Sudes, Noe ha monooncy and ransvy ensure ha all ndfference curves have he same slope a a gven sandard devaon. In pracce a sochasc opmzaon algorhm could be used, or an erave process o fnd an approxmae soluon, as for example descrbed on Sharpe s web se, see hp://web.sanford.edu/~wfsharpe/ma/rr/ma_rr.hm#roles and hp://web.sanford.edu/~wfsharpe/ma/rr/ma_rr.hm#nferrng. 7

8 μ GC G a Solvng for he opmal new exposures o rsky asses gves: μ GC a 1 G Wh 1 beng he nverse of he covarance marx. 7 lernave arge funcon Insead of drecly maxmzng expeced uly, one could also use a arge funcon ha s defned as: 8 XVI. U Er p p If he nvesor apples he mean-varance prncple, her maxmzaon problem can always be expressed as maxmzng a funcon lke XVI. 9 Here, based on equaon II and he symbols nroduced for excess reurns, U can be wren as: XVII. U w w Er p wcr s w and w c are consans, maxmzng U corresponds o maxmzng U 1 defned as follows: p U1 w 7 The opmal allocaon o he rskless asse s deermned by: ca and hence: c a. In Froo and Sen, for he porfolo of exsng exposures, payoff volaly nsead of reurn volaly s used. In he noaon payoff here, ha would mean ha would be replaced wh and hence w n equaon XIII. would have o be payoff 1 μgc replaced wh so ha he opmal exposures are: a 1. Ths would be equaon 1 n Froo and Sen, f he G componens compensang for prced rsk were frs subraced from he expeced reurns (see also foonoe 3 above). Noe ha (excess) payoffs of new exposures n Froo and Sen are expressed per un of a, so ha one dollar can be chosen as un, and excess reurns can be used. 8 See e.g. Sharpe Noe ha as poned ou by Sharpe, he whole effcen froner can be consruced by maxmzng XVI for dfferen values of p, see Sharpe 00, page 78, lef column. For furher nuon magne he nvesor ses, and hen maxmzes he expeced reurn he resul wll be a porfolo on he effcen froner (or he capal allocaon lne, f a rsk-free asse exss.) In realy,, porfolo varance and expeced reurn may have o be defned a he same me dependng on he uly funcon see also he bref dscusson above (page 6). 8

9 Derve he frs order condons for porfolo weghs ha maxmze U 1 : U1 U1w w p 0, or, wh XII: XVIII. w w Opmal weghs can be found based on XVIII analogous o he approach descrbed n he prevous secon. s an advanage of hs alernave dervaon one mgh consder ha he normal dsrbuon assumpon or quadrac uly funcon s no drecly requred for he dervaon. 30 Ths may be helpful where anoher dsrbuon ha mees he crera for mean-varance opmzaon correspondng o uly opmzaon s assumed for porfolo reurns, or where mean varance s consdered a reasonable approxmaon alhough he crera for equvalence wh expeced uly maxmzaon are no fully me. If he condons for equvalence beween expeced uly maxmzaon and mean-varance opmzaon are me, XI and XVIII can be se equal: w w Gw w G so ha he relaonshp beween rsk olerance parameer and Rubnsen rsk averson measure s:. G Sharpe and Tn In he nal Sharpe and Tn seng and n Sharpe 00, he porfolo of nal exposures ncludes only lables, and new exposures are assumed o be all asses. Porfolo surplus a me s defned as - (whch corresponds o wha was above ermed wealh W), and relave surplus (n he followng: r s ) s surplus dvded by : However f he arge funcon s mean o be compable wh expeced uly of wealh maxmzaon, hese assumpons would be necessary. lso noe ha oher dsrbuons besdes he normal lead o equvalence beween expeced uly maxmzaon and mean-varance opmzaon, furher deals and references wll be gven laer n hs blog. 31 The noaon here dffers slghly from he orgnal, as n Sharpe and Tn he value of lables s expressed as a posve number and subraced from. Here nsead a lably s defned as an asse of whch a negave amoun s held n he porfolo. Noe also ha n Sharpe and Tn unradeable asses ge nroduced laer as a model exenson n he same manner as lables. Due o he addvy of he covarance, furher dsncon would be sraghforward e.g. a spl of (subjecvely) unchangeable posons no asse classes or rsk facor exposures, whch may e.g. be presen n dscouned fuure sources of ncome. 9

10 1 r 1 r 1 r 1 r 1 r 1 r rs Smlar o XVII one can defne a uly funcon wh respec o relave surplus: XIX. U Er s Er Er s1 s s s s he nal lables and asse value are consans, maxmzng Us1 s equvalen o maxmzng: 3 XX. U Er s s s Ths s he objecve funcon n Sharpe 00 (equaon 16), whch s a specal case of he fnal objecve funcon n Sharpe and Tn. 33 Equvalence of full surplus opmzaon and porfolo reurn mean varance opmzaon: W s menoned above, +=W so ha r p 1, and he relave surplus can also be wren as: XXI. rs r p 1 Wh expeced value: s E r p 1 nd varance: 3, are here he values of asses and lables before any new posons are enered. s here are no consrans on shor posons n rsky asses and or he rskless asse, he sze of asses and lables can of course change wh new posons. 33 In Sharpe and Tn, surplus ncludes he mporance parameer k: S 1 k 1. s r s here mulpled by k, has k o be mulpled by k as well. Hence, he arge funcon ncludng k s: U ST, whch s he orgnal S S Sharpe and Tn objecve funcon (Sharpe and Tn p. 7 lef column), excep he sgn for lables (see above foonoes 7 and 31). Noe furher ha Sharpe and Tn name he rgh hand erm, k, he lably hedgng cred, HC. The arge funcon o be maxmzed s hen: HC G. Noe ha a HC k, so ha wh defnng HC k one a ges: HC HC. Wh consan rsk olerance, he arge funcon would be maxmzed f he HC s added o each asse s reurn and a mean varance opmzaon s performed usng mean reurns adjused n hs manner (Sharpe and Tn p. 9) smlarly o he adjusmen menoned above (see page 6 and he reference gven n foonoe 3). Noe furher, ha he mporance parameer k can hence be nerpreed as wegh gven o r r relave o he case of full surplus opmzaon (full surplus opmzaon means k=100%,.e. he lably hedgng cred has he same wegh as he expeced asse reurn), or relave o he wegh of asse s expeced excess reurn. 10

11 s p Hence, he uly funcon based on relave surplus n XX. can also be wren as: XXII. U E r p p 1 s fer defnng he rsk olerance parameer for surplus rsk 34 as a funcon of he rsk olerance parameer for porfolo reurn varance conssenly wh he expresson for relave surplus n XXI: XXIII. s, XXII. can be wren as: XXIV. U Er p p 1 nd as and are consans maxmzng XXIV. corresponds o maxmzng: p p, XXV. U Er whch s exacly he same uly funcon as XVII. Hence, he opmaly condon mus be he same as XIX. Replacng n XIX wh s s s w w, (from XXIII) gves: whch s equaon 0 n Sharpe So wh he asse and lably reurn as defned above, he resul of Sharpe 00,.e. he Sharpe and Tn resul for full surplus opmzaon, leads o he same porfolo as opmzng porfolo reurn, and f he condons for equvalence of expeced uly maxmzaon and mean varance opmzaon are me (whch are condons needed o apply he Froo and Sen-based approach descrbed above), hs s he porfolo ha maxmzes expeced uly of fuure wealh,.e. he same porfolo ha can also be found wh he Froo and Sen based approach. sses and lables vs. varable and fxed exposures 34 Sharpe 00 uses he erm surplus rsk on p Noe ha Sharpe 00 uses he ndex p nsead of, as p n Sharpe 00 refers o he asse porfolo whch he dscussed earler, whle here s he oal porfolo. 11

12 s emphaszed by Sharpe 00 36, relave surplus opmzaon n Sharpe 00 and Tn and Sharpe 00 does no nclude consrans on shor posons so ha asses could herefore also be new negave exposures o rsky asses. However a negave poson n an asse s nohng else bu a lably an asse owed o anoher pary. Furher, nal unradeable posons can be asses as well hence Sharpe and Tn exend he model o have boh lables and unradeable asses n he nal porfolo. The comparson wh Froo and Sen shows ha he mporan dsncon s no beween asses and lables, bu beween nal unradeable exposures and new posons o be decded abou as n her model ermnology. However, here are lkely applcaons, e.g. for penson funds, whch are referred o n Sharpe and Tn and Sharpe 00, where exsng posons are only lables, and new posons (ncludng n he rskless asse) are resrced o long exposures (and he opmal porfolo from he unconsraned opmzaon ncludng shor posons may provde a benchmark soluon, bu canno be mplemened perfecly) 37, and appears Sharpe and Tn and Sharpe 00 had such cases n mnd, as opposed o Froo and Sen, who explcly assume fnanced posons and menon for example a radng desk as an applcaon ha ypcally nvolves leverage. 36 p. 78 and Ibd. 1