Risk tolerance and optimal portfolio choice

Transcription

1 Risk oleance and opimal pofolio choice Maek Musiela BNP Paibas London Copoae and Invesmen

2 Join wok wih T. Zaiphopoulou (UT usin) Invesmens and fowad uiliies Pepin 6 Backwad and fowad dynamic uiliies and hei associaed picing sysems: Case sudy of he binomial model Indiffeence picing PUP (5) Invesmen and valuaion unde backwad and fowad dynamic uiliies in a sochasic faco model o appea in Dilip Madan s Fesschif (6) Copoae and Invesmen

3 Conens Invesmen banking and maingale heoy Invesmen banking and uiliy heoy Main weaknesses Dynamic uiliy Value funcion and dynamic uiliy lenaive appoach Opimal pofolio Pofolio dynamics Eplici soluion Eamples Conclusions Copoae and Invesmen 3

4 Invesmen banking and maingale heoy Ideal elaionship Mahemaical logic of he deivaive business pefecly in line wih he heoy Picing by eplicaion comes down o calculaion of an epecaion wih espec o a maingale measue Issues of he measue choice and model specificaion and implemenaion deal wih by he appopiae eseves policy Howeve he moden invesmen banking is no abou hedging (he essence of picing by eplicaion) Indeed i is much moe abou eun on capial - he business of hedging offes he lowes eun Copoae and Invesmen 4

5 Invesmen banking and uiliy heoy Dysfuncional elaionship Mahemaical uiliy heoy fomulaed in a vey absac way and focused on solving poblems of limied pacical impoance Economic uiliy heoy fomulaed and developed in he cone which is no diecly focused on applicaions in invesmen banking When efomulaed in he invesmen cone i faces he difficuly o eplain he inuiive meaning of uiliy Only vey spoadic eamples whee uiliy was used in a picing cone To he bes of my knowledge eemely limied use in he asse allocaion cone Copoae and Invesmen 5

6 Main weaknesses No clea idea how o specify he uiliy funcion The classical o ecusive uiliy is defined in isolaion o he invesmen oppouniies given o an agen Eplici soluions o he opimal invesmen poblems can only be deived unde vey esicive model and uiliy assumpions - dependence on he Makovian assumpion and HJB equaions The geneal non Makovian models concenae on he mahemaical quesions of eisence of opimal allocaions and on he dual epesenaion of uiliy No easy way o develop pacical inuiion fo he asse allocaion Copoae and Invesmen 6

7 Dynamic uiliy U() is an adaped pocess s a funcion of U is inceasing and concave Fo each self-financing saegy he associaed (discouned) wealh saisfies E P ( ( π ) ) ( π U X F U X s) s s s Thee eiss a self-financing saegy fo which he associaed (discouned) wealh saisfies E P ( ( ) ) ( ) π π U X F U X s s s s Copoae and Invesmen 7

8 Value funcion and dynamic uiliy Value funcion V ( ) ( π sup E u X T) P ( ) π F X T π T Dynamic pogamming pinciple V ( ) ( ( ) ) π π X s E V X F s T s P s Dynamic uiliy coincides wih he value funcion U ( ) V ( ) R T Copoae and Invesmen 8

9 Difficulies Dynamic uiliy U() is defined by specifying he uiliy funcion u(t) and hen calculaing he value funcion The uiliy a ime i.e. U() may be vey complicaed and quie uninuiive. I depends songly on he specificaion of he make dynamics The analysis of such uiliies depends songly on he Makovian assumpion fo he asse dynamics and he use of HJB equaions Only vey specific cases of such uiliies like eponenial can be analysed in a model independen way Copoae and Invesmen 9

10 lenaive appoach an eample Sa by defining he uiliy funcion a ime i.e. se U()u() Define an adapive pocess U() by combining he vaiaional and he make elaed inpus o saisfy he popeies of a dynamic uiliy Benefis The funcion u() epesens he uiliy fo oday and no fo say en yeas ahead The vaiaional inpus ae he same fo he geneal classes of make dynamics no Makovian assumpion equied The make inpus have diec inuiive inepeaion The family of such uiliies is sufficienly ich o allow fo hinking abou allocaions in a way which is model and uiliy independen Copoae and Invesmen

11 Vaiaional inpus Uiliy equaion u u u Risk oleance equaion ( ) u u ( ) ( ) Copoae and Invesmen

12 Make inpus Invesmen univese of one iskless and k isky secuiies Geneal Io ype dynamics fo he isky secuiies Sandad d-dimensional Bownian moion diving he dynamics of he aded asses Taded asses dynamics ds db i ( i i µ d σ dw ) i S i... k B d Copoae and Invesmen

13 Make inpus Using mai and veco noaion assume eisence of he make pice fo isk pocess which saisfies Benchmak pocess d Views (consains) pocess Subodinaion pocess µ σ T ( d dw ) σσ δ δ δ dz Z φ dw Z d σ σ ( φ ) δ d Copoae and Invesmen 3

14 4 Copoae and Invesmen lenaive appoach an eample Unde he above assumpions he pocess U() defined below is a dynamic uiliy I uns ou ha fo a given self-financing saegy geneaing wealh X one can wie ( ) Z u U ( ) ( ) X R d R R X Z u dw U XZ u Z u X du du φ σσ δ σπ φ δ σπ

15 5 Copoae and Invesmen Opimal pofolio The opimal pofolio is given by Obseve ha The opimal wealh he associaed isk oleance and he opimal allocaions ae benchmaked The opimal pofolio incopoaes he inveso views o consains on op of he make equilibium The opimal pofolio depends on he inveso isk oleance a ime. ( ) ( ) ( ) X R R R X φ δ σ π

16 6 Copoae and Invesmen Pofolio dynamics ssume ha he following pocesses ae coninuous veco-valued semimaingales Then he opimal pofolio uns ou o be a coninuous veco-valued semimaingale as well. Indeed φ σ σ δ σ ( ) R R X φ σ σ δ σ π

17 Wealh and isk oleance dynamics The dynamics of he (benchmaked) opimal wealh and isk oleance ae given by d X R dr X ( σ σ ( φ ) δ ) ( ( δ ) d dw ) d X Obseve ha eo isk oleance anslaes o following he benchmak and geneaing pue bea eposue. In wha follows we assume ha he funcion () is sicly posiive fo all and Copoae and Invesmen 7

18 Canonical vaiables The wealh and isk oleance dynamics can be wien as follows d dm X Obseve ha ( σ σ ( φ ) δ ) (( δ ) d dw ) R dm dr d M d X dr dm Inoduce he pocesses Copoae and Invesmen X ( ) () ( ) () R ( ) w() M ( ) 8

19 Canonical dynamics The pevious sysem of equaions becomes d d () () dw () ( ) () ( () ) () dw () ( ) ( ) y I uns ou ha i can be solved analyically Copoae and Invesmen 9

20 Linea equaion Le h() be he invese funcion of ( u ) du h ( ) ( u ) du ( ) I uns ou ha h() solves he following linea equaion h h ( ) h ( ) ( u ) du h ( ) Copoae and Invesmen

21 Eplici epesenaion Soluion o he sysem of equaions is given by ( ) h( ( ) ) () h ( () ) ( () ) h ( ) ( s ) ds w( ) One can easily eve o he oiginal coodinaes and obain he eplici epessions fo X R Copoae and Invesmen

22 Copoae and Invesmen Opimal wealh The opimal (benchmaked) wealh can be wien as follows ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) d M d d dw d dm M d h du u h h X s s δ φ σ σ δ δ φ σ σ

23 3 Copoae and Invesmen Risk oleance The isk oleance pocess can be wien as follows ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) d M d d dw d dm M d h du u h h R s s δ φ σ σ δ δ φ σ σ

24 Bea and alpha Fo an abiay isk oleance he inveso will geneae pue bea by fomulaing he appopiae views on op of make equilibium indeed σ σ ( φ ) δ d dr X To geneae some alpha on op of he bea he inveso needs o oleae some isk bu may also fomulae views on op of make equilibium Copoae and Invesmen 4

25 No benchmak and no views The opimal allocaions given below ae epessed in he discouned wih he iskless asse amouns π R σ d σ σ d ( X ) ( ) ( ) They depend on he make pice of isk asse volailiies and he inveso s isk oleance a ime. R Copoae and Invesmen Obseve no diec dependence on he uiliy funcion and he link beween he disibuion of he opimal (discouned) wealh in he fuue and he implici o i cuen isk oleance of he inveso 5

26 No benchmak and hedging consain The deivaives business can be seen fom he invesmen pespecive as an aciviy fo which i is opimal o hold a pofolio which eans iskless ae By fomulaing views agains make equilibium one akes a isk neual posiion and allocaes eo wealh o he isky invesmen. Indeed δ φ π Ohe consains can also be incopoaed by he appopiae specificaion of he benchmak and of he veco of views Copoae and Invesmen 6

27 7 Copoae and Invesmen No iskless allocaion Take a veco such ha Define The opimal allocaion is given by I pus eo wealh ino he iskless asse. Indeed σ ν ( ) φ σ σ δ ν ν σ σ φ ( ) X φ σ π X X ν ν σ σ σ π

28 Seps o follow Specify he invesmen univese and is equilibium dynamics Deemine he cuen isk oleance of an inveso elaively o ha univese (could y o imply i fom he specificaion of fuue wealh disibuion) Specify a benchmak and views o consains Solve he FDE o ecove he funcion () Deemine he vaiaional inpu u() of he uiliy funcion namely solve y uu u u( ) ep ddy ( ) We se he uiliy of eo wealh a ime eo o be eo and he slope of he uiliy a ime eo fo eo wealh o be equal o one. Of couse ohe choices ae possible Copoae and Invesmen 8

29 Seps o follow Specify he dynamic fowad uiliy by combining he vaiaional inpu wih he choice of a benchmak views o consains The opimal pofolio is opimal wih espec o his uiliy Recove he funcion h() which is he invese of he funcion du ( u ) Specify he opimal wealh and isk oleance pocesses nalyse he oucome and poenially ecalibae Copoae and Invesmen 9

30 Disclaime The views epessed in his pesenaion ae hose of he auho and no necessaily hose of BNP Paibas Copoae and Invesmen 3