When does 'All Eggs in One Risky Basket' Make Sense?

Pape p epa ed o he I ish Economics Associa ion’s Annual Con e ence 2005 When does ‘All Eggs in One Risky Baske ’ Make Sense? GERRY BOYLE and DENIS CONNIFFE Depa men o Economics, NUI, Maynoo h Abs ac : In an impo an pape compa ing expec ed u ili y and mean- a iance analysis, Felds ein (1969) examined a simple po olio p oblem in ol ing jus wo asse s, one iskless and one isky. He concluded he e could easily be ‘plunging’, ha is, in es men in he isky asse alone. His backg ound assump ions we e ha he isky asse ’s yield was log no mally dis ibu ed and ha he in es o ’s a i ude o isk was exp essible by a loga i hmic u ili y. We look a how conclusions a e a ec ed by choice o dis ibu ion and u ili y unc ion. While conclusions can depend on choice o dis ibu ion, hey a e ema kably obus o choice wi hin he ange o plausible posi i e dis ibu ions. In con as , conclusions a e sensi i e o choice o u ili y unc ion and we ind he key de e minan o be how much he in es o ’s ela i e isk a e sion di e s om uni y and in wha di ec ion. Based on his o ical s ock ma ke e u ns, ou analysis implies ha he p e alence o di e si ica ion ha is obse ed is consis en wi h a ela i e isk a e sion coe icien o abou 2.5. I INTRODUCTION Felds ein (1969) published an impo an and much ci ed pape c i icising he use o mean- a iance (o   ) analysis a he han expec ed u ili y in he heo y o economic beha iou unde unce ain y. He based his analysis on a simple po olio p oblem in ol ing jus wo asse s, one iskless and one isky. Bu since he isky asse could i sel be conside ed o be he op imal, o ma ke , po olio o isky asse s, he sepa a ion heo em o mean- a iance po olio analysis sugges s he impo ance o he p oblem. The e we e h ee componen s o Felds ein’s pape . The i s ou lined he possibili y ha he    indi e ence cu es o a isk a e e migh no be con ex downwa ds. The second a gued ha an in es o choosing a combina ion o a iskless and a isky asse migh be much mo e likely o op o ‘plunge’ o a isk-only po olio han p e ious analysis had sugges ed, so ha , con a y o Tobin’s (1958) analysis, di e si ica ion be ween money and bonds need no gene ally ollow om isk a e sion. The hi d c i icised he use o    analysis o mul i-asse po olios. Subsequen 2 discussion o Felds ein’s pape has concen a ed on he i s and hi d componen s. The second, which appea s o ha e ecei ed ela i ely li le a en ion, p o ides he mo i a ion o his pape . Felds ein assumed a log no mal dis ibu ion1 o he e u n on he isky asse and a loga i hmic u ili y unc ion. He demons a ed ha a u ili y-maximising s a egy would lead o plunging o he isky asse gi en alues o he pa ame e s o he log no mal ha he conside ed a om unlikely. We will e iew his esul s in sec ion II along wi h some discussion o subsequen commen s. We con inue o examine how much plunging depends on he speci ica ion o he p obabili y dis ibu ion and u ili y unc ion. We show in sec ion III ha i he loga i hmic u ili y unc ion is e ained, Felds ein’s esul s a e ema kably obus ac oss a ange o dis ibu ions. Howe e , in sec ion IV, we show ha he si ua ion is e y di e en wi h ega d o choice o u ili y unc ion and we ela e he a ia ions in he likelihood o plunging o he deg ee o isk a e sion embedded in he u ili y unc ions. Finally, in sec ion V we conside some implica ions o hese indings. II FELDSTEIN’S ANALYSIS Felds ein, ollowing Tobin (1958), conside ed an in es o wi h ini ial weal h A, alloca ing a p opo ion p o he isky asse , so ha , a e one pe iod, weal h becomes y = (1-p)A + pAx, whe e he iskless asse is assumed non-in e es bea ing and x is andom wi h a mean p esumed g ea e han uni y2. Felds ein ook x as log no mal wi h pa ame e s *  and *  , ha is, a densi y o 2* 2 2 *)(log 1 * 1 2 1 )(        x e x x ,    x0. (1) Then he mean o x 2** 2 1    e and he a iance o x   1 22** 22     ee . Fo in es men in he isky asse o ha e any a ac ion, Felds ein u he assumed ha he in es o is isk a e se wi h a loga i hmic u ili y unc ion. So he in es o wan s o choose p so as o 1 Since, as is well known, expec ed u ili y and    analysis a e compa ible gi en no mali y, Felds ein needed o assume some non-no mal dis ibu ion. Howe e , log no mali y, o o he posi i e dis ibu ions, seem mo e plausible han he no mal and ha e mo e empi ical suppo . 2 Ob iously, he ma hema ical na u e o he p oblem is unchanged by a aching a non-s ochas ic yield o he iskless asse and adjus ing he yield o he isky asse acco dingly. 3 maximise }])1[log{()}({ pAxApEyuE    )}]1(1[log{log    xpEA . (2) The de i a i e            )1(1 1)( xp x E p uE is ob iously posi i e a p = 0, so ha he e will ce ainly be some in es men in he isky asse . A p = 1 i is dxx xx E)( 1 1 1 1         (3) which is 2** 1 2 1   e=         2 2 1 1 1    . So he de i a i e is s ill posi i e, implying plunging, ha is, all unds in es ed in he isky asse , i         2 2 1    . (4) Felds ein main ained his would o en be he case, gi ing he example o 05.1   , when plunging occu s unless  is well mo e han ou imes he expec ed bond yield o 5%. He con as ed his si ua ion wi h he ubiqui y o di e si ica ion implied by Tobin’s (1958) mean a iance analysis o he same wo asse model and linked i o non con exi y o indi e ence cu es, so as o emphasise ha mean a iance analysis is no always an adequa e subs i u e o expec ed u ili y maximisa ion. While many subsequen pape s, whe he de ending o u he c i icising mean- a iance analysis, ha e discussed Felds ein’s pape (including, among o he s, Tobin (1969), Tsiang (1972), Bie wag (1974), Bo ch (1974), Le y (1974), Maysha (1978) Fels ein (1978) and Meye (1987)), he plunging phenomenon i sel has no ea u ed p ominen ly. Maysha accep ed he co ec ness o Felds ein’s analysis o plunging, bu main ained i was eally un ela ed o issues o he con exi y o indi e ence cu es, o alidi y o mean- a iance analysis. One o his a gumen s was ha Felds ein had analysed indi e ence cu es gene a ed by when x is assumed log no mal, bu in he )}({ xuE po olio p oblem conside ed a a iable )}1(1(    xpAy , which being a ansla ion o a log 4 no mal has a somewha di e en dis ibu ion. Ano he was he mo e gene al poin ha i he in es o es ic ed o choosing alues o p and will ob ain he same alue o x i espec i e o choice o p, hen wha e e he dis ibu ion o x, he dis ibu ions o he a iables is )( )()( p ppy y y    a e ob iously iden ical, whe e )}1(1{)(      pAp y and   App y  )( . Then wha e e he is ibu ion and wha e e he (conca e) u ili y unc ion, he op imal p, assuming i < 1, could be b ained om mean a iance analysis in e ms o d yy   and o. While mo e oundabou han di ec e e mina ion by equa ing he de i a i e o wi h espec o p o ze o, i is ce ainly ompa ible wi h i 3. Bu whe he p is < 1, o whe he plunging occu s is igno ed in his a gumen nd i is no a all clea ha i should be. The e would seem no sense in using mean a iance analysis hen p=1 and, i plunging is possible, in e es should ocus on how dis ibu ion and u ili y unc ion ec i s occu ence. Ce ainly, condi ion (4) was de i ed assuming log no mali y and log u ili y. o e oo ha i p = 1, y is log no mally dis ibu ed i x is. In con as o Maysha , Meye (1987) ispu ed he alidi y o Felds ein’s esul on plunging, claiming i esul ed om an implici bo owing s ic ion in he model. Bu i he in es o could bo ow wi hou cos , his would jus inc ease A wi h lunging s ill occu ing. o no de ailed examina ion seems o ha e been conduc ed in o how he occu ence o lunging migh change wi h he dis ibu ion o how i migh depend on he p ecise u ili y unc ion hosen. III VARYING THE PROBABILITY DISTRIBUTION e aining, o he p esen , he loga i hmic u ili y, o mulae (2) and (3) emain unchanged. lunging will occu i )}({ yuEd c a w a N d e p S p c R P 0 1 1       x E, (5) hile a mix u e o asse s will be op imal i i is nega i e. I x could ake nega i e alues, he xpec a ion o he ecip ocal o x may no exis , which can usually be in e p e ed as , plying ha plunging canno occu . So, o example, assuming a no mal dis ibu ion o x would w E e im 3 O he au ho s ha e made his poin including (implici ly) Bie wag (1974) and Meye (1987). 5 exclude plunging. Bu i we assume ha he mos ha can be los by in es men in a isky asse is he moun in es ed, a non-nega i e dis ibu ion is app op ia e o x. aking x as posi i ely dis ibu ed , he well known del a me hod o app oxima ion o xpec a ions p oceeds a T e                                         ...1 1 1 111 3 3 2 2 1          EEE x E ... 1 4 3 3 2      , (6) he e 3  w deno es he hi d momen abou he mean. Now i we igno e e ms wi h denomina o s ol ing powe s o  in g ea e han 3, his gi es               2 2 1 11    x E, which implies (4) again as he de e minan o plunging. Admi edly, his app oxima ion assumes ha e coe icien o a ia ion is no la ge, bu no e ha i he dis ibu ion is posi i ely skew, implying 3  h posi i e, his educes (6) making i mo e likely ha (5) is posi i e and plunging occu s. owe e , we need no ely on app oxima ion a gumen s o show ha plunging can occu o he han he log no mal. Suppose x ollows a Pea son Type 3 dis ibu ion (o en called he wo a ame e gamma) H o p     x exx    1 )( 1 )(    , x0 he e he mean and a iance can be shown o be    w and . Then 22       2 1 )1( 11           x E. his will be less han uni y and plunging will occu i . (7) o plunging is qui e plausible unless a ia ion is e y la ge, al hough no qui e as likely as wi h a log since T  /1 2  S no mal 6                           2 2 4 4 2 2 1 2 2 21 1 ...1 1 1 11               . Howe e , aking 05.1  as in Felds ein’s example, (7) shows plunging occu s unless 22.  , again mo e han leas ou imes he expec ed bond yield o 5%. Again, suppose x ollows a Weibull dis ibu ion, which, like he log no mal and Gamma dis ibu ions, has been ound in a ious empi ical s udies o p o ide a good i o weal h dis ibu ions. The densi y is                         x e x x 1 )( ,    x0. I is e iden ha (x)/x ends o in ini y as x goes o ze o o 2   , implying ha he expec a ion goes o in ini y also and ha plunging canno occu . Fo 2  exac in eg a ion gi es              1 1 11 x E. (8) The mean o a Weibull is         1 1 (9) and so o la ge  , since , he igh hand side o (8) becomes he ecip ocal o 1)1(   . This mus be less han uni y, as  was p esumed g ea e han uni y, so plunging mus hen occu . P ac ically plausible alues o  a e bes judged om he coe icien o a ia ion. The a iance o a Weibull is                    1 1 2 1222 and, ob iously, la ge  implies small  . The squa e o he coe icien o a ia ion is 1 1 1 2 1 2 2 2                  . (10) So o any  and  ,  can be deduced om (10) and hen  om (9), so ha (8) can show i plunging occu s. Taking he case o 05.1   and  = .2, o ou imes he expec ed yield, we ind (8) akes he alue .996 so ha plunging occu s e en wi h ha much a ia ion, which is essen ially he 7 same as Felds ein ound o he log no mal dis ibu ion. Simila indings ollow o o he plausible posi i e dis ibu ions o which algeb aic exac solu ions a e ob ainable. The mul iple o he expec ed yield ha a ia ion mus exceed be o e plunging is uled ou can a y somewha , bu he o e all si ua ion is clea . Plunging ough no o be an in equen p edic ion wi h dis ibu ions ha a e usually conside ed plausible candida es o weal h, a leas when we belie e an in es o ’s isk a e sion can be ep esen ed by loga i hmic u ili y. IV VARYING THE UTILITY FUNCTION To poin up he di e ence ha choice o u ili y unc ion can make, we i s con as loga i hmic u ili y wi h he ex emely popula nega i e exponen ial u ili y , wi h y eyu   1)( 0  , whe e, as be o e, y = (1-p)A + pAx. This is o en aken in conjunc ion wi h a no mal dis ibu ion o x and hen i is well known ha hen he op imal p is 2 1    A p  which mus be less han uni y i A is la ge enough and so he amoun in es ed in he isky asse is Ap, cons an i espec i e o he amoun a ailable o in es men . Howe e , his absence o plunging also occu s wi h he dis ibu ions conside ed in he p e ious sec ions. Taking he wo-pa ame e gamma, o example, he expec ed u ili y is dxexeyuE x xpA         1)}1(1{ )( 1 1)}({ . dxxee ex Apx pA 1 ) 1 ( )1( )( 1            . This can be in eg a ed exac ly o gi e     1 1)1(    Ap epA . Di e en ia ing wi h espec o p gi es           1 1 1 )1(       Ap Ap Ae pA 8 which, i A is la ge, will ob iously be nega i e a p = 1. The op imum in es men Ap is 2 )1(1         Ap , again a cons an , i espec i e o A. The con as be ween esul s o he loga i hmic u ili y and he nega i e exponen ial u ili y mus be due o he di e en deg ees o isk a e sion hey embody. The coe icien s o absolu e isk a e sion and ela i e isk a e sion a e y R1  and , o 1 Ryu log  and  R and yR   , o . y eu   1 Fu he mo e is unbounded and yu log  as y  , while he nega i e exponen ial u ili y 1 as y . The e is so much less o be gained om la ge inc emen s in weal h wi h he  nega i e exponen ial u ili y. Fo a ine explo a ion o he in luence o isk a e sion on plunging, we ake he u ili y unc ion , wi h     yu 11    . Fo  nega i e his is unbounded as y    and o  posi i e i is bounded as y . 1  In ac , 0  co esponds o and, ob iously, yu log1    gi es isk neu ali y. The coe icien s o isk a e sion a e y R1   and 1    R, showing dec easing absolu e isk a e sion and cons an ela i e isk a e sion. So his u ili y unc ion is no a all as isk a e se as he nega i e exponen ial u ili y, bu i can be ei he less isk a e se han he loga i hmic u ili y (i  is nega i e) o mo e isk a e se (i  is posi i e). So i should be a sensi i e c i e ion o examining plunging. Fo any dis ibu ion densi y (x), he expec ed u ili y is .  dxx xpAyuE )()}1(1{1)}({     Di e en ia ing wi h espec o p and se ing p = 1 gi es . (11) dxx xxA )()1( 12     I his is posi i e plunging occu s. I can be e alua ed by in eg a ion o any o he posi i e dis ibu ions men ioned ea lie . Fo (1), he log no mal densi y, (11) becomes 9       2**2* 2 *) 2 1 ( 2 21      eeA . So plunging occu s i 2** ) 2 1 (   , o exp essed in o  and  1 2 2 1              , (12) which educes o Felds ein’s c i e ion (4) i  = 0. Fo  nega i e, plunging is e en mo e likely han wi h loga i hmic u ili y, in he sense ha i will occu in spi e o e e g ea e a iabili y. A  = -1 plunging is ce ain. Con e sely, o  posi i e and inc easing, plunging becomes less likely and will no occu i  is la ge enough. So, since dec easing absolu e isk a e sion and cons an ela i e isk a e sion holds o his u ili y unc ion wha e e he alue o  , i is no hese cha ac e is ics in hemsel es ha de e mine plunging. Ra he , since 1    R, i is he di e ence om uni y (and di ec ion) o he ela i e isk a e sion, which o cou se is he (nega i e) elas ici y o he slope o he u ili y unc ion. High implies absence o plunging. R These indings a e no dependen on he log no mal alone. Re u ning o (11), bu assuming (x) ollows he wo pa ame e gamma, gi es dxexxx Ax           11 2)1( )( . In eg a ion gi es  )1()( )( 1 2         A.  1)1( )( )1( 1 2          A So plunging occu s i 1)1(     , o in e ms o  and   11 2      . (13)