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)