A C T A U N I V E R S I T A T I S L O D Z I E N S I S
F O L IA O E C O N O M IC A 194, 2005
G ra ży n a T rzp io t*
EFFECTIV EN ESS OF STO CH ASTIC D O M IN A N C E IN FINANCIAL ANALYSIS
Abstract
I o rtfo lio analysis can be regarded as a p ro b lem o f ch oosing the best investm ent project lro m all possible investm ents. I his choice dep en d s on, the u n iq u e for each investor, utility fu n ctio n an d the d istrib u tio n оГ the re tu rn оГ the investm ent p ro ject. U nlike M V criterio n , SD c riterio n is o p tim a l fo r a class o f utility fu n ctio n and a d d itio n ally we e la b o rate with all value o f the re tu rn o f the investm ent project. W e will p resen t the resu lts o f analysis the p ro p e rties ol the op tim al efficient set according S I) criteria fo r asy m m etric d istrib u tio n .
Key words: asym m etric d istrib u tio n , stochastic do m in an ce criterio n , efficient set.
I. IN T R O D U C T IO N
P o rtfolio analysis poses the problem o f choosing o f the best prospect from all possible alternative random prospects (portfolio). T his selection depends on the investo r’s utility function and on prob ab ility d istribution o f the prospects. In general the analysis proceeds in tw o steps. F irst for a g ro u p o f investor having the sam e class o f utility fun ctio n all possible alternative random prospects we divide in two sets: efficient set and inefficient set. T h e set arc constructed so th a t for any p ro sp ect G in the inefficient set exist at least one prospect F in efficient set w ith the p ro p erty th a t no investor prefers G to F and there is a t least one investor w ho prefers F to G. Secondly, an individual investor chooses his m ost preferred portfolio from an efficient set according his individual utility function.
In this p ap e r we deal w ith characterizatio n o f optim al efficient set. T he efficient set is optim al when it is a subset (n ot necessarily p ro p e r) o f every
Professor, D e p a rtm e n t o f Statistics, U niversity o f E conom ics in K atow ice.
possible efficient set. Each no p roper subset o f an optim al efficient set is a subset (n o t necessarily proper) o f every possible efficient set. M arkow itz (1952) and T o b in (1958) introduce M V criterion (M V - m ean variance) for characterize an efficient set. A ccording to this criterion, prospect belong to efficient set if there no o th er prospect with the sam e or larger m ean and a sm aller variance o r the sam e o r sm aller variance and a larger m ean. T he efficient set is optim al if cither the class o f utility function is q u ad ratic o r prospects arc norm ally distributed. L im itations o f q u a d ra tic utility function have been discussed by: P ra tt (1964), A rrow (1965), M anoch and Levy (1970). T h e rcccnt em pirical study (M andelb ro t, 1963; F am a, 1965) suggests th a t the d istrib u tio n o f stock price, the area in which p ortfo lio analysis has been applied - arc essentially non-norm al.
S tochastic d om inance as a criterion in po rtfolio analysis was introduced by Q u irk and S aposnik (1962), F ish b u rn (1964), H a d a r and Russelll (1969), H an o ch and Levy (1969, 1970) and m any m ore. In co n tra st to efficient set for criterion M V and the efficient set for SD criterion is optim al for w hole class o f utility functions (n ot only for q u a d ra tic one). A d dition ally SD criterion ulilizes all inform ation in th e probability d ist ributions. T h ere arc m any em pirical w orks describing relatio n sh ip between the op tim al efficient set for [ /, and U 2, the efficient set for criterion MV and the efficient set for SD criterion. In this paper we derive p aram etric criteria fo r optim al efficient set when we have a set o f prospects with asym m etric distributions.
II. S O M E T H E O R E M S O N S T O C H A S T IC D O M IN A N C E
Definition 1. F o r two random variables X and У with d istributions F and G, we say th a t X F S D У if and only if
F ( x ) ^ G ( x ) for all x e R .
Definition 2. F o r two random variables X and У with d istrib ution s F and G, we say th a t X SSD У if and only if
t t
J F (x )d x ^ J G (x)dx lo r all t e R ,
— CO — 00
if b o th integrals exists.
Definition 3. U l is th e set o f all n o n -d c crc asin g u tility fu n c tio n [ /, = {u: u! 0}.
Definition 4. U 2 is the set o f all non-decreasing and concave utility function U 2 = {m: u' ^ 0, u" ^ 0}.
Definition 5. F o r tw o prospects X and У with d istrib u tio n s F and G, an investor with utility function и prefers F if and only if
Е (« (Х ))> Е (ы (У )).
Definition 6. F o r tw o prospects X and У with d istrib u tio n s F an d G, X dom in ates У in U, a class o f utility function if and only if
Е (Ы(Х ))^ Е (Ы (У )),
with a strict inequality for som e u.
Definition 7. A n efficient set for a class o f utility fu nction U is defined as a set o f prospects with the property th a t for any p rosp ect G outside the set, th ere exists a prospect F in the set, which do m in ates G in U.
Definition 8. A n efficient set is optim al if and only if n o p ro p e r subset o f it is efficient.
Definition 9. T h e M V efficient set is defined as a set o f prospects with the prop erty th a t for any prospect G outside the set, there exists a prospect F in the set such th a t
Е ( Х ) ^ Е ( У ) and И * К И У )
w ith a t least one strict inequality (the existence o f integrals is assum ed). Definition 10. T h e F S D efficient set is defined as a set o f prospects w ith the prop erty th a t for any prospect G outside the set, there exists a prospect F in th e set such th a t F F S D G.
Definition 11. T h e SSD efficient set is defined as a set o f prospects with the p ro p e rty th a t fo r any p rospect G outside the set, th ere exists a prospect F in the set such th a t F SSD G.
T he follow ing theorem s can be used to characterize th e optim al efficient set for U 1 and U 2.
Theorem 1. F o r tw o ran d o m variables X and У with d istrib u tio n s F and G. R an d o m variable X dom inates У by first stochastic d o m in ance (X F S D
У), if E ^ X ) ) ^ Е(ы(У)) in U i (the existence o f integrals is assum ed). Theorem 2. F o r tw o ran d o m variables X and У w ith d istribu tio ns F and G. R an d o m variable X dom inates У by second stochastic d o m in an ce (X SSD У), if E(u(Ar)) ^ Е (ы (У )) in U 2 (the existence o f integrals is assum ed).
Ш. EFFECT!V EN N ES ANALYSIS KOR ASYMMETRIC DISTRIBUTIONS
Wc derive param etric criteria for optim al efficient set when wc have a set o f prospects w ith asym m etric d istribu tio ns. Wc spccify efficient sets for b o th criterion and then we com pare it. T h e follow ing theorem specifics the optim al efficient set for gam m a distrib utio n.
Theorem 3. Let F aj3 be a family o f gam m a d istrib u tio n with positive p aram eters a and ß the density functions
fa ,ß(x) = (a.ß/ r ß ) e ' axx ß~ l , x > О T hen:
(a) F ^ p F S D F ^ j if and only if a 1 < a2, (a') F Iup SSD F ^ p if and only if ol1 <ol2, (b) F ei/Ji FSD F aJj if and only if ß l > ß 2, (b') Faßi SSD F aJi if and only if ß t > ß 2,
(c) FaiJi F S D Fai ßj if and only if atj ^ a 2 and ß { ~^ß2 w ith at least one strict inequality,
(d) J*"«,,/», SSD F I|JSj if and only if ß l /ß 2 ^ m ,d x ( l,a .l /oc2) with strict inequality a t least when a j a 2 = 1.
L et the prospect have gam m a distribu tion with param eters (a, ß) then any prosp ect can be identified by corresp on din g values (a, ß). A ccording to a) and a') for tw o prospects with gam m a d istrib ution s, which differs only by p aram eter a the one with the sm aller a is preferable. Similarly according to b) and b') for two prospects with gam m a d istribu tio ns, which differs only by p aram eter ß, and a is the sam e for b o th prospects, the larger ß is preferable. U tility function, on which the preference is based, can be chosen arbitrarily from U l or U 2. As the m ean and the variance o f a prospects with gam m a distributions with param eters (a, ß) arc respectively E X = ß/u. and D 2X = Д/а2, it follows th a t in either case - prospects differing in a o r ß, the preferred prospects has the larger m ean and the larger variance. F o r any risk averters interpreting variance as a m easure o f risk, increasing m ean o f the prospect com pensates larger risk.
If we have tw o prospects differing in b oth p aram eters, then from c) we have th a t for all non-decreasing utility function, the preferred prospect should n o t have either the larger a or sm aller ß. I f one o f them has both the larger a and the larger ß, then no preference can be established. These conditio ns can be relaxed if we consider a risk averse utility function (a class oí D A R A functions). In this ease form d), prospect with th e sm aller ß is
never preferred o r prospect with the larger a is preferred only w hen it is com pensated by increased ß.
The optimal and MV cfficicnt set
If the prospects differs only in a, then from a) and a') the F S D efficient set and the SSD efficient set consist only the prospects with the sm allest a. H ence, the optim al efficient set for utility function from U t o r U 2 consist only one prospect. As m ean and variance o f the prospect with param eters (a, //) arc respectively E X = /У/а and D 2X = /ľ/ а 2, so M V efficient set consist all possible prospect.
Sim ilarly, if the prospects differs only in ß from b) and b'), the optim al cfficicnt set for utility function from i / 1 o r U 2 con tain only one prospect with the largest ß, but all prospects belong to efficient set for M V criterion. In case, if prospects differ only one p aram eters, a or ß, optim al efficient set for t/ ,j o r U 2 arc identical and they are a subset o f M V efficient set.
S uppose th a t prospects differs b oth in a and ß. T h en according c) p rospect belong to the FSD efficient set if and only if, there is no other prospect with the sam e o r the sm aller a and the larger ß o r the sam e or the larger ß and the sm aller a. T hus, the optim al efficient set for U l can co n tain m o re th an one prospect. The optim al cfficicnt set is a subset of the M V efficient set.
Let as consider the g roup o f investors from U 2 an d the prospects differs in both a and ß. T hen according part d) SSD cfficicnt set can be characterised in the follow ing way. F o r any tw o d istribution s F a< ßi an d Fa ß , F^ ß eli m inates Fat'P2 from SSD efficient set if and only if
ß j ß 2 > m a x (l, a j a 2) with strict inequality if a j a 2 = 1.
As
E FJ X ) = ß l * and
VFJ X ) = ßja.2 F aißt elim inates F ^ ßj from the M V efficient set if
/V « i > A 2/ a 2 and
ß l l* 2l> ß 2l a 2
with a t least one strict inequality. H ow ever, if' these co n d itio n hold, we m ust have:
ß i l ^ i ^ ß z l^ i anc^ a i > a 2 o r
ß ilß 2 > m a x (l, a j / a j ) and a l > a 2.
F rom criterion M V we have SSD criterion, but we have n o t the inversion. W e can observe this relation on the example when < a 2 and ß 1/ß 2 '^ m a .x (\, a j a j ) . T h e M V efficient set contains the optim al efficient set for U 2. As an exam ple we choose a prospect with gam m a d istrib u tio n w ith p aram eters (a, ß) defined as:
0 < a 0 < a < < oc, and
0 < ß o < ß < ß l < c c .
Efficient set for SSD criterion contains only one p ro sp ect F aaßi, bu t the M V efficient set contains prospect such a th a t
/? = /?i, a 0 < a < a j , and
a = a , , ß Q < ß < ß v
T h e M V efficient set includes no t only th e optim al efficient set for U 2, bu t can be larger.
Theorem 4. Let F ap be a family o f beta d istrib u tio n with positive p aram eters a an d ß the density functions
Then:
(a) p F S D FajJ if and only if < a 2, (a') F ^ ß SSD Faj ß if and only if a l < a 2, (b) F aJt F S D F a>ßi if and only if ß Y > ß 2, (b') FaPi SSD F aJi if and only if ß t > ß 2,
(c) Fai ßt F S D F„ußi if and only if a , < a 2 and ß 1 ~^ß2 with at least one strict inequality,
(d) Fai ßt SSD F X iPj if and only if ß i lß 2 ^ m a x ( l , a , / a 2) with strict inequality a t least when a j a 2 = 1.
T he m eth o d o f analysis between the various optim al and the M V efficient set are the sam e when the prospect has gam m a d istrib u tio n . T his theorem describes efficient set for F S D and SSD criterion. As an ap plication of theorem 3 we have the follow ing results:
Theorem 5. Let F„i/} be a family o f %2 d istrib u tio n in real positive param eters ß the density functions
f ß(x) = x > 0
Then:
(a) F Pi F S D F ft if and only if ß l > ß 2, (b) F ßi SSD F ßj if and only if ß t > ß 2.
T h e p ro o f is based on following relation th a t X / 2 is a gam m a distribution w ith param eters a ' = l, ß' = ß/2. A ccording to this theo rem , either U l o r U 2, if the p rospect has x 2 distribution w ith p aram eter ß, then the p ro sp e c t w ith th e larg est ß is th e m o st p re fera b le. O p tim a l efficient sets fo r l / j an d U 2 have only one pro sp ect. T h e p ro sp e ct w ith the largest p aram eter ß has the largest m ean and variance. A ccording the M V criterion all prospects belong to this efficient set. T h e M V efficient set includes the optim al efficient set.
T h e m eth o d used before we can a d o p t for the follow ing th eorem , which characterizes the optim al efficient set for the prospect w ith F -distribu tion .
Theorem 6. L et Flß be a family o f F -d istrib u tio n in real positive param eters a an d ß the density functions
= Г {(а + /Э/2}«{(«//У)х}<«/2>-1 Г(а/2)Г(Д/2)/Г{ 1 + (a//?)x}W2)/2 A ssum e th a t E FaJj(X ) exists ( ß > 2 ) then:
(a) F Xi'ß SSD FX]J if and only if < x ,> a 2, (h) F ^ Pi SSD FaJi if and only if ß v > ß 2.
If prospect have F distribution with param eters a and ß, then we c a n ’t in sim ple w aypoint ou t the conditions for the optim al sets cither [ / , or U 2. H ow ever, if we consider only U 2, the class o f risk avertcrs, and to the prospects with the sam e ß then the largest a is preferable (p art a)) and considering prospects with the same at, the sm allest ß is preferable (p art b)). It can be show n th a t the prospects with the largest a and ß, when a = ß, have the sam e m ean as an o th er prospects, but prospects the smallest variance. So, if prospects have the sam e p aram eter ß, the M V efficient set and optim al efficient set for U 2 arc identical. In c o n tra st, if prospects have the sam e p aram eter a, th en the one with the sm allest p aram eter ß, which belong to optim al efficient set for U 2, has not only the largest m ean, but also the largest variance. T he M V efficient set co ntain s all prospects and it is m uch largest th an optim al efficient set. In the case th en ß arc bigger th an 4 the variance does not exist and this co m parison are n o t valid.
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Grażyna Trzpiot
A N A L IZ A E F E K T Y W N O Ś C I D O M IN A C JI S T O C H A S T Y C Z N Y C H W Z A S T O S O W A N IA C H F IN A N S O W Y C H
Streszczenie
A n aliza p o rtfelo w a staw ia problem w yboru najlepszego spo śró d m ożliw ych losow ych p ro jek tó w inw estycyjnych. W ybór len zależy od, jedynej d la każd eg o in w esto ra, funkcji użyteczności o ra z od ro z k ła d u p raw d o p o d o b ień stw a ro zw ażan ej inw estycji. W niniejszym o p raco w an iu sk o n cen tro w an o się n a sc harakteryzow aniu zbioru o p ty m aln y ch efektyw nych inwestycji. W o d ró żn ien iu od zbioru efektyw nych inwestycji zgodnego z k ry teriu m m om entów M V , zb ió r efektyw nych inwestycji zgodny z kryterium SD jes t o p ty m aln y d la całych ogólnych klas funkcji użyteczności (nie ty lk o d la funkcji k w adratow ej). D o d a tk o w o k ry teriu m SD w ykorzystuje w szystkie w artości ro zkładu p raw d o p o d o b ień stw a p ro jek tu inw estycyjnego. Wiele p ra c em pirycznych o m aw ia zależności pom iędzy zbiorem efektyw nych inw estycji z kryterium m o m en tó w M V a zbiorem efektyw nych inwestycji zgodnym z k ry teriu m SD . W tym arty k u le p rzed staw io n e zostały wyniki a naliz w ybranych typów ro z k ła d ó w asym etrycznych.
