Preference Relations in Ranking Multivalued Alternatives in Finance Using Stochastic Dominance

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

FO L IA O E C O N O M IC A 175, 2004

G r a ż y n a T r zp io t *

P R E F E R E N C E R E L A T IO N S IN R A N K IN G M U L T IV A L U E D A L T E R N A T IV E S IN F IN A N C E U S IN G S T O C H A S T IC D O M IN A N C E

Abstract. T his study used stochastic d om inan ce tests for ranking alternatives under am biguity, to build an efficient set o f assets for a different class o f investors. We propose a two-step procedure: first test for multivalued stochastic dom inance and next calculate the value o f preference relations.

Key words: am biguity, stochastic dom inance, efficiency criteria, preference relations.

1. IN T R O D U C T IO N

W hile S tochastic D o m in an ce has been em ployed in variou s fo rm s as early as 1932, it has been since 1969-1970 developed and extensively em ployed in the area o f econom ics, finance and o p e ra tio n research. In this study the first, second and third ord er stochastic d o m in an ce rules are discussed for ra n k in g alternatives und er am biguity w ith an em phasis on th e developm ent in the area o f financial issues. T h e first p a rt o f p ap e r reviews the S tochastic D o m in an ce properties. W hile the second p a rt o f th e p ap e r deals with the effectiveness o f the various S tochastic D om in an ce rules in financial app lication.

2. S T O C H A S T IC D O M IN A N C E

In decision situ atio n s we have to co m pare m an y alternatives. W hen alternatives tak e u n ce rtain ch a rac te r we can evaluate the p erfo rm an ce o f alternatives only in a probabilistic way. In finance, fo r exam ple, problem s arise w ith stock selection w hen we need to co m p are re tu rn d istrib u tio n s. T he construction o f a local preference relation already requires the com parison

o f two p ro b ab ility d istrib u tio n s. S tochastic d o m in ance is based o n a m o ­ del o f risk averse preferences, which was d o n e by P. C. F i s h b u r n (1964) and was extended by II. L e v y and K. S a r n a t (1984), H . L e v y (1992).

Definition 1. Let F (x ) and G(x) be the cum ulative d istrib u tio n s o f two

distinct uncertain alternativ es X and У, with su p p o rt b o un ded by [a, i i ] c J ! and F (x ) Ф G(x) fo r som e x e [ a , h ] cz R. X do m in ates Y by first, second and third stochastic d o m in an ce (F S D , SSD, T S D ) if an d only if

I I t ( x ) = F ( x ) - G(x) 0 for all x e [ a , b] (F F S D G) (1)

X

I I2(x) = \ I I t (y )dy ^ 0 for all x e [ a , b] (F SSD G) (2)

Я3(х) = j H2(y)dy < 0 for all x e [ a , b] (F T S D G) (3) a

F o r definition o f F S D and SSD sec J. H a d a r and W. K. R u s s e l l (1969), G. H a n o c h a nd H. L e v y (1969) and L. J. R o t h s c h i l d and

J. E. S t i g l i t z (1970). G . A. W h i t m o r e (1970) suggested the criterion

for T S D . T h e re la tio n sh ip betw een the three stoch astic d om in an ce rules can be sum m arised by the follow ing diagram : F S D => SSD => T S D , which m eans th a t d om in an ce by F S D implies dom in ance by SSD and do m in an ce by SSD in tu rn im plies dom in an ce by T SD .

W hen, in decision situ atio n s, we have an am biguity o n value o f ran k in g uncertain alternatives, th en we m ap a p o in t prob ab ility to an am bigu ou s outcom e. P ro b ab ility d istrib u tio n m ap s probabilities to outcom es described by intervals. P ro b ab ility m ass, sum m ing to one, is d istrib u ted over the subintervals o f the o u tco m e space. T he outcom e space is co n tin u o u s, X is an interval in R and p(Ay) den o te the prob ability m ass a ttrib u te d to the subinterval o f the outco m es space, with no future basis for estab lishin g the likelihood o f a specific value in th a t subinterval. A m biguities in outcom es can be represented by a set o f probability d istrib u tio n s. E ach fam ily has two extrem e probability d istributions on outcom e space X . Low er probability d istrib u tio n is identified by p ro b ab ility m ass co ncen trated o n to m inim um elem ent or value in the subset or interval A y U pper p ro b ab ility d istrib u tio n is identified by p ro b a b ility m ass co ncentrated o n to m ax im um elem ent or value in th e subset or interval A r

Definition 2. L o w er p ro b ab ility d istrib u tio n for all values

xteX,

P * ( x i ) = T . P ( A j ) (4)

J : x , - m l n l y . y e A , )

A ccording to this definition we have: £/>*(•*<) = E i

Definition 3. U p p e r p ro b ab ility d istrib u tio n for all values x , e X , we say

P*(x i) = Z p ( ^ ) (5)

j : x , ~ m u i { y : y e A , }

N ow we also have: YjP*(x í) = 1-i

In ease o f the p o in t values o f ran d o m variable b o th d istrib u tio n s (low er and u p p e r p ro b a b ility d is trib u tio n s ) are ex actly th e sam e:

p*(x,

p*(xt)

Example 1. W e d eterm ine low er and upper p ro b a b ility d istrib u tio n s for

ran d o m variable X, which outcom es are m ultiv alu ed , include in som e intervals Aj.

T a b l e 1

Probability distribution for random variable X

[2 , 4] [3, 4] И, 5] [5, 6]

M D 0.5 0.2 0.2 0.1

A ccording to th e Def. 2 and 3 we have low er an d up per p ro b ab ility distrib u tio n s fo r ra n d o m variable X.

T a b l e 2

Lower and upper probability distributions for random variable X

x l 2 3 4 5 6

P.(x j) 0.5 0.2 0.2 0.1

O ur ap proach now is to use stochastic dom inance fo r rank ing m ultivalued alternatives by using low er and upper pro bability d istrib u tio n s o f each alternative.

Definition 4. Let tw o distinct uncertain m ultivalued alternatives X and

Y have low er p ro b a b ility d istrib u tio n s respectively F *(x) and Gt (x), upper probability d istrib u tio n s respectively F*(x) and G*(x), w ith su p p o rt bound ed by [a, b ] c R and F ,(x ) Ф G*(x) for som e x e [ a , b ] c R. We have m ultivalued first, second and th ird stochastic d om inance if and only if

H i (x ) = F , ( x ) - G \ x ) ^ 0 for all x e [ a , b\ (X F S D Y) (6)

I I2( x) = j l l y W y s i O for all x e [ a , b] ( X SSD Y) (7) a

H3(x) = X\ H2( y ) d y ^ 0, for all x e [ a , b] (X T S D Y) (8) a

E x a m p l e 2 T r z p i o t (1998a). Let take the ran d o m variables С and

D w hose outcom es are m u ltivalued, include in som e intervals A j as follows: T a b l e 3

Probability distributions for random variables С and D

A J (0, 1] [1, 2] [2, 31 [3, 4]

P(C) 0.2 0.4 0.4

P( D) 0.3 0.15 0.55

-W e can determ in e low er and up p er p robability d istrib u tio n s for ra n d o m variables С and D and next we can check th a t C T S D D (third degree m ultivalued stochastic dom inance).

3. S T O C H A S T IC D O M IN A N C E R U LES IN PO R T FO L IO SE L E C T IO N

W e have an a p p ro p ria te investm ent criteria fo r the th ree altern ativ e risk-choice situ atio n s. S tochastic d om inance theorem s assum e th a t a given class o f utility fu n ctio n can describe a decisio n-m aker’s preference stru cture. W e initially assum e th a t no in fo rm atio n is available o n the shap e o f the utility function, a p a rt from the fact th a t it is non-decreasing. A n efficiency

criterion is a decision rule fo r dividing all p o ten tial investm ent alternatives into tw o m u tu ally exclusive sets: an efficient set an d an inefficient set. Firstly, using stochastic dom inance tests we reduce th e n u m b er o f investm ent alternatives by co n stru c tin g an efficient set o f alternatives a p p ro p ria te for a given class o f investors. A t the second step, we can m ak e the final choice o f the alternatives in acco rd an ce to p artic u la r preferences o f th e investor.

T h e F S D ru le places no restrictions on the form o f the utility fu nction beyond the usual req u irem en t th a t it be nondecreasing. T h u s this criterion is a p p ro p ria te fo r risk av e rtcrs and risk lovers alik e since th e u tility function m ay co n tain concave as well as convex segm ents. O w ing to its generality, the F S D perm its a prelim inary scream ing o f investm ent alternatives elim inating those alternatives which no ra tio n a l investor (ind ep en d en t o f his a ttitu d e tow ard risk) will ever choose.

T h e SSD is the a p p ro p ria te efficiency criterion for all risk av erters. H ere we assum e the utility function to be concave. T his criterion is based on stronger assum p tion s and therefore, it perm its a m o re sensitive selection o f investm ents. O n the o th er h an d , the SSD is applicable to a sm aller g ro u p o f investors. T h e SSD efficient set m ust be a subset o f the F S D efficient set; this m eans th a t all the alternatives included in the F S D efficient set, but no t necessarily vice versa.

T h e JTSD rule is a p p ro p ria te for a still sm aller g ro u p o f investors. In addition to the risk aversion assum ption o f SSD , the T S D also assum es decreasing ab so lu te risk aversion. T h e p o p u latio n o f risk av erters w ith decreasing absolute risk aversion is clearly a subset for all risk averters, and the T S D efficient set is correspondingly a subset o f the SSD efficient set: all T S D efficient portfolios are SSD efficient, b u t n o t vice versa.

T h e th ree sto c h a stic d o m in an ce crite ria, F S D , SSD an d T S D , are optim al in the sense th a t given the assum ptions regarding the investors preferences (describing as a class o f utility functions), th e ap p lica tio n o f the corresp o n d in g stochastic dom inance criterion ensures a m inim al efficient set o f investm ent alternatives. F o r a m o re detailed d escrip tio n o f utility functions belong to th e th ree classes o f the utility fu n ctio n divided all investors to gro u p s by stochastic do m inance test see J. P. Q u i r k and R. S a p o s n i k (1962), H . L e v y and Y. K r o l l (1970), H . L e v y (1992), A. L a n g e w i s c h a nd F. C h o o b i n e h (1996).

4. PR E FER EN C E R E L A T IO N S IN RANKING M U L T IV A L U E D A L T E R N A T IV E S U S IN G ST O C H A S T IC D O M IN A N C E

W hen we verified som e o f the stochastic dom in an ce we also observed additionally th a t th e d o m in an ce is n o t equivalent. C o m p arin g results o f

ranking alternatives we can observe, th a t in one type o f stochastic dom inance the overlapping area o f the tw o com p aring d istrib u tio n s are ch an gin g but the type o f sto ch astic dom in an ce is still the same. F o r the investor, when we co m pare the re tu rn d istrib u tio n s, it can be a differen t situ atio n , so we need the m etho d for ra n k in g preference inside o f on e type o f stochastic dom inance. W e present preference relations th a t could help globally ra n k in g alternatives. W hen one o f the type o f stochastic d om in an ce is verified, we can calculate the degree o f the decision m ak er preference by using the preference relation.

Definition 5. F o r tw o distinct uncertain alternatives X and У, / ( x ) and

g( x ) are the density functions, for x e [ a , b] cz R, F(x) and G(x) arc the cum ulative d istrib u tio n s, n f and are the m eans o f the altern ativ es X and

У, we define the index

A ccording to the type o f dom in ance this index m ay take differen t values in [0, 1]. T hese values should rcflect a certain degree o f the decisio n -m ak cr’s preference relatively to the considered attrib u te. T h e clarification o f the level o f the decision m a k e r’s preference im pose us to in tro d u ce tw o o th er functions w ith values in [0, 1].

Definition 6. F o r tw o distinct uncertain alternatives X and У, / ( x ) and

g(x) are the density functions (pj(x) and pg(x) are p ro b ab ility d istrib u tio n s for the discrete case, respectively for X and У), for x e [ a , h ] c R , F(x ) an d G(x) are the cum ulative d istrib u tio n s, SV^ and SVg arc sem i-variances o f the alternatives X and У then we define:

(9)

ÍI

1 — Jm in(/"(x), g(x))dx, in the co n tin o u s cade 1 — X m i n i p / x ) , pg(x)), in the discrete case

a

a x

(10)

(1 1)

b ro m these th ree fu n ctio n s it is possible to define a degree o f credibility o f the preference relatio n o f the alternative X to the altern ativ e У.

Definition 7. F o r tw o d istin ct uncertain alternatives X and У, w ith

respect to Def. 5 and 6, we define the preference relatio n o f the altern ativ e X to the altern ativ e У as:

S(f, g) =

W , g ) ,

if f s d

^(f, g) Q(f, g),

W ,

g) •

’

0, otherw ise

T h e degree o f preference decreases progressively as we go from the dom inance F S D to the d om inance T S D . T his degree o f credibility o f the preference relation will allow us to know the n a tu re o f th e preference relation betw een tw o alternatives X and У basis o f th e characteristic obtained for three fu n ctio n s by type o f dom inan ce, in the case o f each dom inance. T h e im p o rta n t pro perties o f S are: antireflexivity, asym m etry and transivity ( M a r t e l , A z o n d e k o n , Z a r a s 1994). It is easy to apply this relation fo r ra n k m ultivalued outcom es, w hich we firstly ra n k by m ultivalued stochastic dom inance.

Example 3. L et tak e the ran d o m variables A, В and С w hose o utcom es

are m ultiv alued, include in som e intervals Aj as follows:

T a b l e 4 Probability distribution for random variable A

Aj [0,1] [1, 2] [2, 3]

P (A ) 0.2 0.4 0.4

T a b l e 5

Probability distribution for random variable В

Л> [1, 2] [2, 3] [3, 4]

P(Aj) 0.1 0.65 0.25

T a b l e 6 Probability distribution for random variable С

[1,2] [2, 3] [3, 4]

A ccording to the Def. 2 and 3 we have low er and u p p er p ro b ab ility d istrib u tio n s for this ra n d o m variables.

T a b l e ?

Lower and upper probability distributions for random variables A , В and С XJ 0 1 2 3 4 P.(A) 0.2 0.4 0.4 - -P*(A) - 0.2 0.4 0.4 -P .(B ) - 0.1 0.65 0.25 -H B) - - 0.1 0.65 0.25 p.(Q - 0.1 0.7 0.2 -p4(Q - - 0.1 0.7 0.2

N ow we can verify the stochastic dom inance. W e observed th a t B„ T S D A* and C , T S D A , (T ab. 8). So we have questio n if th a t d o m i­ nances are equivalent. F o r the investor, when we co m p are the re tu rn distributions, it ca n be a different situ atio n , so we calculate the degree o f th e d ec isio n m a k e r p refere n ce by usin g th e p re fere n ce re la tio n (Tab. 9). A ccording these results for the investor th e b etter is to choose С th an B.

T a b l e 8

R esults the analysis o f the set o f random variables A, В and С by stochastic dom inance

D om inance A, A , B, в» C , _{c .} A , X A , F S D X F S D T SD X F S D B. F S D F SD FSD X F S D F S D C , F SD T SD X C , F SD F SD FSD F S D X

T a b l e 9

R esults o f analysis o f the set o f random variables Л, В and С by the preference relations S Ф V 0 S B , T S D A* 0.2 0.25 0.1964063 0.0982 C . T S D A* 0.4 0.3 1.1173333 0.1408 5. E M PIR IC A L A P P L IC A T IO N O F M U L T IV A L U E D ST O C H A S T IC A P P R O X IM A T IO N S: E V ID E N C E FRO M T H E W ARSAW ST O C K E X C H A N G E

C o n tin u o u s o b serv atio n s o f the price o f assets from the W arsaw Stock E xchange are th e em p irical exam ple o f m u ltiv alu ed ra n d o m v ariab les. Values o f the price o f the asset are from an interval: from m inim al price to m axim al price, cach day. Daily we have empirical realisation o f m ultivalued ran d o m variables. As an exam ple o f ap plication o f the th eo ry from the previous points we m ad e an analysis o f the daily ra te o f re tu rn assets from the W arsaw Stock E xchange in Ju n e 1997. W e determ ined m u ltivalued rates o f retu rn for the set o f assets from the W arsaw S tock E xchange, an d th en we applied the m ultivalued stochastic d om inance for ra n k in g alternatives. W c can co m p are alternatives used stochastic dom in an ce tests for ra n k in g alternatives u n d er am biguity, to establish an efficient set o f asset. T h e next step o f the pro ced u re is to apply to an efficient set o f asset a preference relation ô to m ak e the final ra n k in g o f the set o f assets.

Wc started by tak in g the price o f a g ro u p o f 14 asset: A N 1M E X , B PH , B RE, BSK, B U D IM E X , D Ę B IC A , E L E K T R 1M , M O S T O S T A L E X P , O K O ­ C IM , O P T IM U S , R O L IM P E X , S T A L E X P O R T , U N IV E R S A L , W B K , which were observed a t W arsaw Stock E xchange in Ju n e 1997. F ro m the set o f in fo rm atio n a b o u t price we co u n t the m ultiv alu ed ra te o f re tu rn . In financial ap p lica tio n we have cach value from tim e series, in o u r analysis - the ra te o f re tu rn , in the sam e p rob ab ility 1/n, acco rd in g to th e tim e o f observation s (see L e v y a nd S a r n a t 1984). So we are able to build low er and upper p ro b a b ility d istrib u tio n s for the set o f assets and next we can apply the m u ltivalu ed stoch astic dom inan ce for ra n k in g alternatives.

R esults the analysis o f the set o f assets from the Warsaw Stock Kxchange in June 1997 by stochastic dom inance

W c determ ined m ultivalued rates o f return for the set o f assets from the W arsaw S tock E xch an g e in Ju n e 1997, an d th en we app lied the m u ltiv alu ed sto c h a stic d o m in a n c e fo r ra n k in g alte rn a tiv e s. F o r w hole analysis o f all 14 assets, we should m atch each o f tw o assets. W c present the results o f analysis in T ab . 4, wc read this tab le from left to the top, for E xam ple 2 SSD 3 ( T r z p i o t 1998b).

F ro m these results we have the im plications th a t ST A L E X P O R T was dom in ated by all assets. A ccording to stochastic do m in an ce rule in p o rtfo lio selection the investors can choose different assets to th eir efficient set. T he investor n eu tra l to the risk can add to efficient set: E L E K T R 1M (bccausc o f FSD ). T h e investor with aversion to th e risk can ad d to efficient set: B PH , B U D IM E X , W BK (because o f SSD). W e ca n notice th a t in ou r research period o f tim e was no t T S D th a t m eans th a t it was difficult tim e for invest for investors with decreasing aversion to the risk.

M ost o f the observed stochastic dom inance is SSD, so we need to com pare the quality o f these relations. We can calculate value o f the preference relations Ö for lower and upper d istrib u tio n s, which were im p o rta n t for m ultivalued stochastic d om inance tests. T he degree o f preference decreases progressively as we go from the d o m in an ce F S D to the d o m in ance SSD. T h is degree o f credibility o f the preference relation will allow us to know in the case o f each dom inance, the natu re o f the preference relation between tw o com paring assets based on the type o f dom inance. We present the results o f analysis in T ab . 11, read this tab le from left to the to p , for exam ple <5(2, 3) = 0.5378.

T a b l e 11 Results o f analysis o f the set o f assets from the Warsaw Stock Exchange in June 1997 by

the preference relations S

6 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 - _0.4229 2 0.5378 0.4320 0.4371 3 - _0.4203 4 - _0.4405 5 0.3295 - _0.4229 6 - _0.4225 7 0.4138 - 0.4560 0.4148 К - 0.4242 9 - _0.4545 10 - _0.4540 11 - 0.4736 12 -13 0.4238 14 0.5000 0.5504 1 - A N 1M E X , 2 - B P H , 3 - BRE, 4 - BSK, 5 - B U D IM E X , 6 - D E B IC A , 7 - ELEK - I'RIM, 8 - M O ST O ST A L E X P, 9 - O K O C IM , 10 - O PT IM U S, 11 - R O L IM P E X , 12 - ST A L E X P O R T , 13 - U N IV E R S A L , 14 - W BK.

N ow we have ad d itio n al in fo rm atio n by value o f preference re la tio n s S. As an exam ple wc ca n notice th a t all assets in d ifferent degree d o m in ate ST A L E X P O R T . W e can p ro p o se for the investor w ith aversion to th e risk efficient set (it was ch o o sin g by SSD ) with the higher value o f ô: B PH , W B K , and R O L 1M P E X (the n u m b er o f assets depends o n how m a n y assets we w ant to ta k e to the portfolio).

A fter these tw o steps o f analysis: test for m ultivalued stochastic dom inance and calculating value o f preference relations Ö, the investor can ch o ose an efficient set o f assets, acco rd in g to individual preferences. N ext he can choose a m eth o d for creatin g an individual p o rtfolio .

6. C O N C L U SIO N

M u ltivalued stoch astic ap p ro x im a tio n s have an ap p lica tio n in this class o f problem s w hen the classical p o in t o f view from ra n d o m v ariables is n o t enough, w hen we h av e a set as an outcom es o f ra n d o m variables. T h e area o f app licatio n s is very wide. W hen we determ in e m u ltiv alu ed sto chastic variables, we can do som e em pirical applications. W e can define m u ltivalued stochastic do m in an ce, an d then we can d o som e analysis on th e stock exchange. W e can use th e sam e m eth o d as in classical sto ch astic d o m in an ce and calculate the value o f preference relations ô, which help in ra n k in g the

set o f assets. T h e em pirical exam ples are the illu stratio n o f the fact, th a t wc have a n u m b er o f n o n d o m in atcd alternatives. In th e situ atio n , where d om inan ce c a n n o t be show n, the investors m ay be satisfied by in fo rm atio n a b o u t any o f n o n d o m in a tc d altern ativ e s, o r they m ay look fo r som e additional in fo rm atio n and rep eat analysis.

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G r a ż y n a T r z p io t

Z W IĄ ZK I PR E FE R E N C JI W R A N K IN G O W A N IU W IELO W A R TO ŚC IO W Y C H A LTER NATY W

W F IN A N SA C H PR ZY U Ż Y C IU D O M IN A C JI ST A T Y S T Y C Z N Y C H

W artykule w ykorzystano testy stochastycznej dominacji dla rangowanych hipotez alter­ natywnych w warunkach dw oistości w celu zbudowania efektyw nego zbioru aktyw ów dla różnych klas inwestorów. Z aproponow ano procedurę składającą się z dw óch kroków. Pierwszym jest test dla w iclow artościowej dominacji stochastycznej. W następnym kroku obliczona jest wartość dla pow iązań preferowanych.