Dipartimento di Matematica

S. ORTOBELLI, F. PELLEREY

MARKET STOCHASTIC BOUNDS AND

APPLICATIONS TO PORTFOLIO THEORY

Rapporto interno N. 22, agosto 2007

Politecnico di Torino

Corso Duca degli Abruzzi, 24-10129 Torino-Italia

MARKET STOCHASTIC BOUNDS AND

APPLICATIONS TO PORTFOLIO THEORY

Sergio OrtobelliW

Dipartimento di Matematica, Statistica, Informatica e Applicazioni

Universitá di Bergamo

Via dei Caniana, 2

24127 Bergamo, ITALY

e-mail: sol@unibg.it

Franco Pellerey

Dipartimento di Matematica

Politecnico di Torino

c.so Duca degli Abruzzi 24

10129 Torino, ITALY

e-mail: franco.pellerey@polito.it

Franco Pellerey: Partially supported by PRIN 2006 "Metodologie a supporto di problemi di

ottimizzazione e di valutazione e copertura di derivati Þnanziari "

*Corresponding author: Sergio Ortobelli: Partially supported by EX-MURST 60% 2005,

2006, 2007.

1

MARKET STOCHASTIC BOUNDS AND

APPLICATIONS TO PORTFOLIO THEORY

Abstract: This paper explores and analyzes the implications and the advantages of safety Þrst

analysis in portfolio choice problems. In particular, we describe the market bounds in order to

examine the evolution of investor�s choices corresponding to the market trend behavior. Thus, Þrst

we prove the uniqueness in distribution of market bounds when limited short sales are allowed.

Then we propose an empirical comparison among optimal choices strategies based on a portfolio

function of the upper market bound. Finally, we analytically determine the link between stochastic

choices and market bounds when the returns are elliptical distributed and no short sales are allowed.

Key words: Stochastic bounds, e!cient frontier, stochastic dominance, safety Þrst optimal port-

folio, elliptical distribution.

JEL ClassiÞcation: G11, G14

2

1 Introduction

The aim of this paper is to provide a comprehensive presentation of the measurement and appli-

cations of the market stochastic bounds in order to understand and forecast the investors� choices

changes. In particular, the paper proposes some new tools to manage return portfolios considering

the evolution of optimal choices respect to the market trend.

The theory of portfolio choice is based on the assumption that investors allocate their wealth

across the available assets in order to maximize their expected utility. One of the Þrst rigorous

approximating result to the portfolio selection problem was given in terms of the mean and the

variance by Markowitz and Tobin. Mean variance theory found its foundation and justiÞcation in

arbitrage theory and in stochastic dominance analysis. In particular, stochastic dominance analysis

justiÞes the partial consistency of mean-variance framework with expected utility maximization

when the portfolios are elliptically distributed.

Almost in contradiction to maximization of expected utility approach, there has been the grad-

ual development of a theory of decision making which has focused on agents who seek to attain

some aspiration or target level of outcome through their actions. However, it is possible to prove

that the two approaches are related and in some cases equivalent even if the economic reasons and

justiÞcations are di�erent (see Bordley, LiCalzi. (2000) Castagnoli, LiCalzi (1996), Ortobelli and

Rachev (2001)). In portfolio literature, Roy (1952), Tesler (1955/6), Bawa (1976, 1978), suggest

the safety Þrst rules as a criterion for decision making under uncertainty. In such models, a subsis-

tence or disaster level of returns is identiÞed. The objective is taken to be the maximization of the

probability that the returns are above the disaster level (or some other variation on this theme).

Further extensions and developments have followed these primary works (see, among others, the

discrete time extensions of Roy (1995), Li, Chan, Ng (2000) or the continuous time extensions of

Majumdar and Radner (1991) and Dutta (1995)).

This paper describes some management tools to forecast the behavior of future choices. In

particular, we study the evolution of investors� optimal choices depending on the market stochastic

bounds when no short sales are allowed. The distribution functions of market stochastic bounds

are the envelopes of the portfolio distribution functions and the upper stochastic bound represents

optimal choices of all admissible investors. Using the upper stochastic bound in an approximating

linear model, we can study and analyze the market trend and we can give a Þrst naive interpretation

to safety Þrst analysis in markets with short sale restrictions. The metodology suggests a way to

determine an average portfolio of optimal choices.Thus with a Þrst ex-post empirical analysis we

compare the forecasting power of these tools. Moreover, when distribution functions of returns

are elliptical, we can analytically describe how choices change with respect to the upper market

stochastic bound even if the process requires to rewrite the Markowitz e!cient frontier in terms

of the mean. So, Þrst we describe an alternative algorithm to Markowitz� one in order to rewrite

recursively the e!cient frontier when no short sales are allowed. The algorithm, determines the

e!cient frontier as function of the return mean. In such way we can describe analytically the

3

evolution of optimal choices as a function of the upper stochastic bound that is generated by the

optimal choices of the e!cient frontier.

In the next section, section 2, we deÞne the stochastic bounds of the market. The third section

introduces the analysis of market trend. Fourth section proposes an empirical comparison among

the dominating portfolios and the portfolio that maximizes the Sharpe ratio. The Þfth section

describes market bounds when returns are elliptically distributed. Sixth section brießy summarizes

the paper, while all the proofs of the results presented here are described in the appendix.

2 Stochastic Bounds

Suppose we have a frictionless market without arbitrage opportunities where all investors act as

price takers. Suppose that all investors have the same temporal horizon at a Þxed date in the future.

Given q risky securities, we indicate with ul the rate of return of the lwk security and ]l = 1+ul the

respective gross return. The random vector of the gross returns is deÞned on the polish probability

space ( >=> S ):

] = []1> ===> ] q]0> (1)

In a market with limited short selling opportunities every admissible vector of portfolio weights

{ = [{1> ===>{ q]0 belongs to the compact set

W =

(| 5 <q@

X

l

|l = 1 ; jl � |l � kl

)>

where (j1> == = > j q) and (k1> = = = >k q) are Þxed vectors belonging to <q. In this case the portfolios

are stochastically bounded, i.e., there exist two random variables \X and \O such that for every

admissible vector of portfolio weights { = [{1> ===> { q]0 belonging to the set we have \X IVG{ 0]

IVG\ O.1

In this case, we can analytically express the distribution functions of the optimal bounds. As

we can observe in the following theorem these bounds are related to the portfolio weights

{X (�) 5 argµinf{MW

IP({0] � �)

¶and {O(�) 5 arg

µsup{MW

IP({0] � �)

¶

1We recall that X Þrst stochastically dominates Y (X FSD Y or [ D1\ ) if and only if E(!([)) D E(!(\ )) for

every non-decreasing function ! such that the two expectations exist, i.e., if and only if I[(w) $ I\ (w) for every real

t. More generally, we say that X dominates Y with respect to the � ( � D 1) stochastic dominance order ( [ D�\ )

i� I(�)[ (w) $ I

(�)\ (w) for every real t where

I(�)[ (w) =

1

K(�)

w]

3"

(w3 |)�31gI[(|) =E�(w3[)�31+

�

K(�)

(see Fishburn (1980)).

4

and to the functions

IX (�) = inf{MW

IP({0] � �) and IO(�) = T(�+) = lim�<�+

T(�) = sup{MW

IP({0] � �)=

These two functions represent the envelopes of all portfolio distribution functions and are themselves

distribution functions as proved in the following theorem.

Theorem 1 Assume limited short sales are allowed in the market. Then IX is the "smallest"

cumulative distribution (in FSD sense) which FSD dominates all portfolios, and it has support

[a> b] where d = sup{MW

f({)> e = sup{MW

g({), and

f({) = sup©f 5 < | IP({0] � f) = 0

ª

g({) = inf©g 5 < | IP({0] Ag ) = 0

ª=

Similarly, IO is the "greatest" cumulative distribution (in FSD sense) which is FSD dominated by

all portfolios, and it has support [ed>ee] where ed = inf{MW

f({), and ee = inf{MW

g({)=

Moreover, when a riskless gross return }0 is allowed, then d � }0 and;e � }0=

Note that two cases are possible:

i) there exists a portfolio {0] that FSD dominates all the others (or is FSD dominated by all the

others), then it would have IX as distribution (it would have IO as distribution);

ii) it does not exist a portfolio {0] in the market that FSD dominates (or is FSD dominated by)

all the others.

Next, we will concentrate our attention on case ii), that is the most common one.

When only limited short sales are allowed, we call IX the stochastically dominating distribution

of all admissible portfolios and IO the stochastically dominated distribution. Let \X and \O be

two random variables (unique in distribution) deÞned on a given probability space (e > e=> eS ), withrespectively the stochastically dominating distribution IX and the stochastically dominated distri-

bution IO. Generically, we deÞne \X and \O as the stochastic bounds of all admissible portfolios

and in particular we call \X preferential market growth because it represents the maximum factor

of growth of future wealth. In some sense, \X is the maximum price that we can give at the future

risky wealth for an unity of wealth invested today. Observe that by deÞnition

\X IVG{ 0] IVG\ O

for every portfolio weight { belonging to the bounded set of admissible choices. The above discussion

can be easily extended to a given category of investors. As a matter of fact, when limited short

selling opportunities are allowed, we can consider the non-dominated choices with respect to a given

��stochastic order (see Fishburn (1980)) belonging to W�> closure of the set

5

W� =

½{(�)(w) 5 W

Áw 5 U> {(�)(w) 5 arg

µinf{MW

E¡(w� {0])�31+

¢¶¾=

Note that W�is a compact subspace of T. Thus, we can easily prove the following theorem.

Theorem 2 Assume limited short sales are allowed in the market. Then

IX>(�)(�) = inf|MW�

IP(|0] � �)

is the "smallest" distribution (in FSD sense) which FSD dominates all portfolios in W�= Similarly,

IO>(�)(�) = T(�)(�+)> where T(�)(�) = sup

{MW�IP({0] � �)>

is the "greatest" cumulative distribution (in FSD sense) which is FSD dominated by all portfolios

in W�=

As for the stochastic bounds of all admissible portfolios, we can deÞne on a given probability

space (e > e=> eS ) two random variables \X>(�) and \O>(�) (unique in distribution) with respectively thestochastically dominating distribution IX>(�) and the stochastically dominated distribution IO>(�).

Therefore, for every random variable G1g6= \X>(�) which FSD dominates every portfolio {

0] in W�,

then G1 IVG\ X>(�), and for every random variable G2g6= \O>(�) which is FSD dominated by

every portfolio {0] in W�, then \O>(�) IVGG 2= Similarly, we can extend these considerations to

the portfolios not dominated with respect to any given order of preferences. For example we could

compute stochastic bounds and optimal portfolios of non satiable risk lover investors, belonging to

WUO

> closure of the set WUO =

½{(UO)(w) 5 W

Áw 5 U> {(UO)(w) 5 arg

µsup{MW

E (({0] � w)+)

¶¾=

3 Market trend

In this and next sections we will consider portfolio choice among q risky assets with returns

] = []1> ===> ] q]0 when no short sales are allowed, i.e. portfolio weights

{ 5 V =

(| 5 Uq : |l � 0;

qX

l=1

|l = 1

)=

Even if this hypothesis can be relaxed allowing limited short sales. We also assume that it does

not exist a portfolio {0] in the market that FSD dominates (or is FSD dominated by) all the

others, and we examine stochastic bounds for all admissible portfolios. First of all we discuss the

uniqueness, for every � belonging to [d> e]> of the portfolio weight

{X (�) 5 argµinf{MVIP({0] � �)

¶

6

when limited short sales are allowed. Actually, the uniqueness is not always satisÞed in presence of

riskless assets, as shown in the following counterexample.

Counterexample: Let }0 be the return of a riskless asset, and suppose that the vector ] of the

other assets is jointly Gaussian distributed. Under this assumptions, we generally share the risky

components from the riskless one for notation, i.e., for every | 5 eV =n| 5 Uq+1 : |l � 0;

Pq+1l=1 |l = 1

o

we consider | = ({> 1�{0h); { 5 <q> h = [1> ===> 1]0= Then every portfolio (1�{0h)}0+{0]> is Gaussiandistributed with mean (1� {0h)}0+ {0E(]) and variance {0T{, where T is the variance covariance

matrix of ]> and 1� {0h � 0> {l � 0 for every l = 1> ===> q= When there exist a risky portfolio with

mean greater than }0, non-satiable risk averse investors maximize the Sharpe ratio

(1� {0h)}0 + {0E(])� }0s{0T{

={0 (E(])� h}0)s

{0T{

(see Sharpe (2004)), and all optimal choices belong to a linear combination between the riskless

return }0 and the market portfolio e{0], where e{ is the solution of the optimization problem

sup{:{lD0;

Sql=1 {l=1

{0E(])� }0s{0T{

=

Thus in this case, when � = }0> we get that

�e{ 5 argµ

inf{:{lD0;

Sql=1 {l$1

IP((1� {0h)}0 + {0] � �)

¶

= arg

Ãsup

{:{lD0;Sq

l=1 {l$1

(1� {0h)}0 + {0E(])� �s{0T{

!

= arg

Ãsup

{:{lD0;Sq

l=1 {l$1

{0 (E(])� h}0)s{0T{

!

for very � 5 (0> 1], i.e., {X (�) is not unique. ¥

As a matter of fact, the above counterexample can be easily extended to a scalar and translation

invariant family of unbounded distribution functions � = {I[} (i.e., such that if I[ 5 � then alsoI�[ and I[+w belong to � for any real u,w and any � A 0) that is uniquely determined by the Þrst

k moments= In fact, the following statement holds.

Corollary 1 Let the portfolios of risky gross returns {0] belong to a scalar and translation invariant

family of unbounded distribution functions absolutely continuous and uniquely determined by the

the mean � and the Þrst k central moments �2> �3> ===>� n= Suppose a riskless portfolio is allowed

with gross return }0 and suppose there exist a risky portfolio with mean greater than }0= Then {X (�)

with � = }0 is not unique.

The above Corollary can be further extended to more general scalar and translation invariant

families of distribution functions (see Ortobelli (2001)). On the other hand we can guarantee the

uniqueness when the risk-free asset is not allowed and the risky returns are elliptically distributed

7

(see Ortobelli and Rachev (2001)). For this reason in all the following considerations we assume

that risk-free assets are not allowed.

Since the stochastic bounds are the market limits, then the market trend is implicitly described

by them and it has sense studying the evolution of investor�s choices with respect to the market

trend. Assume now that {X (�) = arg

µinf{MVIP({0] � �)

¶is a measurable vectorial function. Then,

by the deÞnition of the stochastically dominating distribution IX it follows that

I\X (�) = IX(�) = IP({0

X (�)] � �) ;� 5 <=

We know by Castagnoli and LiCalzi (1996) that maximization of the expected utility is equivalent

to minimization of the probability of being under a given target. Since at any time the investor

know of being lower the preferential market growth, it is intuitive to suppose that every non satiable

investor tries to minimize the probability that his/her choice is lower a given value of the preferential

market growth. Next, we will refer to this assumption as the based-target axiom. Assuming that the

based-target axiom holds, optimal choices are well represented by the random vectorial surjective

function [X deÞned on (e > e=> eS ) by

[X (e$) = {X (\X (e$)) = argµinf{MVIP({0] � \X (e$))

¶for every e$ 5 e =

We call the random vectorial function [X the dominating portfolio of all admissible choices.

The dominating portfolio is a random indicator of the investors� optimal choices, since it represents

all the choices that minimize the probability of losses respect to the upper stochastic bound \X .

Therefore, it could be important to understand how optimal allocations varies among the q

risky assets under preferential market growth changes. That is, the implicit question is: �How

does [X change when \X changes ?� or, more suggestively, �Where does the market go?�. These

questions have not easy solutions even when returns are elliptical distributed, because if no short

sales are allowed it is still di!cult to describe the Markowitz Tobin e!cient frontier. Moreover,

the portfolio distributions are not necessarily elliptical distributed. Thus, when \X admits Þnite

variance �2\X , we can consider the best linear approximation between [X and \X with the least

square estimator (LSE). Thus, we can assume that the following model holds:

[X>l = E([X>l) +fry([X>l> \X )

�2\X(\X �E(\X )) + %l> l = 1> = = = >q> (2)

where H(%l%m) = 0 for all l 6= m> H(%m) = 0 for all m = 1> = = = >q , andPl=1

q%l = 0=

In this model we can distinguish di�erent market indicators that can be interpret by an economic

point of view:

a) The vector of the expected values of the dominating portfolio

{G = E([X ) =

Z

hl{X � \Xg eS =

Z

[d>e]{X (�)gIX (�)>

8

which represents the portfolio weights where the investors� preferences collapse in average.

Thus it is the most logic approximation of investors� choices. In addition, when the set of all

optimal safety Þrst portfolios

X =

½{ : { = arg

µinf{MVIP({0] � �)

¶> � 5 <

¾

is convex, then {G belongs to the portfolio weight e!cient frontier X , since it is a convex

combination of optimal portfolios.

b) The average of the upper stochastic bound

;� = E(\X ) =

Z

hl\Xg eS =

Z

[d>e]�gIX (�)>

which is another interesting indicator of the market growth, since \X is the Þrst random

variable which dominates all portfolios. Then;� can be considered as the average of the

maximum future price of risky wealth for a unity of wealth invested today.

c) Moreover, we can interpret the portfolio

|G =E([X\X)

;�

=

R[d>e] �{X (�)gIX (�)

;�

as the value that we give �today� to the di�erently allocated unitary wealth capitalized

�tomorrow� with the maximum future price of risky wealth. Thus, |G represents in some

sense how we look �today� at the evolution of market portfolio �tomorrow�.

The model (2) is well deÞned: in fact [X remains a random vector of portfolio allocations

because if we consider the vector fry([X > \X ) then we see that fry([X > \X ) = E([X\X)�;�{G =

;�(|G � {G), and therefore

Pql=1[X>l = 1=

We observe that the dependence of the lwk component of [X with \X is determined by the sign

of fry([X>l> \X )= Therefore, the above relations can be interpreted in the following way:

1. If |G>l � {G>l is greater than zero then we can think that component {X>l(\X ) has the same

trend than \X = Thus, if the market is growing then non satiable investors tend to increase

their position on the lwk investment.

2. If |G>l�{G>l is lower than zero then we can think that component {X>l(\X ) has opposite trend

than \X = Thus, if the market is growing we have that non satiable investors tend to disinvest

their position on the lwk investment.

3. Opposite conclusions follow when market is downing.

9

This analysis is coherent with the economic interpretation given to the portfolios |G and {G,

and could be potentially useful to address the future risk management choices. In addition, all the

indicators;�> |G and {G can be numerically estimated, and note that, even if we do not express

explicit distributional assumptions on the returns, we can get linear approximations between [X

and \X = Clearly, this approximations can be only used to understand indicatively the trend of the

market growth by the point of view of non-satiable investors.

Finally, observe that all the above considerations can be extended to the stochastic bounds of

portfolio belonging to any compact set of optimal choices contained in T=S (such as W�or W

UO).

For example, let us suppose we have N observations with probability tn (k=1,...,N ) of n risky assets.

Typically, we can assume tn = (1 � j1@Q )j(Q3n)@Q + j@Q> where j 5 (0> 1]> so that when g=1 weimplicity assume the observations are i.i.d. otherwise we also consider an exponential wheighted

behavior of the historical observations (see, among others, Longerstaey and Zangari (1996)). Then,

in order to determine optimal portfolios for non satiable risk averse investors belonging to W2, we

have to solve the following linear optimization problem for di�erent values of w:

min|

QX

n=1

yn

qX

l=1

|l = 1; |l � 0; l = 1> ===>q

yn � 0; yn �Ãw�

qX

l=1

|l}l>n

!tn n = 1> ===>Q

where }l>n is the k -th observation of the i-th gross return. Then we can estimate the indica-

tors relative to non satiable risk averse IX>(2)(�) = inf|MW 2

IP(|0} � �)>;�(2) = E(\X>(2))> |G>(2) =

E([X>(2)\X>(2));

�(2)

and {G>(2) = E([X>(2)) for the class of non-satiable risk averse investors, where

[X>(2)(�$) = arg

Ãinf{MW 2

IP({0] � \X>(2)(�$))

!for every �$ 5 e deÞnes the dominating portfolio of

non satiable and risk averse choices. Analogously, we can estimate optimal choices of non-satiable

risk lover investors solving a similar optimization problem where we have the maximization instead

of minimization and the constrains yn � tn

µqPl=1

|l}l>n � w

¶; n = 1> ===> Q instead of constrains

yn � tn

µw�

qPl=1

|l}l>n

¶; n = 1> ===>Q . Similarly, we can also estimate the indicators

;�(UO)> |G>(UO)>

{G>(UO) of non satiable risk lover investors.

4 A Þrst empirical analysis of stochastic bounds

In this section we propose an empirical analysis of the stochastic bounds presented above. Since

we do not know the distribution functions of the asset returns, then we approximate the indicators

10

{G,;� and |G using the empirical distributions of portfolio returns. The simplest way to get an

estimation for these quantities is by taking a partition of [d> e]> say [�1 = d> �2> ===> � i = e], and

considering the approximations

;� = �1IX (�1) + �2(IX (�2)� IX (�1)) + ===

===+ �i (IX (�i )� IX (�i31))>(3)

{G = {(�1)IX (�1) + {(�2)(IX (�2)� IX (�1)) + ===

+{(�i )(IX (�i )� IX (�i31))>(4)

and|G =

1;

�[�1{(�1)IX (�1) + �2{(�2)(IX (�2)� IX (�1))+

===+ �i{(�i )(IX (�i )� IX(�i31))] =(5)

In order to test the forecasting power of these indicators, we propose an ex-post analysis. In

particular, we use daily returns quoted on the market from January 1995 to May 2005. By assuming

that short selling is not allowed, we examine optimal allocation among 10 stock returns components

of the Dow Jones Industrials2: Altria Group, Citigroup, General Electric, Home de pot, Intel,

Johnson and Johnson, Microsoft, PÞzer ,United Technologies, and Wal Mart Stores. We propose a

performance comparison with two di�erent assumptions: in the Þrst we assume i.i.d. observations

and we make use of a window of 250 daily observations to approximate the optimal portfolios; in

the second we use a window of 750 daily observations and we assume the k-th observation has the

probability tn = (1 � j1@Q )j(Q3n)@Q + j@Q> .with j = 1034. In particular, we compare the Þnal

wealth processes obtained with the portfolios {G> and |G with the ex-post wealth process that an

investor could obtain investing in the portfolio maximizing the Sharpe ratio {0E(])3}0I{0T{

(assuming a

null riskless return, i.e., }0 = 0).

Suppose the decision makers choose to invest every day their wealth in the portfolios {G> and |G>

we compute the resulting ex-post sample paths of the Þnal wealth processes. For it, we approximate

{G and |G> by means of formulaes (3), (4) and (5), computing 100 optimal portfolios {(�l), with

�l 5 [d> e]> l = 1> ===> 100, assuming also that investors have an initial wealth Zw0 = 1= Thus, once we

approximated the optimal portfolios {G>(wn)> and |G>(wn) at each time wn> we calculate the ex-post

Þnal wealths Z({)wn+1 := Z({)wn ({G>wn)0 ](wn+1) and Z(|)wn+1 capitalized at time wn+1 with the ex-

post observed gross returns ](wn+1)= In the Þrst four Þgures we compare the di�erent results from

18th December 1995 till 24th May 2005 assuming i.i.d. observations (i.e., tn = 1@250)> while in

the last Þgure we compare the ex-post Þnal wealths when historical observations are exponential

weighted.

Figure 1 presents the comparison among the ex-post Þnal wealth processes. Note that, in prac-

tice, there is no di�erence between the sample path of the Þnal wealth obtained with portfolios |G

2We use data from DATASTREAM

11

Figure 1: This figure presents the ex-post final wealth processes (from December 1995 till May 2005) obtained investing daily either in the portfolio Dx , or in Dy or in

the portfolio that maximizes the Sharpe ratio.

Figure 2: This figure presents the daily difference (from December 1995 till May 2005) between the components of portfolios Dy and Dx .

12

Figure 3: This figure presents the daily expected upper and lower bounds for all non satiable investors valued from December 1995 till May 2005.

and the one obtained considering portfolios {G even if both portfolios {G and |G give a better per-

formance than that obtained considering the Sharpe ratio. The reason of this equality is explained

by Figure 2, that reports the sample path of the di�erences |G>l � {G>l between the components of

{G and |G. These di�erences are of order 1033, and do not imply a substantial di�erence in the

Þnal wealth.

In Figure 3 we compare the behaviors over time of the expected upper and lower bounds. In

particular, we observe that during the market crises the distance between the two bounds is higher

since the market is much more volatile. The geometric mean of the expected upper and lower bounds

E(\X ) and E(\O) during the period of observation are respectively 1.00612809 and 0.99615493.

Thus, probably, there exist many strategies that could give better performances than that obtained

with the expected dominating portfolio for non satiable investors. For example, let us compute

the ex-post Þnal wealth sample paths that one can obtains considering the expected dominating

portfolios {G>(2) and {G>(UO) of non satiable and respectively risk averse and risk lover investors

(analogous results can be obtained considering the portfolios |G>(2) and |G>(UO), respectively).

Figure 4 shows the comparison among the ex-post Þnal wealth processes. As we could expect, the

strategies of non satiable risk lover investors sometimes give better performance than the other

strategies. However, these strategies loss much more than the others during the period around

September 11th 2001 and subsequent crises, and among all strategies the strategy based on the

expected dominating portfolio of all non satiable investors gives better performance. Finally, in

Figure 5 we compares the performance of all ex-post Þnal wealth processes from 17 th October 2000

till 24th May 2005 when we also assume an exponential probability of the historical observations

(i.e., we use j = 1034> N=750).

Even in this case we observe the better performance of the expected dominant portfolio of all

13

Figure 4: This figure presents the ex-post final wealth sample paths from December 1995 till May 2005 when we assume i.i.d. observations. We compare the ex-post final wealth processes obtained investing daily either in the portfolio Dx , or in ,(2)Dx , or in ,( )D RLx , or in the

portfolio that maximizes the Sharpe ratio.

Figure 5: This figure presents the ex-post final wealth sample paths from October 2000 till May 2005 when we assume exponential weighted observations. We com pare the ex post final wealth obtained investing daily either in the portfolio Dx , or in ,(2)Dx , or in ,( )D RLx , or in the

portfolio that maximizes the Sharpe ratio.

14

non satiable investors. In particular, we believe that further discussions and extensions born from

this Þrst empirical analysis, since probably:

a) there exist an opportune value j 5 (0> 1] such that the valuation of the historical observationsgives better performance;

b) we should consider dynamic strategies taking into account the intertemporal dependence of

historical observation;

c) we should examine non dominating strategies with respect to behavioral orderings (see Levy

and Levy (2002)).

However, a more general empirical analysis with further studies and comparisons of the above

model does not enter in the objective of this paper.

5 Market trend analysis with elliptically distributed returns when

the risk-free asset is not allowed

It is well-known that asset returns are not normally distributed, since the excess kurtosis found in

many empirical investigations led them to reject the normality assumption and to propose alterna-

tive distributional assumption for asset returns (see Mandelbrot (1963) and Fama (1965)). There

exist many elliptical distribution families which present heavier tails than Gaussian distribution, for

example stable sub-Gaussian distributions and t-student families (see, among others, Mittnik and

Rachev (2000)). This is the main reason because we suppose that the vector ] of risky returns ad-

mits a joint unbounded elliptical distribution (see Owen and Rabinovitch (1983) and Chamberlain

(1983)). Thus every risky portfolio return {0] is an unbounded random variable with cumulative

distribution:

IP({0] � �) =

�Z

3"

1

F0({0T{)1@2j

µ(w� {0�)2

{0T{

¶gw> (6)

where j is a non-negative integrable function, � = E(]) is the mean of the vector ], T is the

positive deÞnite dispersion matrix of the primary asset returns, and

F0 :=

+"Z

3"

1

({0T{)1@2j

µ(w� {0�)2

{0T{

¶gw=

Under this distributional assumption, the vectorial application

{X (�) = arg

µinf{MVIP({0] � �)

¶= arg

µinf{MV

�� {0�

({0T{)1@2

¶

is a measurable vectorial function. Clearly, in order to understand the relation between [X and

\X we need to Þnd the function {X (�) because [X = {X (\X )= In some special cases we can easily

represent the frontier X as we evince by the following proposition.

15

Proposition 1 Assume the primary gross returns to be unbounded random variables jointly ellip-

tically distributed with non singular dispersion matrix and with distribution function (6). Suppose

that there are risky assets traded in a frictionless economy where only limited short sales are al-

lowed. Thus, if all the components of portfolio weights on the Markowitz-Tobin e!cient frontier

are positive and the return portfolio with global maximum mean {0W] admits also global maximum

dispersion, then

[X =

(\XT

31h3T31�\XF3E if \X � E({0W])E3D

E({0W])F3E

{W if \X A E({0W])E3DE({0W])F3E

(7)

and

I\X (�) =

;AAA?

AAA=

R(3">�)

�F3EF0(�2F32�E+D)1@2

j

µ(w�2F33wE3�E+D)

2

(�2F32�E+D)

¶gw> if � ? E({0W])E3D

E({0W])F3E

�R3"

1F0({0WT{W)

1@2 j³(w3{0W�)2{0WT{W

´gw> if � � E({0W])E3D

E({0W])F3E>

where D = �0T31�> E = h0T31�; F = h0T31h and {W = hl (i.e., the vector with 1 in the i-th

component and 0 in the other components assuming that the i-th component corresponds to the

gross return with maximum mean and maximum dispersion).

Note that under the assumptions of the previous Proposition 1 the choices of non-satiable

investors are the same as those of non-satiable risk averse investors (see Bawa (1976)), thus \X =

\X>(2) and [X = [X>(2)= Generally, Markowitz- Tobin e!cient frontier presents switching points

on the components of optimal portfolios, thus the analysis of the dependence between [X and \X

is more complex. Traditional wisdom says that each switching point corresponds to a kink, while

Dybvig (1984) has shown that there is a kink at a risky portfolio only if that portfolio is a portfolio

of securities of equal expected return. Thus, if all securities have di�erent expected return, a kink

on the frontier can occurs only if a single security is held at that point.

Markowitz (1959, 1987) proposed an algorithm to determine recursively the e!cient frontier.

However we observed that the Markowitz Tobin e!cient frontier was not expressed by Markowitz

in terms of the mean and of a dispersion measure as we do in the next subsection. Moreover,

the following extension permits us to describe the e!cient frontier using, for example, the stable

sub-Gaussian dispersion measure instead of the variance (see Ortobelli et al (2004)), i.e., using a

more realistic approximation of return distributions.

5.1 A general analytical formulation for [X(\X) and IX

Let us assume that the vector of returns z is ordered in the sense of the mean, i.e. H(]1) ?

H(]2) ? ===? H (]q). Therefore, we assume that the securities have di�erent expected returns in

order to limit the presence of kinks on the e!cient frontier (see Markowitz (1959, 1986), Dybvig

(1984, 1985), Vörös (1986, 1987)). Markowitz has shown that when:

16

- there are not redundant assets and variance covariance matrix T is deÞnite positive;

- there is one portfolio for each mean-variance combination;

then the mean-variance e!cient frontier obtained solving the optimization problem

min{ {0T{

subject to

{l � 0; {0h = 1> {0� = p>

(8)

for di�erent values of the mean p> is described by some parabolic arcs where the variance is an

increasing function of the mean. The same analysis holds in a mean-dispersion plane when returns

are elliptically distributed and matrix T is a deÞnite positive dispersion matrix (see Ortobelli and

Rachev (2001)).

In particular, let us denote with ln 5 Ln = {l 5 L = {1> 2> ===> q } | {l = 0} the null components

of the optimal portfolio { for the n-th parabolic arc. Then the optimal solutions of n-th parabolic

arc are univocally determined by the optimization problem

min{12

¡{(n)

¢0T(n){(n)

subject to¡{(n)

¢0h(n) = 1>

¡{(n)

¢0�(n) = p>

(9)

for some opportune values of the mean p> where T(n) is the matrix obtained by T eliminating

every ln-th row and every ln-th column ;ln 5 Ln, while h(n)> {(n) and �(n) are the vectors obtained

eliminating every ln-th element by h = [1> ===> 1]0> { = [{1> ===>{ q]0 and � = E(]).

Problem (9) can be analytically described using the Merton�s derivation (Merton (1972)), thus

the portfolio weights of the n-th arc are given, for p 5 (pn>pn+1], by

{(n) =

³F(n)

¡T(n)

¢31�(n) �E(n)

¡T(n)

¢31h(n)

´p

D(n)F(n) �¡E(n)

¢2 +D(n)

¡T(n)

¢31h(n) �E(n)

¡T(n)

¢31�(n)

D(n)F(n) �¡E(n)

¢2 > (10)

where D(n) =¡�(n)

¢0 ¡T(n)

¢31�(n); E(n) =

¡h(n)

¢0 ¡T(n)

¢31�(n) and F(n) =

¡h(n)

¢0 ¡T(n)

¢31h(n),

whilepn is the mean in a switching point i.e., it is the mean of a portfolio where the components are

changing (see the Appendix on how to get the means pn and the sets Ln). Such {(n) represents the

portfolio weights of positive portfolio components, and we denote with e{(n) the previous portfoliovalued in p = pn. Since we assume that all securities have di�erent mean, then portfolio e{(n)represents a kink in the mean-dispersion frontier only if a single security is held at that point. Here

n varies from 1 to n, the value p1 is the mean of the global minimum dispersion portfolio {min, and

pn = E(]q) is mean of the return with the greatest mean. Moreover, if the asset return with the

greatest mean has also the greatest dispersion, then the e!cient frontier of non satiable investors

is equal to the frontier of non-satiable risk averse investors (see Bawa (1976)). Otherwise, we have

to consider further optimal portfolios in order to obtain the optimal choices of all non-satiable

investors.

17

For this last case, recall that Bawa (1978) has shown that the optimal portfolios for non-satiable

investors is equivalent to X =

½{ : { = arg

µinf{MVIP({0] � �)

¶> � 5 <

¾when returns� distributions

are elliptical, and these portfolios are given by:

a) the Markowitz-Tobin e!cient frontier;

b) some parabolic arcs of maximum dispersion portfolios obtained as the linear combination of

couple of risky returns with dispersion greater than �q (dispersion associated to the maximum

mean return).

Thus, in order to describe the frontier X we can use Bawa�s recursive algorithm. In partic-

ular, in each additional arc of the non-satiable e!cient frontier we obtain that the components

of portfolio are all null except for two assets (see Bawa (1976)). At the extremes of these arcs

we have portfolio composed by an unique security that create a kink. If we distinguish with

ls 5 Ms = {l 5 L = {1> 2> ===> q } | {l = 0} the null components of the optimal portfolio { for the s-th

parabolic arc, then the optimal solutions of the s-th additional parabolic arc are {l = 0 ;ls 5 Ms

and, for ls @5 Ms,

{(s) =

³F(s)

¡T(s)

¢31�(s) �E(s)

¡T(s)

¢31h(s)´p

D(s)F(s) �¡E(s)

¢2 +D(s)

¡T(s)

¢31h(s) �E(s)

¡T(s)

¢31�(s)

D(s)F(s) �¡E(s)

¢2 >

ifp 5 (ps+1>ps], whereT(s) is the 2×2matrix obtained byT eliminating every ls-th row and every

ls-th column, ;ls 5 Ms; h(s)> {(s) and �(s) are the vectors obtained eliminating every ls-th element

from h = [1> ===> 1]0> { = [{1> ===> { q]0 and � = E(]), respectively, while D(s) =

¡�(s)

¢0 ¡T(s)

¢31�(s),

E(s) =¡h(s)¢0 ¡

T(s)¢31

�(s) and F(s) =¡h(s)¢0 ¡

T(s)¢31

h(s). We denote with e{ps the portfolio

composed by an unique security with mean ps.

The procedure concludes with the asset return having the greatest dispersion (that we assume is

represented by portfolio {W). Note that in order to determine ps+1 we can apply Bawa�s algorithm.

This can be summarized in the following theorem, where the dominating portfolio [X is ex-

pressed as function of \X . In the statement the means pn and the sets Ln and Lmin (set of the

null components of the global minimum dispersion portfolio {min = [{min1 > ===> { minq ]) are assumed

to be known, but a procedure to get them is described in the proof of this theorem given in the

Appendix.

Theorem 3 Assume that the primary gross returns are unbounded random variables jointly el-

liptically distributed with non singular dispersion matrix. Suppose that there are risky assets with

di�erent expected returns H(]1) ? H(]2) ? ===?H (]q) traded in a frictionless economy where no

short sales are allowed. Thus, we can obtain the following analytical formulations for the vector

[X as function of \X .

¦ If Lmin 6= L1 then

- [X>l(\X ) = 0> ;l 5 Lmin and [X>l(\X ) = {minl ;l @5 Lmin> for \X � p1E(1)3D(1)p1F(1)3E(1)

;

18

- [X>l(\X ) = 0> ;l 5 L1 and [X>l(\X ) =\X(T(1))

31

(l)h(1)3(T(1))

31

(l)�(1)

\XF(1)3E(1);l @5 L1>

for p1E(1)3D(1)pnF(1)3E(1)

? \X � p2E(1)3D(1)p2F(1)3E(1) ,

¦ If Lmin = L1 then

- [X>l(\X ) = 0> ;l 5 L1 and [X>l(\X ) =\X(T(1))

31

(l)h(1)3(T(1))

31

(l)�(1)

\XF(1)3E(1);l @5 L1> for \X � p2E(1)3D(1)

p2F(1)3E(1)=

Then for n = 2> ===> n � 1 (Lmin = L1 or Lmin 6= L1)

- [X>l(\X ) = 0> ;l 5 Ln and [X>l(\X ) =e{(n)l > ;l 5 L�Ln> for pnE(n31)3D(n31)

pnF(n31)3E(n31)? \X � pn+1E

(n)3D(n)pn+1F(n)3E(n) ;

- [X>l(\X ) = 0> ;l 5 Ln and [X>l(\X ) =\X(T(n))

31

(l)h(n)3(T(n))

31

(l)�(n)

\XF(n)3E(n) > ;l 5 L � Ln>

for pnE(n)3D(n)

pnF(n)3E(n)? \X � pn+1E

(n)3D(n)pn+1F(n)3E(n) .

¦ When the portfolio with the greatest mean is also the portfolio with the greatest dispersion, then

the last step is given by :

- [X (\X ) = [0> 0,..,0> 1]0, for \X AE(]q)E(n)3D(n)

E(]q)F(n)3E(n)

Otherwise, we have maximum dispersion arcs and:

- [X>m(\X ) = 0> ;m 5 Ms and [X>m(\X ) =\X(T(s))

31

(m)h(s)3(T(s))

31

(m)�(s)

\XF(s)3E(s) ;m @5 Ms,

forpsE(s)3D(s)psF(s)3E(s)

? \X � ps+1E(s)3D(s)ps+1F(s)3E(s) ;

- [X (\X ) = e{ps forpsE(s31)3D(s31)psF(s31)3E(s31)

? \X � psE(s)3D(s)psF(s)3E(s)

and the last step is given by:

- [X>m(\X ) = 0> ;m 6= mW and [X>mW(\X ) = 1, for \X A pWE(w)3D(w)pWF(w)3E(w)

,

where pW is the mean of the primary gross return with the greatest dispersion.

¦ We also have the following analytical expression for the distribution function of IX :

IX (�) =

;AAAAAAAAAAAAAAAAAAAAAA?

AAAAAAAAAAAAAAAAAAAAAA=

�R3"

1F0(({min)0T{min)1@2

j³(w3({min)0�)2({min)0T{min

´gw> if Lmin 6= L1 and � � p1E(1)3D(1)

p1F(1)3E(1) ;

�R3"

�F(1)3E(1)

F0(�2F(1)32�E(1)+D(1))1@2 j

µ(w�2F(1)3wE(1)3�E(1)+D(1))

2

(�2F(1)32�E(1)+D(1))

¶gw>

if Lmin = L1 and � � p2E(1)3D(1)p2F(1)3E(1)

;�R

3"

1F0((h{(n))0Th{(n))1@2 j

³(w3(h{(n))0�)2(h{(n))0Th{(n)

´gw> if pnE

(n31)3D(n31)pnF(n31)3E(n31)

? � � pnE(n)3D(n)

pnF(n)3E(n);

�R3"

�F(n)3E(n)

F0(�2F(n)32�E(n)+D(n))1@2 j

µ(w�2F(n)3wE(n)3�E(n)+D(n))

2

(�2F(n)32�E(n)+D(n))

¶gw>

if pnE(n)3D(n)

pnF(n)3E(n)? � � pn+1E

(n)3D(n)pn+1F(n)3E(n)

;�R

3"

1F0({0WT{W)

1@2 j³(w3{0W�)2{0WT{W

´gw> if � � pWE(s

W)3D(sW)pWF(s

W)3E(sW) =

Note that with Theorem 3 we implicitly obtain an analytical formulation of the vector [X>(2)

19

as a function of \X>(2), since \X>(2) = \X for \X ?E(]q)E(n)3D(n)

E(]q)F(n)3E(n)= Thus, [X>(2)(\X>(2)) = [X (\X ) for

\X>(2) ?E(]q)E(n)3D(n)

E(]q)F(n)3E(n)and [X>(2)(\X>(2)) = [0> 0,..,0> 1]

0 if \X>(2) �E(]q)E(n)3D(n)

E(]q)F(n)3E(n).

Example. Suppose we have three risky assets which are Gaussian distributed with vector of means

� and variance covariance matrix T given by

� =

3

EC1

3

4

4

FD and T =

3

EC0=1 0 0

0 1=1 2

0 2 4=1

4

FD =

As observed by Dybvig (1984), under these assumptions the mean variance frontier with no short

sales consists of two arcs: the Þrst arc given by convex combinations of assets 1 and 2 and the

second one by convex combinations of assets 2 and 3 (being p2 = 3 the frontier between these

two arcs) 3. Solving the optimization problem (8) one can get the portfolio with global minimum

variance, which is {min = [0=9167> 0=0833> 0], having mean p1 = 1=1667 (and variance equal to

0.091667). Since for this case it holds L0 = L1 = {3}> L2 = {1}, and since

D(1) = 18=1819> E(1) = 12=7273> F(1) = 10=9091>

and

D(2) = 12=7451> E(2) = 5=2941> F(2) = 2=3529>

(as one can easily verify), we get

p2E(1) �D(1)

p2F(1) �E(1)= 1>

p2E(2) �D(2)

p2F(2) �E(2)= 1=7778 and

p3E(2) �D(2)

p3F(2) �E(2)= 2=0476=

Thus we can subdivide the support of \X in the sets

V1 = (�4> 1]> eV1 = (1> 1=7778]> V2 = (1=7778> 2=0476] and eV2 = (2=0476>+4)>

obtaining the following expression for [X (\X):

[X (\X ) =

;AAAAAAAAAAAAAAAAAAAAAAAA?

AAAAAAAAAAAAAAAAAAAAAAAA=

3

EC

10\X31010>9091\X312>72730>9091\X32>727310>9091\X312>7273

0

4

FD if \X 5 V1>

3

EC0

1

0

4

FD if \X 5 eV1>

3

EC

04>1176\X38=43142=3529\X35=294131>7647\X+3=13732=3529\X35=2941

4

FD if \X 5 V2>

3

EC0

0

1

4

FD if \X 5 eV2=

3Observe that this example does not satisfy the condition of Propostion 1, since the components of portfolio

weights on the Markowitz-Tobin e!cient frontier are not all positive.

20

¥

Once we determined the mean pn for k=1,..., n, we have determined a closed form solution for

the e!cient frontier, and the algorithm described in the Appendix can be furthermore extended to

the most general case analyzed by Markowitz. In addition, we can easily determine the portfolio on

the risky e!cient frontier that maximizes the Sharpe ratio E({0])3}0I{0T{

when no short sales are allowed

and there are risky assets with return mean greater than }0. As a matter of fact, when unlimited

short sales are allowed then the portfolio weights that maximize the Sharpe ratio are given by

e{ = T31(�3}0h)E3}0F with mean p = D3}0E

E3}0F and dispersion � =

sD32E}0+F}20E3F}0 = Thus the portfolio that

maximizes the Sharpe ratio when no short sales are allowed should be either a tangent portfolio or

a portfolio corresponding to a kink. For it, recall that the "tangent" portfolio obtained with the

assets belonging to L � Ln is given by

y(n) =

¡T(n)

¢31 ¡�(n) � }0h

(n)¢

E(n) � }0F(n)(n = 1> ===> n)>

and it presents the Sharpe ratio

E(y0(n)])� }0qy0(n)Ty(n)

=qD(n) � 2E(n)}0 +F(n)}20 =

This portfolio is on the k -th arc if E(y0(n)]) =D(n)3}0E(n)E(n)3}0F(n)

5 (pn>pn+1)> otherwise y(n) presents

negative components. Let VUn denotes the maximum Sharpe ratio on the k -th arc, then

VUn =

;?

=

qD(n) � 2E(n)}0 + F(n)}20 if D(n)3}0E(n)

E(n)3}0F(n)5 (pn>pn+1)

max³pn3}0�n

>pn+13}0�n+1

´othewise

>

where �n is dispersion of the optimal portfolio with mean pn= Then the maximum Sharpe ratio is

given by max1?n$n

(VUn) and the optimal portfolio weights that maximize the Sharpe ratio should be

the corresponding arguments.

6 Concluding remarks

In the paper we show how we can use the market stochastic bounds to study markets with short sale

restrictions. In particular, this analysis is the starting point to study and understand how investor�s

choices change respect to the market trend. Studying the distribution law of the upper bound, we

can understand which is the non-satiable investors� behavior in the market. In particular we can

estimate its average and two portfolios average of the non-satiable investors� choices for any given

class of optimal choices. From these average values we can understand the behavior that non-

satiable investors should have with respect to their position on the lwk investment. Analogously,

we can also use the above relations as control variables of the trend of a given optimal portfolio.

21

We Þrst examine a static model considering also the case of elliptical distributed returns. Under

this distributional assumption, we describe analytically the mean variance e!cient frontier when

no short sales are allowed. This algorithm permits us to analytically describe the mean variance

e!cient frontier when no short sales are allowed.

The results of this paper can be discussed and studied in a more general context where some deci-

sion making agents have to choose a parametric (no degenerate) random variable [� (with � being a

parameter belonging to a compact convex set � � <q). If the criterion of choice consists in maximiz-

ing the agent�s expected utility or choosing a non dominated random value [�> then we can repeat

exactly the same previous analysis. Thus we can estimate fIX (�) = inf�MXIP([� � �)> and the market

indicators;� = E(\X)> |G = E(�X\X )

;

�and {G = E(�X ), where �X (e$) = arg

µinf�MXIP([� � \X (e$))

¶

(for every e$ 5 e ) is the dominating parameter of non satiable choices.

7 Appendix

Proof of Theorem 1. In order to prove that IX and IO are not decreasing, let us consider

�1 � �2= Thus

IX (�1) = IP({0

X (�1)] � �1) � IP({0X (�2)] � �1) � IP({0X (�2)] � �2) = IX (�2) >

where the Þrst inequality is a consequence of the deÞnition of {X (�1)> while the second inequality

follows from the properties of every distribution function (which is a non decreasing function). We

see that IO is right continuous by deÞnition, and that IO is not decreasing if and only if T(�) is

not decreasing. Since,

T (�1) = IP({0

O(�1)] � �1) � IP({0O(�1)] � �2) � IP({0O(�2)] � �2) = T (�2) >

we get that IO is a not decreasing function. In addiction, IX is right continuous since if �q & �0,

we have that

IX (�0) � limq<+"

IX (�q) = limq<+"

IP({0X(�q)] � �q) � limq<+"

IP({0X (�0)] � �q) = IX (�0) >

where the Þrst inequality is a consequence of the fact that IX is a non decreasing function, the second

inequality is due to the deÞnition of {X (�q)> while the Þnal equality follows from the properties of

every distribution function (which is a right continuous function)= Hence, limq<+"

IX (�q) = IX (�0) =

From deÞnition of d (or;d), for every � ? d , we have IX (�) = 0, and for every � A d (or

;d), it is IX (�) A 0 (or IO (�) A 0). Moreover, by right continuous property it is also IX (d) � 0(or IO(

;d) � 0)= Similarly, from deÞnition of e (or

;e), for every � ? e (or

;e) we have IX (�) ? 1

(or IO (�) ? 1)> and, from deÞnition of;e for every � A

;e , IO (�) = 1. Considering the random

portfolios {0(e)]> | 0(;e)]> {0X (d)] and {0O(

;d)]> then

lim�<+"

IP({0O(;e)] � �) = lim

�<+"IP({0X (e)] � �) = 1

22

and

0 = lim�<3"

IP({0O(;d)] � �) = lim

�<3"IP({0X (d)] � �)=

Thus, IX and IO are distribution functions.

Finally, for every portfolio weight vector { we get

IX(�) � IP({0] � �) � IO(�)

for every � and the inequality is strict for some �= If we consider portfolio weights that includes a

weight for the riskless gross return }0, we have IP(}0 = }0) = 1 and for every � ? }0, IP(}0 � �) = 0.

Then in this case IX (�) = 0> which implies that d � }0= Similarly, for every � A }0, IP(}0 � �) = 1

and IO(�) = 1> which implies that;e � }0. ¥

Proof of Theorem 2. Similar to the proof of Theorem 1. ¥

Proof of Corollary 1. Let us call i(w> �({0])> �2({0])> �3({0])> ===> � n({

0])) the density of the

portfolio {0] valued in the point t. Then

IP((1� {0h)}0+{0] � �) =

=

�Z

3"

i(w> (1� {0h)}0 + {0E(])> �2({0])> �3({0])> ===>� n({

0]))gw

=

�3(13{0h)}03�({0])

�({0])Z

3"

i(x> 0> 1>�3({

0])

�3({0])> ===>

�n({0])

�n({0]))gx

since the parameters �l({0])�l({0])

are scalar and translation invariant. Thus, when there exists a portfolio

with mean �({0]) = (1 � {0h)}0 + {0E(]) greater than �> then all the portfolios that minimize

IP((1 � {0h)}0 + {0] � �) also maximize (13{0h)}0+{0E(])3�I{0T{

for some given ratios �l({0])

�l({0])= Let us

consider now a portfolio weights e{> with components in the risky assets e{l � 0;Pq

l=1 e{l � 1 thatminimizes the probability IP((1 � {0h)}0 + {0] � }0), and assume it has mean e�> and the othercentral moments e�2> e�3> ===> e�n. This portfolio maximizes (13{

0h)}0+{0E(])3�I{0T{

with � = }0 for the Þxed

ratios h�3({0])

h�2({0]) = However, all the portfolios (1��e{0h)}0+�e{0]> (with � 5 (0> 1Sql=1 h{l

)) have portfolio

weights belonging to V =©({> 1� {0h) 5 Uq+1 : { 5 Uq; 1� {0h � 0> {l � 0

ªhave the same ratios

h�3({0])h�2({0]) and maximize

{0(E(])3h}0)I{0T{

= Hence they belong to arg inf IP((1� {0h)}0 + {0] � }0). ¥

Proof of Proposition 1. In any arc of the e!cient frontier we can use the analytical form

deducted by the case in which short sales are allowed (see Bawa (1978) and Ortobelli and Rachev

(2001)). Since the e!cient frontier does not present kinks> we know that

{X (�) =

;AA?

AA=

�T31h3T31��F3E if � ? E({0W])E3D

E({0W])F3E

{W if � � E({0W])E3DE({0W])F3E

>

23

while formulation of

IX (�) =

�Z

3"

1

F0({0X (�)T{X (�))1@2

j

µ(w� {0X (�)�)

2

{0X (�)T{X (�)

¶gw

holds by the substitution of the mean and dispersion formulas

E({0X (�)}) =�E �D

�F �Eand �2(�) = {0X (�)T{X (�) =

�2F � 2�E +D

(�F �E)2=

Thus, we can easily deduce the desired results. ¥

Proof of Theorem 3. In order to prove Theorem 3 we Þrst need to explain how one can compute

recursively the values pn+1. For it, assume that pn is known (see below on how to compute p1).

The composition of the optimal portfolio with mean pn+1 could change:

1. because some strictly positive components of the optimal portfolio weights become null;

2. because some components of the optmal portfolio weights contribute to minimize the dispersion

becoming strictly positive.

As observed by Markowitz (1959-87), one can prosecute recalling that the limits of each arc are

determined by the constrains:

a) {l � 0> l = 1> ===>q ;

b) COC{l

= 0 if {l A 0 (i.e., for l 5 L � Ln) andCOC{l� 0 if {l = 0 (i.e., for l 5 Ln), where

O =1

2{0T{� �1

¡{0h� 1

¢� �2

¡{0��p

¢=

Since the e!cient portfolios could change their components that are strictly positive, we want

to know what components (among those strictly positive, i.e., l 5 L � Ln) become null when the

mean is pn+1. Thus from the constraints a) we have that

pn+1 � p3n+1 = minlML3Ln

;?

=wl A pn | wl =

E(n)¡T(n)

¢31(l)

�(n) �D(n)¡T(n)

¢31(l)

h(n)

F(n)¡T(n)

¢31(l)

�(n) �E(n)¡T(n)

¢31(l)

h(n)

<@

>>

where¡T(n)

¢31(l)is the i-th row of the matrix

¡T(n)

¢31and wl is the mean derived by equating to

zero the portfolio weight {l as expressed in the Merton�s formulation (10).

In order to get those new components of the optimal portfolio weights that contribute to mini-

mize the dispersion becoming strictly positive, we consider the constraint b). In fact:

- from the condition COC{l

= 0 ;l 5 L � Ln> we can calculate the Lagrangian coe!cients

�1 =D(n) �pE(n)

D(n)F(n) �¡E(n)

¢2 and �2 =pF(n) �E(n)

D(n)F(n) �¡E(n)

¢2

24

(this derivation follows because the expected returns are di�erent for di�erent assets, see Vörös

(1987) and Dybvig (1984));

- the conditions COC{l� 0> ;l 5 Ln, can be used to determine if an additional asset has to be

considered in optimal e!cient frontier. As a matter of fact, we can consider the partial derivativesCOC{l, ;l 5 Ln, evaluated on an optimal portfolio with mean belonging to [pn>pn+1]= It holds

CO

C{l= T(l){� �1 � �2E(]l) =

X

sML3Ln

�sl{s �D(n) �pE(n)

D(n)F(n) �¡E(n)

¢2 �pF(n) �E(n)

D(n)F(n) �¡E(n)

¢2E(]l)>

where {s =

�F(n)(T(n))

31

(s)�(n)3E(n)(T(n))

31

(s)h(n)

�p

D(n)F(n)3(E(n))2 +

D(n)(T(n))31

(s)h(n)3E(n)(T(n))

31

(s)�(n)

D(n)F(n)3(E(n))2 is determined by

the Merton�s formulation (10) letting {l = 0 ;l 5 Ln. If the i-th asset, l 5 Ln, enters in the compo-

sition of the optimal portfolio when the portfolio mean is given by p = pn+1, then it should beCOC{l

= 0. Thus we can get the mean wl of this optimal portfolio by equating to zeroCOC{l

> ;l 5 Ln,

and observing that

pn+1 � p+n+1 = minlMLn

(wl A pn | wl =

D(n)3E(n)E(}l)+S

sML3Ln�sl�E(n)(T(n))

31

(s)�(n)3D(n)(T(n))

31

(s)h(n)

�

E(n)3F(n)E(}l)+S

sML3Ln�sl�F(n)(T(n))

31

(s)�(n)3E(n)(T(n))

31

(s)h(n)

�

)=

Now we can set

pn+1 = min©p3n+1>p

+n+1

ª>

observing that if pn+1 = p3n+1 ? p+n+1 then Ln+1 = Ln ^ L3n , where

L3n = arg

3

C minlML3Ln

;?

=wl A pn | wl =

E(n)¡T(n)

¢31(l)

�(n) �D(n)¡T(n)

¢31(l)

h(n)

F(n)¡T(n)

¢31(l)

�(n) �E(n)¡T(n)

¢31(l)

h(n)

<@

>

4

D >

while, if pn+1 = p+n+1 ? p3n+1> then Ln+1 = Ln � L+n where

L+n = arg

ÃminlMLn

(wl A pn | wl =

D(n)3E(n)E(}l)+S

sML3Ln�sl�E(n)(T(n))

31

(s)�(n)3D(n)(T(n))

31

(s)h(n)

�

E(n)3F(n)E(}l)+S

sML3Ln�sl�F(n)(T(n))

31

(s)�(n)3E(n)(T(n))

31

(s)h(n)

�

)!=

In the particular case that pn+1 = p+n+1 = p3n+1 then we consider

eLn+1 = Ln � L+n and we deÞne

ep3n+1 = minlML3hLn+1

;AA?

AA=wl � pn+1 | wl =

gE(n)³gT(n)

´31(l)

g�(n) �gD(n)³gT(n)

´31(l)

gh(n)

gF(n)³gT(n)

´31(l)

g�(n) �gE(n)³gT(n)

´31(l)

gh(n)

<AA@

AA>>

where gT(n)> g�(n)> gh(n)> are, respectively, the matrix or vectors obtained eliminating every ln-th row

and every ln-th column, ;ln 5 eLn+1, in the matrix T and in the vectors � and h, respectively, while

the values gD(n)> gE(n)> gF(n) are deÞned in the usual way from the matrix gT(n) and the vectors g�(n)andgh(n) (observe that the matrix gT(n) is dimensionally greater than matrix T(n), since it considersthe apport of the new components in the optimal portfolios). Now, it could be that ep3n+1 A pn+1;

in this case we set Ln+1 = eLn+1. Otherwise we assign pn+1 = ep3n+1 and Ln+1 = eLn+1 ^ eL3n , where

25

eL3n = arg

3

C minlML3hLn+1

;?

=wl � pn+1 | wl =

jE(n)�jT(n)

�31(l)

j�(n)3jD(n)�jT(n)

�31(l)

jh(n)

jF(n)�jT(n)

�31(l)

j�(n)3jE(n)�jT(n)

�31(l)

jh(n)

<@

>

4

D =

Then the set Ln+1 is given by

Ln+1 =

;AAAA?

AAAA=

Ln ^ L3n if pn+1 = p3n+1 ? p+n+1

Ln � L+n if pn+1 = p+n+1 ? p3n+1

Ln � L+n if pn+1 = p+n+1 = p3n+1 ? ep3n+1¡

Ln � L+n¢^ eL3n if pn+1 = p+

n+1 = p3n+1 = ep3n+1 =

Considering that the e!cient frontier of non satiable risk averse investors is bounded from below by

the global minimum dispersion portfolio, and from above by risky return with the greatest mean,

then we can determine the e!cient frontier with the following recursive algorithm.

Step 1 First we determine the global minimum dispersion portfolio {min = [{min1 > ===>{ minq ] of the

e!cient frontier, which is the solution of the system

min{ {0T{

subject to

{l � 0; {0h = 1=

Let Lmin =©l � L = {1> 2> ===>q }

¯̄{minl = 0

ªdenotes the set of the null components of {min, and

let lmin denotes the elements of Lmin. We can easily compute:

� the dispersion matrix T(min) obtained eliminating every lmin-wk row and every lmin-wk column

from T, for all lmin 5 Lmin;

� the vectors h(min)> {(min) and �(min), obtained eliminating every lmin-th element from h = [1> ===> 1]0>

{ = [{1> ===>{ q]0 and � = E(]);

� the values D(min) =¡�(min)

¢0 ¡T(min)

¢31�(min), E(min) =

¡h(min)

¢0 ¡T(min)

¢31�(min) and F(min) =¡

h(min)¢0 ¡

T(min)¢31

h(min);

� the mean of the global minimum dispersion portfolio, which is given by p1 =E(min)

F(min)=

Since we do not know if {min is associated or not to a switching point we have to compute the

portfolio weights e|(1) solution of the problem

min{ {0T{

subject to

{l � 0; {0h = 1> {0� = p1 + %=

for an opportune little % A 0, observing, as a consequence, the set L1 =nl 5 L

¯̄¯e|(1)l = 0

o= Clearly,

the global minimum portfolio is a point where the optimal portfolio weights change its components

only if L1 6= Lmin= ¦

Step 2 Once we know L1> then we should calculate p2 = min©p32 >p

+2

ªas mentioned above. The

26

portfolio weights of the Þrst arc that are given, for p 5 (p1>p2], by

{(1) =

³F(1)

¡T(1)

¢31�(1) �E(1)

¡T(1)

¢31h(1)´p

D(1)F(1) �¡E(1)

¢2 +D(1)

¡T(1)

¢31h(1) �E(1)

¡T(1)

¢31�(1)

D(1)F(1) �¡E(1)

¢2 >

where T(1)> �(1)> h(1)> are respectively the matrix and vectors obtained eliminating every l1-th row

and every l1-th column ;l1 5 L1 in the matrix T and in the respective vectors, while the values

D(1)> E(1)> F(1) are obtained by the matrix T(1)> and vectors �(1)> h(1)= In the case L1 6= Lmin> then

D(1)> E(1)> F(1) are generally di�erent by D(min), E(min) and F(min)= The new set of the new null

components is given by L2 =

;AAAA?

AAAA=

L1 ^ L31 if p2 = p32 ? p+2

L1 � L+1 if p2 = p+2 ? p32

L1 � L+1 if p2 = p+2 = p32 ? ep32¡

L1 � L+1¢^ eL31 if p2 = p+

2 = p32 = ep32 =¦

Step 3 We can proceed in the same way in order to Þnd the other arcs of the e!cient frontier,

until, after a Þnite number n of recursive steps, we arrive at the last portfolio of the Markowitz-

Tobin e!cient frontier, that corresponds to the asset with the greatest expected return ]q (since

the vector of returns Z is ordered in the sense of the mean, i.e. E(]1) ? E(]2) ? ===? E(]q)). This

part of the e!cient frontier represents the non-satiable risk averse investors� choices. If the portfolio

with the greatest mean is also the portfolio with the greatest dispersion then the e!cient choices

of non-satiable investors are the same choices of non-satiable risk averse investors, otherwise we

need to compute the arcs with maximum dispersion using Bawa�s algorithm in order to get optimal

choices of non-satiable investors who are not necessarily risk averse. ¦

Once the means pn and the sets Ln are known, it is possible to get the analytical formulation

for [X(\X ) and IX . In fact, when returns are elliptically and unbounded distributed then every

portfolio {X (�) 5 argµinf|MV

IP(|0} � �)

¶is on the non-satiable e!cient frontier and, for each mean-

dispersion arc, it holds {(n)(�) =�(T(n))

31h(n)3(T(n))

31�(n)

�F(n)3E(n) with pn(�) =�E(n)3D(n)�F(n)3E(n) (see Ortobelli

and Rachev (2001)). Thus, � as function of the mean is given by �(p) = pE(n)3D(n)pF(n)3E(n) =

Therefore:

{m(�) = 0> ;m 5 Lmin and {l(�) = {minl > ;l 5 L � Lmin > when � � p1E(1)3D(1)p1F(1)3E(1)

(if L1 6= Lmin);

{m(�) = 0> ;m 5 L1 and {l(�) =�(T(1))

31

(l)h(1)3(T(1))

31

(l)�(1)

�F(1)3E(1) ;l 5 L�L1 for p1E(1)3D(1)p1F(1)3E(1) ? � � p2E(1)3D(1)

p2F(1)3E(1),

(or for � � p2E(1)3D(1)p2F(1)3E(1)

> if L1 = Lmin)

and, for every 2 � n ? n,

{m(�) =e{(n)m when pnE(n31)3D(n31)

pnF(n31)3E(n31) ? � � pnE(n)3D(n)

pnF(n)3E(n);

{m(�) = 0> ;m 5 Ln and {l(�) =�(T(n))

31

(l)h(n)3(T(n))

31

(l)�(n)

�F(n)3E(n) ;l 5 L � Ln forpnE

(n)3D(n)pnF(n)3E(n) ? � �

27

pn+1E(n)3D(n)

pn+1F(n)3E(n).

If the portfolio with the greatest mean is also the portfolio with the greatest dispersion, then

the last step is given by:

{m(�) = 0> ;m ?q and {q(�) = 1 when � � pnE(n)3D(n)

pnF(n)3E(n)

, where pn = E(]q)=

Otherwise we have that in any maximum dispersion arc

{m(�) = 0> ;m 5 Ms and {l(�) =�¡T(s)

¢31(l)

h(s) �¡T(s)

¢31(l)

�(s)

�F(s) �E(s);l 5 L � Ms

whenpsE(s)3D(s)psF(s)3E(s)

? � � ps+1E(s)3D(s)ps+1F(s)3E(s)

, while {(�) =e{ps ifpsE(s31)3D(s31)psF(s31)3E(s31)

? � � psE(s)3D(s)psF(s)3E(s)

= In

this last case after sW steps we arrive to the mW-th risky return with the greatest dispersion. Thus

the last step is given by

{m(�) = 0> ;m 6= mW and {mW(�) = 1

when � � pWE(w)3D(w)pWF(w)3E(w)

, where pW = E(]mW).

Observe that intervalspsE(s)3D(s)psF(s)3E(s) ? � � ps+1E(s)3D(s)

ps+1F(s)3E(s)do never intersect with the others (see

Bawa (1976) and Ortobelli and Rachev (2001)).

Thus, from deÞnition of {X(�), and similarly as we did in the proof of Proposition 1, we get

the expression of [X (\X) and of the distribution function of preferential market growth stated in

the assertion. ¥

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30