Conditions for the Existence of Absolutely Optimal Portfolios

: Let ∆ n be the n -dimensional simplex, ξ = ( ξ 1 , ξ 2 , . . . , ξ n ) be an n -dimensional random vector, and U be a set of utility functions. A vector x * ∈ ∆ n is a U -absolutely optimal portfolio if E ( u ( ξ T x ∗ )) ≥ E ( u ( ξ T x )) for every x ∈ ∆ n and u ∈ U . In this paper, we investigate the following problem: For what random vectors, ξ , do U -absolutely optimal portfolios exist? If U 2 is the set of concave utility functions, we ﬁnd necessary and sufﬁcient conditions on the distribution of the random vector, ξ , in order that it admits a U 2 -absolutely optimal portfolio. The main result is the following: If x 0 is a portfolio having all its entries positive, then x 0 is an absolutely optimal portfolio if and only if all the conditional expectations of ξ i , given the return of portfolio x 0 , are the same. We prove that if ξ is bounded below then CARA-absolutely optimal portfolios are also U 2 -absolutely optimal portfolios. The classical case when the random vector ξ is normal is analyzed. We make a complete investigation of the simplest case of a bi-dimensional random vector ξ = ( ξ 1 , ξ 2 ). We give a complete characterization and we build two dimensional distributions that are absolutely continuous and admit U 2 -absolutely optimal portfolios.


Introduction. Defining the Problem
Let (Ω, K, P) be a fixed probability space and ξ: Ω → R n be an n-dimensional random vector. The random vector ξ = (ξ 1 , ξ 2 , . . . , ξ n ) will be called the vector of return rates (or the financial market). Let S be a positive number. A portfolio of sum S is a vector x ∈ R n + with the property that x 1 + . . . + x n = S. The set of all portfolios of sum S will be denoted by ∆ n (S). A standard portfolio is a portfolio of sum 1, that is, an element of standard simplex ∆ n (1).
We shall also use the set P n−1 (S) = {x ∈ R n−1 |x i ≥ 0 ∀ i = 1, . . . , n − 1 and x 1 + . . . + x n−1 ≤ S}. Instead of P n−1 (1), we shall write P n−1 . We shall denote by e the n-dimensional vector with all entries equal to one, that is e = (1, 1, . . . , 1). The return of portfolio x is the random variable Y(x) = x T ξ = x 1 ξ 1 + . . . + x n ξ n A utility function is a continuous non-decreasing function u: I → R, where I ⊆ R is an interval. Usually, the interval is either [0,∞) or the whole real line. A utility function of the form u(x) = e rx , x ∈ I (with r > 0) or u(x) = − e −rx , x ∈ I (r > 0) is called a CARA-utility.
The set of CARA-utilities will be denoted with U 3 . The main problem in the portfolio theory is to find "the best" portfolio. How do you compare two portfolios in order to decide which of them is better? In the framework of expected utility theory, any decision maker has a utility function U: I → R and he tries to maximize the expected utility of a portfolio: that is, he tries to maximize V(x, ξ; u) = Eu x T ξ (2) for all portfolios x ∈ ∆ n (S), provided that x T ξ ∈ I. From the point of view of expected utility theory, a decision maker with utility u prefers the portfolio y to portfolio x if V(x, ξ; u) ≤ V(y, ξ; u) ⇔ Eu x T ξ ≤ Eu y T ξ To avoid trivialities, we shall also suppose that the relation (1.2) makes sense; more precisely, we shall always assume that we deal with utilities having the property u(x T ξ) ∈ L 1 (Ω, K, P) for any x ∈ R n + . A decision maker is called risk-avoiding if the utility is concave. The reason for this is that if u is concave then, by Jensen's inequality, Eu(Y) ≤ u(EY) for any random variable Y. The meaning of this is that such a decision maker always prefers a sure amount of money to any random one having the same expected value.
We shall assume in the sequel that all the decision makers are risk-avoiding-after all, they will not be interested in portfolio theory otherwise.
Let U 2 (S, ξ) denote the set of all concave utilities, u having the property that u (x T ξ) makes sense and, moreover, u (x T ξ) ∈ L 1 (Ω, K, P) for all x ∈ ∆ n (S).
Let ξ be a fixed random vector from R n and u ∈ U 2 (S, ξ) be a fixed utility. Consider the problem: P(S): max{Eu(x T ξ): x ∈ ∆ n (S)} The meaning of the above optimization problem is that that any rational decision maker wants to maximize his expected utility. Notice that the mapping x → Eu(x T ξ), x ∈ ∆ n (S) is concave and continuous. The concavity is obvious and the continuity comes from Lebesgue's domination principle: if x n → x, the sequence of random variables (u(x n T ξ)) n converges to u(x T ξ) and is dominated by the random variable max i (|u(S ξ i )|) ∈ L 1 (Ω K,P) As ∆ n (S) is a compact set and the mapping h is continuous, the problem P(S) always has solutions. Moreover, the set of solutions is a closed convex subset from ∆ n (S). This is called the set of optimal portfolios and is denoted by Opt(ξ,S;u). A portfolio x ∈ ∆ n (S) is called an equal weight portfolio if all its entries are equal, that is x = S n , . . . ., S n . Our study was motivated by the following well known fact: Theorem 1. If ξ i , I = 1, 2, . . . , n are independent and identically distributed (i.i.d.) random variables, ξ = (ξ 1 , ξ 2 , . . . , ξ n ) then the equal weight portfolio is optimal for the problem P(S): In other words, any risk-avoiding decision maker agrees that the equal weight portfolio is optimal, provided that the random variables (ξ i ) 1≤i≤n are i.i.d. Put another way, if the return of the assets has the same distribution and they are independent, then the investor's choice to put the same amount of money in each asset is optimal from the point of view of the expected utility of the return.
Samuelson [1] showed that if the distribution of ξ is symmetrical, then the equal weight portfolio is optimal for the problem P(S).
Hadar and Russell [2] gave an alternative proof of Samuelson's result based on stochastic dominance. In [3], Tamir found a more general condition than that belonging to Samuelson. He showed that, if F ξ o σ = F ξ for σ = (2, 3, . . . , n, 1), then the equal weight portfolio is optimal for the problem P(S). Thus, it is enough that F (ξ 1 ,ξ 2 ,...,ξ n ) = F (ξ 2 ,...,ξ n ,ξ 1 ) in order that the equal weight portfolio be optimal no matter the concave utility, u. In other words, the equal weight portfolio is absolutely optimal.
Another direction of research was initiated in Deng and Li [4]. They found necessary and sufficient conditions for the optimality of equal weight portfolios for the minimum variance problem.
An interesting problem is to find larger classes of matrices A with the following properties: (ii) The equal weight portfolio is optimal for the problem P(S); Tamir's result [3] for cyclically symmetry of F ξ is the most general condition we know so far on F ξ . Definition 1. We state that x o is an absolutely optimal portfolio with sum S if x o ∈ Opt(ξ,S;u) for every utility u ∈ U 2 (S, ξ). We state that ξ (or better F ξ ) has the absolute optimal portfolio property (AOP) if an absolutely optimal portfolio does exist.
In the following, we shall write 'simply absolute portfolio' instead of 'absolutely optimal portfolio'.
We investigate the following problem: For what random vectors, ξ, do U -absolute portfolios exist?
In the second section, we recall some facts about stochastic dominance relations and we prove some auxiliary results.
In Section 3, we solve the problem, if U = U 2 , what is the set of concave utility functions; we find necessary and sufficient conditions in order that F ξ has the AOP property, i.e., F ξ admits an absolute portfolio. The main result is the following: If x 0 is a portfolio having all its entries positive, then x 0 is an absolute portfolio if and only if all the conditional expectations of ξ i , given the return of portfolio x 0 , are the same.
In Section 4, we prove that, if ξ is bounded below, then CARA-absolute portfolios are also U 2 -absolute portfolios. In the case when ξ i are rates of returns of assets in a financial market, then all ξ i are bounded below, hence the above assertion holds.
In Section 5, the classical case when the random vector ξ is normal is analyzed. We make a complete investigation of the simplest case of a bi-dimensional random vector ξ = (ξ 1 , ξ 2 ) in Section 6. We give a complete characterization and we build two dimensional distributions that are absolutely continuous and admit U 2 -absolute portfolios.
The AOP property is connected with the stochastic ordering; actually, this means that the set of probability distributions of the random variables x T ξ, x ∈ P n (S) has a maximum in the increasing concave order.

Stochastic Dominance
The increasing concave order, denoted by "icv", is extensively used in economics, where it is called second-order stochastic dominance: see Belzunce [5], Levy [6], Sriboonchita [7], or Shaked and Shantikumar [8]. Firstly, let us recall the definition of the increasing concave order. Definition 2. Let X,Y be two integrable random variables. We state that X is dominated by Y in the increasing concave order if: Eu(X) ≤ Eu(Y) for every concave utility u such that u(X), u(Y) ∈ L 1 We shall denote this domination relation by "X ≺ icv Y". If the inequality (4) holds for any concave function, u, not necessarily non-decreasing, we say that X is concavely dominated by Y and write "X ≺ cv Y". It is known that X ≺ cv Y if, and only if, X ≺ icv Y and EX = EY. Notice that the relation "≺ icv " is an order relation between the probability distributions F and G of X and Y and can be better stated as: In this case, we can restate (3) as: Proof. Let X = (x o ) T ξ = S ξ 1 +ξ 2 +...+ξ n n . We confirm that E(ξ i |X) = E(ξ 1 |X). Indeed, E(ξ i |X) is a random variable of the form g i (X) having the property that E(ξ i h(X)) = E(g i (X)h(X)) for every bounded measurable function h:R→R. As ξ i are i.i.d., The last equality holds because of the equality dF ⊗n (x σ(1) , . . . ,x σ(n) ) = dF ⊗n (x 1 , . . . ,x n ). It follows that E(ξ i |X) = E(ξ 1 |X) ∀ i = 2 , . . . , n. Consequently X = E(X|X) = E(S ξ 1 +ξ 2 +...+ξ n n |X) = S n (E(ξ 1 |X) + E(ξ 2 |X) + . . . + E(ξ n |X)) = S n ·n·E(ξ 1 |X) = SE(ξ 1 |X). It follows that E(ξ 1 |X) = X/S. Let x ∈ ∆ n (S) be arbitrary and v(x) = u(x/S). Then, by Jensen's inequality, Therefore, Eu(x T ξ) ≤ Eu(X) = Eu((x o ) T ξ); hence, x o is indeed optimal.
Remark. Taking into account the fact that ξ i have the same expectation (recall that they are identically distributed) we could say even more: x T ξ ≺ cv (x o ) T ξ for every x ∈ ∆ n (S). Moreover, from the above proof, one can easily see that the assumption that the entries ξ i were i.i.d. was not essential. What we really used was the fact that the distribution F ⊗n is symmetrical (A random vector ξ = (ξ 1 , ξ 2 , . . . , ξ n ) is said to be symmetrically distributed if all its permutations (ξ σ(1) , . . . ,ξ σ(n) ) have the same distribution as ξ). Thus, the result from Proposition 1 holds for every symmetrically distributed random vector.
It is important to keep in mind the following simple properties:
Or, even better. Let F be a probability distribution on R n , L x : R n → R be the linear mappings L x (y) = x T y, and F x = Fo(L x ) −1 and D(F) = {F x |x ∈ ∆ n (S)}. When does D(F) admit the greatest element with respect to the order "≺ icv "?
Then Proposition 1 states that, when F = G ⊗n with G a probability distribution on the real line, then D(F) admits the greatest element. Or, more generally, that if F is symmetric, it also has the greatest element.
A portfolio x o ∈ ∆ n (S), with the property that F x o is the greatest element of D(F), will be called an absolutely optimal portfolio of sum S associated to F. We chose this name because all risk-avoiding decision makers will agree that this maximal element is the best among all portfolios of sum S.
We restate the Definition 1. in stochastic order terms: Definition 3. We state that the random vector ξ has the absolute optimum property (AOP) for sum S if the family D(ξ) has greatest element with respect to the order "≺ icv ". If an absolute optimal portfolio corresponding to this greatest element is x o ∈ ∆ n (S), we will write that ξ has the property AO (x o ) or that ξ ∈ AO (x o ). In terms of distributions, we state that F ∈ AO (x o ). Precisely: Or, in terms of distributions: where L x is the linear form L x (y) = x T y, y ∈ R n . Then, for a given S > 0, the class AO of distributions on R n is defined by:

Properties of Class AO
At this point, we do not know if the AO class contains other distributions on R n beside the symmetric ones. However, the following properties of the class AO are immediate. We shall state them both in terms of random vectors and in terms of distributions. Proposition 3. Invariance Properties. Let S > 0 be fixed.
. Here e = (1, 1, . . . , 1) ∈ R n (invariance with respect to scaling); (ii) If ξ 1 , ξ 2 are independent and both belong to AO (x o ), then ξ 1 + ξ 2 ∈ AO (x o ) (invariance with respect to convolutions); (iii) If ξ 1 , ξ 2 are independent, A ∈ K is independent on both of them and ξ 1 , ξ 2 ∈AO (x o ), then the random vector ξ = ξ 1 1 A + ξ 2 1 A c also belongs to AO (x o ) (invariance with respect to mixtures); (iv) If (ξ n ) n is a sequence of non-negative random vectors and ξ n → ξ (in distribution), then . Then x o /S is absolutely optimal among all the portfolios of sum 1, or the same thing in terms of distributions.
This means that when dealing with distributions of class AO we always can assume that S = 1. Proof.
Important Remark Based on the above proposition, namely on point (v), it suffices to assume that S = 1. In the sequel we shall agree that always S = 1. Thus, ∆ n will mean ∆ n (1). Now we give a necessary and sufficient condition in order that a distribution on R n , F be in the class AO(x o ), provided that x 0 i > 0 for all i = 1, . . . , n.
Proof. The easy part is "⇐". Indeed, let x ∈ ∆ n and let u ∈ U 2 (1, ξ). Then hence, x o is an absolute optimum. We prove now "⇒". Let u ∈ U 2 (1, ξ). Consider the function h u :P n−1 → R defined by: Suppose that u is differentiable, then h u is also differentiable. Let g = Grad(h u ). Thus, Let y o = x 0 1 , x 0 2 , . . . , x 0 n−1 ∈ P n−1 . We know that h u attains its maximum at y o and y o is an interior point in P n−1 . Then: Recall that u is non-decreasing and concave; hence, u' is decreasing, continuous, and non-negative. Conversely, any function g:R → [0,∞) continuous and decreasing is the derivative of some concave utility. Taking a sequence of this kind of functions that monotonically converges to 1 (−∞,a] , we arrive at the conclusion that: Denote by B (R) the σ-algebra of the Borel sets on the real line.
. According to (9), C contains all the intervals (−∞, a]. Moreover, it fulfills the following conditions: Property (i) follows if we let a → ∞. Property (ii) results from the following series of equalities: follows from the Lebesgue's domination principle.
Such a family of sets is called a Dynkin − system or a U-system. SoC is a U-system that contains the intervals (−∞, a]. It then contains the U-system generated by these intervals, but it is standard knowledge that this U-system is the Borel σ-algebra on the real line, B (R). Thus, C contains all the Borelian sets. The conclusion is that: By a standard argument (if (10) holds for indicators, then it holds for simple functions and it holds for positive measurable functions). It follows that: The conclusion is that E(ξ i |X 0 ) = E(ξ n |X 0 ). Indeed, E(ξ i |X 0 ) is a random variable of the form ϕ i (X 0 ) with the property that E(ϕ i (X 0 )f (X 0 )) = E(ξ i f (X 0 )) for any f that is measurable and bounded. Hence, E(ϕ i (X 0 )f (X 0 )) = E(ϕ n (X 0 )f (X 0 )) for any f that is measurable and bounded; therefore, ϕ i (X 0 ) = ϕ n (X 0 ). We claim that, in this case, E(ξ i |X 0 ) = X 0 . Indeed, As a consequence, we find a necessary condition in order that ξ may have the property AO. Proof. According to Proposition 3.2, E(ξ 1 |X 0 ) = E(ξ 2 |X 0 ) = . . . = E(ξ n |X 0 ). If we average the preceding sequence of equalities, we obtain the conclusion of the Corollary.
What if ξ∈ AO (x o ) but some of the components of x o are equal to 0? Is that possible?
In order to answer this question, let us suppose that the components of ξ are ordered in such a way that EX 1 ≤ EX 2 ≤ . . . ≤ EX n . It is possible that some of the expectations coincide. In general, we shall use the convention: Denote by INC(k), the set of all random vectors ξ = (ξ 1 , ξ 2 , . . . , ξ n ) with the above property.
For k = 0, we imply that all the expectations coincide and for k = n − 1 imply that µ n−1 < µ n . Then Corollary 1. states that, if ξ has an absolute optimal portfolio with all the components positive, then ξ ∈ INC(0).
If ξ has an absolute optimal portfolio x o , then In words: If an absolute optimal portfolio does exist, it should both maximize the expectation and minimize the variance.
Proof. Let X 0 = (x o ) T ξ. Suppose, ad absurdum, that x o i > 0 for some i ≤ k. Then EX 0 < EX n ; hence, X 0 cannot be optimal, according to Proposition 2. (i), which states that EX n ≤ EX 0 . Therefore, the possible absolute optima must be of the form x o = 0, 0, . . . , x 0 k+1 , . . . , x 0 n . All the portfolios x with x j = 0 ∀ j ≤ k have the property that E(x T ξ) = µ n = max 1≤j≤n µ j . The absolute optimal portfolio x o , provided that it does exist at all, should dominate them in the "≺ icv " order. As it has the same expectation, µ n , it should dominate them in the concave order "≺ cv "; according to Proposition 2 (ii), Var((x o ) T ξ) ≤ Var(x T ξ) for all such x. If another portfolio, say y ∈ ∆ n , would exist such that Var((x o ) T ξ) > Var(y T ξ), then (x o ) T ξ could not dominate y T ξ in the "≺ icv " order.
Proof. Let u be a concave differentiable utility, z ∈ P n−1 (k), defined as: , then the concave differentiable function, h u , attains its maximum at z. Let g = Grad(h u ). According to Lemma 1, g j (z) ≤ 0 ∀ j ≤ k, g j (z) = 0 ∀ j > k or: In the same way as we did in the proof of Proposition 4., we can take u' to be 1 (−∞, a] . Thus, (14) becomes: which implies (13), by the same argument used in the proof of Proposition 4.: which implies that: Conversely, if the conditions from the right hand of (13) hold, then E(ξ i u'(X o )) ≤ E(ξ n u'(X o )) ∀ j ≤ k and E(ξ i u'(X 0 )) = E(ξ n u'(X 0 )) ∀ j > k for any u concave that is differentiable and non-decreasing. By Lemma 1., the function h u attains its maximum at x o for any differentiable concave utility. Thus Eu((x o ) T ξ) ≥ Eu((x) T ξ) for any x ∈ ∆ n , u differentiable concave that is non-decreasing. However, the restriction that u be differentiable is not a serious one: any utility is the uniform limit of differentiable ones.

Corollary 2. Let x o ∈ ∆ n,k . A sufficient condition in order that
concave, and non-decreasing, such that the integral makes sense.

The Class CARAAO
In this section, we shall deal only with short tailed random vectors. A random vector ξ = (ξ 1 ,...,ξ n ) is short tailed if the moment generating functions, m i (t) = E[exp(tξ i )], are finite in a neighborhood of 0. If we consider only CARA utilities of the form u(x) = − e −rx , we can give the following. Definition 4. Let x o ∈ ∆ n and ξ be a random vector. We say that ξ has the property CARAAO(x o ) (and write ξ ∈ CARAAO(x o )) if: for any r > 0 such that both the expectations are finite. The union of all the classes CARAAO(x o ) is defined to be the class CARAAO.
The fact is that, in some cases, the classes CARAAO and AO coincide. Some conditions under which the equality of classes holds are presented in the following proposition. The function h is concave and attains its maximum at y o , which is an interior point of the compact P n−1 . Let X 0 = (x o ) T ξ. Then: Which further implies: E(ξ j f (Z)) = E(ξ n f (Z)) for any polynomial function f, ∀ j = 1,...,n − 1 As the polynomials are dense in C([0.e −m ]), equality (19) holds for any bounded continuous function, f. The standard procedure is: Approximate the indicators 1 (a,b] with continuous functions and check that (19) implies the equality E(ξ j 1 (a,b] (Z)) = E(ξ n 1 (a,b] (Z)) ∀ j = 1,...,n ∀ a < b ∈ R. The conclusion is that E(ξ j |Z) = E(ξ n |Z) for any j = 1,...,n − 1. As Z and X 0 generate the same σ-algebra; we see that condition (5) is satisfied. The proof is finalized by applying Proposition 4.

Remark.
In the case when ξ I are rates of returns of assets in a financial market, we have ξ i ≥ −1 for all i. The hypothesis that ξ i are bounded below from the preceding proposition is verified.
Example. Suppose that ξ j~G amma(a j ,a j ), j = 1,...,n with a j > 0 and (ξ j ) 1≤j≤n are independent. Then ξ ∈ CARAAO (x o ) with: Indeed, we have to minimize the function: h(x) = a 1 ln a 1 a 1 + rx 1 +a 2 ln a 2 a 2 + rx 2 +... + a n ln a n a n + sx n with After some calculus, one obtains the gradient g = Grad(h). Its components are: Therefore, x o = a 1 S , a 2 S , . . . , a n S . From the above proposition, x o is an absolute optimal portfolio.
Remark. Even for n = 2, it is not true that if ξ 1 and ξ 2 are independent and Eξ 1 = Eξ 2 , then (ξ 1 ,ξ 2 ) has the property AO. Suppose that ξ 1~U niform(−a,a) and ξ 2~U niform(−b,b) for some a,b > 0, a = b. Then the minimum point of the function depends strongly on r; hence, ξ cannot belong to CARAAO.
Counterexample. Proposition 7 says that if the CARA-absolute optimal portfolio is in the interior of the simplex ∆ n , then in many cases it is an absolutely optimal portfolio as well. However, if x o is not an interior point, if it belongs to the face ∆ n,k , k ≥ 1 of the simplex, that fails to be true. Let x 1 = (0;1), x 2 = (0;3), x 3 = (1;3), and x 4 = (3;2) four points in the plane, and suppose that (ξ 1 ,ξ 2 )~ The condition that x o = (0;1) is the absolute optimal portfolio is that: One can easily check that x o = (0;1) is the absolute optimal portfolio if and only if β ≤ α; x o = (0;1) is an absolute CARA-optimal portfolio if and only if [ β 2α ≤ 1 and 3γ + 2δ ≥ β − α] or [ β 2α > 1 and 3γ + 2δ ≥ β Thus, there exists no implications that "x 0 is an absolute CARA-optimal portfolio, ⇒ x 0 is an absolute optimal portfolio" or even "ξ 1 ≺ icv ξ 2 ⇒ x o = (0;1) is an absolute CARA-optimal portfolio", as we thought.

The Normal Case
This is the classic case. The literature concerning it is extensive. The unidimensional normal distributions are very convenient because the "≺ icv " order is easy to establish: Normal (µ,σ 2 ) ≺ icv Normal (µ',σ' 2 ) ⇔ µ ≤ µ', σ ≥ σ'. Moreover, if ξ~Normal (µ,C) is a n-dimensional random vector and x ∈ ∆ n (S) is a portfolio of sum S, then x T ξ~Normal(x T µ, x T Cx). Therefore, it is very easy to compare two portfolios x and y: Because these relations are homogeneous, the sum S does not matter; hence, we always can restrict ourselves to consider only portfolios from ∆ n .
admits at least solution.
If k ≥ 1 then ξ ∈ AO if at least one solution x o of the problem has the property that x 0 j = 0 ∀ j ≤ k. A sufficient condition for that to happen is that c i,j ≥ c n,j ∀ j = k + 1,...,n ∀ i = 1,...,k Proof.
(i) There is nothing to prove: all the portfolios have the same expectation.
Corollary 9. Let C be a non-negative defined matric. Let c i,· be its i'th row of C. A sufficient condition that ξ~N(µ,C), ξ ∈ INC(k) belong to AO is that c i,· ≥ c n,· ∀ i = 1,2,...,k.
Example. Take  One sees that x o = (0, 1 2 ). This is the absolute optimum. Hence, X 0~N ormal(b,1), in spite of the fact that neither the first row nor the second one are greater than the third row of the matrix.

The Case n = 2
In this section, we consider the case when the random vector ξ is bi-dimensional. This is the next simple case. Now ξ = (X,Y) and instead of writing OA(s,t) with s,t ≥ 0, s + t = 1 we write OA(s). We shall give the method of how to construct probability distributions on R 2 with property OA(s), s ∈ [0,1). We shall assume that ξ either has a density f or has a discrete distribution on the set of integers. The same letter f will denote the density in the first case and the probability law in the second one: f (x,y) will denote P(X = x,Y = y). We shall state the conditions in both cases.
The case s > 0. We want the absolute optimum return to be Z = sX + (1 − s)Y. According to Proposition 4, the condition is that E(X|sX In terms of densities, the condition means that: Or, explicitly written: We shall focus on the absolute continuous case. After the substitution u = x − a in (27) one obtains: Let us put ρ(a) = 1 1−s f (a + u, a − λu)du. Then ρ is a probability density. Let also Then p a is also a probability density and (28) becomes: The meaning of (30) is that, if Z a are random variables with density p a , then EZ a = 0. This is very easy to construct; take any random variables and center them. The conclusion is that we may construct as many distributions F ∈ OA(s) as follows: (i) Take a probability density on [0,∞), denoted by ρ; (ii) Take a family of densities on the real line, p a , with property (30); (iii) Set f (a + u, a − λu) = (1 − s)p a (u)ρ(a) or substituting a + u by x and a − λu by y, we obtain: where λ 2 is the Lebesgue measure in the plane. Then F ∈ OA(s).
Notice, as a particular case, that if s = 1 2 , then f (x,y) = 1 2(x+y) 1 ∆ (x, y) with A(0,0), B(0,2), C(2,0); and now the distribution F is symmetric. The reason for this is that all the densities, p a , are symmetrical. The general form of a density of a distribution F ∈ OA( 1 2 ) is: which, of course, may not be symmetric. A sufficient condition that the equality (28) holds is that the density, f, satisfies the condition: f (a + x,a − λx) = f (a − x, a + λx) for any a,x This condition is similar to the symmetry. We can construct many such densities, starting with a symmetric density. It is enough to take a symmetric density, call it g, (meaning that g(x,y) = g(y,x) ∀ x,y !). The reader can check that: is a density that satisfies (32). For s = 1 2 we discover again the symmetric densities because (33) becomes f (x,y) = 4g(2y,2x).
This is the uniform distribution in the interior of the parallelogram ABCD with A(0,0), B(t, (s − t)/2), C(t,s), D(0, 1 2 ). Here t = 1 − s. If g would be the uniform density in the interior of the unity circle, then f would be the density for the uniform density in the interior of some ellipse, etc.
The case s = 0. The condition is that E(X; Y ≤ a) ≤ E(Y; Y ≤ a) ∀ a. A sufficient condition is that E(X|Y) ≤ Y. Otherwise written, the pair (Y,X) should be a sub-martingale (or rather the first two terms of a sub-martingale). It is very easy to construct such distributions.

Conclusions
Our work was motivated by a result from mathematical folklore, which states that in the case of a financial market where the asset rates of return are i.i.d., the equal weight portfolio was an optimal portfolio for all risk-averse investors. Our attention was focused on finding necessary and sufficient conditions for the distribution of a financial market so that the equal weight portfolio is the optimal portfolio for all risk-averse investors. We generalized the previous problem by replacing the equal weight portfolio with a given portfolio. The necessary and sufficient conditions found were formulated using the conditional mean. In case the financial market has two assets, we have provided an algorithm for construction of the financial market that admits a given optimal portfolio. It is a challenge to find algorithms for the construction of probability distributions with property AOP in the case n > 2. The study in this paper could be developed considering sets of utility functions whose derivatives have a constant sign. For example, one can study existence conditions for absolute portfolios when the set of utility functions is composed from U 2 utility functions that have a non-negative third derivative. More generally, one can investigate existence conditions for absolute portfolios in the case where the set U of utility functions is composed of functions u with the property that (−1) i u (i) ≤ 0, i = 1, 2, . . . , k.