Conditions for the Existence of Absolutely Optimal Portfolios

Abstract

Let Δ_n be the n-dimensional simplex, ξ = (ξ₁, ξ₂,…, ξ_n) be an n-dimensional random vector, and $\mathcal{U}$ be a set of utility functions. A vector x^* $\in$ Δ_n is a $\mathcal{U}$ -absolutely optimal portfolio if $\mathrm{E}\left(u\left({\mathsf{\xi }}^{\mathbf{T}}{x}^{*}\right)\right)\ge \mathrm{E}\left(u\left({\mathsf{\xi }}^{\mathbf{T}}x\right)\right)$ for every x $\in$ Δ_n and u ∈ $\mathcal{U}$. In this paper, we investigate the following problem: For what random vectors, ξ, do $\mathcal{U}$-absolutely optimal portfolios exist? If ${\mathcal{U}}_2$ 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 ${\mathcal{U}}_2$-absolutely optimal portfolio. The main result is the following: If x⁰ is a portfolio having all its entries positive, then x⁰ is an absolutely optimal portfolio if and only if all the conditional expectations of ${\xi }_i$, given the return of portfolio x⁰, are the same. We prove that if ξ is bounded below then CARA-absolutely optimal portfolios are also ${\mathcal{U}}_2$-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 = (ξ₁, ξ₂). We give a complete characterization and we build two dimensional distributions that are absolutely continuous and admit ${\mathcal{U}}_2$-absolutely optimal portfolios.

1. Introduction. Defining the Problem

Let (Ω,K,P) be a fixed probability space and ξ: Ω → ℝⁿ be an n–dimensional random vector. The random vector ξ = (ξ₁, ξ₂, …, 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 x₁ + … + x_n = S. The set of all portfolios of sum S will be denoted by Δ_n(S). A standard portfolio is a portfolio of sum 1, that is, an element of standard simplex Δ_n(1).

We shall also use the set P_n−1(S) = {x ∈ ℝⁿ⁻¹|x_i ≥ 0 ∀ i = 1, …, n − 1 and x₁ + … + 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 $e=\left(1,1,\dots ,1\right)$. The return of portfolio x is the random variable

Y(x) = x^Tξ = x₁ξ₁ + … + x_nξ_n

(1)

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 ${\mathcal{U}}_3$.

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

$$V\left(x,\mathit{\xi };u\right)=\mathrm{E}u\left(x^T\mathit{\xi }\right)$$

(2)

for all portfolios x ∈ Δ_n(S), provided that $x^T\mathit{\xi }$ ∈ I.

From the point of view of expected utility theory, a decision maker with utility u prefers the portfolio y to portfolio x if

$$V\left(\mathbf{x},\mathit{\xi };u\right)\le V\left(\mathbf{y},\mathit{\xi };u\right)\iff \mathrm{E}u\left(x^T\mathit{\xi }\right)\le \mathrm{E}u\left(y^T\mathit{\xi }\right)$$

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^Tξ) ∈ L¹(Ω, K,P) for any $x\in {ℝ}_{+}^n$.

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 ${\mathcal{U}}_2$(S, ξ) denote the set of all concave utilities, u having the property that u (x^Tξ) makes sense and, moreover, u (x^Tξ) ∈ L¹(Ω, K,P) for all x ∈ Δ_n(S).

Let ξ be a fixed random vector from ℝⁿ and u ∈ ${\mathcal{U}}_2$(S, ξ) be a fixed utility. Consider the problem:

P(S): max{Eu(x^Tξ): x ∈ Δ_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 $x\to$ Eu(x^Tξ), x ∈ Δ_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^Tξ))_n converges to u(x^Tξ) and is dominated by the random variable

max_i(|u(S ξ_i)|) ∈ L¹ (Ω K,P)

As Δ_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 Δ_n (S). This is called the set of optimal portfolios and is denoted by Opt(ξ,S;u). A portfolio x ∈ Δ_n (S) is called an equal weight portfolio if all its entries are equal, that is x = $\left(\frac{S}{n},....,\frac{S}{n}\right)$.

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 = (ξ₁, ξ₂,…, ξ_n) then the equal weight portfolio is optimal for the problem P(S):

$${\mathbf{x}}^{\mathbf{o}}=\left(\frac{S}{n},....,\frac{S}{n}\right)\in \mathbf{Opt}(\xi ,S;u) \mathrm{for} \mathrm{every} u \in {\mathcal{U}}_2(S, \mathsf{\xi })$$

(3)

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.

In [1], Samuelson generalized the above result. He showed that the hypothesis “ξ_i, I = 1, 2, …, n are i.i.d. random variables” may be replaced by a more general condition:

Denote by F_ξ the distribution of the random vector ξ, that is F_ξ (B) = P(ξ ∈ B). Call F_ξ symmetrical if F_ξoσ = F_ξ for any permutation σ of the set {1, 2,…, n}. Here, ξ σ means the vector (ξ_σ(1), ξ_σ(2),…, ξ_σ(n)).

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_ξoσ = 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 ${\mathit{F}}_{\left({\mathit{\xi }}_{\mathbf{1}},{\mathit{\xi }}_{\mathbf{2}},\dots ,{\mathit{\xi }}_{\mathit{n}}\right)}={\mathit{F}}_{\left({\mathit{\xi }}_{\mathbf{2}},\dots ,{\mathit{\xi }}_{\mathit{n}},{\mathit{\xi }}_{\mathbf{1}}\right)}$ 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:

(i)

$F_{\xi }\left(Ax\right)=F_{\xi }\left(x\right)$ for every $x\in {ℝ}^n$;

(ii)

The equal weight portfolio is optimal for the problem P(S);

Tamir’s result [3] for cyclically symmetry of $F_{\xi }$ is the most general condition we know so far on $F_{\xi }$.

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 ∈${\mathcal{U}}_2$(S, ξ). We state that ξ (or better$F_{\xi }$) 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 $\mathcal{U}$ -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 $\mathcal{U}={\mathcal{U}}_2$, what is the set of concave utility functions; we find necessary and sufficient conditions in order that $F_{\xi }$ has the AOP property, i.e., $F_{\xi }$ admits an absolute portfolio. The main result is the following: If x⁰ is a portfolio having all its entries positive, then x⁰ is an absolute portfolio if and only if all the conditional expectations of ${\xi }_i$, given the return of portfolio x⁰, 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 = (ξ₁, ξ₂) in Section 6. We give a complete characterization and we build two dimensional distributions that are absolutely continuous and admit ${\mathcal{U}}_2$-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^Tξ, x ∈ P_n(S) has a maximum in the increasing concave order.

2. 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:

$$\mathrm{E}u\left(X\right)\le \mathrm{E}u\left(Y\right) \mathrm{for} \mathrm{every} \mathrm{concave} \mathrm{utility} u \mathrm{such} \mathrm{that} u\left(X\right),u\left(Y\right)\in L^1$$

(4)

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:

$$F{\prec }_{\mathrm{icv}}G\iff \int udF\le \int udG \mathrm{for} \mathrm{every} u\in {\mathcal{U}}_2 \mathrm{such} \mathrm{that} \int \left|u\right|dF,\int \left|u\right|dG<\infty$$

In this case, we can restate (3) as:

Proposition 1.

Let ξ = (ξ₁, ξ₂,…, ξ_n) be a random vector with i.i.d. integrable entries and x^o =$\left(\frac{S}{n},....,\frac{S}{n}\right)$with S > 0. Then x^Tξ ≺_icv (x^o)^Tξ for every x ∈ Δ_n(S).

Proof. 

Let X = (x^o)^Tξ = $S\frac{{\xi }_1+{\xi }_2+\dots +{\xi }_n}{n}$. We confirm that E(ξ_i|X) = E(ξ₁|X). Indeed, E(ξ_i|X) is a random variable of the form g_i(X) having the property that E(ξ_ih(X)) = E(g_i(X)h(X)) for every bounded measurable function h:ℝ→ℝ. As ξ_i are i.i.d.,

$$\begin{gathered} \mathrm{E}({\xi }_{\mathrm{i}}h(X))=\int x_{i}h(\frac{S}{n}(x_1+\dots +x_n))d{F}^{\otimes n}\left(x_1,x_2,\dots ,x_n\right) \\ =\int x_{1}h(\frac{S}{n}(x_1+\dots +x_n))d{F}^{\otimes n}\left(x_1,x_{2 },\dots , x_n\right)=\mathrm{E}({\xi }_{1}h(X)) \end{gathered}$$

The last equality holds because of the equality dF^⊗n(x_σ(1),…,x_σ(n)) = dF^⊗n(x₁,…,x_n).

It follows that E(ξ_i|X) = E(ξ₁|X) ∀ i = 2 ,…, n.

Consequently X = E(X|X) = E($S\frac{{\xi }_1+{\xi }_2+\dots +{\xi }_n}{n}$|X) = $\frac{S}{n}$ (E(ξ₁|X) + E(ξ₂|X) + … + E(ξ_n|X)) = $\frac{S}{n}$⋅n⋅E(ξ₁|X) = SE(ξ₁|X). It follows that E(ξ₁|X) = X/S.

Let x ∈ Δ_n(S) be arbitrary and v(x) = u(x/S). Then, by Jensen’s inequality, Eu(x^Tξ) = E(E(u(x^Tξ)|X)) = $\mathrm{E}\left(\mathrm{E}u(S\left(\frac{x_1{\xi }_1+\dots +x_n{\xi }_n}{S}\right)\left|X)\right.\right)$ = $\mathrm{E}\left(\mathrm{E}v\left(\frac{x_1{\xi }_1+\dots +x_n{\xi }_n}{S}\right)\left|X)\right.\right)$ ≤ $\mathrm{E}\left(v(\mathrm{E}\left(\frac{x_1{\xi }_1+\dots +x_n{\xi }_n}{S}\right)\left|X)\right.\right)$ = E(v(E(ξ₁|X))) = E(v(X/S)) = Eu(X).

Therefore, Eu(x^Tξ) ≤ Eu(X) = Eu((x^o)^Tξ); 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^Tξ ≺_cv (x^o)^Tξ for every x ∈ Δ_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 = (ξ₁, ξ₂,…, 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:

Proposition 2.

(i)

If X ≺_icv Y then EX ≤ EY;

(ii)

If X ≺_cv Y then Var(X) ≥ Var(Y);

(iii)

If f:ℝ →ℝ is non-decreasing and concave, then X ≺_icv Y ⇒ f(X) ≺_icv f(Y);

(iv)

(Invariance with respect to convolution): If X_j ≺_icvY_j, j = 1,2 and (X₁, X₂) are independent and (Y₁, Y₂) are independent, then X₁ + X₂ ≺_icv Y₁ + Y₂;

(v)

(Invariance with respect to mixture): If F_j≺_icvG_j, j = 1,2 and 0 ≤ λ ≤ 1, then (1 − λ)F₁ + λF₂ ≺_icv (1 − λ)G₁ + λG₂;

(vi)

(Invariance with respect to weak convergence): If F_n ≺_icvG_n ∀ n, Supp(F_n) ⊆ [0, ∞), ∀ n. Supp(G_n) ⊆ [0, ∞) ∀ n and F_n ⇒ F, G_n ⇒ G then F ≺_icv G.

Proof. 

(i)

Take the utility u(x) = x;

(ii)

Take the concave function f(x) = − x²;

(iii)

Notice that if u,f are non-decreasing and concave, then u◦f is also non-decreasing concave;

(iv)

See, for instance [5,6,7,8];

(v)

Obvious;

(vi)

We know that $\int ud{F}_n\le \int ud{G}_n$ for any concave utility u:[0,∞) → ℝ and we want to check that $\int udF\le \int udG$. We claim that $\int ud{F}_n\to \int udF,\int ud{G}_n\to \int udG$. The reason for this is that:

$$\begin{gathered} \left|\int udF-\int ud{F}_n\right|\le \left|\int udF-\int \min (u,M)dF\right|+ \\ \left|\int \min (u,M)dF-\int \min (u,M)d{F}_n\right|=I_1(M)+I_2(M,n) \end{gathered}$$

Let ε > 0. Then there exists M_ε > 0, such that I₁(M_ε) < ε/2. The function f = min(u,M_ε) is continuous and bounded; hence, I₂(M_ε,n) → 0 as n $\to \infty$. Thus, for n that is great enough, $\left|\int udF-\int ud{F}_n\right|$ < ε. Therefore, $\int ud{F}_n\to \int udF,\int ud{G}_n\to \int udG$.

Thus, our problem becomes:

Let ξ = (ξ₁, ξ₂,…, ξ_n) be a random vector, S > 0, F_x = P○(x^Tξ)⁻¹ be the distribution of the random variable x^Tξ, and let D(ξ) = {P○(x^Tξ)⁻¹|x ∈ Δ_n(S)}. When does the family D(ξ) have the greatest element with respect to the order “≺_icv”?

Or, even better.

Let F be a probability distribution on ℝⁿ, L_x: ℝⁿ → ℝ be the linear mappings L_x(y) = x^Ty, and F_x = Fo(L_x)⁻¹ and D(F) = {F_x|x ∈ Δ_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 ∈ Δ_n(S), with the property that ${\mathit{F}}_{{\mathit{x}}^o}$ 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 ∈ Δ_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:

ξ ∈ AO (x^o) ⇔ x^Tξ ≺_icv (x^o)^Tξ for every x ∈ Δ_n(S)

Or, in terms of distributions:

$$F \in \mathit{AO} ({\mathbf{x}}^o) \iff \mathit{F}\circ {\left({\mathit{L}}_x\right)}^{-\mathbf{1}}{\prec }_{\mathrm{icv}}\mathit{F}\circ {\left({\mathit{L}}_{x^{\mathit{o}}}\right)}^{-\mathbf{1}} \mathrm{for} \mathrm{every} \mathrm{x} \in {\Delta }_n(S)$$

where L_x is the linear form L_x (y) = x^T y, y ∈ ℝⁿ.

Then, for a given S > 0, the class AO of distributions on ℝⁿ is defined by:

$$\mathit{A}\mathit{O}=\underset{{\mathit{x}}^{\mathit{o}}\in {\Delta }_n\left(S\right)}{{\displaystyle \cup }}\mathit{A}\mathit{O}\left({\mathit{x}}^{\mathit{o}}\right)$$

3. Properties of Class AO

At this point, we do not know if the AO class contains other distributions on ℝⁿ 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.

(i)

If ξ ∈ AO (x^o), a ∈ ℝ, λ > 0 then a⋅e + λξ ∈AO (x^o). Here e = (1, 1,…, 1) ∈ ℝⁿ (invariance with respect to scaling);

(ii)

Ifξ₁, ξ₂are independent and both belong to AO (x^o), then ξ₁ + ξ₂ ∈ AO (x^o) (invariance with respect to convolutions);

(iii)

Ifξ₁, ξ₂ are independent, A ∈ K is independent on both of them and ξ₁, ξ₂ ∈AO (x^o), then the random vector $\xi ={\xi }_1{1}_A+{\xi }_2{1}_{A^c}$ 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 ξ_n ∈ AO(x^o) ∀ n implies ξ ∈ AO(x^o) (invariance with respect to weak convergence of nonnegative distributions);

(v)

Let ξ ∈ AO(x^o). Then x^o/S is absolutely optimal among all the portfolios of sum 1, or the same thing in terms of distributions.

(i)

F ∈ AO(x^o), a ∈ ℝ, λ > 0 ⇒ δ_a⋅1 ∗ Fo(h_λ)⁻¹ ∈ AO(x^o); here, $h_{\lambda }$ is the homothety $h_{\lambda }\left(x\right)=\lambda x$;

(ii)

F₁, F₂ ∈ AO(x^o) ⇒ F₁∗F₂ ∈ AO(x^o);

(iii)

F₁, F₂ ∈ AO(x^o), 0 ≤ λ ≤ 1 ⇒ (1 − λ)F₁ + λF₂ ∈ AO(x^o);

(iv)

Supp(F_n) ⊆ ${ℝ}_{+}^n$ ∀ n, F_n ⇒ F, F_n ∈ AO(x^o) ∀ n ⇒ F ∈ AO(x^o);

(v)

Suppose thatx^o ∈ Δ_n(S) and F ∈ AO(x^o), then F ∈ AO(x^o/S) as well. This means that when dealing with distributions of class AO we always can assume that S = 1.

Proof. 

(i)

The assumption is that x^Tξ ≺_icv (x^o)^Tξ ∀ x ∈ Δ_n(S). Notice that x^T(a⋅e + λξ) = aS + λx^Tξ and (x^o)^T(a⋅e + λξ) = aS + λ(x^o)^Tξ. Let X = x^Tξ and X^o = (x^o)^Tξ. We know that X ≺_icv X^o. Then aS + λX ≺_icv aS + λX^o, because of Proposition 2., the function x ↦ aS + λx is increasingly concave.

(ii)

Now we know that x^Tξ₁ ≺_icv (x^o)^Tξ₁ ∀ x ∈ Δ_n(S) and that x^Tξ₂ ≺_icv (x^o)^Tξ₂. By Proposition 2. (iv), we see that x^T(ξ₁ + ξ₂) ≺_icv (x^o)^T(ξ₁ + ξ₂).

(iii)

Let F_j = Po(ξ_j)⁻¹ with j = 1,2 and 1 − λ = P(A). Then the distribution of ξ is (1 − λ)F₁ + λF₂. We assumed that F_j ∈ AO(x^o) ⇔ $F_j\circ {\left(L_x\right)}^{-1}{\prec }_{\mathrm{icv}}{F}_j\circ {\left(L_{x^o}\right)}^{-1}$∀ x ∈ Δ_n(S), j = 1,2. Then, by Proposition 2. (v), we have $\left(1-\lambda \right)\left(F_1\circ {\left(L_x\right)}^{-1}\right)+\lambda \left(F_2\circ {\left(L_x\right)}^{-1}\right){\prec }_{\mathrm{icv}}\left(1-\lambda \right)\left(F_1\circ {\left(L_{x^o}\right)}^{-1}\right)+\lambda \left(F_2\circ {\left(L_{x^o}\right)}^{-1}\right)$ or $\left(\left(1-\lambda \right)F_1+\lambda F_2\right)\circ {\left(L_x\right)}^{-1}{\prec }_{\mathrm{icv}}\left(\left(1-\lambda \right)F_1+\lambda F_2\right)\circ {\left(L_{x^o}\right)}^{-1}$ for every x ∈ Δ_n(S), hence (1 − λ)F₁ + λF₂ ∈ AO(x^o).

(iv)

A consequence of Proposition 2 (v).

(v)

Obvious: x^Tξ ≺_icv (x^o)^Tξ ∀ x ∈ Δ_n(S) ⇒ x^Tξ/S ≺_icv (x^o)^Tξ/S ∀ x ∈ Δ_n(S) ⇔ y^Tξ ≺_icv (x^o/S)^Tξ ∀ x ∈ Δ_n(1). □

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, Δ_n will mean Δ_n(1). Now we give a necessary and sufficient condition in order that a distribution on ℝⁿ, F be in the class AO(x^o), provided that$x_i^0$> 0 for all i = 1, …, n.

Proposition 4.

Let x^o ∈ Δ_n be such that$x_i^0$> 0 ∀ i =1,…,n. Let also$X^0$= (x^o)^Tξ. Then

$$\mathsf{\xi } \in AO ({\mathbf{x}}^o) \iff \mathrm{E}({\xi }_1|X^0)=\mathrm{E}({\xi }_2|X^0)=\dots =\mathrm{E}({\xi }_n|X^0)=X^0$$

(5)

Proof. 

The easy part is “⇐”. Indeed, let x ∈ Δ_n and let u ∈ ${\mathcal{U}}_2$(1, ξ). Then

$$\begin{gathered} \mathrm{E}u({\mathbf{x}}^T\xi )=\mathrm{E}[\mathrm{E}u({\mathbf{x}}^T\mathsf{\xi }|X^0)] \le \mathbf{E}[u(\mathrm{E}({\mathbf{x}}^T\mathsf{\xi }|X^0)]=\mathrm{E}[u(x_1\mathrm{E}({\xi }_1|X^0)+x_2\mathrm{E}({\xi }_2|X^0)+\dots + \\ {x}_n\mathrm{E}({\xi }_n|X^0))]=\mathrm{E}u(x_1{X}^0+x_2{X}^0+\dots +x_n{X}^0)=\mathrm{E}u(X^0)=\mathrm{E}u(({\mathrm{x}}^{\mathrm{o}}{)}^{\mathit{T}}\xi ); \end{gathered}$$

hence, x^o is an absolute optimum. We prove now “⇒”. Let u ∈ ${\mathcal{U}}_2$(1, ξ). Consider the function h_u: P_n−1 → ℝ defined by:

h_u(x₁,…,x_n−1) = Eu(x₁ξ₁ + … + x_n−1ξ_n−1 + (1 − x₁ − … − x_n−1)ξ_n)

(6)

□

Suppose that u is differentiable, then h_u is also differentiable. Let g = Grad(h_u). Thus,

$$\begin{gathered} g_i\left(x_1,\dots ,x_{n-1}\right)=\frac{\partial h_u}{\partial x_i}\left(x_1,\dots ,x_{n-1}\right) \\ =E[\left({\xi }_1-{\xi }_n\right)u\prime \left(x_1{\xi }_1+\dots +x_{n-1}{\xi }_{n-1}+\left(1-x_1-\dots -x_{n-1}\right){\xi }_n\right] \end{gathered}$$

(7)

Let y^o =$\left(x_1^0,x_2^0,\dots ,x_{n-1}^0\right)$∈ 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:

$$\mathbf{g}({\mathbf{y}}^{\mathbf{o}})=0 \iff \mathrm{E}[({\xi }_i-{\xi }_n)u\prime (X^0)]=0 \forall i=1,\dots ,n-1 \forall u \in {\mathcal{U}}_2(\mathbf{1},\mathsf{\xi })$$

(8)

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:

$$\mathrm{E}[({\xi }_i-{\xi }_n); X^0 \le a]=0 \mathrm{for} \mathrm{every} i=1,\dots ,n-1 \mathrm{and} a \in ℝ$$

(9)

Denote by $\mathcal{B}$ (ℝ) the σ-algebra of the Borel sets on the real line.

Let C = {A ∈ $\mathcal{B}$ (ℝ): E((ξ_i − ξ_n)1_A($X^0$) = 0}. According to (9), C contains all the intervals (−∞, a]. Moreover, it fulfills the following conditions:

(i)

ℝ ∈ C;

(ii)

A ∈ C ⇒ A^c ∈ C;

(iii)

A_n ∈ C and (A_n)_n are disjoint ⇒ ${\displaystyle \underset{n}{\cup }{A}_n}$ ∈ C.

Property (i) follows if we let a → ∞. Property (ii) results from the following series of equalities: 0 = E(ξ_i − ξ_n) − E((ξ_i − ξ_n)1_A($X^0$) = E((ξ_i − ξ_n)(1 − 1_A)($X^0$). Property (iii) 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, $\mathcal{B}$ (ℝ). Thus, C contains all the Borelian sets. The conclusion is that:

$$\mathrm{E}[({\xi }_i-{\xi }_n) 1_A(X^0)]=0 \forall I=1 ,\dots , n-1 \forall A\in \mathcal{B} (ℝ)$$

(10)

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:

$$\mathrm{E}[{\xi }_i f(X^0)]=\mathrm{E}[{\xi }_n f(X^0)] \forall i=1,\dots ,n-1 \forall f:ℝ \to ℝ \mathrm{measurable} \mathrm{bounded}$$

(11)

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_if($X^0$)) 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,

$$X^0=\mathrm{E}(X^0|X^0)=\mathrm{E}({x^{\mathrm{o}}}_1{\xi }_1+\dots +{x^{\mathrm{o}}}_n{\xi }_n|X^0)=x_1^0\mathrm{E}({\xi }_1|X^0)+\dots +x_n^0\mathrm{E}({\xi }_n|X^0)=(x_1^0+\dots +x_n^0)\mathrm{E}({\xi }_n|X^0)=\mathrm{E}({\xi }_n|X^0)$$

Hence:

$$\mathrm{E}({\xi }_i|X^0)=\mathrm{E}(X_n|X^0)=X^0$$

As a consequence, we find a necessary condition in order that ξ may have the property AO.

Corollary 1.

If ξ ∈ AO (x^o) and$x_i^0$> 0 ∀ I = 1,…,n, then Eξ₁ = Eξ₂ =…= Eξ_n.

Proof. 

According to Proposition 3.2, E(ξ₁|$X^0$) = E(ξ₂|$X^0$) =…= 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 EX₁ ≤ EX₂ ≤…≤ EX_n. It is possible that some of the expectations coincide. In general, we shall use the convention:

μ₁ ≤ μ₂ ≤…≤ μ_k < μ_k+1 = μ_k+2 =…= μ_n

Denote by INC(k), the set of all random vectors ξ = (ξ₁, ξ₂,…, 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).

Proposition 5.

Let ξ ∈ INC(k) with k ≥ 1.

If ξ has an absolute optimal portfolio x^o, then $x_1^0$ =…= $x_k^0$ = 0. Moreover, Var((x^o)^Tξ) ≤ Var(x^Tξ) ∀ x ∈ Δ_n.

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)^Tξ. Suppose, ad absurdum, that x^o_i > 0 for some i ≤ k. Then E$X^0$ < EX_n; hence, $X^0$ cannot be optimal, according to Proposition 2. (i), which states that EX_n ≤ E$X^0$. Therefore, the possible absolute optima must be of the form x^o =$\left(0,0,\dots ,x_{k+1}^0,\dots ,x_n^0\right)$. All the portfolios x with x_j = 0 ∀ j ≤ k have the property that E(x^Tξ) = 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)^Tξ) ≤ Var(x^Tξ) for all such x. If another portfolio, say y ∈ Δ_n, would exist such that Var((x^o)^Tξ) > Var(y^Tξ), then (x^o)^Tξ could not dominate y^Tξ in the “≺_icv” order. □

Let us denote P_n−1(k) = {y ∈ P_n−1: y₁ =…= y_k = 0, y_j > 0 ∀ j > k }. Note that P_n−1(0) is the interior of P_n−1. Let Δ_n,k = {x ∈ Δ_n: x₁ =…= x_k = 0, x_j > 0 ∀ j > k }. Note that Δ_n,0 = {x ∈ Δ_n: x_j > 0 ∀ j}. We want to find necessary and sufficient conditions in order that ξ ∈ AO (x^o), x^o ∈ Δ_n,k. We shall use the following particular case of Kuhn–Tucker conditions, which probably is well known:

Lemma 1.

Let f: P_n−1 → ℝ be concave and differentiable and let g = Grad(f). Let a ∈ P_n−1 (k). Then:

f(a) ≥ f(y) ∀ y ∈ P_n−1 ⇔ g_j(a) ≤ 0 ∀ j ≤ k, g_j(a) = 0 ∀ j > k

(12)

Proof. 

“⇒”: Let a,y ∈ P_n−1 and φ_a,y: [0,1] → ℝ be defined as φ_a,y(t) = f((1 − t)a + ty), t ∈ [0,1]. Then, φ_a,y is differentiable, concave, and its maximum is at t = 0. Therefore, it is non-increasing, hence (φ_a,y)’ (0) ≤ 0. As a ∈ P_n−1(k), and we see that:

$$({\mathsf{\phi }}_{\mathbf{a},\mathbf{y}})\prime (0)={{\displaystyle \sum }}_{j=1}^{n-1}\left(y_j-a_j\right)g_j\left(\mathit{a}\right)={{\displaystyle \sum }}_{j=1}^k{y}_j{g}_j\left(\mathit{a}\right)+{{\displaystyle \sum }}_{j=k+1}^{n-1}\left(y_j-a_j\right)g_j\left(\mathit{a}\right) \le 0 \forall \mathbf{y} \in P_{n-1}$$

If we choose y ∈ P_n−1(k), the condition becomes ${{\displaystyle \sum }}_{j=k+1}^{n-1}\left(y_j-a_j\right)g_j\left(\mathit{a}\right)$ ≤ 0, which of course implies g_j(a) = 0 ∀ j > k (we can choose y = a ± εe_j with ε > 0 small enough). Thus, we get:

$${{\displaystyle \sum }}_{j=1}^k{y}_j{g}_j\left(a\right) \le 0 \forall \mathbf{y}\in P_{n-1} \Rightarrow g_j(\mathbf{a}) \le 0 \forall j \le k$$

Conversely, if g_j(a) ≤ 0 ∀ j ≤ k, g_j(a) = 0 ∀ j > k, it is obvious that (φ_a,y)’ (0) ≤ 0; as φ_a,y is concave, this fact implies that φ_a,y is non-increasing, hence φ_a,y (0) ≥ φ_a,y (1) ⇔ f(a) ≥ f(y). □

Proposition 6.

Let x^o ∈ Δ_n,k and X^o = (x^o)^Tξ. Then ξ ∈ AO (x^o) ⇔ E(ξ_j/$X^0$≤ a) ≤ E(ξ_n/$X^0$≤ a) ∀ j ≤ k, a ∈ ℝ and:

$$\mathrm{E}({\xi }_j|X^0)=X^0 \forall j > k$$

(13)

Proof. 

Let u be a concave differentiable utility, z ∈ P_n−1(k), defined as:

$$\mathbf{z}=\left(0,0,\dots ,x_{k+1}^0,\dots ,x_{n-1}^0\right) \mathrm{and} h_u(\mathbf{y})=\mathrm{E}u(y_1{\xi }_1+...+y_{n-1}{\xi }_{n-1}+(1-y_1 –...-y_{n-1}){\xi }_n)$$

If ξ ∈AO (x^o), 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:

$$\mathrm{E}[({\xi }_j-{\xi }_n)u^{\prime }(X^0)] \le 0 \forall j \le k,\mathrm{E}[({\xi }_j-{\xi }_n)u^{\prime }(X^0)]=0 \forall j > k \forall u \in {\mathcal{U}}_2 \mathrm{differentiable}$$

(14)

□

In the same way as we did in the proof of Proposition 4., we can take u’ to be 1_{(−∞, a]}. Thus, (14) becomes:

$$\mathrm{E}[({\xi }_j-{\xi }_n)1_{(-\infty ,a]}(X^0)] \le 0 \forall j \le k,\mathrm{E}[({\xi }_j-{\xi }_n) 1_{(-\infty ,a]} (X^0)]=0 \forall j > k \forall a \in ℝ$$

(15)

which implies (13), by the same argument used in the proof of Proposition 4.:

$$\mathrm{E}({\xi }_j|X^0)=\mathrm{E}({\xi }_n|X^0) \forall j>k$$

which implies that:

$$X^o=E(X^o\left|X^o)=\right.{\displaystyle \sum }_{j=k+1}^n{x}_j^{o}E({\xi }_j\left|X^o)\right.=\left({\displaystyle \sum }_{j=k+1}^n{x}_j^o\right)E({\xi }_n\left|X^o)\right.=E({\xi }_n\left|X^o)\right.$$

Hence, E(ξ_j|$X^o$) = $X^o$ for every j > k.

Conversely, if the conditions from the right hand of (13) hold, then E(ξ_iu’(X^o)) ≤ E(ξ_nu’($X^o$)) ∀ j ≤ k and E(ξ_iu’($X^0$)) = E(ξ_nu’($X^0$)) ∀ 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)^Tξ) ≥ Eu((x)^Tξ) for any x ∈ Δ_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 ∈ Δ_n,k. A sufficient condition in order that ξ ∈ AO (x^o) is

$$\mathrm{E}[({\xi }_j-{\xi }_n)|X^0] \le 0 \forall j \le k,\mathrm{E}[({\xi }_j-{\xi }_n)|X^0)]=0 \forall j > k$$

(16)

Proof. 

Obviously, E[(ξ_j − ξ_n)|$X^0$] ≤ 0 implies E(ξ_iu’($X^0$)) ≤ E(ξ_nu’($X^0$)) ∀ u differentiable, concave, and non-decreasing, such that the integral makes sense. □

4. The Class CARAAO

In this section, we shall deal only with short tailed random vectors. A random vector ξ = (ξ₁,...,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 ∈ Δ_n and ξ be a random vector. We say that ξ has the property CARAAO(x^o) (and write ξ ∈ CARAAO(x^o)) if:

$$E{e}^{-r{(x^o)}^T\xi }\le E{e}^{-r{x}^T\xi }$$

(17)

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.

It is obvious that, if ξ ∈ AO(x^o), then ξ ∈ CARAAO(x^o)

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.

Proposition 7.

Suppose that ξ is bounded below, in the sense that there exists m ∈ ℝ such that ξ_i ≥ m a.s. ∀ i = 1,...,n. Then, CARAAO(x^o) = AO(x^o) for every x^o ∈ Δ_n,0. Or, in words: if all the components of x^o are positive, then the classes CARAAO(x^o) and AO(x^o) coincide.

Proof. 

Let x^o ∈ Δ_n,0 and y^o = $\left(x_1^0,x_2^0,\dots ,x_{n-1}^0\right)$ ∈ P_n−1. Let h: P_n−1 → ℝ defined by

$$h\left(x\right)=e^{-r\left(x_1{\xi }_1+\dots +x_{n-1}{\xi }_{n-1}+\left(1-x_1-\dots -x_{n-1}\right){\xi }_n\right)}, \mathbf{x} \in P_{n-1}$$

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)^Tξ. Then:

(Grad h)(y^o) = 0 ⇔ E[(X_i − X_n)e^−rξ] = 0 ∀ r ≥ 0, i = 1,..,n − 1

Let $Z=e^{-X^0}$ ≤ e ^–m a.s. The condition (Grad h)(y^o) = 0 becomes:

$$\mathrm{E}({\xi }_j{Z}^r)=\mathrm{E}({\xi }_n{Z}^r) \forall r > 0 \forall j=1,...,n-1$$

(18)

Which further implies:

E(ξ_jf(Z)) = E(ξ_nf(Z)) for any polynomial function f, ∀ j = 1,...,n − 1

(19)

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 $\ge -1$ for all i. The hypothesis that ξ_i are bounded below from the preceding proposition is verified.

Example. 

Suppose that ξ_j ~ Gamma(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:

$$x^o=\left(\frac{a_1}{S},\frac{a_2}{S},\dots ,\frac{a_n}{S}\right),S={\displaystyle \sum }_{j=1}^n{a}_j$$

Indeed, we have to minimize the function:

$$h(x_1,...,x_{n-1})=\ln \left[\mathrm{E}{e}^{-r\left(x_1{\xi }_1+\dots +x_{n-1}{\xi }_{n-1}+\left(1-x_1-\dots -x_{n-1}\right){\xi }_n\right)}\right] \mathrm{with} \mathbf{x} \in P_{n-1}$$

As (ξ_j)_j are independent,

$$h(\mathbf{x})=a_{1}ln\frac{a_1}{a_1+r{x}_1} +a_{2}ln\frac{a_2}{a_2+r{x}_2}+...+a_{n}ln\frac{a_n}{a_n+s{x}_n} \mathrm{with} x_n=1-x_1 -...- x_{n-1}$$

After some calculus, one obtains the gradient g = Grad(h). Its components are:

$$g_i(\mathbf{x})=\frac{r^2\left(a_i{x}_n-a_n{x}_i\right)}{\left(a_i+r{x}_i\right)\left(a_n+r{x}_n\right)} \mathrm{thus}\mathrm{Grad}(h)=\mathbf{0} \iff x_i=\frac{a_1}{a_n}{x}_n$$

Therefore, x^o = $\left(\frac{{\mathit{a}}_{\mathbf{1}}}{\mathit{S}},\frac{{\mathit{a}}_{\mathbf{2}}}{\mathit{S}},\dots ,\frac{{\mathit{a}}_{\mathit{n}}}{\mathit{S}}\right)$. From the above proposition, x^o is an absolute optimal portfolio.

Remark. 

Even for n = 2, it is not true that if ξ₁ and ξ₂ are independent and Eξ₁ = Eξ₂, then (ξ₁,ξ₂) has the property AO. Suppose that ξ₁ ~ Uniform(−a,a) and ξ₂ ~ Uniform(−b,b) for somea,b > 0, a ≠ b. Then the minimum point of the function

$$h\left(x\right)=E{e}^{-rx{\xi }_1-r\left(1-x\right){\xi }_2}=\frac{sinh(arx)sinh(br\left(1-x\right)}{ab{r}^{2}x\left(1-x\right)}, 0 \le x \le 1$$

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 Δ_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 Δ_n,k, k ≥ 1 of the simplex, that fails to be true. Let x₁ = (0;1), x₂ = (0;3), x₃ = (1;3), and x₄ = (3;2) four points in the plane, and suppose that (ξ₁,ξ₂) ~$\left(\begin{array}{cccc}{\mathit{x}}_1& {\mathit{x}}_2& {\mathit{x}}_3& {\mathit{x}}_4\\ \alpha & \gamma & \delta & \beta \end{array}\right)$. Then ξ₁ ~$\left(\begin{array}{cccc}0& & 1& 3\\ \alpha +\gamma & & \delta & \beta \end{array}\right)$, ξ₂ ~$\left(\begin{array}{cccc}1& & 2& 3\\ \alpha & & \beta & \gamma +\delta \end{array}\right)$.

The condition that x^o = (0;1) is the absolute optimal portfolio is that:

E(ξ₁|ξ₂ ≤ a) ≤ E(ξ₁|ξ₂ ≤ a) ∀ a

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 [$\frac{\beta }{2\alpha }$ ≤ 1 and 3γ + 2δ ≥ β − α] or [$\frac{\beta }{2\alpha }$> 1 and 3γ + 2δ ≥ ${\left(\frac{\beta }{2\alpha }\right)}^2$]. ξ₁ ≺_icv ξ₂ ⇔ Eξ₁ ≤ Eξ₂ ⇔ 3γ + 2δ ≥ β − α.

Thus, there exists no implications that “x⁰ is an absolute CARA-optimal portfolio, ⇒ x⁰ is an absolute optimal portfolio” or even “ξ₁ ≺_icv ξ₂ ⇒ x^o = (0;1) is an absolute CARA-optimal portfolio”, as we thought.

5. 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,σ²) ≺_icv Normal (μ’,σ’²) ⇔ μ ≤ μ’, σ ≥ σ’.

Moreover, if ξ ~ Normal (μ,C) is a n-dimensional random vector and x ∈ Δ_n(S) is a portfolio of sum S, then x^Tξ ~ Normal(x^Tμ, x^TCx). Therefore, it is very easy to compare two portfolios x and y:

x^Tξ ≺_icv y^Tξ ⇔ x^Tμ ≤ y^Tμ and x^TCx ≥ y^TCy

(20)

Because these relations are homogeneous, the sum S does not matter; hence, we always can restrict ourselves to consider only portfolios from Δ_n.

We can now answer the question: when does ξ ∈ AO provided that ξ ~ Normal (μ,C)?

Assuming the convention INC(k), we agree that μ₁ ≤ μ₂ ≤…≤ μ_k < μ_k+1 = μ_k+2 =…= μ_n.

Proposition 8.

Let ξ ~ Normal (μ,C). Then ξ ∈ AO if the Pareto problem

$$\left\{\begin{array}{c}\mathrm{maximize} \left({\mathbf{x}}^T\mathsf{\mu }\right)\\ \mathrm{minimize} \left({\mathbf{x}}^T\mathit{C}\mathit{x}\right)\\ {\mathit{x}}^T·\mathit{e}=1, \mathit{x}\ge 0,\end{array}\right.$$

(21)

admits at least solution.

Proof. 

Obvious from (20). □

Proposition 9.

Let ξ ~ Normal (μ,C) ∩ INC(k). Then

(i)

If k = 0, then ξ ∈ AO;

(i)

If k ≥ 1 then ξ ∈ AO if at least one solution x^o of the problem

$$\left\{\begin{array}{c} minimize \left(x^{T}Cx\right)\\ x^T·\mathbf{e}=1, \mathbf{x}\ge 0, \end{array}\right.$$

(22)

has the property that $x_j^0$ = 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

(23)

Proof. 

(i)

There is nothing to prove: all the portfolios have the same expectation.

(ii)

(Replace x_n with 1 − x₁ − x₂ − … − x_n−1 and let f(x) = Var(x^TCx). Then:

$$f\left(x\right)={\displaystyle \sum }_{1\le i,j\le n-1}{c}_{i,j}{x}_i{x}_j+2{\displaystyle \sum }_{1\le i\le n-1}{c}_{i,n}{x}_i\left(1-x_1-\dots -x_{n-1}\right)+c_{n,n}{\left(1-x_1-\dots -x_{n-1}\right)}^2$$

(24)

Its gradient g = Grad(f) has the components:

$$g_i\left(x\right)={\displaystyle \sum }_{j=1}^n\left(c_{i,j}-c_{n,j}\right)x_j,1\le i\le n-1$$

(25)

We claim that the inequalities (23) imply that f attains a minimum at points of the form x^o = (0,..., 0, x_k+1,..., x_n−1). Notice first that (23) implies that g_i(x) ≥ 0 ∀ 1 ≤ i ≤ k for points of form x = (0,..., 0, x_k+1,..., x_n−1), because $g_i\left(x\right)={{\displaystyle \sum }}_{j=k+1}^n\left(c_{i,j}-c_{n,j}\right)x_j$. In order to prove our claim, let us consider the convex function h: [0,∞) → ℝ, h(s) = f(sx₁,...,sx_k, x_n-k+1,...,x_n−1).

Note that h’(s) = ${{\displaystyle \sum }}_{i=1}^k{x}_i{g}_i\left(s{x}_1,\dots ,s{x}_k,x_{k+1},\dots x_{n-1}\right)$; hence:

$$h^{\prime }(0)={{\displaystyle \sum }}_{i=1}^k{x}_i{g}_i\left(0,\dots ,0,x_{k+1},\dots x_{n-1}\right) \ge 0$$

However, h is convex, hence its derivative is non-decreasing, hence it is non-negative, therefore h is non-decreasing itself. It follows that:

h(0) ≤ h(1) ⇔ f(0,..., 0, x_k+1,..., x_n−1) ≤ f(x₁,..., x_k, x_k+1,..., x_n−1)

In conclusion, f can attain its minimum only in points of the form (0,...,0,x_k+1,...,x_n−1). $g_i\left(x\right)={{\displaystyle \sum }}_{j=1}^n\left(c_{i,j}-c_{n,j}\right)x_j,1\le i\le n-1$. □

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 μ =$\left(\begin{array}{c}1\\ 2\\ 2\end{array}\right)$ and C = $\left(\begin{array}{ccc}6& 3& 2\\ 3& 2& 1\\ 2& 1& 2\end{array}\right)$

Note that ξ ~ N(μ,C) ∈ AO(x^o) for some x^o from Δ_3,1 because its first row is greater than the third one (each component of the first row is greater than the corresponding component of the third row).

One sees that x^o = (0, ½, ½) implies $X^0$ ~ N(2, ${\sqrt{\frac{3}{2}}}^2$). This is the absolute optimum.

Remark. 

The conditions from Corollary 3. are only sufficient, not necessary. For instance, the vector ξ ~ N(μ,C) where μ =$\left(\begin{array}{c}1\\ a\\ b\end{array}\right)$and C =$\left(\begin{array}{ccc}9& 0& 1\\ 0& 9& 1\\ 1& 1& 1\end{array}\right)$, 1 ≤ a ≤ b admits x^o = (0, 0, 1). Hence, $X^0$~ Normal(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.

Remark. 

If we want the absolute optimum porfolio to be of the form x^o = (0,x₂,...,x_n), then the necessary and sufficient condition is that c_1,2 ≥ c_n,2,..., c_1,n ≥ c_n,n. This is the same as the condition c_1,⋅ ≥ b_n,⋅ Indeed, because c_i,j = σ_iσ_jρ_i,j with Var(ξ_i) = σ_i² and ρ_i,j is the correlation coefficient between ξ_i and ξ_j, then the inequality c_1,i ≥ c_n,i is equivalent to σ₁ρ_1,i ≥ σ_nρ_n,i ∀ i = 2,...,n; hence, σ₁ ≥ σ_n$max\left(\frac{{\rho }_{n,2}}{{\rho }_{1,2}},\frac{{\rho }_{n,3}}{{\rho }_{1,3}},\dots ,\frac{{\rho }_{n,n}}{{\rho }_{1,n}}\right)$and this clearly implies c_1,1 ≥ c_1,n ⇔ σ₁ ≥ σ_nρ_1,n. However, for k ≥ 2, this is not true anymore. For instance, for k = 2, the conditions “c_i,j ≥ c_n,j ∀ j = 3,...,n ∀ i = 1,2” and “c_1,⋅≥ c_n,⋅, c_2,⋅ ≥ c_n,⋅” are not the same; the second one is stronger.

For n = 4, an example could be C = $\left(\begin{array}{cccc}16& 0& 4& 2\\ 0& 9& 1& 2\\ 4& 1& 4& 1\\ 2& 2& 1& 1\end{array}\right)$.

Here, σ₁ = 4, σ₂ = 3, σ₃ = 2, σ₁ = 1. The minimum point is of form x^o = (0,0,α,1 − α), in spite of the fact that neither the first row nor the second one are greater than the fourth one. Indeed, the components of the gradient are:

g₁(x) = 14x₁ − 2x₂ + 3x₃ + x₄, g₂(x) = − 2x₁ + 7x₂ + x₄

and has the property that g₁(x) ≥ 0, g₂(x) ≥ 0 for x ∈Δ_4,2. The reader may check that x^o = e₄. The optimal portfolio is e₄ = (0,0,0,1) whenever ξ~N(μ,C) with μ₁ ≤ μ₂ < μ₃ = μ₄.

6. 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 ℝ² 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 + (1 − s)Y) = E(Y|sX + (1 − s)Y).

In terms of densities, the condition means that:

$${{\displaystyle \int }}_{\left\{sx+\left(1-s\right)y=a\right\}}\left(x-y\right)f\left(x,y\right)dx=0\hspace{0.33em}\forall a \left(\mathrm{Absolutely} \mathrm{continuous} \mathrm{case}\right)$$

(26)

$${\displaystyle \sum }_{\left\{sx+\left(1-s\right)y=a\right\}}\left(x-y\right)f\left(x,y\right)=0\hspace{0.33em}\forall a \left(\mathrm{Discrete} \mathrm{case}\right)$$

Or, explicitly written:

$$\int \left(x-\frac{a-sx}{1-s}\right)f\left(x,\frac{a-sx}{1-s}\right)dx=0\hspace{0.33em}\forall a\hspace{1em}\left(\mathrm{Absolutely} \mathrm{continuous} \mathrm{case}\right)$$

(27)

$${\displaystyle \sum }_{\left\{x\right\}}\left(x-\frac{a-sx}{1-s}\right)f\left(x,\frac{a-sx}{1-s}\right)\hspace{0.33em}\forall a \left(\mathrm{Discrete} \mathrm{case}\right)$$

We shall focus on the absolute continuous case. After the substitution u = x − a in (27) one obtains:

$$\int uf\left(a+u,a-\lambda u\right)du=0\hspace{0.33em}\forall a \left(\mathrm{with} \lambda =\frac{s}{1-\mathrm{s}}\right)$$

(28)

Let us put ρ(a) = $\frac{1}{1-s}\int f\left(a+u,a-\lambda u\right)du$. Then ρ is a probability density. Let also

$$p_a(u)=\frac{f\left(a+u,a-\lambda u\right)}{\left(1-s\right)\rho \left(a\right)}$$

(29)

Then p_a is also a probability density and (28) becomes:

$$\int u{p}_a\left(u\right)du=0 \forall a$$

(30)

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 $f\left(a+u,a-\lambda u\right)=\left(1-s\right)p_a\left(u\right)\rho \left(a\right)$ or substituting a + u by x and $a-\lambda u$ by y, we obtain:

$$f(xy)=(1-\mathrm{s})p_{sx+\left(1-s\right)y}((1-s)(xy))\mathsf{\rho }(sx+(1-s)y) \mathrm{and} F=f\cdot {\lambda }^2$$

(31)

where ${\lambda }^2$ is the Lebesgue measure in the plane.

Then F ∈ OA(s).

Example 1.

Take p_a =$\frac{1}{2a}{1}_{\left[-a,a\right]}$and ρ = 1_{[0, 1]}. Then:

$$f(x,y)=(1-s)p_{sx+\left(1-s\right)y}((1-s)(y-x))\rho (sx+(1-s)y)=\frac{1-s}{2\left(y-s\left(y-x\right)\right)}{1}_{\Delta }\left(x,y\right)$$

where Δ is the interior of the triangle ABC with A(0,0), B(0,$\frac{1}{1-s})$, C($\frac{1+2s}{2s},\frac{1-4s^2}{4s\left(1-s\right)}$).

Notice, as a particular case, that if s = ½, then f(x,y) = $\frac{1}{2\left(x+y\right)}{1}_{\Delta }\left(x,y\right)$ 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(½) is:

$$f(x,y)=\frac{1}{2}{p}_{\frac{x+y}{2}}\left(\frac{x-y}{2}\right)\rho \left(\frac{x+y}{2}\right)$$

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

(32)

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:

$$f(x,y)=\frac{2}{1-s}g\left(\frac{2s-1}{1-s}x+2y,\frac{x}{1-s}\right)$$

(33)

is a density that satisfies (32). For s = ½ we discover again the symmetric densities because (33) becomes f(x,y) = 4g(2y,2x).

Example 2.

g(x,y) = 1_{[0, 1]×[0, 1]}(x,y) ⇒ f(x,y) =$\frac{2}{1-s}$1_D(x,y) where D =$\left\{\left(x,y\right)\in {ℝ}^2:0\le x\le 1-s,0\le \left(2s-1\right)x+\left(2-2s\right)y\le 1-s\right\}$.

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, ½). 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.

7. 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 ${\mathcal{U}}_2$ 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 $\mathcal{U}$ of utility functions is composed of functions u with the property that ${\left(-1\right)}^i{u}^{\left(i\right)}\le 0$, i = 1, 2,…, k.