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Weak Arbitrage Theorem Incorporating Loss Aversion

Submitted:

04 November 2024

Posted:

06 November 2024

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Abstract

We provide an extension of the “Arbitrage Theorem” with state-dependent linear utility functions for monetary gains and losses allowing for “loss aversion” in each possible state of nature.

Keywords: 
arbitrage theorem
;  
loss aversion
Subject: 
Business, Economics and Management  -   Economics
For some positive integer L ≥ 2, let {1, 2, …, L} denote the finite set of states of nature exactly one of which will be realized at a future date. We will refer to a state of nature as SON and its plural as SONs.
A row vector x∈ R L where for each j∈{1, …, L}, the jth coordinate of x denoted by xj denotes the monetary return in SON j, from investment worth one unit of money, is said to be a unit return vector.
A vector p∈ R + L satisfying ∑ j = 1 L p j = 1, such that for j∈{1, …, L}, pj is the probability of occurrence of SON j, is a probability vector.
A pair (x, p) where x is a unit return vector and p is a probability vector is called a unit risky investment pair.
Given x, y∈ R L , let yTx denote ∑ j = 1 L y j x j .
The expected value of a unit risky investment pair denoted E(x, p) is pTx = ∑ j = 1 L p j x j .
The seminal contribution of Kahneman and Tversky (Kahneman and Tversky (1979)) noted the experimentally verified observation that agents tend to have a marginal utility of loss that is no less- if not higher-than the marginal utility of gain. Incorporating this idea in our analysis, we define the following.
A linear utility profile is an array in ( R + + 2 ) L , i.e., u = <( u j − , u j + )| j = 1, …, L>, satisfying u j − ≥ u j + , for all j = 1, …, L.
The expected utility of a unit risky investment pair denoted by Eu(x, p) = ∑ j = 1 L p j [ u j −   m i n { x j , 0} + u j + max{ x j , 0}].
For some positive integer K, let <x(k)| k = 1, …, K> be an array of unit return vectors.
An investment plan is a (column) vector α∈ R K where for each k∈{1, …, K}, the kth coordinate of α denoted αk is the amount of money invested in the kth unit return vector.
Given an array of unit return vectors <x(k)| k = 1, …, K> the return from an investment plan α in SON j is ∑ k = 1 K α k x j k .
The well-known version of the Arbitrage Theorem that is applicable for a linear utility profile satisfying u j − = u j + , for all j = 1, …, L is available as Theorem 4.4.1 in Ross (2004). In the following theorem we extend the result- to the extent possible- when linear utility profiles may exhibit loss aversion.
Weak Arbitrage theorem incorporating loss aversion: Let <x(k)| k = 1, …, K> be an array of unit return vector.
(i)
Given a linear utility profile u = <( u j − , u j + )| j = 1, …, L> if there does not exist a probability vector p such that for all k = 1, …, K, Eu(x(k), p) = 0, then there exists an investment plan α such that utility of return from α is strictly positive in SON j for all j∈{1, …, L}.
(ii)
If there exists an investment portfolio α such that the return from α is strictly positive in SON j for all j∈{1, …, L}, then for each j∈{1, …, L} there exists a non-degenerate left-closed right open interval Ij with min{a|a∈Ij} = 1, such that for any linear utility profile u = <( u j − , u j + )| j = 1, …, L> satisfying u j − u j + ∈Ij for all j∈{1, …, L}, the following holds: there does not exist any probability vector p such that for all k = 1, …, K, Eu(x(k), p) = 0.
Proof: (i) Let A be the (K+1)×L matrix whose (k, j)th term for k∈{1, …, K} and j∈{1, …, L} is u j − min ⁡ x j k ,   0 + u j + min ⁡ x j k ,   0 and for j∈{1, …, L} the (K+1, j)th term is 1.
Then by Farkas’s lemma, the “non-existence” of p∈ R + L satisfying Ap = 0 1 where 0 is the K dimensional column vector all whose entries are 0, implies that there exists α∈ R K and a real number β such that ∑ k = 1 K α k [ u j − min ⁡ x j k ,   0 + u j + min ⁡ x j k ,   0 ] + β ≥ 0 for all j∈{1 , …, L}, β < 0, but never both.
Thus, there exists α∈ R K such that ∑ k = 1 K α k [ u j − min ⁡ x j k , 0 + u j + max ⁡ x j k , 0 ] > 0 for all j ∈{1, …, L}.
Since, u j − ≥ u j + , for all j = 1, …, L, ∑ k = 1 K α k [ u j − min ⁡ x j k , 0 + u j + max ⁡ x j k , 0 ] ≤ ∑ k = 1 K α k [ u j + min ⁡ x j k , 0 + u j + max ⁡ x j k , 0 ] = u j + ( ∑ k = 1 K α k [ min ⁡ x j k , 0 + max ⁡ x j k , 0 ] ) = u j + ( ∑ k = 1 K α k x j k ) for all j∈{1 , …, L}.
Thus, ∑ k = 1 K α k [ u j − min ⁡ x j k , 0 + u j + max ⁡ x j k , 0 ] > 0 for all j∈{1 , …, L} implies u j + ( ∑ k = 1 K α k x j k ) > 0 for all j∈{1 , …, L}.
Since, u j + > 0, for all j∈{1 , …, L}, it must be the case that ∑ k = 1 K α k x j k > 0 for all j∈{1 , …, L}.
(ii) Now suppose, there exists an investment portfolio α such that ∑ k = 1 K α k x j k > 0 for all j∈{1, …, L}.
Thus, ∑ k = 1 K α k [ min ⁡ x j k , 0 + max ⁡ x j k , 0 ] = ∑ k = 1 K α k min ⁡ x j k , 0 + ∑ k = 1 K α k max ⁡ x j k , 0 > 0 for all j∈{1 , …, L}.
Hence, ∑ k = 1 K α k max ⁡ x j k , 0 > - ∑ k = 1 K α k min ⁡ x j k , 0 ≥ 0 for all j∈{1 , …, L}.
For j∈{1 , …, L}, let Ij = [1, + ∞) if - ∑ k = 1 K α k min ⁡ x j k , 0 = 0 and Ij = [1, ∑ k = 1 K α k max ⁡ x j k , 0 − ∑ k = 1 K α k min ⁡ x j k , 0 ) if - ∑ k = 1 K α k min ⁡ x j k , 0 > 0.
Clearly Ij is a non-degenerate left closed and right open interval in R + + satisfying min{a|a∈Ij} = 1, for all j∈{1 , …, L}.
For j∈{1 , …, L}, let u j − , u j + > 0 be such that u j − u j + ∈ Ij. Thus u j − ≥ u j + , for all j∈{1 , …, L}.
Let u = <( u j − , u j + )| j = 1, …, L> be a linear utility profile.
Thus, u j + ( ∑ k = 1 K α k max ⁡ x j k , 0 ) > u j − ( ∑ k = 1 K α k min ⁡ x j k , 0 ) for all j∈{1 , …, L}, i.e., u j + ( ∑ k = 1 K α k max ⁡ x j k , 0 ) + u j − ( ∑ k = 1 K α k min ⁡ x j k , 0 ) > 0, for j∈{1 , …, L}.
Let β < 0 be such that - β = min{ u j + ( ∑ k = 1 K α k max ⁡ x j k , 0 ) + u j − ( ∑ k = 1 K α k min ⁡ x j k , 0 )| j∈{1 , …, L}}.
Thus, β < 0 and u j + ( ∑ k = 1 K α k max ⁡ x j k , 0 ) + u j − ( ∑ k = 1 K α k min ⁡ x j k , 0 ) + β ≥ 0 for all j∈{1 , …, L}, i.e., β < 0 and ∑ k = 1 K α k [ u j − min ⁡ x j k , 0 + u j + min ⁡ x j k , 0 ] + β ≥ 0 for all j∈{1 , …, L}.
Let A be the (K+1)×L matrix whose (k, j)th term for k∈{1, …, K} and j∈{1, …, L} is u j − min ⁡ x j k ,   0 + u j + min ⁡ x j k ,   0 and for j∈{1, …, L} the (K+1, j)th term is 1.
Then, by Farkas’s lemma, there does not exist a column vector p∈ R + L such that Ap = 0 1 where 0 is the K dimensional column vector all whose entries are 0, i.e., there does not exist any probability vector p such that for all k = 1, …, K, Eu(x(k), p) = 0. Q.E.D.

References

  1. Kahneman, D.; Tversky, A. Prospect theory: An analysis of decision under risk. Econometrica 1979, 47, 263–291. [Google Scholar] [CrossRef]
  2. Ross, S. M. (2004): Topics in Finite and Discrete Mathematics. Cambridge University Press.
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