If engagement with Choice is robust, necessarily it trades off two countervailing parameters

What is it about?

Suppose there is uncertainty as to which of two paths (equivalently, assets), α or β is best to embark upon (equivalently, is best to acquire). Presence of uncertainty implies neither α nor B can be inferred to be unequivocally better than the other. In Financial Economics parlance, neither α nor β 'Pareto dominates' the other, equivalently neither α nor β is unconditionally strictly preferred to the other. Under the implicit assumption that the trade-offs (the pros and cons) that subsist between α and β are time invariant, the prior literature in Finance posited that the choice between α and β boils down to a computation of what is termed, the 'Information Ratio', equivalently the 'price of risk (λ)'. Let 'μ' denote the anticipated (expected) returns to either α or β. Further, let 'σ' denote the risk (standard deviation of the expected returns) that is characteristic of either α or β. The price of risk is computed as, 'λ=(μ/σ)'. If then μ(α)/σ(α) > μ(β)/σ(β), a rational agent chooses path α over path β, equivalently asset α over asset β. Consider, however, that under the said choice rubric, that ideally asset β is obsoleted, because always asset α is posited to be more desirable than asset β. The evidence in the Finance literature (see for example, Jegadeesh and Titman 2001) that feasibly an asset α that outperforms an alternate asset β in the current period can be predicted to underperform the very same asset in some future period demonstrated the non-robustness of the adoption of a 'time invariant λ' as the metric for the choice between different assets. The systemic nature of the empirical evidence is evident in the finding that there does not exist any asset pricing model (see for example, Fama and French 1995, 1996) that is able to exclude the said pattern. With the phenomenon that is, 'α is preferred to β in the current period' is succeeded by 'β is preferred to α in some future period' then demonstrated to be systemic, as opposed to, ad hoc, there is arrival at the exclusion of a time invariant λ. In an attempt to get around the non-robustness of the postulate of a time invariant λ, some prior studies (the 'relevant prior studies') assume that risk directly spans the anticipated (expected) returns, as such, assume that, μ=σ. However, if always, μ=σ, qualitatively all assets are equally risky. Always, as such, a rational agent chooses the asset that furnishes the highest μ of which he/she has awareness. The rationale is straightforward, namely if risk materializes, it affects all assets equally. But losing the same proportion, θ, of a return of 30% is better than losing the same proportion, θ, of a return of 15%. So then all rational agents invest in the highest risk asset of which they have awareness, an asset that then is termed a 'numeraire asset'. Yet again, the evidence in the foregoing, namely that there do not exist any 'numeraire assets' in markets rendered the relevant prior studies non-robust to the real world. In aggregate, prior to my study that I here discuss, neither Finance nor Economics had arrived at a robust theory in respect of the parameters that ought to be engaged with whenever agents have to choose between two competing paths (assets), neither of which can be asserted to be unequivocally better than the other. In stated respect, absent an Asset 1 dominating an Asset 2 with reference to some Parameter 1, but simultaneously the Asset 2 dominating the Asset 1 with reference to some countervailing Parameter 2, rather unequivocally one or the other of the two competing assets dominates the other. Choice then is non-trivial only whenever none of the assets that are under consideration dominates others with reference to all feasible parameters.

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Photo by Victoriano Izquierdo on Unsplash

Photo by Victoriano Izquierdo on Unsplash

Why is it important?

In the absence of a robust theory for the implementation of rational choice, man is prone to errors in decision making. In Simon (1976, 1978), any such robust theory has characterization as a 'Procedural Rationality Rule (PRR)'. A PRR furnishes principles with respect to, 'how to implement choice', does not tell a rational agent 'what to do'. Consider then that whereas man has been discussing Rationality, namely, 'how exactly best to implement choice' since Adam Smith (Smith 1776), that up until my study that I here discuss, neither Finance nor Economics had arrived at any robust PRR for the implementation of Choice. How then does my study resolve the snafu? Building on one of my prior publications, Obrimah (2023), my new open access publication, Obrimah (2025) furnishes formal theoretical evidence that whereas μ is concave in the uncertainty that necessitates choice - μ increases with uncertainty, but at a decreasing rate - contrarily, σ is strictly convex in the selfsame uncertainty - σ increases with uncertainty, but at an increasing rate. Applying the evidence, the price of risk, λ, is not time invariant (is not static), rather it decreases with uncertainty. Since typically uncertainty varies with time, there is arrival at the insight that λ is inherently time varying. Applying the foregoing, for the very first time in the Finance and Economics literature, my newly published study resolves the phenomenon that has description as, 'α outperforms β in the current period', is succeeded by, 'β outperforms α in a future period' as follows. Suppose that, in the current period, λ(α) > λ(β). Suppose also that μ(α) = μ(β) in the current period. Let the uncertainty that bounds asset α's activities increase significantly between the current and future period. Applying study findings, σ(α), which is strictly convex in uncertainty increases much faster than μ(α). With the ratio, λ(α)=[μ(α)/σ(α)] then decreasing, if it falls far enough, whereas there is arrival at, μ(α) > μ(β), simultaneously, there is arrival also at, λ(α) < λ(β). If the resulting distance, [(λ(β) - λ(α)) > 0] becomes large enough, the probability, ϕ that either μ(α) or μ(β) - both of which are uncertain - are realized satisfies, ϕ(α) < ϕ(β). For concreteness, let π denote the returns that are more likely to be realized by either α or β. Mathematically, feasibly there is arrival at each of, ϕ_d[ π < μ(α)] > [ϕ_d(π < μ(β)] and ϕ_i[π > μ(α)] < ϕ_i[π > μ(β)]. With ϕ_d governing the relatively 'worse' states and ϕ_i governing the relatively 'good' states of the world, the two equations assert that, regardless whether the future turns out to be either a relatively worse or good state, more likely than not, asset β outperforms asset α. Conditional, as such, on the specific estimates for each of ϕ_d and ϕ_i, feasibly 'α is preferred to β in the current period' is succeeded by, 'β is preferred to α in the future period'. The formal theory in my new publication shows, under certain evolutionary conditions, that the enumerated phenomenon is guaranteed to occur, hence the evidence that it is systemically realized in markets. In essence, my newly published open access study shows that choice is robustly implemented only whenever the anticipated (expected) returns are juxtaposed with the probability that they will be realized, a probability that inherently decreases with the size of the expected return. In stated respect, with λ decreasing whenever μ is increasing and with the shape of the trade-off between λ and μ inherently time varying, always there exists a strictly positive probability that the best asset or path in the current period ceases to be the best path or asset in some future period.

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