A reminder that, in first-best equilibria, risk premia are not directly spanned by any of uncertainty risk, volatility risk, or aggregate risk

• Seminally, risk premiums ( RP ) are determined by risk and the price of risk ( PoR ). • More recently, risk premiums have been deemed to be solely spanned by risk. • This study shows risk is unable to solely span RP , because. • PoR that are decreasing in risk are necessary conditions for the spanning of RP . • As such, the modeling of risk is not a substitute for the modeling of PoR . With the assumption in tow that asset expected returns ( μ ) are directly spanned by asset risk ( σ ), resulting in the fixed point, μ = z ( σ ) = σ , some prior studies model either asset valuations or asset choice. With contrarily, uncertainty, υ , as the sole native parameter of markets, this study's formal theory infers the non-robustness of the restriction, μ = z ( σ ) = σ . In stated respect, newly in this study, whereas the risk premium function, f ( υ ) is inferred to be increasing and concave, contrarily the risk function, g ( υ ) is restricted to be increasing and strictly convex. Composed, the two functions span a decreasing and concave price of risk, λ (the ratio of expected returns to risk); a single crossing point, μ * = f ( υ * ) = g ( υ * ) = σ * ; and for all υ ≠ υ * , span f ( υ ) ≠ g ( υ ) , i.e. μ ≠ σ . Evincing the robustness and pragmatism of study findings, the formal theory infers the parameter-free numerical breakpoints which span the uniquely specified equilibrium distribution for λ . In stated respect, λ τ = τ 2 τ , τ = 1 , . . , T ; antecedently, the distribution of risk premiums, ϑ , is spanned by the construct, ϑ τ = λ τ / λ τ + 1 . In aggregate, absent any a priori assumption to the effect, this study infers a choice-theoretic structure that spans each of a monotone decreasing λ , ϑ ′ > 0 , and ϑ ″ < 0 ; regardless absent any artificial restrictions, f uniquely maps υ to ϑ , that is, is endogenously restricted to be strongly, not weakly monotone. Importantly, if all agents are assumed to be rational, the formal theory infers that the sequence, f > g then f < g , equivalently the sequence, ‘valuation premiums, then valuation discounts’ is rational.