Risk aversion recovered from an optimal two-state payoff ($R=\log (Z_u/Z_d)/\log (W_d/W_u)$)

For an interior optimal payoff in a complete two-state market under CRRA utility, $W_u^{-R}/W_d^{-R}=Z_u/Z_d$. If the two positive payoffs differ, taking logarithms gives the displayed coefficient. The multiplier can then be recovered from either state. A positive coefficient requires the price-density and payoff orders to be compatible with decreasing marginal utility.