I need help with understanding risk tolerance as described in Jeffrey W. Herrmann's "Engineering Decision Making and Risk Management," 2015. On page 118 it describes a lottery scenario in which there is a 50% chance of a gain of $Y and a 50% chance of loss of $Y/2 (expected value comes to $Y/4). It continues to let M be the maximum value--making the lottery desirable but then it goes on to say that the expected utility of the lottery with Y = M should be 0. The setup is followed by translating the question into a utility function and asks to solve for R such that 0.5(U(M) + U(-M/2)) = 0. That all makes sense (except I expect the utility of the maximum value to be 1 and not 0).

The section goes on to say that the only feasible answer is an equation for R with respect to M, resulting in a value of approximately 1.039M. There is no explanation on how to get from the utility to the R value. What are the steps to do that? Exercise 5.16 asks the same question.

Please help me to understand how to prove that the value of R is approximately 1.039M.