Prospect Theory as a Piecewise Quadratic Value Function
In risk-variance analysis of decision theory, we deal with total wealth. On the other hand, in prospect theory, we deal with changes to that wealth. This subtle difference implies that we have different risk aversion levels in gains than in losses. Additionally, it has been proven that investors value losses more sharply than the equivalent level in the gains domain . Such behavior is explained with the piecewise quadratic value function (not value function), which might be complicated to visualize and even more complicated to explain. I share this tool to assist those dealing with these matters.
In prospect theory, to make the choice of the behavioral investor compatible with the mean-variance framework for any return distribution, one can use a piecewise quadratic value function.
For the idea of a behavioral investor to be compatible with that of a rational investor, the function accomplishes the following: the risk aversion parameter in gains is positive for the function to be concave in the gains domain. For similarity, the risk aversion parameter in the losses is negative for the function to be concave in the losses domain. Additionally, there is a loss aversion parameter that sharpens the decay of the function in the losses domain.
Since quadratic functions decrease in the positive domain, a constant value is used after the corresponding singularity. Its financial motivation is that an investor cannot differentiate in the domain of gains far beyond the reference point and is therefore constant. Although mathematically not necessary, the same motivation and tweak is used in the losses domain.
 T. Hens and K. Bachmann, Behavioural Finance for Private Banking, Chichester, England: Wiley, 2008.