# Limit of Certainty Equivalent for Cumulative Prospect Theory

I found a piece of code from a former colleague who seems to have copied it from another source (perhaps internet) that confused me a bit. Therein, the Certainty Equivalent (CE henceforth) for Cumulative Prospect Theory (here CPT) is calculated as

Function CEpos(cpt As Double, alpha As Double) As Double
If alpha > 0 Then
CEpos = cpt ^ (1 / alpha) 'OK
ElseIf alpha = 0 Then
CEpos = Exp(cpt)
Else
CEpos = ((1 - cpt) ^ (1 / alpha)) - 1
End If
End Function

Function CEneg(cpt As Double, alpha As Double, lambda As Double) As Double
If alpha > 0 Then
CEneg = (-1) * (((-1) * cpt / lambda) ^ (1 / alpha))
ElseIf alpha = 0 Then
CEneg = (-1) * Exp((-1) * cpt / lambda)
Else
CEneg = 1 - (1 + cpt / lambda) ^ (1 / alpha)
End If
End Function


I am confused as the CE of CPT is supposed to converge to $$\lim_{\alpha\rightarrow{0}} (CE)=e^{CPT}$$. Perhaps this is too simple but i'm stuck as my guess is that for $${\alpha\rightarrow{0}}$$ the agent is insensitive to variations in gains/losses, thus the CE is zero.

My idea was that for $$CPT={CE}^{\alpha}$$ the CPT approaches 1 as i could expand the CPT using a Taylor Series and evalute at $$\alpha=0$$ such that $$$$\label{eq:1} CPT=1+\alpha*\log(CE)+\frac{1}{2}*{\alpha^{2}}*\log^{2}(CE)+O(CE^{3})$$$$ and thus CPT is 1 for all gains (similar for losses, there shifted by the loss aversion parameter).

Would anyone mind helping me out how to get to the expression above from the code? I don't see it yet (and yes: it seems simple but as i said i'm stuck..)