Can the Certainty Equivalent be negative? I am questioning if the CE of a lottery can be negative? For me it doesn't make much sense by definition.
I encountered this problem on the following exercise:
Imagine a case where we have a lottery(like the EuroMillions) with a price ticket of €2.5. By construction we win €13m with a probability of 1/139,838,160 and we loose with probability (1- 1/139,838,160).
Imagine that the utility function is $U(x)=-e^{(-x)}$.
I compute the CE as you can find in the picture. Is this wrong? How can I interpret it?

For the lottery with the prize of €13m, and let $n=139,838,160$:
$$U(x) = \left(\frac{1}{n} \cdot e^{-(13,000,000-2.5)}\right) + \left(\frac{n-1}{n} \cdot e^{-(-2.5)}\right) = -12.182$$
Therefore
\begin{align*}
U(CE)=-e^{-CE}&amp;=-12.182\\
\ln\left(e^{-CE}\right)&amp;=\ln(12.182)\\
CE&amp;=-2.49999
\end{align*}

I hope it is clear.
Thanks a lot!
If you start out with €0, then the certainty equivalent of losing €2.5 with probability 1 is -€2.5.
Your exercise basically asks you to calculate what difference winning the lottery with a small probability makes. Given this utility function, not much.