Can the Certainty Equivalent be negative?

Question

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}&=-12.182\\ \ln\left(e^{-CE}\right)&=\ln(12.182)\\ CE&=-2.49999 \end{align*}

I hope it is clear. Thanks a lot!

Answer 1

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.