I am struggling to find the best way to introduce uncertainty into the comparison of two net present values (NPVs).
The following setting: I would like to compare two annuity plans and find the discount rate ($\delta$) that would make a risk averse individual indifferent between the two plans. Let's assume for simplicity that we are in month $t=0$ and the individuals starts receiving the annuity in $T-s=R$, where $T$ is the end of the annuity payment and $R>0$ and $s>0$.
Plan A: This plan is for free and pays a deterministic annuity $A_1$.
Plan B: This plan has monthly costs of $C$ and pays an annuity $E(A_2)=a+E(x)$, where a is deterministic and $x>=0$ is (log-)normally distributed. How I think about this is: $E(A_2)>A_1>a$.
If I am correct, in the case where $x$ is deterministic ($E(A_2)=A_2=a+x)$, the NPVs are as follows:
$NPV_1=A_1\frac{1-(1+\delta)^{R-T}}{\delta(1+\delta)^R}$
$NPV_2=A_2\frac{1-(1+\delta)^{R-T}}{\delta(1+\delta)^R}-C\frac{1-(1+\delta)^{-R}}{\delta}$.
Setting $NPV_1-NPV_2=0$ and solving for $\delta$ gives the discount rate that makes the individual indifferent between the two annuity plans.
Now I wonder what would be an intuitive and simple way to introduce uncertainty here. I though about replacing $E(x)$ with it's certainty equivalent (CE) and parametrize it as follows: $CE_2=\mu-\frac{\gamma}{2}\sigma^2$, where $\gamma=-\frac{u''(A)}{u'(A)}$ is the coefficient of absolute risk aversion following from assuming a CRRA utility function of the form $u(A)=\frac{-exp^{-\gamma A}}{\gamma}$. However, I am uncertain(!) whether I mix up concepts here by introducing a utility function here without actually comparing the utilities of the two NPVs. I would be happy about clarifications or other suggestions!
