In Barro's (2009) Rare Disaster Model in AER: How to derive equation (5) In Barro (2009) http://piketty.pse.ens.fr/files/Barro2009.pdf
My question is reference to equation #5, whereby Barro is deriving the reciprocal of the market value 1/v, and I am trying to derive this equation, and having trouble. In particular, without further specifications of distribution of v.
So far, I have:

But can't seem to arrive at the same result, without further information/assumptions on the distribution of the disasters.
Barro's Result for equation (5)

Any help would be greatly appreciated.
There is probably an error in your formula for $\sum_0^s v_i$.
You can use a trick to directly compute $E_0 e^{(1-\gamma)v_i}$ using
\begin{align}
E_0 e^{(1-\gamma)v_i}&amp; = \Pi_0^s E_0e^{(1-\gamma)v_i}.
\end{align}
Notice $e^{(1-\gamma)v_i}$ is a random variable equal to 1 with probability $1-p$ and $(1-b)^{1-\gamma}$ with probability $p$. Therefore,
\begin{align}
E_0 e^{(1-\gamma)v_i}&amp; = \Pi_0^s (1-p+pE_0(1-b)^{1-\gamma}),\\
&amp; = (1-p+pE_0(1-b)^{1-\gamma})^s.
\end{align}
The term $E(1-b)^{1-\gamma}$ now appears your expression.