 Small multiplicative risk premium

Compare expected utility of $W(1+\varphi )$, where $\mathbb{E}\varphi =0$, with the certainty equivalent $W(1-\psi )$. A second-order Taylor expansion gives $U(W)+W^2{U}^{\prime \prime }(W)\mathbb{E}{\varphi }^2/2$ for the risky side and $U(W)-W{U}^{\prime }(W)\psi$ for the sure side. Equality gives the displayed proportional risk premium. This requires a genuinely small-shock family with controlled remainders; small variance alone does not control rare large wealth shocks.