Correlated inputs

For the 6.3/6.7 company, take $W \sim N(0.12, 0.01^2)$ and $g \sim N(0.15, 0.04^2)$ with correlation $\rho$, all else fixed. (a) Using a first-order (delta-method) expansion, derive $\operatorname{Var}(V)$ in terms of $\partial V/\partial W$, $\partial V/\partial g$, the standard deviations and $\rho$. (b) With $\partial V/\partial W = -3.3575\times10^5$ and $\partial V/\partial g = 8.8811\times10^4$, evaluate $\operatorname{sd}(V)$ for $\rho = -0.5, 0, 0.5$. (c) Explain why the Monte Carlo standard deviations (about 6,680, 5,270 and 3,630) exceed the delta-method values, and which economic scenario corresponds to $\rho < 0$.