Minimum-variance hedging, properties

(a) Prove that $\operatorname{Var}(\Delta S - h\Delta F) - \operatorname{Var}(\Delta S - h^*\Delta F) = \sigma_F^2(h - h^*)^2$. (b) Show that a one-for-one hedge ($h = 1$) leaves more variance than no hedge at all if and only if $h^* < 1/2$. (c) A ₹2 crore mid-cap portfolio has monthly return volatility 6%, Nifty futures have monthly return volatility 5%, and the correlation is 0.8 (hypothetical). Compute the return-based $h^*$, the number of Dec Nifty lots for a minimum-variance hedge, the hedge effectiveness, and the standard deviation of the monthly P&L in rupees before and after hedging.