Metallgesellschaft: stack-and-roll hedge ratio

A firm has promised to deliver 1 barrel at the end of each month for 120 months at a fixed price. It hedges with front-month futures. Assume a one-factor model in which a shock $\Delta$ to the front-month price moves the forward price for maturity $\tau$ by $e^{-\kappa\tau}\Delta$, with $\kappa = 0.5$ a year (illustrative), and discount at $r = 5\%$. (a) Show that the present value of the obligation changes by $-\Delta\sum_{i=1}^{120} e^{-(\kappa + r)\tau_i}$ with $\tau_i = i/12$, and hence that the minimum-variance stack is $N^* = \sum_i e^{-(\kappa+r)\tau_i}$ barrels of front-month futures. Compute it and the ratio to the one-for-one stack of 120. (b) For a firm with 160 million barrels of commitments, compute the immediate variation margin per $1 fall in the front-month price under the one-for-one stack and under the minimum-variance stack. (c) Under the one-for-one stack, what is the firm's net exposure to the front-month price? Relate your answer to the claim that MGRM's position was partly speculative. (d) Separately, explain how the sign of $F_{\text{far}} - F_{\text{near}}$ affected MGRM's monthly roll, and why liquidity, not value, decided the outcome.