Case analysis: leveraged volatility products

(a) A product promises $L$ times the daily return of an index, rebalancing at each close. Show that after an index return $R$ it must trade $LA\,R(L-1)$ of index exposure, where $A$ is its assets at the previous close. Evaluate for $L=-1$ and $L=+2$. (b) XIV's closing indicative value went from 108.3681 on 2 Feb 2018 to 4.22 on 5 Feb 2018. What daily return of its underlying futures index does that imply? (c) Suppose (hypothetically) inverse products had assets of $1 bn at the close on 2 Feb. What VIX-futures exposure would they have had to buy at the close of 5 Feb under (a)? Who else must buy at the same time? (d) Using 9.9, explain why the outcome was a structural feature of the product, not bad luck.