Leveraged index funds and the drag

A fund holds $L$ times the Nifty, rebalanced continuously, with zero financing cost. Nifty follows GBM with arithmetic drift $\mu = 10.8\%$ and volatility $\sigma = 20.2\%$ a year (the Nifty sample estimates of 2.2). (a) Show that the fund's value follows GBM with drift $L\mu$ and volatility $L\sigma$, and hence that its long-run growth rate is $g(L) = L\mu - L^2\sigma^2/2$. (b) Compute $g$ for $L = 1, 2, 3$ and find the $L$ that maximises $g$. What is the maximum growth rate? (c) Explain why $g(L)$ falls below $g(1)$ for large $L$ even though the arithmetic mean $L\mu$ keeps rising. (This is the Kelly criterion of M03 in disguise.)