This exercise explores the sensitivity of \(\log\)-optimality to the rebalancing frequency. Consider a market consisting of a risk-free asset with zero rate of interest and a stock that over 1 year either increases by a factor \(a>1\) or decreases by a factor of \(1 / a\) with equal probabilities.

(a) Find the log-optimal portfolio for this situation.

(b) Now suppose that the portfolio is rebalanced only every 2 years. Find the log-optimal portfolio in this case.

(c) For the case \(a=2\), what are the growth rates in parts

(a) and (b)? [The growth rate is the yearly expected value of the \(\log\) of the growth.]