Monte Carlo Backtest of Kelly and Fractional Kelly Strategies

You run a Monte Carlo backtest of a Kelly-style betting strategy on a sequence of independent favorable bets. Each bet wins with probability $p > 1/2$ and pays net odds of $b:1$ on a win (you gain $b$ times your bet on a win and lose your bet on a loss). (a) Derive the Kelly-optimal fraction $f^{*}$ of wealth to bet each round to maximize expected log-wealth. (b) In practice, you bet a fraction $cf^{*}$ with $c \in (0, 1)$ ("fractional Kelly"). How do the long-run expected log-growth rate and the volatility of log-wealth depend on $c$? Derive explicit expressions. (c) Describe how you would use Monte Carlo simulation to compare the distribution of terminal wealth across different choices of $c$ over a finite horizon of $T$ rounds. Why do interviewers (and practitioners) emphasize the tradeoff between theoretical optimality and finite-horizon risk?