Estimation risk

(a) Derive (3.13) and (3.14). (b) A backtest of 4 years shows an annual Sharpe ratio of 1.2. Compute $t$ and $k^*$, treating the backtest Sharpe as the truth. (c) A colleague argues that, after accounting for the number of variants tried, the true Sharpe ratio is closer to 0.6. Recompute. What happens to expected excess growth if you bet the plug-in Kelly fraction ($k = 1$) when the true $t$ is 0.8?