PSR and minimum track record

(a) Derive (4.16) from (4.15). (b) An Indian portfolio-management service reports four years of monthly returns with an annualised Sharpe ratio of 1.2, skewness $-0.8$ and kurtosis 5. Compute the monthly $\hat S$, its standard error (non-normal and normal), $\mathrm{PSR}(0)$, and $\mathrm{PSR}(S^*)$ for an annual benchmark Sharpe ratio of 0.5. (c) Compute MinTRL at 95% against $S^*=0$ and against the 0.5 benchmark. (d) Compute MinTRL (normal returns, $S^*=0$, 95%) for an annual Sharpe ratio of 1 on monthly data and of 2 on daily data.