Student-$t$ Expected Shortfall

(a) Derive (14.12) for $\alpha = 0.975$ and verify it for $\nu = 4$ by numerical integration. (b) A ₹50 lakh Nifty position has daily loss standard deviation 1.3% and zero mean. Compute the 99% VaR and the 97.5% ES in rupees under the normal and under a variance-matched Student-$t$ with $\nu = 4$. (c) By what factor does the $t$ ES exceed the normal ES? Why is the factor for 99% VaR smaller?