This paper conducts a comparative statistical analysis of daily log returns for the Nifty 50 index and four NSE constituent stocks (RELIANCE, HDFCBANK, ADANIENT, BAJFINANCE) over 2015–2024. The central contribution is a GARCH(1,1) residual decomposition that separates observed excess kurtosis into two structurally distinct sources: volatility clustering (removable by variance modelling) and fat-tailed innovations (irreducible). Residual excess kurtosis remains statistically significant after GARCH filtering for all series, establishing that leptokurtosis in NSE 50 returns reflects both time-varying volatility and genuinely fat-tailed innovations. The paper further introduces a tractable Misspecification Tax proxy quantifying the annualised tail-risk cost of Gaussian portfolio construction on NSE 50 data. Gaussian VaR systematically underestimates empirical 99.9% tail risk by up to 169% for high-volatility constituents. Full analysis code is included as a supplementary file.