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Formula sheet/NISM Series XV: Research Analyst/Fundamentals of Risk and Return
[ FUNDAMENTALS OF RISK AND RETURN ]

Sample Standard Deviation & Beta

Sample Standard Deviation(s)=∑(Xi−Xs)2/(n−1)\text{Sample Standard Deviation} (s) = \sqrt{ \sum (X_{i} - Xs)^{2} / (n - 1) }
Stock Beta(βi)=Cov(Ri,Rm)/Var(Rm)=ρi,m×(σi/σm)\text{Stock Beta} (\beta_{i}) = Cov(R_{i}, R_{m}) / Var(R_{m}) = \rho_{i,m} \times ( \sigma_{i} / \sigma_{m} )
Variables

s: Measure of total volatility (systematic + unsystematic). β\beta: Measure of systematic market sensitivity. β\beta > 1 (high beta / aggressive); β\beta < 1 (low beta / defensive). XiX_{i}: Individual observation; Xs: Sample mean; n: Sample size; Cov(RiR_{i}, RmR_{m}): Covariance between the stock's and market's returns; Var(RmR_{m}): Variance of market returns; ρi,m\rho_{i,m}: Correlation between stock and market returns; σi\sigma_{i}, σm\sigma_{m}: Standard deviation of stock and market returns.

Exam note

Expected return under CAPM: E(RiR_{i}) = RfR_{f} + βi\beta_{i} × [ E(RmR_{m}) - RfR_{f} ].

Worked example
Risk-Adjusted Performance: Sharpe, Treynor, and Jensen's Alpha

Mutual Fund Scheme X generated an annualized return of 18.5% with a standard deviation of 16.0% and Beta of 1.15. The benchmark market return is 13.0%, and the risk-free rate is 6.5%. Calculate the Sharpe Ratio, Treynor Ratio, and Jensen's Alpha.