Derive Annualized Portfolio Volatility

# Derive Annualized Portfolio Volatility A stock portfolio has weight vector `w`. Each stock has factor exposures, the factors have a covariance matrix, and each stock has an idiosyncratic variance. Derive the portfolio's predicted volatility and explain how to annualize it without mixing return horizons or units. ### Constraints & Assumptions - `B` is the stock-by-factor exposure matrix and `F` is the factor covariance matrix. - `D` is diagonal with nonnegative idiosyncratic variances. - Factor and idiosyncratic returns are uncorrelated in the model. - `B`, `F`, `D`, and `w` refer to the same universe, currency treatment, and return horizon. ### Clarifying Questions to Ask - Are weights fractions of capital, dollar holdings, or active weights? - Is covariance daily, monthly, or already annualized? - Do cash, leverage, hedges, and currency exposures belong in the model? - Are idiosyncratic covariances truly assumed to be zero? ### What a Strong Answer Covers - Total covariance as factor risk plus specific risk - Equivalent stock-space and factor-exposure formulas - Square-root-of-time annualization and its assumptions - Unit checks, numerical checks, and risk attribution ### Follow-up Questions 1. How does the formula change if specific returns are correlated? 2. Why can multiplying volatility by 252 be wrong? 3. How would you identify which factor contributes most to total risk? ```hint Project holdings into factor space Compute the portfolio's factor exposure once, then combine its factor variance with the weighted sum of specific variances. ```