Parametric VaR
Parametric VaR (also called variance-covariance VaR or analytical VaR) is a Value at Risk methodology that estimates the maximum expected portfolio loss at a given confidence level over a specified time horizon by assuming that portfolio returns follow a normal distribution, characterized only by the portfolio's mean and variance (standard deviation).
Key takeaways
- Parametric VaR = Portfolio Value × z-score × Portfolio Volatility × √Time, where z-score reflects the confidence level (1.645 for 95%, 2.326 for 99%).
- The normal distribution assumption makes parametric VaR computationally fast and intuitive but systematically underestimates tail risk.
- Correlation effects across positions are captured through the portfolio covariance matrix, enabling decomposition of VaR by risk factor.
- Parametric VaR is most appropriate for linear portfolios (no options); non-linear exposures require delta-gamma approximations or simulation methods.
- Backtesting parametric VaR regularly reveals exceedance rates higher than expected, confirming that financial returns have fat tails.
Explanation
Parametric VaR was the first widely adopted quantitative risk measurement framework for financial institutions, popularized by JP Morgan's RiskMetrics publication in 1994. Its core appeal is analytical tractability: given assumptions about the distribution of portfolio returns (normal, fully characterized by mean and standard deviation) and estimates of the covariance matrix among individual positions, VaR can be calculated in closed form without simulation — a significant advantage in the 1990s when computing power was expensive.
The methodology works in three steps. First, the standard deviation (volatility) of each position or risk factor is estimated — typically using an exponentially weighted moving average of recent returns (giving more weight to recent observations) or a GARCH model. Second, correlations between positions are estimated to build a full covariance matrix. Third, portfolio volatility is calculated as the square root of the quadratic form: σ_portfolio = √(wᵀΣw), where w is the vector of portfolio weights and Σ is the covariance matrix. Finally, VaR = Portfolio Value × z-score × σ_portfolio × √T, where T is the holding period in days (for daily VaR, T = 1).
The critical limitation of parametric VaR is the normal distribution assumption. As extensively documented in the empirical finance literature, daily portfolio returns exhibit excess kurtosis (fat tails), negative skewness, and volatility clustering — all of which cause actual tail losses to exceed the normal distribution's prediction. A 99% parametric VaR is designed to be exceeded only 1% of the time (2.5 days per year); in practice, actual exceedances for equity portfolios can be 2–4 times as frequent during normal market conditions and dramatically more frequent during crises.
For portfolios containing options or other instruments with non-linear payoff profiles, parametric VaR requires delta-gamma approximation: the portfolio P&L is expanded to include first-order (delta) and second-order (gamma) terms, capturing the curvature of the payoff function. Even with this correction, the normal distribution assumption is suspect for options portfolios where the true P&L distribution is heavily asymmetric.
The covariance matrix also presents practical challenges. For a large portfolio with 500 positions, the full covariance matrix has 500 × 499 / 2 = 124,750 unique correlation pairs — estimates that are noisy and can produce a matrix that is not positive definite (mathematically necessary for valid VaR calculation). Factor-based approaches (projecting all positions onto a smaller set of risk factors such as equity market factors, interest rate factors, and credit spreads) reduce the dimensionality problem and provide a more stable covariance structure. The Cholesky decomposition of the factor covariance matrix is a standard numerical tool used to extract the square root of the matrix needed for VaR attribution calculations.
Formula
Parametric VaR = Portfolio Value × z × σ_portfolio × √T; σ_portfolio = √(wᵀΣw); z = 1.645 (95%), 2.326 (99%)
Example
A fixed income fund holds a portfolio with the following positions: $50M in 10-year U.S. Treasuries (daily volatility = 0.65%), $30M in investment-grade corporate bonds (daily vol = 0.85%), and $20M in high-yield bonds (daily vol = 1.40%). Correlation matrix: Treasuries-IG = 0.75, Treasuries-HY = 0.45, IG-HY = 0.70. Portfolio variance = (0.5 × 0.65%)² + (0.3 × 0.85%)² + (0.2 × 1.40%)² + 2 × 0.5 × 0.3 × 0.75 × 0.65% × 0.85% + 2 × 0.5 × 0.2 × 0.45 × 0.65% × 1.40% + 2 × 0.3 × 0.2 × 0.70 × 0.85% × 1.40% = 0.1056% + 0.0650% + 0.0784% + 0.1236% + 0.0819% + 0.0996% = 0.5541%. Portfolio daily volatility = √0.5541% = 0.7444%. 1-day 99% parametric VaR = $100M × 2.326 × 0.7444% = $1.73M. The fund backtests this VaR over 500 trading days and finds that losses exceed $1.73M on 11 occasions — a 2.2% exceedance rate versus the expected 1.0%, confirming that the normal distribution underestimates tail risk.
Related terms
Cholesky Decomposition Climate Risk Correlation Correlation Matrix Covariance Covariance Matrix Delta Documentation Risk Downside Risk Equity Fat Tails Gamma