4.2.2 Parametric VaR

Basic Prerequisites: Linear Algebra & Covariance and Econometrics & Time Series.

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1. Intuition & Practical Objective

Parametric VaR (also called variance–covariance or delta-normal) is the fastest and most elegant of the three VaR methods, and it is the one RiskMetrics/JP Morgan made famous. The idea in one line: assume the portfolio’s P&L is (multivariate) normal, estimate its volatility from the covariance matrix of the risk factors, and read the quantile off a normal table.

The practical objective is a closed form you can evaluate in microseconds: $\text{VaR}=z_{\alpha }\sqrt{w^T\Sigma w}\sqrt{h}$. For a $N=2000$-asset position that is a 2000×2000 covariance multiply - trivially fast compared to historical or MC. That is precisely why banks ran parametric VaR for liquidity-hit capital: you can recompute it continuously, all day, on a few hundred thousand positions.

Its two defining assumptions, stated up front so you know what you are buying:

• (A1) Normality: the factor returns, and hence portfolio P&L, are jointly normal.
• (A2) Delta (linearity): the portfolio P&L is linear in the factors (a stock or hedge position). Options violate this - their convexity needs the delta–gamma extension in 06.

The covariance matrix $\Sigma$ is the engine: it packages every asset’s own volatility and every pair’s co-movement into one object, so portfolio risk is a single quadratic form $w^T\Sigma w$.

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2. Mathematical Ground Truth & Derivations

Setup. Let $w\in {\mathbb{R}}^N$ be the current dollar positions in the $N$ risk factors and $\Sigma$ the $N\times N$ covariance matrix of one-period factor returns (per-day, if daily VaR). The portfolio’s one-period P&L is

$$\Delta V\approx \sum _i{w}_i{R}_i=w^{T}R,$$

where $R\sim N(0,\Sigma )$ is the factor-return vector. (The mean is usually taken to 0 for a 1-day horizon; over longer horizons drop in a drift $w^T\mu$.)

Portfolio variance (Hull Ch 22, eq. 22.3/22.4; RiskMetrics). Because $R$ is a random vector with covariance $\Sigma$,

$$\mathbb{V}[\Delta V]=w^T\Sigma w=\sum _{i,j}{w}_i{w}_j{\rho }_{ij}{\sigma }_i{\sigma }_j=\sum _{i,j}{w}_i{w}_j\text{Cov}(R_i,R_j).$$

This double sum is the covariance-matrix approach: each pair of assets contributes its covariance, weighted by both positions. Correlations enter as ${\rho }_{ij}=\text{Cov}(R_i,R_j)/({\sigma }_i{\sigma }_j)$.

Delta-normal VaR. $\Delta V$ is a linear combination of normals ⇒ normal with that variance. The VaR is $z_{\alpha }$ portfolio standard deviations:

$$\boxed{{\text{VaR}}_{\alpha }=z_{\alpha }{\sigma }_p\sqrt{h},{\sigma }_p=\sqrt{w^T\Sigma w}}$$

where $z_{\alpha }=N^{-1}(\alpha )$ (e.g. $z_{0.99}=2.3263$). Under i.i.d. returns the $h$-day version multiplies by $\sqrt{h}$ (Hull Ch 22: “N-day VaR = 1-day VaR × √N”).

Extra drift term (long horizon). With a nonzero expected return, param VaR becomes $\text{VaR}=-w^T\mu +z_{\alpha }\sqrt{w^T\Sigma w}\sqrt{h}$ - the flat-file form. For 1-day equity/IR risk $w^T\mu$ is usually negligible and dropped.

The covariance engine - estimating $\Sigma$ (Hull Ch 23). You need the daily vols and correlations:

• Plain sample covariance from $m$ days of demeaned returns.
• EWMA (RiskMetrics): ${\sigma }_n^2=\lambda {\sigma }_{n-1}^2+(1-\lambda )u_{n-1}^2$ with $\lambda =0.94$ for daily data - reacts to recent vol spikes, has no mean reversion (Hull eq. 23.7; verified).
• GARCH(1,1): ${\sigma }_n^2=\omega +\alpha u_{n-1}^2+\beta {\sigma }_{n-1}^2$ with long-run level $V_L=\omega /(1-\alpha -\beta )$ - the “sticky” EWMA that does mean-revert (Hull eq. 23.8/23.9; verified). Both feed a covariance matrix that must stay positive-semidefinite (Hull eq. 23.17).

The choice of $\Sigma$ is where parametric VaR quietly becomes good or terrible - a stale flat covariance understates risk in a vol regime change (see 05).

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3. Computational Implementation - the closed form, stdlib only

The entire method is the covariance machinery. We verify the 3,396.15 hub number and show the double sum by hand.

(The 99% 1-day figure 3,396.15 is the hub/verified cross-method baseline.)

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4. Failure Modes & First-Principles Breakdowns

1. Normality is the whole model (A1). $\text{VaR}=z_{\alpha }\sigma \sqrt{h}$ is the normal quantile. Real returns are fat-tailed (see EVT): a variance-matched Student-$t(4)$ breaches this 99% VaR at 1.56%, not 1.00% - ~56% more tail losses than promised (verified in 05).
2. Linearity/delta fails for options (A2). P&L of an option is convex in the underlying; delta-normal uses only the linear term and misprices the tail. For a deep-OTM put the delta-only VaR was a ~3.6× over-estimate vs full revaluation (verified in 06) - the delta–gamma extension fixes most of it.
3. A stale/flat covariance lies. $\Sigma$ estimated over a calm window understates vol when a regime shifts. EWMA ($\lambda =0.94$) reacts fast to vol spikes; plain sample covariance does not. Estimating $\Sigma$ well is most of the hard work of parametric VaR.
4. Correlations break down in crises. ${\rho }_{ij}$ rises toward 1 when markets crash (“correlations go to one”), so a calm-period $\Sigma$ understates joint tail risk exactly when it matters - a fundamental failure of the covariance-matrix approach itself, not just its estimation.
5. The $\sqrt{h}$ rule assumes i.i.d. With GARCH vol clustering the true $h$-day VaR scales sub/super-linearly in $\sqrt{h}$ (Hull Ch 23 term-structure formula). Basel’s 10-day scaling is an approximation of convenience.

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5. References

• Hull, Options, Futures, and Other Derivatives
• RiskMetrics / J.P. Morgan: *RiskMetrics
• Glasserman, Monte Carlo Methods in Financial Engineering

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6. Connected Graph Bridges

• Base: Linear Algebra & Covariance (quadratic forms, PSD) · Econometrics & Time Series (EWMA/GARCH).
• Back: 01 · From Zero.
• Forward: 03 · Historical Simulation · Index Hub.
• Extension: 06 · Delta–Gamma · Stress Testing.