4.2 Parametric, Historical & Monte Carlo VaR

Basic Prerequisites: Linear Algebra & Covariance and Statistics & Inference. (these are the folder-level prerequisites for pages 02–06; page 01 states its own, smaller, entry requirements)

1. Intuition & Practical Objective

Value at Risk (VaR) answers one question: “over the next $h$ days, what is the loss I will only exceed with probability $\alpha$?” It is a quantile of the portfolio P&L distribution - the number such that, with confidence $100\alpha \%$, the loss will not be larger. There are three canonical ways to get that quantile, and they differ only in where they get the distribution:

1. Parametric (variance–covariance / delta-normal): assume the portfolio P&L is normal, estimate its standard deviation from a covariance matrix, and read the quantile off the normal table: $\text{VaR}=z_{\alpha }\sigma \sqrt{h}$. Fast, closed-form, lovely algebra - but the normality assumption is doing all the work.
2. Historical simulation: take the empirical distribution of past daily P&L and read off its quantile directly - no normality assumption, fat tails for free, but you only see crises that are in your window, and a single old crash can linger like a ghost.
3. Monte Carlo: simulate the risk factors (a few thousand or a few hundred thousand scenarios), fully revalue the portfolio under each, and read the quantile of the simulated loss distribution. The most general - handles options, path-dependence, nonlinearity - but computationally heavy and only as good as its factor model and revaluation engine.

All three compute the same number (the P&L quantile); they disagree because they model the P&L distribution differently. This folder is the topic-hub for the method family. It gives you the method-lookup table below (job #1), and routes you to six sub-pages that walk you from raw intuition through each method, the failure modes, and the extensions (delta–gamma and backtesting).

The one-sentence essence. “VaR is one quantile of the loss distribution; parametric assumes the distribution, historical samples it from the past, and Monte Carlo builds it by simulation - and you must backtest whichever you pick, because a VaR number nobody checks is a guess wearing a confidence level.”

2. Mathematical Ground Truth & the Method Lookup

Quick-Reference Lookup (job #1). All formulas are transcribed from Hull Ch 22 (VaR & ES, verified), Glasserman Ch 9 (MC for risk, verified), and cross-checked against the corpus tail/backtesting literature (Kupiec 1995, Christoffersen 1998, BCBS 1996). The numbers in the comparison column were re-executed and reproduced exactly from the verified corpus on a shared 2-asset portfolio ($40,000$ + $60,000$ exposures, daily vols $1.2\%$/$2.0\%$, $\rho =0.40$, 1-day, 99% VaR).

Notation: $L=-\Delta V$ the one-period loss; $\alpha$ the confidence level; $z_{\alpha }=N^{-1}(\alpha )$ the normal quantile ($z_{0.99}=2.3263$); $w\in {\mathbb{R}}^N$ portfolio weights/positions; $\Sigma$ the covariance matrix of factor returns; ${\sigma }_p$ portfolio daily volatility; $1-\alpha$ the violation rate.

The three agree here on purpose. When returns are genuinely normal, parametric, historical, and MC VaR all estimate the same normal quantile and land within sampling noise of each other ($\approx 3,300$–$3,400$). The entire reason to study the differences is the failure modes in 05 · Failure Modes, where fat tails make them diverge by factors.

Backtesting statistics (the “check” that turns a number into a model):

• Kupiec (1995) Proportion-of-Failures (POF): count breaches $x$ of the VaR over $T$ days; under $H_0:p=1-\alpha$,

Reject at 5% if ${\text{LR}}_{\text{POF}}>3.841$. Verified: 11 breaches / 1000 days of a well-calibrated 99% VaR → LR=0.098 (accept); an under-stated-vol model with 115 breaches / 1000 → LR=363.29 (reject).

• Christoffersen (1998) independence: a test that breaches are not clustered (which Kupiec misses) via the likelihood ratio over the $0/1$ violation sequence. Verified: same well-calibrated series → LR=0.245 (accept).

3. Computational Implementation - all three methods + a backtest, stdlib only

This runs on the standard library only (math.erf, random). It reproduces every verified number in §2 on the shared portfolio, then backtests a VaR model. (Each method also gets its own deeper example on its sub-page.)

Note the deliberate sign convention: $L=-\Delta V$ (loss positive), so a historical rank k of the worst P&L is the k-th smallest of sorted(pnl) and VaR = -pnl[k-1]; MC sorts loss directly and takes the $\alpha$-quantile. Mixing these two signs is the single most common implementation bug.

4. Failure Modes & First-Principles Breakdowns

Hub signposts - the folder’s failure-mode analysis lives in 05 · Failure Modes & Practice. In one line each:

1. The normality assumption is doing the whole job (parametric). Delta-normal VaR = $z_{\alpha }\sigma \sqrt{h}$ is the normal quantile; the moment real returns have fat tails this understates true tail risk. Verified on the same-sigma comparison: a Student-$t(4)$ distribution (variance-matched to normal) breaches the normal 99% VaR at a 1.56% rate, not 1.00%.
2. The historical window can lie (historical). You only see the crises in your window. A single 1987-style crash raises historical VaR while it is in the window and drops it out the moment it scrolls off - the “ghost effect” (verified: VaR 3,465 with the crash inside, 3,196 after it exits).
3. MC is only as good as its factor model (Monte Carlo). Simulating under a wrong correlation or a normal (vs fat-tailed) factor process just makes a fancier version of the same mistake. And full revaluation is slow - hence the delta–gamma shortcut in 06 · Advanced Extensions.
4. VaR is not subadditive. $\text{VaR}(X+Y)$ can exceed $\text{VaR}(X)+\text{VaR}(Y)$, so summing desk VaRs can understate firm VaR - the reason regulators moved to Expected Shortfall (see VaR & Expected Shortfall).

5. References

• Hull, John C.: Options, Futures, and Other Derivatives (11th ed.)
• Glasserman, Paul: Monte Carlo Methods in Financial Engineering (2004)
• Kupiec, Paul H.: Techniques for Verifying the Accuracy of Risk Measurement Models, Journal of Derivatives 3(2):73–84 (1995)
• Christoffersen, Peter F.: Evaluating Interval Forecasts, International Economic Review 39(4):841–862 (1998)
• BCBS: Supervisory Framework for the Use of Backtesting in Conjunction with the Internal Models Approach to Market Risk Capital Requirements (1996)
• McNeil & Frey: Estimation of Tail-Related Risk Measures for Heteroscedastic Financial Time Series (2000)

6. Connected Graph Bridges

• Foundational base: Linear Algebra & Covariance · Statistics & Inference · Econometrics & Time Series · Numerical Methods.
• Sibling topic: VaR & Expected Shortfall (CVaR) (the measure this folder’s methods estimate).
• Forward (tails & regulation): Extreme Value Theory & Fat Tails · Stress Testing & Scenario Analysis.
• Sub-pages (in-folder): 01 From Zero · 02 Parametric VaR · 03 Historical Simulation · 04 Monte Carlo VaR · 05 Failure Modes · 06 Advanced Extensions (delta–gamma & Backtesting).

Beginner: start at 01 · Practitioner: start at 05