Value at Risk
One sentence to summarise a portfolio's risk: "we are 99% confident we won't lose more than $X tomorrow." Learn to build that number three different ways, average the tail beyond it, scale it across time, backtest it against reality — and see how it blew up in 2008.
How banks and funds turn a whole portfolio's risk into a single number — Value at Risk (VaR) and its sharper cousin Expected Shortfall (CVaR). The three ways to compute them (historical, parametric, Monte Carlo), how to scale across time, how to backtest them, and exactly where they lie to you.
A bank holds thousands of tangled positions, but the board, the regulator and the risk desk want exactly one number in dollars — “we are 99% confident we won’t lose more than $4.2 million tomorrow.” That number is Value at Risk (VaR): compact, seductive, written into the Basel rules, and quietly capable of lying to you.
This topic builds VaR from scratch, then teaches you to distrust it intelligently. You’ll learn:
- What VaR actually says — its three ingredients (horizon, confidence, amount) and the crucial things it deliberately doesn’t tell you.
- Historical simulation — replay the past to read the loss straight off the data.
- Parametric / variance–covariance — assume a bell curve and get VaR with algebra.
- Monte Carlo — simulate thousands of synthetic tomorrows, each with its own assumptions and failure modes.
- Time scaling — stretch VaR across horizons with the square-root-of-time rule, and spot when it breaks.
- Expected Shortfall (CVaR) — VaR’s sharper successor, answering when you breach the line, how bad is it on average?
- Backtesting — count the days reality punched through, run the Kupiec test and the Basel traffic light, and confront the fat tails that humbled VaR in 2008.
This is the keystone of the quant-risk ladder — Monte Carlo finance, extreme-value theory and risk of ruin all sharpen a blade you first pick up here. By the end you’ll compute a VaR three ways, scale it, backtest it, and explain with numbers exactly when it’s lying to you.
In this topic
- 1 What Value at Risk Is Value at Risk compresses a whole portfolio's risk into one number — a horizon, a confidence level, and a loss amount — plus the dangers it deliberately ignores. 9 min
- 2 Historical Simulation VaR Estimate VaR by replaying actual past returns, sorting the realised P&L, and reading off the percentile — no bell curve assumed. Strengths, traps, and worked numbers. 9 min
- 3 Parametric VaR The variance–covariance method: assume normal returns and read VaR straight off z-scores, scale across horizons with the √t rule, and combine assets through correlation. 10 min
- 4 Monte Carlo VaR Estimate Value at Risk by simulating thousands of synthetic tomorrows from a chosen model, fully revaluing the portfolio in each, and reading the percentile of the simulated losses. 9 min
- 5 Expected Shortfall (CVaR) Expected Shortfall (CVaR) is the average loss in the tail beyond VaR. Why it's a coherent risk measure, why VaR isn't, and why regulators switched to it. 9 min
- 6 Backtesting VaR and Its Limits Validate a VaR model by counting exceptions against the expected rate — the Kupiec test and Basel traffic light — then confront fat tails, procyclicality and how VaR broke in 2008. 9 min
- 7 Value at Risk — Final Exam The graded final exam for Value at Risk: the VaR sentence, historical, parametric and Monte Carlo methods, square-root-of-time scaling, Expected Shortfall, and backtesting. 15 min
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