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📈 Valuation and Risk Models 25 min readWeight 30%

Value at Risk (VaR): A Comprehensive Guide

Everything you need to know about VaR — parametric, historical simulation, Monte Carlo, backtesting, Expected Shortfall, and the strengths and weaknesses of each approach.

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Loss Tail Versus Expected Loss

Valuation and risk-model questions often hinge on the difference between a normal day and a stressed tail outcome.

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Why it matters

Value at Risk gives a cutoff; expected shortfall tells you what happens after the cutoff is breached.

What Is VaR?Parametric VaRHistorical SimulationMonte Carlo VaRExpected ShortfallBacktestingLimitations

Value at Risk (VaR): A Comprehensive Guide

What Is VaR?

Value at Risk is a statistical measure that estimates the maximum potential loss on a portfolio over a specified time period, at a given confidence level.

Definition: "The VaR at the X% confidence level is the loss that will be exceeded only (100−X)% of the time."

Example: A 10-day 99% VaR of $5 million means: "There is a 1% probability that the portfolio will lose more than $5 million over the next 10 trading days."

VaR Has Three Parameters

  1. Confidence level (typically 95% or 99%)
  2. Time horizon (1 day, 10 days, etc.)
  3. The portfolio being measured

Why VaR Became Standard

  • Single number summarizing risk — easy for executives and boards to understand
  • Comparable across asset classes and business units
  • Required by regulators (Basel framework uses VaR for market risk capital)
  • Can be aggregated across portfolios (with correlation assumptions)

Parametric (Variance-Covariance) VaR

The simplest approach. Assumes portfolio returns are normally distributed.

Formula

VaR = z_α × σ_P × √T

Where:

  • z_α = standard normal quantile at confidence level α (e.g., 2.326 for

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