# VaR Introduction I: Parametric VaR

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1 VaR Introduction I: Parametric VaR Tom Mills FinPricing

2 VaR Definition VaR Roles VaR Pros and Cons VaR Approaches Parametric VaR Parametric VaR Methodology Parametric VaR Implementation VaR Scaling VaR Backtest Summary

3 Value at Risk (VaR) Definition The maximum likely loss on a portfolio for a given probability defined as x% confidence level over N days Pr(Loss > VaR(x%)) < 1- x%

4 VaR Roles Risk measurement Risk management Risk control Financial reporting Regulatory and economic capital

5 VaR Pros & Cons Pros Regulatory measurement for market risk Objective assessment Intuition and clear interpretation Consistent and flexible measurement Cons Doesn t measure risk beyond the confidence level: tail risk Non sub-additive

6 Three VaR Approaches Parametric VaR Historical VaR Monte Carlo VaR The presentation focuses on parametric VaR.

7 Assumption Parametric VaR Pros Asset returns follow normal distribution Fast and simple calculation Intuitive Cons Poor accuracy for non-linear products Second order approximation Hard to incorporate stress test

8 Parametric VaR Methodology Assuming an asset return/valuechange follows normal distribution, the quantile of 99% confidence level is 2.326σ where σ is standard derivation If absolute return is normally distributed, the 99% worse change of X is The VaR is given by where is the delta Similarly for a relative return, the VaR can be expressed as

9 Parametric VaR Implementation For each asset/instrument/riskfactor, calibrate volatility σ i based on daily return For each risk factor pair, calibrate correlation ρ ij Calculate the variance of a portfolio value change ρ 11 ρ 1n V 2 p = (P 1 )σ 1 (P n )σ n ρ n1 ρ nn (P 1 )σ 1 (P n )σ n The portfolio VaR is V p 2

10 VaR Scaling Normally firms compute 1-day 99% VaR Regulators require 10-day 99% VaR Under IID assumption, 10-day VaR = 10 VaR 1 day

11 VaR Backtest The only way to verify a VaR system is to backtest At a certain day, compute hypothetic P&L. If (hypothetic P&L > VaR) breach, otherwise, ok Hypothetic P&L is computed by holding valuation date and portfolio unchanged In one year period, If number of breaches is 0-4, the VaR system is in Green zone If number of breaches is 5-9, the VaR system is in Yellow zone If number of breaches is 10 or more, the VaR system is in Red zone

12 Thanks!

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