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FIN 685: Risk Management Topic 6: VaR Larry Schrenk, Instructor

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Types of Risks Value-at-Risk Expected Shortfall

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Types of Risk

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Market Risk Credit Risk Liquidity Risk Operational Risk

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VaR

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J. P. Morgan Chairman, Dennis Weatherstone and the 4:14 Report 1993 Group of Thirty 1994 RiskMetrics

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Probable Loss Measure Multiple Methods Comprehensive Measurement Interactions between Risks

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There is an x percent chance that the firm will loss more than y over the next z time period.”

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Correlation Historical Simulation Monte Carlo Simulation

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Historical Prices – Various periods Values Portfolio in Next Period Generate Future Distributions of Outcomes

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Variance-covariance – Assume distribution, use theoretical to calculate – Bad – assumes normal, stable correlation Historical simulation – Good – data available – Bad – past may not represent future – Bad – lots of data if many instruments (correlated) Monte Carlo simulation – Good – flexible (can use any distribution in theory) – Bad – depends on model calibration Finland 2010

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Basel Capital Accord – Banks encouraged to use internal models to measure VaR – Use to ensure capital adequacy (liquidity) – Compute daily at 99 th percentile Can use others – Minimum price shock equivalent to 10 trading days (holding period) – Historical observation period ≥1 year – Capital charge ≥ 3 x average daily VaR of last 60 business days Finland 2010

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At 99% level, will exceed 3-4 times per year Distributions have fat tails Only considers probability of loss – not magnitude Conditional Value-At-Risk – Weighted average between VaR & losses exceeding VaR – Aim to reduce probability a portfolio will incur large losses Finland 2010

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E.G. RiskMetrics Steps 1.Means, Variances and Correlations from Historical Data Assume Normal Distribution 2.Assign Portfolio Weights 3.Portfolio Formulae 4.Plot Distribution

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Assuming normal distribution 95% Confidence Interval – VaR -1.65 standard deviations from the mean 99% Confidence Interval – VaR -2.33 standard deviations from the mean

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Two Asset Portfolio AssetReturnVarWeightCov A20%0.0450%0.02 B12%0.0350%

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= 0.1658 5% tail is 1.65*0.1658 = 0.2736 from mean Var = 0.16 - 0.2736 =-0.1136 There is a 5% chance the firm will loss more than 11.35% in the time period

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= 0.1658 1% tail is 2.33*0.1658 = 0.3863 from mean Var = 0.16 - 0 0.3863 =-0.2263 There is a 1% chance the firm will loss more than 22.63% in the time period

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Steps 1.Get Market Data for Determined Period 2.Measure Daily, Historical Percentage Change in Risk Factors 3.Value Portfolio for Each Percentage Change and Subtract from Current Portfolio Value

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Steps 6.Rank Changes 7.Choose percentile loss 95% Confidence – 5 th Worst of 100 – 50 th Worst of 1000

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1. Model changes in risk factors – Distributions – E.g.r t+1 = r t + + r t + t 2. Simulate Behavior of Risk Factors Next Period 3. Ranks and Choose VaR as in Historical Simulation

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One Number Sub-Additive Historical Data No Measure of Maximum Loss

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Holding period – Risk environment – Portfolio constancy/liquidity Confidence level – How far into the tail? – VaR use – Data quantity

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Benchmark comparison – Interested in relative comparisons across units or trading desks Potential loss measure – Horizon related to liquidity and portfolio turnover Set capital cushion levels – Confidence level critical here

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Uninformative about extreme tails Bad portfolio decisions – Might add high expected return, but high loss with low probability securities – VaR/Expected return, calculations still not well understood – VaR is not Sub-additive

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A sub-additive risk measure is Sum of risks is conservative (overestimate) VaR not sub-additive – Temptation to split up accounts or firms

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Market Risk VaR: Historical Simulation Approach

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