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

 Types of Risks  Value-at-Risk  Expected Shortfall

Types of Risk

 Market Risk  Credit Risk  Liquidity Risk  Operational Risk

VaR

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

 Probable Loss Measure  Multiple Methods  Comprehensive Measurement  Interactions between Risks

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

 Correlation  Historical Simulation  Monte Carlo Simulation

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

 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

 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

 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

 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

 Assuming normal distribution  95% Confidence Interval – VaR standard deviations from the mean  99% Confidence Interval – VaR standard deviations from the mean

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

  =  5% tail is 1.65* = from mean  Var = =  There is a 5% chance the firm will loss more than 11.35% in the time period

  =  1% tail is 2.33* = from mean  Var = =  There is a 1% chance the firm will loss more than 22.63% in the time period

 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

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

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

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

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

 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

 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

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