6_VaR_II - VALUE AT RISK (VAR) II Model-building approach...

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VALUE AT RISK (VAR) II J. Wei, Department of Management, U of T 1 MGTD78 Model-building approach (MBA) Basic methodology (single-, two-, multi-asset cases) Handling interest rates The linear model and options Historical simulation approach (HSA) Methodology Accuracy Comparison between historical simulation approach and model-building approach
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MODEL-BUILDING APPROACH (MBA) J. Wei, Department of Management, U of T 2 MGTD78 An alternative to the historical simulation approach Essence: assume a model for the joint distribution of changes in market variables and calculate VaR analytically. Foundation: Harry Markowitz’s mean-covariance analysis
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MODEL-BUILDING APPROACH (MBA) J. Wei, Department of Management, U of T 3 MGTD78 Daily versus annual volatility - In option pricing, we usually use annual volatility - Assuming 252 trading days in a year, we have Daily return and standard deviation of market variables - Compared with std, average daily return is very small - We therefore assume zero daily mean for most cases - Example: IBM stock, annual return = 18%, std = 35%. Then, daily return = 18% / 252 = 0.071%, whereas the daily std is 35% / sqrt(252) = 2.205%. 252 σ σ day annual =
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MBA BASIC METHODOLOGY: 1 ASSET J. Wei, Department of Management, U of T 4 MGTD78 Portfolio consists of one asset: IBM stock, worth $5 million. What is the 10-day 99% VaR? Parameters: annual = 35% σ daily = 2.205% (still assume σ zero mean) Solution: assume normal distribution for returns one-day std in dollars: 5,000,000 x 0.02205 = $110,250 99% confidence level N(-x) = 0.01, x = 2.326 meaning: there is a 1% chance that the change is bigger than 2.326 σ 1-day 99% VaR = 2.326 x 110,250 = $256,441.50 10-day 99% VaR = 256,441.50 x 10 = $810,939
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MBA BASIC METHODOLOGY: 1 ASSET J. Wei, Department of Management, U of T 5 MGTD78 Portfolio consists of one asset: Home Depot stock, worth $20 million. What is the 10-day 99% VaR? Parameters: annual = 25% σ daily = 1.575% σ Solution (similar to IBM): assume normal distribution for returns one-day std in dollars: 20,000,000 x 0.01575 = $315,000 1-day 99% VaR = 2.326 x 315,000 = $732,690 10-day 99% VaR = 732,690 x 10 = $2,316,969
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MBA BASIC METHODOLOGY: 2 ASSETS J. Wei, Department of Management, U of T 6 MGTD78 Portfolio consists of two assets: IBM stock, worth $5 million. Home Depot stock, worth $20 million. Same std as before, and = 0.35. ρ What is the 10-day 99% VaR? Recap parameters: IBM: σ 1 = 5,000,000 x 0.02205 = $110,250 Home Depot: σ 2 = 20,000,000 x 0.01575 = $315,000 Harry Markowitz result: Solution: similar to one-asset case assume joint-normal distribution for returns one-day std in dollars: 1-day 99% VaR = 2.326 x 368,361 = $856,808 10-day 99% VaR = 856,808 x 10 = $2,709,465 ρ σ σ σ σ 2 1 2 2 2 1 portfolio + + = 368,361 35) 315000)(0. 2(110250)( 315000 110250 2 2 = + +
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6_VaR_II - VALUE AT RISK (VAR) II Model-building approach...

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