MarketRiskVaR:ModelBuildingApproachChapter 15RiskManagementandFinanciallnstitutions3e,Chapter15,CopyrightJohnC.Hull2012
Risk Management and Financial Institutions 3e, Chapter 15, Copyright © John C. Hull 2012 Chapter 15 Market Risk VaR: ModelBuilding Approach 1
The Model-Building ApproachThe main alternative to historical simulation is tomake assumptions about the probabilitydistributions of the returns on the marketvariablesThis is known as the model building approach(or sometimes the variance-covarianceapproach)2RiskManagementandFinancialInstitutions3e,Chapter15,CopyrightJohnC.Hull2012
Risk Management and Financial Institutions 3e, Chapter 15, Copyright © John C. Hull 2012 The Model-Building Approach ⚫ The main alternative to historical simulation is to make assumptions about the probability distributions of the returns on the market variables ⚫ This is known as the model building approach (or sometimes the variance-covariance approach) 2
Microsoft Example (page 323-324)We have aposition worth $10 million inMicrosoft shares The volatility of Microsoft is 2% per day(about 32% per year)We use N=10 and X=993RiskManagementandFinancialInstitutions3e,Chapter15,CopyrightJohnC.Hull2012
Risk Management and Financial Institutions 3e, Chapter 15, Copyright © John C. Hull 2012 Microsoft Example (page 323-324) ⚫ We have a position worth $10 million in Microsoft shares ⚫ The volatility of Microsoft is 2% per day (about 32% per year) ⚫ We use N=10 and X=99 3
Microsoft Example continued The standard deviation of the change inthe p0rtfolio in 1 day is $200,000 The standard deviation of the change in 10days is200,000/10 = $632,456RiskManagementandFinancialInstitutions3e,Chapter15,CopyrightJohnC.Hull20124
Risk Management and Financial Institutions 3e, Chapter 15, Copyright © John C. Hull 2012 Microsoft Example continued ⚫ The standard deviation of the change in the portfolio in 1 day is $200,000 ⚫ The standard deviation of the change in 10 days is 200,000 10 = $632,456 4
Microsoft Examplecontinued.We assume that the expected change inthe value of the portfolio is zero (This isOK for short time periods) We assume that the change in the value ofthe portfolio is normally distributed Since N(-2.33)=0.01, the VaR is2.33 x 632,456 = $1,473,621RiskManagementandFinancialInstitutions3e,Chapter15,CopyrightJohnC.Hull20125
Risk Management and Financial Institutions 3e, Chapter 15, Copyright © John C. Hull 2012 Microsoft Example continued ⚫ We assume that the expected change in the value of the portfolio is zero (This is OK for short time periods) ⚫ We assume that the change in the value of the portfolio is normally distributed ⚫ Since N(–2.33)=0.01, the VaR is 2.33 632,456 = $1,473,621 5
AT&TExampleConsidera positionof $5 million in AT&TThe daily volatility of AT&T is 1% (approx16% per year)The SD per 10 days is50.000/10 = $158,144TheVaR is158,114 x 2.33 = $368,4056RiskManagementandFinancialInstitutions3e,Chapter15,CopyrightJohnC.Hull2012
Risk Management and Financial Institutions 3e, Chapter 15, Copyright © John C. Hull 2012 AT&T Example ⚫ Consider a position of $5 million in AT&T ⚫ The daily volatility of AT&T is 1% (approx 16% per year) ⚫ The SD per 10 days is ⚫ The VaR is 50,000 10 = $158,144 158,114 2.33 = $368,405 6
Portfolio (page 325) Now consider a portfolio consisting of bothMicrosoft and AT&T Suppose that the correlation between thereturns is 0.37RiskManagementandFinancialInstitutions3e,Chapter15,CopyrightJohnC.Hull2012
Risk Management and Financial Institutions 3e, Chapter 15, Copyright © John C. Hull 2012 Portfolio (page 325) ⚫ Now consider a portfolio consisting of both Microsoft and AT&T ⚫ Suppose that the correlation between the returns is 0.3 7
S.D. of Portfolio A standard result in statistics states that0x+y=/o+o+2pox0yIn this case x = 200,000 and oy = 50,000and p = 0.3. The standard deviation of thechange in the portfolio value in one day istherefore 220.2278RiskManagementandFinancialInstitutions3e,Chapter15,CopyrightJohnC.Hull2012
Risk Management and Financial Institutions 3e, Chapter 15, Copyright © John C. Hull 2012 S.D. of Portfolio ⚫ A standard result in statistics states that ⚫ In this case sX = 200,000 and sY = 50,000 and r = 0.3. The standard deviation of the change in the portfolio value in one day is therefore 220,227 X Y X Y X Y s + = s + s + 2rs s 2 2 8
VaR for Portfolio The 10-day 99% VaR for the portfolio is220,227 × V10 ×2.33 = $1,622,657The benefits of diversification are(1,473,621+368,405)-1,622,657=$219,369 What is the incremental effect of the AT&Tholding on VaR?9RiskManagementandFinancialInstitutions3e,Chapter15,CopyrightJohnC.Hull2012
Risk Management and Financial Institutions 3e, Chapter 15, Copyright © John C. Hull 2012 VaR for Portfolio ⚫ The 10-day 99% VaR for the portfolio is ⚫ The benefits of diversification are (1,473,621+368,405)–1,622,657=$219,369 ⚫ What is the incremental effect of the AT&T holding on VaR? 220,227 10 2.33 = $1,622,657 9
The Linear ModelWe assume The daily change in the value of a portfoliois linearly related to the daily returns frommarket variablesThe returns from the market variables arenormally distributed10RiskManagementandFinancialInstitutions3e,Chapter15,CopyrightJohnC.Hull2012
Risk Management and Financial Institutions 3e, Chapter 15, Copyright © John C. Hull 2012 The Linear Model We assume ⚫ The daily change in the value of a portfolio is linearly related to the daily returns from market variables ⚫ The returns from the market variables are normally distributed 10