《金融风险管理》课程PPT教学课件（Risk Management and Financial Institutions）Chapter 14 Market Risk VaR - Historical Simulation Approach

Market RiskVaR:HistoricalSimulationApproachChapter 14RiskManagementandFinanciallnstitutions3e,Chapter14,CopyrightJohnC.Hull2012

Chapter 14 Risk Management and Financial Institutions 3e, Chapter 14, Copyright © John C. Hull 2012 1 Market Risk VaR: Historical Simulation Approach

Historical Simulation Collect data on the daily movements in allmarket variables.The first simulation trial assumes that thepercentage changes in all market variables areas on the first dayThe second simulation trial assumes that thepercentage changes in all market variables areas on the second dayand so on2RiskManagementandFinancial Institutions3e,Chapter14,CopyrightJohnC.Hull2012

Historical Simulation ⚫ Collect data on the daily movements in all market variables. ⚫ The first simulation trial assumes that the percentage changes in all market variables are as on the first day ⚫ The second simulation trial assumes that the percentage changes in all market variables are as on the second day ⚫ and so on Risk Management and Financial Institutions 3e, Chapter 14, Copyright © John C. Hull 2012 2

Historical Simulation continuedSupposeweusendaysofhistorical datawithtoday being day nLet y, be the value of a variable on day iThere are n-1 simulation trialsThe ith trial assumes that the yalue of themarket variable tomorrow (i.e., on day n+1) isV3RiskManagementandFinancialInstitutions3e,Chapter14,CopyrightJohnC.Hull2012

Historical Simulation continued ⚫ Suppose we use n days of historical data with today being day n ⚫ Let vi be the value of a variable on day i ⚫ There are n-1 simulation trials ⚫ The ith trial assumes that the value of the market variable tomorrow (i.e., on day n+1) is Risk Management and Financial Institutions 3e, Chapter 14, Copyright © John C. Hull 2012 3 i−1 i n v v v

Example:Portfolio on Sept252008(Table14.1,page304)IndexAmountInvested ($ooos)DJIA4,000FTSE1003,000CAC 401,000Nikkei2252,000Total10,000RiskManagementandFinancialInstitutions3e,Chapter14,CopyrightJohnC.Hull20124

Example: Portfolio on Sept 25, 2008 (Table 14.1, page 304) Risk Management and Financial Institutions 3e, Chapter 14, Copyright © John C. Hull 2012 4 Index Amount Invested ($000s) DJIA 4,000 FTSE 100 3,000 CAC 40 1,000 Nikkei 225 2,000 Total 10,000

U.S.DollarEquivalent of StockIndices (Table14.2,page305)DayDJIAFTSENikkeiDateCAC 40011,219.3811,131.846,373.89131.77Aug 7, 200616,378.16134.38Aug 8, 200611,173.5911,096.28211,185.356,474.04135.94Aug 9, 200611.076.183Aug 10, 200611,124.3711,016.716,357.49135.44499Sep 24,200810,825.179,438.586,033.93114.26500112.82Sep 25, 200811,022.069,599.906,200.40RiskManagementandFinancialInstitutions3e,Chapter14,CopyrightJohnC.Hull20125

U.S. Dollar Equivalent of Stock Indices (Table 14.2, page 305) Risk Management and Financial Institutions 3e, Chapter 14, Copyright © John C. Hull 2012 5 Day Date DJIA FTSE CAC 40 Nikkei 0 Aug 7, 2006 11,219.38 11,131.84 6,373.89 131.77 1 Aug 8, 2006 11,173.59 11,096.28 6,378.16 134.38 2 Aug 9, 2006 11,076.18 11,185.35 6,474.04 135.94 3 Aug 10, 2006 11,124.37 11,016.71 6,357.49 135.44 . . . . . . 499 Sep 24, 2008 10,825.17 9,438.58 6,033.93 114.26 500 Sep 25, 2008 11,022.06 9,599.90 6,200.40 112.82

Scenarios (Table14.3,page305)11,173.5911.022.0611.219.38CACScenarioDJIAFTSENikkeiLossPortfolioValueNumber110.977.0826,204.55115.059,569.2310,014.334-14.334210.925.979,676.966,293.60114.1310,027.481-27,481311,070.019.455.166,088.77112.409,946.73653,2644996,051.94113.859,857.465142.53510,831.439,383.4950011,222.539,763.97111.4010,126.439-126.4396,371.45RiskManagementandFinancialInstitutions3e,Chapter14,CopyrightJohnC.Hull20126

Scenarios (Table 14.3, page 305) Risk Management and Financial Institutions 3e, Chapter 14, Copyright © John C. Hull 2012 6 Scenario Number DJIA FTSE CAC Nikkei Portfolio Value Loss 1 10,977.08 9,569.23 6,204.55 115.05 10,014.334 -14.334 2 10,925.97 9,676.96 6,293.60 114.13 10,027.481 -27,481 3 11,070.01 9,455.16 6,088.77 112.40 9,946.736 53,264 . . . . . . . 499 10,831.43 9,383.49 6,051.94 113.85 9,857.465 142.535 500 11,222.53 9,763.97 6,371.45 111.40 10,126.439 -126.439 11,219.38 11,173.59 =11,022.06

Losses (Table 14.4, page 307)Scenario NumberLoss ($oo0s)494477.841339345.435349282.204329277.041487253.385227217.974131205.256One-day99%VaR=$253,3857RiskManagementandFinancialInstitutions3e,Chapter14,CopyrightJohnC.Hull2012

Losses (Table 14.4, page 307) Risk Management and Financial Institutions 3e, Chapter 14, Copyright © John C. Hull 2012 7 Scenario Number Loss ($000s) 494 477.841 339 345.435 349 282.204 329 277.041 487 253.385 227 217.974 131 205.256 One-day 99% VaR=$253,385

Accuracy (page 308)Suppose that x is the gth quantile of the lossdistribution when it is estimated from n observationsThe standard errorofxis1q(1-q)f(x) Vnwhere f(x) is an estimate of the probability density ofthe loss at the gth quantile calculated by assuming aprobability distributionforthelossRiskManagementandFinancialInstitutions3e,Chapter14,CopyrightJohnC.Hull20128

Accuracy (page 308) Suppose that x is the qth quantile of the loss distribution when it is estimated from n observations. The standard error of x is where f(x) is an estimate of the probability density of the loss at the qth quantile calculated by assuming a probability distribution for the loss n q q f x (1 ) ( ) 1 − Risk Management and Financial Institutions 3e, Chapter 14, Copyright © John C. Hull 2012 8

Example 14.1 (page 308)We estimate the 0.01-quantile from 500 observations as$25 millionWe estimate f(x) by approximating the actual empiricaldistribution with a normal distribution mean zero andstandard deviation $10 millionThe 0.01 quantile of the approximating distribution isNORMINV(0.01,0,10) = 23.26 and the value of f(x) isNORMDIST(23.26,0,10,FALSE)=0.0027The estimate of the standard error is therefore10.01×0.99-1.675000.0027RiskManagementandFinancialInstitutions3e,Chapter14,CopyrightJohnC.Hull20129

Example 14.1 (page 308) ⚫ We estimate the 0.01-quantile from 500 observations as $25 million ⚫ We estimate f(x) by approximating the actual empirical distribution with a normal distribution mean zero and standard deviation $10 million ⚫ The 0.01 quantile of the approximating distribution is NORMINV(0.01,0,10) = 23.26 and the value of f(x) is NORMDIST(23.26,0,10,FALSE)=0.0027 ⚫ The estimate of the standard error is therefore Risk Management and Financial Institutions 3e, Chapter 14, Copyright © John C. Hull 2012 9 1.67 500 0.01 0.99 0.0027 1 =  

Extension 1 Let weights assigned to observationsdecline exponentially as we go back intimeRank observations from worst to bestStarting at worst observation sum weightsuntil the required quantile is reached10RiskManagementandFinancialInstitutions3e,Chapter14,CopyrightJohnC.Hull2012

Extension 1 ⚫ Let weights assigned to observations decline exponentially as we go back in time ⚫ Rank observations from worst to best ⚫ Starting at worst observation sum weights until the required quantile is reached Risk Management and Financial Institutions 3e, Chapter 14, Copyright © John C. Hull 2012 10