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# Brandeis FIN 285a - Fin285a:Computer Simulations and Risk Assessment

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OverviewBackground and historyRisk measuresValue-at-Risk (VaR)VaR advantagesHistoryVaR methodsDefine Value-at-Risk (VaR)Value-at-Risk (VaR): general definitionMove to simpler (continuous) distributionsValue-at-Risk (VaR): continuousValue-at-Risk (VaR): continuousNow, some mathVaR p=0.05, Q is N(0,=10)VaR detailsVaR parametersVaR examplePortfolio value densityPortfolio p=0.05 quantileVaR calculationPortfolio p=0.01 quantileVaR calculationMore VaR detailsVaR detailsTwo annoyancesCan VaR be negativeNegative VaR(0.05)=-(106.8-100)=-6.8p versus (1-p)Choosing VaR parametersVaR usesOverviewFin285a:Computer Simulations andRisk AssessmentSection 4.1Value-at-Risk (VaR) BasicsDan´ıelson, 4.1-4.3Over viewFall 2014: LeBaron Fin285a: 4.1 – 2 / 32Back ground and historyDefine Value-at-Risk (VaR)More VaR detailsBackground and histor yBackground andhistoryDefine Value-at-Risk(VaR)More VaR detailsFall 2014: LeBaron Fin285a: 4.1 – 3 / 32Risk measuresFall 2014: LeBaron Fin285a: 4.1 – 4 / 32•Quantify risk•Examples⇒ Variance⇒ Beta⇒ Duration⇒ Value-at-R isk (VaR)⇒ Expected shortfall, Expected tail loss, CVaRValue-at-Risk (VaR)Fall 2014: LeBaron Fin285a: 4.1 – 5 / 32•Probalistic worst caseValue-at-Risk (VaR)Fall 2014: LeBaron Fin285a: 4.1 – 5 / 32•Probalistic worst case•Almost “perfect storm”Value-at-Risk (VaR)Fall 2014: LeBaron Fin285a: 4.1 – 5 / 32•Probalistic worst case•Almost “perfect storm”•1 in 100 year flood levelVaR advantagesFall 2014: LeBaron Fin285a: 4.1 – 6 / 32•Risk → single numberVaR advantagesFall 2014: LeBaron Fin285a: 4.1 – 6 / 32•Risk → single number•Firm wide summary⇒ No problems with derivatives and other co mplexcontractsVaR advantagesFall 2014: LeBaron Fin285a: 4.1 – 6 / 32•Risk → single number•Firm wide summary⇒ No problems with derivatives and other co mplexcontracts•Relatively model freeVaR advantagesFall 2014: LeBaron Fin285a: 4.1 – 6 / 32•Risk → single number•Firm wide summary⇒ No problems with derivatives and other co mplexcontracts•Relatively model free•Easy to explainVaR advantagesFall 2014: LeBaron Fin285a: 4.1 – 6 / 32•Risk → single number•Firm wide summary⇒ No problems with derivatives and other co mplexcontracts•Relatively model free•Easy to explain•Handles deviations from normal dis tributionsHistoryFall 2014: LeBaron Fin285a: 4.1 – 7 / 32•Late 1980’s: trading portfolios⇒ Trading desk: FX⇒ High frequency⇒ High liquidityHistoryFall 2014: LeBaron Fin285a: 4.1 – 7 / 32•Late 1980’s: trading portfolios⇒ Trading desk: FX⇒ High frequency⇒ High liquidity•JP Morgan, 1990’s⇒ 4:15 and VaR⇒ RiskMetrics, 1994Now part of MSCIVaR methodsFall 2014: LeBaron Fin285a: 4.1 – 8 / 32•Delta normal•Historical•Monte-carlo•BootstrapDefine Value-at-Risk (VaR)Background andhistoryDefine Value-at-Risk(VaR)More VaR detailsFall 2014: LeBaron Fin285a: 4.1 – 9 / 32Value-at -Risk (VaR): general definitionFall 2014: LeBaron Fin285a: 4.1 – 10 / 32Definition: VaR(p) is the smallest value such that theprobability that your los ses, Q, exceed −VaR(p) is les sthan or equal to p.Pr(Q < −VaR(p)) ≤ p (4.1.1)Move to simpler (continuous) distri butionsFall 2014: LeBaron Fin285a: 4.1 – 11 / 32•This definition of VaR can get confusing/complicated•Easier when distributions are continuous(At least in the reg ion where VaR is defined.)•Inequalities b ecome equalities•Often most relevant to the real worldValue-at-Risk (VaR): continuousFall 2014: LeBaron Fin285a: 4.1 – 12 / 32Definition (continuous): VaR(p) is the value such thatprobability that your profts/losses, Q, exceed −VaR(p) isp.Pr(Q ≤ −VaR(p)) = p (4.1.2)Sometimes you will see this defined in reverse:Value-at-Risk (VaR): continuousFall 2014: LeBaron Fin285a: 4.1 – 12 / 32Definition (continuous): VaR(p) is the value such thatprobability that your profts/losses, Q, exceed −VaR(p) isp.Pr(Q ≤ −VaR(p)) = p (4.1.2)Sometimes you will see this defined in reverse:Definition (continuous): I am confident w ith probability1 − p that my portfolio will not lose mor e than −VaR(p).Pr(Q ≥ −VaR(p)) = (1 − p) (4.1.3)Value-at-Risk (VaR): continuousFall 2014: LeBaron Fin285a: 4.1 – 13 / 32•VaR is easier with continuous distributions. Start herewhen un derstanding VaR.•Many textbooks simply start with this as sumption. (Thisincludes Dan´ıelson.) It makes the initial understandingmuch easier, and applies to mos t practical cases.Now, some mathFall 2014: LeBaron Fin285a: 4.1 – 14 / 32fq(x) = pro bability den sity for P/LFq(x) = Cumulative distribution function for P/L (CDF)Pr(Q ≤ −V aR(p)) = pp =Z−VaR(p)−∞fq(x)dx (4.1.4)p = Fq(−V aR(p)) (4.1.5)VaR p = 0.05, Q is N(0, σ = 10)Fall 2014: LeBaron Fin285a: 4.1 – 15 / 32-40 -30 -20 -10 0 10 20 30050010001500200025003000-16.4Q = P/LObservations(from 100000)VaR detailsFall 2014: LeBaron Fin285a: 4.1 – 16 / 32How do we really calculate VaR?VaR parametersFall 2014: LeBaron Fin285a: 4.1 – 17 / 32•Horizon (1 day)•Prob of loss (p)VaR exampleFall 2014: LeBaron Fin285a: 4.1 – 18 / 32•Portfolio value today: 100•Time horizon: 1 month•Distribution of value over month⇒ Normal distribution⇒ Mean 100⇒ Std. 10•Loss probability, p = 0.05Portfolio value densityFall 2014: LeBaron Fin285a: 4.1 – 19 / 3250 100 15001000200030004000500060007000Frequency (out of 100,000)Portfolio value in 1 month, mean = 100, Std. = 10Portfolio p = 0. 05 quantileFall 2014: LeBaron Fin285a: 4.1 – 20 / 3260 70 80 90 100 110 120 130 1400100020003000400050006000 83.6Portfolio value in 1 month, today = 100, mean = 100, Std. = 10Frequency (out of 100,000)VaR calculationFall 2014: LeBaron Fin285a: 4.1 – 21 / 32•Profit/Loss (P/L) = Q = (future value) − 100•Loss at 0.05 = 83. 6 − 100 = −16.4VaR calculationFall 2014: LeBaron Fin285a: 4.1 – 21 / 32•Profit/Loss (P/L) = Q = (future value) − 100•Loss at 0.05 = 83. 6 − 100 = −16.4•Flip sign (loss) : 16. 4VaR calculationFall 2014: LeBaron Fin285a: 4.1 – 21 / 32•Profit/Loss (P/L) = Q = (future value) − 100•Loss at 0.05 = 83. 6 − 100 = −16.4•Flip sign (loss) : 16. 4•Pr(Q ≤ −VaR(0. 05)) = Pr(Q ≤ −(16.4)) = 0.05 = pVaR calculationFall 2014: LeBaron Fin285a: 4.1 – 21 / 32•Profit/Loss (P/L) = Q = (future value) − 100•Loss at 0.05 = 83. 6 − 100 = −16.4•Flip sign (loss) : 16. 4•Pr(Q ≤ −VaR(0. 05)) = Pr(Q ≤ −(16.4)) = 0.05 = p•Basic

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