Probabilistic Models
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Probabilistic Models • Value-at-Risk (VaR) • Chance constrained programming – Min variance – Max return s.t. Prob{function≥target}≥α – Max Prob{function≥target} – Max VaR Finland 2010
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Probabilistic Models. Value-at-Risk (VaR) Chance constrained programming Min variance Max return s.t. Prob{function≥target}≥ α Max Prob{function≥target} Max VaR. Value at Risk. Maximum expected loss given time horizon, confidence interval. - PowerPoint PPT Presentation
Transcript of Probabilistic Models
Probabilistic Models
• Value-at-Risk (VaR)• Chance constrained programming
– Min variance– Max return s.t. Prob{function≥target}≥α– Max Prob{function≥target}– Max VaR
Finland 2010
Value at Risk
Maximum expected loss given time horizon,
confidence interval
Finland 2010
VaR = 0.64expect to exceed 99% of time in 1 year
Here loss = 10 – 0.64 = 9.36
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Use• Basel Capital Accord
– Banks encouraged to use internal models to measure VaR
– Use to ensure capital adequacy (liquidity)– Compute daily at 99th 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
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VaR Calculation Approaches• Historical simulation
– Good – data available– Bad – past may not represent future– Bad – lots of data if many instruments (correlated)
• Variance-covariance– Assume distribution, use theoretical to calculate– Bad – assumes normal, stable correlation
• Monte Carlo simulation– Good – flexible (can use any distribution in theory)– Bad – depends on model calibration
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Limits
• 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
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Optimization
Maximize f(X)Subject to: Ax ≤ bx ≥ 0
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Minimize VarianceMarkowitz extreme
Min Var [Y]Subject to: Pr{Ax ≤ b} ≥ α∑ x = limit = to avoid null solutionx ≥ 0
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Chance Constrained Model
• Maximize the expected value of a probabilistic function
Maximize E[Y] (where Y = f(X))Subject to: ∑ x = limitPr{Ax ≤ b} ≥ αx ≥ 0
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Maximize Probability
Max Pr{Y ≥ target}Subject to: ∑ x = limitPr{Ax ≤ b} ≥ αx ≥ 0
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Minimize VaR
Min LossSubject to: ∑ x = limitLoss = initial value - z1-α √[var-covar] + E[return]
where z1-α is in the lower tail, α= 0.99
x ≥ 0
• Equivalent to the worst you could experience at the given level
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Demonstration DataStock S Bond B SCIP G
Average return 0.148 0.060 0.152Variance 0.014697 0.000155 0.160791Covariance with S
0.000468 -0.002222
Covariance with B
-0.000227
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Maximize Expected Value of Probabilistic Function
• The objective is to maximize return:Expected return = 0.148 S + 0.060 B + 0.152 G• subject to staying within budget:Budget = 1 S + 1 B + 1 G ≤ 1000Pr{Expected return ≥ 0} ≥ αS, B, G ≥ 0
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SolutionsProbability{return≥0}
α Stock Bond Gamble Expected return
0.50 0 - - 1000.00 152.000.80 0.253 379.91 - 620.09 150.480.90 0.842 556.75 - 443.25 149.770.95 1.282 622.18 - 377.82 149.510.99 2.054 668.92 - 331.08 149.32
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Minimize Variance
Min 0.014697S2 + 0.000936SB - 0.004444SG + 0.000155B2 - 0.000454BG + 0.160791G2
st S + B + G 1000 budget constraint 0.148 S + 0.06 B + 0.152 G ≥ 50 • S, B, G ≥ 0
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SolutionsSpecified
GainVariance Stock Bond Gamble
≥50 106.00 - 825.30 3.17≥100 2,928.51 406.31 547.55 46.14≥150 42,761 500.00 - 500.00≥152 160,791 - - 1,000.00
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Max Probabilityα Stock Bond Gamble Expected
return3 157.84 821.59 20.57 75.784 73.21 914.93 11.86 67.53
4.5 406.31 547.55 46.14 64.174.8 500.00 - 500.00 61.48
4.9 and up - - - 0
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Real Stock Data – Student-t fit
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Logistic fit
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Daily Data: Gains
Ford IBM Pfizer SAP WalMart XOM S&P
Mean 1.00084 1.00033 0.99935 0.99993 1.00021 1.00012 0.99952
Std. Dev 0.03246 0.02257 0.02326 0.03137 0.02102 0.02034 0.01391
Min 0.62822 0.49101 0.34294 0.81797 0.53203 0.51134 0.90965
Max 1.29518 1.13160 1.10172 1.33720 1.11073 1.17191 1.11580
Cov(Ford) 0.00105 0.00019 0.00014 0.00020 0.00016 0.00015 0.00022
Cov(IBM) 0.00051 0.00009 0.00016 0.00013 0.00012 0.00018
Cov(Pfizer) 0.00054 0.00011 0.00014 0.00014 0.00014
Cov(SAP) 0.00098 0.00010 0.00016 0.00016
Cov(WM) 0.00044 0.00011 0.00014
Cov(XOM) 0.00041 0.00015
Cov(S&P) 0.00019
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ResultsModel Ford IBM Pfizer SAP WM XOM S&P Return Stdev
Max Return 1000.000
- - - - - - 1000.84 32.404
Min Variance - 45.987 90.869 30.811 127.508 116.004 588.821 999.76 13.156
NormalPr{>970}>.95
398.381 283.785
- - 222.557 95.277 - 1000.49 18.534
t Pr{>970}>.95
607.162 296.818
- - 96.020 - - 1000.63 23.035
t Pr{>970}>.95Pr{>980}>.9
581.627 301.528
- - 116.845 - - 1000.61 22.475
t Pr{>970}>.95Pr{>980}>.9Pr{>990}>.8
438.405 279.287
- - 220.254 62.054 - 1000.51 19.320
Max Pr{>1000}
16.275 109.867
105.586 38.748 174.570 172.244 382.711 999.91 13.310
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