1. 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

2. Value at Risk Maximum expected loss given time horizon, confidence interval Finland 2010

3. VaR = 0.64 expect to exceed 99% of time in 1 year Here loss = 10 – 0.64 = 9.36 Finland 2010

4. Use Basel Capital Accord – Banks encouraged to use internal models to measure VaR – Use to ensure capital adequacy (liquidity) – Compute daily at 99 th 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 Finland 2010

5. 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 Finland 2010

6. 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 Finland 2010

7. Optimization Maximize f(X) Subject to: Ax ≤ b x ≥ 0 Finland 2010

8. Minimize Variance Markowitz extreme Min Var [Y] Subject to: Pr{Ax ≤ b} ≥ α ∑ x = limit= to avoid null solution x ≥ 0 Finland 2010

9. Chance Constrained Model Maximize the expected value of a probabilistic function Maximize E[Y] (where Y = f(X)) Subject to: ∑ x = limit Pr{Ax ≤ b} ≥ α x ≥ 0 Finland 2010

10. Maximize Probability Max Pr{Y ≥ target} Subject to: ∑ x = limit Pr{Ax ≤ b} ≥ α x ≥ 0 Finland 2010

11. Minimize VaR Min Loss Subject to: ∑ x = limit Loss = initial value - z 1-α √[var-covar] + E[return] where z 1-α is in the lower tail, α= 0.99 x ≥ 0 Equivalent to the worst you could experience at the given level Finland 2010

12. Demonstration Data Stock SBond BSCIP G Average return0.1480.0600.152 Variance0.0146970.0001550.160791 Covariance with S 0.000468-0.002222 Covariance with B -0.000227 Finland 2010

13. 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 ≤ 1000 Pr{Expected return ≥ 0} ≥ α S, B, G ≥ 0 Finland 2010

14. Solutions Probability {return≥0} αStockBondGambleExpected return 0.500--1000.00152.00 0.800.253379.91-620.09150.48 0.900.842556.75-443.25149.77 0.951.282622.18-377.82149.51 0.992.054668.92-331.08149.32 Finland 2010

15. Minimize Variance Min 0.014697S 2 + 0.000936SB - 0.004444SG + 0.000155B 2 - 0.000454BG + 0.160791G 2 st S + B + G  1000budget constraint 0.148 S + 0.06 B + 0.152 G ≥ 50 S, B, G ≥ 0 Finland 2010

16. Solutions Specified Gain VarianceStockBondGamble ≥50106.00-825.303.17 ≥1002,928.51406.31547.5546.14 ≥15042,761500.00- ≥152160,791--1,000.00 Finland 2010

17. Max Probability αStockBondGambleExpected return 3157.84821.5920.5775.78 473.21914.9311.8667.53 4.5406.31547.5546.1464.17 4.8500.00- 61.48 4.9 and up---0 Finland 2010

18. Real Stock Data – Student-t fit Finland 2010

19. Logistic fit Finland 2010

20. Daily Data: Gains FordIBMPfizerSAPWalMartXOMS&P Mean1.000841.000330.999350.999931.000211.000120.99952 Std. Dev0.032460.022570.023260.031370.021020.020340.01391 Min0.628220.491010.342940.817970.532030.511340.90965 Max1.295181.131601.101721.337201.110731.171911.11580 Cov(Ford)0.001050.000190.000140.000200.000160.000150.00022 Cov(IBM)0.000510.000090.000160.000130.000120.00018 Cov(Pfizer)0.000540.000110.00014 Cov(SAP)0.000980.000100.00016 Cov(WM)0.000440.000110.00014 Cov(XOM)0.000410.00015 Cov(S&P)0.00019 Finland 2010

21. Results ModelFordIBMPfizerSAPWMXOMS&PReturnStdev Max Return1000.00 0 ------1000.8432.404 Min Variance-45.98790.86930.811127.508116.004588.821999.7613.156 Normal Pr{>970}>.95 398.381283.785--222.55795.277-1000.4918.534 t Pr{>970}>.95 607.162296.818--96.020--1000.6323.035 t Pr{>970}>.95 Pr{>980}>.9 581.627301.528--116.845--1000.6122.475 t Pr{>970}>.95 Pr{>980}>.9 Pr{>990}>.8 438.405279.287--220.25462.054-1000.5119.320 Max Pr{>1000} 16.275109.867105.58638.748174.570172.244382.711999.9113.310 Finland 2010