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Economic Scenario Generator Ahmed Blanco, Caylee Chunga, Branden Diniz, Brittany Mowe, Bowei Wei Advisors: Jon Abraham Barry Posterro.

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Presentation on theme: "Economic Scenario Generator Ahmed Blanco, Caylee Chunga, Branden Diniz, Brittany Mowe, Bowei Wei Advisors: Jon Abraham Barry Posterro."— Presentation transcript:

1 Economic Scenario Generator Ahmed Blanco, Caylee Chunga, Branden Diniz, Brittany Mowe, Bowei Wei Advisors: Jon Abraham Barry Posterro

2 Worcester Polytechnic Institute Overview What is an Economic Scenario Generator? Basics ─ Regime Switching (Markov Chains) ─ Inverse Transform Methods Calibration ─ Maximum Likelihood Estimation ─ Covariance Matrix ─ Cholesky Decomposition Our ESG ─ Differences ─ Results ─ Recommendations

3 Worcester Polytechnic Institute What is an Economic Scenario Generator? Model that simulates correlated returns of multiple assets Life Insurance Companies ─ Asset Liability Management Property and Casualty Insurance Companies ─ Dynamic Financial Analysis Banks ─ Balance Sheet Management

4 Worcester Polytechnic Institute Regime Switching (Markov Chains) A system of multiple states that switch based on fixed probabilities Growing regime and falling regime Movement between states is determined by random numbers and an application of inverse transform method Starting in Regime 1 Starting in Regime 2 Ending in Regime 1.8.65 Ending in Regime 2.2.35 Transition Matrix Sample Regime Switching Starting in Regime Random Number Ending in Regime 1.769981 1.828372 2.217922 2.571011 ………

5 Worcester Polytechnic Institute Regime Switching (Markov Chains) A system of multiple states that switch based on fixed probabilities Growing regime and falling regime Movement between states is determined by random numbers and an application of inverse transform method Starting in Regime 1 Starting in Regime 2 Ending in Regime 1.8.65 Ending in Regime 2.2.35 Transition Matrix Sample Regime Switching Starting in Regime Random Number Ending in Regime 1.769981 1.828372 2.217922 2.571011 ………

6 Worcester Polytechnic Institute Regime Switching (Markov Chains) A system of multiple states that switch based on fixed probabilities Growing regime and falling regime Movement between states is determined by random numbers and an application of inverse transform method Starting in Regime 1 Starting in Regime 2 Ending in Regime 1.8.65 Ending in Regime 2.2.35 Transition Matrix Sample Regime Switching Starting in Regime Random Number Ending in Regime 1.769981 1.828372 2.217922 2.571011 ………

7 Worcester Polytechnic Institute Simulated Returns

8 Worcester Polytechnic Institute Inverse Transform Methods (Continuous) Cumulative Distribution Function (CDF) of the Normal Distribution

9 Worcester Polytechnic Institute Inverse Transform Methods (Continuous) Cumulative Distribution Function (CDF) of the Normal Distribution Begin with a uniform random number on (0,1) and use the Inverse Transform method to develop a random number that is normally distributed.

10 Worcester Polytechnic Institute Cumulative Distribution Function (CDF) of the Normal Distribution Inverse Transform Methods (Continuous) µ=6 0.80 F -1 (0.8)=9 This is the random number in the Normal Distribution resulting from a uniform random number of 0.80

11 Worcester Polytechnic Institute Inverse Transform Methods Starting in Regime Random Number for Regime Ending in Regime Random Number for Return Transformed Number 1.769981.65868.05660 1.828372.27794.02139 2.917322.35738.02179 2.571011.66318.04371 ………… … Regime 1Regime 2.05-.02.01.05 Starting in Regime 1 Starting in Regime 2 Ending in Regime 1.8.65 Ending in Regime 2.2.35

12 Worcester Polytechnic Institute Inverse Transform Methods Starting in Regime Random Number for Regime Ending in Regime Random Number for Return Transformed Number 1.769981.65868.05660 1.828372.27794.02139 2.917322.35738.02179 2.571011.66318.04371 ………… … Regime 1Regime 2.05-.02.01.05 Starting in Regime 1 Starting in Regime 2 Ending in Regime 1.8.65 Ending in Regime 2.2.35

13 Worcester Polytechnic Institute Inverse Transform Methods Starting in Regime Random Number for Regime Ending in Regime Random Number for Return Transformed Number 1.769981.65868.05660 1.828372.27794.02139 2.917322.35738.02179 2.571011.66318.04371 ………… … Regime 1Regime 2.05-.02.01.05 Starting in Regime 1 Starting in Regime 2 Ending in Regime 1.8.65 Ending in Regime 2.2.35

14 Worcester Polytechnic Institute Definition: A method of estimating the parameters of a model given data. ─ In other words, finding the values of the parameter set with the highest probability of resulting in the observations. Calibration (Maximum Likelihood Estimation)

15 Worcester Polytechnic Institute Calibration (Maximum Likelihood Estimation)

16 Worcester Polytechnic Institute The set up for our MLE: Calibration (Maximum Likelihood Estimation)

17 Worcester Polytechnic Institute Calibration (Maximum Likelihood Estimation)

18 Worcester Polytechnic Institute Calibration (Maximum Likelihood Estimation)

19 Worcester Polytechnic Institute Calibration (Maximum Likelihood Estimation)

20 Worcester Polytechnic Institute Calibration (Maximum Likelihood Estimation) Now we begin the recursion… Currently in Regime 1Currently in Regime 2 Previously in R1Previously in R2Previously in R1Previously in R2

21 Worcester Polytechnic Institute Probability of being in the previous regime ─ Regime 1’s contribution to the pdf Regime switching probability Calibration (Maximum Likelihood Estimation)

22 Worcester Polytechnic Institute Multiply by the normal pdf… Calibration (Maximum Likelihood Estimation)

23 Worcester Polytechnic Institute Now in Regime 1, Previously in Regime 2 Calibration (Maximum Likelihood Estimation)

24 Worcester Polytechnic Institute Now in Regime 2, Previously in Regime 1 Calibration (Maximum Likelihood Estimation)

25 Worcester Polytechnic Institute Now in Regime 2, Previously in Regime 2 Calibration (Maximum Likelihood Estimation)

26 Worcester Polytechnic Institute Calibration (Maximum Likelihood Estimation) Estimated pdf values

27 Worcester Polytechnic Institute Calibration (Maximum Likelihood Estimation) Natural log of estimated pdf values

28 Worcester Polytechnic Institute Calibration (Maximum Likelihood Estimation) Parameters which we solved for using Excel’s built-in “Solver” Metric we maximized to by solving for the parameters

29 Worcester Polytechnic Institute Where Next?

30 Worcester Polytechnic Institute Covariance Matrix

31 Worcester Polytechnic Institute Cholesky Decomposition Theorem: every symmetric positive definite matrix can be decomposed into a product of a unique lower triangular matrix (the Cholesky factor) and its transpose.

32 Worcester Polytechnic Institute Cholesky Example A = Decompose the matrix to get L and L T L =and L T = 0.0150.009 0.048 0.1220 0.0730.206 0.1220.073 00.206

33 Worcester Polytechnic Institute Cholesky Example A = Decompose the matrix to get L and L T L =and L T = X = 0.0150.009 0.048 0.1220 0.0730.206 0.1220.073 00.206 Stock 1Stock 2 2.3072.629 0.710-0.796 1.614-0.163 -0.2650.494 0.1220.073 00.206

34 Worcester Polytechnic Institute Cholesky Example A = Decompose the matrix to get L and L T L =and L T = Covariance Matrix X =  0.0150.009 0.048 0.1220 0.0730.206 0.1220.073 00.206 Stock 1Stock 2 2.3072.629 0.710-0.796 1.614-0.163 -0.2650.494 0.1220.073 00.206 Stock 1Stock 2 0.2830.712 0.087-0.112 0.1980.085 -0.0320.082 0.0150.009 0.048

35 Worcester Polytechnic Institute The ESG A 1 =L 1 T = A 2 =L 2 T = X OR = 0.0150.009 0.048 0.1220.073 00.206 RegimeStock 1Stock 2 12.3072.629 10.710-0.796 21.614-0.163 2-0.2650.494 0.1220.073 00.206 RegimeStock 1Stock 2 10.2830.712 10.087-0.112 2 0.1610.025 2 -0.0270.064 0.1000.030 00.145 0.0100.003 0.022 0.1000.030 00.145

36 Worcester Polytechnic Institute The ESG A 1 =L 1 T = A 2 =L 2 T = X OR = 0.0150.009 0.048 0.1220.073 00.206 RegimeStock 1Stock 2 1 2.3072.629 10.710-0.796 21.614-0.163 2-0.2650.494 0.1220.073 00.206 RegimeStock 1Stock 2 1 0.2830.712 10.087-0.112 2 0.1610.025 2 -0.0270.064 0.1000.030 00.145 0.0100.003 0.022 0.1000.030 00.145

37 Worcester Polytechnic Institute The ESG A 1 =L 1 T = A 2 =L 2 T = X OR = 0.0150.009 0.048 0.1220.073 00.206 RegimeStock 1Stock 2 12.3072.629 10.710-0.796 2 1.614-0.163 2-0.2650.494 0.1220.073 00.206 RegimeStock 1Stock 2 10.2830.712 10.087-0.112 2 0.1610.025 2 -0.0270.064 0.1000.030 00.145 0.0100.003 0.022 0.1000.030 00.145

38 Worcester Polytechnic Institute Our ESG Uses 3 regimes instead of 2 ─ 3 rd regime represents an economic crash and occurs rarely ─ µ 3, σ 3, third Covariance Matrix Uses 10 Exchange Traded Funds (ETFs) ─ ETFs track groups of stocks Outputs Daily Returns

39 Worcester Polytechnic Institute Results: Mean & St. Dev. Difference Regime 1μ 1 Parameterμ 1 Simulationσ 1 Parameterσ 1 Simulation SPY 0.000800.000790.007510.00726 VXX -0.00679-0.006740.025170.03496 EFA 0.000640.000630.008990.00987 OIL 0.00004 0.014660.01812 FEZ 0.000570.000550.012590.01379 EEM 0.000280.000270.011380.01149 HYG 0.00015 0.003280.00409 TLT -0.00005-0.000060.012100.00871 IWM -0.00043-0.000450.024310.01007 GLD -0.00034-0.000330.014740.01019

40 Worcester Polytechnic Institute Results: Covariance Difference Simulated Covariance Matrix Actual Covariance Matrix Regime 1 SPYVXXEFAOILFEZEEMHYGTLTIWMGLD SPY 0.000053-0.0001910.0000600.0000470.0000800.0000650.000017-0.0000250.0000640.000008 VXX -0.0001910.001222-0.000223-0.000145-0.000297-0.000232-0.0000680.000092-0.000241-0.000021 EFA 0.000060-0.0002230.0000970.0000710.0001250.0000930.000022-0.0000290.0000730.000023 OIL 0.000047-0.0001450.0000710.0003280.0000900.0000840.000020-0.0000350.0000600.000053 FEZ 0.000080-0.0002970.0001250.0000900.0001900.0001160.000028-0.0000420.0000950.000024 EEM 0.000065-0.0002320.0000930.0000840.0001160.0001320.000024-0.0000280.0000800.000030 HYG 0.000017-0.0000680.0000220.0000200.0000280.0000240.000017-0.0000040.0000210.000006 TLT -0.0000250.000092-0.000029-0.000035-0.000042-0.000028-0.0000040.000076-0.0000290.000010 IWM 0.000064-0.0002410.0000730.0000600.0000950.0000800.000021-0.0000290.0001010.000013 GLD 0.000008-0.0000210.0000230.0000530.0000240.0000300.0000060.0000100.0000130.000104 Regime 1 SPYVXXEFAOILFEZEEMHYGTLTIWMGLD SPY 0.000053-0.0001910.0000600.0000470.0000800.0000650.000017-0.0000250.0000640.000008 VXX -0.0001910.001224-0.000224-0.000146-0.000298-0.000233-0.0000680.000092-0.000241-0.000021 EFA 0.000060-0.0002240.0000970.0000700.0001250.0000930.000022-0.0000290.0000730.000023 OIL 0.000047-0.0001460.0000700.0003280.0000900.0000840.000020-0.0000350.0000600.000053 FEZ 0.000080-0.0002980.0001250.0000900.0001900.0001160.000028-0.0000420.0000950.000024 EEM 0.000065-0.0002330.0000930.0000840.0001160.0001320.000024-0.0000280.0000800.000030 HYG 0.000017-0.0000680.0000220.0000200.0000280.0000240.000017-0.0000040.0000210.000005 TLT -0.0000250.000092-0.000029-0.000035-0.000042-0.000028-0.0000040.000076-0.0000290.000010 IWM 0.000064-0.0002410.0000730.0000600.0000950.0000800.000021-0.0000290.0001010.000013 GLD 0.000008-0.0000210.0000230.0000530.0000240.0000300.0000050.0000100.0000130.000104

41 Worcester Polytechnic Institute Results: Covariance Difference Simulated Covariance Matrix Actual Covariance Matrix Regime 1 SPYVXXEFAOILFEZEEMHYGTLTIWMGLD SPY 0.000053-0.0001910.0000600.0000470.0000800.0000650.000017-0.0000250.0000640.000008 VXX -0.0001910.001222-0.000223-0.000145-0.000297-0.000232-0.0000680.000092-0.000241-0.000021 EFA 0.000060-0.0002230.0000970.0000710.0001250.0000930.000022-0.0000290.0000730.000023 OIL 0.000047-0.0001450.0000710.0003280.0000900.0000840.000020-0.0000350.0000600.000053 FEZ 0.000080-0.0002970.0001250.0000900.0001900.0001160.000028-0.0000420.0000950.000024 EEM 0.000065-0.0002320.0000930.0000840.0001160.0001320.000024-0.0000280.0000800.000030 HYG 0.000017-0.0000680.0000220.0000200.0000280.0000240.000017-0.0000040.0000210.000006 TLT -0.0000250.000092-0.000029-0.000035-0.000042-0.000028-0.0000040.000076-0.0000290.000010 IWM 0.000064-0.0002410.0000730.0000600.0000950.0000800.000021-0.0000290.0001010.000013 GLD 0.000008-0.0000210.0000230.0000530.0000240.0000300.0000060.0000100.0000130.000104 Regime 1 SPYVXXEFAOILFEZEEMHYGTLTIWMGLD SPY 0.000053-0.0001910.0000600.0000470.0000800.0000650.000017-0.0000250.0000640.000008 VXX -0.0001910.001224-0.000224-0.000146-0.000298-0.000233-0.0000680.000092-0.000241-0.000021 EFA 0.000060-0.0002240.0000970.0000700.0001250.0000930.000022-0.0000290.0000730.000023 OIL 0.000047-0.0001460.0000700.0003280.0000900.0000840.000020-0.0000350.0000600.000053 FEZ 0.000080-0.0002980.0001250.0000900.0001900.0001160.000028-0.0000420.0000950.000024 EEM 0.000065-0.0002330.0000930.0000840.0001160.0001320.000024-0.0000280.0000800.000030 HYG 0.000017-0.0000680.0000220.0000200.0000280.0000240.000017-0.0000040.0000210.000005 TLT -0.0000250.000092-0.000029-0.000035-0.000042-0.000028-0.0000040.000076-0.0000290.000010 IWM 0.000064-0.0002410.0000730.0000600.0000950.0000800.000021-0.0000290.0001010.000013 GLD 0.000008-0.0000210.0000230.0000530.0000240.0000300.0000050.0000100.0000130.000104

42 Worcester Polytechnic Institute Results: Covariance Difference Simulated Covariance Matrix Actual Covariance Matrix Regime 1 SPYVXXEFAOILFEZEEMHYGTLTIWMGLD SPY 0.000053-0.0001910.0000600.0000470.0000800.0000650.000017-0.0000250.0000640.000008 VXX -0.0001910.001222-0.000223-0.000145-0.000297-0.000232-0.0000680.000092-0.000241-0.000021 EFA 0.000060-0.0002230.0000970.0000710.0001250.0000930.000022-0.0000290.0000730.000023 OIL 0.000047-0.0001450.0000710.0003280.0000900.0000840.000020-0.0000350.0000600.000053 FEZ 0.000080-0.0002970.0001250.0000900.0001900.0001160.000028-0.0000420.0000950.000024 EEM 0.000065-0.0002320.0000930.0000840.0001160.0001320.000024-0.0000280.0000800.000030 HYG 0.000017-0.0000680.0000220.0000200.0000280.0000240.000017-0.0000040.0000210.000006 TLT -0.0000250.000092-0.000029-0.000035-0.000042-0.000028-0.0000040.000076-0.0000290.000010 IWM 0.000064-0.0002410.0000730.0000600.0000950.0000800.000021-0.0000290.0001010.000013 GLD 0.000008-0.0000210.0000230.0000530.0000240.0000300.0000060.0000100.0000130.000104 Regime 1 SPYVXXEFAOILFEZEEMHYGTLTIWMGLD SPY 0.000053-0.0001910.0000600.0000470.0000800.0000650.000017-0.0000250.0000640.000008 VXX -0.0001910.001224-0.000224-0.000146-0.000298-0.000233-0.0000680.000092-0.000241-0.000021 EFA 0.000060-0.0002240.0000970.0000700.0001250.0000930.000022-0.0000290.0000730.000023 OIL 0.000047-0.0001460.0000700.0003280.0000900.0000840.000020-0.0000350.0000600.000053 FEZ 0.000080-0.0002980.0001250.0000900.0001900.0001160.000028-0.0000420.0000950.000024 EEM 0.000065-0.0002330.0000930.0000840.0001160.0001320.000024-0.0000280.0000800.000030 HYG 0.000017-0.0000680.0000220.0000200.0000280.0000240.000017-0.0000040.0000210.000005 TLT -0.0000250.000092-0.000029-0.000035-0.000042-0.000028-0.0000040.000076-0.0000290.000010 IWM 0.000064-0.0002410.0000730.0000600.0000950.0000800.000021-0.0000290.0001010.000013 GLD 0.000008-0.0000210.0000230.0000530.0000240.0000300.0000050.0000100.0000130.000104

43 Worcester Polytechnic Institute Recommendations Create a user-friendly interface Alternative platforms ─ Matlab would allow implementation on the WPI supercomputer Output results to a.txt file ─ Avoids Excel’s row limitations Introduce an automatic results checker More experimentation with regime 3 Improve run time

44 Thank you for listening! Questions?

45 Worcester Polytechnic Institute Appendix I – Our 3 rd Regime Mean: ─ Twice the mean of Regime 2 Standard Deviation: ─.5 times the Std. Dev. of Regime 2 Covariance: ─.5 times the Covariances of Regime 2

46 Worcester Polytechnic Institute Regime 1μ 1 Parameterμ 1 SimulationDifferencesσ 1 Parameterσ 1 SimulationDifferences SPY 0.000800.000797.37E-060.007510.00726-0.00024 VXX -0.00679-0.00674-4.2E-050.025170.034960.009786 EFA 0.000640.000631.11E-050.008990.009870.000874 OIL 0.00004 -6E-060.014660.018120.003458 FEZ 0.000570.000551.7E-050.012590.013790.001194 EEM 0.000280.000271.07E-050.011380.011490.000107 HYG 0.00015 4.33E-070.003280.004090.000821 TLT -0.00005-0.000064.95E-060.012100.00871-0.0034 IWM -0.00043-0.000451.24E-050.024310.01007-0.01424 GLD -0.00034-0.00033-7.5E-060.014740.01019-0.00455 Results: Mean & St. Dev. Difference With Differences Appendix II – Mean & St. Dev. Differences

47 Worcester Polytechnic Institute Minimum: 1.81E-09 Maximum: 1.5E-06 Regime 1SPYVXXEFAOILFEZEEMHYGTLTIWMGLD SPY 0.000000040.000000240.000000040.000000100.00000001 0.000000000.00000001 0.00000020 VXX 0.000000240.000001510.000000510.000000130.000000950.000000490.000000070.000000390.000000370.00000014 EFA 0.000000040.000000510.000000080.000000250.000000070.00000002 0.000000030.000000000.00000022 OIL 0.000000100.000000130.000000250.000000500.000000570.000000020.000000010.000000020.000000110.00000019 FEZ 0.000000010.000000950.000000070.000000570.000000060.00000001 0.000000000.000000050.00000032 EEM 0.000000010.000000490.00000002 0.000000010.000000080.000000060.000000010.000000000.00000021 HYG 0.000000000.000000070.000000020.00000001 0.000000060.000000010.000000070.000000020.00000007 TLT 0.000000010.000000390.000000030.000000020.000000000.000000010.000000070.000000010.000000160.00000010 IWM 0.000000010.000000370.000000000.000000110.000000050.000000000.000000020.000000160.000000030.00000025 GLD 0.000000200.000000140.000000220.000000190.000000320.000000210.000000070.000000100.000000250.00000007 Appendix III – Covariance Differences

48 Worcester Polytechnic Institute Appendix IV – ETFs Used ETF/ETNSPYIWMTLTHYGGLD Underlying Index/ Commodity S&P 500Russel 2000 Barclays U.S. 20+ Year Treasury Bonds Markit iBoxx USD Liquid High Yield Gold bullions spot price Features of the Index Largest 500 U.S. companies Smallest 2000 companies in the Russel 3000 index of small-cap equities U.S. Treasury Bonds that will not reach maturity for twenty or more years High yield corporate bonds for sale in the U.S. Bars of gold with a purity of 99.5% or higher ETF/ETNEFAVXXOILFEZEEM Underlying Index/ Commodity MSCI EAFE S&P 500 VIX Short-Term Futures S&P GSCI Crude Oil Total Return EURO STOXX 50 MSCI Emerging Markets Features of the Index Large-cap and medium-cap equities CBOE Volatility Index which measures the volatility of S&P 500 futures Returns of oil futures contracts with West Texas Intermediate 50 of the largest and most liquid Eurozone stocks Medium-cap and large-cap equities from emerging markets

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