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

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

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

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 Ending in Regime Transition Matrix Sample Regime Switching Starting in Regime Random Number Ending in Regime ………

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 Ending in Regime Transition Matrix Sample Regime Switching Starting in Regime Random Number Ending in Regime ………

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 Ending in Regime Transition Matrix Sample Regime Switching Starting in Regime Random Number Ending in Regime ………

Worcester Polytechnic Institute Simulated Returns

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

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.

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

Worcester Polytechnic Institute Inverse Transform Methods Starting in Regime Random Number for Regime Ending in Regime Random Number for Return Transformed Number ………… … Regime 1Regime Starting in Regime 1 Starting in Regime 2 Ending in Regime Ending in Regime

Worcester Polytechnic Institute Inverse Transform Methods Starting in Regime Random Number for Regime Ending in Regime Random Number for Return Transformed Number ………… … Regime 1Regime Starting in Regime 1 Starting in Regime 2 Ending in Regime Ending in Regime

Worcester Polytechnic Institute Inverse Transform Methods Starting in Regime Random Number for Regime Ending in Regime Random Number for Return Transformed Number ………… … Regime 1Regime Starting in Regime 1 Starting in Regime 2 Ending in Regime Ending in Regime

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)

Worcester Polytechnic Institute Calibration (Maximum Likelihood Estimation)

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

Worcester Polytechnic Institute Calibration (Maximum Likelihood Estimation)

Worcester Polytechnic Institute Calibration (Maximum Likelihood Estimation)

Worcester Polytechnic Institute Calibration (Maximum Likelihood Estimation)

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

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

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

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

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

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

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

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

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

Worcester Polytechnic Institute Where Next?

Worcester Polytechnic Institute Covariance Matrix

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.

Worcester Polytechnic Institute Cholesky Example A = Decompose the matrix to get L and L T L =and L T =

Worcester Polytechnic Institute Cholesky Example A = Decompose the matrix to get L and L T L =and L T = X = Stock 1Stock

Worcester Polytechnic Institute Cholesky Example A = Decompose the matrix to get L and L T L =and L T = Covariance Matrix X =  Stock 1Stock Stock 1Stock

Worcester Polytechnic Institute The ESG A 1 =L 1 T = A 2 =L 2 T = X OR = RegimeStock 1Stock RegimeStock 1Stock

Worcester Polytechnic Institute The ESG A 1 =L 1 T = A 2 =L 2 T = X OR = RegimeStock 1Stock RegimeStock 1Stock

Worcester Polytechnic Institute The ESG A 1 =L 1 T = A 2 =L 2 T = X OR = RegimeStock 1Stock RegimeStock 1Stock

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

Worcester Polytechnic Institute Results: Mean & St. Dev. Difference Regime 1μ 1 Parameterμ 1 Simulationσ 1 Parameterσ 1 Simulation SPY VXX EFA OIL FEZ EEM HYG TLT IWM GLD

Worcester Polytechnic Institute Results: Covariance Difference Simulated Covariance Matrix Actual Covariance Matrix Regime 1 SPYVXXEFAOILFEZEEMHYGTLTIWMGLD SPY VXX EFA OIL FEZ EEM HYG TLT IWM GLD Regime 1 SPYVXXEFAOILFEZEEMHYGTLTIWMGLD SPY VXX EFA OIL FEZ EEM HYG TLT IWM GLD

Worcester Polytechnic Institute Results: Covariance Difference Simulated Covariance Matrix Actual Covariance Matrix Regime 1 SPYVXXEFAOILFEZEEMHYGTLTIWMGLD SPY VXX EFA OIL FEZ EEM HYG TLT IWM GLD Regime 1 SPYVXXEFAOILFEZEEMHYGTLTIWMGLD SPY VXX EFA OIL FEZ EEM HYG TLT IWM GLD

Worcester Polytechnic Institute Results: Covariance Difference Simulated Covariance Matrix Actual Covariance Matrix Regime 1 SPYVXXEFAOILFEZEEMHYGTLTIWMGLD SPY VXX EFA OIL FEZ EEM HYG TLT IWM GLD Regime 1 SPYVXXEFAOILFEZEEMHYGTLTIWMGLD SPY VXX EFA OIL FEZ EEM HYG TLT IWM GLD

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

Thank you for listening! Questions?

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

Worcester Polytechnic Institute Regime 1μ 1 Parameterμ 1 SimulationDifferencesσ 1 Parameterσ 1 SimulationDifferences SPY E VXX E EFA E OIL E FEZ E EEM E HYG E TLT E IWM E GLD E Results: Mean & St. Dev. Difference With Differences Appendix II – Mean & St. Dev. Differences

Worcester Polytechnic Institute Minimum: 1.81E-09 Maximum: 1.5E-06 Regime 1SPYVXXEFAOILFEZEEMHYGTLTIWMGLD SPY VXX EFA OIL FEZ EEM HYG TLT IWM GLD Appendix III – Covariance Differences

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