Applying Stochastic Programs to Improve Investor Performance

1. Applying Stochastic Programs to Improve Investor Performance Professor John M. Mulvey Bendheim Center for Finance Department of Operations Research & Financial Engineering Princeton University Senior Consultant: Towers Perrin–Tillinghast, Mt. Lucas Management Hedge Fund, Rydex Investments June 27, 2007

2. Outline • Advanced Portfolio Theory Alternative optimization frameworks Success stories Advantages of multi-period portfolio models Optimize overlay strategies in a global hedge fund Novel sources for diversification (momentum?) • Optimize Pension Trust and Sponsoring Company Status of pension trusts in U.S. ALM model equations Hybrid approach Link stochastic program <-> policy simulator • Future Research • Active managers and momentum strategies • Other issues

3. Alternative Frameworks for Optimization A. Dynamic stochastic control Discretize state space Solve via dynamic program B. Multi-stage stochastic program Discretize uncertainties (scenario tree) C. Policy simulation (with optimization) Given a policy rule (s) Apply across scenarios via Monte Carlo simulation D. Hybrid strategies Stylized stochastic program (discover sound rules) Evaluate in a comprehensive policy simulator

4. Success Stories: Leading U.S. University Endowments • Harvard ($29b), Yale ($18b), Princeton ($13b) 16-22 % annual return over past decade • How? • Stress private markets (e.g. private equity, venture capital, hedge funds) • Wide diversification • Novel asset classes (timber, tips, structured products) • Uncorrelated return patterns (when possible) • Employ leverage with careful risk management • Reference: David Swensen Pioneering Portfolio Management, Free Press, 2000, CIO-Yale U.

5. Princeton Policy Portfolio (2005)

6. Success Stories:Global (Re) Insurance Companies • Enterprise risk management (ERM/DFA) • AXA (Paris) • Renaissance Reinsurance (Bermuda) • Geo Vera (California and Florida) • How? • Add new insurance activities to gain diversification benefits and increased profit (higher returns and reduced risks) • Search for businesses across the global • wide diversification • Lower capital requirement due to reduced loss tail • Greater profits

7. Success Stories:Defined Benefit Pension Trusts (few and far between) • Goal: Maintain fund surplus and grow assets with superior performance • Example: Kodak Pension Plan (School of Hard Knocks, R. Olson) • How? Discover assets with relatively high volatility and good performance during economic downturns -strips (zero coupon government bonds) Rebalance portfolio to achieve rebalancing gains Advanced concept – apply overlay strategies

8. Why Dynamic (Multi-Period) Portfolio Models?(J. of Portfolio Management, Winter 2003, Summer 2004) • Advantages • Greater realism (transaction costs, contribution, borrowing) • Addresses temporal issues (short vs. long horizons) • Greater performance (rebalancing gains)

9. Dynamic Diversification • Given assets with identical growth =15%/year and volatility = 20%, independent • Combine ten assets (equal weights, rebalanced monthly) • Total portfolio return = 15% + 1.8% (excess rebalancing gains) • If Vol = 40%  portfolio return = 15 + 7.2% • If Vol = 60%  portfolio return = 15 + 16.2% Luenberger (1998) “Volatility is not the same as risk. Volatility is an opportunity.”

10. The Rebalancing Decision Initial investment mix Equity up 65% Nikkei, 35% Bonds Option: to sell and purchase assets back to original mix? What should be done? 50% Nikkei, 50% Bonds 45% Nikkei, 55% Bonds Equity down 6 Months Active Rebalancing

11. Durable Policy Rule: Fixed Mix • Target mix each period = Li = % of wealth in asset i, each period • Example: 70% stock, and 30% government bonds (70/30) • Active rebalancing each period (e.g. monthly) • Address Transaction and market impact costs • No-trade-zone: Fixed-Mix Optimization requires Non-convex Solver (or approximation NLP)

12. Simple Example A: Enhance Performance • Typical target portfolio: • 70% Equity • 30% bonds • How to Improve? • Traditional approach – adjust equity/bond mix depending upon forecast (say due to interest rate, volatility, or other triggers) • Add diversifying assets • Additional equity markets, bond categories, real estate, etc. • Overlay strategy via futures contracts • Trend following • Go long or short based on position relative to moving average • Rebalance regularly to get volatility pumping

13. Historical Returns of Asset Classes

14. Advantages of Wide Diversification

15. Example B: Enhance Performance by Non Traditional Strategies • Core portfolio: • 70% U. S. Equity • 30% U. S. Government bonds • Geometric returns = 9.73% (1991 to 2005) • How to Improve? • Traditional approach – adjust equity/bond mix depending upon forecast on scenario tree (stochastic program) • Non-traditional approach – add overlay strategy via futures contracts • Trend following • Go long or short based on position relative to moving average • (rebalance each month)

16. $16.00 Wheat CPI $14.00 $12.00 $10.00 $8.00 $6.00 $4.00 $2.00 $0.00 Jan-96 Jan-97 Jan-98 Jan-99 Jan-00 Jan-01 Jan-02 Jan-03 Jan-04 Jan-61 Jan-62 Jan-63 Jan-64 Jan-65 Jan-66 Jan-67 Jan-68 Jan-69 Jan-70 Jan-71 Jan-72 Jan-73 Jan-74 Jan-75 Jan-76 Jan-77 Jan-78 Jan-79 Jan-80 Jan-81 Jan-82 Jan-83 Jan-84 Jan-85 Jan-86 Jan-87 Jan-88 Jan-89 Jan-90 Jan-91 Jan-92 Jan-93 Jan-94 Jan-95 Do Commodity Prices Always Rise? Commodities and Inflation

17. Why trend follow?Wheat Price Chart 09/1990 – 08/2004

18. MLM Index provides Diversification and Equity like Returns

19. Results of Wide Diversification and Overlay Strategy

20. The Fixed-Mix Rule and an Equity Momentum Strategy (a) • The dynamic diversification strategy is • a fixed mix portfolio of • long-only industry-level momentum strategies with various parameters (evaluation period, holding period) • across five markets that cover the most of the world stock market (US, EU, Europe ex. EU, Japan, Asia ex. Japan). • Over the last 27 years (1980~2006), the performance of the dynamic diversification strategy has been outstanding.

21. Portfolio of Equity Momentum Strategies (b)

22. The Fixed-Mix Rule and Equity Momentum Strategies (c)

23. Actively Managed Funds and the Momentum Strategies (a) • Many actively managed funds have become highly correlated with the momentum strategy after 1992. • A considerable amount of the active funds seem to be adopting the momentum strategy as their stock selection rules, although a deep style analysis is required to conclude so. • Interestingly, the similarity to the momentum strategy is proportionate to fund performance. • Even for the value-oriented funds, which are not supposed to employ aggressive strategies as the momentum strategy, the best performance group has shown similar return patterns as the momentum strategy during the last decade. • Trivia: The key paper on the momentum strategy by Jegadeesh and Titman, “Returns to buying winners and selling losers: implications for stock market efficiency”, was published on 1993!

24. Actively Managed Funds and the Momentum Strategies (b)

25. Actively Managed Funds and the Momentum Strategies (c)

26. Actively Managed Funds and the Momentum Strategies (d)

27. Actively Managed Funds and the Momentum Strategies (e)

28. Actively Managed Funds and the Momentum Strategies (f)

29. Surplus 2. Pension Trusts: ALM Issues A S S E T S L I A B • Pension planning in the US • Defined benefit vs. Defined contribution • S&P 500 DB Pension Plans (Dec 1999  May 2003) • Net surplus of $239 bn  Net deficit of $252 bn • Three primary causes • decline in equity markets  decrease in pension plan assets • decrease in interest rates  increase in pension liabilities • poor planning

30. Pension Trust Stakeholders(Defined Benefit – DB) Pension System (A-L or A/L) Sponsoring company contributions Pay retirees PBGC The public

31. Status of Industries in S&P500

32. Optimize Assets and Liabilities/Goals • Investors seek to maximize the growth of their wealth (capital) -- optimal asset allocation • Financial organizations manage their products (banks, insurance companies) Assets (i) Products (liabilities, j)

33. Modeling Preliminaries • Decisions: • asset mix (proportions of equity, bonds, etc.) • Contributions (from investors or outsiders) • Payment policy (how much to pay and to whom) • Uncertainties: • Returns on assets • Amount and timing of future cashflows • Length of the horizon (insurance or pensions) • Key Decision Variables: • x(i, t, s) amount invested in asset i, time t, scenario s • y(j,t,s) amount of liability or business activity

34. 1 2 3 4 ... T time Horizon Structure of Multi Stage Portfolio Models:Developing an Investment Policy Project state of enterprise across multi-period horizon • Decisions at beginning each stage • Uncertainties occur between decision points • Policy rules or model recommendations guide system • Iterate over all scenarios {S} Decisions

36. CAP:Link : Cascade of Stochastic Processes TreasuryYield Curve General PriceInflation Currencies Real Yields Expected Inflation Wage Inflation Fixed Income Returns Dividend Yields Stock Dividend Growth Rate Stock Returns Other Asset Classes

37. Fundamental Asset Equations(for every scenario) asset j sales purchases cash

38. Integrated Framework

39. Anticipatory Integrated Corporate/Pension Planning Model t=1 t=2 t=T Company CP CP CP cash i=1 i=1 i=1 Pension Plan i=2 i=2 i=2 i=M i=M i=M

40. The Integrated Pension-Corporate Financial Planning Problem as a Multi-stage Stochastic Program

41. Model Structure • The integrated pension and corporate financial planning problem as a multi-stage stochastic program.

42. Empirical Analysis to Assist Pensions Regain Financial Health

43. 1000-Scenario Tree 5000-Scenario Tree Number of variables 68, 000 340,000 Number of constraints 77, 000 385,000 Point on the efficient frontier Max Compromise Min Max Compromise Min Solve Time (seconds) 260 217 178 1380 311 334 Optimization Model for U.S. Department of Labor(Multi-stage stochastic program) * Solver: CPLEX Algorithms: Dual Simplex for LP and QP Barrier

44. Analysis of DB Pension System in S&P 500 • Identify industries with potential problems • Evaluate pension funding and investment decisions • Current conditions and simulation inputs:

45. Analysis of DB Pension System in S&P 500: Pockets of Severe Difficulty

46. Linking Stochastic Program and Policy Simulator Stochastic Program Scenario Tree Model Uncertainties Calibrate and Sample Set out benchmarks Explore improved policy rules Scenarios Policy simulator Complex details

47. Refine Policy Rules For Target Industries Recommendations under Adverse Conditions New Policy Rules Stochastic Program Examine Design Test Improved Policy Rules Policy Simulator

48. Refine Policy Rules For Telecom Services

49. Policy Rules For Telecom Services Switching investment strategies under adverse conditions: • Reduces excessive contributions considerably: - on average: dropping from $5 billion to $3.5 billion - worst case: from $10.6 billion to $7.5 billion • Maintains or improves all other objectives. • Much better results than alternatives (CPPI, etc.)

50. Recommendations for DB Pension Trusts in U.S. • Encourage healthy companies to remain in the DB system • Keep expected costs reasonable • Allow smoothing, if risk is deemed safe (for large companies) • Regulatory oversight • Anticipating failures without placing too much burden of existing system (conditional regulations) • Pay attention to companies with large ratios of pension assets to market capitalization • Increase performance • wide diversification, leverage, and private investments