Exploring Financial Instability Through Agent-based Modeling Part 2: Time Series, Adaptation, and Survival

Transcription

1 Mini course CIGI-INET: False Dichotomies Exploring Financial Instability Through Agent-based Modeling Part 2: Time Series, Adaptation, and Survival Blake LeBaron International Business School Brandeis University

2 Where are we going? Part 1: What are agent-based models? Simple models from finance Part 2: Adaptation and time series Heterogeneous gain learning Part 3: Current directions in agent design and applications Empirical validation Instability and macro connections LeBaron CIGI/INET November / 66

3 Overview Learning and Time Series Model Structure Basic Simulation Plots Large Swings From Fundamentals Momentum, Risk, and Return Consumption Summary LeBaron CIGI/INET November / 66

4 Learning and Time Series Model Structure Basic Simulation Plots Large Swings From Fundamentals Momentum, Risk, and Return Consumption Learning and Time Series Summary LeBaron CIGI/INET November / 66

5 Time series and populations Time series features Strategy populations LeBaron CIGI/INET November / 66

6 Financial empirical summary Short term Volatility persistence Leptokurtic (fat tailed) return distributions Uncorrelated returns Long term Volatility persistence Return predictability fundamental mean reversion Momentum Risk and return relationships Consumption and returns LeBaron CIGI/INET November / 66

7 Features and dynamics We will go through these time series features, demonstrating how they are connected to underlying learning dynamics. LeBaron CIGI/INET November / 66

8 Models generating long range features Boswijk et al. (2007) Simple, few type Estimation Long horizon only LeBaron (2013 forthcoming) More complicated All horizons Richer evolutionary story LeBaron CIGI/INET November / 66

9 Readings H. Peter Boswijk, Cars H. Hommes, and Sebastiano Manzan. Behavioral heterogeneity in stock prices. Journal of Economic Dynamics and Control, 31(6): , 2007 Blake LeBaron. Heterogeneous gain learning and long swings in asset prices. In Roman Frydman and Edmund S. Phelps, editors, Rethinking Expecations: The Way Forward For Macroeconomics. Princeton University Press, 2013 forthcoming LeBaron CIGI/INET November 2012 note 1 of slide 8

10 Learning and Time Series Model Structure Basic Simulation Plots Large Swings From Fundamentals Momentum, Risk, and Return Consumption Model Structure Summary LeBaron CIGI/INET November / 66

11 Key features Core strategies Adaptive and fundamental types Data perspectives (gain levels) Short and long term views of history Learning (or not learning) stationarity Learning about return and risk Trader/strategy survival LeBaron CIGI/INET November / 66

12 Connections to other economic ideas Limits to arbitrage Noise trader risk Minsky instability LeBaron CIGI/INET November / 66

13 Model components 1.Economy (simple) 2.Agents (risk/return) 3.Forecasting rules 4.Trading 5.Evolution LeBaron CIGI/INET November / 66

14 Economy Stock Stochastic dividend Random walk (growth and volatility calibrated) Finite supply Endogenous price, P t Risk free R f = 0 Infinite supply LeBaron CIGI/INET November / 66

15 Agents: portfolio choice E i,t (R t+1 ) = Expected return ˆσ 2 i,t+1 = Expected variance α i,t = f(e i,t (R t+1 ),ˆσ 2 i,t+1) = Fraction of savings in stock α i,t = E i,t(r t+1 ) R f γˆσ 2 i,t+1 LeBaron CIGI/INET November / 66

16 Agents: portfolio choice S i,t = α i,t(1 λ)w i,t P t = Shares B i,t = (1 α i,t )(1 λ)w i,t = Cash C i,t = λw i,t = Consumption LeBaron CIGI/INET November / 66

17 Forecast rules 1.Adaptive expectations (technical/trend following) 2.Mean reverting (fundamental) 3.Noise (short term forecasts) 4.Buy and hold (long range) 5.Variance forecasts LeBaron CIGI/INET November / 66

18 Adaptive expectations f t+1,j = (1 g j )f t,j +(g j )R t g i = Gain LeBaron CIGI/INET November / 66

19 Mean reverting (fundamental) pd t = log( P t D t ) f t+1,j = R+β t,j (pd t pd) Estimation: Recursive least squares Constant gain, g j LeBaron CIGI/INET November / 66

20 Other rules Noise Short range linear forecasts Recursive least squares Buy and hold Long range expected returns and variance Relatively passive LeBaron CIGI/INET November / 66

21 Variance forecasts ǫ t,j = (R t f t,j ) ˆσ t+t,j 2 = (1 gσ j )ˆσ2 t,j +gσ j ǫ2 t,j Adaptive Riskmetrics/GARCH(1,1) Similar across agents Differ in gain gj σ levels Gain: Horizon/memory Signal/noise LeBaron CIGI/INET November / 66

22 Evolution: rule selection Agents Rules LeBaron CIGI/INET November / 66

23 Gain range Discrete gain levels Half life experiments All gain: [50,18,7,2.5,1] half-lives in years Low gain: 50 years only High gain: 1 5 years only LeBaron CIGI/INET November / 66

24 Decreasing weights into the past (1 g) j 1.00 ŷ t = m (1 g) j y t j j= g = 0.05 g = 0.10 j Lag LeBaron CIGI/INET November / 66

25 Forecast state space (by gain) Forecast(g j ) Variance(g σ j ) 5 x 5 5 x 5 Adaptive Fundamental 5 x 5 1 x 1 Noise Buy and Hold LeBaron CIGI/INET November / 66

26 Forecast state space (by gain) Forecast(g j ) Variance(g σ j ) 5 x 5 5 x 5 Agent Adaptive Fundamental 5 x 5 1 x 1 Noise Buy and Hold LeBaron CIGI/INET November / 66

27 Trading z i (P t ) = a i(p t )(1 λ)w t,i P t I z i (P t ) = 1 i=1 LeBaron CIGI/INET November / 66

28 Brief parameter table Parameter Value γ (CRRA) 3.5 (W/C)(λ) 40 Agents Rules 4000 µ D 0.02 σ D 0.07 Gain values [50,18,7,2.5,1] Note: Gain values are in annualized half lives. LeBaron CIGI/INET November / 66

29 Learning and Time Series Model Structure Basic Simulation Plots Large Swings From Fundamentals Momentum, Risk, and Return Consumption Basic Simulation Plots Summary LeBaron CIGI/INET November / 66

30 Simulation summary 2 Price Weekly returns Trading Volume Years LeBaron CIGI/INET November / 66

31 Time series features Asymmetry in up and down markets Volatility/volume increases in falling markets LeBaron CIGI/INET November / 66

32 Data sets Sources CRSP-VW ( ) (monthly/daily) Shiller ( ) Schwert ( ) (daily returns) Measuring worth (risk free ) Series used Returns, excess returns Price/dividend (P/D) ratios Monthly realized volatility (daily returns) Simulations 100,000 week burn in 100,000 week sample ( 1900 years) LeBaron CIGI/INET November / 66

33 S&P Price/dividend ratio S&P Price/Dividend ratio Year LeBaron CIGI/INET November / 66

34 S&P Price/earnings ratio S&P Price/Earnings ratio Year LeBaron CIGI/INET November / 66

35 Weekly return distributions 800 Weekly CRSP VW Simulation Weekly return/std. LeBaron CIGI/INET November / 66

36 Weekly return autocorrelations Simulation CRSP VW Return autocorrelation Autocorrelation lag (weeks) LeBaron CIGI/INET November / 66

37 Monthly volatility autocorrelation U.S. merged Simulation 0.5 Autocorrelation Lag (months) LeBaron CIGI/INET November / 66

38 Annual return summary statistics Excess Returns R e,t = R t R f,t Series Merged U.S. Simulation years Mean excess return Std Autocorrelation LeBaron CIGI/INET November / 66

39 Learning and Time Series Model Structure Basic Simulation Plots Large Swings From Fundamentals Momentum, Risk, and Return Consumption Summary Large Swings From Fundamentals LeBaron CIGI/INET November / 66

40 Simulation price/dividend ratios 40 P/D ratio 20 Weekly returns Trading Volume Years LeBaron CIGI/INET November / 66

41 Long range return forecasts R t+1 R f,t+1 = α+βlog(p t /D t ) Series β R-squared CRSP (quarterly) (0.008) CRSP (annual) (0.015) Simulation (quarterly) (0.004) Simulation (annual) (0.015) See: Lettau and Ludvigson (2010), Cochrane (2011) LeBaron CIGI/INET November / 66

42 Readings Martin Lettau and Sydney Ludvigson. Measuring and modeling variation in the riskreturn trade-off. In Yacine Ait-Shalia and Lars Peter Hansen, editors, Handbook of Financial Econometrics, volume 1, pages Elsevier, 2010 John H. Cochrane. Discount rates. Journal of Finance, 66(4): , 2011 LeBaron CIGI/INET November 2012 note 1 of slide 40

43 CRSP P/D and volatility σ x,y = Log(variance) Price/dividend LeBaron CIGI/INET November / 66

44 Long range volatility forecasts log(σ 2 t+1) = α+β 1 log(p t /D t )+β 2 log(σ 2 t) Series β 1 β 2 R-squared S&P (0.13) (0.09) (0.03) Simulation (0.03) (0.04) (0.01) LeBaron CIGI/INET November / 66

45 Strategy forecasts 2 Price Weekly returns Return forecast(%) Adaptive Fundamental Years LeBaron CIGI/INET November / 66

46 Strategy portfolios 40 P/D Fraction Adaptive Fundamental Annual volatility Low var gain High var gain Years LeBaron CIGI/INET November / 66

47 Population dynamics Adaptive Fundamental Noise Buy and Hold 0.5 Wealth Fractions Weeks x 10 5 Wealth shift = 4 % / year LeBaron CIGI/INET November / 66

48 Fundamental strategy by var gain 40 P/D Equity fraction Annual volatility Low var gain High var gain Low var gain High var gain Years LeBaron CIGI/INET November / 66

49 Wealth across forecast gains 0.4 Adaptive 0.4 Fundamental Fraction 0.2 Fraction Gain: Min < > Max Gain: Min < > Max 0.8 Noise 0.6 Fraction Gain: Min < > Max LeBaron CIGI/INET November / 66

50 Wealth across variance gains 0.4 Adaptive 0.4 Fundamental Fraction 0.2 Fraction Gain: Min < > Max Gain: Min < > Max 0.4 Noise 0.3 Fraction Gain: Min < > Max LeBaron CIGI/INET November / 66

51 Explanations for swings Limits to arbitrage Not enough wealth in stabilizing strategies Fundamental strategies don t perform well in risk/return space Some fundamental strategies get scared (also too aggressive in rising markets) Risk perceptions attenuate fundamental trading LeBaron CIGI/INET November / 66

52 Explanation for downside instability Aggregate share demand: S t = I i=1 Which can be split into two parts, α i (P t )(S t 1,i (P t +D t )+B t 1,i ) P t S t = I i=1 α i (P t )S t 1,i + I i=1 + α i (P t ) (S t 1,iD t +B t 1,i ) P t LeBaron CIGI/INET November / 66

53 Who determines prices? Marginal trader Many agents are inframarginal Prices not averages of all beliefs Can swing quickly Some trading mechanical (rebalancing) LeBaron CIGI/INET November / 66

54 Learning and Time Series Model Structure Basic Simulation Plots Large Swings From Fundamentals Momentum, Risk, and Return Consumption Momentum, Risk, and Return Summary LeBaron CIGI/INET November / 66

55 Conditional returns Empirical methodology: Kernel regression Bandwidth: Randomized cross validation on simulation series Structure exploration (shared nonlinear features) Relationships Momentum forecasts Lagged 6 months, 1 month wait, forecast 6 months (6 1 6) Current volatility and future excess returns (risk/return) LeBaron CIGI/INET November / 66

56 Momentum (6,1,6) Month Strategy U.S. merged Simulation 0.07 Future 6 month return Lagged 6 month return LeBaron CIGI/INET November / 66

57 Dow firm momentum 25/50/75 quantiles 0.1 Expected future (6 month + 1) return Lagged (6 month) return LeBaron CIGI/INET November / 66

58 Current volatility and future returns U.S. merged Simulation Future excess return (quarter) Lagged variance (quarter) LeBaron CIGI/INET November / 66

59 Commentary Diffusion of beliefs through the population Information processing moving across gains Transition from momentum to mean reversion Asymmetric price dynamics (rise slowly/fall fast) LeBaron CIGI/INET November / 66

60 Learning and Time Series Model Structure Basic Simulation Plots Large Swings From Fundamentals Momentum, Risk, and Return Consumption Consumption Summary LeBaron CIGI/INET November / 66

61 Consumption C i,t = λŵi,t K Ŵ i,t = ω k W i,t k k=1 Series E (log(c t )) σ( log(c t )) Annual % Annual % U.S. ( ) C i,t = λw i,t C i,t = λŵi,t LeBaron CIGI/INET November / 66

62 Learning and Time Series Model Structure Basic Simulation Plots Large Swings From Fundamentals Momentum, Risk, and Return Consumption Summary Summary LeBaron CIGI/INET November / 66

63 Common to most agent-based markets Instability (experiments) Short run Fat tail return distributions Volatility persistence Mechanisms Interactions between adaptive and fundamental types Wealth adjusting to successful strategies Leverage not necessary LeBaron CIGI/INET November / 66

64 Features of this market Survival of multiple gain levels Importance of risk in strategy fitness Calibrated (small) aggregate shocks Small shifts in wealth across strategies Interpretable strategies (basis?) Fast versus slow learning Blake LeBaron. Wealth dynamics and a bias toward momentum trading. Financial Review Letters, 9:21 28, 2012 LeBaron CIGI/INET November / 66

65 Interesting counterfactuals Passive wealth (too large?) Magnitude of predictability/mean reversion Strong counter cyclical volatility Consumption Interest rates? LeBaron CIGI/INET November / 66

66 More thoughts on the dynamics Volatility and risk perceptions 1.Some traders overreact to recent volatility 2.Discount P t, and increases forecast heterogeneity Volatility forecasts take time to sweep through heterogeneous beliefs Key mistake: Short term volatility perspective Changing risk perception versus risk aversion LeBaron CIGI/INET November / 66

67 Summary Model structure Statistical features Short horizon Long horizon Evolutionary questions Why can t stabilizing strategies gain more wealth? Why can t long memory strategies gain more wealth? LeBaron CIGI/INET November / 66

68 Overview Learning and Time Series Model Structure Basic Simulation Plots Large Swings From Fundamentals Momentum, Risk, and Return Consumption Summary LeBaron CIGI/INET November / 66