Difficult Empirical Facts from Finance

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Difficult Empirical Facts from Finance

• Blake LeBaron
• International Business School
• Brandeis University
• www.brandeis.edu/blebaron

SFI CSSS, 2007 Santa Fe, NM
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Empirical Facts

• Robust over time
• Robust over markets
• Equity
• Bonds
• FX
• Commodities
• Highly significant

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Empirical Challenges

• Two big features
• Long memory
• Fat tails
• Complexity connections
• Length scales (time or space)
• Learning connections
• Coevolution

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Finance Facts/puzzles

• Price time series
• Near martingale behavior
• Volatility persistence
• Fat tails (leptokurtosis)
• Technical trading
• Nonlinear features/predictability??
• Prices relative to something
• Deviations from fundamentals
• Equity premium
• Trading volume
• Microstructure facts

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Near Martingale Prices

• r(t1) uncorrelated, and difficult to
  directionally forecast (including nonlinear)
• Many horizons (better short)
• Many series (almost any liquid asset)
• Theory EMH (Efficient Market Hypothesis)

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Volatility Persistence

• Return magnitudes persistent
• Very persistent!!
• History
• Mandelbrot
• ARCH/GARCH
• Stochastic volatility
• Realized volatility

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Data Introduction

• Dow Jones Industrials
• Jan 1897-Sep 2004 (29602 obs)
• British Pound
• June 1973 - Feb 2006
• IBM Daily returns and volume

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Dow Jones Daily Returns1897-2004
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Dow Volatility Persistence
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Long Memory Stochastic Volatility
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Fractionally Integrated Process (Long Memory)
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Autocorrelation Comparisons
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ACF Comparison
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Dow Versus Long Memory Volatility Process
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Daily British Pound Returns 1973-2006
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Volatility persistence Pound versus Long Memory
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Long Memory in Volatility

• Present in almost all financial series
• Best estimates
• Realized volatility
• 0.35 lt d lt 0.50
• Causes
• Adding short memory processes
• Regime shifts
• Nonlinearities
• Other

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Finance Facts/puzzles

• Price time series
• Near martingale behavior
• Volatility persistence
• Fat tails (leptokurtosis)
• Technical trading
• Nonlinear features/predictability??
• Prices relative to something
• Deviations from fundamentals
• Equity premium
• Trading volume
• Microstructure facts

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Fat Tailed Return Distributions

• Returns at the lt monthly frequency are not
  normally distributed
• Fat tails
• Leptokurtic
• Power laws

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Dow Returns and Gaussian
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Returns and Student-t(3)
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Normal Quantile Comparisonsupper Dow, lower
BP
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Normal Quantile Comparisonsupper Dow, lower
long memory (d0.45)
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Normal Quantile Comparisonsupper Dow daily,
lower Dow monthly
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Return Summary Statistics
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Approximate Power-law TailsShape Parameter
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Dow Tail Probabilities
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Practical Implications

• Higher (gt3) moment failure??
• If true, problems for variance (not mean)
  estimation
• Possibly other problems

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Variance Estimation

• Increasing sample sizes
• 20, 60, 250, 1250, 2500, 5000
• 5000 length monte-carlo
• Record quantiles
• (0.01, 0.05, 0.5, 0.95, 0.99)

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Daily Monte-carlo Series

• Gaussian, mean 0, variance 1
• Student-ts, mean 0, variance 1
• DF 3, 4, 5

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1 Day Variance EstimateGaussian versus
Student-t(3) Returns
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Fine Sampling Frequency

• Use higher frequency data to improve precision
• Doesnt help for means
• Good for variances and covariances

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Monthly/daily Volatility EstimatesGaussian
Returns
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Monthly/daily Volatility EstimatesStudent-t(3)
Returns
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Why Should We Care About Tails?

• Large events / risk
• Portfolio construction
• Robust learning
• Variance estimation
• Filtering and parameter updating

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Finance Facts/puzzles

• Price time series
• Near martingale behavior
• Volatility persistence
• Fat tails (leptokurtosis)
• Technical trading
• Nonlinear features/predictability??
• Prices relative to something
• Deviations from fundamentals
• Equity premium
• Trading volume
• Microstructure facts

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Technical Trading Patterns

• Simple trend following rules have some predictive
  abilities
• Both equity and fx markets

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Moving Average Trading Rules (Simplest)
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Simple Rule Test
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Dow MA lengthsReturns and T-tests
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BP MA lengthsReturns and T-tests
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Dow Total Annual ReturnLong/short strategy (150
day MA)
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Total Annual Return BP (long/short, 150 day MA)
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Nonlinear Features

• Leverage effect
• Volume/volatility and autocorrelations
• Key question
• Stability and reliability

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Finance Facts/puzzles

• Price time series
• Near martingale behavior
• Volatility persistence
• Fat tails (leptokurtosis)
• Technical trading
• Nonlinear features/predictability??
• Prices relative to something
• Deviations from fundamentals
• Equity premium
• Trading volume
• Microstructure facts

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Fundamentals

• Dividend price ratios
• Interest rate differentials
• Real exchange rates

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Real SP Level and Shillers Dividend Discounted
Price
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SP Dividend Yield
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Dividend Yield Autocorrelations
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US Dollar - British PoundInterest Differential
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Interest Differential Autocorrelation
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Equity Premium

• Equity real return 7-8 per year
• Bond real return 1
• Spread of 6 difficult to explain
• Explanations
• Tails and risk estimates
• Learning (premium is falling over time)

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Finance Facts/puzzles

• Price time series
• Near martingale behavior
• Volatility persistence
• Fat tails (leptokurtosis)
• Technical trading
• Nonlinear features/predictability??
• Prices relative to something
• Deviations from fundamentals
• Equity premium
• Trading volume
• Microstructure facts

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Trading Volume

• Persistence
• Trading Time

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IBM Trading Volume
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Detrended IBM Trading Volume
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IBM Volume Autocorrelations
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IBM Volatility/volume Cross Correlation
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Mixtures of Distributions(Clock/calendar time)

• Clark(1973, Econometrica)
• Ann and Geman, (J of Fin, 2000)
• Martens and van Dijk (2006, Erasmus)
• Gillemot, Farmer, and Lillo (2005)
• Theres more to volatility than volume

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Finance Facts/puzzles

• Price time series
• Near martingale behavior
• Volatility persistence
• Fat tails (leptokurtosis)
• Technical trading
• Nonlinear features/predictability??
• Prices relative to something
• Deviations from fundamentals
• Equity premium
• Trading volume
• Microstructure facts

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Microstructure Facts

• Seasonalities in spreads and volume
• Order flows
• Evans and Lyons (2002, Journal of Political
  Economy, and others)
• Predictability
• Lillo and Farmer (2004) (and others)
• Long memory
• Schulmeister (2006, Financial Research Letters)
• Technical trading and order flows

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More Microstructure Facts

• Book matters
• Large moves
• Osler, Stop loss orders and price cascades
• Farmer et. al. What really causes large price
  changes?, SNDE (2004).
• Depth/liquidity
• Trading time/mixtures of distributions

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High Frequency Example

• EBS Electronic trading system
• /Euro exchange rates (high frequency quotes and
  deals)
• 12/28/02 - 03/03/06
• Clock versus event (deal) time
• High/low range volatility estimate
• (H-L)/( 0.5(HL) )
• 1 Hour / 100 deal windows

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1 Hour /Euro Volatility ACF
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1 Hour /Euro Volatility ACFTime of Day Effects
Removed
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100 Event /Euro Volatility ACF
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100 Event /Euro Volatility ACFTime of Day
Effects Removed
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Finance Facts/puzzles

• Price time series
• Near martingale behavior
• Volatility persistence
• Fat tails (leptokurtosis)
• Technical trading
• Nonlinear features/predictability??
• Prices relative to something
• Deviations from fundamentals
• Equity premium
• Trading volume
• Microstructure facts

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Explanations

• Many facts hard to explain with traditional
  modeling approaches
• Fat tails
• Volatility persistence
• Deviations from fundamentals
• Agent-based approaches