Fat tails and 4th Moments: Practical Problems of Variance Estimation Blake LeBaron International Business School Brandeis University www.brandeis.edu/~blebaron QWAFAFEW May 2006
Asset Returns and Fat Tails
Asset Returns Return distributions are fat tailed Noah effect How fat are they? Variances (2nd moments) Skewness (3rd moments) Kurtosis?? (4th moments)???? (5th moments) What if we lived in a world without 4th moments? Would it really matter?
Coauthors Ritirupa Samanta State Street Global Advisors Extreme value theory and fat tails in equity markets (2005) Stephen Cecchetti International Business School, Brandeis University
Outline Return tail properties and higher moments Monte-carlo experiments Variance estimation Min variance portfolios Optimal portfolio weights Robust improvements Conclusions and bigger picture
Dow Returns and Gaussian
Approximate Power-law Tails Shape Parameter Pr(X > x) A x α log(pr(x > x)) log(a) + α log( x ) E( X m ) = m α
A Very Short Fat Tail History Stable distributions and no variances Mandelbrot and Fama (1960 s) Mixtures and volatility Clark(1973), Engle(1982) Variance existence Loretan and Phillips(1994) Econophysics α = 3 Mechanisms Farmer and Geanakoplos (2004)
LeBaron and Samanta (2005): Quick summary Shape parameter estimation Various methods Shape parameter near 3 Left and right tail symmetry Emerging versus developed market differences
Return Distribution Pictures Daily Dow: Jan 1897 - Sept 2004 Sample size = 29,601days Tail region = 10 percent
Dow Returns and Gaussian
Returns and Student-t(3)
Returns and Student-t(5)
Return Tails: Dow 1 day returns
What happens without 4th moments? μ r = E( r ) t σ r2 = E(r t μ r ) 2 T 1 r T t N(μ r, σ 2 r t=1 T ) T 1 (r T t μ r ) 2??? t=1 E(r t μ r ) 4
Student-t Reminders All moments, m < df, exist df = degrees of freedom Example: t(3) has mean, variance, but nothing else t(4) has mean, variance, skew = 0, but no kurtosis t(5) has all moments <5
Outline Return tail properties and higher moments Monte-carlo experiments Variance estimation Min variance portfolios Optimal portfolio weights Robust improvements Conclusions and bigger picture
Variance Estimation (1/T) variance estimates 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)
Daily Monte-carlo Series Gaussian, mean 0, variance 1 Student-t s, mean 0, variance 1 DF = 3, 4, 5 Daily Dow: Jan 1897 - Sept 2004 sample = 29,601 AR(1) residuals, mean = 0, variance = 1 IID bootstrap with replacement
1 Day Variance Estimate: Gaussian Returns
1 Day Variance Estimate: Gaussian Returns - Central Limit Adjusted
1 Day Variance Estimate: Gaussian versus t(3) Returns
1 Day Variance Estimate: Gaussian versus t(4) Returns
1 Day Variance Estimate: Gaussian versus t(5) Returns
1 Day Variance Estimate: t(4) versus t(5) Returns
1 Day Variance Estimate: t(3) versus bootstrap returns
1 Day Variance Estimate: t(4) versus bootstrap returns
20 Day (Monthly) Variance Estimates Pseudo monthly returns (sums of daily) Built off same daily distributions Since 2nd moments exist, closer to Gaussian Variance improvements at higher frequencies (daily) Use higher frequency data (daily) to get better estimates of lower frequencies (monthly)
Monthly Variance Estimate: Gaussian versus t(3) returns
Monthly Variance Estimate: Gaussian versus t(4) returns
Monthly Variance Estimate: Gaussian versus IID Bootstrap
Fine Sampling Frequency Use higher frequency data to improve precision Doesn t help for means Good for variances and covariances
Monthly Variance Estimate: Gaussian monthly versus daily returns
Monthly Variance Estimate: t(3) monthly versus daily returns
Daily/Monthly Quantile Range Ratios Gaussian = 1/sqrt(20) = 0.22
Monthly Variance Estimate: Daily/Monthly precision ratio σ d = 1 μ 4 d 1 σ m n d μ m4 1
Multivariate Data Descriptions MSCI indices World and Emerging market indices Daily 3/1996-3/2006 Procedures AR(1) residuals Calibrate to var/cov matrix Expected returns = 9% and 12% Risk free = 3% Monte-carlo: Gaussian and Student-t Bootstrap
Minimum Variance Portfolio w 1 = σ 22 σ 1,2 σ 12 + σ 22 2σ 1,2
1 Day Min Variance: Gaussian versus t(3) ( true = 0.61)
1 Day Min Variance: Gaussian versus t(4)
1 Day Min Variance: t(4) versus Bootstrap
Monthly Min Variance: t(3) versus bootstrap
Monthly Min Variance: Gaussian monthly versus daily variance estimates
Monthly Min Variance: t3 monthly versus daily returns
Monthly Min Variance: t4 monthly versus daily returns
Optimal Portfolio Weight w = 1 E(r r γ (Ω) 1 f ) Ω=E(r j E(r j ))(r i E(r i )) γ = 4 Record w 1
Optimal Portfolio Weight: Daily Gaussian versus t(3) ( True value = 0.34)
Monthly Optimal Portfolio Weight: Gaussian monthly versus daily returns
Robust Estimators Build variance estimates off mean absolute deviations Covariances from Huber robust methods
1 Day Min Variance: Gaussian: Variance versus robust estimate
1 Day Min Variance : t(3): Variance versus robust estimate
Summary No compelling evidence 4th moments exist Give some weight to a world where 4th moments don t exist In a world without 4th moments: Variance difficult to estimate High frequency data less useful Data/kurtosis tradeoff Realized x Min variance portfolios difficult to estimate Optimal portfolios Expected return uncertainty dominates variances Robust estimators have value
Future Questions Variance persistence estimates?? Bootstrap performance?? Risk measures requiring 4th moments
Big Picture Hard problems in financial data Fat tails - moment failure Long memory (extreme persistence) Implications Practical problems in forecasting and strategy design Building agent-based financial markets Agents worry about large losses Agents confused about time scales Both help to self-generate these same features