Fat tails and 4th Moments: Practical Problems of Variance Estimation
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1 Fat tails and 4th Moments: Practical Problems of Variance Estimation Blake LeBaron International Business School Brandeis University QWAFAFEW May 2006
2 Asset Returns and Fat Tails
3 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?
4 Coauthors Ritirupa Samanta State Street Global Advisors Extreme value theory and fat tails in equity markets (2005) Stephen Cecchetti International Business School, Brandeis University
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
6 Dow Returns and Gaussian
7 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
8 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)
9 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
10 Return Distribution Pictures Daily Dow: Jan Sept 2004 Sample size = 29,601days Tail region = 10 percent
11 Dow Returns and Gaussian
12 Returns and Student-t(3)
13 Returns and Student-t(5)
14 Return Tails: Dow 1 day returns
15 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
16 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
17 Outline Return tail properties and higher moments Monte-carlo experiments Variance estimation Min variance portfolios Optimal portfolio weights Robust improvements Conclusions and bigger picture
18 Variance Estimation (1/T) variance estimates Increasing sample sizes 20, 60, 250, 1250, 2500, length monte-carlo Record quantiles (0.01, 0.05, 0.5, 0.95, 0.99)
19 Daily Monte-carlo Series Gaussian, mean 0, variance 1 Student-t s, mean 0, variance 1 DF = 3, 4, 5 Daily Dow: Jan Sept 2004 sample = 29,601 AR(1) residuals, mean = 0, variance = 1 IID bootstrap with replacement
20 1 Day Variance Estimate: Gaussian Returns
21 1 Day Variance Estimate: Gaussian Returns - Central Limit Adjusted
22 1 Day Variance Estimate: Gaussian versus t(3) Returns
23 1 Day Variance Estimate: Gaussian versus t(4) Returns
24 1 Day Variance Estimate: Gaussian versus t(5) Returns
25 1 Day Variance Estimate: t(4) versus t(5) Returns
26 1 Day Variance Estimate: t(3) versus bootstrap returns
27 1 Day Variance Estimate: t(4) versus bootstrap returns
28 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)
29 Monthly Variance Estimate: Gaussian versus t(3) returns
30 Monthly Variance Estimate: Gaussian versus t(4) returns
31 Monthly Variance Estimate: Gaussian versus IID Bootstrap
32 Fine Sampling Frequency Use higher frequency data to improve precision Doesn t help for means Good for variances and covariances
33 Monthly Variance Estimate: Gaussian monthly versus daily returns
34 Monthly Variance Estimate: t(3) monthly versus daily returns
35 Daily/Monthly Quantile Range Ratios Gaussian = 1/sqrt(20) = 0.22
36 Monthly Variance Estimate: Daily/Monthly precision ratio σ d = 1 μ 4 d 1 σ m n d μ m4 1
37 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
38 Minimum Variance Portfolio w 1 = σ 22 σ 1,2 σ 12 + σ 22 2σ 1,2
39 1 Day Min Variance: Gaussian versus t(3) ( true = 0.61)
40 1 Day Min Variance: Gaussian versus t(4)
41 1 Day Min Variance: t(4) versus Bootstrap
42 Monthly Min Variance: t(3) versus bootstrap
43 Monthly Min Variance: Gaussian monthly versus daily variance estimates
44 Monthly Min Variance: t3 monthly versus daily returns
45 Monthly Min Variance: t4 monthly versus daily returns
46 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
47 Optimal Portfolio Weight: Daily Gaussian versus t(3) ( True value = 0.34)
48 Monthly Optimal Portfolio Weight: Gaussian monthly versus daily returns
49 Robust Estimators Build variance estimates off mean absolute deviations Covariances from Huber robust methods
50 1 Day Min Variance: Gaussian: Variance versus robust estimate
51 1 Day Min Variance : t(3): Variance versus robust estimate
52 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
53 Future Questions Variance persistence estimates?? Bootstrap performance?? Risk measures requiring 4th moments
54 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
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