Using Fat Tails to Model Gray Swans

Swans: White, Black, & Gray The Black Swan is a metaphor for a rare event with extreme impact All swans white accepted truth until first black one seen Example: Stock market crash on 10/19/87 Such events tell us that our models are seriously flawed Term popularized in books by Nassim Taleb Gray Swans are events of considerable nature which are far too big for the bell curve, which are predictable, and for which one can take precautions Benoit Mandelbrot (inventor of fractal geometry) We seem to have a once-in-a-lifetime crisis every three or four years Leslie Rahl (founder of Capital Market Risk Advisors) 2

If the Bell Curve is Such a Bad Model, Why Do We Use It? Central Limit Theorem Sums of independent & identically distributed (i.i.d.) random variables with finite variance tend towards a normal distribution, regardless of underlying distribution Application to forecasting cumulative wealth W T = W 0 (1+R 1 )(1+R 2 ) (1+R T ) lnw T = lnw 0 +ln(1+r 1 )+ln(1+r 2 )+ +ln(1+r T ) So if log-returns are i.i.d., with finite variance, log of cumulative wealth tends to a normal distribution & cumulative wealth tends to a lognormal distribution 3

What if Variance is Infinite? In the 1960s, Mandelbrot & his student, Eugene Fama, explored a model in which extreme events occur at realistic frequencies Log of return relative, ln(1+r), has infinite variance Generalized Central Limit Theorem concludes that log of cumulative wealth has stable distribution Stable distribution have very fat tails Until recently, Mandelbrot-Fama work largely ignored Hard math Most portfolio theory does not work with infinite variance Difficult to estimate model from data 4

Parameters of Stable Distributions Alpha Fatness of Tails 0<alpha 2 (normal) if alpha 1, mean of distribution infinite Beta Skewness (if alpha<2) (fully left skewed) -1 beta 1 (fully right skewed) if beta = 0, distribution symmetric Gamma Scale Gamma>0 If alpha=2, gamma 2 = variance/2 Delta Location if alpha>1, delta = mean of distribution 5

Alpha & The Fatness of Tails 1 Alpha = 0.5 Alpha = 1.5 Alpha = 1.0 Alpha = 2.0 0-5 -4-3 -2-1 0 1 2 3 4 5 6

Beta & Skewness Beta = +0.75 Beta = 0 Beta = -0.75 0-5 -4-3 -2-1 0 1 2 3 4 5 7

Scaling Property of Gamma & Delta 12 11 10 9 8 Gamma 7 6 Alpha = 0.5 Alpha = 1.0 Alpha = 1.5 Alpha = 2.0 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 11 12 Delta 8

What About Kurtosis? How much of the variance is due to infrequent extreme deviations from the mean, rather than frequent modest deviations Often described as a measure of fat tailness No variance, no kurtosis either Stable distributions have fatter tails than distributions with finite kurtosis 9

Normal Distributions: Thin Tails 40% 35% 30% Probability of Loss of at Least 2x 25% 20% 15% 10% 5% 0% 1 2 4 8 16 32 64 128 256 Given Loss of at Least x 10

Finite Kurtosis: Dieting Tails 40% 35% 30% Probability of Loss of at Least 2x 25% 20% 15% 10% 5% 0% 1 2 4 8 16 32 64 128 256 Given Loss of at Least x 11

Stable Distributions: Fat Tails 40% 35% 30% Probability of Loss of at Least 2x 25% 20% 15% 10% 5% 0% 1 2 4 8 16 32 64 128 256 Given Loss of ar Least x 12

Alpha & Conditional Tail Probability 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Alpha 13 Probability of Loss Being at Least Twice of What Is Known

Log-Stable Model of Returns & Cumulative Wealth Lognormal model: ln(1+r) has a normal distribution Log-stable model: ln(1+r) has a stable distribution Parameters: ln(1+r) lnw T Normal Stable Normal Stable 2 alpha 2 alpha - beta - beta sigma/ä2 gamma T sigma/ 2 T 1/alpha gamma mu delta T mu T delta 14

EnCorr Application: Histogram Overlay in Analyzer Procedure Calculate historical log-returns Fit parameters of stable distribution to historical log-returns Draw resulting distribution of returns over histogram Results Distribution curve that fits data better than lognormal model Example: Monthly returns on S&P 500 Jan. 1926 Dec. 2007 Parameter Alpha Beta Gamma Delta Lognormal 2.000-0.0366 0.0084 Log-Stable 1.6653-0.4491 0.0283 0.0066 15

Histogram of S&P 500 with Lognormal Overlay -0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 200 200 180 180 160 160 140 140 120 120 100 100 80 80 60 60 40 40 20 20 0-29% -21% -13% -5% 3% 11% 19% 27% 35% 43% 0 16

Histogram of S&P 500 with Log-Stable Overlay -30% -20% -10% 0% 10% 20% 30% 40% 200 200 180 180 160 160 140 140 120 120 100 100 80 80 60 60 40 40 20 20 0-29% -21% -13% -5% 3% 11% 19% 27% 35% 43% 0 17

Applications in EnCorr Optimizer: Forecasting & Simulation Use log-stable model in place of lognormal model Setting the four parameters Use lognormal model value for E[ln(1+R)] for delta Fit parameters of stable distribution to historical log-returns Use fitted values of alpha & beta Annualize fitted values of gamma using scaling property 18

Example: Lognormal Model of S&P 500 Set delta using SBBI data Historical equity premium = 7.05% Recent Treasury yield = 4.50% E[R] = 7.05% + 4.50% = 11.56% Historical standard deviation = 19.97% Lognormal model gives E[ln(1+R)] = 0.0936 = delta Set alpha & beta from fit of monthly historical data Alpha = 1.6653 Beta = -0.4491 Set gamma by annualizing monthly fitted value Gamma A = 12 1/alpha gamma M = 0.1260 19

Wealth Forecasting Using Lognormal Model 5000 2000 1000 500 200 95th Percentile Wealth Index 100 50 20 75th Percentile 50th Percentile 10 5 2 25th Percentile 5th Percentile 1-0.69315 0.5 0 5 10 15 20 25 30 35 40 45 50 Years into Future 20

Wealth Forecasting Using Log-Stable Model 5000 2000 1000 500 95th Percentile 200 Wealth Index 100 50 20 75th Percentile 50th Percentile 10 5 25th Percentile 2 1 5th Percentile -0.69315 0.5 0 5 10 15 20 25 30 35 40 45 50 Years into Future 21

Monte Carlo Simulation with Log-Stable Distributions Example: Drawdown Problem Start with $1,000,000 Invest in S&P 500 Assume returns follow log-stable distribution Withdraw $80,000 per year for 50 years Use Monte Carlo Simulation Run 1,000 simulations of 50 years each Calculate percentiles of wealth for each year Calculate probability of not running out of money each year 22

Simulated Wealth Paths: 10 th Percentile $1,000,000 $500,000 $200,000 Wealth $100,000 Log-Stable Lognormal $50,000 $20,000 $10,000 4 0 5 10 15 20 25 30 35 40 45 50 Future Year 23

Probability of Not Running Out of Money 100% 95% 90% Probability of Not Running Out of Money 85% 80% 75% 70% 65% Log-Stable Lognormal 60% 55% 50% 0 5 10 15 20 25 30 35 40 45 50 Future Year 24

Summary Significant market events occur far more frequently than predicted by the lognormal model of returns The basis of the lognormal normal model is the Central Limit Theorem which assumes that variance is finite If variance is infinite, the log-stable model follows from the Generalized Central Limit Theorem Stable distributions have parameters for fatness of tails, skewness, scale, & location The log-stable model will be an alternative to the lognormal model in the EnCorr Analyzer & Optimizer 25