Fat Tail Analysis and Package Fattailsr

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Fat Tail Analysis and Package FatTailsR

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9th R/Rmetrics Workshop 27June2015

Villa Hatt – Zürich

[email protected] Fat Tail Analysis and Package FatTailsR 9th R/Rmetrics Workshop 27June2015 1 / 26 Outline

1 Introduction

2 Mathematics

3 Fat tails: Application in ﬁnance

4 Conclusion

[email protected] Fat Tail Analysis and Package FatTailsR 9th R/Rmetrics Workshop 27June2015 2 / 26 About InModelia

Since 2009: a small company located in Paris Consulting services in data analysis Design of experiments Multivariate nonlinear modeling Neural networks Times series Software development in R

June 2014: 8th R/Rmetrics workshop First version of R package FatTailsR introducing Kiener distributions with symmetric (3 parameters) and asymmetric (4 parameters) fat tails

June 2015: 9th R/Rmetrics workshop Second version of R package FatTailsR including 3 functions for parameter estimation R package FatTailsRplot including advanced plotting functions

[email protected] Fat Tail Analysis and Package FatTailsR 9th R/Rmetrics Workshop 27June2015 3 / 26 Fat tails

It is now well established that ﬁnancial markets exhibit fat tails. It occurs on all markets.

SP500 janv. 1957 − déc. 2013 SP500 janv. 1957 − déc. 2013 100 µ F(X) > 0.995 14349 points ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●● ●● ● ● ● X − 14350 points daily ●●●●●●●●●●●●●● 1.0 ●●●●●●●● ●●●●●● ●●●● ● ●●● ●●●●●●●● ●●●●● ● ●● ● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●● ●●●●●●●●● ●●● ●●●● ●● 2975 points weekly ● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●● ●●●●●●●●●●●●●●● ●●●●● ●●● ● ●●●●●● ● ●● ●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●● ●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●● ● ●● ●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●● ●●●●●●●●●●●● ●●●●●● ●●●●●●●●● ● ●● ●●●●●●●●●●●●●●●●●●● ●●●●●●●● ●●●●●●●●●●●●● ●●●● ●●●●●● ● ●● ●●●●●●●●●●●●●●●●● ●●●●●● ●●●●●●●●●●●● ●●●●●● ●●●●●●●● ● ●● ●●●●●●●●●●●●●●●●● ●●●●●● ●●●●●●●●●● ●●●●● ●●●● ● ● ●●●●●●●●●●●●●●●● ●●●●●● ●●●●●●●● ●●● ●●●●● ● ● ●●●●●●●●●●●●●●●● ●●●●● ●●●●●●● ●●●● ●●●●● ● ● ●●●●●●●●●●●●●●●● ●●●● ●●●●●●●● ●●● ●●● 685 points monthly ● ● ●●●●●●●●●●●●●● ●●●●● ●●●●●●●● ●●● ●●● ● ● ●●●●●●●●●●●●●● ●●●●● ●●●● ●●●● ●●●●●● ● ● ●●●●●●●●●●●●●● ●●●● ●●●●●●● ●●● ●●●● ● ● ●●●●●●●●●●●●● ●●●● ●●●●●● ●●● ●●●●● ● ● ●●●●●●●●●●●● ●●●●● ●●●●● ●●● ●●● ● ● ●●●●●●●●●●●● ●●●●● ●●●●● ●●● ●●● ● ● ●●●●●●●●●●●● ●●●● ●●●●● ●● ●●●● ● ● ●●●●●●●●●●● ●●●●● ●●●● ●●● ●●●● ● ● ●●●●●●●●●● ●●● ●●●●● ●●● ●●●●● ● ● − ●●● ●●●●● ●●● ●●●● ●● ●●● 0.999 ● ● 1 ●●● ●●●● ●●● ●●● ● ●●● ● ● ●●●●●●●●●● ●● ●●●● ●● ● ● ● ●●●●●●●●● ●●●● ●●●● ●● ●●● ● ● ●●● ●●●●● ●●● ●●●● ●● ● ● ● 10 ●●● ●●●●● ●●● ●●● ●● ●● ● ● ●● ●●● ●● ●●● ● ●● ● ● ●●● ●●●● ●●● ●●● ● ●● ● ● ●●● ●●●● ●●● ●●●● ●● ●●● ● ● ●●● ●●●●● ●●● ●●● ●● ●● ● ● ●●● ●●●●● ●● ●●●● ● ●●● ● ● ●●● ●●●●● ●● ●●● ●● ●●● ● ● ●●●● ●●●● ●●● ●●●●● ●● ● ● ● ●●● ●●●●● ●●● ●●● ●● ●● ● ● ●●● ●●●● ●● ●●●● ●● ● ● ● ●●● ●●●● ●● ●●● ●● ● ● ● ●●●●●●● ●● ●●●● ●● ●● ● ● ●● ●●●● ●● ●●● ● ● ● ● ● ●● ●●● ●●● ●● ●● ● ● ● ●●● ●●● ●●● ●●● ● ● ● ● ●●●●●● ●● ●● ● ● ● ● ●●●●●● ●● ●●● ● ●● ● ● ●●●●●●● ●● ●● ●● ● ● ● ●●●●● ●● ●● ● ●● ● ● ●●●●● ●● ●● ●● ● ● ● ●●●●●● ●● ●● ● ● ● ● ●●●●● ●●● ●●●● ● ● ● ● ●●●●● ● ● ● ● ● ● ●●●●● ●● ●● ● ● ● ● ●●●●● ●● ●● ● ● 0.997 ● ● ●●●●● ●● ●● ● ● ● ● ●●●●●● ● ●● ● ● ● ●●●●●● ●●● ●● ● ● ● ● ●●●●● ●● ●● ● ● ● ● ●●●●● ● ● ● ● ● ● ●●●●● ●● ●● ● ● ● ● ●● ● ●● ● ● ● ●●●●● ●● ●●● ● ● ● ● ●●●● ● ● ● ● ● ●●●● ● ●● ● ● ● − ●●●●●● ●● ● ● ● ● 2 ●●●● ● ● ● ● ● ●●●●●● ● ●● ● ● ● ● ●● ●● ●● ● ● ● ●●●●● ●● ● ● ● ● ● 10 ●●●● ● ●● ● ● ●● ● ● ● ● ●●●●● ● ● ● ● ●●●●● ● ● ● ● ● ● ●●● ● ● ● ● F(X) < 0.005 ●●●● ● ●● ● ●●● ● ● ● ● ● ●●●●● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ●●●● ● ● ● ● ●●●● ● ● 0.995 ● ● ●●● ● ● ● ●●●● ● ● ● ● ● ● ●●●● ● 4 6 8 10 ● ● ●● ● ● ● ●●● ● ● ● ● ●●● ● ● ● ●●● ● ● ●●● ● ● ● ● ● ●● ● ● 0.5 ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ●● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ●● ●● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● − ● ● ● ● ● 3 ● ● ● 0.004 ● ● ● ● ● ● ● ● ● 10 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.002 ● ● ● ● − ● ● 4 ● ● ● ● 10 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ●● ● ●● ● ● ●● ● ●● ● ●●● ● ●●● ● ●●● ● ●●● ●●●● ●●●● ●●●●●●● ●●●●●●●●● ●●●●●●●●●● 0.000 0.0 ●●● ● ● ●●●● ● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● −20 −15 −10 −5 − 10 5 −1 0 1 2 −10 −5 0 5 10 10 10 10 10

Figure : (a) SP500 daily log-returns (b) SP500 daily, weekly and monthly log-returns in log-log scale black dots = negative returns, white circles = positive returns

α ω lim F (x) x − lim 1 F (x) x − α, ω 3.5 x = | | x − = ≈ →−∞ →+∞

[email protected] Fat Tail Analysis and Package FatTailsR 9th R/Rmetrics Workshop 27June2015 4 / 26 Symmetric distributions K1. Asymmetric distributions K2, K3, K4

Last year 1, we introduced a family of four probability distributions which deal with symmetric and asymmetric fat tails. These distributions were named Kiener distributions or, simply, distributions K1, K2, K3, K4. They combine two log-logistic quantile functions, one for each tail. The parameters are: 2αω 1 1 ³ 1 1 ´ median µ, scale γ, left tail α, right tail ω, shape parameter κ α ω or harmonic mean κ 2 α ω = + = + ³ ´ distorsion δ and eccentricity ² deﬁned by δ ² 1 1 1 with bonds 1 δ 1 and 1 ² 1 = κ = 2 − α + ω − κ < < κ − < <

Quantile functions of distributions K2(µ,γ,α,ω), K3(µ,γ,κ,δ), K4(µ,γ,κ,²), K1(µ,γ,κ)

  µ ¶ 1 µ ¶ 1 Ã logit(p) logit(p) ! p − α p ω (K2) x(p) µ γκ  µ γκ e− α e ω = + − 1 p + 1 p = + − + − −   µ ¶ 1 µ ¶ 1 µ ¶δ µ ¶ p κ p κ p logit(p) δlogit(p) (K3) x(p) µ γκ −  µ 2γκsinh e = + − 1 p + 1 p 1 p = + κ − − −  1 1  ² µ p ¶ κ µ p ¶ κ µ p ¶ κ µ logit(p) ¶ ² logit(p) (K4) x(p) µ γκ −  µ 2γκsinh e κ = + − 1 p + 1 p 1 p = + κ − − −  1 1  µ p ¶ κ µ p ¶ κ µ logit(p) ¶ (K1) x(p) µ γκ −  µ 2γκsinh µ γkogit(p, κ) = + − 1 p + 1 p = + κ = + − −

1P. Kiener, Explicit models for bilateral fat-tailed distributions and applications in ﬁnance with the package FatTailsR, 8th R/Rmetrics Workshop, Paris, 2014. http://www.inmodelia.com/fattailsr-en.html [email protected] Fat Tail Analysis and Package FatTailsR 9th R/Rmetrics Workshop 27June2015 5 / 26 Distribution K1 and some properties

³ x µ ´ Probability and density functions of distribution K1(µ,γ,κ), t asinh − = 2γκ 1 1 1 F (x) ³ x µ ´ f (x) f (0) = κ asinh − = 4γcosh(t)(1 cosh(κt)) = 8γ 1 e− 2γκ + +

Logit-probability: Density as a function of the probability: ³ x µ ´ 1 logit(p) logit(F (x)) κasinh − f (x) sech( )p (1 p) = 2γκ = 2γ κ −

Conversion between parameters α, ω, κ, δ, ²

1 1 µ 1 1 ¶ ² 1 µ 1 1 ¶ 1 1 1 1 α ω κ κ δ δ δ ² − α ω κ = 2 α + ω = κ = 2 − α + ω α = κ − ω = κ + = α ω = 1 ² = 1 ² + − +

Property 1: Pareto aymptotic functions ¯ ¯ α ¯ ¯ ω ¯ x µ ¯− ¯ x µ ¯− lim F (x) ¯ − ¯ lim 1 F (x) ¯ − ¯ x = ¯ γκ ¯ x − = ¯ γκ ¯ →−∞ →+∞ Property 2: Karamata theorem about slowly α-varying and ω-varying functions x f (x) x f (x) lim α lim ω x F (x) = − x 1 F (x) = →−∞ →+∞ −

[email protected] Fat Tail Analysis and Package FatTailsR 9th R/Rmetrics Workshop 27June2015 6 / 26 Comparaison with other distributions

Distributions K1, K2, K3, K4 are similar, but more tractable and easier to use than 2 Generalized Tukey lambda distribution and Tadikamalla and Johnson LU distribution .

K2 and K4 distributions (Kiener, 2014):

" µ ¶ 1/α µ ¶1/ω# µ ¶ p − p logit(p) ² logit(p) x(p) µ γκ µ 2γκsinh e κ = + − 1 p + 1 p = + κ − −

Generalized Tukey lambda distribution (Ramberg et Schmeiser, 1972):

1 h λ λ i x(p) λ1 (1 p) 4 p 3 = + λ2 − − +

LU distribution (Tadikamalla and Johnson, 1982):

µ γ 1 ¶ x(p) ξ λ sinh logit(p) = + − δ + δ

2Non-central Student distribution and Cauchy distribution were already discussed last year [email protected] Fat Tail Analysis and Package FatTailsR 9th R/Rmetrics Workshop 27June2015 7 / 26 Figures related to symmetric distribution K1

Quantiles : K1 ( µ = 0, γ = 1, κ = ... ) Probabilités : K1 ( µ = 0, γ = 1, κ = ... ) + Gauss ( σ = 3.2 ) Densités : K1 ( µ = 0, γ = 1, κ = ... ) + Gauss ( σ = 3.2 ) 20 1.0 0.15 κ κ κ 0.6 0.6 0.6 1 1 1 1.5 1.5 1.5 10 2 2 2 3.2 3.2 3.2 0.10 10 10 10 Gauss Gauss

0 0.5

0.05

−10 Kogit −20 0.0 0.00 0.0 0.5 1.0 −15 −10 −5 0 5 10 15 −15 −10 −5 0 5 10 15

Dérivées des quantiles : K1 ( µ = 0, γ = 1, κ = ... ) Logit(Proba) : K1 ( µ = 0, γ = 1, κ = ... ) + Logistique + Gauss (σ = 3.2 ) LogDensités : K1 ( µ = 0, γ = 1, κ = ... ) + Logistique + Gauss (σ = 3.2 ) 200 6.9 κ κ −2 0.6 0.6 1 4.6 1 1.5 1.5 2 2 3.2 2.9 3.2 10 10 −4 Logistique Gauss 100 0.0 κ 0.6 logit(q.999) = 6.9 1 −6 −2.9 logit(q.99) = 4.6 1.5 logit(q.95) = 2.9 2 logit(q.50) = 0 3.2 −4.6 logit(q.05) = −2.9 10 logit(q.01) = −4.6 Logistique logit(q.001) = −6.9 Gauss 0 −6.9 −8 0.0 0.5 1.0 −15 −10 −5 0 5 10 15 −15 −10 −5 0 5 10 15

Figure : Distribution K1 : (a) Quantiles - (b) Cumulative functions (c) Densities (d) Quantile derivatives - (e) Logit of the cumulative functions - (f) Logdensities

[email protected] Fat Tail Analysis and Package FatTailsR 9th R/Rmetrics Workshop 27June2015 8 / 26 Figures related to asymmetric distributions K2, K3, K4

Quantiles : K2 ( µ = 0, γ = 1, α = 2, ω = ... ) Quantiles + Probabilités : K2 ( µ = 0, γ = 1, α = 2, ω = ... ) Quantiles + Densités : K2 ( µ = 0, γ = 1, α = 2, ω = ... ) 15 1.0 0.200 ω ω ω 0.6 0.6 0.6 10 1 1 1 1.5 1.5 1.5 2 2 2 3.2 3.2 3.2 5 10 10 0.125 10

0 0.5

−5

−10

−15 0.0 0.000 0.0 0.5 1.0 −15 −10 −5 0 5 10 15 −15 −10 −5 0 5 10 15

Dérivées des quantiles : K2 ( µ = 0, γ = 1, α = 2, ω = ... ) Quantiles + Logit(Probabilités) : K2 ( µ = 0, γ = 1, α = 2, ω = ... ) Quantiles + Log−Densités : K2 ( µ = 0, γ = 1, α = 2, ω = ... ) 200 6.9 ω ω −1 0.6 0.6 1 4.6 1 1.5 1.5 2 2 3.2 2.9 3.2 10 10 −3

100 0.0

ω logit(q.999) = 6.9 −2.9 logit(q.99) = 4.6 −5 0.6 logit(q.95) = 2.9 1 logit(q.50) = 0 1.5 −4.6 logit(q.05) = −2.9 2 logit(q.01) = −4.6 3.2 logit(q.001) = −6.9 10 0 −6.9 −7 0.0 0.5 1.0 −15 −10 −5 0 5 10 15 −15 −10 −5 0 5 10 15

Figure : Distribution K2 : (a) Quantiles - (b) Cumulative functions (c) Densities (d) Quantile derivatives - (e) Logit of the cumulative functions - (f) Logdensities

[email protected] Fat Tail Analysis and Package FatTailsR 9th R/Rmetrics Workshop 27June2015 9 / 26 Limit points

When α or ω , the quantile function becomes a single log-logistic quantile function with left or right= ∞ limit points= ∞

Property 4: Left limit point (α ), Right limit point (ω ) = ∞ = ∞ 1 µ p ¶ 1/ω α , κ 2ω, δ , ² 1 x(p) µ 2γω 2γω − µ 2γω = ∞ = = 2ω = ⇒ = − + 1 p −−−−→p 0 − − → 1 µ p ¶ 1/α ω , κ 2α, δ , ² 1 x(p) µ 2γα 2γα − µ 2γα = ∞ = = − 2α = − ⇒ = + − 1 p −−−−→p 1 + − →

Limit points: Probability K2 ( µ = 0, γ = 1, α = ∞, ω = ... ) Limit points: Density K2 ( µ = 0, γ = 1, α = ∞, ω = ... ) Limit points: Density K2 ( µ = 0, γ = 1, α = ..., ω = ∞ ) Limit points: Probability K2 ( µ = 0, γ = 1, α = ..., ω = ∞ ) 1.0 0.200 0.200 1.0 ω ω α α 0.6 0.6 0.6 0.6 1 1 1 1 1.5 1.5 1.5 1.5 2 2 2 2 3.2 3.2 3.2 3.2 10 0.125 10 0.125 10 10 α = ∞ α = ∞ ω = ∞ ω = ∞ 0.5 0.5

0.0 0.000 0.000 0.0 −15 −10 −5 0 5 10 15 −15 −10 −5 0 5 10 15 −15 −10 −5 0 5 10 15 −15 −10 −5 0 5 10 15

Figure : Cumulative functions K2: (a) α (b) ω Density functions K2: (c) α (d) ω = ∞ = ∞ = ∞ = ∞

[email protected] Fat Tail Analysis and Package FatTailsR 9th R/Rmetrics Workshop 27June2015 10 / 26 Raw and central moments — K2, K1

Mean exists if min(α, ω) 1. Variance exists if min(α, ω) 2. > > Skewness exists if min(α, ω) 3. Kurtosis exists if min(α, ω) 4. > > Distribution K2(µ,γ,α,ω)

µ µ 1 1 ¶ µ 1 1 ¶¶ m E(X) µ γκ β 1 ,1 β 1 ,1 µ γκν 1 = = + − − α + α + − ω + ω = + ³ ´ · µ 2 2 ¶ µ 2 2 ¶ µ 1 1 1 1 ¶¸ µ E (X E(X))2 γ2κ2 β 1 ,1 β 1 ,1 2β 1 ,1 2 = − = − α + α + − ω + ω − + α − ω − α + ω · µ 1 1 ¶ µ 1 1 ¶ µ 1 1 ¶ µ 1 1 ¶¸ γ2κ2 β2 1 ,1 β2 1 ,1 2β 1 ,1 β 1 ,1 + − − α + α − − ω + ω + − α + α − ω + ω P3 Pi ¡3¢¡i¢ 3 i j ³ j i j j i j ´ ( ν) ( 1) β 1 − ,1 − µ3 i 0 j 0 i j − − − + α − ω − α + ω β = = 1 = 3/2 = 3/2 (µ ) hP2 Pi ¡2¢¡i¢ j ³ j i j j i j ´i 2 ( ν)2 i ( 1) β 1 − ,1 − i 0 j 0 i j − − − + α − ω − α + ω = = P4 Pi ¡4¢¡i¢ 4 i j ³ j i j j i j ´ ( ν) ( 1) β 1 − ,1 − µ4 i 0 j 0 i j − − − + α − ω − α + ω β 2 2 = = 2 = (µ ) = hP2 Pi ¡2¢¡i¢ j ³ j i j j i j ´i 2 ( ν)2 i ( 1) β 1 − ,1 − i 0 j 0 i j − − − + α − ω − α + ω = =

Distribution K1(µ,γ,κ)

³ 4 4 ´ ³ 2 2 ´ µ µ 2 2 ¶ ¶ β 1 κ ,1 κ 4β 1 κ ,1 κ 3 m µ µ 2γ2κ2 β 1 ,1 1 β 0 β − + − − + + 1 = 2 = − κ + κ − 1 = 2 = ³ ³ ´ ´2 2 β 1 2 ,1 2 1 − κ + κ −

[email protected] Fat Tail Analysis and Package FatTailsR 9th R/Rmetrics Workshop 27June2015 11 / 26 2 Pearson chart (β1, β2)

2 We plot the skewness β1 and kurtosis β2 of distributions K1 and K2, K3, K4 in chart (β1, β2). K1 is on the vertical β2 axis. K2, K3 and K4 are located under the log-logistic curve. Some remarkable points at the limits:

(α , ω ) (κ , ² 0) (β1 0, β2 4.2) = ∞ = ∞ ⇒ = ∞ = ⇒ = = (α 4.1, ω 4.1) (κ 4.1, ² 0) (β1 0, β2 64.8) = = ⇒ = = ⇒ = = 2 (α , ω 4.1) (κ 8.2, ² 1) (β1 3.97, β 15.8, β2 340) = ∞ = ⇒ = = ⇒ = 1 = =

b 2 2 - b -1 4.2 1 = 0

8 log-normal

log-logistic b2

16 domain K2, K3, K4

24 2 0 1 2 3 4 5 b1

Figure : Skewness β1 and kurtosis β2 of distributions K1 and K2, K3, K4

The domain is identical to the domain associated to Tadikamalla and Johnson LU distribution.

[email protected] Fat Tail Analysis and Package FatTailsR 9th R/Rmetrics Workshop 27June2015 12 / 26 Experimental and theoretical variance

Large datasets are required to reach the theoretical variance or the standard deviation K1 r q 1 PN 2 ³ 2 2 ´ σ (x(pi ) µ) (coloured lines) vs (black line) σ γκ 2β 1 ,1 2 = N i − = − κ + κ −

³ 1 ³ i ´´ x(pi ) µ 2γκ sinh κ logit N 1 with µ 0, γ 1, N (20, 50, 100, 250, 1000, 10.000) = + + = = =

Size of the dataset and convergence to theoretical variance 6.0 sqrt.mu_2 N = 10000 N = 1000 N = 250 5.5 N = 100 N = 50 N = 20

5.0 − σ N 1∑(x(i) − µ)2 x(i) = K1(i (N + 1), κ) σ

4.5

4.0

3.5

3.0

2 4 6 κ 8 10

Figure : Theoretical variance and experimental variance related to the size of the dataset

Explanation: There is a lot of information carried by the tails and not seen in small datasets! In most cases, standard deviation should not be used with fat tails.

[email protected] Fat Tail Analysis and Package FatTailsR 9th R/Rmetrics Workshop 27June2015 13 / 26 Parameter estimation

Mathematics in FatTailsR package Plot and ﬁgures in FatTailsRplot package

2 algorithms and 3 functions are proposed in FatTailsR to estimate the parameters

1 Nonlinear regression Functions regkienerLX ( list), ﬁtkienerLX ( data.frame) Work with K1, K2, K3, K4→ → Use the whole dataset of size N i Perform a nonlinear regression from logit(F (x)) to x with F (xi ) pi N 1 Nonlinear least squares = = + Logit(Proba) : K1 ( µ = 0, γ = 1, κ = ... ) + Logistique + Gauss (σ = 3.2 ) Levenberg-Marquardt algorithm (package minpack.lm) 6.9 κ 0.6 Examples: slide 16, videos of slide 17 4.6 1 1.5 2 2.9 3.2 10 Logistique 2 Quantile estimation Gauss 0.0 Function estimkienerX ( data.frame) logit(q.999) = 6.9 → −2.9 logit(q.99) = 4.6 Works with K2, K3, K4 logit(q.95) = 2.9 logit(q.50) = 0 −4.6 logit(q.05) = −2.9 logit(q.01) = −4.6 Requires 5 quantiles only! logit(q.001) = −6.9 −6.9 Very fast! −15 −10 −5 0 5 10 15 Examples: slides 19–20

[email protected] Fat Tail Analysis and Package FatTailsR 9th R/Rmetrics Workshop 27June2015 14 / 26 Fat tail characterization

Pair (κ, δ) describes the tails. Let us decompose the quantile function x(p) as: the median µ and scale γ linear part =⇒ the tail as a function of κ and δ nonlinear part, asymmetric curvature =⇒ K3, C3 — Corrective tail function C(p, κ,δ), g logit(p) = κ ³ g ´ C(p,κ,δ) sinh exp(g δ) x(p) µ 2logit(p) γ C(p, κ, δ) = g κ ⇐⇒ = + ⇑ ⇑

C(p, κ, δ) acts as a multiplier of the logistic asymptote and incorporates the imbalance between negative and positive tails. For simplicity, let us consider: c01 C (0.01, κ, δ) Risk over long periods = =⇒ c05 C (0.05, κ, δ) Risk during trading = =⇒

1.8 κ 1.8 κ 6.9 lg.999 = 6.9 6.9 lg.999 = 6.9 lg.99 = 4.6 1.5 lg.99 = 4.6 1.5 lg.95 = 2.9 2 lg.95 = 2.9 2 4.6 lg.50 = 0 3 4.6 lg.50 = 0 3 lg.05 = −2.9 1.6 4 lg.05 = −2.9 1.6 4 lg.01 = −4.6 6 lg.01 = −4.6 6 2.9 lg.001 = −6.9 10 2.9 lg.001 = −6.9 10 logistic logistic

ε 1.4 ε ε 1.4 ε 0.0 0 0 0.0 −0.15 −0.15

κ κ 1.2 1.2 −2.9 1.5 −2.9 1.5 2 2 3 3 −4.6 4 −4.6 4 6 6 10 1.0 10 1.0 −6.9 logistic −6.9 logistic −15 −10 −5 0 5 10 15 −6.9 −4.6 −2.9 0.0 2.9 4.6 6.9 −15 −10 −5 0 5 10 15 −6.9 −4.6 −2.9 0.0 2.9 4.6 6.9

Figure : (a) Logit-Proba K4(0, 1, κ, ² 0) (b) C4(p, κ, ² 0) in logit scale = = (c) Logit-Proba K4(0, 1, κ, ² 0.15) (d) C4(p, κ, ² 0.15) in logit scale = − = −

[email protected] Fat Tail Analysis and Package FatTailsR 9th R/Rmetrics Workshop 27June2015 15 / 26 Example 1: SP500

SP500 from 1 January 1957 to 31 December 2013: Days: 14349 points, Weeks: 2975 points, Months: 685 points

SP500 janv. 1957 − déc. 2013 SP500 janv. 1957 − déc. 2013 100 µ F(X) > 0.995 14349 points ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●● ●● ● ● ● X − 14350 points ●●●●●●●●●●●●●● 1.0 ●●●●●●●● ●●●●●● ●●●● ● ●●● ●●●●●●●● ●●●●● ● ●● ● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●● ●●●●●●●●● ●●● ●●●● ●● 2975 points ● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●● ●●●●●●●●●●●●●●● ●●●●● ●●● ● ●●●●●● ● ●● ●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●● ●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●● ● ●● ●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●● ●●●●●●●●●●●● ●●●●●● ●●●●●●●●● ● ●● ●●●●●●●●●●●●●●●●●●● ●●●●●●●● ●●●●●●●●●●●●● ●●●● ●●●●●● ● ●● ●●●●●●●●●●●●●●●●● ●●●●●● ●●●●●●●●●●●● ●●●●●● ●●●●●●●● ● ●● ●●●●●●●●●●●●●●●●● ●●●●●● ●●●●●●●●●● ●●●●● ●●●● ● ● ●●●●●●●●●●●●●●●● ●●●●●● ●●●●●●●● ●●● ●●●●● ● ● ●●●●●●●●●●●●●●●● ●●●●● ●●●●●●● ●●●● ●●●●● ● ● ●●●●●●●●●●●●●●●● ●●●● ●●●●●●●● ●●● ●●● 685 points ● ● ●●●●●●●●●●●●●● ●●●●● ●●●●●●●● ●●● ●●● ● ● ●●●●●●●●●●●●●● ●●●●● ●●●● ●●●● ●●●●●● ● ● ●●●●●●●●●●●●●● ●●●● ●●●●●●● ●●● ●●●● ● ● ●●●●●●●●●●●●● ●●●● ●●●●●● ●●● ●●●●● ● ● ●●●●●●●●●●●● ●●●●● ●●●●● ●●● ●●● ● ● ●●●●●●●●●●●● ●●●●● ●●●●● ●●● ●●● ● ● ●●●●●●●●●●●● ●●●● ●●●●● ●● ●●●● ● ● ●●●●●●●●●●● ●●●●● ●●●● ●●● ●●●● ● ● ●●●●●●●●●● ●●● ●●●●● ●●● ●●●●● ● ● − ●●● ●●●●● ●●● ●●●● ●● ●●● 0.999 ● ● 1 ●●● ●●●● ●●● ●●● ● ●●● ● ● ●●●●●●●●●● ●● ●●●● ●● ● ● ● ●●●●●●●●● ●●●● ●●●● ●● ●●● ● ● ●●● ●●●●● ●●● ●●●● ●● ● ● ● 10 ●●● ●●●●● ●●● ●●● ●● ●● ● ● ●● ●●● ●● ●●● ● ●● ● ● ●●● ●●●● ●●● ●●● ● ●● ● ● log% k2 k2 k2 ●●● ●●●● ●●● ●●●● ●● ●●● ● ● ●●● ●●●●● ●●● ●●● ●● ●● ● ● ●●● ●●●●● ●● ●●●● ● ●●● ● ● ●●● ●●●●● ●● ●●● ●● ●●● ● ● ●●●● ●●●● ●●● ●●●●● ●● ● ● ● ●●● ●●●●● ●●● ●●● ●● ●● ● ● ●●● ●●●● ●● ●●●● ●● ● ● ● ●●● ●●●● ●● ●●● ●● ● ● ● ●●●●●●● ●● ●●●● ●● ●● ● ● ●● ●●●● ●● ●●● ● ● ● ● ● ●● ●●● ●●● ●● ●● ● ● ● ●●● ●●● ●●● ●●● ● ● ● ● ●●●●●● ●● ●● ● ● ● ● x.4 = 30.9 8.7 5.5 ●●●●●● ●● ●●● ● ●● ● ● ●●●●●●● ●● ●● ●● ● ● ● ●●●●● ●● ●● ● ●● ● ● ●●●●● ●● ●● ●● ● ● ● ●●●●●● ●● ●● ● ● ● ● ●●●●● ●●● ●●●● ● ● ● ● x.3 = −1 −0.6 −0.7 ●●●●● ● ● ● ● ● ● ●●●●● ●● ●● ● ● ● ● ●●●●● ●● ●● ● ● 0.997 ● ● ● ● ● ● ● ● ●●●●●●● ● ● ● ● ● ● ●●●●● ●● ●●● ● ● ● σ = ●●●●●● ●●● ● ● ● ● ● 0.999 2.18 4.27 ●●●●●● ●● ●● ● ● ● ●●●● ● ● ● ● ● ● ●●●●● ●● ●● ● ● ● ● ●● ● ●● ● ● ● ●●●●● ●● ●●● ● ● ● ● ●●●● ● ● ● ● ● ●●●● ● ●● ● ● ● − m = 0.026 0.124 0.539 ●●●●●● ●● ● ● ● ● 2 ●●●● ● ● ● ● q.9999 = 10.8 ● ●●●●●● ● ●● ● ● ● ● ●● ●● ●● ● ● ● ●●●●● ●● ● ● ● ● ● 10 ●●●● ● ●● ● ● ●●● ● ● ● ● ●●●● ● ● ● ● q.99 = 2.6 q.999 = 5.4 ● ●●●● ● ● ● ● ● ●●● ● ● ● ● F(X) < 0.005 ●●●● ● ●● ● µ = ●●● ● ● ● ● ● ●●●●● ● ● ● 0.044 0.3 0.937 ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ●●●● ● ● ● ● γ = ●●●● ● ● 0.995 ● ● ● ● ● ●●● ● ● ● 0.213 0.522 1.11 ●●● ● ● ● ● ● ● ●●● ● ● 4 6 8 10 ● q.001 = −6.2 q.01 = −2.8 ●●● ● ● ● ●● ● ● ● α = ●● ● ● ● ● 3.1 3.7 5 ●●● ● ● ●● ● ● ● ● q.0001 = −13.3 ● ●● ● 0.5 ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ω = ●● ●● ● ● ● 3.3 4.7 8.5 ● ●● log% k2 ● ● ● ●● ● ● ● ● ●● ●● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● − ● ● ● ● ● 3 ● ● ● 0.004 ● ● ● ● ● ● ● ● ● 10 ● ● ● ● x.4 = 30.9 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● q.999 = 5.4 9.5 14.9 ● ● ● x.3 = −1 ● ● ● ● ● ● ● ● ● ● σ = ● ●● q.99 = 2.6 5.5 10.1 ● ● 0.999 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● q.95 = 1.4 3.4 6.9 ● m = 0.026 ● ● ● ● ● ● ● ●● ● ● q.5 = 0.044 0.3 0.937 ● ● ● ● ● ● ● ● ● ● q.05 = −1.4 −3.3 −6.8 µ = ● ● 0.044 ● 0.002 ● ● ● ●● − ● ● 4 γ = ● ● q.01 = −2.8 −6.3 −12.6 0.213 ● ● 10 ● ● ● ● ● ● q.001 = −6.2 −13 −24 α = ● ● 3.1 ● ● ● ● ● ● ●● ● ω = ●● ● 3.3 ●● ● ●● ● ● ●● ● ●● ● ●●● ● ●●● ● ●●● ● ●●● ●●●● ●●●● ●●●●●●● ●●●●●●●●● ●●●●●●●●●● 0.000 0.0 ●●● ● ● ●●●● ● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● −20 −15 −10 −5 − 10 5 −1 0 1 2 −10 −5 0 5 10 10 10 10 10

Figure : (a) SP500 daily log-returns (b) SP500 daily, weekly and monthly log-returns in log-log scale black dots = negative returns, white circles = positive returns

[email protected] Fat Tail Analysis and Package FatTailsR 9th R/Rmetrics Workshop 27June2015 16 / 26 Example 2: Videos SP500 — Rolling windows and Garch

SP500 from January 2005 to December 2013, rolling windows 252 points rebalanced every month (left) without Garch (right) K2 applied on Garch(1,1) residuals

Figure : (a) SP500 log-returns + K2 (b) SP500 log-returns + Garch(1,1) + K2

Click on the links to watch the videos online: url: SP500 url: SP500-Garch url: VIX url: VIX-Garch

[email protected] Fat Tail Analysis and Package FatTailsR 9th R/Rmetrics Workshop 27June2015 17 / 26 SP500 in 2011 and 2012 — Legend of next ﬁgures

Rolling windows 21, 41, 101 days rebalanced every day on SP500 index

Plots: Left: Year 2011 Right: Year 2012

Plots:

1 Top: Prices + rolling median (dots = 0)

2 Middle top: Rolling parameters c05 and γ

3 Middle bottom: Rolling parameters 1/κ and δ

4 Bottom: Returns + rolling quantiles at 5% and 95 % justiﬁes c05 rather than c01 ⇒

Colours: Black: Prices and returns on a daily basis Green: Rolling windows 101 days rebalanced every day Blue: Rolling windows 41 days rebalanced every day Red: Rolling windows 21 days rebalanced every day Cyan: Moving median 21 days rebalanced every day

[email protected] Fat Tail Analysis and Package FatTailsR 9th R/Rmetrics Workshop 27June2015 18 / 26 Example 3: SP500 in 2011 and 2012 — Rolling 21, 41, 101 days

1400 2011−01−03 SP500 2011−12−30 2012−01−04 SP500 2012−12−31 1350 1450

1300 1400 1250

Prix 1200 Prix 1350

1150 1300 1100

1250 1050

janv. mars mai juil. sept. nov. janv. janv. mars mai juil. sept. nov. janv.

1.5 1.5

1.25 1.25 c05 c05

1 1

0.78 0.78 gamma gamma 0.09 0.09

janv. mars mai juil. sept. nov. janv. janv. mars mai juil. sept. nov. janv.

1/2 1/2

1/3 1/3 1/4 1/4 1/kappa 1/kappa 1/10 1/10

0.07 0.07 delta delta

−0.07 −0.07

janv. mars mai juil. sept. nov. janv. janv. mars mai juil. sept. nov. janv.

4 4

2 2

0 0 Rendement Rendement −2 −2

−4 −4

janv. mars mai juil. sept. nov. janv. janv. mars mai juil. sept. nov. janv.

[email protected] FigureFat : TailSP500 Analysis during and Package years 2011 FatTailsR and 2012. 9th R/Rmetrics Workshop 27June2015 19 / 26 SP500 — Evolving parameters c05, q05. Rolling 41 days

2011−08−15 1.001.081.151.251.351.50 d, 1/k, c05 − SP500 − 41 days 1.001.081.151.251.351.50 d, 1/k, c05 − SP500 − 41 days 2011−01−03 − 2011−12−23 2012−01−04 − 2012−12−31 1/1.5 1/1.5 1.5 1.35 1.5 1.35 2011−01−03 1.25 1.25 2011−08−29 2011−01−19

1.15 2012−10−26 1.15

1/2 1/2 2011−03−04 2012−12−14 2012−09−28 2011−02−02 1.08 1.08 1 2012−01−20 2012−09−14 1 2012−11−30 2012−08−15 1/k 1/k 2012−10−12

2011−09−14 2012−12−31 2012−06−04 2012−02−17 2012−11−14 2012−02−03 2011−04−01 2012−08−29 1/3 1/3 2011−04−15 2012−04−19 2011−03−18 2011−05−032011−02−16 2012−03−20 2012−03−06 1/4 1/4 2012−07−02 2012−07−18 2012−08−01 2011−09−28 2012−06−18 2011−12−23

2012−05−17 1/6 1/6 2012−05−03 2011−05−17 2012−01−042012−04−03 2011−06−02 1/10 1/10 2011−06−162011−07−182011−11−092011−10−122011−10−262011−06−30 2011−08−012011−11−232011−12−09

1/20 1/20 −0.10 −0.05 0.00 d 0.05 0.10 −0.10 −0.05 0.00 d 0.05 0.10

0.661.302.103.304.906.90 g, c05, q05 − SP500 − 41 days 0.661.302.103.304.906.90 g, c05, q05 − SP500 − 41 days 2011−01−03 − 2011−12−23 2012−01−04 − 2012−12−31

1.80 1.80 ● 2011−08−29

● 6.9 6.9 c05 ● ● c05 2012−12−14 2011−01−032011−03−04 ● ●● 2011−01−19 2012−11−142012−11−30 1.50 ● 1.50 2011−04−01

● ● 2011−02−02 ● ● 2012−10−26 1.35 2011−03−18 1.35 ● ● 2011−09−14 ● 2011−02−162011−05−03● 2011−04−15 ● ● ● ● 1.25 2011−06−16 ● 1.25 2011−06−022011−07−18 ●● ● 2012−12−312012−08−15● ● 2011−09−28 ● 2011−10−122011−10−262011−11−09 2012−06−04● 1.15 2011−06−30 1.15 2012−04−19 2012−02−03● ● ● ● ● ● ● 2012−01−20 1.08 ● 1.08 2012−06−182012−07−18 2011−05−17 ● 2012−03−06 2012−07−02 2011−08−01 2012−02−17● 2011−12−23 ● ●●● ● 4.9 2012−05−03 4.9 1.00 1.00 ● ● 2011−11−23 2012−09−282012−09−142012−10−12 ● 2011−12−09 2012−03−20● ● 2012−08−01 2012−05−17 2012−01−04 0.90 0.90 2012−04−03● 2012−08−29

1.3 1.3 0.80 0.66 0.80 0.66 2.1 2.1 3.3 3.3

0.2 0.4 g 0.6 0.8 0.2 0.4 g 0.6 0.8

Figure : Parameters c05 (top) and q05 (bottom) from January 2011 (left) to December 2012 (right). SP500: Rolling periods of 41 days rebalanced every 10 days

[email protected] Fat Tail Analysis and Package FatTailsR 9th R/Rmetrics Workshop 27June2015 20 / 26 Evolving parameter c01 — Rolling 88 days

³ ´ c01(κ,δ) κ sinh 4.6 exp( 4.6 δ) changes over time. We can track it using plot (δ,1/κ). = 4.6 κ −

1.0 1.2 1.4 1.6 1.9 2.3 d, 1/k, c01 − DJIA 1.0 1.2 1.4 1.6 1.9 2.3 d, 1/k, c01 − CAC40 janv.10 − nov.13 janv.10 − nov.13 1/1.5 1/1.5 2.3 2.3

1.9 1.9

1/2 1/2 1.6 1.4 mai10 1.6 1.4

1/k mai10mars11 janv.11 1.2 1/k 1.2 mars12 nov.12 1 1 1/3 1/3 mars13sept.13 janv.13 janv.13 juil.10 nov.10 1/4 sept.11 sept.10 1/4 mars10 juil.13nov.12 sept.12 nov.13 sept.12 nov.10 janv.12 1/6 1/6 sept.10 mars10mars12 1/10 janv.10 1/10 sept.11 juil.13mai11mai13nov.11sept.13mars13juil.11 mai12nov.13 juil.12 mai12 janv.10mai13mai11juil.11janv.11nov.11juil.10mars11juil.12 janv.12 1/20 1/20 −0.10 −0.05 0.00 d 0.05 0.10 −0.10 −0.05 0.00 d 0.05 0.10

1.0 1.2 1.4 1.6 1.9 2.3 d, 1/k, c01 − GOLD 1.0 1.2 1.4 1.6 1.9 2.3 d, 1/k, c01 − SOCGEN 1.0 1.2 1.4 1.6 1.9 2.3 d, 1/k, c01 − VIVENDI janv.10 − nov.13 janv.10 − nov.13 janv.10 − nov.13 1/1.5 1/1.5 1/1.5 2.3 2.3 2.3 mai10 1.9 1.9 1.9 mai10 mars12 mai13 1/2 1/2 1/2 1.6 1.4 1.6 1.4 1.6 1.4 juil.10 juil.10 mai12 1/k 1.2 1/k 1.2 1/k 1.2 sept.13 mai11 sept.10 mai10 janv.10 1 1 1 1/3 1/3 1/3 juil.10 mai12 sept.11 juil.11juil.12 nov.12 juil.13 sept.11 1/4 1/4 1/4 nov.11 mai11 nov.11 mars10 nov.13 sept.12 sept.12juil.13 janv.11 sept.13mars13nov.10 1/6 mars11 1/6 sept.10 1/6 juil.11 sept.10 1/10 1/10 1/10 janv.13sept.13nov.12mars13nov.13mars12janv.12nov.10 mars10 mars13janv.10mai11janv.13mai13mars12mai12nov.10janv.12mars11janv.11juil.12nov.13 juil.11 juil.13 sept.11mai13janv.13juil.12janv.11sept.12nov.11nov.12mars10mars11 janv.12 janv.10 1/20 1/20 1/20 −0.10 −0.05 0.00 d 0.05 0.10 −0.10 −0.05 0.00 d 0.05 0.10 −0.10 −0.05 0.00 d 0.05 0.10

Figure : Parameter c01 from October 2009 (January 2010) to November 2013. Rolling periods of 4 months ( 88 days) rebalanced every 2 months: SP500, DJIA, CAC40, Gold, Société Générale, Vivendi. ≈

[email protected] Fat Tail Analysis and Package FatTailsR 9th R/Rmetrics Workshop 27June2015 21 / 26 Evolving parameter c01 — Table

1.0 1.2 1.4 1.6 1.9 2.3 c01 − 4m − 2m mai07 − nov.13

SP500

DJIA

CAC40

EURUSD

VIVENDI

SOCGEN

GOLD

mai07 nov.07 mai08 nov.08 mai09 nov.09 mai10 nov.10 mai11 nov.11 mai12 nov.12 mai13 nov.13

Figure : Parameter c01 from February 2007 (May 2007) to November 2013. Rolling periods of 4 months ( 88 days) rebalanced every 2 months: SP500, DJIA, CAC40, Gold, Société Générale, Vivendi. ≈

[email protected] Fat Tail Analysis and Package FatTailsR 9th R/Rmetrics Workshop 27June2015 22 / 26 Conclusion (1)

A lot of knowledge on distributions with fat tails has been gained during the last 12 months

K1, K2, K3, K4 distributions are good candidates for describing distributions with fat tails They compare favourably to Tukey Generalized Lambda distribution, Tadikamalla and Johnson distribution, Student distribution, Cauchy distribution (see presentation last year for these last two). ³ 2 ´ They cover a vast domain under the log-logistic curve in Pearson β1, β2 plot. Their moments fall exactly at the expected values of the tail parameters α, ω, κ. Their parameters are well separated: median µ, scale γ, shape κ, distorsion δ or eccentricity ². Their parameters can be estimated by 2 diﬀerent methods: nonlinear regression, quantile estimation.

Eﬀective tools to measure risk Risk is split in scale risk (γ ampliﬁcation) and asymmetric tail risk (c01, c05). Approximation is good for datasets of various sizes: from 21 points to ten of thousands points. SP500 has skewed and fat tails!

Potential developments Market risk (Bâle III, bcbs240, Jan-Fev.2013) Include K3 distribution in the regulation? Portfolio management Extend the results to=⇒ multidimensional distributions. =⇒ Derivatives New process µγκδ-Garck (Garck to diﬀerentiate from Gaussian Garch)? Bayesian change=⇒ points Combine K1..K4 distributions with BCP? =⇒ Stability index Combine K1..K4 distributions (κ, c01, c05) with SI? =⇒

Two R packages: FatTailsR (license GPL-2, on CRAN) and FatTailsRplot (contact me)

[email protected] Fat Tail Analysis and Package FatTailsR 9th R/Rmetrics Workshop 27June2015 23 / 26 Conclusion (2)

A remark rather than a conclusion:

During this presentation, I have talked a lot about risks but I never used the words « volatility » and « standard deviation » in the restrictive Gaussian meaning.

Distributions K1, K2, K3 and K4 introduce a new way of thinking of risk. They deal with a 4 dimension space which is much larger than the 2 dimension space of the Gaussian distribution.

With these new distributions, standard deviation is not the appropriate measure of risk.

Please, think about it.

[email protected] Fat Tail Analysis and Package FatTailsR 9th R/Rmetrics Workshop 27June2015 24 / 26 References

Literature 1 N. L. Johnson, S. Kotz, N. Balakrishnan, Continuous univariate distributions, 2nd ed., Wiley, vol. 1, 1994 and vol. 2, 1995. 2 P. Kiener, Explicit models for bilateral fat-tailed distributions and applications in ﬁnance with the package FatTailsR, 8th R/Rmetrics workshop and summer school, Paris, 27 June 2014, http://www.inmodelia.com/fattailsr-en.html 3 C. Ley, Flexible modelling in statistics: past, present and future, J. Soc. Fran. Stat., Vol. 156, No. 1, 76-96, 2015. http://journal-sfds.fr/index.php/J-SFdS/article/download/415/394 4 The R project for statistical computing, http://www.r-project.org

Software Package FatTailsR, version 1.2-0, 18 June 2015, available on CRAN, license GPL-2, http://cran.r-project.org/web/packages/FatTailsR/index.html Package FatTailsRplot, version 1.2-0, 18 June 2015, http://www.inmodelia.com/fattailsr-en.html

To cite and download this document P. Kiener, Fat tail analysis and package FatTailsR, 9th R/Rmetrics workshop and summer school, Zurich, 27 June 2015, http://www.inmodelia.com/fattailsr-en.html

[email protected] Fat Tail Analysis and Package FatTailsR 9th R/Rmetrics Workshop 27June2015 25 / 26 .

Thank you for your attention !

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Tél. : +33.9.53.45.07.38

[email protected] Fat Tail Analysis and Package FatTailsR 9th R/Rmetrics Workshop 27June2015 26 / 26