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Slides univ-van-amsterdam

  1. 1. Arthur CHARPENTIER - Nonparametric quantile estimation. Estimating quantiles and related risk measures Arthur Charpentier arthur.charpentier@univ-rennes1.fr Universiteit van Amsterdam, January 2008 joint work with Abder Oulidi, IMA Angers 1
  2. 2. Arthur CHARPENTIER - Nonparametric quantile estimation. Agenda • General introduction Risk measures • Distorted risk measures • Value-at-Risk and related risk measures Quantile estimation : classical techniques • Parametric estimation • Semiparametric estimation, extreme value theory • Nonparametric estimation Quantile estimation : use of Beta kernels • Beta kernel estimation • Transforming observations A simulation based study 2
  3. 3. Arthur CHARPENTIER - Nonparametric quantile estimation. Agenda • General introduction Risk measures • Distorted risk measures • Value-at-Risk and related risk measures Quantile estimation : classical techniques • Parametric estimation • Semiparametric estimation, extreme value theory • Nonparametric estimation Quantile estimation : use of Beta kernels • Beta kernel estimation • Transforming observations A simulation based study 3
  4. 4. Arthur CHARPENTIER - Nonparametric quantile estimation. Risk measures and price of a risk Pascal, Fermat, Condorcet, Huygens, d’Alembert in the XVIIIth century proposed to evaluate the “produit scalaire des probabilit´s et des gains”, e n < p, x >= pi xi = EP (X), i=1 based on the “r`gle des parties”. e For Qu´telet, the expected value was, in the context of insurance, the price that e guarantees a financial equilibrium. From this idea, we consider in insurance the pure premium as EP (X). As in Cournot (1843), “l’esp´rance math´matique est donc le juste prix des chances” e e (or the “fair price” mentioned in Feller (1953)). 4
  5. 5. Arthur CHARPENTIER - Nonparametric quantile estimation. Risk measures : the expected utility approach Ru (X) = u(x)dP = P(u(X) > x))dx where u : [0, ∞) → [0, ∞) is a utility function. Example with an exponential utility, u(x) = [1 − e−αx ]/α, Ru (X) = 1 log EP (eαX ) . α 5
  6. 6. Arthur CHARPENTIER - Nonparametric quantile estimation. Risk measures : Yarri’s dual approach Rg (X) = xdg ◦ P = g(P(X > x))dx where g : [0, 1] → [0, 1] is a distorted function. Example – if g(x) = I(X ≥ 1 − α) Rg (X) = V aR(X, α), – if g(x) = min{x/(1 − α), 1} Rg (X) = T V aR(X, α) (also called expected shortfall), Rg (X) = EP (X|X > V aR(X, α)). 6
  7. 7. Arthur CHARPENTIER - Nonparametric quantile estimation. Distortion of values versus distortion of probabilities 0.0 0.2 0.4 0.6 0.8 1.0 Calcul de l’esperance mathématique 0 1 2 Fig. 1 – Expected value 3 4 xdFX (x) = 5 6 P(X > x)dx. 7
  8. 8. Arthur CHARPENTIER - Nonparametric quantile estimation. Distortion of values versus distortion of probabilities 0.0 0.2 0.4 0.6 0.8 1.0 Calcul de l’esperance d’utilité 0 1 2 3 Fig. 2 – Expected utility 4 5 6 u(x)dFX (x). 8
  9. 9. Arthur CHARPENTIER - Nonparametric quantile estimation. Distortion of values versus distortion of probabilities 0.0 0.2 0.4 0.6 0.8 1.0 Calcul de l’intégrale de Choquet 0 1 2 3 Fig. 3 – Distorted probabilities 4 5 6 g(P(X > x))dx. 9
  10. 10. Arthur CHARPENTIER - Nonparametric quantile estimation. Distorted risk measures and quantiles Equivalently, note that E(X) = Rg (X) = 1 0 1 0 −1 FX (1 − u)du, and −1 FX (1 − u)dgu. A very general class of risk measures can be defined as follows, 1 Rg (X) = 0 −1 FX (1 − u)dgu where g is a distortion function, i.e. increasing, with g(0) = 0 and g(1) = 1. Note that g is a cumulative distribution function, so Rg (X) is a weighted sum of quantiles, where dg(1 − ·) denotes the distribution of the weights. 10
  11. 11. Arthur CHARPENTIER - Nonparametric quantile estimation. 0.2 0.4 0.6 0.8 1.0 0.8 0.0 qqqqqqqqqq qqqqqqqqq qqqqqqqqq qqqqqqqqq qqqqqqqqq 0.0 0.2 0.4 0.6 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq Distortion function, cdf 1.0 Distortion function, TVaR (expected shortfall) − cdf 1.0 1.0 Distortion function, VaR (quantile) − cdf 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 1 − probability level Distortion function, TVaR (expected shortfall) − pdf 0.8 1.0 0.8 1.0 1 − probability level Distortion function, VaR (quantile) − pdf 1.0 1 − probability level Distortion function, pdf 0.0 0.2 0.4 0.6 1 − probability level 0.8 1.0 0 qqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq 0 0.0 1 1 0.2 2 2 0.4 3 3 4 0.6 5 4 0.8 6 5 q 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 1 − probability level 0.4 0.6 1 − probability level Fig. 4 – Distortion function, g and dg 11
  12. 12. Arthur CHARPENTIER - Nonparametric quantile estimation. Agenda • General introduction Risk measures • Distorted risk measures • Value-at-Risk and related risk measures Quantile estimation : classical techniques • Parametric estimation • Semiparametric estimation, extreme value theory • Nonparametric estimation Quantile estimation : use of Beta kernels • Beta kernel estimation • Transforming observations A simulation based study 12
  13. 13. Arthur CHARPENTIER - Nonparametric quantile estimation. Using a parametric approach −1 If FX ∈ F = {Fθ , θ ∈ Θ} (assumed to be continuous), qX (α) = Fθ (α), and thus, a natural estimator is qX (α) = F −1 (α), θ (1) where θ is an estimator of θ (maximum likelihood, moments estimator...). 13
  14. 14. Arthur CHARPENTIER - Nonparametric quantile estimation. Using the Gaussian distribution A natural idea (that can be found in classical financial models) is to assume Gaussian distributions : if X ∼ N (µ, σ), then the α-quantile is simply q(α) = µ + Φ−1 (α)σ, where Φ−1 (α) is obtained in statistical tables (or any statistical software), e.g. u = −1.64 if α = 90%, or u = −1.96 if α = 95%. Definition 1. Given a n sample {X1 , · · · , Xn }, the (Gaussian) parametric estimation of the α-quantile is qn (α) = µ + Φ−1 (α)σ, 14
  15. 15. Arthur CHARPENTIER - Nonparametric quantile estimation. Using a parametric models Actually, is the Gaussian model does not fit very well, it is still possible to use Gaussian approximation If the variance is finite, (X − E(X))/σ might be closer to the Gaussian distribution, and thus, consider the so-called Cornish-Fisher approximation, i.e. Q(X, α) ∼ E(X) + zα V (X), (2) where −1 zα = Φ 2 ζ1 −1 2 ζ2 −1 3 ζ1 −1 (α)+ [Φ (α) −1]+ [Φ (α) −3Φ (α)]− [2Φ−1 (α)3 −5Φ−1 (α)], 6 24 36 where ζ1 is the skewness of X, and ζ2 is the excess kurtosis, i.e. i.e. E([X − E(X)]3 ) E([X − E(X)]4 ) ζ1 = and ζ2 = − 3. E([X − E(X)]2 )2 E([X − E(X)]2 )3/2 (3) 15
  16. 16. Arthur CHARPENTIER - Nonparametric quantile estimation. Using a parametric models Definition 2. Given a n sample {X1 , · · · , Xn }, the Cornish-Fisher estimation of the α-quantile is 1 qn (α) = µ + zα σ, where µ = n n Xi and σ = i=1 1 n−1 n 2 (Xi − µ) , i=1 and 2 ζ2 −1 3 ζ1 ζ1 −1 2 zα = Φ−1 (α)+ [Φ (α) −1]+ [Φ (α) −3Φ−1 (α)]− [2Φ−1 (α)3 −5Φ−1 (α)], 6 24 36 (4) where ζ1 is the natural estimator for the skewness of X, and ζ2 is the natural √ n(n − 1) n n (Xi − µ)3 i=1 estimator of the excess kurtosis, i.e. ζ1 = and n − 2 ( n (Xi − µ)2 )3/2 i=1 ζ2 = n−1 (n−2)(n−3) (n + 1)ζ2 + 6 where ζ2 = n ( n 4 i=1 (Xi −µ) n 2 2 i=1 (Xi −µ) ) − 3. 16
  17. 17. Arthur CHARPENTIER - Nonparametric quantile estimation. Parametrics estimator and error model Density, theoritical versus empirical Theoritical Student Fitted lStudent Fitted Gaussian 0.0 0.0 0.2 0.1 0.4 Theoritical lognormal Fitted lognormal Fitted gamma 0.2 0.6 0.3 0.8 Density, theoritical versus empirical 0 1 2 3 4 5 −4 −2 0 2 4 Fig. 5 – Estimation of Value-at-Risk, model error. 17
  18. 18. Arthur CHARPENTIER - Nonparametric quantile estimation. Using a semiparametric models Given a n-sample {Y1 , . . . , Yn }, let Y1:n ≤ Y2:n ≤ . . .≤ Yn:n denotes the associated order statistics. If u large enough, Y − u given Y > u has a Generalized Pareto distribution with parameters ξ and β ( Pickands-Balkema-de Haan theorem). If u = Yn−k:n for k large enough, and if ξ> 0, denote by βk and ξk maximum likelihood estimators of the Genralized Pareto distribution of sample {Yn−k+1:n − Yn−k:n , ..., Yn:n − Yn−k:n }, Q(Y, α) = Yn−k:n + βk ξk n (1 − α) k − ξk −1 , (5) An alternative is to use Hill’s estimator if ξ > 0, Q(Y, α) = Yn−k:n n (1 − α) k − ξk 1 , ξk = k k log Yn+1−i:n − log Yn−k:n . (6) i=1 18
  19. 19. Arthur CHARPENTIER - Nonparametric quantile estimation. On nonparametric estimation for quantiles −1 For continuous distribution q(α) = FX (α), thus, a natural idea would be to −1 consider q(α) = FX (α), for some nonparametric estimation of FX . Definition 3. The empirical cumulative distribution function Fn , based on n 1 sample {X1 , . . . , Xn } is Fn (x) = 1(Xi ≤ x). n i=1 Definition 4. The kernel based cumulative distribution function, based on sample {X1 , . . . , Xn } is 1 Fn (x) = nh n x k i=1 −∞ Xi − t h 1 dt = n n K i=1 Xi − x h x where K(x) = k(t)dt, k being a kernel and h the bandwidth. −∞ 19
  20. 20. Arthur CHARPENTIER - Nonparametric quantile estimation. Smoothing nonparametric estimators Two techniques have been considered to smooth estimation of quantiles, either implicit, or explicit. • consider a linear combinaison of order statistics, The classical empirical quantile estimate is simply −1 Qn (p) = Fn i n = Xi:n = X[np]:n where [·] denotes the integer part. (7) The estimator is simple to obtain, but depends only on one observation. A natural extention will be to use - at least - two observations, if np is not an integer. The weighted empirical quantile estimate is then defined as Qn (p) = (1 − γ) X[np]:n + γX[np]+1:n where γ = np − [np]. 20
  21. 21. Arthur CHARPENTIER - Nonparametric quantile estimation. The quantile function in R q q qq qq qq q type=1 type=3 type=5 type=7 qq qq qqq qq qqqq qq qq q qqqqq qqqq qq qqqq qqq q qqqq q qq q qqq qqq q qqqq qq qqq q qq qq qqqq qq 5 qqq q qq qqq qq qq 3 4 q 4 quantile level 6 6 q q q q 2 2 quantile level 8 type=1 type=3 type=5 type=7 7 The quantile function in R q q 0.0 0.2 0.4 0.6 probability level 0.8 1.0 qq qq 0.0 0.2 0.4 0.6 0.8 1.0 probability level Fig. 6 – Several quantile estimators in R. 21
  22. 22. Arthur CHARPENTIER - Nonparametric quantile estimation. Smoothing nonparametric estimators In order to increase efficiency, L-statistics can be considered i.e. n Qn (p) = n Wi,n,p Xi:n = i=1 i n −1 Wi,n,p Fn i=1 1 −1 Fn (t) k (p, h, t) dt = (8) 0 where Fn is the empirical distribution function of FX , where k is a kernel and h a bandwidth. This expression can be written equivalently n Qn (p) = i n k i=1 (i−1) n t−p h n dt X(i) = IK i=1 i n −p h − IK i−1 n −p h X(i) (9) x where again IK (x) = k (t) dt. The idea is to give more weight to order −∞ statistics X(i) such that i is closed to pn. 22
  23. 23. 0 1 2 3 Arthur CHARPENTIER - Nonparametric quantile estimation. 0.0 0.2 0.4 0.6 0.8 1.0 quantile (probability) level Fig. 7 – Quantile estimator as wieghted sum of order statistics. 23
  24. 24. 0 1 2 3 Arthur CHARPENTIER - Nonparametric quantile estimation. 0.0 0.2 0.4 0.6 0.8 1.0 quantile (probability) level Fig. 8 – Quantile estimator as wieghted sum of order statistics. 24
  25. 25. 0 1 2 3 Arthur CHARPENTIER - Nonparametric quantile estimation. 0.0 0.2 0.4 0.6 0.8 1.0 quantile (probability) level Fig. 9 – Quantile estimator as wieghted sum of order statistics. 25
  26. 26. 0 1 2 3 Arthur CHARPENTIER - Nonparametric quantile estimation. 0.0 0.2 0.4 0.6 0.8 1.0 quantile (probability) level Fig. 10 – Quantile estimator as wieghted sum of order statistics. 26
  27. 27. 0 1 2 3 Arthur CHARPENTIER - Nonparametric quantile estimation. 0.0 0.2 0.4 0.6 0.8 1.0 quantile (probability) level Fig. 11 – Quantile estimator as wieghted sum of order statistics. 27
  28. 28. Arthur CHARPENTIER - Nonparametric quantile estimation. Smoothing nonparametric estimators E.g. the so-called Harrell-Davis estimator is defined as n Qn (p) = i=1 i n (i−1) n Γ(n + 1) y (n+1)p−1 (1 − y)(n+1)q−1 Xi:n , Γ((n + 1)p)Γ((n + 1)q) • find a smooth estimator for FX , and then find (numerically) the inverse, The α-quantile is defined as the solution of FX ◦ qX (α) = α. If Fn denotes a continuous estimate of F , then a natural estimate for qX (α) is qn (α) such that Fn ◦ qn (α) = α, obtained using e.g. Gauss-Newton algorithm. 28
  29. 29. Arthur CHARPENTIER - Nonparametric quantile estimation. Agenda • General introduction Risk measures • Distorted risk measures • Value-at-Risk and related risk measures Quantile estimation : classical techniques • Parametric estimation • Semiparametric estimation, extreme value theory • Nonparametric estimation Quantile estimation : use of Beta kernels • Beta kernel estimation • Transforming observations A simulation based study 29
  30. 30. Arthur CHARPENTIER - Nonparametric quantile estimation. Kernel based estimation for bounded supports Classical symmetric kernel work well when estimating densities with non-bounded support, 1 fh (x) = nh n k i=1 x − Xi h , where k is a kernel function (e.g. k(ω) = I(|ω| ≤ 1)/2). If K is a symmetric kernel, note that 1 E(fh (0) = f (0) + O(h) 2 30
  31. 31. Arthur CHARPENTIER - Nonparametric quantile estimation. 1.0 0.8 0.0 0.2 0.4 0.6 Density 0.6 0.4 0.2 0.0 Density 0.8 1.0 1.2 Kernel based estimation of the uniform density on [0,1] 1.2 Kernel based estimation of the uniform density on [0,1] 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Fig. 12 – Density estimation of an uniform density on [0, 1]. 31
  32. 32. Arthur CHARPENTIER - Nonparametric quantile estimation. Kernel based estimation for bounded supports Several techniques have been introduce to get a better estimation on the border, ¨ – boundary kernel (Muller (1991)) – mirror image modification (Deheuvels & Hominal (1989), Schuster (1985)) – transformed kernel (Devroye & Gyrfi (1981), Wand, Marron & Ruppert (1991)) – Beta kernel (Brown & Chen (1999), Chen (1999, 2000)), see Charpentier, Fermanian & Scaillet (2006) for a survey with application on copulas. 32
  33. 33. Arthur CHARPENTIER - Nonparametric quantile estimation. Beta kernel estimators A Beta kernel estimator of the density (see Chen (1999)) - on [0, 1] is 1 fb (x) = n n x 1−x k Xi , 1 + , 1 + b b i=1 , x ∈ [0, 1], uα−1 (1 − u)β−1 where k(u, α, β) = , u ∈ [0, 1]. B(α, β) If {X1 , · · · , Xn } are i.i.d. variables with density f0 , if n → ∞, b → 0, then Bouzmarni & Scaillet (2005) fb (x) → f0 (x), x ∈ [0, 1]. This is the Beta 1 estimator. 33
  34. 34. Arthur CHARPENTIER - Nonparametric quantile estimation. Beta kernel, x=0.10 0 0 2 5 4 6 10 8 10 15 12 Beta kernel, x=0.05 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.6 0.8 1.0 0.8 1.0 Beta kernel, x=0.45 0 0 2 2 4 4 6 6 8 8 10 Beta kernel, x=0.20 0.4 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 Fig. 13 – Shape of Beta kernels, different x’s and b’s. 34
  35. 35. Arthur CHARPENTIER - Nonparametric quantile estimation. Improving Beta kernel estimators Problem : the convergence is not uniform, and there is large second order bias on borders, i.e. 0 and 1. Chen (1999) proposed a modified Beta 2 kernel estimator, based on   k t , 1−t (u) , if t ∈ [2b, 1 − 2b]  b b  k2 (u; b; t) = 1−t (u) k , if t ∈ [0, 2b)  ρb (t), b   t k b ,ρb (1−t) (u) , if t ∈ (1 − 2b, 1] where ρb (t) = 2b2 + 2.5 − t 4b4 + 6b2 + 2.25 − t2 − . b 35
  36. 36. Arthur CHARPENTIER - Nonparametric quantile estimation. Non-consistency of Beta kernel estimators Problem : k(0, α, β) = k(1, α, β) = 0. So if there are point mass at 0 or 1, the estimator becomes inconsistent, i.e. fb (x) = = = ≈ 1 n 1 n x 1−x k Xi , 1 + , 1 + , x ∈ [0, 1] b b x 1−x k Xi , 1 + , 1 + , x ∈ [0, 1] b b Xi =0,1 n − n0 − n1 1 n n − n0 − n1 Xi =0,1 x 1−x k Xi , 1 + , 1 + b b , x ∈ [0, 1] (1 − P(X = 0) − P(X = 1)) · f0 (x), x ∈ [0, 1] and therefore Fb (x) ≈ (1 − P(X = 0) − P(X = 1)) · F0 (x), and we may have problem finding a 95% or 99% quantile since the total mass will be lower. 36
  37. 37. Arthur CHARPENTIER - Nonparametric quantile estimation. Non-consistency of Beta kernel estimators ´ Gourieroux & Monfort (2007) proposed (1) fb (x) = fb (x) 1 0 , for all x ∈ [0, 1]. fb (t)dt It is called macro-β since the correction is performed globally. ´ Gourieroux & Monfort (2007) proposed 1 (2) fb (x) = n n i=1 kβ (Xi ; b; x) 1 0 , for all x ∈ [0, 1]. kβ (Xi ; b; t)dt It is called micro-β since the correction is performed locally. 37
  38. 38. Arthur CHARPENTIER - Nonparametric quantile estimation. Transforming observations ? In the context of density estimation, Devroye and Gy¨ ’orfi (1985) suggested to use a so-called transformed kernel estimate Given a random variable Y , if H is a strictly increasing function, then the p-quantile of H(Y ) is equal to H(q(Y ; p)). An idea is to transform initial observations {X1 , · · · , Xn } into a sample {Y1 , · · · , Yn } where Yi = H(Xi ), and then to use a beta-kernel based estimator, if H : R → [0, 1]. Then qn (X; p) = H −1 (qn (Y ; p)). In the context of density estimation fX (x) = fY (H(x))H (x). As mentioned in ¨ Devroye and Gyorfi (1985) (p 245), “for a transformed histogram histogram estimate, the optimal H gives a uniform [0, 1] density and should therefore be equal to H(x) = F (x), for all x”. 38
  39. 39. Arthur CHARPENTIER - Nonparametric quantile estimation. Transforming observations ? a monte carlo study Assume that sample {X1 , · · · , Xn } have been generated from Fθ0 (from a familly F = (Fθ , θ ∈ Θ). 4 transformations will be considered – H = Fθ (based on a maximum likelihood procedure) – H = Fθ0 (theoritical optimal transformation) – H = Fθ with θ < θ0 (heavier tails) – H = Fθ with θ > θ0 (lower tails) 39
  40. 40. 0.0 0.2 0.4 0.6 0.8 1.0 1.0 0.8 0.6 0.4 0.2 0.0 q q qq qq qq qq qq qq q q q q q q q q q q q q q q qq qq q q q q q qq qq qq q q q q q qq q q q q q q qq qq qq q q qq qq qq qq qq q q q q qq qq qq qq q qq qq qq q q q q q q q q q q q q q q q qq q q q q q q q q q qq qq qq qq qq qq q q q qq q qq q q q qq qq q q q q q q q q q q qq q qq qq qq qq q q q q q q qq qq qq qq q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q qq q q q qq qq qq qq q q q q q qq qq qq qq q q qq q qq q q Transformed observations 0.6 0.4 0.2 0.0 Transformed observations 0.8 1.0 Arthur CHARPENTIER - Nonparametric quantile estimation. q q q q q q q q q qq q q q q q q q q q q q q q q q qq q qq q q q q q q qq q q qq qq qq qq qq qq q q q q q q q q q q q qq qq q q q q q q q q q qq qq qq qq q q q q q q q qq q q q q q q q q q qq qq qqq qqq qqq qq q q q qq qq qq qq q q qq qq qq q q q q qq q q q q q qq q qq q q q q q q q q q q q q q q q q q q q q qq qq q q q q q q qq q qq q q q q q qq qq q q q q q q q q q q q q q q q q qq qq q qq q q q q q q qq qq qq qq qq qq q q q q q q q q qq 0.0 0.2 0.4 0.6 0.8 1.0 Fig. 14 – F (Xi ) versus Fθ (Xi ), i.e. P P plot. ˆ 40
  41. 41. 1.4 1.2 1.0 q 0.6 0.8 Estimated density 1.2 1.0 0.8 0.6 Estimated density 1.4 Arthur CHARPENTIER - Nonparametric quantile estimation. 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Fig. 15 – Nonparametric estimation of the density of the Fθ (Xi )’s. ˆ 41
  42. 42. Arthur CHARPENTIER - Nonparametric quantile estimation. Estimated optimal transformation 4.0 Estimated optimal transformation q 3.5 5 q q 4 q q 2.5 q q q q q q 3 Quantile q q qq 0.80 q qq qq qq qq qq q q q q q q q 1 qq q qq q qq q qq q q q 2 2.0 q 1.5 1.0 Quantile 3.0 q 0.85 0.90 Probability level 0.95 1.00 qqq 0.80 qqq qq qqq qq 0.85 qq qq q q qq qq qq qq 0.90 q q q q q q 0.95 1.00 Probability level −1 Fig. 16 – Nonparametric estimation of the quantile function, Fθ (q). ˆ 42
  43. 43. 0.0 0.2 0.4 0.6 0.8 1.0 1.0 0.8 0.6 0.4 0.2 0.0 q q q q q qq qq qq qq q q q q q q q q qq q q qq qq qq q q q q q q q qq q qq q q q q q q q q q q q q q qq qq q q q q qq qq qq qq q q q q qq qq qq qq qq qq q qq q q qq qq q q q q q qq q q qq q q q q q q q q q q q q q q q q q q q q qq qq q q q q q q q qq q q q q qq qq q q q q q q q q q q qq q q qq q qq q q q qq qq q q q q q q q q q q q q q q q q qq q q qq q qq qq q q q q q qq qq q qq qq q q q qq qq qq qq q q qq qq q qq qq qq qq qq qq q qq q q q Transformed observations 0.6 0.4 0.2 0.0 Transformed observations 0.8 1.0 Arthur CHARPENTIER - Nonparametric quantile estimation. q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q qq q qq qq qq qq qq qq qq q q q q q q q q q qq qq qq q q q q q qq qq qq qq q q q qq q q q q q q q q q q q q q q q q q q q qqq qqq qqq qqq q q q q qq qq qq qq q qq qq qq q q q q q qq q q q q q qq q qq q q q q q q q q q q q q q q q q q q q q qq qq q q q qq qq q q q q q q q q q q qq qq q q q q q q q q q q q q q q q qq qq qq qq q q q q qq qq qq q qq qq q qq q q q q q q qq qq 0.0 0.2 0.4 0.6 0.8 1.0 Fig. 17 – F (Xi ) versus Fθ0 (Xi ), i.e. P P plot. 43
  44. 44. 1.4 1.2 1.0 q 0.6 0.8 Estimated density 1.2 1.0 0.8 0.6 Estimated density 1.4 Arthur CHARPENTIER - Nonparametric quantile estimation. 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Fig. 18 – Nonparametric estimation of the density of the Fθ0 (Xi )’s. 44
  45. 45. Arthur CHARPENTIER - Nonparametric quantile estimation. Estimated optimal transformation Estimated optimal transformation q 4 4 q q q 3 q q q q 3 Quantile q q q q q 0.80 qqq qq q qq qq q qq q q q q q q q q 1 qqq q qq q qq q qq q q 2 2 q 1 Quantile q 0.85 0.90 Probability level 0.95 1.00 qqq 0.80 qqq qq qq q qq 0.85 qq qq qq q q qq q q q 0.90 q q q q q q 0.95 1.00 Probability level −1 Fig. 19 – Nonparametric estimation of the quantile function, Fθ0 (q). 45
  46. 46. 0.0 0.2 0.4 0.6 0.8 1.0 1.0 0.8 0.6 0.4 0.2 0.0 q q q q q q q q qq q q q q q q q qq qq qq qq qq qq qq q q qq qq q q q qq q qq qq qq q q q q q q q qq qq qq q qq qq qq qq q q qq q qq q q q q q q q q q q q q qq qq qq qq q q q qq qq q qq q q qq qq qq qq qq qq qq q q q qq q q q q q q qq qq qq qq q qq qq qqq qqq qqq qq q q q q q q qq q q q q q q qq qq qq q q q q qq q q q q q q qq qq qq q q q q q q q q q qq qq qq qq q q q q q q q qq qq qq qq q q qq qq qq qq q q q q q q q q q q q q Transformed observations 0.6 0.4 0.2 0.0 Transformed observations 0.8 1.0 Arthur CHARPENTIER - Nonparametric quantile estimation. q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q qq q qq qq qq qq qq qq qq q q q q q q q q q qq qq qq q q q q q qq qq qq qq q q q qq q q q q q q q q q q q q q q q q q q q qqq qqq qqq qqq q q q q qq qq qq qq q qq qq qq q q q q q qq q q q q q qq q qq q q q q q q q q q q q q q q q q q q q q qq qq q q q qq qq q q q q q q q q q q qq qq q q q q q q q q q q q q q q q qq qq qq qq q q q q qq qq qq q qq qq q qq q q q q q q qq qq 0.0 0.2 0.4 0.6 0.8 1.0 Fig. 20 – F (Xi ) versus Fθ (Xi ), i.e. P P plot, θ < θ0 (heavier tails). 46
  47. 47. 1.4 1.2 1.0 q 0.6 0.8 Estimated density 1.2 1.0 0.8 0.6 Estimated density 1.4 Arthur CHARPENTIER - Nonparametric quantile estimation. 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Fig. 21 – Estimation of the density of the Fθ (Xi )’s, θ < θ0 (heavier tails). 47
  48. 48. Arthur CHARPENTIER - Nonparametric quantile estimation. Estimated optimal transformation 12 Estimated optimal transformation q 10 10 12 q q q 8 q q 6 Quantile 6 q q q q q qqqqq 0.80 qq 0.85 qqq qqq qq q qq q q q q q 2 qqq qqqq q qq q q 4 4 q 2 Quantile 8 q qqqqq 0.90 Probability level 0.95 1.00 0.80 qq qqq qqqq 0.85 qqq qqq qq q q qq 0.90 qq q q q q q 0.95 q 1.00 Probability level −1 Fig. 22 – Estimation of quantile function, Fθ (q), θ < θ0 (heavier tails). 48
  49. 49. 0.0 0.2 0.4 0.6 0.8 1.0 1.0 0.8 0.6 0.4 0.2 0.0 q qq qq qq q qq q q q qq qq qq q q q q qq qq q q q q q q q q q q q qq q qq qq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq qq qq qq q q q q q q qq qq qq q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q qq qq q q q q qq q q q q qq qq qq qq q q q qq qq qq q q q q qq qq qq qq q q qq qq qq qq q q q qq q q q qq q q q q q qq q q q qq q qq qq qq qq q qq qq qq q q q qq q qq qq qq q q q q q q q Transformed observations 0.6 0.4 0.2 0.0 Transformed observations 0.8 1.0 Arthur CHARPENTIER - Nonparametric quantile estimation. q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q qq q qq qq qq qq qq qq qq q q q q q q q q q qq qq qq q q q q q qq qq qq qq q q q qq q q q q q q q q q q q q q q q q q q q qqq qqq qqq qqq q q q q qq qq qq qq q qq qq qq q q q q q qq q q q q q qq q qq q q q q q q q q q q q q q q q q q q q q qq qq q q q qq qq q q q q q q q q q q qq qq q q q q q q q q q q q q q q q qq qq qq qq q q q q qq qq qq q qq qq q qq q q q q q q qq qq 0.0 0.2 0.4 0.6 0.8 1.0 Fig. 23 – F (Xi ) versus Fθ (Xi ), i.e. P P plot, θ > θ0 (lighter tails). 49
  50. 50. 1.4 1.2 1.0 q 0.6 0.8 Estimated density 1.2 1.0 0.8 0.6 Estimated density 1.4 Arthur CHARPENTIER - Nonparametric quantile estimation. 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Fig. 24 – Estimation of density of Fθ (Xi )’s, θ > θ0 (lighter tails). 50
  51. 51. Arthur CHARPENTIER - Nonparametric quantile estimation. Estimated optimal transformation Estimated optimal transformation q 3.5 3.5 q q 3.0 3.0 q q q q 2.0 q qq 0.80 qq q qq qq qq qq qq q q q q q q 0.85 0.90 Probability level q 2.0 q q q q 1.0 qq q qq q qq q q q 1.5 1.5 q 2.5 Quantile 2.5 q q q 1.0 Quantile q 0.95 1.00 qq qq 0.80 qq q q qq qq qq 0.85 qq q qq qq qq q q q 0.90 q q q q q q 0.95 1.00 Probability level −1 Fig. 25 – Estimation of quantile function, Fθ (q), θ > θ0 (lighter tails). 51
  52. 52. Arthur CHARPENTIER - Nonparametric quantile estimation. A universal distribution for losses BNGB considered the Champernowne generalized distribution to model insurance claims, i.e. positive variables, α (y + c) − cα Fα,M,c (y) = where α > 0, c ≥ 0 and M > 0. α α (y + c) + (M + c) − 2cα The associated density is then fα,M,c (y) = α (y + c) α−1 α α ((M + c) − cα ) α ((y + c) + (M + c) − 2 2cα ) . 52
  53. 53. Arthur CHARPENTIER - Nonparametric quantile estimation. A Monte Carlo study to compare those nonparametric estimators As in ....., 4 distributions were considered – normal distribution, – Weibull distribution, – log-normal distribution, – mixture of Pareto and log-normal distributions, 53
  54. 54. Arthur CHARPENTIER - Nonparametric quantile estimation. Box−plot for the 11 quantile estimators Density of quantile estimators (mixture longnormal/pareto) 0.30 R benchmark qqqqqq qq qqq qqqqq q qqqq q qq q qq qq qq qq qq q qqq qqqqq qq qqq q q qq q qq qq qqq q q qq qqq q qqq q q q q q q qqq q qq q q q qq q qq qq q HD Harrell Davis q qqqqq qqqqq q qqq qq qq qqqq qqq q qqq q qq qq PDG Padgett q PRK Park 0.15 0.20 Benchmark (R estimator) HD (Harrell−Davis) PRK (Park) B1 (Beta 1) B2 (Beta 2) Beta1 0.10 density of estimators 0.25 qq qqqq qqqqqq qqqqq q qqq q qqq qqq qq qq qq qq qq qq q qq qq qqqqqqqq q qq qqq q qqq q qq q qq qqq q qqq qq q qq q q q qq q q q q q q q q q q q E Epanechnikov q q q q qq q q q qq qq qq qq q q q qq qqqqq q q qqqq q q qqqqqq qqq q q q qq qq q q q q q q q qq q qq q q qq q q q qq q q q q qq q q q q MACRO Beta1 q q qqq q q qqq q qqq q qqq qqq qq qq qq qq q 0.05 0.00 q q q q q q q q q q q q q q q q qqqq q q qqqqq q qqq qq q qq qqq q qq qqq q qq qq q q qq q q qqqqq q q qq q q qq qqq qqq qqq qqq qqq q qqqqqq q q q qqqqq q qqq qq qqq q q qq qq q qq qqq qq q q qq q q qq q qqqqq q qq qqq qqqq qq qq qqqq qqq q q qq qq q q Beta2 q q q qqqqq qqq q q q q qq q q q qqqqqq qq q qq q q qq qqq qq q q qq qq q q q q q qq q q q qq q q q q q q q q q qqqqqqq q q qqqq qqqq qqqq qqqq qqq qqq qq q MICRO Beta1 q q q qq q q qqqqq q q q q q q q q q q q q qq q q q q q qq q q q q q q q qq q qq q q q q qqqq qqqqq q qq q qqqq qqqqq q q qq q qq qqqq qqqqq q qq q qqq qq qqq qqqq q qq q q q q q q q q q q qq qqqqq q qqqqq qqqqq qqqq qqqq qq qq qq qq q qqqq q qq q q q qq q qq q qqq q q qqq qq q q q qqq qq q qqq qq q q q q q q q q q MACRO Beta2 q qqqq q qqqq q qqqq q qqqq qqq qqq qq qq q qqqqqqqqqq qqqqqqqq q qqq qq q q q qqq qqq q q qqq q q qqq q q qqq q q q q q q 5 10 15 20 MICRO Beta2 qqqqq q qqqq q qqqq qqqq qqq qqq qq q qqq qq q qqq qqqq qqq q q qqqq qqq qq qq qqqq q qq qq qqq qq qqqq q q q qq q q q q q q q q Estimated value−at−risk 5 10 15 20 Fig. 26 – Distribution of the 95% quantile of the mixture distribution, n = 200, and associated box-plots. 54
  55. 55. Arthur CHARPENTIER - Nonparametric quantile estimation. n= 50 n=100 n=200 n=500 q 1.5 q 1.5 1.5 q n= 50 n=100 n=200 n=500 2.0 MSE ratio, normal distribution, MACB1 (MACRO−Beta1) 2.0 MSE ratio, normal distribution, B1 (Beta1) 2.0 MSE ratio, normal distribution, HD (Harrell−Davis) n= 50 n=100 n=200 n=500 q q q q q q q q q q 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 q 0.0 0.2 0.4 0.6 0.8 1.0 Probability, confidence levels (p) MSE ratio, normal distribution, B1 (Beta1) MSE ratio, normal distribution, PRK (Park) MSE ratio, normal distribution, MACB1 (MACRO−Beta1) q q q 1.5 q q n= 50 n=100 n=200 n=500 q n= 50 n=100 n=200 n=500 q 1.5 q 2.0 q 2.0 2.0 q q q Probability, confidence levels (p) q n= 50 n=100 n=200 n=500 q q q q q 0.2 0.4 0.6 0.8 Probability, confidence levels (p) 1.0 q q 1.0 q q q q q q q 0.5 0.0 0.5 0.0 0.5 q 0.0 q q q q q q MSE ratio 1.0 q q MSE ratio 1.0 MSE ratio q 0.0 q q 0.5 q Probability, confidence levels (p) 1.5 q q q 0.0 0.5 0.0 0.5 q 0.0 q q q q q q 1.0 q 1.0 q q MSE ratio q MSE ratio 1.0 MSE ratio q q q 0.0 0.2 0.4 0.6 0.8 Probability, confidence levels (p) 1.0 q 0.0 0.2 0.4 0.6 0.8 1.0 Probability, confidence levels (p) Fig. 27 – Comparing MSE for 6 estimators, the normal distribution case. 55
  56. 56. Arthur CHARPENTIER - Nonparametric quantile estimation. q q q 2.0 1.5 q q q q q q q n= 50 n=100 n=200 n=500 q 1.0 1.5 q q MSE ratio q q n= 50 n=100 n=200 n=500 1.0 q q q q MSE ratio 1.5 q n= 50 n=100 n=200 n=500 1.0 MSE ratio MSE ratio, Weibull distribution, MICB1 (MICRO−Beta1) 2.0 MSE ratio, Weibull distribution, MACB1 (MACRO−Beta1) 2.0 MSE ratio, Weibull distribution, HD (Harrell−Davis) q q q q q q q q q q q q 0.0 0.2 0.4 0.6 0.8 1.0 0.5 0.0 0.5 0.0 0.0 0.5 q q q 0.0 0.2 0.4 0.6 0.8 1.0 q 0.0 0.2 0.4 0.6 0.8 1.0 Probability, confidence levels (p) Probability, confidence levels (p) MSE ratio, Weibull distribution, PRK (Park) MSE ratio, Weibull distribution, MACB1 (MACRO−Beta1) MSE ratio, Weibull distribution, MICB1 (MICRO−Beta1) 2.0 q 1.0 q q q q q q q 1.5 q 1.5 q n= 50 n=100 n=200 n=500 q q n= 50 n=100 n=200 n=500 q 1.0 q q MSE ratio q 1.0 MSE ratio 1.5 q n= 50 n=100 n=200 n=500 MSE ratio 2.0 q 2.0 Probability, confidence levels (p) q q q q q q q q q q q q 0.0 0.2 0.4 0.6 0.8 Probability, confidence levels (p) 1.0 0.5 0.0 0.5 0.0 0.0 0.5 q q q 0.0 0.2 0.4 0.6 0.8 Probability, confidence levels (p) 1.0 q 0.0 0.2 0.4 0.6 0.8 1.0 Probability, confidence levels (p) Fig. 28 – Comparing MSE for 6 estimators, the Weibull distribution case. 56
  57. 57. Arthur CHARPENTIER - Nonparametric quantile estimation. 2.0 q n= 50 n=100 n=200 n=500 q q q 1.5 n= 50 n=100 n=200 n=500 1.5 q 1.5 1.5 q n= 50 n=100 n=200 n=500 MSE ratio, lognormal distribution, B1 (Beta1) 2.0 MSE ratio, lognormal distribution, MICB1 (MICRO−Beta1) 2.0 MSE ratio, lognormal distribution, MACB1 (MACRO−Beta1) 2.0 MSE ratio, lognormal distribution, HD (Harrell−Davis) n= 50 n=100 n=200 n=500 q q q q q q q q q q 1.0 q q q q q MSE ratio q q q 1.0 q q 1.0 q q q q MSE ratio q MSE ratio 1.0 MSE ratio q q q q q q q q q q q q 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.5 0.5 q 0.0 0.0 q 0.0 0.0 0.5 0.0 0.0 0.5 q q 0.0 0.2 0.4 0.6 0.8 1.0 q 0.0 0.2 0.4 0.6 0.8 1.0 MSE ratio, lognormal distribution, PRK (Park) MSE ratio, lognormal distribution, MACB2 (MACRO−Beta2) MSE ratio, lognormal distribution, MICB2 (MICRO−Beta2) MSE ratio, lognormal distribution, B2 (Beta2) 0.4 0.6 0.8 Probability, confidence levels (p) 1.5 1.5 q q q q 1.0 q q q q q 0.2 0.4 0.6 0.8 Probability, confidence levels (p) 1.0 q q q q q q q q q q q 0.0 0.0 q 0.0 1.0 1.0 q q q MSE ratio q q q 0.5 q MSE ratio q q 0.5 0.2 q n= 50 n=100 n=200 n=500 q 0.0 0.5 q 0.0 q 0.5 1.0 q q q n= 50 n=100 n=200 n=500 q q 0.0 MSE ratio q n= 50 n=100 n=200 n=500 1.0 q q MSE ratio 1.5 q 1.5 q n= 50 n=100 n=200 n=500 2.0 Probability, confidence levels (p) 2.0 Probability, confidence levels (p) 2.0 Probability, confidence levels (p) 2.0 Probability, confidence levels (p) q 0.0 0.2 0.4 0.6 0.8 Probability, confidence levels (p) 1.0 q 0.0 0.2 0.4 0.6 0.8 1.0 Probability, confidence levels (p) Fig. 29 – Comparing MSE for 9 estimators, the lognormal distribution case. 57
  58. 58. Arthur CHARPENTIER - Nonparametric quantile estimation. q MSE ratio, mixture distribution, MICB1 (MICRO−Beta1) n= 50 n=100 n=200 n=500 q 1.5 q 1.5 1.5 q n= 50 n=100 n=200 n=500 MSE ratio, mixture distribution, B1 (Beta1) 2.0 2.0 2.0 q 2.0 MSE ratio, mixture distribution, MACB1 (MACRO−Beta1) n= 50 n=100 n=200 n=500 q 1.5 MSE ratio, mixture distribution, HD (Harrell−Davis) n= 50 n=100 n=200 n=500 q q q q q q q q q q q 1.0 q q MSE ratio q 1.0 q q MSE ratio q q 1.0 q q MSE ratio 1.0 MSE ratio q q q q q q q q q q q q 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.5 0.5 q 0.0 0.0 q 0.0 0.0 0.5 0.0 0.0 0.5 q q 0.0 0.2 0.4 0.6 0.8 1.0 q 0.0 0.2 0.4 0.6 0.8 1.0 MSE ratio, mixture distribution, PRK (Park) MSE ratio, mixture distribution, MACB2 (MACRO−Beta2) MSE ratio, mixture distribution, MICB2 (MICRO−Beta2) MSE ratio, mixture distribution, B2 (Beta2) q q q q q q q q q q q q q q q 1.0 q q q q 1.0 q MSE ratio q 1.0 MSE ratio 1.0 q q q q q q q q q 0.2 0.4 0.6 0.8 Probability, confidence levels (p) 1.0 0.2 0.4 0.6 0.8 Probability, confidence levels (p) 1.0 0.5 0.5 q 0.0 0.0 q 0.0 0.0 0.0 0.5 q 0.5 0.0 MSE ratio q n= 50 n=100 n=200 n=500 q q MSE ratio q n= 50 n=100 n=200 n=500 1.5 q n= 50 n=100 n=200 n=500 1.5 q 1.5 1.5 q q n= 50 n=100 n=200 n=500 2.0 Probability, confidence levels (p) 2.0 Probability, confidence levels (p) 2.0 Probability, confidence levels (p) 2.0 Probability, confidence levels (p) q 0.0 0.2 0.4 0.6 0.8 Probability, confidence levels (p) 1.0 q 0.0 0.2 0.4 0.6 0.8 1.0 Probability, confidence levels (p) Fig. 30 – Comparing MSE for 9 estimators, the mixture distribution case. 58
  59. 59. Arthur CHARPENTIER - Nonparametric quantile estimation. Portfolio optimal allocation Classical problem as formulated in Markowitz (1952), Journal of Finance,    ω ∗ ∈ argmin{ω Σω}  ω ∗ ∈ argmax{ω µ} convex ⇔  u.c. ω µ ≥ η and ω 1 = 1  u.c. ω Σω ≤ η and ω 1 = 1 Roy (1952), Econometrica,“the optimal bundle of assets (investment) for investors who employ the safety first principle is the portfolio that minimizes the probability of disaster”.   ω ∗ ∈ argmin{VaR(ω X, α)}  u.c. E(ω X) ≥ η and ω 1 = 1 nonconvex   ω ∗ ∈ argmax{E(ω X)}  u.c. VaR(ω X, α) ≤ η ,ω 1 = 1 59
  60. 60. Arthur CHARPENTIER - Nonparametric quantile estimation. Empirical data, Eurostocks 0.05 0.05 −0.05 q q q qqqq q q q qq q qq qq q qqq qq q q q q q q q q qq q q q q qq q qqq qq qqqqq q qq qqqqq q qqqqq qq qq qqq q qqqqqq q qqq q qq qqq qq qq q q qqqqq qq q qq q q qq q qqq qq q qqqq qqqq qqqqq q q qqqqqqq qqqqq q qq q qqqq qq qqqqqq qq qqqqq q qq q qqqqqq q qq qq q qqqq qqq qq q q qq qqqq q qq qq qq qq q q qq qq q q q qq q q q q qq qq qq q q qq qq q qq q q q q q qq q q q qq q q qq q q q q q q q q qq qq q q q qq q SMI (2) q q q qq q q qq qq q q qq q qq q q qq q qq q q q q qq q q q q qq qq qqq q q q q qq q q q qq qqqqqqq qq q q qq qqq q q q qq qq q qqqq q qq q qqq q qqq qq q q qqqq q q qq q q qq qq qqq qq qq qq q qqqq q q q qqqq qqqq q q qq q qqqqq qqqq qq q qqqqqq qqqq q qq qq q qqqqq qqq qq q qq q q qqqqq q qq q qq qq qq qq q q qq q qqqq qqq q qq qq q q qq qqqq q q q q qq q q q qq qq qq q q q qq q qq q qq q q qq q q q qq q q q q 0.05 −0.05 q −0.05 0.05 q q q q q q q q qqq q q q qq q q qq q qq q q qq q q q q q qq q q qq q q q q qqq q qqq q qq q qq q q q q qqqqqq qqq qq qqq q q qqqq q q qqqq qq qqqq qqqq qqqq qqqq q q qqqqqq qqqq q qqq q qqq q q q qqqq qqqq q qq q q q q q q qqq qqqqq qqqqq q qq q qqq q qq qqq q q q q qqqq q qq q qq q q q q q q qq qqq q qqqq q q qq q qq q qqqq q q qqqq q qqqqq q q q q q qqqq q qq q qqq q q q qq qqq q q q q qq q q q q qqq q q q q q q q q q qq q q q q qq q q q q qq q qq q q qq q qq q q q qq q q qq q q qq q q q qq qq q q qq q q qq q qqq q q q qqq q qq q q q qqqqq qqq qqqqqq q qqqqqq qq q qqqq q qqq q qqqqq q qq q qq qq q qq q qq q q qq q qq q q qqqq qq qq q qqqqq q q q qqqq q qqq q q qqqqq q q q qq q qq qqq q q qqq q qqqqq q q qqq qq q q q q q q qqq qq q q q q qq q q qq qqqqq q qqq q qqqqq q q q qqqq q q qqq qqqqqq qqqq qq qq q qq qq q q qq q q q qq qqq q q qq q q q qq q q qq q q q q q q q q q q q qq q q q q q qq qq qq qq q q q q qqqqq qq q q q qq q q q qq qq q qqqqq qq q q qq qq q qqqqqq q qq qq q q qqq qqq qqq q q qq q q qq qq q qqqq qqq qqqq q q q q qq qq qqqqq q q q qq q qqq q qqqqqq q q qqqqq qqq qqqqq qqqq qqqqq qq qq q qqqqqq q qqqq qqqqq q qqqqq q q q q qqqq qq qqqq q q q q qq q q qq q qqqqq qq qq qqqq qq qq qqq q qqqq qq qqq q q qq q q q q q q qq q q q q q qq q q q qq q q q q q q q q q q q q q q qq qq q q qq q qq q qqq qq q q q qq qq qq q q q q q qq q qq qq q qq q q q qqq q q q qq q qqq qqq qqqqq qqq qqq q qqqqqq qq qqq q q qqqqqqq q qqq q qqq q qqqqqq qq qqqqq q qqq q q q q q q qqq q qqq q qqq q q q q q q q qq q qqqqq q qqqq qq q qqqqqq q qq qqq q qq qqqq qqq q q qqqq q q qq q q q qqq q q qq qq q qqqq qq q q qqq q qq q qq q qq qq q qqqqq q q qq q q q q q q qq qqq qqq q qqq q q q qq q q q qq q q q q q qq q q q q q q q q q q q qq qqq q qqq q qq q qq q q qqq q q q qq q q q q qqqq q q q q qq q qq qqq q qqqqq qq qq q q q q q qqq q qqqqqq qqqq q q qqq q q q q qq qq q q q q qq q q qqqqqq q qq q q qq qq qqqqq qq q q qqqqqq q q qqqq qqqq q q qq q qq q qqq q q q q qqqqqq qq qqqq q qqqqqq qq q q qqqq q qqqq q q qq qqq q q qqqqq q q q q qqqq qq qq qqqq qqqq q qqqq q qq q q qqq q q qq q q qq q q qq qqqq q qq q qqqqq qq q q q qq q q q q q 0.05 q q q q q qq qq q q q qq q qqq q q q qqq q q q q q q qq q q q qqqq q q q q q q qq qq q q q q qq qq q qqq qq qq qqqq q qq q q q q qqq q qq qqqq qq qqq q qqq q qq q qqq q qqqqqqq q q q qqqqqqq qq q qq q qqqqq q q q qq qq qq q qqqqqq q qq q q qqqqq q qqq q qqqq q q q qqqq q qq q qqq qq q q q qqqq q q qq qqq qq qqq qq qq qqq q q q qq qqqqqqq q qq q q q q q qqq q q qq qqq q qqq q q qq q q q qq q qqqq q qq q q q qq q qq q qqq qq qq q qq q CAC (3) q q q qq q q q q q qq q q q qq q q qq q q qq qqqqq q qq q qq q qqq qqq q qq qqqqqqq qq q qqq qq qqqqqq q q q qqqq q q qqq qq q qq q q qq qq q qqq qq q qqqqqq q q qq qqqqq qqq q q q qqqqq q qqqqqq q q qq qq q q qqqqqqqq q qqq qq q qq qqq q q q qq qq q q qqq q qqqqq q qqqqq q qq qqqqq qqq q q q qq qq qq qqq qq qq q q qqqqq qq q qqqq q qq q q q qqqqq qq qq q q q qq q qq qq q qq q q q q q qqq q qq q q q q q q q q q q q q q qq q q q q qq q qqq qq qq q qq q q qqqqq q qq qqqqq qq q qqq qqq q qq qqq qqqq qqqqq q qqq q q q qqqqq qq qq qqq qq q q qqq q q qqqqqq q q q q qq q q q q qqq q q q q q qqq q qq qq qq qqq qq q qq qqq q q q q qqqqq qqqq qq qqqq q qq q qq q qqqqqq q qqqq q qq q qqqq qq q q qq q q q qqqqq q q qq q q q q qqqq q q qq q q qqqqq q qq q qqqq q q q qq q q qq q q qq q qqq qq q qq q q qq qq qq q q q q q q q qq q q q q qq q qq q q q q qq q q qq q q qqq q q q q q qqq q q q q q qqq q qq q q q qq qq qq qq qq qq q qq q qqqqq q qqqqq q q qq q q qqqq q qq qq qqqqq qq qq qq qqqqqq qqqq qq q q qqqq qq q qqqqq q qqq qq qqqq q qq qqqqq qqqq qqq q q q qqq qq qq q qqq qq q qqqq q qqq q q qqqq qqqqq q q q qqqq q qqq q qqqqqqqq qqqqqqq q q q qq qq qq qq q qqqqq q q q qq q q qqq q q q q q qq q qq q q qqq q q qqq q qq qq q q q q q q −0.05 q q DAX (1) 0.05 0.10 q q q q q qq q qq qq q qq q q q q q qq q q qq qq q q qq q q q q qqq q q q q q q qqqqqqq q q qqqqqqq q qqqq q qqqq qq qqqq q qqqq qq qqqqqqq qqqq qqq qqq qqq q qq qq q q qqqqq q qq qq q qq qq q qqqq q q q qqqq qqq q q q q qq q q qqqqq qqq q q qqqqqq q q qqqqq qqq qq qq qq qqqq qq qq q qqqqq q qqqqq qqq q q qqqqq qqq qq q q qqq qqq q q q qq qqq q qq qqqq q q qq q q q qq q q q q q q q qq q qq q q q q q q q q q q q q q −0.05 −0.10 0.00 −0.05 q q −0.10 0.00 FTSE (4) 0.10 Fig. 31 – Scatterplot of log-returns. 60
  61. 61. Arthur CHARPENTIER - Nonparametric quantile estimation. Value−at−Risk (75%) on the grid 0 −1 −2 on t) −3 ati asse loc n (4 th Al catio (3 rd a ss et) Quantile Allo Allocation (4th asset) 1 2 Value−at−Risk (75%) on the grid −3 −2 −1 0 1 2 Allocation (3rd asset) Value−at−Risk (97,5%) on the grid t) 0 −1 −3 on ati asse Al loc catio n (4 th −2 (3 rd as s et) Quantile Allo Allocation (4th asset) 1 2 Value−at−Risk (97.5%) on the grid −3 −2 −1 0 1 2 Allocation (3rd asset) Fig. 32 – Value-at-Risk for all possible allocations on the grid G (surface and level curves), with α = 75% on the left and α = 97.5% on the right. 61
  62. 62. Arthur CHARPENTIER - Nonparametric quantile estimation. q q q q q q q q q q q q q q q q q q q q q q q 97.5% q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 95% 92.5% q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 90% 87.5% q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 85% 82.5% 80% 77.5% 75% q q q q q q q q q 0.0 0.5 1.0 1.5 2.0 q −1.0 q q q q q q q q q q q q q Optimal allocation (asset 2) weight of allocation 0.0 0.5 1.0 1.5 2.0 −1.0 weight of allocation Optimal allocation (asset 1) q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 97.5% 95% 92.5% q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 90% 87.5% 85% 82.5% 80% q q q q q q q q q q q 77.5% q q q q q 97.5% q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 92.5% 90% 87.5% 85% q q q q q q q q q q q 95% q q q q q q q q q q q q q q q q q q q q q q q q q q 82.5% q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 80% q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 77.5% 75% q Probability level (97.5%−75%) 0.0 0.5 1.0 1.5 2.0 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q −1.0 −1.0 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q weight of allocation 0.0 0.5 1.0 1.5 2.0 q q 75% Optimal allocation (asset 4) q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Probability level (97.5%−75%) Optimal allocation (asset 3) weight of allocation Probability level (97.5%−75%) q q q q q q q q 97.5% 95% 92.5% 90% 87.5% 85% 82.5% 80% 77.5% 75% Probability level (97.5%−75%) Fig. 33 – Optimal allocations for different probability levels (α = 75%, 77.5%, 80%, ..., 95%, 97.5%), with allocation for the first asset (top left) up 62 to the fourth asset (bottom right).
  63. 63. Arthur CHARPENTIER - Nonparametric quantile estimation. mean 75% 77.5% 80% 82.5% 85% 87.5% 90% 92.5% 95% 97 variance asset 1 0.2277 0.222 (0.253) 0.206 (0.244) 0.215 (0.259) 0.251 (0.275) 0.307 (0.276) 0.377 (0.241) 0.404 (0.243) 0.394 (0.224) 0.402 (0.214) 0. (0. asset 2 0.5393 0.550 (0.141) 0.558 (0.136) 0.552 (0.144) 0.530 (0.152) 0.500 (0.154) 0.460 (0.134) 0.444 (0.135) 0.448 (0.124) 0.441 (0.121) 0. (0. asset 3 −0.2516 −0.062 (0.161) −0.083 (0.176) −0.106 (0.184) −0.139 (0.187) −0.163 (0.215) −0.196 (0.203) −0.228 (0.163) −0.253 (0.141) −0.310 (0.184) −0 (0. asset 4 0.4846 0.289 (0.162) 0.319 (0.179) 0.339 (0.191) 0.357 (0.204) 0.357 (0.221) 0.359 (0.205) 0.380 (0.175) 0.410 (0.153) 0.466 (0.170) 0. (0. Tab. 1 – Mean and standard deviation of estimated optimal allocation, for different quantile levels. 1. raw estimator Q(Y, α) = Y[α·n]:n n 2. mixture estimator Q(Y, α) = λi (α)Yi:n , which is the standard quantile i=1 estimate in R (see [?]), 3. Gaussian estimator Q(Y, α) = Y + z1−α sd(Y , where sd denotes the empirical standard deviation, 63
  64. 64. Arthur CHARPENTIER - Nonparametric quantile estimation. 4. Hill’s estimator, with k = [n/5], Q(Y, α) = Yn−k:n 1 ξk = k k log i=1 n (1 − α) k −ξk , where Yn+1−i:n (assuming that ξ > 0), Yn−k:n 5. kernel based estimator is obtained as a mixture of smoothed quantiles, derived as inverse values of a kernel based estimator of the cumulative n λi (α)F −1 (i/n). distribution function, i.e. Q(Y, α) = i=1 64
  65. 65. Arthur CHARPENTIER - Nonparametric quantile estimation. q q Optimal allocation (asset 2) Optimal allocation (asset 1) q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Est. 4 Hill q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Est. 5 Kernel q q q q q q q q q −2 q q q q Est. 3 q Gaussian q q q 1 q Est. 2 mixture Est. 1 raw 2 Est. 5 Kernel 0 q q q q q q q q q q q q Est. 4 Hill −1 q q Est. 3 q Gaussian weight of allocation q q q q q q q q q −1 0 q q q q q q q q q q q q q q −2 weight of allocation q q q q q q Est. 2 mixture q 1 2 Est. 1 raw q −3 −3 q q Quantile estimator Quantile estimator Optimal allocation (asset 3) q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 2 1 1 0 −1 −2 weight of allocation q q q q q q q q q q q q q q q q Est. 1 raw Est. 2 mixture q q q q Optimal allocation (asset 4) Est. 5 Kernel q q q q q q q q q q 0 q Est. 4 Hill −1 Est. 3 Gaussian q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Est. 3 q Gaussian q q q q q q q q q q q q q q q q q q q q q q q q q q q Est. 4 Hill Est. 5 Kernel q q q q q q q q q q q q q q q q q q q q q q q q q q q q −2 Est. 2 mixture weight of allocation 2 Est. 1 raw Quantile estimator −3 −3 q q Quantile estimator Fig. 34 – Optimal allocations for different 95% quantile estimators, with allocation for the first asset (top left) up to the fourth asset (bottom right). 65
  66. 66. Arthur CHARPENTIER - Nonparametric quantile estimation. Some references Charpentier, A. & Oulidi, A. (2007). Beta Kernel estimation for Value-At-Risk of heavy-tailed distributions. in revision Journal of Computational Statistics and Data Analysis. Charpentier, A. & Oulidi, A. (2007). Estimating allocations for Value-at-Risk portfolio optimzation. to appear in Mathematical Methods in Operations Research. Chen, S. X. (1999). A Beta Kernel Estimator for Density Functions. Computational Statistics & Data Analysis, 31, 131-145. ´ Gourieroux, C., & Montfort, A. 2006. (Non) Consistency of the Beta Kernel Estimator for Recovery Rate Distribution. CREST-DP 2006-32. 66

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