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Journal of Statistical Computation and Simulation iFirst, 2011, 1–16
A computational study of a quasi-PORT methodology for VaR based on second-order reduced-bias estimation Fernanda Figueiredoa , M. Ivette Gomesb *, Lígia Henriques-Rodriguesc and M. Cristina Mirandad de Economia, Universidade do Porto, Porto, and CEAUL, Lisboa, Portugal; b FCUL, DEIO and CEAUL, Universidade de Lisboa, Lisboa, Portugal; c Instituto Politécnico de Tomar, Tomar, and CEAUL, Lisboa, Portugal; d ISCA, Universidade de Aveiro, Aveiro, and CEAUL, Lisboa, Portugal
a Faculdade
(Received 20 May 2010; final version received 8 December 2010 ) In this paper, we deal with the estimation, under a semi-parametric framework, of the Value-at-Risk (VaR) at a level p, the size of the loss occurred with a small probability p. Under such a context, the classical VaR estimators are the Weissman–Hill estimators, based on any intermediate number k of top-order statistics. But these VaR estimators do not enjoy the adequate linear property of quantiles, contrarily to the PORT VaR estimators, which depend on an extra tuning parameter q, with 0 ≤ q < 1. We shall here consider ‘quasi-PORT’ reduced-bias VaR estimators, for which such a linear property is obtained approximately. They are based on a partially shifted version of a minimum-variance reduced-bias (MVRB) estimator of the extreme value index (EVI), the primary parameter in Statistics of Extremes. Due to the stability on k of the MVRB EVI and associated VaR estimates, we propose the use of a heuristic stability criterion for the choice of k and q, providing applications of the methodology to simulated data and to log-returns of financial stocks. Keywords: statistics of extremes; value-at-risk; semi-parametric estimation; heuristics 2000 AMS Subject Classifications: 62G32; 65C05
1.
Introduction, preliminaries and scope of the paper
We shall place ourselves under a semi-parametric framework, to refer the estimation of a positive extreme value index (EVI), denoted γ , the primary parameter in Statistics of Extremes and the basis for the estimation of the Value-at-Risk (VaR) at a level p, denoted VaRp , the size of the loss occurred with a small probability p. In other words, we are interested in the estimation of a (high) quantile, χ1−p := F ← (1 − p), of a probability distribution function (d.f.) F , with F ← (y) := inf{x : F (x) ≥ y}, the generalized inverse function of F . Let us denote U (t) := F ← (1 − 1/t), t ≥ 1, a reciprocal quantile function such that χ1−p ≡ VaRp = U (1/p). We shall thus consider heavy-tailed parents, quite common in the most diversified areas of application, like insurance
*Corresponding author. Email:
[email protected]
ISSN 0094-9655 print/ISSN 1563-5163 online © 2011 Taylor & Francis DOI: 10.1080/00949655.2010.547196 http://www.informaworld.com
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and finance, i.e. parents such that, as t → ∞,
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U ∈ RVγ ⇐⇒ 1 − F ∈ RV−1/γ , where, as usual, the notation RVα stands for regularly varying functions with an index of regular variation equal to α, i.e. positive measurable functions g(·) such that for any x ≥ 0, g(tx)/g(t) → x α , as t → ∞. We are then working in DM (EVγ >0 ), the domain of attraction for maxima of EVγ , γ > 0, with EVγ denoting the general extreme value distribution (EVD), given by exp(−(1 + γ x)−1/γ ), 1 + γ x > 0 if γ = 0 EVγ (x) = (1) exp(− exp(−x)), x ∈ R if γ = 0, with γ the EVI. The EVD, in Equation (1), is one of the crucial models in the field of extreme value theory (EVT). Indeed, all possible non-degenerate weak limit distributions of the normalized partial maxima Xn:n , of i.i.d. random variables X1 , . . . , Xn , from an underlying parent F , are EVDs, i.e. if there are normalizing constants an > 0, bn ∈ R and some non-degenerate d.f. G such that, for all x, Xn:n − bn ≤ x = G(x), lim P n→∞ an we can redefine the constants in such a way that G(x) = EVγ (x), in Equation (1), for some γ ∈ R [1]. We then write F ∈ DM (EVγ ). Another seminal result in the field of EVT is the one due to Balkema and de Haan [2] and Pickands [3]. Independently, they proved that, under adequate conditions, the generalized Pareto distribution (GPD), 1 − (1 + γ x)−1/γ , 1 + γ x > 0, x ≥ 0 if γ = 0 GPγ (x) = (2) 1 − exp(−x), x ≥ 0 if γ = 0, is the limit distribution of scaled excesses over high thresholds. More precisely, consider the excess function, Fu (x) := P [X − u ≤ x|X > u]. Denoting x F := U (∞), the right endpoint of F , F ∈ DM (Gγ ) if and only if there exists a positive real function σ (u) such that lim |Fu (σ (u)x) − GPγ (x)| = 0
u→x F
(see, for instance, Embrechts et al. [4, Section 3.4], and Reiss and Thomas [5, Section 1.4], for more details). Such a limiting result enabled the development of the so-called maximum likelihood (ML) EVI estimators. We here refer the peaks over threshold methodology of estimation [6] as well as the methodology used by Drees et al. [7], named PORT (peaks over random threshold) in Araújo Santos et al. [8]. For heavy-tailed parents and given a sample Xn = (X1 , . . . , Xn ), the classical EVI estimator is the Hill estimator [9], denoted H ≡ Hn (k) and given by 1 {ln Xn−i+1:n − ln Xn−k:n }, Hn (k) ≡ Hn (k; Xn ) := k i=1 k
1 ≤ k < n,
(3)
the average of the k log-excesses over a high random threshold Xn−k:n . Consistency of the estimator in Equation (3) is achieved if Xn−k:n is an intermediate order statistic (o.s.), i.e. if k → 0, as n → ∞. (4) n The Hill estimator in Equation (3) is scale-invariant but not location invariant, as often desired, and this contrarily to the PORT-Hill estimators, recently introduced in Araújo Santos et al. [8] and k = kn → ∞ and
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further studied in Gomes et al. [10]. The class of PORT-Hill estimators is based on a sample of excesses over a random threshold Xnq :n , nq := [nq] + 1, with [x] denoting, as usual, the integer part of x, i.e. it is based on
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X(q) n := (Xn:n − Xnq :n , Xn−1:n − Xnq :n , . . . , Xnq +1:n − Xnq :n ),
nq = [nq] + 1.
(5)
We need to have 0 < q < 1, for d.f.’s with an infinite left endpoint xF := inf{x : F (x) > 0} (the random threshold is an empirical quantile). We can also have q = 0, provided that the underlying model has a finite left endpoint xF (the random threshold is then the minimum). These new classes (q) of EVI estimators are the so-called PORT-Hill estimators, denoted by Hn , and, for 0 ≤ q < 1 and k < n − nq , they are given by Hn(q) (k) := Hn (k; X(q) n )=
k 1 Xn−i+1:n − Xnq :n ln , k i=1 Xn−k:n − Xnq :n
(6)
i.e. they have the same functional form of the Hill estimator in Equation (3), but with the original sample Xn = (X1 , . . . , Xn ) replaced by the sample of excesses X(q) n in Equation (5). These estimators are now invariant for both changes of scale and location in the data, and depend on the tuning parameter q, which provides a highly flexible class of EVI estimators. Provided that we adequately choose the tuning parameter q, the PORT-Hill estimators may even compare favourably with the second-order minimum-variance reduced-bias (MVRB) EVI estimators, recently introduced in the literature and briefly discussed in the following. Indeed, due to the high bias of the Hill estimator, in Equation (3), for moderate up to large k, several authors have dealt with bias reduction in the field of extremes, working then in a slightly more strict class than DM (EVγ >0 ), the class of models U (·) such that A(t) U (t) = C t γ 1 + + o(t ρ ) , A(t) = γβt ρ , (7) ρ as t → ∞, where ρ < 0 and β = 0. This means that the slowly varying function L(t) in U (t) = t γ L(t) is assumed to behave asymptotically as a constant C. Note that to assume Equation (7) is equivalent to saying that we can choose A(t) = γβt ρ , ρ < 0, in the more general second-order condition ln U (tx) − ln U (t) − γ ln x xρ − 1 lim = . (8) t→∞ A(t) ρ The asymptotic bias, i.e. as n → ∞, √ Hill estimator, in Equation (3), reveals usually a high k(Hn (k) − γ ) is asymptotically normal with variance γ 2 and a non-null mean value, equal √ to λA /(1 − ρ), whenever k A(n/k) → λA = 0, finite, with A(·) the function in Equation (8). √ This non-null asymptotic bias, together with a rate of convergence of the order of 1/ k, leads to sample paths with a high variance for small k, a high bias for large k, and a very sharp meansquared error (MSE) pattern, as a function of k. A simple class of second-order MVRB EVI estimators is the one in Caeiro et al. [11], used for a semi-parametric estimation of ln VaRp in Gomes and Pestana [12]. This class, here defined b H¯ ≡ H¯ n (k), depends upon the estimation of the second-order parameters (β, ρ) in Equation (7). Its functional form is ρˆ ˆ β(n/k) H¯ n (k) ≡ H¯ n (k; Xn ) ≡ H¯ β, , (9) ˆ ρˆ (k) := Hn (k) 1 − 1 − ρˆ ˆ ρ) with Hn (k) the Hill estimator in Equation (3), and where (β, ˆ needs to be an adequate consistent estimator of (β, ρ). Algorithms for the estimation of (β, ρ) are provided in Gomes and Pestana [12,13], among others.
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Let us now think about semi-parametric high quantile estimation. With Q standing for quantile function, the classical Weissman–Hill Var p estimator,
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Qp|H (k) := Xn−k+1:n ckHn (k) ,
ck ≡ ck,n,p =
k , np
(10)
has been introduced in Weissman [14]. The MVRB VaR-estimator Qp|H¯ , with Qp|H given in Equation (10), was studied in Gomes and Pestana [12]. However, for any positive real δ, and with Qp|• denoting either Qp|H or Qp|H¯ , Qp|• (k; δ Xn ) = δ Qp|• (k; Xn ), as desirable, but contrarily to the linear property for quantiles, χp (δX + s) = δχp (X) + s for any real s and positive real δ, we no longer have Qp|• (k; s1n + δXn ) = s + δ Qp|• (k; Xn ), with 1n denoting, as usual, a vector with n unit elements. Araújo Santos et al. [8] have developed a class of high quantile estimators based on the sample of excesses over a random threshold Xnq :n , provided in Equation (5), and, among others, they propose the so-called PORT-Weisman–Hill VaRp -estimators, (q)
(q)
Qp|H (k) := (Xn−k:n − Xnq :n ) ckHn
(k)
+ Xnq :n ,
(11)
(q)
where Hn (k) is the Hill estimator of γ , made location/scale invariant by using the transformed (q) sample X(q) n , i.e. Hn (k) is the estimator in Equation (6). They consequently obtain exactly the above-mentioned linear property for the quantile estimators, but they still get a high bias for moderate values of k, induced by PORT-Hill EVI estimation. We get to know that the second-order MVRB EVI estimators in Equation (9) are not location invariant, but they are ‘approximately’ location invariant. However, if we merely replace, in (q) Equation (11), Hn (k) by H¯ n (k) in Equation (9), we have practically no improvement comparatively with the MVRB-estimator Qp|H¯ , introduced and studied in Gomes and Pestana [12]. With (q) Hn (k) defined in Equation (6), we shall consider here the ‘quasi-PORT’ EVI estimator, ρˆ ˆ β(n/k) (q) (q) (q) ˆ ρ) ˆ := Hn (k) 1 − (12) H¯ n (k) ≡ H¯ n (k; β, 1 − ρˆ and the associated ‘quasi-PORT’ Var p -estimator, with the functional form (q)
¯ (q) (k)
Qp|H¯ (k) := (Xn−k:n − Xnq :n )ckHn
+ Xnq :n .
(13)
In Section 2 of this paper, after a brief discussion on the estimation of the second-order parameters β and ρ, we describe the results associated with a Monte-Carlo simulation study of the new VaR-estimators, in Equation (13), making also a short remark on the use of the GPD approximation for high quantile estimation. Finally, in Section 3, due to the stability on k of the MVRB estimates H¯ , in Equation (9), and Qp|H¯ , with Qp|H provided in Equation (10), as well as the new VaR-estimates in Equation (13), we propose the use of a heuristic stability criterion for the choice of k and q, providing applications of the methodology to simulated data and to log-returns of financial stocks.
2.
Finite sample behaviour: a Monte-Carlo simulation
In this section, for p = 1/n and q = 0, 0.1 and 0.25, we are interested in the finite-sample (q) behaviour of the new VaR-estimators, Qp|H¯ (k), in Equation (13), comparatively with the classical Weissman–Hill VaR-estimator, Qp|H (k), in Equation (10), the associated PORT-Weissman–Hill
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(q)
VaR-estimators, Qp|H (k), in Equation (11), and the MVRB VaR-estimator Qp|H¯ (k), with Qp|H (k) given in Equation (10). The overall simulation is based on a multi-sample simulation with size 5000 × 20, i.e. 20 replicates with 5000 runs each. For details on multi-sample simulation, refer to Gomes and Oliveira [15]. The patterns of mean value (E) and root mean squared error (RMSE) are based on the first replicate only. We have considered the following underlying parents:
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(A) the Burr model, with d.f. Bγ ,ρ (x) = 1 − (1 + x −ρ/γ )1/ρ , x ≥ 0, for γ = 0.25 and ρ = −0.5; (B) Student’s tν -model, with a probability density function ftν (t) =
((ν + 1)/2)[1 + t 2 /ν]−(ν+1)/2 , √ πν (ν/2)
t ∈ R (ν > 0),
for ν = 4 degrees of freedom. We then have γ = 1/ν = 0.25 and ρ = −2/ν = −0.5; (C) the general EV model, with d.f. EVγ (x) in Equation (1), for γ = 0.5 (ρ = −0.5). All reduced-bias EVI estimators, like the ones in Equation (9) and in Equation (12), as well as associated VaR-estimators, require the estimation of scale and shape second-order parameters, (β, ρ), in Equation (7). Such an estimation will be briefly discussed in Section 2.1. Bias-reduction is neatly needed for models with |ρ| ≤ 1, the most common in practical situations, and the type of models also considered in this simulation study. On the basis of the algorithms proposed before in papers like Gomes and Pestana [12] and Gomes et al. [16], we shall use now the tuning parameter τ = 0, in the ρ-estimators discussed in Fraga Alves et al. [17], and the associated β-estimators in Gomes and Martins [18]. 2.1. Estimation of second-order parameters As mentioned above, we shall here consider the estimator 3(Tn (k) − 1) ρˆ0 (k) := min 0, , Tn (k) − 3
k < n,
(14)
with Tn (k) :=
ln Mn(1) (k) − ln(Mn(2) (k)/2)/2 ln(Mn(2) (k)/2)/2 − ln(Mn(3) (k)/6)/3
,
Mn(j ) (k) :=
k Xn−i+1:n j 1 ln , k i=1 Xn−k:n
for j = 1, 2, 3, a particular member of the class of estimators in Fraga Alves et al. [17]. Other interesting alternative classes of ρ-estimators have recently been introduced in Goegebeur et al. [19], Ciuperca and Mercadier [20] and Goegebeur et al. [21]. Distributional properties of the estimators in Equation (14) can be found in Fraga Alves et al. [17]. Consistency is achieved in the class of models in Equation (7), for intermediate √ k-values, denoted by k1 , such that apart from condition (4), with k replaced by k1 , we have k1 A(n/k1 ) → ∞, as n → ∞. We have here decided for the choice k1 = [n1− ],
= 0.001,
(15)
both in simulations and in case studies. √ Remark 2.1 With the choice of k1 in Equation (15), and whenever k1 A(n/k1 ) → ∞, we get ρˆ − ρ := ρˆ0 (k1 ) − ρ = op (1/ ln n), a condition needed, in order not to have any increase in the asymptotic variance of the new bias-corrected Hill estimator in Equation (9), comparatively with the Hill estimator in Equation (3). Note that with the choice of k1 in Equation (15), we get
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√
k1 A(n/k1 ) → ∞ if and only if ρ > 1/2 − 1/(2) = −499.5, an irrelevant restriction, from a practical point of view.
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For the estimation of the scale second-order parameter β, in Equation (7), we shall consider ρˆ dρˆ (k) D0 (k) − Dρˆ (k) k ˆ βρˆ (k) := , k < n, (16) n dρˆ (k) Dρˆ (k) − D2ρˆ (k) dependent on the estimator ρˆ = ρˆ0 (k1 ), with ρˆ0 (k) and k1 given in Equations (14) and (15), respectively, and where, for any α ≤ 0, k 1 i −α dα (k) := k i=1 k
k 1 i −α and Dα (k) := Ui , k i=1 k
with Ui := i(ln Xn−i+1:n − ln Xn−i:n ), 1 ≤ i ≤ k, the scaled log-spacings. Moreover, we shall compute βˆρˆ (k), in Equation (16), at the value k1 , in Equation (15), and work with βˆ = βˆρˆ (k1 ). Details on the distributional behaviour of the estimator in Equation (16) can be found in Gomes and Martins [18] and more recently in Gomes et al. [16] and Caeiro et al. [22]. √ Consistency is achieved for models in Equation (7), k values such that Equation (4) holds and kA(n/k) → ∞, as n → ∞, and estimators ρˆ of ρ such that ρˆ − ρ = op (1/ ln n). Alternative estimators of β can be found in Caeiro and Gomes [23] and Gomes et al. [24]. 2.2.
Mean value and RMSE patterns
We shall consider the following normalized V aRp -estimators, Qp|H (k)/VaRp , Qp|H¯ (k)/VaRp , (q) (q) Qp|H (k)/VaRp and Qp|H¯ (k)/VaRp . For the sake of simplicity, we denote these quotients by QH , QH¯ , QH |q and QH¯ |q , respectively. In Figures 1–3, for the models in (A), (B) and (C), respectively, we show the simulated patterns of mean value, E(Q• ), and RMSE, RMSE(Q• ), of these normalized estimators. These parents were chosen just to illustrate the fact that: • the quasi-PORT VaR-estimators are sometimes unable to improve the performance of QH¯ , as happens also with the PORT-Weissman–Hill estimators when compared with the Weissman– Hill estimator QH (see Figure 1, associated with the above-mentioned Burr model); • we can often find a value of q that provides the best estimator of VaRp through the use of the (q) new class of estimators Qp|H¯ (k), in Equation (13) (like the value q = 0.25, both in Figures 2 and 3).
Figure 1.
Underlying the Burr parent with (γ , ρ) = (0.25, −0.5).
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Journal of Statistical Computation and Simulation
Figure 2.
Underlying Student’s t4 parent (γ = 0.25, ρ = −0.5).
Figure 3.
Underlying extreme value parent with γ = 0.5 (ρ = −0.5).
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2.3. Relative efficiency and bias indicators at optimal levels Given a sample Xn = (X1 , . . . , Xn ), let us denote S(k) = Sn (k) any statistic dependent on k, the number of top o.s.’s used in an inferential procedure related with a parameter of extreme events ξ . The optimal sample fraction for S(k) is denoted k0S (n)/n, with k0S (n) := arg mink MSE(Sn (k)), the so-called optimal level for the estimation of the parameter ξ . We have computed, for n = 200, 500, 1000, 2000 and 5000, and with • denoting H or H¯ or H |q or H¯ |q, the simulated optimal sample fraction (k0• /n), mean values (E0• ) and relative efficiencies (REFF•0 ) of Q• , at their optimal levels. The search of the minimum MSE has been performed over the region of k-values between 1 and [0.95 × n]. For a certain Q• , the REFF•0 indicator is given by
MSE{QH (k0H )} RMSEH 0 REFF•0 := . (17) =: MSE{Q• (k0• )} RMSE•0 As an illustration, we provide Table 1, for the Student underlying parent in (B), where among the estimators considered, and for all n, the one providing the smallest squared bias and smallest MSE, or equivalently, the highest REFF is underlined and in bold. The MSE of QH (k0H ), denoted H by MSEQ 0 , is also provided so that it is possible to recover the MSE of any other quantile estimator. Extensive tables, for the models in (A) and (C), as well as other heavy-tailed models, are available from the authors. For a visualization of the obtained results, we present, in Figure 4, the REFF-indicators, in Equation (17), of the new VaR-estimators, in Equation (13), as well as of the PORT-Weissman–Hill VaR-estimators, at optimal levels, comparatively with the classical VaR-estimator, in Equation (10), also at its optimal level. Figure 5 is equivalent to Figure 4, but with the simulated mean value of Q•0 = Q• (k0• ).
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Table 1. Simulated optimal sample fractions (k0• /n), mean values (E0• ), MSE of QH0 , and relative efficiency measures (REFF•0 ), at optimal levels, together with corresponding 95% confidence intervals, for a Student’s t4 parent (γ = 0.25, ρ = −0.5). 200
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n
500
1000
k0• /n
2000
5000
QH QH |0.1 QH |0.25 QH¯ QH¯ |0.1 QH¯ |0.25
0.0475 ± 0.0021 0.0980 ± 0.0033 0.0693 ± 0.0027 0.0890 ± 0.0046 0.1630 ± 0.0040 0.1850 ± 0.0088
(and 95% confidence intervals) 0.0312 ± 0.0016 0.0228 ± 0.0012 0.0774 ± 0.0023 0.0644 ± 0.0017 0.0491 ± 0.0027 0.0393 ± 0.0017 0.1122 ± 0.0043 0.2897 ± 0.0113 0.1834 ± 0.0650 0.5727 ± 0.0020 0.3508 ± 0.0050 0.3465 ± 0.0029
0.0167 ± 0.0007 0.0537 ± 0.0015 0.0304 ± 0.0010 0.1287 ± 0.0065 0.5393 ± 0.0012 0.2468 ± 0.0044
0.0110 ± 0.0006 0.0442 ± 0.0009 0.0221 ± 0.0008 0.0461 ± 0.0016 0.4881 ± 0.0014 0.1413 ± 0.0038
QH QH |0.1 QH |0.25 QH¯ QH¯ |0.1 QH¯ |0.25
1.0987 ± 0.0043 1.0812 ± 0.0027 1.0893 ± 0.0030 0.9032 ± 0.0053 0.8892 ± 0.0027 0.8941 ± 0.0039
E0• (and 95% confidence intervals) 1.0892 ± 0.0037 1.0846 ± 0.0037 1.0701 ± 0.0027 1.0607 ± 0.0020 1.0805 ± 0.0044 1.0773 ± 0.0035 0.9220 ± 0.0030 0.9776 ± 0.0028 0.9093 ± 0.0048 0.9752 ± 0.0019 0.9417 ± 0.0018 1.0038 ± 0.0017
1.0814 ± 0.0035 1.0498 ± 0.0020 1.0712 ± 0.0023 1.0337 ± 0.0014 0.9899 ± 0.0011 1.0128 ± 0.0015
1.0765 ± 0.0037 1.0354 ± 0.0015 1.0632 ± 0.0025 1.0564 ± 0.0015 0.9968 ± 0.0007 1.0194 ± 0.0015
0.1026 ± 0.0019
0.0727 ± 0.0012
0.0460 ± 0.0006
0.0341 ± 0.0003
1.5841 ± 0.0087 1.2239 ± 0.0050 1.8198 ± 0.0160 2.8835 ± 0.0300 2.5464 ± 0.0216
1.8079 ± 0.0069 1.2800 ± 0.0049 1.5306 ± 0.0095 3.7644 ± 0.0298 2.6082 ± 0.0200
Q
MSE0 H QH |0.1 QH |0.25 QH¯ QH¯ |0.1 QH¯ |0.25
1.2732 ± 0.0081 1.1318 ± 0.0067 1.3098 ± 0.0129 1.5208 ± 0.0153 1.4769 ± 0.0151
REFF•0
0.0575 ± 0.0007
(and 95% confidence intervals) 1.3585 ± 0.0066 1.4569 ± 0.0064 1.1561 ± 0.0062 1.1877 ± 0.0039 1.4804 ± 0.0134 1.8808 ± 0.0156 1.6520 ± 0.0139 2.2205 ± 0.0237 1.9905 ± 0.0221 2.5165 ± 0.0145
Figure 4. REFF-indicators for a Burr model with γ = 0.25 and ρ = −0.5 (left), a Student’s t4 model (centre) and an EV model, with γ = 0.5 (right).
Regarding the REFF-indicators, we would like to draw the following comments: • For models like the Burr, or any other underlying parent with a left endpoint equal to zero, we cannot achieve any improvement with the shifted estimators. • For a model like Student’s tν , here illustrated for ν = 4, as well as for an underlying EV model, we reach a clear improvement in the estimation of a high quantile, whenever we consider the quasi-PORT VaR estimators, in Equation (13). • Note however that, for these models, even the PORT-estimators based on the Hill estimator provide REFF-indicators higher than one for all n, with the highest indicator associated with q = 0.1, for the t4 parent, and q = 0 (a shift induced by the minimum of the available sample), for the EV model.
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Figure 5. BIAS-indicators for a Burr model with γ = 0.25 and ρ = −0.5 (left), a Student’s t4 model (centre) and an EV model, with γ = 0.5 (right).
• For these same underlying parents, and when we consider the quasi-PORT VaR-estimators, the pattern of the REFF-indicators is not so clear-cut, as the one of PORT-Hill estimators. Anyway, there is, for all n, a value of q providing the highest efficiency. For the t4 parent, the best performance of the quasi-PORT VaR-estimators has been attained for q = 0.25, if n = 500, 1000 and for q = 0.1, for other values of n. For the EV0.5 parent, the best performance has been attained for q = 0.1, if n ≤ 500 and n = 5000, and for q = 0.25, if n = 1000 and 2000. It is also clear from the comparison of Figures 4 and 5 that there is not a full agreement between REFF and BIAS indicators, but the discrepancies are moderate. Regarding bias at optimal levels, we can draw the following comments: • For the simulated Burr model, the MVRB VaR-estimators exhibit the smallest bias for all n, but not a long way from the quasi-PORT VaR-estimator associated with q = 0, as expected. • For Student’s t4 model, the quasi-PORT VaR-estimator based on both q = 0.1 and q = 0.25 have interesting bias patterns, particularly for n ≥ 1000. • For an EV model with γ = 0.5 and for all n, the smallest bias is achieved by the quasi-PORT quantile estimator based on the shifted version of H¯ , for q = 0.25. Note also the over-estimation achieved by the quantile estimators based on the Hill, as a counterpart of an under-estimation achieved by the quantile estimators based on the MVRB EVI-estimator, for small n. In summary, we may draw the following conclusions: (1) If the underlying model has a finite left endpoint at zero, the PORT or quasi-PORT estimators can never beat the original estimators regarding efficiency. (2) For parents with an infinite left endpoint, like the Student parents, or a left endpoint different from zero, like the EV parents, the best performance regarding efficiency is attained by the new estimators for an adequate value of q. Such a q depends on the underlying model and on the sample size n. A similar comment applies to bias reduction. 2.4. A brief remark on ML estimation based on excesses As briefly mentioned before, in Section 1, one of the most common location/scale invariant EVI estimators is the so-called ML estimator, obtained on the basis of the excesses Vik := Xn−i+1:n − Xn−k:n , 1 ≤ i ≤ k, which are approximately distributed as the k top-order statistics associated with a sample of size k from a GPD, GPγ (x/σ ), σ > 0, with GPγ (x) given in Equation (2). The estimation of a quantile of probability 1 − p, with p small can then be done on the basis of the
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Figure 6.
Underlying Student’s t4 parent (γ = 0.25, ρ = −0.5).
ML estimates (γˆk , σˆ k ) of (γ , σ ), through the equation γˆk ˜ p|ML (k) := Xn−k:n + σˆ k (k/(np)) − 1 . Q γˆk
(18)
This is the estimator considered, among others, in McNeil and Frey [25], in a conditional EVT estimation of tail-related risk measures for heteroscedastic financial time series. In order to emphasize the better performance of the quasi-PORT VaRp estimators, in Equation (13), comparatively with the VaRp estimator, in Equation (18), we present Figure 6, similar to Figure 2, but with a zoom in the region where we obtained convergence and a reasonable stability for the ML estimates. Similar results have been obtained for all other simulated models. The simulations suggest that for estimating high quantiles, the ML method based on the excesses, often called the GPD method, clearly outperforms the Weissman–Hill quantile estimation, as mentioned in MacNeil and Frey [25], but the quasi-PORT method overcomes the GPD method for all k, essentially in terms of MSE.
3. A heuristic choice of tuning parameters and case studies 3.1. An algorithm for the heuristic choice of k and q (q) With the notation X0:n = 0, and with H¯ n and H¯ n , given in Equations (9) and (12), respectively, (q) we can consider that H¯ n ≡ H¯ n for q = −1/n (nq = 0). Our interest lies now on the estimation (q) of VaRp through Qp|H¯ , in Equation (13). On the basis of the stability of the sample paths of the estimators under study, as a function of k, we propose now the following adaptive heuristic estimation of VaRp .
Algorithm 1 (adaptive estimation of VaRp ) (1) Consider q = −1/n, 0(0.05)0.25; (q) (2) Compute the observed values of Qp|H¯ (k), k = 1, 2, . . . , n − [nq] − 1; (q)
(3) Obtain j0 , the minimum value of j , a non-negative integer, such that ak (j ) = [Qp|H¯ (k) × 10j ], k = 1, 2, . . . , n − [nq] − 1, has distinct elements (in this case, we were led to j0 = 0 for all data sets considered). Choose q in the following way: for each q consider as possible estimates (q) (q) (q) of V aRp the values Qp|H¯ (k), kmin ≤ k ≤ kmax , to which is associated the largest run, with a (q)
(q)
size lq = kmax − kmin + 1. Choose q0 := arg maxq lq ;
Journal of Statistical Computation and Simulation (q )
(q )
11
(q )
0 (4) Consider all those estimates, Qp|0H¯ (k), kmin0 ≤ k ≤ kmax , now with an extra decimal place,
i.e. Qp|0H¯ (k) = ak (j0 + 1)/10j0 +1 . Count the frequencies associated to these estimates and obtain the mode of these values, considering them with an extra decimal figure. Let us denote K∗ the set of k-values corresponding to those estimates. Take k0 as the maximum of K∗ (in order to minimize the variance). (q )
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3.2. Applications to a simulated data set In order to have some indication about the way the algorithm performs and to motivate its use prior to its validation through simulation, a topic outside of the scope of this paper, we first apply it to an arbitrarily simulated sample, with size n = 1762 from a Student’s tν model, with ν = 4 degrees of freedom (γ = 0.25, ρ = −0.5). The choice of such a model was due to the specificity of the samples of log-returns of financial time series analysed in Section 3.3, all with a size n = 1762. Indeed, these log-returns are often modelled through a Student’s t d.f. or its skewed extensions [26,27]. The number of positive elements in the generated sample is n0 = 904. We have then been led to the ρ-estimate ρˆ ≡ ρˆ0 = −0.72, obtained at the level k1 = [n0.999 ] = 897. 0 ˆ ˆ The associated β-estimate is β ≡ β0 = 1.02. (q) In Figure 7, we present the estimates of VaRp , provided by Qp|H , Qp|H¯ and Qp|H¯ , for p = (q)
1/(4n), q = 0.1, 0.2, with Qp|H and Qp|H¯ given in Equations (10) and (13), respectively. In Step (1) of the algorithm in Section 3.1, the value q = 0 was excluded, due to the inconsistency of these estimators whenever the left endpoint of the underlying parent is infinite [10].
Figure 7.
Quantile estimates for the Student tν data, with ν = 4 (γ = 0.25).
Regarding the VaR-estimation, note that whereas the Weissman–Hill estimator Qp|H (k), in Equation (10), is unbiased for the estimation of VaRp when the underlying model is a strict Pareto model, it exhibits a relevant bias when we have only Pareto-like tails, as happens here, and may (q) be seen from Figure 7. The quasi-PORT estimators, Qp|H¯ , in Equation (13), based on shifted MVRB EVI-estimators, which can be ‘asymptotically unbiased’, have a smaller bias, exhibit more stable sample paths as function of k, and enable us to take a decision upon the estimate of γ and VaR to be used, with the help of any heuristic stability criterion, like a ‘largest run’ method, written algorithmically in Section 3.1. In this case, the largest run, in Step (3) of the Algorithm, is equal to 190 and was attained by the estimate 12. We have then been led to the choice q = 0.2. Next, in Step (4) of the algorithm, we were led to the choice of an estimate 12.1 (with an associated frequency equal to 63). We finally came to the choice k = 770 and to the final
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(0.2) estimate VaRp|H¯ (0.2) := Qp| (770) = 12.10, quite close to the real value of VaRp , equal to 11.92. H¯ Similar results were obtained for other simulated samples.
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3.3. Applications to data in the field of finance We have next considered the performance of the non-adaptive and adaptive VaR-estimators studied in this paper, when applied to the analysis of the log-returns associated with two of the four sets of finance data considered in Gomes and Pestana [12]. Those sets of data, collected over the same period, i.e. from 4 January 1999 through 17 November 2005, were the daily closing values of the Dow Jones Industrial Average In (DJI) and Microsoft Corp. (MSFT). Additionally, we have considered over the same period the Euro-GB Pound (EGBP) daily exchange rates. All these samples have a size n = 1762. The VaR, defined as a large quantile of negative log-returns, i.e. of Li = − ln(Si+1 /Si ), 1 ≤ i ≤ n − 1, with Si , 1 ≤ i ≤ n, a sample of consecutive close prices, is a common risk measure for large losses. For details about VaR see, for instance, Holton [28], among others. Here, since we are interested in the analysis of the risk of holding short positions, we have dealt with the positive log-returns, i.e. with Pi = ln(Si+1 /Si ) = −Li , 1 ≤ i ≤ n − 1. Although there is some increasing trend in the volatility, stationarity and weak dependence are assumed, under the same considerations as in Drees [29]. All the above-mentioned semi-parametric estimators are then still consistent and asymptotically normal for adequate k, although with different asymptotic variances and slightly different dominant components of asymptotic bias. Indeed, whenever confronted with weakly dependent processes with an extremal index θ < 1, we heuristically expect a shrinkage of maximum values and a mean size approximately equal to 1/θ for the size of the clusters of exceedances of high levels. Asymptotically, we thus expect an increase in the variance of a factor proportional to 1/θ , a high increase in the MSE of the estimators, but small changes in the bias. The presence of clustered volatility, or equivalently conditional heteroscedasticity, seems not to be highly crucial in the performance of semi-parametric reducedbias estimators. See Gomes et al. [30] and Gomes and Miranda [31], among others. Note, however, that the possible presence of clustered volatility is a question of particular relevance to applied financial research, as extensively discussed in McNeil and Frey [25], where VaR is estimated for heteroscedastic return series, through an approach combining quasi-likelihood fitting of a GARCH model to estimate the current volatility and EVT for estimating the tail of the innovations’ distribution of the GARCH model. We now advance with the possible use of an approach similar to the one in McNeil and Frey [25], but with the combination of the quasi-likelihood fitting of a GARCH model to estimate the current volatility, together with the methodology in this paper for the estimation of the tail of the innovations’ distribution of such a GARCH model. A comparison of such a technique with the one carried out in this paper, as well as with the methodology in McNeil and Frey [25], deserves further research, both from a theoretical as well as from an applied point of view, to be dealt with in the near future. For all data sets, we present essentially two figures. In the first one, we picture a box-andwhiskers’ plot (left) and a histogram (right) of the available data. It is clear from all the graphs that all sets of data have heavy left and right tails, and we have thus eliminated the estimators associated with q = 0, due to their inconsistency. In the second figure, we present, for p = 1/(2n), (q) (q) the estimates of VaRp , provided by the Qp|H , Qp|H¯ and Qp|H¯ , q = 0.1, 0.2, with Qp|H and Qp|H¯ given in Equations (10) and (13), respectively. 3.3.1.
DJI data
From Figure 8, we immediately see that the underlying model has heavy left and right tails. The number of positive elements in the available sample of log-returns is n0 = 867. We have been
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Density
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-6
-4
-2
0 2 DJI Log-returns
4
6
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
Histogram of DJI
-5
0 DJI Log-returns
Figure 8.
Box-and-whiskers (right) and histogram (left) associated with the DJI data.
Figure 9.
Quantile estimates for the DJI data.
5
led to the ρ-estimate ρˆ ≡ ρˆ0 = −0.72, obtained at the level k1 = [n00.999 ] = 861. The associated β-estimate is βˆ ≡ βˆ0 = 1.03. In this case, the largest run, in Step (3) of the algorithm, is equal to 327 and was attained by the estimate 8. We have then been led to the choice q = 0.1. Next, in Step (4) of the algorithm, we were led to the choice of an estimate 7.9 (with an associated frequency equal to 75). We finally (0.1) came to the choice k = 645 and to the final estimate VaR1/(2n)|H¯ (0.1) := Q1/(2n)| (645) = 7.89. H¯ 3.3.2. MSFT data Figures 10 and 11 are similar to Figures 8 and 9, respectively, now for the MSFT data. As can be inferred from Figure 10, both tails are again heavy. The number of positive elements in the available sample of log-returns is now n0 = 882. Step (3) of the algorithm here presented led us to the ρ-estimate ρˆ ≡ ρˆ0 = −0.72, obtained at the level k1 = [n00.999 ] = 876. The associated β-estimate is βˆ ≡ βˆ0 = 1.02. In this case, the largest run, in Step (3) of the algorithm is equal to 113 and was attained by the estimate 19. We have then been led to the choice q0 = 0.2. Next, in Step (4) of the algorithm, we were led to the choice of an estimate 19.1 (with an associated frequency equal to 24). We finally (0.2) came to the choice k = 749 and to the final estimate VaR1/(2n)|H¯ (0.2) := Q1/(2n)| (749) = 19.13, H¯ the values pictured in Figure 11.
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0.06 0.04 0.02 0.00
-15
-10
-5 0 5 MSFT Log-returns
10
15
-20
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0 10 MSFT Log-returns
Figure 10.
Box-and-whiskers (right) and histogram (left) associated with the MSFT data.
Figure 11.
Quantile estimates for the MSFT data.
3.3.3.
20
EGBP data
Figures 12 and 13 are again similar to Figures 8 and 9, respectively, now for EGBP data, and similar conclusions can be drawn. The number of positive elements in the available sample of log-returns is now n0 = 835. We have been led to the ρ-estimate ρˆ ≡ ρˆ0 = −0.67, obtained at the level k1 = [n00.999 ] = 829. The associated β-estimate is βˆ ≡ βˆ0 = 1.03.
0.4 0.2
Density
0.6
0.8
Histogram of STERLING
0.0
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Density
0.08
Histogram of MSFT
-2
Figure 12.
-1
0 1 2 Euro-GB Pound Log-returns
3
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-1
0 1 2 Euro-GB Pound Log-returns
Box and whiskers (right) and histogram (left) associated with the EGBP data.
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Journal of Statistical Computation and Simulation
Figure 13.
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Quantile estimates for the EGBP data.
In this case, the largest run, in Step (3) of the algorithm is equal to 886 and was attained by the estimate 3. We have then been led to the choice q0 = 0.1. Next, in Step (4) of the algorithm, we were led to the choice of an estimate 3.0 (with an associated frequency equal to 226). We finally (0.1) came to the choice k = 494 and to the final estimate VaR1/(2n)|H¯ (0.1) := Q1/(2n)| (494) = 2.97, H¯ the values now pictured in Figure 13. Acknowledgements Research partially supported by FCT/OE and PTDC/FEDER. We also would like to thank a referee for the valuable comments on a first version of this paper.
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