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Multivariate affine generalized hyperbolic distributions: An empirical investigation
International Review of Financial Analysis 18 (2009) 174–184
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International Review of Financial Analysis
Multivariate affine generalized hyperbolic distributions: An empirical investigation José Fajardo a,⁎, Aquiles Farias b a b
IBMEC Business School, Av. Presidente Wilson 118, 1108, 20030020. Rio de Janeiro, RJ., Brazil Banco Central do Brasil
a r t i c l e
i n f o
Article history: Received 7 March 2008 Received in revised form 27 May 2009 Accepted 29 June 2009 Available online 6 July 2009
a b s t r a c t The aim of this paper is to estimate multivariate affine generalized distributions (MAGH) using market data. We use the Ibovespa, CAC, DAX, FTSE, NIKKEI and S&P500 indexes. We estimate the univariate distributions, bi-variate distributions and six-dimensional distribution. Then we assess their goodness of fit using Kolmogorov distances. As an application we study the efficient frontier. © 2009 Elsevier Inc. All rights reserved.
Keywords: Generalized hyperbolic distributions Multivariate distributions Affine transformation Fat tails
1. Introduction In the past decade a class of distributions called generalized hyperbolic distributions (GH) has been suggested to fit financial data. The development of these distributions is due to Barndorff-Nielsen (1977). He applied the hyperbolic subclass to fit the grain size of sand subjected to continuous wind. Barndorff-Nielsen extended this work to multivariate generalized hyperbolic distributions (MGH). These distributions have been used in different fields of knowledge, such as physics, biology, agronomy and others (see Blæsild &Sørensen, 1992). Eberlein and Keller (1995) were the first to apply these distributions to finance. They used univariate hyperbolic subclasses to fit German data. Keller (1997) studied derivatives pricing with GH and Prause (1999) extended the results of Eberlein and Keller (1995) by fitting financial data using the MGH class. He also priced derivatives and measured value at risk. Using the GH class one can capture fat tails and the skewness observed in asset returns. Blæsild and Sørensen (1992) were the first to develop a computer program, called Hyp, to estimate the parameter of the hyperbolic subclass up to three dimensions. Prause (1999) developed a program to estimate the MGH class. Unfortunately, the MGH parameter estimation procedure is computationally intense and has some shortcomings. Improvement of this procedure would be an important contribution, in particular from a practical point of view. More recently, Schmidt, Hrycej, and Stützle (2006) introduced multivariate affine generalized hyperbolic distributions (MAGH). Their main goal was to develop a multivariate distribution that can
⁎ Corresponding author. Tel.: +55 21 4503 4162; fax: +55 21 4503 4168. E-mail addresses: [email protected] (J. Fajardo), [email protected] (A. Farias). 1057-5219/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.irfa.2009.06.002
capture fat tails, skewness and tail dependence and at the same time is simple to estimate. Another way to capture tail dependence is to use copulas. That is, given the univariate marginals, one can obtain a determined multivariate distribution, having these univariate distributions as their marginals, by using a copula. Reciprocally, having a copula and given marginals, one can obtain a multivariate distribution having these marginals. Then, given an empirical distribution, one can give a very precise description of the observed dependence structure by choosing the appropiate copula from the available copula families. Of course, the goodness of fit will depend strongly on the available families of copulas and on the distribution dimension. For example, for n ≥ 3 the lower Fréchet bound is not a copula. In that sense, this approach presents many estimation problems in higher dimensions. Again, there is the trade-off between accuracy and computational effort. The use of pair copulas might provide some help, as pointed out by Mendes, Leal, and Semeraro (2008). For more applications of copulae in finance, see Cherubini, Luciano, and Vecchiato (2004) and Mendes, Kolev, and Anjos (2006). It is worth noting that the tail dependence property of MAGH, allows us modeling dependences of extreme events, which is a key ingredient in the computation of portfolios' value at risk (VaR) and in many other applications in risk management (see Embrechts, Lindskog, & McNeil, 2003). This dependence cannot be captured with MGH. As a result, the VaR calculations using MGH will be unappropiate in multidimensional settings. More comparisons between MAGH and MGH can be found in Section 4 in Schmidt et al. (2006). Some applications to the Brazilian market have been performed to analyze the use of univariate GH. Using the Hyp software Fajardo, Schuschny, and Silva (2001) studied the goodness of fit of the
J. Fajardo, A. Farias / International Review of Financial Analysis 18 (2009) 174–184 Table 1 Sample.
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Table 2 Descriptive statistics.
Asset
Ticker
Start
End
Index
Mean (%)
Std Deviation (%)
Skewness
Kurtosis
Min (%)
Max (%)
Bovespa Cac40 Dax FTSE Nikkei Standard and Poors
BVSP CAC DAX FTSE NIKK SP500
08/01/1994 08/01/1994 08/01/1994 08/01/1994 08/01/1994 08/01/1994
10/20/2005 10/20/2005 10/20/2005 10/20/2005 10/20/2005 10/20/2005
BVSP CAC DAX FTSE NIKK SP500
0.0668 0.0256 0.0277 0.0174 − 0.0147 0.0322
2.4107 1.3766 1.4967 1.0680 1.3625 1.0737
0.5901 − 0.0988 − 0.1415 − 0.1238 − 0.0997 − 0.1002
16.9409 5.7326 5.6244 5.9075 5.1872 6.4608
− 17.2082 − 7.6781 − 6.4999 − 5.5888 − 7.2340 − 7.1127
28.8325 7.0023 7.5527 5.9038 7.6553 5.5744
hyperbolic subclass. Fajardo and Farias (2004) and Fajardo, Farias, and Ornelas (2005) extended those results and priced some derivatives using the univariate GH class. In this paper we generalize Fajardo and Farias (2004) to the multidimensional case by using MAGH. We assess the goodness of fit of MAGH with international financial data and the Brazilian index Ibovespa. As an application, we derive the efficient frontier for two and six assets. The paper is organized as follows: Section 2 presents generalized hyperbolic distributions; Section 3 presents multivariate affine generalized distributions; Section 4 describes our sample; Section 5 describes the MAGH estimation procedures; Section 6 presents the empirical results; and Section 7 analyzes the efficient frontier. The last sections have the conclusions and an appendix. 2. Generalized hyperbolic distributions
3. Multivariate affine generalized hyperbolic distributions The n-dimensional MAGH consists of the following stochastic representation: d
1 λ− 2
ghðx; α; β; δ; μ; λÞ = aðλ; α; β; δÞðδ + ðx−μÞ Þ
2
2
Kðλ; α; β; δ; μÞ
where μ is a location parameter, δ is a scale factor, compared to Gaussian σ in Eberlein (2000), α and β determine the distribution shape, λ defines the tail fatness (Barndorff-Nielsen & Blaesid, 1981) and therefore the subclasses of GH, and Kðλ; α; β; δ; μÞ = K
Proposition 1. We can write GH(x;ω,δ,μ) as an affine transformation of a canonical GH: GH(x;ω,1,0), where ω: = (α̃ ,β̃,λ).
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 1 ðα δ + ðx−μÞ Þexpðβðx−μÞÞ
where A is an upper triangular matrix ∈Rnxn such that A′A = Σ is a positive definite and the random vector Y∈Rn consists of n mutually independent one-dimensional canonical GH(ω,1,0) (for more details see Schmidt et al., 2006). This definition eases the estimation procedure. M is the location parameter and Σ is a scaling factor. The family of n-dimensional multivariate affine generalized hyperbolic distributions is denoted by MAGMn(ω,Σ,M), where ω: = (ω1,…,ωn) and ωi: = (λi,αi,βi), i = 1,…,n. Table 3 Correlation matrix.
λ−
2
where, 2
2
λ
ðα −β Þ2 aðλ; α; β; δÞ = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi λ−1 λ 2 δ K ðδ α2 −β2 Þ 2πα λ
is a norming factor to make the curve area equal to 1 and 1 ∞ λ−1 1 −1 exp − xðy + y Þ dy Kλ ðxÞ = ∫0 y 2 2
BVSP CAC DAX FTSE NIKK SP500
is the modified Bessel function of the third kind with index λ. The domains of the parameters are: μ; λ∈R −α b β b α δ; α N 0:
1
λ−1
expðð∓α + βÞxÞ as x→F∞
For more details about Bessel functions, see Abramowitz and Stegun (1968).
CAC
DAX
FTSE
NIKK
SP500
1.0000 0.2682 0.2756 0.2735 0.1136 0.4277
0.2682 1.0000 0.7902 0.7875 0.2434 0.4366
0.2756 0.7902 1.0000 0.7215 0.2315 0.4983
0.2735 0.7875 0.7215 1.0000 0.2551 0.4311
0.1136 0.2434 0.2315 0.2551 1.0000 0.1138
0.4277 0.4366 0.4983 0.4311 0.1138 1.0000
Index BVSP
CAC
DAX
The GH have semi-heavy tails. This name is due to the fact that their tails are heavier than Gaussian's, but they have finite variance, which is observed by the following approximation:
BVSP
Table 4 GH and its subclass estimation parameters.
1
ghðx; λ; α; β; δÞ∼jx j
′
X= AY +M
The probability density function of the one-dimensional GH is defined by:
2
Many distributions can be obtained as subclasses of or limiting distributions of GH. Some examples are the Gaussian distribution, Student's T and the normal inverse Gaussian. See Barndorff-Nielsen (1978) and Prause (1999) for detailed descriptions. We can alternatively write the GH density as an affine transformation of a canonical form, with scale 1 and position 0.
FTSE
NIKK
SP500
GH NIG HYP GH NIG HYP GH NIG HYP GH NIG HYP GH NIG HYP GH NIG HYP
α̃
β̃
λ
δ
μ
Loglike
0.6350 0.8862 0.4917 1.0065 1.0410 1.0511 0.9814 0.9726 1.0531 0.9851 0.9851 1.0543 1.0405 1.0376 1.0470 1.0444 1.0414 1.0529
− 0.0998 − 0.0957 − 0.0437 − 0.0819 − 0.0091 − 0.0041 − 0.0863 − 0.1003 − 0.0057 − 0.0804 − 0.0804 − 0.0075 − 0.0055 0.0000 0.0000 − 0.0075 − 0.0076 − 0.0053
− 1.0149 − 0.5000 1.0000 − 0.9964 − 0.5000 1.0000 − 0.0100 − 0.5000 1.0000 − 0.4989 − 0.5000 1.0000 0.5046 − 0.5000 1.0000 0.0026 − 0.5000 1.0000
2.3911 2.1473 0.7420 1.6317 1.3874 0.8504 1.2092 1.4476 0.9167 1.0388 1.0388 0.6541 0.9902 1.4066 0.8489 0.9030 1.0751 0.6600
0.2131 0.2693 0.2162 0.1194 0.0520 0.0372 0.1776 0.1733 0.0454 0.0975 0.0975 0.0318 0.0056 − 0.0067 − 0.0085 0.0471 0.0477 0.0424
− 6387.0 − 6392.8 − 6408.0 − 4919.8 − 4920.6 − 4927.0 − 5141.6 − 5142.2 − 5153.9 − 4153.2 − 4153.2 − 4163.7 − 4935.9 − 4937.7 − 4936.8 − 4180.3 − 4180.4 − 4186.7
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Fig. 1. NIKK PDF.
Fig. 3. BVSP PDF.
Fig. 2. NIKK log-PDF.
Fig. 4. BVSP log-PDF.
The mean and covariance matrix of the MAGH can be easily calculated: ′
cii = Rλi ;1
′
E½X = E½A Y + M = A E½Y + M
where E[Y], by independence, is a vector containing in each row the q mean of the univariate Y i ~ GH(ω i ,1,0), given by E½Yi = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rλi ;1 ðα2i ð1−β2i ÞÞαi βi and ′
Cov½X = A CA; Table 5 Log-likelihood ratio test. Index BVSP CAC DAX FTSE NIKK SP500
where C = diag(c11,…,cnn) with qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðα2i ð1−β2i ÞÞ + Rλi ;2 ðα2i ð1−β2i ÞÞ−Rλi ;1 ðα2i ð1−β2i ÞÞ
½
β 1−β 2 i
2 i
If cii = c for all i = 1,…,n, then Cov[X] = cΣ. This distribution is extremely flexible because the parameters λ and α can be defined for each margin, improving the fit even if the margins have very different tail fatness, because, λ and α are directly responsible for that phenomenon. Furthermore, if Σ is a diagonal matrix, the margins are independent, which is important in some scenarios. Table 6 Kolmogorov–Smirnov tests.
NIG
Hyp
Index
Stats
P-value
Stats
P-value
11.5260 1.7523 1.2142 0.0005 3.7139 0.2018
6.86E−04 0.1856 0.2705 0.9820 0.0540 0.6533
41.9750 14.4540 24.4810 20.9120 1.9122 12.7990
9.24E−11 1.44E−04 7.50E−07 4.81E−06 0.1667 3.47E−04
BVSP CAC DAX FTSE NIKK SP500
Normal
GH
NIG
Hyp
KS
p-value
KS
p-value
KS
p-value
KS
p-value
0.0683 0.0525 0.0680 0.0571 0.0486 0.0549
2.7E−12 1.9E−07 3.4E−12 9.9E−09 2.0E−06 4.2E−08
0.0129 0.0098 0.0167 0.0143 0.0112 0.0149
0.7134 0.9414 0.3871 0.5875 0.8576 0.5360
0.0115 0.0085 0.0173 0.0142 0.0107 0.0144
0.8318 0.9841 0.3413 0.5917 0.8902 0.5731
0.0141 0.0139 0.0307 0.0219 0.0122 0.0200
0.6049 0.6244 0.0080 0.1199 0.7744 0.1924
:
J. Fajardo, A. Farias / International Review of Financial Analysis 18 (2009) 174–184 Table 7 Anderson–Darling distances.
Bovespa CAC40 Dax FTSE Nikkei SP500
Table 9 Log-likelihood ratio tests.
Normal
GH
NIG
Hyp
50077.4824 8.3298 0.6159 2.8125 3.2669 334.6462
0.0470 0.0355 0.0374 0.0416 0.0388 0.1211
0.1080 0.0530 0.0375 0.0418 0.0351 0.0991
0.3071 0.0786 0.0858 0.0868 0.0563 0.2245
Assets
The facilitation of estimation is due to a simple procedure that allows, instead of a simultaneous process of acquiring parameters, estimation using n univariate estimations, where n represents the number of dimensions. 4. Sample Our empirical evaluation uses the Ibovespa, CAC 40, Dax 100, FTSE 100, Nikkei 225 and S&P 500 indexes. The data consist of the daily logreturns, calculated using:
Ri;t = ln
Pi;t
177
!
NIG
BVSP × CAC BVSP × DAX BVSP × FTSE BVSP × NIKK BVSP × SP500 CAC × DAX CAC × FTSE CAC × NIKK CAC × SP500 DAX × FTSE DAX × NIKK DAX × SP500 FTSE × NIKK FTSE × SP500 NIKK × SP500
Hyp
Stats
P-value
Stats
P-value
3.1351 25.1630 5.4250 7.8421 7.9218 0.8372 0.8429 3.0086 0.2348 0.8951 3.4144 0.4480 3.0308 0.2363 3.3853
0.2086 0.0000 0.0664 0.0198 0.0190 0.6580 0.6561 0.2222 0.8892 0.6392 0.1814 0.7993 0.2197 0.8886 0.1840
57.2950 86.1360 73.2390 49.7060 59.6940 35.4220 30.2700 12.1060 25.3790 29.7720 18.5980 16.9600 16.3950 34.0040 14.8280
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0024 0.0000 0.0000 0.0001 0.0002 0.0003 0.0000 0.0006
which also is to be expected since they are from completely different continents. 5. Estimation algorithm
Pi;t−1
The samples with their tickers and respective periods are in Table 1. The starting date is the date Brazil implemented its currency stabilization plan (Real plan), which brought some stability to the prices, avoiding daily inflation adjustment of asset prices. Since we are talking about different countries, we interpolated the data when a date was not a trading day in all countries. We did not exclude any trading day, not even September 11, 2001. In Table 2 we give the main descriptive statistics of the data, and in Table 3 we have the correlation matrix of the data. These two tables show the most important features of this database: • Highly correlated data. CAC × DAX have a correlation coefficient of 0.7902; • Almost uncorrelated data. BVSP × NIKK have a correlation coefficient of 0.1136; • High amplitude data. BVSP has a minimum of −17.2082 and a maximum of 28.8325, much greater the other indexes. • High kurtosis data. Following the last item, BVSP has kurtosis of 16.9409, implying a much heavier distribution tail. Analysis of the correlation matrix shows that the most correlated data are from the DAX, CAC and FTSE, which is to be expected since all of them are European markets (not Euro zone). The BVSP and NIKK, followed by the SP500 and NIKK, contain the least correlated data,
To estimate the GH parameters we used a slight modification of algorithm presented in Fajardo and Farias (2004) to estimate the affine transformation form of GH(ω,δ2,μ).That algorithm was implemented in Matlab and uses maximum likelihood estimation Freund (2004), Lagarias, Reeds, Wright, and Wright (1998), and Neumaier (2004) all show properties of restricted optimization and Baritompal and Hendrix (2005), Björkman and Holmström (1999), Hart (1994), Iwaarden (1996), Mendivii et al. (1999), and Stützle and Hrycej (2002a) also discuss ways to implement global optimization. Based on them, in order to improve performance and get more reliable estimates, we transformed the restricted parameters into unrestricted parameters: ˜ u = lnðαÞ ˜ α
ð1Þ
δu = lnðδÞ
ð2Þ
˜ = ð1− expð−β ˜ × signðβ ˜ ÞÞÞ × signðβ ˜ Þ; β u u u
ð3Þ
To estimate MAGH parameters we used some propositions in order to simplify the procedure and improve the efficiency of the
Table 8 MAGH and its subclasses estimated parameters. Assets CAC × DAX
GH Nig Hyp
BVSP × NIKK
GH Nig Hyp
α̃
β̃
λ
M
Σ
1.0441 1.0028 1.0441 0.9842 1.0574 1.0191 1.0377 1.0409 1.0474 1.0386 1.0759 1.0470
0.0037 − 0.0878 0.0037 − 0.1058 0.0024 − 0.0699 − 0.0157 − 2E−05 − 0.0151 − 0.0056 − 0.0077 − 1.5E−07
− 0.5036 − 0.0100 − 0.5 − 0.5 1 1 − 1.0173 0.5022 − 0.5 − 0.5 1 1
0.1301 0.1773 0.1310 0.1785 0.1412 0.1912 0.1208 − 0.0011 0.1258 0.0010 0.0975 − 0.0033
1.4936 1.0866 1.8239 1.5411 0.6857 0.5771 7.3805 0.1977 5.3607 0.3984 2.0753 0.1449
Loglike 1.0866 1.4952 1.5411 2.1206 0.5771 0.7940 0.1977 0.9834 0.3984 1.9816 0.1449 0.7206
− 3939.5 − 3965.0 − 3939.5 − 3965.4 − 3949.4 − 3972.8 − 3820.7 − 4033.1 − 3823.1 − 4034.6 − 3844.8 − 4033.8
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Fig. 5. BVSP × NIKK PDF. ρ = 0.11.
estimation. This approach was used by Stützle and Hrycej (2001, 2002a,b,2005,2006), applied in many other distributions, included MAGH. Proposition 2. If X ~ MAGHn(ω,Σ,M), then W = BX is a set of n independent GH(ω,δ,μ) distributions, where B is the inverse Cholesky factorization of the X covariance matrix.
So, the dependence structure of the MAGH is due to the A matrix. Let S be the covariance matrix of X, so we can use Cholesky to factor it as S=B̃ ′B̃. Applying the inverse, we have:
S
−1
= B˜
−1
By definition if X ~ MAGHn(ω,Σ,M) then we can state that: d X = A′Y + M
for some upper triangular matrix A such that A′A = Σ is positive definite and the random vector Y = (Y1,…,Yn)′ consists of mutually independent random variables Yi ~ GH(ωi,1,0).
−1 ′
ð B˜
Þ
Calling (B̃− 1)′ = B we get S− 1 = B′B So, when we let:
W = BX
Fig. 6. CAC × DAX PDF. ρ = 0.79.
J. Fajardo, A. Farias / International Review of Financial Analysis 18 (2009) 174–184
This transforms the correlated X into an uncorrelated W (Horn & Johnson, 1985; Press, Teukolsky, Vetterling, & Flannery, 1992). The question is: Is W ∈ MAGH? ′
′
W = BX = BðA Y + MÞ = BA Y + BM BA′AB′ is clearly positive definite, so W ∈ MAGH and as stated is a set of independent GH(ω,δ,μ) distributions. □ Proposition 3. We can estimate X by a two-step procedure. In above proposition we stated that W is a vector of independent distributions, so we can estimate W by its conditional distributions Wi. After we estimate all Wi we can recover the original X parameters: d Each Wi can be written as Wi = δYi + μi , so: ′
−1
W = BX→ðW1 ; W2 ; :::; Wn Þ = BX→X = B
W
As stated before B− 1 W ~ MAGH(ω,Σ,M) so, A′Y + M = B− 1(DY + μ). Thus: A′ =B− 1D and M =B− 1 μ, where D is the diagonal matrix containing the δi of marginal distributions, and μ is the vector of μi. 5 So, we use the following steps: (1) Find B as the Cholesky series factorization of the inverse sample covariance matrix. (2) Get W = BX, which is a set of independent GH(ωi,δ,μ). (3) Estimate the univariate GH distributions. (4) Translate the univariate parameters into the multivariate parameters using Proposition 3. This procedure takes less computational effort, since it consists of n univariate estimations, requiring 5 parameters at each, instead of 1 multivariate estimation of 4n + n(n + 1) / 2 parameters at once. 6. Empirical results 6.1. Unidimensional estimation Table 4 presents the results of the unidimensional estimation of GH distributions and the normal inverse gaussian (NIG) and hyperbolic (Hyp) subclasses. The GH estimation has to be done carefully, since the presence of λ as a free-parameter may lead to multiple local minimums. The Hyp and NIG subclasses are particularly important because the first one (hyperbolic) is easier to estimate, since the Bessel function (the most computer demanding) is evaluated only once for loglikelihood evaluation,2 while the others need at least n times, where n is the sample size. The second one (normal inverse Gaussian) is more often desirable, especially in derivatives pricing, since it is closed under convolution, a characteristic the other subclasses do not have. Due to this, we performed likelihood ratio tests to check for the possibility of restricting GH parameters to NIG or Hyp subclasses. The test statistics such as their p-values are in Table 5. It can be seen in Table 5 that the developed countries' stock markets can be modeled with NIG instead of GH, but this is not true for the Brazilian market. This happens because of the high kurtosis and standard deviation of the Brazilian stock market, characteristics already explored in Fajardo and Farias (2004). The same table shows that only for NIKK can we restrict the estimation to the Hyp subclass (the null hypothesis is rejected for all other indexes).
2
For more on GH subclasses, see Barndorff-Nielsen (1977, 1978).
179
6.2. Unidimensional goodness of fit In order to evaluate the goodness of fit, we show some figures and Distances calculated. Figs. 1–4 show the estimated vs. empirical distributions of the BVSP and NIKK indexes. We can see in the PDF graphs that the GH distribution fits the kurtosis of the distribution well, reinforced by the log-density, showing that the tails are also well fitted. Table 6 lists all the Kolmogorov distances and the respective p-values (for details see Fajardo and Farias, 2004). The null hypothesis that the empirical distribution is GH/NIG distributed is not rejected. In Hyp case, only the DAX is rejected. Just as a comparative exercise, we show in Table 7 the Anderson and Darling distance (for more details see Fajardo and Farias, 2004). This distance mainly shows the difference in the tails of the distribution. In this case the normal distribution performs very poorly, with the worst distance being 50,077, against 0.0339 for the GH (BVSP) and the best performance being 0.6159 against 0.0288 for the GH (DAX). The question is: How can we model multivariate data, considering the dependence among them? In next section we examine the multivariate affine generalized hyperbolic distribution, an attempt to solve this without intensive computational effort. 6.3. Two-dimensional MAGH estimation Even though the univariate estimates present desirable goodness of fit measures, the correlation between the assets is not negligible, so if we want to model the joint distribution of the assets, for V@R necessities or even derivatives pricing, we have to consider multivariate distributions. First we present the results of the MAGH estimation. In order to assess the results more easily, we first present the estimation of 2 by 2 combinations of the sample. Then we present the full sample treated as a single multivariate distribution. Table 8 contains the estimates of the two assets with highest correlation (CAC and DAX) and the two assets with lowest correlation (BVSP and NIKK). Again, we may be interested in particular subclasses (MANig and MAHyp). Table 9 shows the estimates concerning the restriction of MAGH to one of its main subclasses. The results are quite similar to the univariate case. When the BVSP is one of the distributions, only the BVSP × CAC can be restricted to MANig, but all other two-index combination can be restricted. Once again, the high volatility and kurtosis of the BVSP distribution contributes to this. In the MAHyp case, we cannot restrict any of the samples.
Table 10 Kolmogorov distances for two-dimensional estimations.
BVSP × CAC BVSP × DAX BVSP × FTSE BVSP × NIKK BVSP × SP500 CAC × DAX CAC × FTSE CAC × NIKK CAC × SP500 DAX × FTSE DAX × NIKK DAX × SP500 FTSE × NIKK FTSE × SP500 NIKK × SP500
Normal
GH
Nig
Hyp
0.0065 0.0072 0.0054 0.0062 0.0059 0.0023 0.0017 0.0018 0.0017 0.0019 0.0019 0.0019 0.0015 0.0014 0.0016
0.0016 0.0022 0.0028 0.0016 0.0023 0.0013 0.0010 0.0006 0.0005 0.0008 0.0006 0.0007 0.0005 0.0004 0.0004
0.0017 0.0024 0.0011 0.0017 0.0028 0.0008 0.0006 0.0005 0.0006 0.0006 0.0005 0.0010 0.0005 0.0005 0.0004
0.0022 0.0028 0.0021 0.0017 0.0025 0.0009 0.0006 0.0005 0.0007 0.0009 0.0007 0.0012 0.0005 0.0006 0.0005
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Table 11 MAGH six-dimensional estimations.
α̃ β̃ λ M
Σ
Log-like
BVSP
CAC
DAX
FTSE
NIKK
SP500
1.0420 − 0.0142 − 1.0157 0.1304 6.9439 0.6938 0.7598 0.5495 0.2256 0.7831 − 3846.8
1.0430 0.0017 − 0.5017 0.0510 0.6938 1.8567 1.6821 1.0513 0.2575 0.4564 − 3972.3
1.0423 − 0.0065 − 1.0146 0.0634 0.7598 1.6821 2.5637 1.0277 0.2697 0.5665 − 4041.2
1.0432 − 0.0066 − 0.4980 0.0374 0.5495 1.0513 1.0277 1.0582 0.2086 0.3496 − 3963.3
1.0419 0.0000 0.5028 0.0011 0.2256 0.2575 0.2697 0.2086 0.9832 0.1178 − 4028.8
1.0447 − 0.0079 0.0030 0.0480 0.7831 0.4564 0.5665 0.3496 0.1178 0.8155 − 3972.8
BVSP
CAC
DAX
FTSE
NIKK
SP500
1.0135 − 0.0727 − 0.5000 0.2329 5.3897 0.8962 1.0013 0.7083 0.3862 1.1103 − 3848.3
1.0430 0.0017 − 0.5000 0.0512 0.8962 1.9261 1.6716 1.1646 0.4781 0.6472 − 3972.3
1.0415 − 0.0070 − 0.5000 0.0637 1.0013 1.6716 2.3335 1.1601 0.4933 0.8032 − 4042.7
1.0432 − 0.0066 − 0.5000 0.0375 0.7083 1.1646 1.1601 1.1469 0.3889 0.4958 − 3963.3
1.0382 0.0000 − 0.5000 0.0012 0.3862 0.4781 0.4933 0.3889 1.9640 0.1670 − 4030.3
1.0417 − 0.0077 − 0.5000 0.0483 1.1103 0.6472 0.8032 0.4958 0.1670 1.1563 − 3972.9
BVSP
CAC
DAX
FTSE
NIKK
SP500
1.0726 − 0.0072 1.0000 0.1050 2.1198 0.3279 0.3764 0.2664 0.1432 0.4181 − 3867.9
0.5985 − 0.0013 1.0000 0.0460 0.3279 0.5989 0.6220 0.4382 0.1762 0.2437 − 3974.6
1.0451 0.0000 1.0000 0.0535 0.3764 0.6220 0.8617 0.4365 0.1820 0.3025 − 4041.4
1.0543 − 0.0047 1.0000 0.0311 0.2664 0.4382 0.4365 0.4316 0.1433 0.1867 − 3972.2
1.0470 0.0000 1.0000 − 0.0036 0.1432 0.1762 0.1820 0.1433 0.7203 0.0629 − 4029.8
1.0530 − 0.0060 1.0000 0.0433 0.4181 0.2437 0.3025 0.1867 0.0629 0.4354 − 3979.2
Table 12 MANig six-dimensional estimations.
α̃ β̃ λ M
Σ
Log-like
Table 13 MAHyp six-dimensional estimations.
α̃ β̃ λ M
Σ
Log-like
6.4. Two-dimension goodness of fit We present two kinds of goodness of fit evaluations. First we show the multivariate distribution of the most correlated pair of assets (CAC × DAX) and the least correlated pair (BVSP × NIKK), then we calculate two-dimensional Kolmogorov distances. Figs. 5 and 6 represent the empirical, MAGH fit and normal fit to BVSP × NIKK (correlation = 0.1136) and CAC × DAX (correlation = 0.7902). Once again the kurtosis of the series is better captured with MAGH distributions. Furthermore, we need to calculate the Kolmogorov distance for the multidimensional case. We use the approach of Fasano and Franceschini (1987) and Peacock (1983), which calculates the maximum distance in all possible directions (in this case 4). The number of
sample points used in the distance calculation was 100 for each margin, totaling 10,000 points. The results in Table 10 lead us to conclude that the MAGH really provides better fit to the data. The MAGH distributions and their subclasses consistently have less distance between the theoretical and empirical distributions. 6.5. Six-dimensional MAGH estimation In this section we present the results of the joint estimation of the assets. Tables 11–13 give the results concerning, respectively, MAGH, MANig and MAHyp. The estimation procedure explained in Section 3 makes this estimation easier.
Table 15 Six-Dimensional Kolmogorov distances.
Table 14 Log-likelihood ratio tests. NIG
Hyp
Stats
P-value
Stats
P-value
9.4108
0.1518
80.072
3.4E−15
Distribution
Distance
Normal GH NIG Hyp
0.2394 0.1773 0.1742 0.2312
J. Fajardo, A. Farias / International Review of Financial Analysis 18 (2009) 174–184
Fig. 7. BVSP margin PDF.
Fig. 10. FTSE margin PDF.
Fig. 8. CAC margin PDF.
Fig. 11. NIKK margin PDF.
Fig. 9. DAX margin PDF.
Fig. 12. SP500 margin PDF.
181
182
J. Fajardo, A. Farias / International Review of Financial Analysis 18 (2009) 174–184 Table 17 Portfolio with minimum variance. Distribution
CAC
DAX
Mean
S.D.
MAGH MANig MAHyp Normal
0.6589 0.7012 0.6966 0.5461
0.3411 0.2988 0.3034 0.4539
− 0.0302 − 0.0219 − 0.0254 − 0.0422
1.2943 1.2907 1.2673 1.0601
results in Fig. 13 and the minimum variance portfolio is described in the Table 16. It is clear, in this example, that MAGH offers a better description of the risk-return trade-off than its subclasses. But in general it is not true, as we will see in the next example. 7.2. Example 2: Two indexes Fig. 13. Six index efficient frontier.
Following our script, we show in Table 14 the log-likelihood ratio Test results for subclass restriction. They show that we can restrict to the MANig subclass but not to the MAHyp subclass, reaffirming the previous results (fewer dimensions).
As we have seen in Section 4, the CAC and DAX indexes are the most correlated. In what follows we construct the EF with these two pairs of indexes. Also, we describe the portfolio with the minimum variance for the three distributions in Table 17 (Fig. 14). As we can see MAHyp EF dominates MAGH EF in some parts of the EF. It means that MAGH EF not necessarily dominates the other subclasses in the risk-return trade-off.
6.6. Six-dimensional goodness of fit
7.3. Efficient portfolios
We felt challenged to give some measure of goodness of fit to a sixdimensional data set, since all multidimensional Kolmogorov distances in the literature only go up to four dimensions. We implemented the algorithm mentioned in the two-dimensional case, which calculates the Kolmogorov distance in all possible accumulation directions. In a six-dimensional problem, there are 26−64 possible directions. We used 20-point to evaluate the data in each marginal, giving a total of 64,000,000 evaluation points in each accumulation. Table 15 shows the results, and once again the MAGH distributions obtain the best fits. The sizes of the distances were influenced by the number of data points in each marginal, but the main result remains valid. Another way to show the goodness of fit is showing the behavior of the fit in each marginal. Figs. 7–12 show a visual depiction of the fit in each one of the marginals. We can infer that the MAGH distributions have good fit performance.
In Fig. 15, we describe the composition of 1000 efficient portfolios. MAGH and MAHyp efficient portfolios short-sale most of the time FTSE and NIKK indexes. While MANig and classical efficient portfolios short-sale FTSE, NIKK and SP500 indexes. The exact percentages are presented in Table 18.
7. Application: Efficient Frontier (EF)
8. Conclusions In this paper we evaluated the goodness of fit of multivariate affine generalized hyperbolic distributions to various asset returns and showed that they present a very good fit with real data. As an application we address portfolio management for two- and six-asset portfolios. Interesting extension are applications to multivariate derivatives pricing and value at risk calculations, since MAGH better capture the kurtosis of the data. This will be the topic of future research. Acknowledgements
Now we derive the efficient frontier of a portfolio that consists of a combination of two and all six indexes introduced in our sample. Also, we analyze the composition of the efficient portfolios in the six-index case. To find the efficient portfolios we use the Matlab software and allow short-sales.
The authors thanks an anonymous referee for his/her suggestions. Also, J. Fajardo thanks CNPq Brazil for financial support. The views expressed in this work are those of the authors and do not reflect those of the Banco Central do Brasil or its members. Appendix A
7.1. Example 1: Six indexes Consider our six indexes: BVSP, CAC, DAX, FTSE, NIKK and SP500 and the estimated parameters obtained in Tables 11–13. We compute the efficient portfolios for the three distributions and present the
Proof 1. Proposition 1 Let X be a GH(x; ω, δ, μ). Define Y as: d
Y=
X−μ ; δ
ð4Þ
Table 16 Portfolio with minimum variance. Distribution
BVSP
CAC
DAX
FTSE
NIKK
SP500
Mean
S.D.
GH NIG Hyp Normal
0.0375 − 0.0289 0.0233 − 0.0841
− 0.0412 − 0.0432 − 0.4387 − 0.0833
− 0.0261 − 0.1259 − 0.1243 − 0.0834
0.4201 0.4795 0.6459 0.4168
0.2278 0.2503 0.3175 0.4170
0.3818 0.4682 0.5763 0.4170
0.0268 − 0.0019 0.0066 0.0042
0.7300 0.7994 0.8731 0.0084
J. Fajardo, A. Farias / International Review of Financial Analysis 18 (2009) 174–184
183
Table 18 Percentage of time with short-sales in each index. Distribution
BVSP
CAC
DAX
FTSE
NIKK
SP500
GH NIG Hyp Normal
0 1% 0 4.3%
2.9% 1.4% 9.4% 0
1.6% 5.6% 13.3% 0
84.1% 80.1% 88.1% 73.3%
88.8% 89.2% 86.3% 85.4%
0 86.7% 0 56.2%
Now doing a simple parameter transformation: Fig. 14. CAC–DAX efficient frontier.
α=
˜ ˜ ˜β α α and β = δ δ
ð9Þ
and replacing in Eqs. (9) and (8), we obtain: this leads to: PðY≤ yÞ = P
X−μ ≤ y = PðX≤ δy + μÞ; δ
ð5Þ fY ðyÞ =
then,
˜2 α δ2
˜ ˜2β α − 2 δ
2
λ 2
2
ðδ + δ y Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! K
pffiffiffiffiffiffiα˜ λ−1 = 2 λ ˜2 ˜2 α ˜2β α δ Kλ δ 2 − 2 2π δ
FY ðyÞ = FX ðδy + μÞ
ð6Þ =
Deriving both sides w.r.t y, we have: fY ðyÞ = fX ðδy + μÞδ
ð7Þ
So, using the definition of the GH density, we have: 2
2 λ=2
1 =2 2
λ−
6ðα −β Þ ðδ + ðδyÞ Þ fY ðyÞ = 6 4 pffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ− 2πα 2 δλ Kλ ðδ α2 −β2 Þ 2
2
2
K
1 2
λ−
3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi βδy 7 α δ2 + ðδyÞ2 e 7 5δ ð8Þ
ð
Þ
1 =2 2
λ−
2 2
δ
δ
λ ˜ 2 ÞÞ 2 δ−λ ð1 ˜ 2 ð1− β δðα
˜ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi α˜ βδy ˜ α δ δ2 + δ2 y2 e δ δ
1 λ− 2
1 1 λ− = 2 λ− 2 δ 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 +y Þ ˜ ˜ ˜ −1 δ2 ð1 + y2 Þ e α βy qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K λ−1 αδ ð10Þ 2 2 2 −2 ˜ ˜ ð1− β ÞÞδ δ ðα λ 1 λ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ˜ 2 Þ2 δ−λ ð1 + y2 Þ 2 −4 δλ−1 = 2 ˜ λ ð1− β δα ˜ ˜ ˜ 1 + y2 e α βy = qffiffiffiffiffiffiffiffiffiffiffiffiffiffi K 1 α pffiffiffiffiffiffi λ−1 = 2 −λ + 1 = 2 λ λ− 2 ˜ 2 ˜ ˜ 1− β δ δ Kλ α 2π α λ 1 λ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ˜ 2 Þ2 ð1 + y2 Þ 2 −4 ˜ 1 = 2 ð1− β α ˜ ˜ ˜ 1 + y2 e α βy = K qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 α pffiffiffiffiffiffi λ− 2 ˜ 2 ˜ 1− β 2πK α pffiffiffiffiffiffi λ−1 −λ ˜ 2δ 2π α
1 + 2 δλ Kλ
ð
Þ
ð
λ
ð
Þ
Þ
ð
ð
Þ
ð
Þ
Þ
We get an expression similar to Schmidt et al. (2006).
Fig. 15. Efficient portfolios composition.
□
184
J. Fajardo, A. Farias / International Review of Financial Analysis 18 (2009) 174–184
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Horn, R. A., & Johnson, C. R. (1985). Matrix analysis. Cambridge University Press. Iwaarden, R. 1996. An improved unconstrained global optimization algorithm. Ph.D. thesis, University of Colorado at Denver. Keller, U. 1997. Realistic modelling of financial derivatives. Doctoral thesis, University of Freiburg. Lagarias, J. C., Reeds, J. A., Wright, M. H., & Wright, P. E. (1998). Convergence properties of the Nelder–Mead simplex method in low dimensions. Siam Journal of Optimization, 9(1), 112–147. Mendes, B. V. M., Kolev, N., & Anjos, U. (2006). Copulas: A review and recent developments. Applied stochastic models in business and industry(4), 617–660. Mendes, B.V.M., Leal, R., & Semeraro, M.M. (2008). Optimal portfolios through paircopulas. working paper COPPEAD. Mendivii, F., Shonkwiler, R., & Spruill, M. 1999. Optimization by stochastic methods. Working paper. Neumaier, A. (2004). Acta numerica. Chap. Complete search in continuous global optimization and constraint satisfaction (pp. 271–369). Cambridge University Press. Peacock, J. A. (1983). Two-dimensional goodness-of-fit testing in astronomy. Monthly Notices of the Royal Astronomical Society, 202, 615–627. Prause, K. 1999. The generalized hyperbolic model: Estimation, financial derivatives, and risk measures. Doctoral thesis, University of Freiburg. Press, W., Teukolsky, S., Vetterling, W., & Flannery, B. (1992). Numerical recipes in C. Cambridge: Cambridge University Press. Schmidt, Rafael, Hrycej, Tomas, & Stützle, Eric (2006). Multivariate distribution models with generalized hyperbolic margins. Computational statistics and data analysis, 50, 2065–2096. Stützle, E., & Hrycej, T. (2001). Forecasting of conditional distributions—An application to the spare parts demand forecast. Proceedings of the 2001 IASTED International Conference on Artificial Intelligence and Soft Computing, Cancun, México. Stützle, E., & Hrycej, T. (2002). Estimating multivariate conditional distributions via neural networks and global optimization. Proceedings of the 2002 Ieee International Joint Conference on Neural Networks, Honolulu, Hawaii. Stützle, E., & Hrycej, T. (2002). Modelling future demand by estimating the multivariate conditional distribution via the maximum likelihood principle and neural networks. Proceedings of the 2002 IASTED International Conference on Modelling, Identification and Control, Innsbruck, Austria. Stützle, E., & Hrycej, T. (2005). Numerical method for estimating multivariate conditional distributions. Computational statistics, 20(1), 151–176.
Contents lists available at ScienceDirect
International Review of Financial Analysis
Multivariate affine generalized hyperbolic distributions: An empirical investigation José Fajardo a,⁎, Aquiles Farias b a b
IBMEC Business School, Av. Presidente Wilson 118, 1108, 20030020. Rio de Janeiro, RJ., Brazil Banco Central do Brasil
a r t i c l e
i n f o
Article history: Received 7 March 2008 Received in revised form 27 May 2009 Accepted 29 June 2009 Available online 6 July 2009
a b s t r a c t The aim of this paper is to estimate multivariate affine generalized distributions (MAGH) using market data. We use the Ibovespa, CAC, DAX, FTSE, NIKKEI and S&P500 indexes. We estimate the univariate distributions, bi-variate distributions and six-dimensional distribution. Then we assess their goodness of fit using Kolmogorov distances. As an application we study the efficient frontier. © 2009 Elsevier Inc. All rights reserved.
Keywords: Generalized hyperbolic distributions Multivariate distributions Affine transformation Fat tails
1. Introduction In the past decade a class of distributions called generalized hyperbolic distributions (GH) has been suggested to fit financial data. The development of these distributions is due to Barndorff-Nielsen (1977). He applied the hyperbolic subclass to fit the grain size of sand subjected to continuous wind. Barndorff-Nielsen extended this work to multivariate generalized hyperbolic distributions (MGH). These distributions have been used in different fields of knowledge, such as physics, biology, agronomy and others (see Blæsild &Sørensen, 1992). Eberlein and Keller (1995) were the first to apply these distributions to finance. They used univariate hyperbolic subclasses to fit German data. Keller (1997) studied derivatives pricing with GH and Prause (1999) extended the results of Eberlein and Keller (1995) by fitting financial data using the MGH class. He also priced derivatives and measured value at risk. Using the GH class one can capture fat tails and the skewness observed in asset returns. Blæsild and Sørensen (1992) were the first to develop a computer program, called Hyp, to estimate the parameter of the hyperbolic subclass up to three dimensions. Prause (1999) developed a program to estimate the MGH class. Unfortunately, the MGH parameter estimation procedure is computationally intense and has some shortcomings. Improvement of this procedure would be an important contribution, in particular from a practical point of view. More recently, Schmidt, Hrycej, and Stützle (2006) introduced multivariate affine generalized hyperbolic distributions (MAGH). Their main goal was to develop a multivariate distribution that can
⁎ Corresponding author. Tel.: +55 21 4503 4162; fax: +55 21 4503 4168. E-mail addresses: [email protected] (J. Fajardo), [email protected] (A. Farias). 1057-5219/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.irfa.2009.06.002
capture fat tails, skewness and tail dependence and at the same time is simple to estimate. Another way to capture tail dependence is to use copulas. That is, given the univariate marginals, one can obtain a determined multivariate distribution, having these univariate distributions as their marginals, by using a copula. Reciprocally, having a copula and given marginals, one can obtain a multivariate distribution having these marginals. Then, given an empirical distribution, one can give a very precise description of the observed dependence structure by choosing the appropiate copula from the available copula families. Of course, the goodness of fit will depend strongly on the available families of copulas and on the distribution dimension. For example, for n ≥ 3 the lower Fréchet bound is not a copula. In that sense, this approach presents many estimation problems in higher dimensions. Again, there is the trade-off between accuracy and computational effort. The use of pair copulas might provide some help, as pointed out by Mendes, Leal, and Semeraro (2008). For more applications of copulae in finance, see Cherubini, Luciano, and Vecchiato (2004) and Mendes, Kolev, and Anjos (2006). It is worth noting that the tail dependence property of MAGH, allows us modeling dependences of extreme events, which is a key ingredient in the computation of portfolios' value at risk (VaR) and in many other applications in risk management (see Embrechts, Lindskog, & McNeil, 2003). This dependence cannot be captured with MGH. As a result, the VaR calculations using MGH will be unappropiate in multidimensional settings. More comparisons between MAGH and MGH can be found in Section 4 in Schmidt et al. (2006). Some applications to the Brazilian market have been performed to analyze the use of univariate GH. Using the Hyp software Fajardo, Schuschny, and Silva (2001) studied the goodness of fit of the
J. Fajardo, A. Farias / International Review of Financial Analysis 18 (2009) 174–184 Table 1 Sample.
175
Table 2 Descriptive statistics.
Asset
Ticker
Start
End
Index
Mean (%)
Std Deviation (%)
Skewness
Kurtosis
Min (%)
Max (%)
Bovespa Cac40 Dax FTSE Nikkei Standard and Poors
BVSP CAC DAX FTSE NIKK SP500
08/01/1994 08/01/1994 08/01/1994 08/01/1994 08/01/1994 08/01/1994
10/20/2005 10/20/2005 10/20/2005 10/20/2005 10/20/2005 10/20/2005
BVSP CAC DAX FTSE NIKK SP500
0.0668 0.0256 0.0277 0.0174 − 0.0147 0.0322
2.4107 1.3766 1.4967 1.0680 1.3625 1.0737
0.5901 − 0.0988 − 0.1415 − 0.1238 − 0.0997 − 0.1002
16.9409 5.7326 5.6244 5.9075 5.1872 6.4608
− 17.2082 − 7.6781 − 6.4999 − 5.5888 − 7.2340 − 7.1127
28.8325 7.0023 7.5527 5.9038 7.6553 5.5744
hyperbolic subclass. Fajardo and Farias (2004) and Fajardo, Farias, and Ornelas (2005) extended those results and priced some derivatives using the univariate GH class. In this paper we generalize Fajardo and Farias (2004) to the multidimensional case by using MAGH. We assess the goodness of fit of MAGH with international financial data and the Brazilian index Ibovespa. As an application, we derive the efficient frontier for two and six assets. The paper is organized as follows: Section 2 presents generalized hyperbolic distributions; Section 3 presents multivariate affine generalized distributions; Section 4 describes our sample; Section 5 describes the MAGH estimation procedures; Section 6 presents the empirical results; and Section 7 analyzes the efficient frontier. The last sections have the conclusions and an appendix. 2. Generalized hyperbolic distributions
3. Multivariate affine generalized hyperbolic distributions The n-dimensional MAGH consists of the following stochastic representation: d
1 λ− 2
ghðx; α; β; δ; μ; λÞ = aðλ; α; β; δÞðδ + ðx−μÞ Þ
2
2
Kðλ; α; β; δ; μÞ
where μ is a location parameter, δ is a scale factor, compared to Gaussian σ in Eberlein (2000), α and β determine the distribution shape, λ defines the tail fatness (Barndorff-Nielsen & Blaesid, 1981) and therefore the subclasses of GH, and Kðλ; α; β; δ; μÞ = K
Proposition 1. We can write GH(x;ω,δ,μ) as an affine transformation of a canonical GH: GH(x;ω,1,0), where ω: = (α̃ ,β̃,λ).
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 1 ðα δ + ðx−μÞ Þexpðβðx−μÞÞ
where A is an upper triangular matrix ∈Rnxn such that A′A = Σ is a positive definite and the random vector Y∈Rn consists of n mutually independent one-dimensional canonical GH(ω,1,0) (for more details see Schmidt et al., 2006). This definition eases the estimation procedure. M is the location parameter and Σ is a scaling factor. The family of n-dimensional multivariate affine generalized hyperbolic distributions is denoted by MAGMn(ω,Σ,M), where ω: = (ω1,…,ωn) and ωi: = (λi,αi,βi), i = 1,…,n. Table 3 Correlation matrix.
λ−
2
where, 2
2
λ
ðα −β Þ2 aðλ; α; β; δÞ = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi λ−1 λ 2 δ K ðδ α2 −β2 Þ 2πα λ
is a norming factor to make the curve area equal to 1 and 1 ∞ λ−1 1 −1 exp − xðy + y Þ dy Kλ ðxÞ = ∫0 y 2 2
BVSP CAC DAX FTSE NIKK SP500
is the modified Bessel function of the third kind with index λ. The domains of the parameters are: μ; λ∈R −α b β b α δ; α N 0:
1
λ−1
expðð∓α + βÞxÞ as x→F∞
For more details about Bessel functions, see Abramowitz and Stegun (1968).
CAC
DAX
FTSE
NIKK
SP500
1.0000 0.2682 0.2756 0.2735 0.1136 0.4277
0.2682 1.0000 0.7902 0.7875 0.2434 0.4366
0.2756 0.7902 1.0000 0.7215 0.2315 0.4983
0.2735 0.7875 0.7215 1.0000 0.2551 0.4311
0.1136 0.2434 0.2315 0.2551 1.0000 0.1138
0.4277 0.4366 0.4983 0.4311 0.1138 1.0000
Index BVSP
CAC
DAX
The GH have semi-heavy tails. This name is due to the fact that their tails are heavier than Gaussian's, but they have finite variance, which is observed by the following approximation:
BVSP
Table 4 GH and its subclass estimation parameters.
1
ghðx; λ; α; β; δÞ∼jx j
′
X= AY +M
The probability density function of the one-dimensional GH is defined by:
2
Many distributions can be obtained as subclasses of or limiting distributions of GH. Some examples are the Gaussian distribution, Student's T and the normal inverse Gaussian. See Barndorff-Nielsen (1978) and Prause (1999) for detailed descriptions. We can alternatively write the GH density as an affine transformation of a canonical form, with scale 1 and position 0.
FTSE
NIKK
SP500
GH NIG HYP GH NIG HYP GH NIG HYP GH NIG HYP GH NIG HYP GH NIG HYP
α̃
β̃
λ
δ
μ
Loglike
0.6350 0.8862 0.4917 1.0065 1.0410 1.0511 0.9814 0.9726 1.0531 0.9851 0.9851 1.0543 1.0405 1.0376 1.0470 1.0444 1.0414 1.0529
− 0.0998 − 0.0957 − 0.0437 − 0.0819 − 0.0091 − 0.0041 − 0.0863 − 0.1003 − 0.0057 − 0.0804 − 0.0804 − 0.0075 − 0.0055 0.0000 0.0000 − 0.0075 − 0.0076 − 0.0053
− 1.0149 − 0.5000 1.0000 − 0.9964 − 0.5000 1.0000 − 0.0100 − 0.5000 1.0000 − 0.4989 − 0.5000 1.0000 0.5046 − 0.5000 1.0000 0.0026 − 0.5000 1.0000
2.3911 2.1473 0.7420 1.6317 1.3874 0.8504 1.2092 1.4476 0.9167 1.0388 1.0388 0.6541 0.9902 1.4066 0.8489 0.9030 1.0751 0.6600
0.2131 0.2693 0.2162 0.1194 0.0520 0.0372 0.1776 0.1733 0.0454 0.0975 0.0975 0.0318 0.0056 − 0.0067 − 0.0085 0.0471 0.0477 0.0424
− 6387.0 − 6392.8 − 6408.0 − 4919.8 − 4920.6 − 4927.0 − 5141.6 − 5142.2 − 5153.9 − 4153.2 − 4153.2 − 4163.7 − 4935.9 − 4937.7 − 4936.8 − 4180.3 − 4180.4 − 4186.7
176
J. Fajardo, A. Farias / International Review of Financial Analysis 18 (2009) 174–184
Fig. 1. NIKK PDF.
Fig. 3. BVSP PDF.
Fig. 2. NIKK log-PDF.
Fig. 4. BVSP log-PDF.
The mean and covariance matrix of the MAGH can be easily calculated: ′
cii = Rλi ;1
′
E½X = E½A Y + M = A E½Y + M
where E[Y], by independence, is a vector containing in each row the q mean of the univariate Y i ~ GH(ω i ,1,0), given by E½Yi = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rλi ;1 ðα2i ð1−β2i ÞÞαi βi and ′
Cov½X = A CA; Table 5 Log-likelihood ratio test. Index BVSP CAC DAX FTSE NIKK SP500
where C = diag(c11,…,cnn) with qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðα2i ð1−β2i ÞÞ + Rλi ;2 ðα2i ð1−β2i ÞÞ−Rλi ;1 ðα2i ð1−β2i ÞÞ
½
β 1−β 2 i
2 i
If cii = c for all i = 1,…,n, then Cov[X] = cΣ. This distribution is extremely flexible because the parameters λ and α can be defined for each margin, improving the fit even if the margins have very different tail fatness, because, λ and α are directly responsible for that phenomenon. Furthermore, if Σ is a diagonal matrix, the margins are independent, which is important in some scenarios. Table 6 Kolmogorov–Smirnov tests.
NIG
Hyp
Index
Stats
P-value
Stats
P-value
11.5260 1.7523 1.2142 0.0005 3.7139 0.2018
6.86E−04 0.1856 0.2705 0.9820 0.0540 0.6533
41.9750 14.4540 24.4810 20.9120 1.9122 12.7990
9.24E−11 1.44E−04 7.50E−07 4.81E−06 0.1667 3.47E−04
BVSP CAC DAX FTSE NIKK SP500
Normal
GH
NIG
Hyp
KS
p-value
KS
p-value
KS
p-value
KS
p-value
0.0683 0.0525 0.0680 0.0571 0.0486 0.0549
2.7E−12 1.9E−07 3.4E−12 9.9E−09 2.0E−06 4.2E−08
0.0129 0.0098 0.0167 0.0143 0.0112 0.0149
0.7134 0.9414 0.3871 0.5875 0.8576 0.5360
0.0115 0.0085 0.0173 0.0142 0.0107 0.0144
0.8318 0.9841 0.3413 0.5917 0.8902 0.5731
0.0141 0.0139 0.0307 0.0219 0.0122 0.0200
0.6049 0.6244 0.0080 0.1199 0.7744 0.1924
:
J. Fajardo, A. Farias / International Review of Financial Analysis 18 (2009) 174–184 Table 7 Anderson–Darling distances.
Bovespa CAC40 Dax FTSE Nikkei SP500
Table 9 Log-likelihood ratio tests.
Normal
GH
NIG
Hyp
50077.4824 8.3298 0.6159 2.8125 3.2669 334.6462
0.0470 0.0355 0.0374 0.0416 0.0388 0.1211
0.1080 0.0530 0.0375 0.0418 0.0351 0.0991
0.3071 0.0786 0.0858 0.0868 0.0563 0.2245
Assets
The facilitation of estimation is due to a simple procedure that allows, instead of a simultaneous process of acquiring parameters, estimation using n univariate estimations, where n represents the number of dimensions. 4. Sample Our empirical evaluation uses the Ibovespa, CAC 40, Dax 100, FTSE 100, Nikkei 225 and S&P 500 indexes. The data consist of the daily logreturns, calculated using:
Ri;t = ln
Pi;t
177
!
NIG
BVSP × CAC BVSP × DAX BVSP × FTSE BVSP × NIKK BVSP × SP500 CAC × DAX CAC × FTSE CAC × NIKK CAC × SP500 DAX × FTSE DAX × NIKK DAX × SP500 FTSE × NIKK FTSE × SP500 NIKK × SP500
Hyp
Stats
P-value
Stats
P-value
3.1351 25.1630 5.4250 7.8421 7.9218 0.8372 0.8429 3.0086 0.2348 0.8951 3.4144 0.4480 3.0308 0.2363 3.3853
0.2086 0.0000 0.0664 0.0198 0.0190 0.6580 0.6561 0.2222 0.8892 0.6392 0.1814 0.7993 0.2197 0.8886 0.1840
57.2950 86.1360 73.2390 49.7060 59.6940 35.4220 30.2700 12.1060 25.3790 29.7720 18.5980 16.9600 16.3950 34.0040 14.8280
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0024 0.0000 0.0000 0.0001 0.0002 0.0003 0.0000 0.0006
which also is to be expected since they are from completely different continents. 5. Estimation algorithm
Pi;t−1
The samples with their tickers and respective periods are in Table 1. The starting date is the date Brazil implemented its currency stabilization plan (Real plan), which brought some stability to the prices, avoiding daily inflation adjustment of asset prices. Since we are talking about different countries, we interpolated the data when a date was not a trading day in all countries. We did not exclude any trading day, not even September 11, 2001. In Table 2 we give the main descriptive statistics of the data, and in Table 3 we have the correlation matrix of the data. These two tables show the most important features of this database: • Highly correlated data. CAC × DAX have a correlation coefficient of 0.7902; • Almost uncorrelated data. BVSP × NIKK have a correlation coefficient of 0.1136; • High amplitude data. BVSP has a minimum of −17.2082 and a maximum of 28.8325, much greater the other indexes. • High kurtosis data. Following the last item, BVSP has kurtosis of 16.9409, implying a much heavier distribution tail. Analysis of the correlation matrix shows that the most correlated data are from the DAX, CAC and FTSE, which is to be expected since all of them are European markets (not Euro zone). The BVSP and NIKK, followed by the SP500 and NIKK, contain the least correlated data,
To estimate the GH parameters we used a slight modification of algorithm presented in Fajardo and Farias (2004) to estimate the affine transformation form of GH(ω,δ2,μ).That algorithm was implemented in Matlab and uses maximum likelihood estimation Freund (2004), Lagarias, Reeds, Wright, and Wright (1998), and Neumaier (2004) all show properties of restricted optimization and Baritompal and Hendrix (2005), Björkman and Holmström (1999), Hart (1994), Iwaarden (1996), Mendivii et al. (1999), and Stützle and Hrycej (2002a) also discuss ways to implement global optimization. Based on them, in order to improve performance and get more reliable estimates, we transformed the restricted parameters into unrestricted parameters: ˜ u = lnðαÞ ˜ α
ð1Þ
δu = lnðδÞ
ð2Þ
˜ = ð1− expð−β ˜ × signðβ ˜ ÞÞÞ × signðβ ˜ Þ; β u u u
ð3Þ
To estimate MAGH parameters we used some propositions in order to simplify the procedure and improve the efficiency of the
Table 8 MAGH and its subclasses estimated parameters. Assets CAC × DAX
GH Nig Hyp
BVSP × NIKK
GH Nig Hyp
α̃
β̃
λ
M
Σ
1.0441 1.0028 1.0441 0.9842 1.0574 1.0191 1.0377 1.0409 1.0474 1.0386 1.0759 1.0470
0.0037 − 0.0878 0.0037 − 0.1058 0.0024 − 0.0699 − 0.0157 − 2E−05 − 0.0151 − 0.0056 − 0.0077 − 1.5E−07
− 0.5036 − 0.0100 − 0.5 − 0.5 1 1 − 1.0173 0.5022 − 0.5 − 0.5 1 1
0.1301 0.1773 0.1310 0.1785 0.1412 0.1912 0.1208 − 0.0011 0.1258 0.0010 0.0975 − 0.0033
1.4936 1.0866 1.8239 1.5411 0.6857 0.5771 7.3805 0.1977 5.3607 0.3984 2.0753 0.1449
Loglike 1.0866 1.4952 1.5411 2.1206 0.5771 0.7940 0.1977 0.9834 0.3984 1.9816 0.1449 0.7206
− 3939.5 − 3965.0 − 3939.5 − 3965.4 − 3949.4 − 3972.8 − 3820.7 − 4033.1 − 3823.1 − 4034.6 − 3844.8 − 4033.8
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Fig. 5. BVSP × NIKK PDF. ρ = 0.11.
estimation. This approach was used by Stützle and Hrycej (2001, 2002a,b,2005,2006), applied in many other distributions, included MAGH. Proposition 2. If X ~ MAGHn(ω,Σ,M), then W = BX is a set of n independent GH(ω,δ,μ) distributions, where B is the inverse Cholesky factorization of the X covariance matrix.
So, the dependence structure of the MAGH is due to the A matrix. Let S be the covariance matrix of X, so we can use Cholesky to factor it as S=B̃ ′B̃. Applying the inverse, we have:
S
−1
= B˜
−1
By definition if X ~ MAGHn(ω,Σ,M) then we can state that: d X = A′Y + M
for some upper triangular matrix A such that A′A = Σ is positive definite and the random vector Y = (Y1,…,Yn)′ consists of mutually independent random variables Yi ~ GH(ωi,1,0).
−1 ′
ð B˜
Þ
Calling (B̃− 1)′ = B we get S− 1 = B′B So, when we let:
W = BX
Fig. 6. CAC × DAX PDF. ρ = 0.79.
J. Fajardo, A. Farias / International Review of Financial Analysis 18 (2009) 174–184
This transforms the correlated X into an uncorrelated W (Horn & Johnson, 1985; Press, Teukolsky, Vetterling, & Flannery, 1992). The question is: Is W ∈ MAGH? ′
′
W = BX = BðA Y + MÞ = BA Y + BM BA′AB′ is clearly positive definite, so W ∈ MAGH and as stated is a set of independent GH(ω,δ,μ) distributions. □ Proposition 3. We can estimate X by a two-step procedure. In above proposition we stated that W is a vector of independent distributions, so we can estimate W by its conditional distributions Wi. After we estimate all Wi we can recover the original X parameters: d Each Wi can be written as Wi = δYi + μi , so: ′
−1
W = BX→ðW1 ; W2 ; :::; Wn Þ = BX→X = B
W
As stated before B− 1 W ~ MAGH(ω,Σ,M) so, A′Y + M = B− 1(DY + μ). Thus: A′ =B− 1D and M =B− 1 μ, where D is the diagonal matrix containing the δi of marginal distributions, and μ is the vector of μi. 5 So, we use the following steps: (1) Find B as the Cholesky series factorization of the inverse sample covariance matrix. (2) Get W = BX, which is a set of independent GH(ωi,δ,μ). (3) Estimate the univariate GH distributions. (4) Translate the univariate parameters into the multivariate parameters using Proposition 3. This procedure takes less computational effort, since it consists of n univariate estimations, requiring 5 parameters at each, instead of 1 multivariate estimation of 4n + n(n + 1) / 2 parameters at once. 6. Empirical results 6.1. Unidimensional estimation Table 4 presents the results of the unidimensional estimation of GH distributions and the normal inverse gaussian (NIG) and hyperbolic (Hyp) subclasses. The GH estimation has to be done carefully, since the presence of λ as a free-parameter may lead to multiple local minimums. The Hyp and NIG subclasses are particularly important because the first one (hyperbolic) is easier to estimate, since the Bessel function (the most computer demanding) is evaluated only once for loglikelihood evaluation,2 while the others need at least n times, where n is the sample size. The second one (normal inverse Gaussian) is more often desirable, especially in derivatives pricing, since it is closed under convolution, a characteristic the other subclasses do not have. Due to this, we performed likelihood ratio tests to check for the possibility of restricting GH parameters to NIG or Hyp subclasses. The test statistics such as their p-values are in Table 5. It can be seen in Table 5 that the developed countries' stock markets can be modeled with NIG instead of GH, but this is not true for the Brazilian market. This happens because of the high kurtosis and standard deviation of the Brazilian stock market, characteristics already explored in Fajardo and Farias (2004). The same table shows that only for NIKK can we restrict the estimation to the Hyp subclass (the null hypothesis is rejected for all other indexes).
2
For more on GH subclasses, see Barndorff-Nielsen (1977, 1978).
179
6.2. Unidimensional goodness of fit In order to evaluate the goodness of fit, we show some figures and Distances calculated. Figs. 1–4 show the estimated vs. empirical distributions of the BVSP and NIKK indexes. We can see in the PDF graphs that the GH distribution fits the kurtosis of the distribution well, reinforced by the log-density, showing that the tails are also well fitted. Table 6 lists all the Kolmogorov distances and the respective p-values (for details see Fajardo and Farias, 2004). The null hypothesis that the empirical distribution is GH/NIG distributed is not rejected. In Hyp case, only the DAX is rejected. Just as a comparative exercise, we show in Table 7 the Anderson and Darling distance (for more details see Fajardo and Farias, 2004). This distance mainly shows the difference in the tails of the distribution. In this case the normal distribution performs very poorly, with the worst distance being 50,077, against 0.0339 for the GH (BVSP) and the best performance being 0.6159 against 0.0288 for the GH (DAX). The question is: How can we model multivariate data, considering the dependence among them? In next section we examine the multivariate affine generalized hyperbolic distribution, an attempt to solve this without intensive computational effort. 6.3. Two-dimensional MAGH estimation Even though the univariate estimates present desirable goodness of fit measures, the correlation between the assets is not negligible, so if we want to model the joint distribution of the assets, for V@R necessities or even derivatives pricing, we have to consider multivariate distributions. First we present the results of the MAGH estimation. In order to assess the results more easily, we first present the estimation of 2 by 2 combinations of the sample. Then we present the full sample treated as a single multivariate distribution. Table 8 contains the estimates of the two assets with highest correlation (CAC and DAX) and the two assets with lowest correlation (BVSP and NIKK). Again, we may be interested in particular subclasses (MANig and MAHyp). Table 9 shows the estimates concerning the restriction of MAGH to one of its main subclasses. The results are quite similar to the univariate case. When the BVSP is one of the distributions, only the BVSP × CAC can be restricted to MANig, but all other two-index combination can be restricted. Once again, the high volatility and kurtosis of the BVSP distribution contributes to this. In the MAHyp case, we cannot restrict any of the samples.
Table 10 Kolmogorov distances for two-dimensional estimations.
BVSP × CAC BVSP × DAX BVSP × FTSE BVSP × NIKK BVSP × SP500 CAC × DAX CAC × FTSE CAC × NIKK CAC × SP500 DAX × FTSE DAX × NIKK DAX × SP500 FTSE × NIKK FTSE × SP500 NIKK × SP500
Normal
GH
Nig
Hyp
0.0065 0.0072 0.0054 0.0062 0.0059 0.0023 0.0017 0.0018 0.0017 0.0019 0.0019 0.0019 0.0015 0.0014 0.0016
0.0016 0.0022 0.0028 0.0016 0.0023 0.0013 0.0010 0.0006 0.0005 0.0008 0.0006 0.0007 0.0005 0.0004 0.0004
0.0017 0.0024 0.0011 0.0017 0.0028 0.0008 0.0006 0.0005 0.0006 0.0006 0.0005 0.0010 0.0005 0.0005 0.0004
0.0022 0.0028 0.0021 0.0017 0.0025 0.0009 0.0006 0.0005 0.0007 0.0009 0.0007 0.0012 0.0005 0.0006 0.0005
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Table 11 MAGH six-dimensional estimations.
α̃ β̃ λ M
Σ
Log-like
BVSP
CAC
DAX
FTSE
NIKK
SP500
1.0420 − 0.0142 − 1.0157 0.1304 6.9439 0.6938 0.7598 0.5495 0.2256 0.7831 − 3846.8
1.0430 0.0017 − 0.5017 0.0510 0.6938 1.8567 1.6821 1.0513 0.2575 0.4564 − 3972.3
1.0423 − 0.0065 − 1.0146 0.0634 0.7598 1.6821 2.5637 1.0277 0.2697 0.5665 − 4041.2
1.0432 − 0.0066 − 0.4980 0.0374 0.5495 1.0513 1.0277 1.0582 0.2086 0.3496 − 3963.3
1.0419 0.0000 0.5028 0.0011 0.2256 0.2575 0.2697 0.2086 0.9832 0.1178 − 4028.8
1.0447 − 0.0079 0.0030 0.0480 0.7831 0.4564 0.5665 0.3496 0.1178 0.8155 − 3972.8
BVSP
CAC
DAX
FTSE
NIKK
SP500
1.0135 − 0.0727 − 0.5000 0.2329 5.3897 0.8962 1.0013 0.7083 0.3862 1.1103 − 3848.3
1.0430 0.0017 − 0.5000 0.0512 0.8962 1.9261 1.6716 1.1646 0.4781 0.6472 − 3972.3
1.0415 − 0.0070 − 0.5000 0.0637 1.0013 1.6716 2.3335 1.1601 0.4933 0.8032 − 4042.7
1.0432 − 0.0066 − 0.5000 0.0375 0.7083 1.1646 1.1601 1.1469 0.3889 0.4958 − 3963.3
1.0382 0.0000 − 0.5000 0.0012 0.3862 0.4781 0.4933 0.3889 1.9640 0.1670 − 4030.3
1.0417 − 0.0077 − 0.5000 0.0483 1.1103 0.6472 0.8032 0.4958 0.1670 1.1563 − 3972.9
BVSP
CAC
DAX
FTSE
NIKK
SP500
1.0726 − 0.0072 1.0000 0.1050 2.1198 0.3279 0.3764 0.2664 0.1432 0.4181 − 3867.9
0.5985 − 0.0013 1.0000 0.0460 0.3279 0.5989 0.6220 0.4382 0.1762 0.2437 − 3974.6
1.0451 0.0000 1.0000 0.0535 0.3764 0.6220 0.8617 0.4365 0.1820 0.3025 − 4041.4
1.0543 − 0.0047 1.0000 0.0311 0.2664 0.4382 0.4365 0.4316 0.1433 0.1867 − 3972.2
1.0470 0.0000 1.0000 − 0.0036 0.1432 0.1762 0.1820 0.1433 0.7203 0.0629 − 4029.8
1.0530 − 0.0060 1.0000 0.0433 0.4181 0.2437 0.3025 0.1867 0.0629 0.4354 − 3979.2
Table 12 MANig six-dimensional estimations.
α̃ β̃ λ M
Σ
Log-like
Table 13 MAHyp six-dimensional estimations.
α̃ β̃ λ M
Σ
Log-like
6.4. Two-dimension goodness of fit We present two kinds of goodness of fit evaluations. First we show the multivariate distribution of the most correlated pair of assets (CAC × DAX) and the least correlated pair (BVSP × NIKK), then we calculate two-dimensional Kolmogorov distances. Figs. 5 and 6 represent the empirical, MAGH fit and normal fit to BVSP × NIKK (correlation = 0.1136) and CAC × DAX (correlation = 0.7902). Once again the kurtosis of the series is better captured with MAGH distributions. Furthermore, we need to calculate the Kolmogorov distance for the multidimensional case. We use the approach of Fasano and Franceschini (1987) and Peacock (1983), which calculates the maximum distance in all possible directions (in this case 4). The number of
sample points used in the distance calculation was 100 for each margin, totaling 10,000 points. The results in Table 10 lead us to conclude that the MAGH really provides better fit to the data. The MAGH distributions and their subclasses consistently have less distance between the theoretical and empirical distributions. 6.5. Six-dimensional MAGH estimation In this section we present the results of the joint estimation of the assets. Tables 11–13 give the results concerning, respectively, MAGH, MANig and MAHyp. The estimation procedure explained in Section 3 makes this estimation easier.
Table 15 Six-Dimensional Kolmogorov distances.
Table 14 Log-likelihood ratio tests. NIG
Hyp
Stats
P-value
Stats
P-value
9.4108
0.1518
80.072
3.4E−15
Distribution
Distance
Normal GH NIG Hyp
0.2394 0.1773 0.1742 0.2312
J. Fajardo, A. Farias / International Review of Financial Analysis 18 (2009) 174–184
Fig. 7. BVSP margin PDF.
Fig. 10. FTSE margin PDF.
Fig. 8. CAC margin PDF.
Fig. 11. NIKK margin PDF.
Fig. 9. DAX margin PDF.
Fig. 12. SP500 margin PDF.
181
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J. Fajardo, A. Farias / International Review of Financial Analysis 18 (2009) 174–184 Table 17 Portfolio with minimum variance. Distribution
CAC
DAX
Mean
S.D.
MAGH MANig MAHyp Normal
0.6589 0.7012 0.6966 0.5461
0.3411 0.2988 0.3034 0.4539
− 0.0302 − 0.0219 − 0.0254 − 0.0422
1.2943 1.2907 1.2673 1.0601
results in Fig. 13 and the minimum variance portfolio is described in the Table 16. It is clear, in this example, that MAGH offers a better description of the risk-return trade-off than its subclasses. But in general it is not true, as we will see in the next example. 7.2. Example 2: Two indexes Fig. 13. Six index efficient frontier.
Following our script, we show in Table 14 the log-likelihood ratio Test results for subclass restriction. They show that we can restrict to the MANig subclass but not to the MAHyp subclass, reaffirming the previous results (fewer dimensions).
As we have seen in Section 4, the CAC and DAX indexes are the most correlated. In what follows we construct the EF with these two pairs of indexes. Also, we describe the portfolio with the minimum variance for the three distributions in Table 17 (Fig. 14). As we can see MAHyp EF dominates MAGH EF in some parts of the EF. It means that MAGH EF not necessarily dominates the other subclasses in the risk-return trade-off.
6.6. Six-dimensional goodness of fit
7.3. Efficient portfolios
We felt challenged to give some measure of goodness of fit to a sixdimensional data set, since all multidimensional Kolmogorov distances in the literature only go up to four dimensions. We implemented the algorithm mentioned in the two-dimensional case, which calculates the Kolmogorov distance in all possible accumulation directions. In a six-dimensional problem, there are 26−64 possible directions. We used 20-point to evaluate the data in each marginal, giving a total of 64,000,000 evaluation points in each accumulation. Table 15 shows the results, and once again the MAGH distributions obtain the best fits. The sizes of the distances were influenced by the number of data points in each marginal, but the main result remains valid. Another way to show the goodness of fit is showing the behavior of the fit in each marginal. Figs. 7–12 show a visual depiction of the fit in each one of the marginals. We can infer that the MAGH distributions have good fit performance.
In Fig. 15, we describe the composition of 1000 efficient portfolios. MAGH and MAHyp efficient portfolios short-sale most of the time FTSE and NIKK indexes. While MANig and classical efficient portfolios short-sale FTSE, NIKK and SP500 indexes. The exact percentages are presented in Table 18.
7. Application: Efficient Frontier (EF)
8. Conclusions In this paper we evaluated the goodness of fit of multivariate affine generalized hyperbolic distributions to various asset returns and showed that they present a very good fit with real data. As an application we address portfolio management for two- and six-asset portfolios. Interesting extension are applications to multivariate derivatives pricing and value at risk calculations, since MAGH better capture the kurtosis of the data. This will be the topic of future research. Acknowledgements
Now we derive the efficient frontier of a portfolio that consists of a combination of two and all six indexes introduced in our sample. Also, we analyze the composition of the efficient portfolios in the six-index case. To find the efficient portfolios we use the Matlab software and allow short-sales.
The authors thanks an anonymous referee for his/her suggestions. Also, J. Fajardo thanks CNPq Brazil for financial support. The views expressed in this work are those of the authors and do not reflect those of the Banco Central do Brasil or its members. Appendix A
7.1. Example 1: Six indexes Consider our six indexes: BVSP, CAC, DAX, FTSE, NIKK and SP500 and the estimated parameters obtained in Tables 11–13. We compute the efficient portfolios for the three distributions and present the
Proof 1. Proposition 1 Let X be a GH(x; ω, δ, μ). Define Y as: d
Y=
X−μ ; δ
ð4Þ
Table 16 Portfolio with minimum variance. Distribution
BVSP
CAC
DAX
FTSE
NIKK
SP500
Mean
S.D.
GH NIG Hyp Normal
0.0375 − 0.0289 0.0233 − 0.0841
− 0.0412 − 0.0432 − 0.4387 − 0.0833
− 0.0261 − 0.1259 − 0.1243 − 0.0834
0.4201 0.4795 0.6459 0.4168
0.2278 0.2503 0.3175 0.4170
0.3818 0.4682 0.5763 0.4170
0.0268 − 0.0019 0.0066 0.0042
0.7300 0.7994 0.8731 0.0084
J. Fajardo, A. Farias / International Review of Financial Analysis 18 (2009) 174–184
183
Table 18 Percentage of time with short-sales in each index. Distribution
BVSP
CAC
DAX
FTSE
NIKK
SP500
GH NIG Hyp Normal
0 1% 0 4.3%
2.9% 1.4% 9.4% 0
1.6% 5.6% 13.3% 0
84.1% 80.1% 88.1% 73.3%
88.8% 89.2% 86.3% 85.4%
0 86.7% 0 56.2%
Now doing a simple parameter transformation: Fig. 14. CAC–DAX efficient frontier.
α=
˜ ˜ ˜β α α and β = δ δ
ð9Þ
and replacing in Eqs. (9) and (8), we obtain: this leads to: PðY≤ yÞ = P
X−μ ≤ y = PðX≤ δy + μÞ; δ
ð5Þ fY ðyÞ =
then,
˜2 α δ2
˜ ˜2β α − 2 δ
2
λ 2
2
ðδ + δ y Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! K
pffiffiffiffiffiffiα˜ λ−1 = 2 λ ˜2 ˜2 α ˜2β α δ Kλ δ 2 − 2 2π δ
FY ðyÞ = FX ðδy + μÞ
ð6Þ =
Deriving both sides w.r.t y, we have: fY ðyÞ = fX ðδy + μÞδ
ð7Þ
So, using the definition of the GH density, we have: 2
2 λ=2
1 =2 2
λ−
6ðα −β Þ ðδ + ðδyÞ Þ fY ðyÞ = 6 4 pffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ− 2πα 2 δλ Kλ ðδ α2 −β2 Þ 2
2
2
K
1 2
λ−
3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi βδy 7 α δ2 + ðδyÞ2 e 7 5δ ð8Þ
ð
Þ
1 =2 2
λ−
2 2
δ
δ
λ ˜ 2 ÞÞ 2 δ−λ ð1 ˜ 2 ð1− β δðα
˜ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi α˜ βδy ˜ α δ δ2 + δ2 y2 e δ δ
1 λ− 2
1 1 λ− = 2 λ− 2 δ 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 +y Þ ˜ ˜ ˜ −1 δ2 ð1 + y2 Þ e α βy qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K λ−1 αδ ð10Þ 2 2 2 −2 ˜ ˜ ð1− β ÞÞδ δ ðα λ 1 λ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ˜ 2 Þ2 δ−λ ð1 + y2 Þ 2 −4 δλ−1 = 2 ˜ λ ð1− β δα ˜ ˜ ˜ 1 + y2 e α βy = qffiffiffiffiffiffiffiffiffiffiffiffiffiffi K 1 α pffiffiffiffiffiffi λ−1 = 2 −λ + 1 = 2 λ λ− 2 ˜ 2 ˜ ˜ 1− β δ δ Kλ α 2π α λ 1 λ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ˜ 2 Þ2 ð1 + y2 Þ 2 −4 ˜ 1 = 2 ð1− β α ˜ ˜ ˜ 1 + y2 e α βy = K qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 α pffiffiffiffiffiffi λ− 2 ˜ 2 ˜ 1− β 2πK α pffiffiffiffiffiffi λ−1 −λ ˜ 2δ 2π α
1 + 2 δλ Kλ
ð
Þ
ð
λ
ð
Þ
Þ
ð
ð
Þ
ð
Þ
Þ
We get an expression similar to Schmidt et al. (2006).
Fig. 15. Efficient portfolios composition.
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184
J. Fajardo, A. Farias / International Review of Financial Analysis 18 (2009) 174–184
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