Financial Institutions Strategy Department, Bank of Thailand

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1 Slug Simple Nonlinearit Enhancement to the -Factor and Copula Models in Finance with Parametric Estimation and Goodness-o-Fit Tests on US and Thai Equit Data Poomjai Nacaskul & Worawut Sabborriboon Abstract A bivariate normal distribution with the attendant non-analticall integrable p.d.. lies at the hearts o man inancial theories. Its derived copula ostensibl does awa with the normalit assumptions onl to retain the linear Pearson s correlation measure implicit to said bivariate normal p.d.. In inancial modelling contet the copula suer rom at least three setbacks namel its inabilit to capture etreme tail asmmetric upside vs. downside and nonlinear diminishing dependenc structures. Noting that various ies have been proposed w.r.t. the ormer two issues i this paper attempts to address the nonlinearit with the proposal o a bivariate Slug distribution ii rom which a derived copula densit unction quite naturall and parsimoniousl captures a particular nonlinear dependenc structure. In addition iii this paper devises a simple intuitive ormulation o copula parameter estimation as a minimisation o a chi-square test statistics iv whose resultant value readil lends itsel to the widel popular statistical goodness-o-it testing. Tests were perormed comparing independent vs. vs. Slug copulas on weekl US and Thai equit market inde and individual stock returns data all available on Reuters.. Introduction A bivariate normal distribution with the attendant non-analticall integrable p.d.. lies at the hearts o man inancial theories: rom the Trenor/Sharpe/Lintner/Mossin Capital Asset Pricing Model CAPM and subsequent -actor market/credit risk models which presuppose joint normalit o market and individual risk actors to the Vasicek/Gord Asmptotic Single Risk Factor ASRF model underling the Basel II Internal-Ratings Based IRB approach to calculating minimal regulator credit- Team Eecutive Quantitative Models & Financial Engineering Team Financial Institutions Strateg Department Bank o Thailand [PoomjaiN@bot.or.th] & Facult MBA Program Mahanakorn Universit o Technolog. Analst Quantitative Models & Financial Engineering Team Financial Institutions Strateg Department Bank o Thailand [WorawutS@bot.or.th]

2 risk capitals to the Duie-Singleton/Li CDO pricing methodolog which b utilising copula [][][0] ostensibl does awa with the normalit assumptions or marginal loss distributions onl to retain the linear Pearson s correlation measure implicit and integral to said bivariate normal p.d.. Indeed rom the advent o Markowitz Modern Portolio Theor MPT o Pareto optimal asset allocation down to contemporar Value-at-Risk VaR measure o market risks in the aggregate questions have been raised and widel controverted since about the assumed normalit o the marginal distributions o asset returns. B and large this line o inquiries has been superseded b misgiving w.r.t. the dependence structure between risk actors as implied b the widel adopted copula. One line o inquir is whether the copula can adequatel capture dependenc in the etreme tails o the marginal distributions. In their seminal paper on testing the copula hpothesis or inancial-asset dependencies Malevergne & Sornette 00 [4] adapted the Kolmogorov as well as Anderson-Darling distances as their distributional test metrics. 4 In particular Mashal & Zeevi00 [5] and similarl Chen Fan Patton004 [] eploited the act that the Student s t distribution is eectivel a heav-tailed generalisation and thereore embeds as a special case o the normal distribution. Söderberg 009 [9] then applied both methods to good eect in testing the copula on the Swedish stock market. 5 Another line o inquir is whether upside dependenc needs be smmetric with the downside leg. For equit this is related to the well-known post-black Monda volatilit skew phenomenon where correlations rise dramaticall during downturns then revert to lower level once recover takes place. A less obvious investigation in the oreign echange market is whether joint appreciation and joint depreciation are indeed asmmetric and whether dependenc relation is time-conditional Patton 00 [8]. In other words joint normalit has two components normal marginals and copula so going beond normal marginals whilst retaining the copula constitutes but a partial generalisation. 4 The epressl highlighted how the Kolmogorov measure is more sensitive to the deviations occurring in the bulk o the distributions while the Anderson-Darling measure is more accurate in the tails o the distributions. 5 Although such tail dependenc issues arose vis-à-vis market returns i.e. random variables with distributional support over the real number line similar issues ma arise w.r.t. random variables with distributional support over the positive interval i.e. dollar values o operational/credit losses.

3 Our line o inquir is whether the inherentl linear dependence structure imposed b the copula is appropriate or inancial modelling. For instance in an emerging equit market dominated b a ew large stocks it is anecdotall observed how initiall these leading stocks would be the ones that move ver quickl in response to perhaps because the are leading the general market trends but as the overall inde moves deeper in the negative territor these core stocks seem to hold value better as long-term bargain hunters come in dismissing what the perceive as overshot market sentiments. Conversel in a strong rall international capital unds ma opt to take earl proits in the core stocks the principall built their cross-countr investment diversiication positions on. In a sense betas or these lead stocks peak in the middle and taper o at either end. This paper is motivated initiall rom this nonlinearit consideration in particular. Although not discussed urther in this paper should one take up the demarcation between high-requenc/low-impact vs. low-requenc/high-impact operational losses and pursue an etended diminishing dependenc hpothesis i.e. distinguishing high-requenc/lowimpact/high-dependenc vs. low-requenc/high-impact/low-dependenc operational losses a similar nonlinear etension to copula modelling would be just as essential i not more so. In particular we go back to the bivariate normal distribution in an attempt to endow it with nonlinear dependenc. First we write out a 5-parameter a bivariate normal p.d.. : ep π The conditional epectation o is a linear unction o thus: The authors are pursuing such investigation especiall in relations to the Basel II Advanced Measurement Approach AMA to calculating minimal regulator operational-risk capitals where dierent tpes o operational-risk events and/or dierent business lines tend to ehibit clear regime preerence i.e. weak dependenc amongst low-requenc high-impact natural catastrophes in contrast with stronger dependenc amongst high-requenc low-impact retail transaction errors and/or rauds which could be traced back to poor control training manpower issues etc.

4 Ε [ ] α β } d The aim o this paper is simpl to ind an analticall simple alternative to and possibl a generalisation o or which an epression derived analogousl to is nonlinear. The resulting bivariate distribution is called a Slug or a reason which will become intuitivel/visuall obvious. This is done in section on top o which section then derives a corresponding copula densit unction. Section then 4 ormulates and proposes a simple intuitive method or optimising a copula it over the relevant parameter space one which readil lends itsel to Pearson s chi-square goodness-oit test. 7 Section 5 demonstrates how the proposed copula is used and tested against weekl US as well as Thai equit market inde and individual stock returns data downloaded rom Reuters. In summar our paper introduces innovations in all our areas: distribution copula estimation and testing.. The Slug Distribution This paper is motivated b the inabilit o linear dependence structure to implement marginall decreasing dependenc i.e. whereb a response to stimulus is positive throughout but the sensitivit peaks and wanes in essence a miture between linear and sigmoidal responses. 8 In other words we want conditional epectation o to be a nonlinear unction o o the orm: 78 } Ε α β γ [ ] linear response response Sigmoidal Indeed there are a number o alternatives available: 7 Incidentall Fermanian 005 [] and Panchenko 005 [7] both proposed a goodness-o-it test or copulas the ormer non-parametricall the latter parametricall. We believe our approach is much simpler than either one. And unlike Malevergne & Sornette 00 [4] there is no simulation involved. 8 Another motivation could be to model the run-awa eect i.e. whereb a response to stimulus is positive throughout but this time the sensitivit keeps increasing at both ends essentiall reversing the roles between the response and the stimulus i.e. instead o it would be etc. 4

5 5 [ ] [ ] [ ] [ ] ln arcsinh arcsinh tanh tanh z z z e e z e iv iii z z ii i β α β α γ β α γ β α Ε Ε Ε Ε 4 It turns out that in order to introduce such nonlinearit while keeping closest to the original unctional orm an epedient choice is to modi the bivariate normal p.d.. via a simple change o variable again or which a number o alternatives are available: L L K a sinh 5!! } {0 h h k h h h iv iii k ii i β β β 5 For simplicit this paper considers the cubic unction hence the ollowing kernel unction: ep π For a bivariate distribution proper a normalising term must be ound. Unortunatel although indeed analticall epressible 9 the epression is rather mess involving gamma and so-called Kummer s conluent hpergeometric unction o the irst kind: 9 We used Mathematica throughout.

6 L Γ!! / 7 b b b a a a b b a a b a b a H H dd π 7 Writing it all out ields the ollowing p.d.. proper: Γ 7 / 5 / ep H dd g π π 8 Unortunatel we have moved rom a distribution which is location-scale invariant w.r.t. both random variates and to a distribution which is merel location invariant w.r.t. the random variates. The trick then is to irst reduce this 5-parameter p.d.. into a -parameter standard Slug distribution: 7 5 /. ep π π Γ g std 9 Then reintroduce the conditional means and variance parameters as a simple linear transormation to obtain the 5-parameter Slug distribution. In summar:

7 7 7 5 /. ep π π Γ std g g 0 We now have one working bivariate distribution whose conditional epectation has the S shape resembling common gastropod mollusc hence a Slug Figure Figure. Figure : Bivariate Normal vs. Slug p.d.. D Plots Figure : Bivariate Normal vs. Slug p.d.. Contour Plots

8 Whereas with a bivariate normal distribution the conditional epectation o is as noted in a linear unction o with our Slug distribution the conditional epectation o i.e. the value o that maimises the conditional distribution given is a cubic root o as desired thus: g * [ ] [ ] Ε * 0 Ε Note how it would now be incorrect to reer to as the correlation parameter as per normal distribution. So when the appear together it would be advisable to write or the old correlation Slug parameter and or the new Slug dependenc parameter. In terms o estimation the advantage to orcing our new distribution to remain location-scale invariant in this orm is that the means and variance estimates are as beore which leaves onl the question o how to estimate the dependenc parameter. Here in analog with simple regression analsis 0 we can use as the itting unction then deine an Ordinar Least Square OLS estimate o asˆ that given the data set { } n i i minimises the it error: i ' itting error' n ˆ i ˆ ˆ ˆ ˆ min ˆ i OLS i i ˆ i. The Slug Copula Recall that a copula unction is constructed as a bivariate standard normal c.d.. o the inverses o univariate normal c.d.. thus: 0 However the analog is not perect as we are not assuming uniorm noise over a cubic root o as would be the case o a nonlinear regression hence the un-weighted error terms assign less weights than perhaps ideal to itting error as goes awa rom its average i.e. as increases. 8 Alternativel given an n-pair sample data { } n i i a Maimum Likelihood Estimator MLE i could be ound b taking the partial derivative o the log-likelihood unction w.r.t.. Unortunatel n i i std. equating the result log g n i i i i i does not ield a simple solution or an arbitrar sample size n. 4 i i i to zero

9 C u v u Φ v ΦΦ v v Φ Φ Φ Φ std. s t ds dt s t st ep dsdt π u u From which the corresponding copula densit is given b: c. Φ u Φ v Φ u Φ v std u v 4 Constructing a Slug copula requires modiication onl w.r.t. the bivariate integrand: C Slug u v G Φ u Φ v Φ v Φ u std. g Φ v Φ u s t dsdt s t st ep 5 / Γ 7 π π dsdt 5 From which the corresponding Slug copula densit is given b: c Slug normalising constant 4748 / Γ u v π 7 std. g Φ u Φ v Φ u Φ v We now have one working bivariate copula whose dependenc relation has the S shape resembling common gastropod mollusc hence a Slug Figure Figure 4. 9

10 Figure : Standard vs. Slug Copula Densit D Plots Figure 4: Standard vs. Slug Copula Densit Contour Plots The net section details our simple intuitive approach to constructing and utilising the chi-square statistics: irstl as the error criterion or optimising a it over the copula densit unction s parameter space and secondl as a goodness-o-it statistics as per the widel popular Pearson s chi-square test. 0

11 4. Chi-Square Error Criterion and Goodness-o-Fit Test or Copulas Our methodolog is as ollows.. Transorm the original bivariate data rom its sample space withinr e.g. Figure 5 Figure let plots to within[ 0] e.g. Figure 5 Figure right plots each ais via its own univariate empirical c.d.. 5% 0% 0.9 5% 0% -5% -0% -5% -0% -5% 0% 5% 0% 5% -5% -0% % 0. -0% 0. -5% -0% Figure 5: Scatter Plots o "NSE Inde" vs. "Coca Cola" returns actual let & [0] right 5% 0% 0.9 5% 0.8 0% 0.7 5% 0% -0% -5% -0% -5% -0% -5% 0% 5% 0% 5% -5% -0% -5% -0% -5% Figure : Scatter Plots o "SET Inde" vs. "Siam Cement" returns actual let & [0] right. Partition each [ 0] transormed sample space into cells i.e. 5 square cells o equal size and perorm sample counts within each cell e.g. Table & Table each with 500 points. Call this the observed requenc matri. Although the method generalises to an copulas our eposition is restricted to bivariate ones.

12 Oij [00.] 0.0.4] 0.40.] 0.0.8] 0.8] 0.8] ] ] ] [00.] Table : Uniormed "NSE Inde" & "Coca Cola" Returns Observed Frequencies Oij [00.] 0.0.4] 0.40.] 0.0.8] 0.8] 0.8] ] ] ] [00.] 4 0 Table : Uniormed "SET Inde" & "Siam Cement" Returns Observed Frequencies. For an bivariate copula integrate the copula densit over the areas corresponding to the cell boundaries. Multipl each cell b the total number o data points. Call this the epected requenc matri i.e. epected cell populations consistent with the speciied copula densit. For eample perorming 5 double integrations w.r.t. the copula densit parameterised b 0. 5 ields Table note: a smmetric matri; whereas perorming 5 double integrations w.r.t. the Slug copula densit parameterised b 0. 5 ields Table note: an asmmetric matri. Note how linearit o dependenc structure now translates to smmetr o the corresponding epected requenc matri; whereas nonlinearit o dependenc structure now translates to asmmetr o the corresponding epected requenc matri. Incidentall the un-parameterised independent copula densit translate into uniorm epected requencies i.e. 500 points divide evenl into 0 data points or each o the 5 cells. Eij [00.] 0.0.4] 0.40.] 0.0.8] 0.8] 0.8] ] ] ] [00.] Table : Consistent with the copula densit with 0. 5 Epected Frequencies

13 Eij [00.] 0.0.4] 0.40.] 0.0.8] 0.8] 0.8] ] ] ] [00.] Table 4: Consistent with the Slug copula densit with 0. 5 Epected Frequencies 4. Calculate the chi-square statistics. Recall that it is given b: 5 5 Observed ij Epected ij ˆ χ ~ χ d. χ j i Epected ij 5. For the purpose o itting the copula densit parameter this χ deines our error criterion to minimise. Thus or our Slug copula: * min χ Epected ij 5 5 Observed ij Epected ij j i ij 0. j 0. j0. 0.i c 0.i0. Epected Slug u v du dv i j K5 8. For the purpose o testing the goodness-o-it o the parameterised copula densit this ˆχ is precisel our test statistics as per Pearson s chi-square test. Thus with degrees o reedom at the α 00% 99% conidence level reject the copula whenever ˆχ reaches or eceeds in value. Where the Ecel unction "CHIINV0.0" ields.9999.

14 5. Testing Slug Copula or Nonlinear Dependencies in Equit Returns For our purpose 4 the -data will represent some kind o market/inde returns while the -data will represent some kind o individual-stock returns. Our US & Thai data consist o 500 weekl Januar nd 000 August nd 009 equit market/inde and individual-stock returns. Three alternative copula densit unctions available or testing are thus: i the un-parameterised independent copula densit ii the -parameterised copula densit and iii our Slug -parameterised Slug densit. Indeed or both and Slug copulas the parametric search/optimisation is rather restricted i.e. Slug 0 so or most cases a simple bisection would suice as an optimisation routine. For illustration we simpl ran through nine values each i.e. calculating the chi-square test statistics or { 0.0. K0.9} and again Slug or { 0.0. K0.9}. In the case o US equit let s now consider the "NSE Inde"/"Coca Cola" pair in particular Figure 7. Right awa the i independent copula uniorm epected requenc matri is emphaticall rejected as its chi-square test statistics is etremel high at The ii copula with onl an eception when 0. 9 ields much better results and at { } cannot be rejected at 99% conidence. The iii Slug copula contrar to what we had hoped does not Slug improve over the standard copula with an onl redeeming act being at 0. it cannot be rejected at 99% conidence either. In the case o Thai equit let s now consider the "SET Inde"/"Siam Cement " pair in particular Figure 8. Right awa the i independent copula uniorm epected requenc matri is emphaticall rejected as its chi-square test statistics is etremel high at 07. The ii copula with no eception ields much better results and at { } cannot be rejected at 99% conidence. The iii Slug copula contrar to what we had hoped does not improve over the Slug standard copula and in act is rejected or all values o { 0.0. K0.9} tried. 4 Again should we wish to capture/test or a run-awa eect the roles would be eectivel reversed i.e. the -data would instead represent some kind o individual-stock returns while the -data would instead represent some kind o market/inde returns. 4

15 In act similar results were observed with a number o other pair returns prompting us to preliminaril concede that the case or enhancing the standard copula models in inance with the kind o nonlinearit we envisioned is et unproven. Nonetheless we believe the test devised and used in this paper while negating our contribution in terms o adding realism to copula modelling is appropriate or testing the goodness-o-it o an copula model and provides a transparent benchmark b which to evaluate an candidate copulas against established ones. "NSE Inde" & "Coca Cola" Returns - Chi-Square Test Statistics or Dierent Copulas Indep. Copula Copula Slug Copula 99% conidence rejection Figure 7: "NSE Inde" vs. "Coca Cola" Returns Chi-Square Test Statistics or Dierent Copulas "SET Inde" & "Siam Cement" Returns - Chi-Square Test Statistics or Dierent Copulas Indep. Copula Copula Slug Copula 99% conidence rejection Figure 8: "SET Inde" vs. "Siam Cement" Returns Chi-Square Test Statistics or Dierent Copulas 5

16 Detailed results o our eperiments is given in Table 5 # Equit Market/Inde vs. Individual-Stock Paired Returns Weekl Data Jan00-Aug09 Pearson's correlation Chi-Square p-value Independent Copula Copula s rho best it Chi-Square p-value Copula Slug Copula s rho best Chi-Square p-value Slug Copula NSE-ATT % % % NSE-CocaCola % % % NSE-EonMobil % % % 4 NSE-GE % % % 5 NSE-IBM % % % NSE-Merck % % % 7 NSE-Microsot % % % 8 NSE-WalMart % % 0..0% 9 SET-BangkokBank % % % 0 SET-SiamCement % % % Table 5: Chi-Square statistics indicates that the copula cannot be rejected at 99% conidence level.. Concluding remarks We set out to construct without introducing additional it parameters nonlinear dependenc structure while keeping as close as possible to the widel popular normal distribution and copula unctions. Our Slug distribution appears unctionall similar to the normal distribution and retains the location-scale invariant propert et manages to transorm the linear epression o the conditional epectation into one that epresses a nonlinear diminishing sensitivit response. Although this is initiall motivated b stlised observation that especiall in emerging market equities individual core stocks tend to react sharpl to general market/inde initiall but retain value better as the market slumps conversel get capped b proit taking when the overall market rallies preliminar evidences rom weekl US and Thai equit data do not support our position. Notwithstanding the parametric search and goodness-o-it methodologies devised and demonstrated in this paper are simple intuitive and transparent. Put in another wa o the our areas we attempted to innovate distribution copula estimation and testing the latter two remain useul despite the ormer two having et to prove their worth.

17 Bibliograph [] Chen Fan Patton 004 Simple Tests or Models o Dependence Between Multiple Financial Time Series with Applications to U.S. Equit Returns and Echange Rates SSRN: [] Cherubini Luciano Vecchiato 004 Copula Methods in Finance Chichester: John Wile & Sons. [] Fermanian Jean-David 005 Goodness-o-Fit Tests or Copulas Journal o Multivariate Analsis vol. 95 no. pp.9-5. [4] Malevergne. & Sornette D. 00 Testing the Copula Hpothesis or Financial Assets Dependences Quantitative Finance vol. pp [5] Mashal R. & Zeevi A. 00 Beond Correlation: Etreme Co-Movements between Financial Assets Working Paper Columbia Universit SSRN: [] Nelsen Roger B. 999 An Introduction to Copulas New ork: Springer. [7] Panchenko Valentn 005 Goodness-o-it Test or Copulas Phsica A vp;. 55. pp [8] Patton Andrew J. 00 Modelling Asmmetric Echange Rate Dependence International Economic Review vol. 47 no. pp [9] Söderberg Jonas 009 Test o the Copula on the Swedish Stock Market Working Paper Series Centre or Labour Market Polic Research no. 9 [0] Wikipedia 7