Financial Risk Forecasting Chapter 3 Multivariate volatility models

Transcription

1 Financial Risk Forecasting Chapter 3 Multivariate volatility models Jon Danielsson 2017 London School of Economics To accompany Financial Risk Forecasting Published by Wiley 2011 Version 3.1, November 2017 Financial Risk Forecasting 2011,2017 Jon Danielsson, page 1 of 81

2 Financial Risk Forecasting 2011,2017 Jon Danielsson, page 2 of 81

3 Volatility The previous chapter focused on the volatility of a single asset In most we hold a portfolio of assets And therefore need to estimate both the volatility of each asset in the portfolio And the correlations between all the assets This means that it is much more complicated to estimate multivariate volatility models than univariate models Financial Risk Forecasting 2011,2017 Jon Danielsson, page 3 of 81

4 The focus of this chapter is on EWMA Orthogonal GARCH CCC and DCC models Estimation comparison Multivariate extensions of GARCH (MV GARCH and BEKK) Financial Risk Forecasting 2011,2017 Jon Danielsson, page 4 of 81

5 Notation Σ t Conditional covariance matrix Y t,k Return on asset k at time t y t,k Sample return on asset k at time t y t = {y t,k } Vector of sample returns on all assets at time t y = {y t } Matrix of sample returns on all assets and dates A and B Matrices of parameters R Correlation matrix D t Conditional variance forecast Financial Risk Forecasting 2011,2017 Jon Danielsson, page 5 of 81

6 Data Amazon and Google daily stock price from date, amzn, goog ,44.52, , , , , , , Financial Risk Forecasting 2011,2017 Jon Danielsson, page 6 of 81

7 Reading data in Matlab data=csvread( amzn goog. csv,1); % the last 1 is to skip first line prices=data (:,2:3); % first column is date y=diff ( log( prices )); % make returns T=length (y); Financial Risk Forecasting 2011,2017 Jon Danielsson, page 7 of 81

8 Matlab estimation It is easy to implement EWMA directly in Matlab For the other models it is generally best to use some library functions The only one I know of is Kevin Sheppard s MFE toolbox Toolbox His documentation lags behind the code and does not mention the multivariate volatility functions But if you download the toolbox you can see his code and each function is documented at the top Financial Risk Forecasting 2011,2017 Jon Danielsson, page 8 of 81

9 Multivariate Volatility Forecasting Financial Risk Forecasting 2011,2017 Jon Danielsson, page 9 of 81

10 Consider the univariate volatility model: Y t = σ t Z t where Y t are returns, σ t is conditional volatility and Z t are random shocks If there are K > 1 assets under consideration, it is necessary to indicate which asset and parameters are being referred to, so the notation becomes more cluttered: Y t,i = σ t,i Z t,i where the first subscript indicates the date and the second subscript the asset Financial Risk Forecasting 2011,2017 Jon Danielsson, page 10 of 81

11 Conditional covariance matrix Σ t The conditional covariance between two assets i and j is indicated by: Cov(Y t,i,y tj ) σ t,ij In the three-asset case (note that σ t,ij = σ t,ji ): σ t,11 Σ t = σ t,12 σ t,22 σ t,13 σ t,23 σ t,33 Financial Risk Forecasting 2011,2017 Jon Danielsson, page 11 of 81

12 Portfolios If w is the vector of portfolio weights The portfolio variance is σ 2 portfolio = w Σw Financial Risk Forecasting 2011,2017 Jon Danielsson, page 12 of 81

13 The curse of dimensionality Number of diagonal elements is K and off diagonal elements K(K 1)/2 so in all K +K(K 1)/2 For two assets it is 2+1, for 3 assets 3+4, for 4 10, etc. The explosion in the number of variance and especially covariance terms, as the number of assets increases, is known as the curse of dimensionality This is one reason why it is more difficult to estimate the covariance matrix Financial Risk Forecasting 2011,2017 Jon Danielsson, page 13 of 81

14 Positive semi-definiteness For univariate volatility we need to ensure that the variance is not negative (σ 2 0) And for a portfolio σ 2 portfolio = w Σw 0 So a covariance matrix should be positive semi-definite: Σ 0 This can be difficult to ensure Financial Risk Forecasting 2011,2017 Jon Danielsson, page 14 of 81

15 What about multivariate (MV) GARCH? For one asset For two σ 2 t+1 = ω +αy2 t +βσ 2 t σ 2 t+1,1 = ω 1 +α 1 y 2 t,1 +β 2σ 2 t,1 +α 2y 2 t,2 +β 2σ 2 t,2 +δ 1σ 2 t+1,1,2 +γ 1y t,1 y t,2 σ 2 t+1,2 = ω 2 +α 3 y 2 t,1 +β 3σ 2 t,1 +α 4y 2 t,2 +β 4σ 2 t,2 +δ 2σ 2 t+1,1,2 +γ 2y t,1 y t,2 σ 2 t+1,1,2 = ω 3 +α 5 y 2 t,1 +β 5σ 2 t,1 +α 6y 2 t,2 +β 6σ 2 t,2 +δ 3σ 2 t+1,1,2 +γ 3y t,1 y t,2 Or 21 parameters to estimate Almost impossible in practice Financial Risk Forecasting 2011,2017 Jon Danielsson, page 15 of 81

16 Numerical issues Stationarity is more important for multivariate volatility For univariate GARCH model, violation of covariance stationarity does not hinder the estimation process with numerical problems A univariate volatility forecast is still obtained even if α+β > 1 This is generally not the case for MV GARCH models A parameter set resulting in violation of covariance stationarity might also lead to unpleasant numerical problems Numerical algorithms need to address these problems, thus complicating the programming process considerably Problems with multiple local minima, flat surfaces and other pathologies discussed in the last chapter Financial Risk Forecasting 2011,2017 Jon Danielsson, page 16 of 81

17 Instead Use some simplification approaches Unfortunately they come with significant trade-offs So MV estimation is much harder and much less accurate than univariate estimation 1. EWMA 2. CCC 3. DCC (perhaps most widely used for portfolios ) 4. OGARCH (perhaps most widely used for combining portfolios) 5. BEKK (not to be recommended except for very small problems, K = 2, perhaps K = 3) 6. MV GARCH (practically impossible except maybe when K = 2) Financial Risk Forecasting 2011,2017 Jon Danielsson, page 17 of 81

18 EWMA Financial Risk Forecasting 2011,2017 Jon Danielsson, page 18 of 81

19 Univariate: A vector of returns is EWMA model The multivariate EWMA is: ˆσ 2 t = λˆσ2 t 1 +(1 λ)y2 t 1 y t = [y t,1,y t,2,...,y t,k ] K 1 ˆΣ t = λˆσ t 1 +(1 λ)y t 1 y t 1 with an individual element given by: ˆσ t,ij = λˆσ t 1,ij +(1 λ)y t 1,i y t 1,j Financial Risk Forecasting 2011,2017 Jon Danielsson, page 19 of 81

20 Properties The same weight, λ, is used for all assets It is pre-specified and not estimated The variance of any particular asset only depends on its own lags Financial Risk Forecasting 2011,2017 Jon Danielsson, page 20 of 81

21 Pros and cons of multivariate EWMA model Usefulness: Straightforward implementation, even for a large number of assets Covariance matrix is guaranteed to be positive semi-definite Financial Risk Forecasting 2011,2017 Jon Danielsson, page 21 of 81

22 Pros and cons of multivariate EWMA model Usefulness: Straightforward implementation, even for a large number of assets Covariance matrix is guaranteed to be positive semi-definite Drawbacks: Restrictiveness: Simple structure The assumption of a single and usually non-estimated λ Financial Risk Forecasting 2011,2017 Jon Danielsson, page 22 of 81

23 Matlab estimation Can use the MFE library riskmetrics (y,lambda) Financial Risk Forecasting 2011,2017 Jon Danielsson, page 23 of 81

24 EWMA = nan(t,3); % create a matrix to hold the covariance matrix lambda = 0.94 S = cov(y) % initial (t=1) covariance matrix Financial Risk Forecasting 2011,2017 Jon Danielsson, page 24 of 81

25 Be careful with the matrix multiplications, getting the transposes right for i = 2:T % loop though the sample S = lambda S + (1 lambda) y(i,:) y(i,:); EWMA(i,:) = S([1,4,2]); % convert matrix to vector end EWMArho = EWMA(:,3)./ sqrt (EWMA(:,1). EWMA(:,2)) % calculate correlations % note the matrix multiplication. Financial Risk Forecasting 2011,2017 Jon Danielsson, page 25 of 81

26 Constant Conditional Correlation Models Financial Risk Forecasting 2011,2017 Jon Danielsson, page 26 of 81

27 Steps De-mean returns Separate out correlation modelling from volatility modelling 1. correlation matrix 2. variances Model volatilities with GARCH or some standard method The correlation matrix can be static (CCC) or dynamic (DCC) Financial Risk Forecasting 2011,2017 Jon Danielsson, page 27 of 81

28 Definitions Let D t be a diagonal matrix where each element is the volatility of each asset D t,ii = σ t,i, i = 1,...,K D t,ij = 0, i j Use univariate GARCH (or some method) to estimate the variance of each asset separately, i.e. to get D t D t,ii = σ t,i = ω i +α i yt 1,i 2 +β iσt 1,i 2, i = 1,...,K Financial Risk Forecasting 2011,2017 Jon Danielsson, page 28 of 81

29 We want the correlations of the residuals ǫ t,i := y t,i D t,i ǫ t := Dt 1 y t K 1 And Then the correlations are ǫ T K ˆR := Cov(ǫ) = ǫǫ t T Financial Risk Forecasting 2011,2017 Jon Danielsson, page 29 of 81

30 Constant conditional correlations (CCC) Then combine these two ˆΣ t = ˆD t ˆR ˆDt Note that while D is time dependent, R is not Financial Risk Forecasting 2011,2017 Jon Danielsson, page 30 of 81

31 Pros and cons Pros Guarantees the positive definiteness of ˆΣ t if ˆR is positive definite Simple model, easy to implement Since matrix ˆD t has only diagonal elements, we can estimate each volatility separately Financial Risk Forecasting 2011,2017 Jon Danielsson, page 31 of 81

32 Pros and cons Pros Guarantees the positive definiteness of ˆΣ t if ˆR is positive definite Simple model, easy to implement Since matrix ˆD t has only diagonal elements, we can estimate each volatility separately Cons The assumption of correlations being constant over time is at odds with the vast amount of empirical evidence supporting nonlinear dependence Financial Risk Forecasting 2011,2017 Jon Danielsson, page 32 of 81

33 Dynamic Conditional Correlation Models Financial Risk Forecasting 2011,2017 Jon Danielsson, page 33 of 81

34 Dynamic conditional correlations (DCC) DCC model is an extension of the the CCC Let the correlation matrix ˆR t be time dependent While one might propose a model like R t = a+bǫ t 1 ǫ t 1 +cr t 1 That will not work since we have to ensure Σ t > 0, i.e. positive definite. For that: D t is positive definite by construction since all elements in D t are positive All elements in R t need to be 1 and 1 Financial Risk Forecasting 2011,2017 Jon Danielsson, page 34 of 81

35 So need more steps Decompose R t into R t = Q t Q tq t Q t is a positive definite matrix that dives the dynamics Qt re-scales Q t to ensure each element q t,ij < 1 1/ q t, / q t,22 0 Q t = / q t,kk Financial Risk Forecasting 2011,2017 Jon Danielsson, page 35 of 81

36 The have Q t follow an ARMA type process Q t = (1 ζ ξ)q +ζǫ t 1 ǫ t 1 +ξq t 1 Q is the (K K) unconditional covariance matrix of ǫ ζ and ξ are parameters Parameter restrictions: Positive definiteness ζ,ξ > 0 Stationarity ζ +ξ < 1 Financial Risk Forecasting 2011,2017 Jon Danielsson, page 36 of 81

37 Pros and cons of DCC Pros Large covariance matrices can be easily be estimated Financial Risk Forecasting 2011,2017 Jon Danielsson, page 37 of 81

38 Pros and cons of DCC Pros Large covariance matrices can be easily be estimated Cons Parameters ζ and ξ are constants So the conditional correlations of all assets are driven by the same underlying dynamics, parameters are same for all assets Financial Risk Forecasting 2011,2017 Jon Danielsson, page 38 of 81

39 Matlab estimation in 2 asset case [p, lik, Ht] = dcc(y,1,1,1,1) Ht = reshape (Ht,4,T) ; % because Ht comes as a time vector of one % period matrixes % so need to convert those % into a T by 4 matrix DCCrho = Ht(:,3)./ sqrt(ht(:,1). Ht(:,4)); Financial Risk Forecasting 2011,2017 Jon Danielsson, page 39 of 81

40 Orthogonal GARCH Financial Risk Forecasting 2011,2017 Jon Danielsson, page 40 of 81

41 Large problems Even a medium-sized financial institutions will have hundreds of thousands or millions of types of assets what is known as risk factors Estimating the covariance matrix for the entire institution is effectively impossible using methods like EWMA or DCC Instead, we can split the risk factors up into subcomponents Estimate the covariance of each And then combine them back Financial Risk Forecasting 2011,2017 Jon Danielsson, page 41 of 81

42 For example Bonds Bank US China Commodities S&P 100 Index Equities S&P 400 Mid Cap Index Bonds Equities Brazil Bonds Equities Financial Risk Forecasting 2011,2017 Jon Danielsson, page 42 of 81

43 The orthogonal approach transforms the observed returns matrix into a set of portfolios with the key property that they are uncorrelated We can forecast their volatilities separately Principal components analysis (PCA) Known as orthogonal GARCH, or OGARCH Because it involves transforming correlated returns into uncorrelated portfolios and then using GARCH to forecast the volatilities of each uncorrelated portfolio separately Financial Risk Forecasting 2011,2017 Jon Danielsson, page 43 of 81

44 Idea We have a matrix of returns With covariance matrix Let Then the correlations are Y T K Σ K K D = diagσ R = D 1 ΣD 1 Financial Risk Forecasting 2011,2017 Jon Danielsson, page 44 of 81

45 Orthogonalizing covariance Making covariance uncorrelated Transform the return matrix Y into uncorrelated portfolios U Calculate the K K matrix of eigenvectors of R Denote the matrix Λ U is defined as: U = Λ Y Financial Risk Forecasting 2011,2017 Jon Danielsson, page 45 of 81

46 Large-scale implementations Large number of assets The method also allows estimates for volatilities and correlations of variables to be generated even when data are sparse (e.g., in illiquid markets) The use of PCA guarantees the positive definiteness of the covariance matrix PCA also facilitates building a covariance matrix for an entire financial institution by iteratively combining the covariance matrices of the various trading desks, simply by using one or perhaps two PCs Financial Risk Forecasting 2011,2017 Jon Danielsson, page 46 of 81

47 Consider an example... Your financial institution has the following securities: Financial Risk Forecasting 2011,2017 Jon Danielsson, page 47 of 81

48 Consider an example... Your financial institution has the following securities: σ 2 E Financial Risk Forecasting 2011,2017 Jon Danielsson, page 48 of 81

49 Consider an example... Your financial institution has the following securities: σe 2 σb 2 Financial Risk Forecasting 2011,2017 Jon Danielsson, page 49 of 81

50 Consider an example... Your financial institution has the following securities: σ 2 E σ 2 B σ 2 E Financial Risk Forecasting 2011,2017 Jon Danielsson, page 50 of 81

51 Consider an example... Your financial institution has the following securities: σ 2 E σ 2 B σ 2 E σ 2 B Financial Risk Forecasting 2011,2017 Jon Danielsson, page 51 of 81

52 Consider an example... Your financial institution has the following securities: σ 2 E σ 2 B σ 2 E σ 2 B where red is UK, blue is Germany, E denotes EQUITY and B is BOND Financial Risk Forecasting 2011,2017 Jon Danielsson, page 52 of 81

53 Consider an example... Your financial institution has the following securities: σ 2 E σ 2 B σ 2 E σ 2 B where red is UK, blue is Germany, E denotes EQUITY and B is BOND Individual covariance matrices of your securities can be combined into a covariance matrix for the entire financial institution: Financial Risk Forecasting 2011,2017 Jon Danielsson, page 53 of 81

54 Consider an example... Your financial institution has the following securities: σ 2 E σ 2 B σ 2 E σ 2 B where red is UK, blue is Germany, E denotes EQUITY and B is BOND Individual covariance matrices of your securities can be combined into a covariance matrix for the entire financial institution: σ 2 E Financial Risk Forecasting 2011,2017 Jon Danielsson, page 54 of 81

55 Consider an example... Your financial institution has the following securities: σ 2 E σ 2 B σ 2 E σ 2 B where red is UK, blue is Germany, E denotes EQUITY and B is BOND Individual covariance matrices of your securities can be combined into a covariance matrix for the entire financial institution: σ 2 E σ 2 B Financial Risk Forecasting 2011,2017 Jon Danielsson, page 55 of 81

56 Consider an example... Your financial institution has the following securities: σ 2 E σ 2 B σ 2 E σ 2 B where red is UK, blue is Germany, E denotes EQUITY and B is BOND Individual covariance matrices of your securities can be combined into a covariance matrix for the entire financial institution: σ 2 E ρ E,B σ 2 B Financial Risk Forecasting 2011,2017 Jon Danielsson, page 56 of 81

57 Consider an example... Your financial institution has the following securities: σ 2 E σ 2 B σ 2 E σ 2 B where red is UK, blue is Germany, E denotes EQUITY and B is BOND Individual covariance matrices of your securities can be combined into a covariance matrix for the entire financial institution: σ 2 E ρ E,B σ 2 B σ 2 E Financial Risk Forecasting 2011,2017 Jon Danielsson, page 57 of 81

58 Consider an example... Your financial institution has the following securities: σ 2 E σ 2 B σ 2 E σ 2 B where red is UK, blue is Germany, E denotes EQUITY and B is BOND Individual covariance matrices of your securities can be combined into a covariance matrix for the entire financial institution: σ 2 E ρ E,B σ 2 B ρ E,E ρ B,E σ 2 E Financial Risk Forecasting 2011,2017 Jon Danielsson, page 58 of 81

59 Consider an example... Your financial institution has the following securities: σ 2 E σ 2 B σ 2 E σ 2 B where red is UK, blue is Germany, E denotes EQUITY and B is BOND Individual covariance matrices of your securities can be combined into a covariance matrix for the entire financial institution: σ 2 E ρ E,B σ 2 B ρ E,E ρ B,E σ 2 E σ 2 B Financial Risk Forecasting 2011,2017 Jon Danielsson, page 59 of 81

60 Consider an example... Your financial institution has the following securities: σ 2 E σ 2 B σ 2 E σ 2 B where red is UK, blue is Germany, E denotes EQUITY and B is BOND Individual covariance matrices of your securities can be combined into a covariance matrix for the entire financial institution: σ 2 E ρ E,B σ 2 B ρ E,E ρ B,E σ 2 E ρ B,B ρ E,B σ 2 B Financial Risk Forecasting 2011,2017 Jon Danielsson, page 60 of 81

61 Consider an example... Your financial institution has the following securities: σ 2 E σ 2 B σ 2 E σ 2 B where red is UK, blue is Germany, E denotes EQUITY and B is BOND Individual covariance matrices of your securities can be combined into a covariance matrix for the entire financial institution: σ 2 E ρ E,B σ 2 B ρ E,E ρ B,E σ 2 E ρ B,B ρ E,B σ 2 B Financial Risk Forecasting 2011,2017 Jon Danielsson, page 61 of 81

62 Consider an example... Your financial institution has the following securities: σ 2 E σ 2 B σ 2 E σ 2 B where red is UK, blue is Germany, E denotes EQUITY and B is BOND Individual covariance matrices of your securities can be combined into a covariance matrix for the entire financial institution: σ 2 E ρ E,B σ 2 B ρ E,E ρ B,E σ 2 E ρ B,B ρ E,B σ 2 B Financial Risk Forecasting 2011,2017 Jon Danielsson, page 62 of 81

63 Consider an example... Your financial institution has the following securities: σ 2 E σ 2 B σ 2 E σ 2 B where red is UK, blue is Germany, E denotes EQUITY and B is BOND Individual covariance matrices of your securities can be combined into a covariance matrix for the entire financial institution: σ 2 E ρ E,B σ 2 B ρ E,E ρ B,E σ 2 E ρ B,B ρ E,B σ 2 B Financial Risk Forecasting 2011,2017 Jon Danielsson, page 63 of 81

64 Matlab estimation [par, Ht] = o mvgarch(y,2, 1,1,1); Ht = reshape (Ht,4,T) ; OOrho = Ht(:,3)./ sqrt(ht(:,1). Ht(:,4)); Financial Risk Forecasting 2011,2017 Jon Danielsson, page 64 of 81

65 Estimation Comparison Financial Risk Forecasting 2011,2017 Jon Danielsson, page 65 of 81

66 Prices 200 MSFT 180 IBM Financial Risk Forecasting 2011,2017 Jon Danielsson, page 66 of 81

67 (b) Returns 15 % MSFT 10 % 5 % 0 % 5 % 10 % 15 % 10 % 5 % 0 % 5 % 10 % 15 % IBM Financial Risk Forecasting 2011,2017 Jon Danielsson, page 67 of 81

68 Correlation estimates with average correlation 49% 80% 60% 40% 20% 0% 20% EWMA Financial Risk Forecasting 2011,2017 Jon Danielsson, page 68 of 81

69 Correlation estimates with average correlation 49% 80% 60% 40% 20% 0% 20% EWMA DCC Financial Risk Forecasting 2011,2017 Jon Danielsson, page 69 of 81

70 Let s focus on 2008, the midst of the crisis, when the correlations of all stocks increased dramatically... Financial Risk Forecasting 2011,2017 Jon Danielsson, page 70 of 81

71 Prices 100 MSFT IBM Jan Mar May Jul Sep Nov Jan Financial Risk Forecasting 2011,2017 Jon Danielsson, page 71 of 81

72 Returns 15 % MSFT 10 % 5 % 0 % 5 % 10 % Jan Mar May Jul Sep Nov Jan 15 % IBM 10 % 5 % 0 % 5 % 10 % Jan Mar May Jul Sep Nov Jan Financial Risk Forecasting 2011,2017 Jon Danielsson, page 72 of 81

73 Correlations with average correlation 49% 80% 60% 40% 20% EWMA Jan Mar May Jul Sep Nov Jan Financial Risk Forecasting 2011,2017 Jon Danielsson, page 73 of 81

74 Correlations with average correlation 49% 80% 60% 40% 20% EWMA DCC Jan Mar May Jul Sep Nov Jan Financial Risk Forecasting 2011,2017 Jon Danielsson, page 74 of 81

75 BEKK Financial Risk Forecasting 2011,2017 Jon Danielsson, page 75 of 81

76 The BEKK model An alternative to the MV-GARCH models The matrix of conditional covariances is Σ t A function of the outer product of lagged returns and lagged conditional covariances Each pre-multiplied and post-multiplied by a parameter matrix Results in a quadratic function that is guaranteed to be positive semi-definite Financial Risk Forecasting 2011,2017 Jon Danielsson, page 76 of 81

77 The two-asset, one-lag BEKK(1,1,2) model is defined as: Σ t = ΩΩ +A Y t 1 Y t 1A+B Σ t 1 B or: ( ) ( )( σt,11 σ Σ t = t,12 ω11 0 ω11 0 = σ t,12 σ t,22 ω 21 ω 22 ω 21 ω 22 ( ) ( α11 α + 12 α 21 α 22 ( β11 β + 12 β 21 β 22 Y 2 t 1,1 Y t 1,2 Y t 1,1 Yt 1,2 2 ) ( σt 1,11 σ t 1,12 σ t 1,12 σ t 1,22 ) )( ) Y t 1,1 Y t 1,2 α11 α 12 α 21 α 22 )( ) β11 β 12 β 21 β 22 Financial Risk Forecasting 2011,2017 Jon Danielsson, page 77 of 81

78 The general BEKK(L 1,L 2,K) model is given by: Σ t = ΩΩ + + K L 1 A i,ky t iy t i A i,k k=1 i=1 K L 2 B j,kσ t j B j,k k=1 j=1 The number of parameters in the BEKK(1,1,2) model is K(5K +1)/2 11 in two asset case 24 in three asset case 42 in four asset case Financial Risk Forecasting 2011,2017 Jon Danielsson, page 78 of 81

79 Pros and cons Pros Allows for interactions between different asset returns and volatilities Relatively parsimonious Financial Risk Forecasting 2011,2017 Jon Danielsson, page 79 of 81

80 Pros and cons Pros Allows for interactions between different asset returns and volatilities Relatively parsimonious Cons Parameters hard to interpret Many parameters are often found to be statistically insignificant, which suggests the model may be overparametrized Can only handle a small number of assets Financial Risk Forecasting 2011,2017 Jon Danielsson, page 80 of 81

81 Matlab estimation [PARAMETERS,LL,HT]=bekk(y,[],1,0,1); This took 33 seconds on my laptop Financial Risk Forecasting 2011,2017 Jon Danielsson, page 81 of 81