Financial Risk Forecasting Chapter 3 Multivariate volatility models

Transcription

1 Financial Risk Forecasting Chapter 3 Multivariate volatility models Jon Danielsson 2016 London School of Economics To accompany Financial Risk Forecasting Published by Wiley 2011 Version 2.1. November 2016 Financial Risk Forecasting 2011,2016 Jon Danielsson, page 1 of 66

2 Financial Risk Forecasting 2011,2016 Jon Danielsson, page 2 of 66

3 The focus of this chapter is on Multivariate volatility forecasting EWMA Orthogonal GARCH CCC and DCC models Estimation comparison Multivariate extensions of GARCH Financial Risk Forecasting 2011,2016 Jon Danielsson, page 3 of 66

4 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 2011,2016 Jon Danielsson, page 4 of 66

5 Multivariate Volatility Forecasting Financial Risk Forecasting 2011,2016 Jon Danielsson, page 5 of 66

6 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,k = σ t,k Z t,k where the first subscript indicates the date; and the second subscript the asset Financial Risk Forecasting 2011,2016 Jon Danielsson, page 6 of 66

7 Conditional covariance matrix Σ t The conditional covariance between two assets i and j is indicated by: Cov(Y t,k,y t,j ) σ t,kj In the three-asset case, the conditional covariance matrix takes the following form (note that σ t,ij = σ t,ji ): Σ t = σ t,11 σ t,12 σ t,13 σ t,12 σ t,22 σ t,23 σ t,13 σ t,23 σ t,33 The explosion in the number of volatility terms, as the number of assets increases, is known as the curse of dimensionality Financial Risk Forecasting 2011,2016 Jon Danielsson, page 7 of 66

8 Problems with multivariate volatility models The curse of dimensionality Positive semi-definiteness For a single-asset volatility we need to ensure that the variance is not negative Similarly, a covariance matrix should be positive semi-definite: Σ 0 To ensure that portfolio variance will always be nonnegative regardless of the underlying portfolio Financial Risk Forecasting 2011,2016 Jon Danielsson, page 8 of 66

9 It is usually very hard to estimate multivariate GARCH models. Alternative methodologies for obtaining the covariance matrix are needed. As a result, present practical multivariate volatility forecasting models separate the forecasting of univariate volatility from correlation forecasting (consider OGARCH) The reason one needs to estimate the entire conditional covariance matrix of returns in one go is that the correlations are not zero Financial Risk Forecasting 2011,2016 Jon Danielsson, page 9 of 66

10 It is also conceptually straightforward to develop multivariate extensions of the univariate GARCH-type models discussed in the last chapter - such as multivariate GARCH (MVGARCH) More difficult in practice because the most obvious model extensions result in the number of parameters exploding as the number of assets increases Simplifying model structure comes at a cost to model flexibility Financial Risk Forecasting 2011,2016 Jon Danielsson, page 10 of 66

11 Numerical problems Multivariate stationarity constraints are much more important for MVGARCH models than in the univariate case For univariate GARCH model, violation of covariance stationarity does not hinder the estimation process with numerical problems A volatility forecast is still obtained even if α+β > 1 This is generally not the case for MVGARCH models discussed in this section 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 Financial Risk Forecasting 2011,2016 Jon Danielsson, page 11 of 66

12 EWMA Financial Risk Forecasting 2011,2016 Jon Danielsson, page 12 of 66

13 Univariate form of the model: ˆσ 2 t = λˆσ 2 t 1 +(1 λ)y 2 t 1 The multivariate form is essentially the same: ˆΣ t = λˆσ t 1 +(1 λ)y t 1y 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 2011,2016 Jon Danielsson, page 13 of 66

14 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 2011,2016 Jon Danielsson, page 14 of 66

15 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 2011,2016 Jon Danielsson, page 15 of 66

16 Orthogonal GARCH Financial Risk Forecasting 2011,2016 Jon Danielsson, page 16 of 66

17 The orthogonal approach linearly 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 2011,2016 Jon Danielsson, page 17 of 66

18 Orthogonalizing covariance Making covariance uncorrelated Transform the return matrix y into uncorrelated portfolios u Denote the sample correlation matrix of y by ˆR Calculate the K K matrix of eigenvectors of ˆR Denote the matrix Λ u is defined as: u = Λ y Financial Risk Forecasting 2011,2016 Jon Danielsson, page 18 of 66

19 Properties of u u has the same dimensions as y Different rows are uncorrelated so we can run a univariate GARCH or a similar model on each row in u separately to obtain its conditional variance forecast, denoted by ˆD t where ˆD t = ˆσ t, ˆσ t,k Financial Risk Forecasting 2011,2016 Jon Danielsson, page 19 of 66

20 Obtain the forecast of the conditional covariance matrix of the returns by: ˆΣ t = ΛˆD t Λ Implies that covariance terms can be ignored when modelling the covariance matrix of u The problem has been reduced to a series of univariate estimations Financial Risk Forecasting 2011,2016 Jon Danielsson, page 20 of 66

21 Large-scale implementations It is possible to construct the conditional covariance matrix for a very 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 2011,2016 Jon Danielsson, page 21 of 66

22 Consider an example... Your financial institution has the following securities: Financial Risk Forecasting 2011,2016 Jon Danielsson, page 22 of 66

23 Consider an example... Your financial institution has the following securities: σ 2 E Financial Risk Forecasting 2011,2016 Jon Danielsson, page 23 of 66

24 Consider an example... Your financial institution has the following securities: σe 2 σb 2 Financial Risk Forecasting 2011,2016 Jon Danielsson, page 24 of 66

25 Consider an example... Your financial institution has the following securities: σ 2 E σ 2 B σ 2 E Financial Risk Forecasting 2011,2016 Jon Danielsson, page 25 of 66

26 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 2011,2016 Jon Danielsson, page 26 of 66

27 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 2011,2016 Jon Danielsson, page 27 of 66

28 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 2011,2016 Jon Danielsson, page 28 of 66

29 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 2011,2016 Jon Danielsson, page 29 of 66

30 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 2011,2016 Jon Danielsson, page 30 of 66

31 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 2011,2016 Jon Danielsson, page 31 of 66

32 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 2011,2016 Jon Danielsson, page 32 of 66

33 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 2011,2016 Jon Danielsson, page 33 of 66

34 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 2011,2016 Jon Danielsson, page 34 of 66

35 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 2011,2016 Jon Danielsson, page 35 of 66

36 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 2011,2016 Jon Danielsson, page 36 of 66

37 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 2011,2016 Jon Danielsson, page 37 of 66

38 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 2011,2016 Jon Danielsson, page 38 of 66

39 Constant Conditional Correlations and Dynamic Conditional Correlations Models Financial Risk Forecasting 2011,2016 Jon Danielsson, page 39 of 66

40 Separate out correlation modelling from volatility modelling within a GARCH-type framework The estimation can be divided into two parts: one for the correlation matrix and the other for the variances Two such models to be discussed: CCC and DCC Financial Risk Forecasting 2011,2016 Jon Danielsson, page 40 of 66

41 Constant conditional correlations (CCC) Model where time-varying covariances are proportional to the conditional standard deviation In the CCC model the conditional covariance matrix ˆΣ t consists of two components that are estimated separately: sample correlations ˆR the diagonal matrix of time-varying volatilities ˆD t Financial Risk Forecasting 2011,2016 Jon Danielsson, page 41 of 66

42 Then, the covariance forecast is given by: ˆΣ t = ˆD t ˆR ˆDt The volatility of each asset σ t,k follows a GARCH process or any of the univariate models discussed in the Chapter 2 Financial Risk Forecasting 2011,2016 Jon Danielsson, page 42 of 66

43 Usefulness of the CCC model: 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 2011,2016 Jon Danielsson, page 43 of 66

44 Usefulness of the CCC model: 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 Drawbacks of the CCC model: The assumption of correlations being constant over time is at odds with the vast amount of empirical evidence supporting nonlinear dependence Consequently, the DCC model below is preferred to the CCC model Financial Risk Forecasting 2011,2016 Jon Danielsson, page 44 of 66

45 Dynamic conditional correlations (DCC) DCC model is an extension of the the CCC Let the correlation matrix ˆΣ t be time dependent, so ˆR t is composed of a symmetric positive definite autoregressive matrix ˆQ t : ˆR t = ˆQ t ˆQ t with ˆQ t given by: ˆQ t = (1 ζ ξ)q +ζy t 1Y t 1 +ξ ˆQ t 1 where Q is the (K K) unconditional covariance matrix of Y; and ζ and ξ are parameters to be estimated such that ζ,ξ > 0 and ζ +ξ < 1 to ensure positive definiteness and stationarity, respectively Financial Risk Forecasting 2011,2016 Jon Danielsson, page 45 of 66

46 Usefulness of the DCC model: Large covariance matrices can be consistently estimated using DCC two-step estimation technique without requiring too much computational power Financial Risk Forecasting 2011,2016 Jon Danielsson, page 46 of 66

47 Usefulness of the DCC model: Large covariance matrices can be consistently estimated using DCC two-step estimation technique without requiring too much computational power Drawbacks of the DCC model: Parameters ζ and ξ are constants, which implies that the conditional correlations of all assets are driven by the same underlying dynamics - often an unrealistic assumption Financial Risk Forecasting 2011,2016 Jon Danielsson, page 47 of 66

48 Estimation Comparison Financial Risk Forecasting 2011,2016 Jon Danielsson, page 48 of 66

49 (a) Prices 200 MSFT 180 IBM Financial Risk Forecasting 2011,2016 Jon Danielsson, page 49 of 66

50 (b) Returns 15 % MSFT 10 % 5 % 0 % 5 % 10 % 15 % 10 % 5 % 0 % 5 % 10 % 15 % IBM Financial Risk Forecasting 2011,2016 Jon Danielsson, page 50 of 66

51 (c) Correlations with average correlation 49% 80% 60% 40% 20% 0% 20% EWMA Financial Risk Forecasting 2011,2016 Jon Danielsson, page 51 of 66

52 (c) Correlations with average correlation 49% 80% 60% 40% 20% 0% 20% EWMA DCC Financial Risk Forecasting 2011,2016 Jon Danielsson, page 52 of 66

53 (c) Correlations with average correlation 49% 80% 60% 40% 20% 0% 20% EWMA DCC OO Financial Risk Forecasting 2011,2016 Jon Danielsson, page 53 of 66

54 Let s focus on 2008, the midst of the crisis, when the correlations of all stocks increased dramatically... Financial Risk Forecasting 2011,2016 Jon Danielsson, page 54 of 66

55 (a) Prices 100 MSFT IBM Jan Mar May Jul Sep Nov Jan Financial Risk Forecasting 2011,2016 Jon Danielsson, page 55 of 66

56 (b) 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 2011,2016 Jon Danielsson, page 56 of 66

57 (c) Correlations with average correlation 49% 80% 60% 40% 20% EWMA Jan Mar May Jul Sep Nov Jan Financial Risk Forecasting 2011,2016 Jon Danielsson, page 57 of 66

58 (c) Correlations with average correlation 49% 80% 60% 40% 20% EWMA DCC Jan Mar May Jul Sep Nov Jan Financial Risk Forecasting 2011,2016 Jon Danielsson, page 58 of 66

59 (c) Correlations with average correlation 49% 80% 60% 40% 20% EWMA DCC OO Jan Mar May Jul Sep Nov Jan Financial Risk Forecasting 2011,2016 Jon Danielsson, page 59 of 66

60 Multivariate Extensions of GARCH Financial Risk Forecasting 2011,2016 Jon Danielsson, page 60 of 66

61 The BEKK model The most widely used alternative to the MVGARCH models available is the BEKK model Engle and Kroner (1995): the matrix of conditional covariances is 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 2011,2016 Jon Danielsson, page 61 of 66

62 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 2011,2016 Jon Danielsson, page 62 of 66

63 The general BEKK(L 1,L 2,K) model is given by: Σ t = ΩΩ + + K L 1 A i,k Y t i Y t ia i,k k=1 i=1 K L 2 B j,k Σ t jb j,k k=1 j=1 Financial Risk Forecasting 2011,2016 Jon Danielsson, page 63 of 66

64 The general BEKK(L 1,L 2,K) model is given by: Σ t = ΩΩ + + K L 1 A i,k Y t i Y t ia i,k k=1 i=1 K L 2 B j,k Σ t jb j,k k=1 j=1 The number of parameters in the BEKK(1,1,2) model is K(5K +1)/2 (i.e., 11 in the two-asset case) Financial Risk Forecasting 2011,2016 Jon Danielsson, page 64 of 66

65 Usefulness of the BEKK model Allows for interactions between different asset returns and volatilities Relatively parsimonious in terms of parameters required Financial Risk Forecasting 2011,2016 Jon Danielsson, page 65 of 66

66 Usefulness of the BEKK model Allows for interactions between different asset returns and volatilities Relatively parsimonious in terms of parameters required Drawbacks of the BEKK model Too many parameters that do not directly represent the impact of lagged squared returns or lagged volatility forecasts on elements of the covariance matrix - parameters may be hard to interpret Many parameters are often found to be statistically insignificant, which suggests the model may be overparametrized Financial Risk Forecasting 2011,2016 Jon Danielsson, page 66 of 66