Principal component analysis and conditional variance

In this chapter the value-at-risk framework that is used to assess freight risk for a shipping portfolio that consists of distinct multi-freight routes is an extension of the work carried out in chapter four for single VaR measures. Furthermore, as this thesis is concerned with studying a variety of single and multi-state conditional volatility methods to capture the dynamics of freight returns and better judge the impact on VaR measures for a portfolio of multi-tanker routes. The use of a powerful statistical tool such as principal component analysis (PCA), which is capable of reducing the dimensions of a system of assets returns to estimate risk factors for a multi-freight routes portfolio, is logical.

117 | P a g e

Models such as PCA are commonly used to assess risk for financial portfolios and hence to provide the risk adjusted performance measures that are used for banker investments, Alexander (2008a). Therefore, a principal component representation of our tanker portfolio is derived from the percentage of return for each tanker freight returns that constitute the portfolio, through an eigenvector analysis of a very large covariance matrix of freight returns within the portfolio. The ability of PCA to express relationship patterns and capture the volatility dynamics of the data set is down to its decomposition technique which is perfect for analysing a correlation structure for a set of assets returns. Furthermore, the attractiveness of this decomposition technique is the fact that it deals with a reduced number of factors that represent a large set of data within a portfolio without a significant loss of information.

In this chapter we make use of a reduced set of principal components with GARCH conditional variance model to extract patterns from a portfolio of tanker freight returns. This is known as the Orthogonal GARCH framework introduced by Alexander and Chibumba (1996) and Alexander (2001b) and extended by van der Weide (2002). Therefore, if we consider a data set of returns with zero mean summarized in a T × n

matrix X and suppose we perform PCA on V the covariance matrix of X. Thus, the principal components of V are the columns of the T × n matrix P defined by:

(5.6)

where the W is the n × n Orthogonal matrix of eigenvectors of V and W that are ordered so that the first column of W is the eigenvector corresponding to the largest eigenvalue of V, and so on. Following the work of Alexander we consider using only a reduced set of principal components, where the first k principal components of freight returns are the first k columns of P, in which these columns are represented in the T × k matrix P*. Thus, a principal component approximation can be represented as:

(5.7)

The variance of 5.7 is:

118 | P a g e

where W* is the n × k matrix in which k columns are given by the first k eigenvectors. The accuracy of the approximation is positively correlated with the value of k. is the

m × m returns conditional covariance matrix at time t and is a k × k diagonal covariance matrix of the conditional variances of the principal components. Hence, according to Alexander the full m × m matrix with different elements is obtained from just k different conditional variance estimates, with where N is the number of Orthogonal transformation. Furthermore, the Orthogonal GARCH model requires estimating k separate univariate GARCH models, one for each principal component conditional variance in . With always positive definite for the O- GARCH matrix and always positive semi-definite and can be expressed as:

(5.9)

where . Since Y is zero for some non-zero X and not strictly positive definite, but positive semi-definite.

Therefore, let us consider that freight returns for the different routes under investigation are included in a vector stochastic process of dimension N × 1 and conditional on past information t-1. Thus, a symmetric O-GARCH model applied to a portfolio of freight returns with a number of principal component vectors k is defined as:

(5.10)

(5.11)

(5.12)

where , with the population variance of and is a matrix of dimension N × m given by:

(5.13)

where being m the largest eigenvalues of the population correlation matrix and covariance matrix of and , respectively. With the N × m matrix of associated eigenvectors and the vector is a random process such that

119 | P a g e

(5.14)

Consequently,

(5.15)

where , and V and are the model parameters and ‟s

and ‟s are the GARCH factors parameters. In practice V and are replaced by their sample counterparts and m is normally chosen by principal component analysis applied to the standardised residuals .

The O-GARCH model is estimated using a constrained maximum likelihood (ML) approach known as a quasi-likelihood function, where a vector stochastic process for is a realisation of the data generating process, with a conditional mean, conditional variance matrix and conditional distribution are respectively ,

and where is a r-dimensional parameter vector and

is the vector that contains the parameters of the distribution of the innovations . Thus, to estimate we maximise the likelihood function for the T

observations with respect to the vector of parameters where:

(5.16)

with

where the density function denotes the auxiliary assumption of

i.i.d for the standardized innovations . is a vector of nuisance parameters and the

likelihood function is expressed as:

(5.16)

With respect to the rejection of the normality assumption in the literature, especially for daily and weekly data, Bollerslev and Wooldridge (1992) have shown that a consistent estimator of may be obtained by maximizing (5.16) with respect to even if the data generating process is not conditionally Gaussian, arguing that the quasi- maximum likelihood (QML) is consistent provided that the conditional mean and the

120 | P a g e

conditional variance are specified correctly. For more details see Alexander (2008a) and the references within.

Finally, in respect to diagnostic tests, in comparison to univariate volatility models, specific tests are limited for multivariate volatility models. Thus, there are two approached to running diagnostic tests. On the one hand, one can choose form the huge body of diagnostic tests devoted to univariate models, where each time series is independently diagnostically tested. On the other hand, one can choose from the few available tests for multivariate models by diagnosing a vector representation of the whole system. In this thesis we feel that the diagnostic tests conducted in chapter four for the different proposed univariate conditional variance models are adequate.