Three complementary dependence modelling approaches are developed in this thesis. The first two approaches address the challenge of modelling the multivariate distribution of a portfolio of asset returns. The third approach developed concerns commodity price dependence modelling where the link between maturities through the term structure of futures prices is considered.
The first approach adopted in this thesis is a statistical framework with a high degree of sophistication, however its fundamental reasoning and justification is indeed analogous in nature to the ideas considered when investigating the
“equity risk premium puzzle” coined by Mehra and Prescott [1985] in the late
80’s. The equity risk premium puzzle effectively refers to the fact that demand for government bonds which have lower returns than stocks still exists and generally remains high. This poses a puzzle for economists to explain why the magnitude of the disparity between the returns on each of these asset classes, stocks versus bonds, known as the equity risk premium, is so great and therefore implies an implausibly high level of investor risk aversion. In the seminal paper written by
Rietz [1988], the author proposes to explain the “equity risk premium puzzle”,
Mehra and Prescott[1985], by taking into consideration the low but still significant
probability of a joint catastrophic event.
Analogously in this thesis, an exploration is presented of the highly leveraged arbitrage opportunities in currency carry trades that arise due to violation of the UIP. However, it is conjectured that if the assessment of the risk associated with such trading strategies was modified to adequately take into account the potential for joint catastrophic risk events accounting for the non-trivial probabilities of joint adverse movements in currency exchange rates, then such strategies may not seem so profitable relative to the risk borne by the investor. A rigorous probabilistic model is proposed in order to quantify this phenomenon and potentially detect when liquidity in FX markets may dry up. This probabilistic measure of depen- dence can then be very useful for risk management of such portfolios but also for making more tractable the valuation of structured products or other derivatives indexed on this specific strategy. To be more specific, one of the principal contri- butions of this thesis is indeed to model the dependences between exchange rates
1. INTRODUCTION
using a flexible family of mixture copulae comprised of Archimedean members. This probabilistic approach allows the joint distribution of the vectors of random variables, in this case vectors of exchange rates log-returns in each basket of currencies, to be expressed as functions of each marginal distribution and the copula function itself.
Whereas in the literature mentioned earlier, the tail thickness resulting from the carry trade has been either treated individually for each exchange rate or through the measurement of distribution moments that may not be adapted to a proper estimation of the tail dependences. In this thesis, it is proposed instead to build, on a daily basis, a set of portfolios of currencies with regards to the interest rate differentials of each currency with the US dollar. Using a mixture of copula functions, a measure of the tail dependences within each portfolio is extracted and finally the results are interpreted. Among the outcomes of this study, it is demonstrated that during the crisis periods, the high interest rate currencies tend to display very significant upper tail dependence. Accordingly, it can thus be concluded that the appealing high return profile of a carry portfolio is not only compensating the tail thickness of each individual component probability distribution but also the fact that they tend to occur simultaneously and lead to a portfolio particularly sensitive to the risk of drawdown. Furthermore, it is also shown that high interest rate currency portfolios can display periods during which the tail dependence gets inverted demonstrating when periods of construction of the aforementioned carry positions are being undertaken by investors.
This thesis also explores the impact of speculative trading behaviour on the dependence structure of currency returns. The ratio of speculative open interest (net non-commercial positions) to total open interest, termed the SP EC factor, is shown to provide a good proxy to the behaviour of carry trade investors via a PCA analysis. A covariance regression modelling approach whereby the influence of observed covariates on the covariance of the multivariate returns of a basket of assets is proposed. In particular, the impact of speculative trading behaviour, i.e. the SP EC factors, on the covariance of carry currencies is investigated. These SP EC factors are shown to hold several orders of magnitude more explanatory
power than the price index factors, DOL and HM LF X, previously suggested in
the literature. Furthermore, it is demonstrated that the time series for the DOL
and HM LF X factors are very close to white noise and as such are essentially
unforecastable. The suggested speculative open interest factors are shown to be amenable to ARIMA model fits and so produce reasonable forecast accuracy.
Thus, time series models for these covariates of interest are built and hence forecasts of the covariance of a basket of currencies can be obtained. Therefore, the inherent heteroskedasticity of the covariance of a basket of currencies can be modelled and forecast whilst maintaining the desirable property of interpretability of the model. This forecasting ability is then useful for risk management, portfolio optimisation and trading strategy development.
A sensitivity analysis of the covariance to the factors is also presented allow- ing the estimation of a confidence interval of the covariance matrix entries as a function of the marginal distribution of each covariate used for the covariance regression. In addition, a regression of the tail dependence measures, obtained from the mixture copula modelling approach, on the SP EC factors illustrates the influence of carry trade speculative behaviour on the extremal joint currency
returns. The DOL and HM LF X are shown to hold little explanatory power in
the joint tails.
In this thesis, I also investigate financial time series dependence structure in commodity markets. The dynamic behaviour of the futures price term structure which combines time series and cross-sectional data has been modelled in this thesis using a so-called Hybrid Multi-Factor (HMF) model. This state-space modelling framework is proposed in order to understand the key factors driving commodity prices. A consistent estimation framework is developed, which builds on the familiar two-factor model of Schwartz and Smith (2000), to allow for an investigation of the influence of observable covariates on commodity prices. Using this novel Hybrid Multi-Factor (HMF) model, it is possible to obtain closed form futures prices under standard risk neutral pricing formulations, and one can incorporate state-space model estimation techniques to consistently estimate both the structural features related to the convenience yield and spot price dynamics (long and short term stochastic dynamics) and also the structural parameters that relate to the influence on the spot price and the futures price term structure of the
1. INTRODUCTION
observed exogenous covariates. Such models can then be utilised to gain significant insight into the futures and spot price dynamics in terms of interpretable observed factors that influence speculators and hedgers differently. This is not attainable with existing modelling approaches.
The proposed HMF modelling framework reconciles two classes of model: the latent multi-factor stochastic differential equation (s.d.e.) models and the alternative class of observable regression econometric factor models, by doing so in a statistically consistent manner from interpretation and estimation perspectives. The novel class of stochastic HMF models developed in this thesis allows for incorporation of exogenous covariate structures in a statistically rigorous manner. Such models are a genuine combination of the two approaches and do not presume any prevalence from one approach or the other. The crux of the matter lies in building a state-space model which allows a one-stage estimation with simultaneous inference of the latent factors dynamic and the covariates coefficients. In order to overcome the estimation error associated to the two-stage approach generally proposed in the literature. In such a two-stage model, typically the latent factor estimates are first extracted in order to later regress as a function of a set of covariates. This conditional estimation of the latent factor suffers from several flaws compared to the conditional estimates proposed in this thesis.
The HMF modelling framework also allows one to consider covariate forecasts in order to extrapolate values for the futures prices along the term structure while considering the confidence interval associated to this estimate. This is particularly convenient in risk management and commodity hedging as one needs to consider not only the amount to invest but also the uncertainty associated to this measurement.
1.4
Thesis Structure
This thesis is structured as follows: Part I introduces the copula modelling
framework and its novel application to investigate asymmetric tail dependence
in currency carry trade portfolios. Part II introduces the covariance regression
framework and its novel application to investigate how observable and interpretable
explanatory factors influence the covariance structure of currency returns. PartIII
introduces portfolio optimisation techniques and then utilises the novel covariance
forecasting approach developed in PartII to investigate portfolio optimisation in
currency carry portfolios. Finally, Part IVintroduces a novel Hybrid Multi-Factor
(HMF) stochastic differential equation (s.d.e.) framework to model the term structure dynamic of commodity futures prices.
1. INTRODUCTION