Application of levy processes to unitised with-profits policies
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The objective of this thesis is to develop more realistic long term asset models based on L´evy processes and discuss their applications to risk management of unitised with-profits policies. We investigate the behaviour of long-term returns of the UK total share return index by testing the common statistical properties for financial data, so-called “stylised facts”. We show that for the monthly U.K. share total return indices, the Gaussian return hypothesis is rejected in series of tests. The local distribution of the returns has higher kurtosis and heavier tails than the Gaussian. In addition, the returns series show significant nonlinear autocorrelation, extreme returns appear in clusters. The first long term asset model purposed in this thesis is the exponential L´evy model with non-Gaussian increment. We describe the Generalised Hyperbolic distributions with their subclasses. They are considered as candidate distributions for the increments of the driving L´evy processes. We estimate model parameters to the UK share gross total return index using two approaches, maximum likelihood (MLE) and Markov Chain Monte Carlo (MCMC) algorithm. Statistical and graphical goodness-of-fit tests demonstrate that these L´evy driven models give more accurate fits to the historical equity index returns data. For the liability model we consider long term participating life insurance products specifically unitised with-profits contracts. The payouts of unitised with-profits policies are simulated under a variety of asset models driven by L´evy processes. At first a basic model policy is considered with limited insurer operations and no risk controls. We look into various risk measures of the maturity loss for the insurer xiii and compare the statistical properties for different non-Gaussian increment L´evy models. It is found that the classical Gaussian model substantially underestimates the risk measures in unitised with-profits policies. The L´evy driven models which have semi-heavy tailed increments are aggregate to normal distributions in the long run. Then we consider different retrospective bonus mechanisms by varying the participating rate and the smoothing period. As a comparison we use a bonus earning power method with deterministic projected maturity asset share and 25 percent terminal bonus cushion. We study the joint distributions of the maturity asset shares and guarantees under these two bonus mechanism. With similar risk measures, there are larger expected maturity guarantees under bonus earning power method than retrospective bonus. Declaring bonuses on a more frequent basis is then tested, which has the desired effect of reducing the risk measures when declaring monthly bonuses using bonus earning power mechanism. We make observations on two different investment strategies, a diversified investment strategy and a hedging based investment strategy. The former method tries to reduce the variance of the invest return distribution while the hedging investment strategy, on the other hand, narrows the left tail of the maturity loss distribution by paying an extra amount of expenses. Finally, the L´evy models are extended by using GARCH(1,1)-m type volatility. Both maximum likelihood estimators and Bayesian estimators using Markov Chain Monte Carlo are presented. The statistical tests on the devolatilised data show that the GARCH model reduces the non-linear autocorrelation in the conditional return processes and furthermore improve the fitting of the asset models. Also, multi-variable models are considered. Stochastic bridges driven by L´evy processes are constructed while the yearly returns follow the Wilkie model.