Bibliography: pages 100-108. / The salient feature under examination in this thesis is the assumption that the error terms, ZD(t) and Zy(t), are normally distributed. This assumption is common to most of the stochastic asset models that are in widespread use within the actuarial profession. An example is the well known Wilkie model (Wilkie (1984, 1995)).

Includes bibliographical references. / A regime-switching model allows a process to switch randomly between different regimes which have different parameter estimates. This study investigates the use of a two regime-switching model for inflation in South Africa as a means of determining a hedging strategy for inflation linked liabilities of a financial institution. Each regime is modeled using an autoregressive process with different parameters and the change in regimes is governed by a two state Markov chain. Once the parameters have been estimated, the predictive validity of the regime-switching process as a model for inflation in South Africa is tested and a hedging strategy is outlined for a set of inflation linked cash flows. The hedging strategy is to invest in inflation linked bonds, the number of which is determined through the use of a Rand-per-point methodology that is applied to the inflation linked cash flows and inflation linked bonds. Over the period from January 2008 to June 2013 this hedging strategy was shown to be profitable.

Stock option valuations have long been studied, being inherently non-linear financial derivatives. These instruments have a ubiquitous presence in institutional investment practice, and present many favourable and unique benefits to an investment portfolio. Neural Networks on the other hand have become a more familiar concept in recent times. They are by design set to deal with complex, non-linear classification and prediction tasks. Using Neural Networks to predict stock option prices has been studied at length, by various authors in the last 30 years. These studies have considered their relative performance against closed-form pricing solutions like the infamous Black-Scholes-Merton model, as well as in real-world settings. The collective conclusion that is deduced from past literature presents a clear case for their use in finance, albeit that there are some notable pitfalls, like the lack of interpretability and the ability to explicitly enforce certain constraints. Constraints such as option price bounds (upper and lower) and the Put-Call parity, that a stock option's value should satisfy have not been considered in many prior studies. This dissertation sets out to study stock option valuations using Neural Networks with techniques to enforce constraints. First, a functional and appropriately performing Neural Network configuration is derived that outputs European call and put option prices under one model. Thereafter, enforcement of the lower, upper and relative bounds (Put-Call parity) is incorporated into the model. Finally, the Neural Network application is extended to the real-world setting. The performance of the Neural Network model is assessed by means of mean error, as well as percentiles.

In developing countries, systems that collect vital statistics are usually inadequate to facilitate the direct estimation of adult mortality. This has necessitated the development of indirect methods such as the orphanhood method. These methods are however limited, i.e., the single-survey approach produces out of date estimates of mortality and the two-survey approach is affected by the differential reporting of orphanhood between two surveys. To avoid these limitations, this research considers an extension of the orphanhood approach pioneered by Chackiel and Orellana (1985) to estimate adult mortality using year-of-death data rather than the conventional form of the orphanhood data. This is because the year-of- death data can be used to produce accurate time locations to which estimates of mortality apply but more important, one can create a synthetic survey from a single survey and hence obtain more recent and accurate estimates of mortality. The single-survey orphanhood method is applied to survey data to obtain estimates of adult mortality and time location. A variation of the two-survey orphanhood method (Timxus 1991b) is also applied to survey data and the synthetic survey that is created from year-of-death data in order to derive estimates of adult mortality. In addition, the age range of respondents is extended down to age O to include year-of-death data from younger respondents on the assumption that underestimating orphanhood due to the adoption effect is minimal. This is done to investigate if the estimates derived from the two-survey method can be improved. Further, a cohort survival method that involves the calculation of a survival ratio for each age group at the first survey and the equivalent older ages groups at the second survey is applied to investigate the possibility of producing useful estimates of adult mortality based on cohort survival. The level and trend in mortality estimates calculated from the single-survey, two - survey and the cohort survival approaches are discussed and compared to the estimates from the Rapid Mortality Surveillance (RMS) which are used as a benchmark for the trend and level of adult mortality in South Africa. The estimates produced using the single-survey method appear too low, while those from the two-survey method appear to be reasonable for the conventional from of the orphanhood data. Extending the two-survey method to include younger respondents produces estimates that are too low indicating that both the conventional form of the orphanhood data and the year-of-death data suffer from the adoption effect. The cohort survival approach produces reasonable estimates that are consistent with the RMS benchmark for both the conventional form of the orphanhood data and year-of-death data.

Child mortality can be used to measure the level of social development as well as the health status of children (Hill 1991). By world regions, sub-Saharan Africa maintains the highest rates of under-five mortality. Current under-five mortality is estimated at 76 deaths per 1,000 live births (Hug, Sharrow, Zhong et al. 2018). In Zambia, under-five mortality reached a peak of 197 in 1996 and is currently estimated at 60 (Hug, Sharrow, Zhong et al. 2018). On the world health agenda, reducing child mortality has been made a priority, especially for low income countries that remain the most affected. Among the targets of the Sustainable Development Goals (SDGs) is reduction of neonatal mortality to at least 12 deaths per 1,000 live births and under-five mortality to 25 deaths by 2030 (United Nations 2015). HIV/AIDS is one of the leading causes of mortality in Zambia and has contributed to the slow decline of under-five mortality (Garenne and Gakusi 2006). Children under the age of five years get infected with HIV mainly through vertical transmission (Fishel, Ren, Barrère et al. 2014). In the absence of treatment, vertical transmission of HIV is high and can range between 15 and 45 per cent, reducing below 5 per cent with effective interventions (Barral, Oliveira, Lobato et al. 2014). Despite vertical transmission being the main pathway through which children get infected with HIV, little research has been done to determine the significance of maternal HIV status on under-five mortality in Zambia. The aim of the study was, therefore, to determine the extent to which mortality of children with HIV-positive mothers differs from that of children with HIV-negative mothers. The Zambia Demographic and Health Survey (ZDHS) data for 2007 and 2014 which contain HIV serotesting data were used. Survival analysis using Poisson regression was used to model the influence of maternal HIV status taking into account confounding factors. The results of the study indicate that maternal HIV status was significantly associated with child mortality in both survey periods but by 2013/14 the influence of maternal HIV status had reduced and was insignificant for children born within one year of the 2013/14 survey. The reduction in the risk of dying between the inter-survey period may be as a result of increased coverage of prevention of mother-to-child transmission (PMTCT) and antiretroviral therapy (ART) services over the years. In order to reach universal coverage, there is need for increased provision of PMTCT and ART treatments and support for HIV strategies such as the 90 90 90 target.

While working as an intern in the corporate pensions department of Rudd and Wisdom, Inc., Consulting Actuaries, I learned about the different kinds of pension plans available to businesses and became familiar with the process of benefit calculation. I performed several benefit calculations for clients under different scenarios and plan specifications. When a participant of a pension plan retires, terminates employment, dies, or becomes disabled, the participant (or his or her beneficiary) may request a calculation to illustrate benefit options. In each case, the methods used to perform such a calculation and the benefit options available to the participant or beneficiary are different. In this report, I will demonstrate a number of scenarios illustrating plan provisions and methods of calculation. / text

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Quota-share and stop-loss/excess-of-loss reinsurances are two important reinsurance strategies. An important question, both in theory and in application, is to determine optimal retentions for these reinsurances. In this thesis, we study the optimal retentions of quota-share and stop-loss/excess-of-loss reinsurances under ruin-related optimization criteria. We attempt to balance the interest for a ceding company and a reinsurance company and employ an optimization criterion that considers the interests of both a cedent and a reinsurer. We also examine the influence of interest, dividend, commission, expense, and diffusion on reinsurance retentions.

Quota-share and stop-loss/excess-of-loss reinsurances are two important reinsurance strategies. An important question, both in theory and in application, is to determine optimal retentions for these reinsurances. In this thesis, we study the optimal retentions of quota-share and stop-loss/excess-of-loss reinsurances under ruin-related optimization criteria. We attempt to balance the interest for a ceding company and a reinsurance company and employ an optimization criterion that considers the interests of both a cedent and a reinsurer. We also examine the influence of interest, dividend, commission, expense, and diffusion on reinsurance retentions.

This thesis was written with the aim of modelling the lactation process in dairy cows and goats by applying a hierarchical Bayesian approach. Information on cofactors that could possibly affect lactation is included in the model through a novel approach using covariates. Posterior distributions of quantities of interest are obtained by means of the Markov chain Monte Carlo methods. Prediction of future lactation cycle(s) is also performed. In chapter one lactation is defined, its characteristics considered, the factors that could possibly influence lactation mentioned, and the reasons for modelling lactation explained. Chapter two provides a historical perspective to lactation models, considers typical lactation curve shapes and curves fitted to the lactation composition traits fat and protein of milk. Attention is also paid to persistency of lactation. Chapter three considers alternative methods of obtaining total yield and producing Standard Lactation Curves (SLACâs). Attention is paid to methods used in fitting lactation curves and the assumptions about the errors. In chapter four the generalised Bayesian model approach used to simultaneous ly model more than one lactation trait, while also incorporating information on cofactors that could possibly influence lactation, is developed. Special attention is paid not only to the model for complete data, but also how modelling is adjusted to make provision for cases where not all lactation cycles have been observed for all animals, also referred to as incomplete data. The use of the Gibbs sampler and the Metropolis-Hastings algorithm in determining marginal posterior distributions of model parameters and quantities that are functions of such parameters are also discussed. Prediction of future lactation cycles using the model is also considered. In chapter five the Bayesian approach together with the Wood model, applied to 4564 lactation cycles of 1141 Jersey cows, is used to illustrate the approach to modelling and prediction of milk yield, percentage of fat and percentage of protein in milk composition in the case of complete data. The incorporation of cofactor information through the use of the covariate matrix is also considered in greater detail. The results from the Gibbs sampler are evaluated and convergence there-of investigated. Attention is also paid to the expected lactation curve characteristics as defined by Wood, as well as obtaining the expected lactation 254 curve of one of the levels of a cofactor when the influence of the other cofactors on the lactation curve has be eliminated. Chapter six considers the use of the Bayesian approach together with the general exponential and 4-parameter Morant model, as well as an adaptation of a model suggested by Wilmink, in modelling and predicting milk yield, fat content and protein content of milk for the Jersey data. In chapter seven a diagnostic comparison by means of Bayes factors of the results from the four models in the preceding two chapters, when used together with the Bayesian approach, is performed. As a result the adapted form of the Wilmink model fared best of the models considered! Chapter eight illustrates the use of the Bayesian approach, together with the four lactation models considered in this study, to predict the lactation traits for animals similar to, but not contained in the data used to develop the respective models. In chapter nine the Bayesian approach together with the Wood model, applied to 755 lactation cycles of 493 Saanen does collected during either or both of two consecutive year, is used to illustrate the approach to modelling and predicting milk yield, percentage of fat and percentage of protein in milk in the case of incomplete data. Chapter ten provides a summary of the results and a perspective of the contribution of this research to lactation modelling.