Temporal aggregation of equity return time-series models

In many economic and financial applications, the problem of temporal aggregation arises when data are observed at a lower frequency than the data generation frequency of the underlying model. The resulting observed data, which is referred to as the aggregate series, contain less information, and may lead to a distorted view of the true model, leading to potential errors in decision making. Therefore, the study of the effects of temporal aggregation is important for making proper decisions that are based on aggregate data. This paper examines the effect of temporal aggregation on five classes of stochastic equity return models that are commonly used in actuarial practice. Analytical formulae for the linkages between some aggregated and disaggregated stochastic models are presented. If a model is closed under temporal aggregation, the parameters of the lower frequency model can be directly implied by the higher frequency (i.e., more data) model. With large volatility observed in stock markets around the world over the last few years, many actuaries are now being urged to employ stochastic models to measure the solvency risk generated from insurance products with equity-linked guarantees. There are a large number of potential stochastic models for equity returns. Insurance regulators, both in Europe and North America, normally do not restrict the use of any stochastic model that reasonably fits the historical baseline data. However, in the U.S. and Canada, the final model must be calibrated to some specified distribution percentiles. The emphasis of the calibration process remains on the tails of the equity return distribution over different holding periods. The results in this paper are particularly useful when we need to study distributions and dynamics of longer term (for instance, more than 10 years) accumulation factors for equity-linked insurance products. The data on 10-year (or longer) accumulation factors is often very limited, even for the mature markets in Europe and North America. The advantages of choosing a closed (under temporal aggregation) class of processes for modelling asset returns and equity-linked guarantees are discussed. Actuarial applications of temporal aggregation using S&P500 total return data are given.