## A theory based stochastic investment model for actuarial use

Master Thesis

2007

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University of Cape Town

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This thesis reviews the origins, development and uses of asset-liability modelling, as well as existing largely stochastic investment models, notably those of the Maturity Guarantees Working Party (1980), Wilkie (1986,1995) and Thomson (1996). A stochastic investment model is developed which describes returns from equities, bonds and cash, as well as inflation and economic growth. The model is consistent with economic theory, adequately fits past data, and is relatively parsimonious compared with other models. A series of assumptions about the causal relationships between inflation, economic growth and interest rates are made based on standard economic theory. It is noted that consensus does not exist on some of the economic theory. Similarly a series of assumptions on the pricing of assets are made based on financial economic theory on market efficiency, expectations and asset pricing. Notably, it is assumed that financial markets are efficient. An economic model is described for inflation, economic growth and interest rates based on the set of assumptions. Each variable is modelled such that its value in one period is a function of its value in the previous period, the value of the other economic variables in the current and previous period, and a normally distributed residual. The model is a mixture of a random walk and autoregressive process that has two special cases of a (non-mean-reverting) pure random walk, and a (mean-reverting) pure autoregressive process. A financial market model is described for bond and equity returns based on the set of assumptions. Expected returns are derived from the expected real interest rate plus a risk premium, where the risk premium is linearly related to the standard deviation of real return. Bond yields are modelled as the sum of expected future short term real interest rates, expected future inflation, and a risk premium. Share prices are modelled as the present value of expected future distributable earnings, discounted at a rate equal to the sum of expected future short term real interest rates, expected future inflation, and a risk premium. The growth in earnings per share is modelled as the sum of inflation, real economic growth and a normal residual, and is also linked to real interest rates. Dividends are modelled as a smoothed function of earnings, with unit-gain from earnings to dividends. Annual data for a 15 year period is used to parameterise the model for the United States, Britain and South Africa respectively. The modelled volatilities of financial market returns, together with the economic data, are used to fit the economic model. The procedure is similar to the method of moments for statistical estimation. Parameters in the economic model that are not statistically significant or are not consistent with the assumptions are excluded. It was found that neither the random walk nor the autoregressive special case models could adequately explain observed volatility in financial markets, so the general case (mixture model) was adopted for economic variables. The parameterised models for the three countries studied exhibited a ""cascade structure"" where all variables are a function of one or two ""driving variables"", without any circularity/""feedback"". The models for the United States and Britain all have inflation as the driving variable, whereas the South African model has both inflation and economic growth as driving variables. The model achieves the objectives of consistency with economic theory as well as parsimony (when compared to Wilkie (1995)). With regards to the criterion of producing reasonable output, the model has advantages over existing models. These include that financial market returns simulated by the model are non-normal and exhibit significant leptokursis (fat-tails) with higher probabilities of severe down-market returns than are predicted by normal or log-normal distributions. Simulated returns also exhibit the weak and slow mean reversion that is observed in markets, and the simulated yield curve exhibits non-parallel shifts and inversions. However, simulated interest rates (particularly nominal interest rates), and even bond yields can become negative, although the probability of negative nominal interest rates is small in the model, and that of negative bond yields is negligible. Two areas where a good fit was not achieved were in the models of risk premiums and dividends. It is recommended that alternative approaches for estimating risk premiums be used. The poor fit to dividend data is not regarded as a significant weakness because modelled equity returns are not dependent on dividends.

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Includes bibliographical references (leaves 73-79).

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Howie, R. 2007. A theory based stochastic investment model for actuarial use. University of Cape Town.