Stochastic modelling of long-term investment risks

There are many risks that individuals, firms, and societies have to face, and among them are the uncertainties of future investment variables, which include inflation (both of prices and earnings), interest rates, exchange rates, and returns on ordinary shares (including both dividend income and changes in capital values). These investment risks affect individuals in their own financial planning; affect companies in planning investment projects and in arrangements for raising capital; affect governments and government institutions that have to borrow in the capital markets; and especially affect investment institutions and intermediaries who take on borrowings, deposits, insurance contracts, or pension fund liabilities on the one hand and invest assets in loans, ordinary shares, property, or other investments on the other. A great deal of work done by financial economists in recent decades has established reasonable models for describing movements of many investment variables in the short run. Typically these models are based on a ‘random walk’ or Gauss–Wiener continuous diffusion process. This sort of model has been particularly valuable to market-makers and other investment participants whose time horizon is short. But these short-term models often do not provide a satisfactory structure for the long term. This presentation will describe some of the author's work in the statistical analysis of long-term investment series, both in the United Kingdom and in other countries, based on statistical time-series analysis of historical data. Although many of the series could be valued using multivariate methods, such as vector autoregressive (VAR) models, preliminary investigation showed that many of the series could be investigated in a ‘cascade’ fashion, with price inflation being put as the initial ‘driver’. A very long historic series shows long periods when changes in prices in successive years could be taken as random, with zero drift, and other periods (including most of this century) when inflation rates in successive years were correlated. A similar pattern has applied in recent years in many other countries. It is postulated that the prices of ordinary shares in aggregate are closely related to the dividends paid on them, so that the ratio between dividend and price, i.e. the dividend yield, is stationary – fluctuating around a constant mean. The dividend-yield series can be described by means of a first-order autoregressive time-series model, while the dividend series can be described by a model that depends on inflation in the current and preceding years, with an appropriate time lag. Interest rates, both long-term and short-term, are first decomposed into an allowance for prospective future inflation and a ‘real’ rate of interest, comparable to the yield on index-linked stocks. The real rate of interest can also be modelled as a mean-reverting autoregressive model. The allowance for future inflation can be derived as a moving average of past inflation rates. In order to link models for different countries, it is necessary to have a model for currency exchange rates. This can be done by postulating a hypothetical ‘purchasing-power parity’ exchange rate, which exactly allows for changes in inflation, and then by modelling the deviation of the actual rate from the hypothetical rate by means of yet another autoregressive model. It is necessary also to keep an appropriate structure for cross rates between any pair of currencies.