Harmonic analysis, time series variations and the distributional properties of financial ratios

This paper models the behaviour of financial ratios using the techniques of continuous time stochastic calculus. Previous work in the area has been restricted to models based on first order stochastic differential equations. However, the present paper models a ratio’s displacement from its long term mean in terms of a second order stochastic differential equation. In this way it shows that higher order equations may be used to provide more flexible modelling procedures than those previously studied. The paper begins by describing a ratio’s evolution in terms of a particular form of second order stochastic differential equation. A solution is then obtained for this equation. It is then shown how the parameters of the model may be estimated from a given set of empirical data. The model is then applied to a data set of four ratios for 118 UK companies covering a period of 37 years. For three of the four ratios, clear evidence of a period emerges. Previous work suggests that these ratios are well described by mean reversion processes. The empirical analysis here, however, suggests there is a significant tendency for adjustments to ‘overshoot’ the targeted long term mean. Taken together with prior work in the area, the paper begins to provide a broad picture of the way in which financial ratios evolve over time.