A hyperbolic model of optimal cash balances

ABSTRACT

We develop a hyperbolic cash management model based on the Pearson Type IV probability density which minimises extreme variations in firm cash balances. Since the moments for the Type IV probability density are in general undefined and maximum likelihood estimation is compromised by the non-algebraic nature of the Type IV normalising constant, parameter estimation is implemented using the minimum method. Empirical analysis shows that the Type IV density is highly compatible with the quarterly cash flow data of a randomly selected sample of 100 large U.S. corporations. In contrast, around 60% of the 100 corporations return Jarque–Bera test statistics which are incompatible with the Gaussian probability density.

KEYWORDS:

• Cash balance
• hyperbolic
• Hamilton–Jacobi–Bellman equation
• Pearson Type IV probability density

Acknowledgements

The authors gratefully acknowledge the very helpful suggestions and criticisms of the Editor and referees.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. Jones and Pewsey (2009, 761–762) refer to the particular functional forms employed in our analysis as the sinh–arcsinh class of transformations.

2. The European sovereign debt crisis under which the European Central Bank provided loans of more than one trillion euro to maintain money flows between European banks, demonstrates the devastating effect that extreme variations in the availability of cash can have on commercial and industrial activities.

3. Mood, Graybill, and Boes (1974, 286–287) refer to this procedure as the “minimum distance method”. Avni (1976) and Berkson (1980) develop its major mathematical properties and, in particular, compare its efficiency with maximum likelihood parameter estimation procedures. Guo et al. (2016) and Chen et al. (2017) give a detailed exposition of how the minimum method may be empirically implemented.

4. The Jarque and Bera (1980, 256) test is based on the null hypothesis of Gaussian distributed data against the alternative hypothesis that the data are generated by one of the Pearson family of probability densities. This means the Jarque–Bera test will have maximum asymptotic power in discriminating between the random walk (that is, Gaussian) distributional assumptions on which the Miller and Orr (1966) cash management model is based (Karlin and Taylor 1981, 342) and the Pearson Type IV probability density on which the cash management model developed here is based.

5. Bartlett (1955, 83) considers the class of stochastic differential equations of the following general form: where β(x) and α(x) are analytic functions and dq(t) is a white noise process with unit variance parameter. Bartlett (1955) then applies Taylor expansions to β(x) and α(x) which can be truncated at any given order of approximation. Our analysis is based on a linear approximation for β(x) and a quadratic approximation for α(x). However, the quadratic approximation for α(x) is stated in such a way as to permit the use of the inverse sinh transformations of Jones and Pewsey (2009) to simplify the Fokker–Planck equation and thereby facilitate the determination of the probability density associated with the truncated interpretation of the stochastic differential equation. The inverse sinh transformation applied in our analysis is in fact a parsimonious interpretation of the Lamperti Transform of Møller and Madsen (2010, 11).

6. See Keynes (1936, 194–199) for a detailed summary of the factors which influence the determination of D. These include what Keynes broadly defined as the transactions, precautionary and the speculative demand for money.

7. The envelope condition is determined from the requirement: Moreover, since the envelope condition represents the abscissa of a minimum.

8. Without this condition, the differential equation will lead to a non-convergent distribution function and a stationary probability density will not exist (Karlin and Taylor 1981, 221).

9. See Heinrich (2004, 4) for a more detailed discussion of this issue.

10. With a few exceptions, 2006 is the earliest date from which quarterly cash and cash equivalent balances data are available on the Compustat file.

11. Note how equation (35) shows that the Cramér–von Mises goodness-of-fit statistic, T₃, is based purely on the vertical distance between the hypothesised distribution function and the empirical distribution function as derived from the ordered random sample of cash and cash equivalent balance observations, w₁, w₂, w₃, … , w_N (Mood, Graybill, and Boes (1974, 286-287). Thus, the determination of the test statistic, T₃, does not require differentiation of the likelihood function (as in maximum likelihood) or the estimation of possibly non-convergent moments (as with the Generalised Method of Moments).

12. Simulation results summarised by Jones and Pewsey (2009, 772–774) show that the Jarque–Bera test “has the best overall performance” in comparison with six other widely used tests of compatibility with the Gaussian probability density.