Knowledge that Transforms

Suresh P. Sethi, Gerald L. Thompson, Applications of Mathematical Control Theory to Finance: Modeling Simple Dynamic Cash Balance Problems, The Journal of Financial and Quantitative Analysis, Vol. 5, No. 4/5 (Dec., 1970), pp. 381-394

In recent years, large commercial banks in the United States have turned to so-called “liability management” as a way to acquire additional funds. The use of Certificates of Deposit (CD's), Eurodollar balances, debentures, and recently the issuing of commercial paper through their holding companies represent an extension of the more traditional secondary reserve asset management approach to bank liquidity and reserve adjustment.

It is often necessary to estimate a frequency function or certain points on a frequency function from very limited data. A usual procedure for this estimation involves two steps. From the set of “well-known” frequency functions, e.g., the normal, poisson, binomial, etc., one chooses that function which seems likely to best “fit” and then uses the available data to estimate the parameters of the chosen distribution. If no one “well-known” function can be chosen a priori, then perhaps several likely candidates are tried and the one which fits best according to some criterion is chosen. For many purposes, this procedure is quite unobjectionable.

The relationship between return on assets and their riskiness is one of the liveliest topics in financial literature. In his 1952 landmark article, Markowitz developed a mathematical model that captured this key financial concept. defined risk as the variance of the rate of return of a portfolio. Later, Sharpe hypothesized a positive linear relationship between expected rate of return on an asset and the risk premium associated with that asset. Subsequently, Sharpe tested this hypothesis empirically and found support for his theory. Although portfolio theory specifies the two parameters of this model as ex ante return and risk, Sharpe used ex post data for testing the risk-return relationship. designated the ex post mean rate of return obtained on an an asset as a proxy for expected return and the standard deviation of ex post annual rates of return as a surrogate for risk.

Recently, several researchers, including Evans, Archer [1], Latané, and Young [2], have performed empirical analyses of the relationship between the number of securities in a portfolio and the reduction in portfolio dispersion. In this note, an exact mathematical relationship between these two factors is presented.

In a recent article, Professors Robichek and Van Horne have noted the importance of considering abandonment in the capital budgeting process. The basic point of their paper is that:… a project should be abandoned at that point in time when its abandonment value exceeds the net-present value of the subsequent expected future cash flows discounted at the cost-of-capital rate.

Researchers and political analysts concerned with the inter-regional flow of mortgage funds have often pointed to the existence of yield differentials as prima facie evidence of misallocation of capital and national resources. Limited information and myopic lending horizons, with market imperfections reinforced by state laws and institutional segmentation, have been postulated. They are regarded as responsible for costly “frictions” in the export of capitalto the fast-growing, generally low-income, states, particularly those of the South. Both federal and state legislative action, intensified private arbitrage, and better secondary market facilities and instruments are then urged to improve inter-regional financial mediation to reduce or eliminate the yield differentials.

In two previous articles [11] and [12] a family of normative models of the individual's economic decision problem under risk was presented. At the same time, certain implications of these models with respect to individual behavior were deduced for a class of utility functions. This paper will show that these models also give rise to an induced theory of the formation and operation of firms under risk for the same class of utility functions.

This paper describes the development and testing of a model of bank portfolio selection that considers variability of both income and gross asset levels as the risks involved in banking. In particular, the model is formulated as maximizing expected profit subject to risk constraints on wealth losses and the availability of liquid assets. The parameters for these constraints include the covariance matrices of rates of return on bank assets and deposit changes. Included in the latter category are fluctuations in business loans which are treated as negative deposits because, due to deposit feedbacks and the like, these can reasonably be considered exogenous to the short-run portfolio decision. Basically, the model is an extension of Markowitz's work [11] to incorporate problems associated with portfolios that include assets having imperfect markets and liabilities that are not completely under the control of the economic unit. As such, it is applicable (with minor institutional modifications) to the entire spectrum of financial intermediaries.

In an earlier paper [2], the sequential decision process was applied to two major facets of credit management: (a) deriving unambiguous decision rules for handling individual credit requests; and (b) devising relevant credit indices for effective management control and evaluation of the system. Usefulness of the model was constrained by its static nature and by exogenous determination of other significant variables, notably, collection efforts and costs.