The purpose of this chapter is to demonstrate how the notion of “proof” can be used to resolve issues of valuation in finance and how the method of simultaneous equations can be applied to determine the value of shares in two firms that hold an investment of shares in one another at the same time. The reader will be introduced to the notion of proof by arbitrage as it is was first pioneered in modern finance by Modigliani and Miller and then its application in providing guidance to practitioners of valuation will be explored.
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“A postulate of sound investing is that an investor does not pay more than for an asset than it’s worth” (Damodaran, 2002, p.1). The price paid should reflect the future cash flows that are expected to be generated from holding the asset. The source of these cash flows are the expected earnings stream that is generated from the operating assets employed by the firm for which the asset has a claim over and, as such when appraising the price, investors should consider these along with the obligations that a firm has to other security holders. A sound methodology in which to evaluate a firm and its securities is therefore required and a wide array of models and techniques are used in practice.
These range from the simple to the complex and each relies on a range of assumptions which lead to outcomes that may vary significantly. For example, the discounted cash flow model relies on estimates of expected future cash flows that are then discounted to a present value. A simple approach to model cash flows is to assume the current level of cash flows will occur in perpetuity, whereas a more complex approach will entail a detailed analyst in which individual items of cash flow together with their sources are considered and then aggregated on an annual basis. In other approaches, such as relative valuation models, estimates of future performance are not required and a value is determined from observation of the price of a comparable asset and its relationship with an accounting variable, such as earnings per share.
As such, the choice of model type and the underlying assumptions will determine the outcome and employing more than one model will lead to a disparity in values. Wishing to avoid paying a price for a security that is more than it is worth, practitioners may adopt techniques that minimise this variation and to provide a verification that the outcome is reasonable.
One approach is to employ a number of these methodologies to be used to act as a cross-check but ultimately the determination of value is a matter of judgement. A framework which enables the user to reconcile the outcome of one method against another and provide a justification for implementing a preferred method or set of assumptions is more useful to investors than comparing the outcomes from a set of disparate assumptions or models. As such this chapter is interested in how mathematics assists in this endeavour.