The Tyranny of Rounding Errors: The Mismatching of APV and the DCF in Perpetuities in Brealey and Myers 6th and 7th Edition of Principles of Corporate Finance

Abstract

In theory, different valuation methods, with consistent assumptions, must give identical results. Numerical examples that purport to illustrate the theory should demonstrate the identical results. Unfortunately, in popular textbooks it is all too easy to find numerical examples that are at odds with the theory. There are several possible explanations for the discrepancies. First, there might be some conceptual confusion about the underlying assumptions. Second, it could simply be "rounding errors." It is intellectual laziness to ascribe the discrepancies to the tyranny of rounding errors when in fact it is easy to show that rounding errors are not the reasons for the discrepancies.

It is common to read that different valuation methods give different results. For instance, Brealey and Myers (2000, 2003) say: "If the company's debt ratio is constant over time, the flow-to-equity method should give the same answer as discounting company cash flows at the WACC and subtracting debt." On the other hand, they say, "If financial leverage will change significantly discounting flows to equity at today's cost of equity will not give the right answer."

Inselbag and Kaufold, 1997, conclude that the APV is better than the DCF when the debt schedule is given. This is misleading in two senses: one, they mix methods because they disregard the possibility to solve the circularity posed by the relationship between value and discount rates and second, as a consequence, they say that "one must already have calculated the firm's value" in order to know the WACC.

In the latest edition of Principles of Corporate Finance (Brealey, Myers and Allen, 2006) the authors use a finite cash flow example to illustrate the valuation procedure for using the Discounted Cash Flow (DCF) method with the free cash flow (FCF) and the Adjusted Present Value (APV). The two firm values obtained are different. They say that the "... difference [...] is not a big deal considering all the lurking risks and pitfalls in forecasting [...] cash flows".

Once more, in this teaching note we show that the two methods give identical values when the proper discount rates are used.

Keywords: Cash flows, free cash flow, cash flow to equity, valuation, levered value, Adjusted Present Value, APV, Discounted Cash Flow, DCF, weighted average cost of capital, WACC, cost of unlevered equity, tax savings

JEL Classification: M21, M40, M46, M41, G12, G31, J33