Earnings growth rates in business valuation models: The impossible quaternity

We develop formulae for earnings growth rates in business valuation models that justify the timing of reinvestment. First, we show that the cross-reference in the calculation between the reinvestment rate (RIR) and return on invested capital (ROIC; i.e., the weighted average of invested capital) introduces valuation errors. We then explain the formulae of earnings growth rates to avoid errors in two situations: reinvestment at the end of each year and at any time. The study also shows the timing of capital reinvestment, where the false growth rate occurring due to cross-referencing will match the actual growth rate. We also study the case where the false growth rate is always higher or lower than the actual growth rate. We use numerical examples to show that our models are correct and highlight arising from cross-referencing. We provide a practitioner's guide for two scenarios: valuers directly estimate earnings growth rates and clients provide future earnings and related information. Finally, and most importantly, our results imply the principle of impossible quaternity for estimating earnings growth in the discounted cash flow (DCF) framework. More specifically, a business valuation cannot have an available expected growth rate, a fixed change in ROIC, an independent reinvestment timing, and a fixed level of actual reinvestment.