We will be using the Discounted Free Cash-Flow to Firm model (FCFF hereafter). As its name suggests, this model effectively discounts all the cash-flows that are/can be distributed to the shareholders and debt holders of the firm. Applying (1) to the firm, we get : \[EV_{t}=\displaystyle\sum_{i=1}^{K}\frac{FCFF_{t+i}}{(1+R)^i}+\frac{EV_{t+K}}{(1+R)^K}\] where \(EV_{t}\) is the enterprise value of the firm generating the free cash-flows (\(FCFF\)). \(R\) is the discount rate, or the return required by shareholders and debt holders. \(R\) is effectively the cost of capital for the firm and is equivalent to the concept of \(WACC\) used by the financial analyst community.
Secondly, we need to define the different components of the model. We follow Damodaran and use his definitions. We are quite aware that there exists numerous definitions for each of the following concepts but Damodaran’s definitions can serve as a starting point, where other definitions are refined versions of these basics definitions. Specifically, Damodaran posts articles on his blog (http://pages.stern.nyu.edu/ adamodar/) that summarize the courses he teaches at the Stern School of Business at New York University. In an article posted in 2013 entitled “A tangled web of values: Enterprise value, Firm Value and Market Cap” Damodaran gives a very clear definition of the accounting concepts we will be using. As a starting point, the balance sheet allows us to write the following accounting identity : \[Cash & Other Non-Operating Assets+Operating Assets=Debt+Equity+Short Term Liabilities\] Operating Assets are comprise of Fixed Assets, Intangible Assets and Working Capital, so that we have : \[Cash & Other Non-Operating Assets+Fixed Assets+Intangible Assets+Working Capital=Debt+Equity\]
If we define Enterprise Value as the market value of Debt and Equity minus the cash (and other non-operating assets) a firm holds, it appears clear from these accounting identities that Enterprise Value is the market value of the operating assets. We define Invested Capital as the book value of the operating assets, also equal to the book value of Debt and Equity. Following the Profitablity-Valuation framework based on earnings and equity, the challenge is to link the FCFF model defined in (9) to a Profitablity-Valuation relationship where valuation is a ratio that relates the market value of operating assets \(EV_{t}\) to the book value of the operating assets \(IC_{t}\).
Thirdly, we need to identify a certain number of accounting identities similar to the ones we used for the RIM in order to link cash-flow generation, the balance sheet and the market value of the balance sheet.
As its name suggests, the RIM hinges on the clean surplus accounting identity, where profits that are not distributed to shareholders are reinvested in the firm thus changing the value of the equity. Similarly, the cash generated by the firm that is not distributed to equity holders and debt holders is reinvested in the firm in net capital expenditures and non-cash working capital : \[FCFF_{t}=NOPAT_{t}-(NetCapex_{t}+\bigtriangleup WC_{t})\]
Where \(NOPAT_{t}\) is the Net Operating Profit After Tax at time \(t\) and \(NetCapex_{t}\) are the Capital Expenditures Net of Depreciation and \(\bigtriangleup WC_{t}\) the change in non-cash Working Capital also at time \(t\). Over one period, the change in Invested Capital \(IC\) is equal to : \[IC_{t}-IC_{t-1}=NetCapex_{t}+\bigtriangleup WC_{t}\] or \[IC_{t}=IC_{t-1}+NetCapex_{t}+\bigtriangleup WC_{t}\] Just as in the RIM approach, we introduce the notion of abnormal operating earnings : \[A_{t}=NOPAT_{t}-WACC \times IC_{t-1}\] where \(A_{t}\) are the abnormal operating earnings and \(WACC\) the Weighted Average Cost of Capital. We define abnormal operating earnings as earnings that are not discounted by shareholders and debt holders (\(WACC \times IC_{t-1}\)).
By combining Eq. 15 and Eq. 17, we get : \[IC_{t}=IC_{t-1}+NOPAT_{t}-FCFF_{t}\] And by replacing \(NOPAT\) by its equivalent in Eq.13 we get : \[IC_{t}=IC_{t-1}+A_{t}+WACC \times IC_{t-1}-FCFF_{t}\] Which gives : \[IC_{t}=IC_{t-1}(1+WACC)+A_{t}-FCFF_{t}\] And \(FCFF\) is thus equal to : \[FCFF_{t}=IC_{t-1}(1+WACC)+A_{t}-IC_{t}\] We can now transform Eq. 9 by replacing the \(FCFF\) with its equivalent identified in Eq. 17. \[EV_{t}=\displaystyle\sum_{i=1}^{K}\frac{FCFF_{t+i}}{(1+R)^i}+\frac{EV_{t+K}}{(1+R)^K}\] becomes \[EV_{t}=\displaystyle\sum_{i=1}^{K}\frac{IC_{t+i-1}(1+WACC)+A_{t+i}-IC{t+i}}{(1+WACC)^i}+\frac{EV_{t+K}}{(1+WACC)^K}\] This can be simplified (with alot of terms cancelling out) so that finally we get : \[EV_{t}=IC_{t}+\displaystyle\sum_{i=1}^{K}\frac{A_{t+i}}{(1+WACC)^i}-\frac{IC_{t+K}}{(1+WACC)^K}+\frac{EV_{t+K}}{(1+WACC)^K}\] If we assume that at a period sufficiently far out in the future \(t+k\), abnormal earnings have been arbitraged away and disappear, then market value of invested capital (\(EV\)) must equal book value (\(IC\)) then the formula becomes : \[EV_{t}=IC_{t}+\displaystyle\sum_{i=1}^{K}\frac{A_{t+i}}{(1+WACC)^i}\] Using the notion of persistence rate allows us to simplify even more the valuation equation. The persistence rate \(\omega\) is defined such that \(A_{t+1}= \omega^i A_{t}\). \(\omega\) is necessarely less than one so that abnormal earnings fade away at speed \(\omega\).
Finally, Eq. becomes : \[EV_{t}=IC_{t}+(\frac{1}{1+WACC-\omega})(NOPAT_{t+1}-WACC\times IC_{t})\] By dividing left and right term by \(IC_{t}\) we have : \[\frac{EV_{t}}{IC_{t}}=(\frac{1}{1+WACC-\omega})(ROIC_{t+1}-WACC)\]
Our equation linking \(\frac{EV}{IC}\) with the \(ROIC\) and the \(WACC\) is completely in line with valuation models and concepts such as EVA. The idea is the same : value is created when a business is able to earn more than its cost of capital (\(ROIC > WACC\)); in this situation, the market value of a business warrants a premium relative to its book value (\(\frac{EV}{IC}\) >1).
Just as we did for the \(DDM\), the \(DC\)F can be transformed and simplified in order to take into account growth dynamics in a very simplified manner. For example, a \(DCF\) version of the Gordon Growth Model (\(GMM\)) would look like : \[EV_{t}=\frac{FCF_{t+1}}{WACC-g}\] Using the clean surplus accounting rule and replacing \(\rho ROIC_{t+1}\) (where \(\rho\) is the proportion of NOPAT converted into Free Cash Flows) by \(ROIC_{t+1}-g\) where \(g\) is the perpetual growth rate in Free Cash Flows, we get : \[\frac{EV_{t}}{IC_{t}}=\frac{ROIC_{t+1}-g}{WACC-g}\]