# Multi-Stage Valuation Multiples

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Valuation Multiple Equational Forms 1

Each of the valuation multiples under consideration in this study include an Observable form of the equation and a Target form.  The Target forms include a Single-Stage Multiple equation reliant on a constant rate of change of the firm’s cash flow variable, cost of capital, and return on capital rate in perpetuity, and a Multi-Stage Multiple equation in which the variables may reflect some change between stages (i.e. the firm’s g may differ between the explicit period and the continuation period).

Further, the Multi-Stage Multiple equation may represent a variety of conditions, such as one in which growth adds value to the firm in the explicit period (short-term or ST), or ROIC > WACC in the short-term, but in the long-term (LT) ROIC = WACC and growth does not add value to the firm (nor does it destroy value).

Alternatively a Multi-Stage Multiple equation may represent the condition in which ROIC > WACC in both the long-run and short-run such that growth adds value to the firm continually, albeit at potentially differing rates of growth as modeled by the equations multiple stages.

Multi-Stage Valuation Multiples1

A multi-stage valuation multiple relaxes the constraining assumptions used in a single-state multiple and extends the multiple to multiple periods of time, each with potentially unique rates of growth (g) and WACC.  It also assumes that ROIC is equal to WACC in the continuation period (perpetuity).

$\frac{EV}{FCF_{OPS}}\,\,=\,\,\frac{ROIC\,\,-\,\,g}{ROIC\,\,(WACC\,\,-\,\,g}\,\,(1\,\,-\,\,T)\,\,x\,\,\big[1\,\,-\,\,\frac{(1\,\,+\,\,g)^{n}}{(1\,\,+\,\,WACC)^{n}}\big])\,\,+\,\,\frac{1}{WACC}\,\,x\,\,\frac{(1\,\,+\,\,g)^{n}}{(1\,\,+\,\,WACC)^{n}}$

The strength of a multi-stage multiple is in its allowance for different rates of growth and WACC in the continuation period.  A potential weakness of this particular form is that it operates under the assumption that there is no value added in the continuation period; that is, ROIC is equal to WACC such that continued growth does not add value to the firm.

We can relax the model’s assumptions further supposing the firm does add value in the continuation period by differentiating the firm’s expected long-term (LT) returns and cost costs of capital

$\frac{EV}{FCF_{OPS}}\,\,=\,\,\frac{ROIC\,\,-\,\,g}{ROIC\,\,(WACC\,\,-\,\,g}\,\,(1\,\,-\,\,T)\,\,x\,\,\big[1\,\,-\,\,\frac{(1\,\,+\,\,g)^{n}}{(1\,\,+\,\,WACC)^{n}}\big])\,\,+\,\,\frac{ROIC_{LT}\,\,-\,\,g_{LT}}{ROIC_{LT}\,\,x\,\,(WACC_{LT}\,\,-\,\,g_{LT})}\,\,x\,\,\frac{(1\,\,+\,\,g)^{n}}{(1\,\,+\,\,WACC)^{n}}$

While a multi-stage model in which rates of growth, ROIC, and WACC can be differentiated yields interesting results, its use warrants a note of caution:

1. Absent some product or process innovation in which the firm continues to wield a competitive advantage over its industry peers, there is no expected gain in the long-term beyond that which the expected cost of capital captures
2. Valuation equations in which a substantial spread between long-term returns and costs of capital are forecast often result in unrealistically high outcomes
3. Perpetuity is a long time and there are no known examples of firms maintaining such an advantage in the long-run.

 Enterprise Multiples Condition Multi-Stage Valuation Multiple . EV/SALES ROICST > WACCST; ROICLT = WACCLT $\frac{ROIC\,\,-\,\,g}{ROIC\,\,(WACC\,\,-\,\,g)}\,(1-T)\,(1-M)\,x\,\,\big[1\,\,-\,\,\frac{(1\,\,+\,\,g)^{n}}{(1\,\,+\,\,WACC)^{n}}\big])\,\,+\,\,\frac{1}{WACC}\,\,x\,\,\frac{(1\,\,+\,\,g)^{n}}{(1\,\,+\,\,WACC)^{n}}$ . ROICST > WACCST; ROICLT > WACCLT $\frac{ROIC\,\,-\,\,g}{ROIC\,\,(WACC\,\,-\,\,g)}\,(1-T)\,(1-M)\,\,x\,\,\big[1\,\,-\,\,\frac{(1\,\,+\,\,g)^{n}}{(1\,\,+\,\,WACC)^{n}}\big])\,\,+\,\,\frac{ROIC_{LT}\,\,-\,\,g_{LT}}{ROIC_{LT}\,\,x\,\,(WACC_{LT}\,\,-\,\,g_{LT})}\,\,x\,\,\frac{(1\,\,+\,\,g)^{n}}{(1\,\,+\,\,WACC)^{n}}$ EV/EBIT ROICST > WACCST; ROICLT = WACCLT $\frac{ROIC\,\,-\,\,g}{ROIC\,\,(WACC\,\,-\,\,g)}\,\,(1-T)\,\,x\,\,\big[1\,\,-\,\,\frac{(1\,\,+\,\,g)^{n}}{(1\,\,+\,\,WACC)^{n}}\big])\,\,+\,\,\frac{1}{WACC}\,\,x\,\,\frac{(1\,\,+\,\,g)^{n}}{(1\,\,+\,\,WACC)^{n}}$ . ROICST > WACCST; ROICLT > WACCLT $\frac{ROIC\,\,-\,\,g}{ROIC\,\,(WACC\,\,-\,\,g)}\,\,(1-T)\,\,x\,\,\big[1\,\,-\,\,\frac{(1\,\,+\,\,g)^{n}}{(1\,\,+\,\,WACC)^{n}}\big])\,\,+\,\,\frac{ROIC_{LT}\,\,-\,\,g_{LT}}{ROIC_{LT}\,\,x\,\,(WACC_{LT}\,\,-\,\,g_{LT})}\,\,x\,\,\frac{(1\,\,+\,\,g)^{n}}{(1\,\,+\,\,WACC)^{n}}$ EV/NOPLAT ROICST > WACCST; ROICLT = WACCLT $\frac{ROIC\,\,-\,\,g}{ROIC\,\,(WACC\,\,-\,\,g)}\,\,x\,\,\big[1\,\,-\,\,\frac{(1\,\,+\,\,g)^{n}}{(1\,\,+\,\,WACC)^{n}}\big])\,\,+\,\,\frac{1}{WACC}\,\,x\,\,\frac{(1\,\,+\,\,g)^{n}}{(1\,\,+\,\,WACC)^{n}}$ .. ROICST > WACCST; ROICLT > WACCLT $\frac{ROIC\,\,-\,\,g}{ROIC\,\,(WACC\,\,-\,\,g)}\,\,x\,\,\big[1\,\,-\,\,\frac{(1\,\,+\,\,g)^{n}}{(1\,\,+\,\,WACC)^{n}}\big])\,\,+\,\,\frac{ROIC_{LT}\,\,-\,\,g_{LT}}{ROIC_{LT}\,\,x\,\,(WACC_{LT}\,\,-\,\,g_{LT})}\,\,x\,\,\frac{(1\,\,+\,\,g)^{n}}{(1\,\,+\,\,WACC)^{n}}$ EV/EBITDA ROICST > WACCST; ROICLT = WACCLT $\frac{ROIC\,\,-\,\,g}{ROIC\,\,(WACC\,\,-\,\,g)}\,(1-T)\,(1-D)\,x\,\,\big[1\,\,-\,\,\frac{(1\,\,+\,\,g)^{n}}{(1\,\,+\,\,WACC)^{n}}\big])\,\,+\,\,\frac{1}{WACC}\,\,x\,\,\frac{(1\,\,+\,\,g)^{n}}{(1\,\,+\,\,WACC)^{n}}$ . ROICST > WACCST; ROICLT > WACCLT $\frac{ROIC\,\,-\,\,g}{ROIC\,\,(WACC\,\,-\,\,g)}\,(1-T)\,(1-D)\,\,x\,\,\big[1\,\,-\,\,\frac{(1\,\,+\,\,g)^{n}}{(1\,\,+\,\,WACC)^{n}}\big])\,\,+\,\,\frac{ROIC_{LT}\,\,-\,\,g_{LT}}{ROIC_{LT}\,\,x\,\,(WACC_{LT}\,\,-\,\,g_{LT})}\,\,x\,\,\frac{(1\,\,+\,\,g)^{n}}{(1\,\,+\,\,WACC)^{n}}$ EV/FCFOPS ROICST > WACCST; ROICLT = WACCLT $\frac{ROIC\,\,-\,\,g}{ROIC\,\,(WACC\,\,-\,\,g)}\,\,(1-T)\,\,x\,\,\big[1\,\,-\,\,\frac{(1\,\,+\,\,g)^{n}}{(1\,\,+\,\,WACC)^{n}}\big])\,\,+\,\,\frac{1}{WACC}\,\,x\,\,\frac{(1\,\,+\,\,g)^{n}}{(1\,\,+\,\,WACC)^{n}}$ . ROICST > WACCST; ROICLT > WACCLT $\frac{ROIC\,\,-\,\,g}{ROIC\,\,(WACC\,\,-\,\,g)}\,\,(1-T)\,\,x\,\,\big[1\,\,-\,\,\frac{(1\,\,+\,\,g)^{n}}{(1\,\,+\,\,WACC)^{n}}\big])\,\,+\,\,\frac{ROIC_{LT}\,\,-\,\,g_{LT}}{ROIC_{LT}\,\,x\,\,(WACC_{LT}\,\,-\,\,g_{LT})}\,\,x\,\,\frac{(1\,\,+\,\,g)^{n}}{(1\,\,+\,\,WACC)^{n}}$ EV/IC ROICST > WACCST; ROICLT = WACCLT $\frac{ROIC\,\,-\,\,g}{(ROIC\,\,-\,\,WACC)}\,\,x\,\,\big[1\,\,-\,\,\frac{(1\,\,+\,\,g)^{n}}{(1\,\,+\,\,WACC)^{n}}\big])\,\,+\,\,\frac{1}{WACC}\,\,x\,\,\frac{(1\,\,+\,\,g)^{n}}{(1\,\,+\,\,WACC)^{n}}$ . ROICST > WACCST; ROICLT > WACCLT $\frac{ROIC\,\,-\,\,g}{(ROIC\,\,-\,\,WACC)}\,\,x\,\,\big[1\,\,-\,\,\frac{(1\,\,+\,\,g)^{n}}{(1\,\,+\,\,WACC)^{n}}\big])\,\,+\,\,\frac{ROIC_{LT}\,\,-\,\,g_{LT}}{ROIC_{LT}\,\,x\,\,(WACC_{LT}\,\,-\,\,g_{LT})}\,\,x\,\,\frac{(1\,\,+\,\,g)^{n}}{(1\,\,+\,\,WACC)^{n}}$ Return to top of page Equity Multiples Condition Multi-Stage Valuation Multiple . PRICE/EARNINGS ROEST > COEST; ROELT = COELT $\frac{ROE\,\,-\,\,g}{ROE\,\,(COE\,\,-\,\,g)}\,\,x\,\,\big[1\,\,-\,\,\frac{(1\,\,+\,\,g)^{n}}{(1\,\,+\,\,COE)^{n}}\big])\,\,+\,\,\frac{1}{COE}\,\,x\,\,\frac{(1\,\,+\,\,g)^{n}}{(1\,\,+\,\,COE)^{n}}$ . ROEST > COEST; ROELT > COELT $\frac{ROE\,\,-\,\,g}{ROE\,\,(COE\,\,-\,\,g)}\,\,x\,\,\big[1\,\,-\,\,\frac{(1\,\,+\,\,g)^{n}}{(1\,\,+\,\,COE)^{n}}\big])\,\,+\,\,\frac{ROE_{LT}\,\,-\,\,g_{LT}}{ROE_{LT}\,\,x\,\,(COE_{LT}\,\,-\,\,g_{LT})}\,\,x\,\,\frac{(1\,\,+\,\,g)^{n}}{(1\,\,+\,\,COE)^{n}}$ PRICE/BOOK VALUE ROEST > COEST; ROELT = COELT $\frac{ROE\,\,-\,\,g}{ROE\,\,(COE\,\,-\,\,g)}\,\,x\,\,\frac{NI}{CE}\,\,x\,\,\big[1\,\,-\,\,\frac{(1\,\,+\,\,g)^{n}}{(1\,\,+\,\,COE)^{n}}\big])\,\,+\,\,\frac{1}{COE}\,\,x\,\,\frac{(1\,\,+\,\,g)^{n}}{(1\,\,+\,\,COE)^{n}}$ . ROEST > COEST; ROELT > COELT $\frac{ROE\,\,-\,\,g}{ROE\,\,(COE\,\,-\,\,g)}\,\,x\,\,\frac{NI}{CE}\,\,x\,\,\big[1\,\,-\,\,\frac{(1\,\,+\,\,g)^{n}}{(1\,\,+\,\,COE)^{n}}\big])\,\,+\,\,\frac{ROE_{LT}\,\,-\,\,g_{LT}}{ROE_{LT}\,\,x\,\,(COE_{LT}\,\,-\,\,g_{LT})}\,\,x\,\,\frac{(1\,\,+\,\,g)^{n}}{(1\,\,+\,\,COE)^{n}}$ PRICE/CASH EARNINGS ROEST > COEST; ROELT = COELT $\frac{ROE\,\,-\,\,g}{COE\,\,-\,\,g}\,\,x\,\,\big[1\,\,-\,\,\frac{(1\,\,+\,\,g)^{n}}{(1\,\,+\,\,COE)^{n}}\big])\,\,+\,\,\frac{1}{COE}\,\,x\,\,\frac{(1\,\,+\,\,g)^{n}}{(1\,\,+\,\,COE)^{n}}$ . ROEST > COEST; ROELT > COELT $\frac{ROE\,\,-\,\,g}{COE\,\,-\,\,g}\,\,x\,\,\big[1\,\,-\,\,\frac{(1\,\,+\,\,g)^{n}}{(1\,\,+\,\,COE)^{n}}\big])\,\,+\,\,\frac{ROE_{LT}\,\,-\,\,g_{LT}}{ROE_{LT}\,\,x\,\,(COE_{LT}\,\,-\,\,g_{LT})}\,\,x\,\,\frac{(1\,\,+\,\,g)^{n}}{(1\,\,+\,\,COE)^{n}}$