Page info: *Author:Mathiesen, H. *Document version:2.6. *Copyright 1997-2019, H. Mathiesen.Legal notice. 


Model: The General Cash Flow Model



1†††††† Introduction


This text presents models and proofs of the cash flow framework for firm valuation. Note that footnotes e.g. '[2]' can be viewed by clicking them.




2†††††† The Cash Flow Model


The different equations below represent most of the existing versions of the cash flow way of calculating firm value in a world of uncertainty:


††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† (1)

††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† <=>

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† =>[1]

††††††††††††††††††††††††††† (2)

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† <=>

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† <=>

††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† =>[2]

††††††††††††††††††††††††††††† (3)

††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† <=>[3]

††††††††††† (4)

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† =>[4]

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† <=>

††††††††††††††††††††††† (5)





      PVF is the estimated present value of the firm, or the value of ownership. This is also called equity value.

      E[FCFt] is the expected nominal free cash flow from the firmís operations for period t. This is, the cash flow free to honor returns to debt and equity holders under the going concern assumption. The latter means that the investment needed to create continuing cash flows must be subtracted the gross cash flow (is included in the FCF). The present value of the FCF () stream is called operating value.

      PVNOA is the present value of all non-operating assets. Among other things it includes value of 1) overfunded pension funds, 2) excess marketable securities. Together the PVNOA and the operating value is called entity value.

      PVNEL is the Present Value of all Non-Equity Liabilities. Among other things it includes value of 1) long term debt, 2) short term debt 3) operating leases, 4) capital leases, 5) preferred stocks, 6) warrants, 7) convertible debt that is unlikely to ever be converted. 8) stock options (this is stock value but when given as payment by the Ďtrueí owners it should be considered as debt since it deludes the value of ownership). PVNEL is also called debt value.

      E[rt] is the expected average cost of capital in period t. It is a discount rate equal to the market cost of financing the capital that generates the FCF. In the finance literature, it is also called the one period forward rate. The one period is normalized to one year in most cash flow models. Furthermore, E[rt] = E[WACCt], the expected weighted average cost of capital.

      E[NFCFn+1] = E[NFCFn](1+g) is the normalized level of free cash flow in period n+1. By normalized is meant that the FCF is free of any extraordinary influence. This will naturally never come true in actual corporate FCF. The reason for using period n+1 is due to the logic of Gordons growth formula.

      g1 is the expected long term growth rate in E[NFCF]. This is reasonably estimated by the average nominal growth in the economy.

      E[r] is the expected average cost of capital. The removal of t is based on the assumption of constant cost of capital for all periods (E[rt] = E[r] for ).

      E[NNOPLATn+1] is the normalized net operating profit less adjusted taxes in period n+1. It is normalized because this earnings measure should be free of any extraordinary influence.

      g2 is the expected long term growth rate in E[NNOPLAT]. This is as g1 reasonable to estimate by the average nominal growth in the economy.

      i is the expected rate of return on net increase in new invested capital (In). The latter is defined as the difference between gross investment (INV) and depreciation expenses (DEP): In = INV - DEP.

      g is the expected long term growth rate. It replaces g1 and g2 because it is assumed that g = g1 = g2. In the long run this assumption is fairly reasonable since g1 Ļ g2 is impossible.


The above equations demonstrated how the equity value is calculated by the cash flow approach. Go to see how the some of the above definitions fit the standard account.




3†††††† Gordons Growth Formula[5]


Proof of Gordons growth formula:


For notational ease let d0 = E[NFCFn] and d1 = E[NFCFn+1], where E[NFCFn+1] = E[NFCFn](1+g). The present value of a cash flow stream that starts growing from d0 at a rate g each period for n periods and discounted with E(r) is:


†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† <=>

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† <=>

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† <=>

PV = ud0(u0+ u1 + u2 + ...+ un-1)

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† <=>


uPV = uud0(u0+ u1 + u2 + ...+ un-1) = ud0(u1 + u2 + ...+ un)

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† (by subtracting the last two equations)††††††† <=>

PV - uPV = ud0(u0 - un) = ud0(1 - un)

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† (solving for PV)†††††††††† <=>


PV(1 - u) = ud0(1 - un)

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† <=>

PV = ud0(1 - un) / (1 - u)

††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† (substituting back u)††††††† <=>

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† (substituting d1 = d0(1+g)††††††††††† <=>

††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† (rearranging the denominator)††††††††††† <=>

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† (rearranging (1 + E[r])††††††† <=>

††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† (simplyfying the denominator)††††††††††† <=>


and finally






The last equation: PV = d1/(E[r] - g) is Gordons growth formula. It calculates the present value of a cash flow stream that starts at a level d0 and grows at a constant rate g for indefinitely many periods.



4 ††††† From Cash Flow to Earnings[6]



Proof that E[NFCFn+1] = E[NNOPLATn+1](1 - g/i):


From the ordinary earnings account, it is known that:



†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† <=>


E[FCF] = E[NOPLAT] - In††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† (6)




      INV is the gross investment from the earnings account.

      DEP is the depreciation expenses from the earnings account.

      In is the net increase in new invested capital over and above replacement capital: In = INV - DEP. The replacement capital is by definition equal to the depreciation expenses. This is, the investment needed to replace depreciated capital in order to keep the stock of capital constant given no growth.


Now, as long as the return on existing capital including replacement capital remains constant, a firmís NOPLAT in any period equals last periodís NOPLAT plus the return it earns on last periodís net increase in new invested capital (i*In). Therefore:


NOPLATt = NOPLATt-1+ i*Int-1

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† <=>

NOPLATt - NOPLATt-1= i*Int-1††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† (7)




      i is the expected rate of return on net increase in new invested capital (In).



Dividing (A7) by NOPLATt-1 and introducing g yields:††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††


g = (NOPLATt - NOPLATt-1)/NOPLATt-1= i*Int-1/NOPLATt-1†††††††††††††††††††††††††††††††††††††††††††††††††††

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† <=>


g = i*Int/NOPLAT

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† <=>

In = NOPLAT (g/i)††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† (8)

and by substituting (A8) into (A6) we get:



E[FCF] = E[NOPLAT] - E[NOPLAT](g/i)††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† <=>


E[FCF] = E[NOPLAT](1 - g/i)



-Copyright 1997-2019, H. Mathiesen. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.Legal notice. 



[1] ††† This step assumes a constant growth rate g1 in the normalized level of free cash flow NFCF from year n + 1.

[2] ††† This step assumes that E[rt] = E[r] for . In words; the one period forward rate is assumed to remain constant over the entire life of the corporation. Gordonís growth model is also applied in this step. This formula is proved in section 3, this appendix.

[3] ††† This step implies that E[NFCFn+1] = E[NNOPLATn+1](1 - g/i). That is so purely by definition and it does not require any new assumptions. The proof is given in section 4 this text. Note further that g = g1 = g2.

[4] ††† This step assumes that the return on net new investment (i) is equal to the weighted average cost of capital (E[r]), that is E[r] = i.

[5] ††† This proof draws on the proof of the formula given in Copeland and Weston [1988, page 847].

[6] ††† This proof draws on the proof of the formula given in Copeland, Koller, and Murrin [1990, page 399].