

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) where, ·
PV^{F} is the estimated present value of the firm, or the value of
ownership. This is also called equity
value. ·
E[FCF_{t}] is the
expected nominal free cash flow from the firm’s 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 nonoperating 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 NonEquity 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 ‘true’ owners it should
be considered as debt since it deludes the value of ownership). PVNEL is also
called debt value. ·
E[r_{t}] 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[r_{t}] = E[WACC_{t}], the
expected weighted average cost of capital. ·
E[NFCF_{n+1}] = E[NFCF_{n}](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. ·
g_{1} 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[r_{t}] = E[r] for _{}). ·
E[NNOPLAT_{n+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. ·
g_{2} is the expected long term growth rate in
E[NNOPLAT]. This is as g_{1} 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 g_{1}
and g_{2} because it is assumed that g = g_{1} = g_{2}.
In the long run this assumption is fairly reasonable since g_{1} ¹ g_{2} 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 d_{0}
= E[NFCF_{n}] and d_{1} = E[NFCF_{n+1}], where E[NFCF_{n+1}] = E[NFCF_{n}](1+g). The
present value of a cash flow stream that starts growing from d_{0} at
a rate g each period for n periods and discounted with E(r) is: _{} <=> _{} <=> _{} <=> PV = ud_{0}(u^{0}+ u^{1}
+ u^{2} + ...+ u^{n1}) <=> uPV = uud_{0}(u^{0}+
u^{1} + u^{2} + ...+ u^{n1}) = ud_{0}(u^{1}
+ u^{2} + ...+ u^{n}) (by
subtracting the last two equations)
<=> PV  uPV = ud_{0}(u^{0}
 u^{n}) = ud_{0}(1  u^{n}) (solving
for PV) <=> PV(1  u) = ud_{0}(1  u^{n}) <=> PV = ud_{0}(1
 u^{n}) / (1  u) (substituting
back u) <=> _{} (substituting d_{1} = d_{0}(1+g) <=> _{} (rearranging
the denominator) <=> _{} (rearranging
(1 + E[r]) <=> _{} (simplyfying the denominator) <=> _{} and finally _{} _{} Q.E.D. The last equation: PV = d_{1}/(E[r]
 g) is Gordons growth formula. It calculates the present value of a cash
flow stream that starts at a level d_{0} and grows at a constant rate
g for indefinitely many periods. 4 From Cash Flow to
Earnings[6] Proof that E[NFCF_{n+1}] = E[NNOPLAT_{n+1}](1  g/i): From the ordinary earnings account, it
is known that: E[FCF] = E[NOPLAT]  [INV 
DEP] <=> E[FCF] = E[NOPLAT]  In
(6) where · 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 firm’s
NOPLAT in any period equals last period’s NOPLAT plus the return it earns on
last period’s net increase in new invested capital (i*In). Therefore: NOPLAT_{t} = NOPLAT_{t1}+
i*In_{t1} <=> NOPLAT_{t}  NOPLAT_{t1}=
i*In_{t1} (7) where · i is the
expected rate of return on net
increase in new invested capital (In). Dividing (A7) by NOPLAT_{t1}
and introducing g yields: g = (NOPLAT_{t}  NOPLAT_{t1})/NOPLAT_{t1}=
i*In_{t1}/NOPLAT_{t1} <=> g = i*In_{t}/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) Q.E.D. 

 Copyright 19972019, H. Mathiesen. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Legal notice. 
[1] This
step assumes a constant growth rate g_{1} in the normalized level of
free cash flow NFCF from year n + 1.
[2] This
step assumes that E[r_{t}]
= E[r] for . In words; the one period forward rate is assumed to remain
constant over the entire life of the corporation. Gordon’s growth model is also applied in this
step. This formula is proved in section 3, this appendix.
[3] This
step implies that E[NFCF_{n+1}] = E[NNOPLAT_{n+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 = g_{1}
= g_{2}.
[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].