Discounted Cash
Flow Valuation
Challenges
 Defining and forecasting CF’s
 Estimating appropriate discount rate
Basic DCF model
An asset’s value is the present value of its
(expected) future cash flows

CFt
V0  
t
t 1 (1  r )
Three alternative definitions of cash
flow
Dividend discount model
Free cash flow model
Residual income model
Free cash flow
Free cash flow to the firm (FCFF) is cash
flow from operations minus capital
expenditures
Free cash flow to equity (FCFE) is cash
flow from operations minus capital
expenditures minus net payments to
debtholders (interest and principal)
FCF valuation
PV of FCFF is the total value of the
company. Value of equity is PV of FCFF
minus the market value of outstanding
debt.
PV of FCFE is the value of equity.
Discount rate for FCFF is the WACC.
Discount rate for FCFE is the cost of
equity (required rate of return for equity).
Intro to Free Cash Flows
Dividends are the cash flows actually paid to
stockholders
Free cash flows are the cash flows available
for distribution.
Defining Free Cash Flow
Free cash flow to equity (FCFE) is the
cash flow available to the firm’s common
equity holders after all operating expenses,
interest and principal payments have been
paid, and necessary investments in
working and fixed capital have been made.
• FCFE is the cash flow from operations minus
capital expenditures minus payments to (and
plus receipts from) debtholders.
Valuing FCFE
 The value of equity can also be found by
discounting FCFE at the required rate of return on
equity (r):
Equity Value 


t 1
FCFE t
(1  r )t
 Since FCFE is the cash flow remaining for equity
holders after all other claims have been satisfied,
discounting FCFE by r (the required rate of return
on equity) gives the value of the firm’s equity.
 Dividing the total value of equity by the number of
Discount rate determination
Jargon
• Discount rate: any rate used in finding the
present value of a future cash flow
• Risk premium: compensation for risk,
measured relative to the risk-free rate
• Required rate of return: minimum return
required by investor to invest in an asset
• Cost of equity: required rate of return on
common stock
Two major approaches for cost of
equity
Equilibrium models:
• Capital asset pricing model (CAPM)
• Arbitrage pricing theory (APT)
Bond yield plus risk premium method
(BYPRP)
CAPM
Expected return is the risk-free rate plus a
risk premium related to the asset’s beta:
E(Ri) = RF + i[E(RM) – RF]
The beta is i = Cov(Ri,RM)/Var(RM)
[E(RM) – RF] is the market risk premium or
the equity risk premium
CAPM
What do we use for the risk-free rate of return?
• Choice is often a short-term rate such as the 30day T-bill rate or a long-term government bond
rate.
• We usually match the duration of the bond rate
with the investment period, so we use the longterm government bond rate.
• Risk-free rate must be coordinated with how the
equity risk premium is calculated (i.e., both
based on same bond maturity).
Equity risk premium
Historical estimates: Average difference
between equity market returns and government
debt returns.
• Choice between arithmetic mean return or
geometric mean return
• Survivorship bias
• ERP varies over time
• ERP differs in different markets
Equity risk premium
Expectational method is forward looking instead
of historical
One common estimate of this type:
• GGM equity risk premium estimate
= dividend yield on index based on year-ahead
dividends
+ consensus long-term earnings growth rate
- current long-term government bond yield
Sources of error in using
models
Three sources of error in using CAPM or
APT models:
• Model uncertainty – Is the model correct?
• Input uncertainty – Are the equity risk
premium or factor risk premiums and riskfree rate correct?
• Uncertainty about current values of stock
beta or factor sensitivities
BYPRP method
The bond yield plus risk premium method
finds the cost of equity as:
BYPRP cost of equity
= YTM on the company’s long-term debt
+ Risk premium
The typical risk premium added is 3-4
percent.
Single-stage, constant-growth
FCFE valuation model
FCFE in any period will be equal to FCFE in the
preceding period times (1 + g):
• FCFEt = FCFEt–1 (1 + g).
The value of equity if FCFE is growing at a
constant rate is FCFE FCFE (1  g )
Equity Value 
1
rg

0
rg
The discount rate is r, the required return on
equity. The growth rate of FCFF and the growth
Computing FCFF from Net Income
This equation can be written more compactly
as
FCFF = NI + Depreciation + Int(1 – Tax rate) – Inv(FC) –
Inv(WC)
 Or
 FCFF = EBIT(1-tax rate) + depreciation – Cap. Expend. – change in
working capital – change in other assets
Forecasting free cash flows
 Computing FCFF and FCFE based upon
historical accounting data is straightforward.
Often times, this data is then used directly in a
single-stage DCF valuation model.
 On other occasions, the analyst desires to
forecast future FCFF or FCFE directly. In this
case, the analyst must forecast the individual
components of free cash flow. This section
extends our previous presentation on computing
FCFF and FCFE to the more complex task of
forecasting FCFF and FCFE. We present FCFF
and FCFE valuation models in the next section.
Forecasting free cash flows
 Given that we have a variety of ways in which to
derive free cash flow on a historical basis, it
should come as no surprise that there are
several methods of forecasting free cash flow.
 One approach is to compute historical free cash
flow and apply some constant growth rate. This
approach would be appropriate if free cash flow
for the firm tended to grow at a constant rate
and if historical relationships between free cash
flow and fundamental factors were expected to
be maintained.
Forecasting FCFE
 If the firm finances a fixed percentage of its capital
spending and investments in working capital with
debt, the calculation of FCFE is simplified. Let DR
be the debt ratio, debt as a percentage of assets. In
this case, FCFE can be written as
 FCFE = NI – (1 – DR)(Capital Spending – Depreciation)
– (1 – DR)Inv(WC)
 When building FCFE valuation models, the logic,
that debt financing is used to finance a constant
fraction of investments, is very useful. This
equation is pretty common.
Forecasting future dividends or
FCFE
Using stylized growth patterns
• Constant growth forever (the Gordon
growth model)
• Two-distinct stages of growth (the twostage growth model and the H model)
• Three distinct stages of growth (the threestage growth model)
Forecasting future dividends
Forecast dividends for a visible time
horizon, and then handle the value of the
remaining future dividends either by
• Assigning a stylized growth pattern to
dividends after the terminal point
• Estimate a stock price at the terminal point
using some method such as a multiple of
forecasted book value or earnings per
share
Gordon Growth Model
Assumes a stylized pattern of growth,
specifically constant growth:
Dt = Dt-1(1+g)
Or
Dt = D0(1 + g)t
Gordon Growth Model
PV of dividend stream is:
D0 (1  g ) D0 (1  g )2
D0 (1  g )n
V0 



2
n
(1  r )
(1  r )
(1  r )
Which can be simplified to:
D0 (1  g )
D1
V0 

rg
rg
Gordon growth model
Valuations are very sensitive to inputs.
Assuming D1 = 0.83, the value of a stock is:
g = 3.45% g = 3.70%
g = 3.95%
r = 5.95%
$33.20
$36.89
$41.50
r = 6.20%
$30.18
$33.20
$36.89
r = 6.45%
$27.67
$30.18
$33.20
Other Gordon Growth issues
Generally, it is illogical to have a perpetual
dividend growth rate that exceeds the
growth rate of GDP
Perpetuity value (g = 0):
D1
V0 
r
Negative growth rates are also acceptable
in the model.
Gordon Model & P/E ratios
If E is next year’s earnings (leading P/E):
P0 D1 / E1 (1  b)


E1
rg
rg
If E is this year’s earnings (trailing P/E):
P0 D0 (1  g ) / E0 (1  b)(1  g )


E0
rg
rg
Using a P/E for terminal value
The terminal value at the beginning of the
second stage was found above with a
Gordon growth model, assuming a longterm sustainable growth rate.
The terminal value can also be found
using another method to estimate the
terminal value at t = n. You can also use a
P/E ratio, applied to estimated earnings at
t = n.
Using a P/E for terminal value
For DuPont, assume
• D0 = 1.40
• gS = 9.3% for four years
• Payout ratio = 40%
• r = 11.5%
• Trailing P/E for t = 4 is 11.0
Forecasted EPS for year 4 is
• E4 = 1.40(1.093)4 / 0.40 = 1.9981 = 4.9952
Using a P/E for terminal value
Time
Value
Calculation
Dt or Vt
1
2
3
4
4
D1
D2
D3
D4
V4
1.40(1.093)1
1.40(1.093)2
1.40(1.093)3
1.40(1.093)4
11  [1.40(1.093)4 / 0.40]
= 11  [1.9981 / 0.40] = 11  4.9952
1.5302
1.6725
1.8281
1.9981
54.9472
Total
Present Values
Dt/(1.115)t or Vt/(1.115)t
1.3724
1.3453
1.3188
1.2927
35.5505
40.88
Three-stage DDM
There are two popular version of the threestage DDM
• The first version is like the two-stage model, only
the firm is assumed to have a constant dividend
growth rate in each of the three stages.
• A second version of the three-stage DDM
combines the two-stage DDM and the H model. In
the first stage, dividends grow at a high, constant
(supernormal) rate for the whole period. In the
second stage, dividends decline linearly as they
do in the H model. Finally, in stage three,
dividends grow at a sustainable, constant rate.
Spreadsheet modeling
Spreadsheets allow the analyst to build
very complicated models that would be
very cumbersome to describe using
algebra.
Built-in functions such as those to find rates
of return use algorithms to get a numerical
answer when a mathematical solution
would be impossible or extremely
complicated.
Strengths of multistage DDMs
Can accommodate a variety of patterns of
future dividend streams.
Even though they may not replicate the
future dividends exactly, they can be a
useful approximation.
The expected rates of return can be
imputed by finding the discount rate that
equates the present value of the dividend
stream to the current stock price.
Strengths of multistage DDMs
Because of the variety of DDMs available,
the analyst is both enabled and compelled to
evaluate carefully the assumptions about the
stock under examination.
Spreadsheets are widely available, allowing
the analyst to construct and solve an almost
limitless number of models.
Weaknesses of multistage
DDMs
Garbage in, garbage out. If the inputs are
not economically meaningful, the outputs
from the model will be of questionable
value.
Analysts sometimes employ models that
they do not understand fully.
Valuations are very sensitive to the inputs
to the models.
Forecasting growth rates
There are three basic methods for
forecasting growth rates:
• Using analyst forecasts
• Using historical rates (use historical
dividend growth rate or use a statistical
forecasting model based on historical
data)
• Using company and industry fundamentals
Finding g
The simplest model of the dividend growth
rate is:
• g = b x ROE
• where g = Dividend growth rate
• b = Earnings retention rate (1 – payout ratio)
• ROE = Return on equity.