ESTIMATING INPUTS FOR VALUATION
ESTIMATION OF DISCOUNT RATES
* Critical ingredient in discounted cashflow valuation. Errors
in estimating the discount rate or mismatching cashflows and discount rates
can lead to serious errors in valuation.
* At an intutive level, the discount rate used should be consistent with
both the riskiness and the type of cashflow being discounted.
I. Cost of Equity
The cost of equity is the rate of return that investors require to make
an equity investment in a firm. There are two approaches to estimating the
cost of equity;
* a risk and return model
* a dividendgrowth model.
Models of Risk and Return
The Capital Asset Pricing Model
* Measures risk in terms on nondiversifiable variance
* Relates expected returns to this risk measure.
* It is based upon several assumptions 
(a) that investors have homogeneous expectations about asset returns and
variances
(b) that they can borrow and lend at a riskfree rate
(c) that all assets are marketable and perfectly divisible
(d) that there are no transactions costs and that there are no restrictions
on short sales.
Risk in the CAPM
Beta: The nondiversifiable risk for any asset can be measured
by the covariance of its returns with returns on a market index, which is
defined to be the asset's beta.
The cost of equity will be the required return,
Cost of Equity = Rf + Equity Beta * (E(Rm)  Rf)
where,
Rf = Riskfree rate
E(Rm) = Expected Return on the Market Index
Using the Capital Asset Pricing Model
* Inputs required to use the CAPM 
(a) the current riskfree rate
(b) the expected return on the market index and
(c) the beta of the asset being analyzed.
Practical issues in using the CAPM 
1. Measurement of the risk premium
* It is generally based upon historical data, and the premium is defined
to be the difference between average returns on stocks and average returns
on riskfree securities over the measurement period.
Magnitude of the risk premium
Historical Period 
Stocks  T. Bills 
Stocks  T.Bonds 

Arithmetic 
Geometric 
Arithmetic 
Geometric 
19261990 
8.41% 
6.41% 
7.24% 
5.50% 
19621990 
4.10% 
2.95% 
3.92% 
3.25% 
19811990 
6.05% 
5.38% 
0.13% 
0.19% 
* Generally, geometric averages provide better estimates of risk premiums
in valuation.
 The risk premiums will vary across markets, depending upon their riskiness.
While historical data can be used to estimate premiums outside the United
States, it is not very reliable. An alternative way of estimating premiums
is to use country bond ratings to estimate these premiums relative to the
U.S. premium. For instance, the risk premiums for South American countries
can be estimated as follows:
Country 
Rating 
Risk Premium 
Argentina 
BBB 
5.5% + 1.75% = 7.25% 
Brazil 
BB 
5.5% + 2% = 7.5% 
Chile 
AA 
5.5% + 0.75% = 6.25% 
Columbia 
A+ 
5.5% + 1.25% = 6.75% 
Mexico 
BBB+ 
5.5% + 1.5% = 7% 
Paraguay 
BBB 
5.5% + 1.75% = 7.25% 
Peru 
B 
5.5% + 2.5% = 8% 
Uruguay 
BBB 
5.5% + 1.6% = 7.1% 
A Warning: If you add a default premium to the risk premium, do not
add a risk premium to the risk free rate. That would be double counting.
II. Riskfree Rate
The longterm government bond rate is the appropriate riskfree rate.
When the long term government bond rate is not available, it may make sense
to look at the rate at which large corporations can borrow in the local
market.
Illustration 3: Using the CAPM to calculate cost of equity
Glaxo Holdings had an estimated beta of 1.10 at the end of 1994. At the
same point in time, the thirtyyear treasury bond rate in the United States
was 8.00%. The estimated cost of equity for Glaxo in December 1994 was ñ
Cost of Equity = 8.00% + 1.10 (5.50%) = 14.05%
This is the cost of equity, if cash flows are estimated in dollars. The
same estimation can be done in British Pounds, using the long term Government
Bond rate in the U.K., which was 8.50% at the same point in time ñ
Cost of Equity (British Pounds) = 8.50% + 1.10 (5.50%) = 14.55%
This difference reflects differences in expected inflation in the two markets.
 Wellcome is planning an acquisition of Glaxo. Wellcome has a beta of
0.65. Which beta would you use in valuing Glaxo?
Determinants of Betas
I. Type of Business:
* The more sensitive a business is to market conditions, the higher is its
beta. Thus cyclical firms can be expected to have higher betas, other things
remaining equal, than noncyclical firms.
* The beta of a firm is the weighted average of the betas of its different
business lines.
 Pharmaceutical companies are increasingly looking at expanding into
biotechnology, either through acquisitions or through projects. Biotechnology
firms have an average beta of 1.8. What would the impact of these actions
be on pharmaceutical firmsí betas?
II. Degree of Operating Leverage:
* DOL is a function of the cost structure of a firm, and is usually defined
in terms of the relationship between fixed costs and total costs.
* A firm which has high operating leverage > higher variability in earnings
before interest and taxes (EBIT) > higher beta
 Sticking again with pharmaceutical firms, there are some firms which
have very high fixed costs, either because they focus heavily on research,
or because of the nature of their business. (Synergen, for instance, had
a beta of 1.65 in 1994. It had research and development expenses which
were 670% of sales in 1994.) These firms will have higher betas than other
firms which have more traditional cost structures. (Wellcome is a good
example.)
III. Financial Leverage:
* An increase in financial leverage will increase the equity beta of a firm.
* If all of the firm's risk are borne by the stockholders, i.e., the beta
of debt is zero, and debt has a tax benefit to the firm, then,
Levered Beta = Unlevered Beta (1 + (1t) (D/E))
where,
t = Corporate tax rate
D/E = Debt/Equity Ratio
Illustration 4: Effects of Financial Leverage on Betas
Glaxo, with a beta of 1.10, had a debt/equity ratio of 4% in 1994. If Glaxo
were to raise its debt/equity ratio to 20%, its beta would be much higher
(Tax rate was 30%) ñ
Current Beta (Levered) = 1.10
Unlevered Beta = 1.10 /(1+0.7*0.04) = 1.07
New Levered Beta = 1.07 (1+0.7*0.2) = 1.22
Other approaches to estimating Betas
I. Using comparable firms:
* Use the betas of publicly traded firms which are comparable in terms of
business risk and operating leverage.
* Correct for differences in financial leverage between the firm being analyzed
and the comparable firms.
Illustration 5: Using comparable firms to estimate betas
Assume that you are trying to estimate the beta for a private firm that
is in the business of manufacturing, selling and servicing fax machines.
The betas of publicly traded firms involved in office equipment and supplies
are as follows (They face an average tax rate of 40%). The private firm
had a debt/equity ratio of 30%.
Firm 
Beta 
Debt/Equity 
General Binding 
1.00 
0.20 
Hunt Manufacturing 
0.80 
0.03 
Moore Coroporation 
0.95 
0.05 
Nashua Company 
0.90 
0.10 
Pitney Bowes 
1.20 
0.45 
Average 
0.97 
1.17 
Unlevered Beta of office supply firms = 0.97 / (1 + (10.4) (0.17)) = 0.88
Beta for private firm involved in office supplies = 0.88 (1 + (10.4) (0.3))
= 1.04
II. Using fundamental factors:
* Combines industry and companyfundamental factors to predict betas.
* Income statement and balance sheet variables are important predictors
of beta
* Following is a regression relating the betas of NYSE and AMEX stocks in
1991 to five variables  dividend yield, coefficient of variation in operating
income, size, debt/equity and growth in earnings.
BETA = 0.9832 + 0.08 CV in Operating Income  0.126 Dividend Yield + 0.15
Debt/Equity Ratio + 0.034 Growth in Earnings per Share  0.00001 Total Assets
where,
CV in Operating Income = Coefficient of Variation in Operating Income
= Standard Deviation in Operating Income/ Average Operating Income
Illustration 6: Using fundamental information to predict betas
Assume that you are trying to estimate the beta for a private firm, with
the following financial characteristics (defined consistently with the regression):
Coefficient of Variation in Operating Income = 2.2
Dividend Yield = 0.04
Debt/Equity Ratio = 0.30
Growth in Earnings per share = 0.30
Total Assets = $ 10,000 (in thousands)
The estimated beta for this firm,
BETA = 0.9832 +0.08*2.2  0.126* 0.04 + 0.15* 0.30 + 0.034 * 0.30  0.00001*10,000
= 1.19
The Arbitrage Pricing Model
* Logic behind the arbitrage pricing model (APM) same as the logic behind
the CAPM, i.e., investors get rewarded for taking on nondiversifiable
risk.
* Measure of this nondiversifiable risk in the APM, however, is not a single
factor but is determined by an asset's sensitivity to various economic factors
that affect all assets.
* The number and the identity of the factors are determined by the data
on historical returns.
* The arbitrage pricing model relates expected returns to economic factors,
with a beta specific to each factor. If these factorspecific betas and
the factor risk premia can be estimated, the cost of equity can also be
estimated.
where,
Rf = Riskfree rate
bj = Beta specific to factor j
E(Rj)  Rf = Risk premium per unit of factor j risk
k = Number of factors
Using the Arbitrage Pricing Model
Illustration 7: Using the APM to estimate the cost of equity
Assume that the parameters for the arbitrage pricing model have been estimated,
and that there are three factors,
Riskfree rate = 3.35%
E(R1)  Rf = 3% : Risk Premium for factor 1
E(R2)  Rf = 4% : Risk Premium for factor 2
E(R3)  Rf = 1.5% : Risk Premium for factor 3
Assume that the betas specific to each of these factors are estimated for
Pepsi Cola in 1992 and that the estimates are as follows:
b1 = 1.20
b2 = 0.90
b3 = 1.10
Substituting into the APM,
Cost of Equity = 3.35% + 1.20 (3%) + 0.90 (4%) + 1.1 (1.5%) = 12.20%
Considerations on the use of the APM
* Capital asset pricing model can be considered to be a specialized case
of the arbitrage pricing model, where there is only one underlying factor
and that underlying factor is completely measured by the market index.
* In general, the CAPM has the advantage of being a simpler model to estimate
and to use, but it will underperform the richer APM when the company is
sensitive to economic factors not well represented in the market index.
Example: Oil companies, which derive most of their risk from oil price movements,
tend to have low CAPM betas.
* The biggest intuitive block in using the arbitrage pricing model is its
failure to identify specifically the factors driving expected returns.
Multifactor Models for risk and return
* Unidentified factors in the arbitrage pricing model are replaced with
macroeconomic variables,
* For example, Chen, Roll and Ross (1986) suggest that the following macroeconomic
variables are highly correlated with the factors that come out of factor
analysis  industrial production, changes in default premium, shifts in
the term structure, unanticipated inflation and changes in the real rate
of return. These variables can then be correlated with returns to come up
with a model of expected returns, with firmspecific betas calculated relative
to each variable.
* Costs of going from the arbitrage pricing model to a macroeconomic multifactor
model can be traced directly to the errors that can be made in identifying
the factors.
* The factors in the model can change over time, as will the risk premia
associated with each economic factor.
* Using the wrong factor(s) or missing a significant factor in a multifactor
model can lead to inferior estimates of cost of equity.
Dividend Growth Model
For a firm which has a stable growth rate in earnings and dividends,
the present value of a share of equity can be written as:
Po = Present Value of expected dividends = DPS1 / (ke  g)
where,
P0 = Price of the stock today
DPS1 = Expected dividends per share next year
ke = Cost of Equity
g = Growth rate in dividends (steady state)
A simple manipulation of this formula yields,
ke = DPS1 / P0 + g
= Expected Dividend Yield + Growth rate in earnings/dividends
More importantly, since the current price is a key input to the model, it
is inappropriate to use this approach to value stock in a firm. There is
a strong element of circular reasoning involved that will lead the analyst
to conclude, using this cost of equity, that equity is fairly valued.
Illustration 8: Using the Dividend Growth Model to estimate the cost
of equity: Southwestern Bell
In 1992, Southwestern Bell paid dividend per share of $2.82 and the stock
traded at $66 in December 1992. The estimated growth rate in dividends is
5.5% and the firm is assumed to be in steady state. The dividend growth
model can then be used to estimate the cost of equity;
Expected Dividends in 1993 = $2.82 *1.055 = $2.98
Cost of Equity = $2.98 / $66 + 5.5%
= 10%
To illustrate the circular reasoning involved in using this cost of equity
to value stock,
Value of Equity = $ 2.98 / (.10  .055) = $66
Not surprisingly, the stock is found to be fairly valued.
Weighted Average Cost of Capital (WACC)
Definition of the Weighted Average Cost of Capital (WACC)
The weighted average cost of capital is defined as the weighted average
of the costs of the different components of financing used by a firm.
WACC = ke ( E/ (D+E+PS)) + kd ( D/ (D+E+PS)) + kps ( PS/ (D+E+PS))
where,
WACC = Weighted Average Cost of Capital
ke = Cost of Equity
kd = Aftertax Cost of Debt
kps = Cost of Preferred Stock
E/(E+D+PS) = Market Value proportion of Equity in Funding Mix
D/(E+D+PS) = Market Value proportion of Debt in Funding Mix
PS/(E+D+PS) = Market Value proportion of Preferred Stock in Funding Mix
Illustration 9: Calculating the Cost of Capital: Genzyme Corporation.
In December 1994, Genzyme Corporation had
* a beta of 1.60. This results in
Cost of equity = 8.00%+1.60 (5.50%) = 16.80%
* an aftertax cost of debt of 6.30%. (Pretax cost of debt=9.00%; Tax rate=30%)
* Equity comprised 85.00% (in market value terms) of the funding mix and
debt made up the remaining 15.00%.
* The cost of capital for Genzyme can then be calculated as follows:
WACC = 16.80% (0.85) + 6.30% (0.15) = 15.23 %
ESTIMATION OF CASH FLOWS
Cash flows to Equity for a Levered Firm
Revenues
 Operating Expenses
= Earnings before interest, taxes and depreciation (EBITDA)
 Depreciation & Amortization
= Earnings before interest and taxes (EBIT)
 Interest Expenses
= Earnings before taxes
 Taxes
= Net Income
+ Depreciation & Amortization
= Cash flows from Operations
 Preferred Dividends
 Capital Expenditures
 Working Capital Needs
 Principal Repayments
+ Proceeds from New Debt Issues
= Free Cash flow to Equity
Levered Firm at Desired Leverage
Net Income
 (1 DR) (Capital Expenditures  Depreciation)
 (1 DR) Working Capital Needs
= Free Cash flow to Equity
For this firm,
Proceeds from new debt issues = Principal Repayments +DR (Capital Expenditures
 Depreciation + Working Capital Needs)
Illustration 10: Estimating the cash flow to equity for a firm at its
desired leverage: WarnerLambert
The following is an estimation of free cash flows to equity for WarnerLambert
in 1994 and for 1995 (projected).
 The company had $1.5 billion in debt outstanding and $ 9 billion in
market value of equity, leading to a debt to capital ratio (ìdebt
ratioî) of 14%. This debt ratio is assumed to be stable. (How does
one know?)
 The company reported net income of $695 million in 1994 and is projected
to have a net income of $765 million in 1995.
 The company reported depreciation of $180 million in 1994 and is expected
to have depreciation of $200 million in 1995.
 The company had capital expenditures of $362 million in 1994 and is
expected to have capital expenditures of $400 million in 1995.
 The companyís working capital increased to $225 million in 1994
from $203 million in 1993. It is expected to maintain working capital at
the same percentage of sales in 1995, and sales are expected to increase
from $6420 million in 1994 to $7100 million in 1995.

1994 
Estd 1995 

Net Income 
$ 695.00 
$ 765.00 

 (Cap Ex  Depreciation) (1DR) 
$ 156.52 
$ 172.00 
(DR = 14%) 
 ( Change in Working Capital) (1DR) 
$ 18.92 
$ 20.50 
(DR = 14%) 
FCFE 
$ 519.56 
$ 572.50 

Proposition : The Free Cash Flow to Equity will increase as the amount of
debt financing used by the firm increases. Thus FCFE will be an increasing
function of _.
Illustration 11: Sensitivity to Debt Ratio  Warner Lambert Corporation
The following are the cash flows to equity for Warner Lambert, using a debt
ratio of 40% instead of a debt ratio of 14%.

1994 
Estd 1995 

Net Income 
$ 695.00 
$ 765.00 

 (Cap Ex  Depreciation) (1DR) 
$ 109.20 
$ 120.00 
(DR = 40%) 
 ( Change in Working Capital) (1DR) 
$ 13.20 
$ 14.30 
(DR = 40%) 
FCFE 
$ 572.60 
$ 630.70 

The following graph illustrates the effect on free cash flows to equity
of changing the debt ratios from 0% to 100% ñ
CASHFLOWS TO THE FIRM
EBIT ( 1  tax rate)
+ Depreciation
 Capital Spending
 Change in Working Capital
= Cash flow to the firm
Illustration 12: Estimating the expected cash flow to the firm  Siemens
Siemens AG, as a Germanbased multinational involved in a wide range of
businesses, reported earnings before interest and taxes of 3,482 million
DM in 1992, prior to general provisions and extraordinary charges. It had
depreciation of 4,613 million DM and capital expenditures of 5,560 million
DM in 1992. In addition, the working capital increased from 14,306 million
DM in 1991 to 15,405 million DM in 1992. The earnings before interest and
taxes is expected to increase to 3,967 million DM in 1993. Capital expenditures,
depreciation and working capital are all expected to increase by 5% in 1993.
The firm had a tax rate of 38% in 1992. The free cashflows to the firm for
1992 and 1993 (estimated) are provided below.
1992 Projected 1993
EBIT (1  tax rate) 2,159 mil DM 2,460 mil DM
+ Depreciation 4,613 mil DM 4,844 mil DM
 Capital Expenditures 5,560 mil DM 5,838 mil DM
 Change in Working Capital 1,098 mil DM 770 mil DM
Free Cashflow to firm 114 mil DM 696 mil DM
INFLATION AND VALUATION
Consistency Principle 1: Nominal cash flows should be discounted
at nominal discount rates; Real cash flows should be discounted at real
discount rates.
Illustration 13: The Effect of Inflation on Cash flows and Discount Rates
Consider a firm which has cash flows to equity currently of $100 million
and is expected to grow, in real terms, at 5% a year for the next three
years and 3% a year after that. The firm has a beta of 1.0 and the current
T.Bond rate is 6.5%. The expected inflation rate is 3% in both the firm's
cash flows and the general economy. The valuation of this firm can be done
on either a real or a nominal basis:
The estimates of growth on a real and nominal basis are done first
Real Nominal
Growth Rate in first 3 years 5% (1.05)(1.03)1 = 8.15%
Growth Rate after year 3 3% (1.03)(1.03)1 = 6.09%
The discount rate can similarly be estimated on a real and nominal basis:
Real Nominal
Discount Rate 1.12/1.03 1 = 8.74% E(R) =6.5% +1(5.5%) =12%
The expected nominal return is estimated using the CAPM, with a historical
premium earned by stocks over T.Bonds of 5.5%.
Using these growth rates the cash flows can be generated in both nominal
and real terms:

Real Cash flows 
Nominal Cash flows 
Year 
CF to Equity 
Terminal Value 
Nominal Cash flows 
Terminal Value 
1 
$105 

$108 

2 
$110 

$117 

3 
$116 
$2,078 
$126 
$2,271 
The terminal values are calculated as follows for real and nominal cash
flows:
Terminal value (Real Cash flows) = $ 116 * 1.03 / (.0874  .03) = $ 2,078
Terminal value (Nominal Cash flows) = $ 126 * 1.0609 / (.12  .0609) = $
2,271
This calculation assumes that the growth rates after year 3 are steady state
growth rates and will continue through infinity.
The present values of the cash flows to equity and the terminal value can
then be calculated using the appropriate discount rate:
Present value (using real cash flows) = $105/1.0874 + $ 110 / 1.08742 +
($116 + $2,078) / 1.08743 = $ 1,896
Present value (using nominal cash flows) = $108/1.12 + $ 117 / 1.122 + ($126
+ $2,271) / 1.123 = $ 1,896
Illustration 13A: The Effect of mismatching cash flows and discount rates

Real Cash flows@ Nominal rate 
Nominal Cash flows@ Real rate 
Year 
CF to Equity 
Terminal Value 
Nominal Cash flows 
Terminal Value 
1 
$105 

$108 

2 
$110 

$117 

3 
$116 
$1,325 
$126 
$5,068 
The terminal values are miscalculated because cash flows and discount rates
are not matched.
Terminal value (using nominal rate and real cash flows) = $ 116 *1.03 /
(.12.03) = $1,325
Terminal value (using real rate and nominal cash flows) = $ 126 * 1.0609
/ (.0874 .0609) = $ 5,068
If real cash flows are discounted at the nominal discount rate, the present
value is:
Present value (real cash flows discounted at nominal rates) = $105/1.12
+ $ 110 / 1.122 + ($116 + $1,325) / 1.123 = $ 1,207 < True value of $1,896
If nominal cash flows are discounted at the real discount rate, the present
value is:
Present value (nominal cash flows discounted at real rates) = $108/1.0874
+ $ 117 / 1.08742 + ($126 + $5,068) / 1.08743 = $4,293 > True value of
$1,896
TAXES AND VALUATION
Consistency Principle 2: Pretax cash flows should be discounted
at pretax discount rates; Aftertax cash flows should be discounted at
aftertax discount rates.
Aftertax version of the Capital Asset Pricing Model
E(Rj) = Rf + d0 + d1 bj
+ d2 (gj  Rf)
where,
E(Rj) = Expected pretax return on asset j
Rf = Riskfree rate
bj = Beta of asset j
gj = Dividend Yield of asset j
d0 = A constant term
d1 = Market premium for systematic risk
d2 = Influence of dividend payout on expected
returns
Illustration 14: Effect of personal taxes on cash flows and discount
rates: Eli Lilly
The expected dividends and the terminal price were estimated for Eli Lilly
at the beginning of 1992 for the next five years (before personal taxes).
The firm had a beta of 1.05 and the treasury bond rate wass 7.15%. The following
example values Eli Lilly on the basis of cash flows after personal taxes
for an investor with a tax rate of 40% for ordinary income and 28% for capital
gains. The cash flows on a pre and post tax basis are as follows:

Before personal taxes 
After personal taxes 
Year 
Dividends per share 
Terminal price 
Dividends per share 
Terminal price 
1 
$2.27 

$1.36 

2 
$2.60 

$1.56 

3 
$2.98 

$1.79 

4 
$3.41 

$2.05 

5 
$3.91 
$93.43 
$2.35 
$84.41 
The discount rate can be estimated before personal taxes and used to calculate
the present value per share:
Discount rate before personal taxes = 7.15% + 1.05 (5.5%) = 12.93%
Present Value per share (based upon pretax cash flows and pretax discount
rates)
=$ 2.27/1.1293 + $ 2.60 / 1.12932 + $2.98/1.12933 + $3.41/1.12934 + ($3.91+$93.45)/1.12935
= $ 62.20
The initial price is assumed to be $62.20. The cash flows after personal
taxes are estimated after personal taxes as follows:
Dividends per share after taxes = Dividends per share before taxes * (1
 Ordinary tax rate)
Terminal price after taxes = Terminal price before taxes  (Terminal price
 Initial price) * Capital Gains tax rate = $ 93.43  ($93.43  $62.20)
* 0.28 = $84.41
The discount rate before personal taxes can be apportioned into dividend
yield and price appreciation:
Expected Dividend Yield = Expected Dividends next year/ Initial Price
=$2.27 / $62.20 = 3.71%
Expected Price Appreciation = Expected Return  Expected Dividend Yield
= 12.93%  3.71% = 9.22%
Discount rate after personal taxes = 3.71% (1  0.40) + 9.22% (1  0.28)
= 8.86%
Present Value per share (based upon aftertax cash flows and aftertax discount
rates)
=$ 1.36/1.0886 + $ 1.56 / 1.08862 + $1.79/1.08863 + $2.05/1.08864 + ($2.35+$84.41)/1.08865
= $ 62.20
Thus the value is unaffected by whether cash flows are before or after personal
taxes, as long as the discount rates are adjusted accordingly.
ESTIMATING GROWTH RATES
I. Estimating and Using Historical Growth Rates
 Historical growth rates can be estimated in a number of different ways

 Arithmetic versus Geometric Averages
 Simple versus Regression Models
 Historical growth rates can be sensitive to
 the period used in the estimation
 In using historical growth rates, the following factors have to be
considered
 how to deal with negative earning
 the effect of changing size
Illustration 15: Using arithmetic average versus geometric average:
Autodesk
The following are the earnings per share at Glaxo Pharmaceuticals, starting
in 1989 and ending in 1994:
Year 
EPS 
Growth Rate 
1989 
$0.66 

1990 
$0.90 
36.36% 
1991 
$0.91 
1.11% 
1992 
$1.27 
39.56% 
1993 
$1.13 
11.02% 
1994 
$1.27 
12.39% 
Arithmetic mean = (36.36%+1.11%+39.56%11.02%+12.39%)/5 = 15.68%
Geometric mean = (1.27/0.66)1/5 1 = 13.99%
Illustration 16: Sensitivity of historical growth rates to the length
of the estimation period: Glaxo
The following table provides earnings per share at Glaxo, starting in 1988
instead of 1989 and uses six years of growth rather than five to estimate
the arithmetic and geometric averages.
Time (t) 
Year 
EPS 
1 
1988 
$0.65 
2 
1989 
$0.66 
3 
1990 
$0.90 
4 
1991 
$0.91 
5 
1992 
$1.27 
6 
1993 
$1.13 
7 
1994 
$1.27 
Arithmetic average = 13.32%
Geometric mean = (1.27/0.65)1/6 1 = 11.81%
Illustration 17: Linear and Loglinear models of growth: Glaxo Inc.
The earnings per share from 1988 until 1994 is provided for Glaxo., and
the linear and log linear regressions are done below:
Time (t) 
Year 
EPS 
ln(EPS) 
1 
1988 
$0.65 
0.43 
2 
1989 
$0.66 
0.42 
3 
1990 
$0.90 
0.11 
4 
1991 
$0.91 
0.09 
5 
1992 
$1.27 
0.24 
6 
1993 
$1.13 
0.12 
7 
1994 
$1.27 
0.24 
Linear Regression : EPS = 0.5171 + 0.1132 t
Loglinear Regression: ln (EPS) = 0.5536 + 0.1225 t
Expected EPS (1995) : linear regression = 0.5171 + 0.1132 (8) = $1.42
Expected EPS (1995): loglinear regression = e (0.55536 + 0.1225 (8))
= $ 1.53
Dealing with negative earnings
Calculating growth rates when earnings become negative is problematic, since
the traditional growth rate measures often fail. For instance, if earnings
per share goes from <$1.00> to $1.00, the traditional growth rate
measures would be ñ
Arithmetic Growth Rate = = EPS_{t}/EPS_{t1} 1 = $1.00/<$1.00>
 1 = 200 %
Geometric growth rate = $1.00/<$1.00> = 100%
There is a solution to this problem,
Modified growth rate =(EPS_{t}EPS_{t1})/Max(EPS_{t},EPS_{t1})
Illustration 18: Dealing with negative earnings: Sterling Chemicals.
The following series lists earnings per share from 1988 to 1994 for Sterling
Chemicalsñ
Time (t) 
Year 
EPS 
log(EPS) 
Growth Rate 
Modified Growth Rate 
1 
1988 
$3.56 
1.27 


2 
1989 
$1.77 
0.57 
50.28% 
50.28% 
3 
1990 
$1.07 
0.07 
39.55% 
39.55% 
4 
1991 
$0.67 
0.40 
37.38% 
37.38% 
5 
1992 
$0.08 
2.53 
88.06% 
88.06% 
6 
1993 
($0.10) 
NMF 
225.00% 
225.00% 
7 
1994 
$0.34 
1.08 
440.00% 
129.41% 
Approach 1: Using the slope coefficient from the linear regression
EPS =3.1114  0.5139 t
Average EPS (198894) = $ 1.06
Growth rate =  0.5139 / 1.06 = 48.48%
Approach 2: Using the minimum or maximum of earnings as the denominator
Arithmetic average, using modified growth rates =  51.81%
Illustration 19: The Effect of size on growth: Amgen
Amgen increased its net income from $19.1 million in 1989 to $430 million
in 1994. The following table shows the growth in net income for Amgen from
1989 to 1994, in both percentage and dollar terms.
Year 
Net Income 
% Growth Rate 
Æ Net Income 
1989 
$19.10 


1990 
$86.20 
351.31% 
$67.10 
1991 
$186.30 
116.13% 
$100.10 
1992 
$306.70 
64.63% 
$120.40 
1993 
$354.90 
15.72% 
$48.20 
1994 
$430.00 
21.16% 
$75.10 
Geometric Average Growth Rate = 86.42%
Assuming that this growth rate continues for the next five years,
Year 
Net Income 
% Growth Rate 
Æ Net Income 
1995 
$801.60 
86.42% 
$371.60 
1996 
$1,494.32 
86.42% 
$692.72 
1997 
$2,785.67 
86.42% 
$1,291.35 
1998 
$5,192.98 
86.42% 
$2,407.31 
1999 
$9,680.63 
86.42% 
$4,487.65 
II. Analystsí Forecasts Of Earnings: How Good Are They?
Studies indicate that analysts do better than mechanical models at forecasting
earnings in the short term, but do not do much better than such models over
the long term.
Mean Relative Absolute Error
Study Analyst Group Analyst Mechanical
Forecasts Models
Collins & Hopwood Value Line Forecasts 31.7% 34.1%
197074
Brown & Rozeff Value Line Forecasts 28.4% 32.2%
197275
Fried & Givoly Earnings Forecaster 16.4% 19.8%
196979
Are some analysts more equal than others?
Some analysts are better at predicting earnings than other analysts. They
are also usually better at picking stocks. Studies examining how effective
analysts are conclude the following ñ
 On average, analysts affect stock prices with their recommendations;
Buy recommendations affect prices less than sell recommendations. The prices
tend to drift back to their original levels.
 The recommendations made by the ëbestí analysts (Institutional
Investorís All American Analysts) have a greater impact on stock
prices (3% on buys; 4.7% on sells). For these recommendations the price
changes are sustained, and they continue to rise in the following period
(2.4% for buys; 13.8% for the sells).
III. Expected Growth And Fundamentals
Retention Ratio and Return on Equity
gt = Retained Earningst1/ NIt1 * ROE
= Retention Ratio * ROE
= b * ROE
Proposition 1: The expected growth rate in earnings for a company cannot
exceed its return on equity in the long term.
Short term changes in ROE
Small changes in return on equity can lead to large shifts in the growth
rate for some firms. The relationship between growth rates and changes in
return on equity is as follows ñ
gt = [BV of Equityt1 * (ROEt  ROEt1) / NIt1]+
b * ROE
Illustration 20: Changes in ROE and growth rates: Merck Inc.
The following table provides information on the retention ratio and the
return on equity for Merck for 1994 and projections for 1995.
1994 1995
BV of Equity 11700
Net Income 3010
Retention Ratio 52.00% 52%
Return on Equity 26.0% 25.5%
Growth Rate = 0.52*0.26 = (11700*(.255.26)/3010)
= 13.52% + 0.52*0.255 = 11.32%
A drop in the return on equity of 0.5% translates into a drop in the growth
rate of 2.20%. If the ROE drops by 2% the expected growth rate in 1995 would
be 4.72%.
ROE and Leverage
ROE = ROA + D/E (ROA  i (1t))
where,
ROA = (Net Income + Interest (1  tax rate)) / BV of Total Assets
= EBIT (1 t) / BV of Total Assets
D/E = BV of Debt/ BV of Equity
i = Interest Expense on Debt / BV of Debt
t = Tax rate on ordinary income
Note that BV of Assets = BV of Debt + BV of Equity.
Return on Assets, Profit Margin and Asset Turnover
ROA = EBIT (1t) / Total Assets
= [EBIT (1t) / Sales] * [Sales/Total Assets]
= Preinterest profit margin * Asset Turnover
Illustration 22: Evaluating the effects of corporate strategy on growth
rate and value: Procter and Gamble
Procter & Gamble decided to reduce prices on their disposable diapers
in April 1993 to compete better with lowprice private label brands. The
preinterest, aftertax profit margin is expected to drop from 7.43% to
7% as a consequence and the asset turnover is expected to increase from
1.6851 to 1.80. The following table provides projections in profit margins,
asset turnover and growth rates after the shift in corporate strategy:
1992 After shift in strategy
EBIT (1 tax rate) $ 2181
Sales $ 29,362
Preinterest, aftertax profit margin 7.43% 7.00%
Total Assets $17,424
Asset Turnover 1.6851 1.80
Return on Assets 12.52% 12.60%
Retention Ratio 58.00% 58.00%
Debt/Equity 0.7108 0.7108
Interest rate on debt (1  tax rate) 4.27% 4.27%
Growth Rate 10.66% 10.74%
Illustration 23: Adjusting inputs for firm type: Neutrogena Inc.
The following table provides estimates of return on assets, retention ratio
and interest rates for Neutrogena, a cosmetics manufacturer. Neutrogena
is expected to grow at an extraordinary rate in the first five years (growth
phase) and at a stable rate after that (steady state).
Growth phase Steady State
Retention Ratio 76% 50%
Return on Assets 19.5% 15%
Debt/Equity 0% 25%
Interest rate on debt 10% 8.00%
Growth Rate 14.82% 8.375%
Illustration 24: Weighting based upon standard deviations: Autodesk Inc.
Consider Autodesk Inc,a software company, with earnings per share available
from 1987 to 1992. The following table also provides analyst forecasts of
expected growth over the next five years from nine analysts following Autodesk.
Historical EPS Analyst Estimates
Year 
EPS 
Growth Rate 

Analyst Number 
Estimated Growth 
1987 
$0.89 


1 
10.00% 
1988 
$1.35 
51.69% 

2 
10.50% 
1989 
$1.91 
41.48% 

3 
12.00% 
1990 
$2.30 
20.42% 

4 
12.50% 
1991 
$2.31 
0.43% 

5 
13.00% 
1992 
$1.98 
14.29% 

6 
16.00% 




7 
16.00% 




8 
18.00% 
Arithmetic mean = 19.95% Consensus Forecast = 14%
Standard Deviation = 27.49% Standard Deviation = 3.45%