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are public information and are easily accessible.5 The third is to consider fundamentals
and to estimate a growth rate based upon a firm™s investment policy. In particular, the
growth in earnings per share of a firm can be written as the product of two variables “ the
percentage of the earnings per share that is retained in the firm to generate future growth
(retention ratio) and a the return earned on equity in these new investments:
Expected Growth Rate = Retention Ratio * Return on Equity
Thus, a firm with a return on equity of 20% and a retention ratio of 70% should have
earnings growth of 14% a year. Reverting back to our discussion of dividend policy in
chapter 10, note that the retention ratio and the payout ratio are two sides of the same
Retention Ratio = 1 “ Payout Ratio
Since the retention ratio cannot exceed 100%, the expected growth in earnings per share
in the long term for a firm cannot exceed its return on equity.
Assuming that we can obtain all three estimates of the growth rate in earnings for
a firm, which one should we use in valuing a company? Historical growth should be
weighted the least, because earnings are volatile and past growth has generally not been

5 I/B/E/S, First Call and Zacks all track equity research analyst forecasts continuously and the consensus
estimate across all analysts is publicly available.


highly correlated with future growth.6 Analyst estimates are useful signposts of what the
investment community thinks about a company and could include information that is not
be in the financial statements. In particular, it could reflect changes in both the
company™s management and strategic plans. However, trusting analysts, no matter how
well informed they may be, to come up with the most important input in a valuation is not
prudent. Ultimately, the fundamental growth equation offers the most promise because it
relates growth back to what the firm does and also constrains us to pay for growth (by
requiring firms to reinvest) as we estimate value.

12.4. ˜: Differences in Growth Rates
The growth rates from historical earnings, analyst projections and fundamentals can often
be very different. These differences can be best explained by:
a. As firms become larger, the differences between growth rates will increase.
b. Analysts are biased towards making optimistic estimates of growth
c. The inputs used to estimate fundamental growth reflect what happened last year
rather than what we expect will happen in the future.
d. All of the above

Illustration 12.2: Growth in Earnings per share: Deutsche Bank
In 2003, Deutsche Bank reported net income of $1,365 million on a book value of
equity of $29,991 million at the end of 2002. The resulting return on equity for the firm is
Return on Equity = Net Income2003/ Book Value of Equity2002 = 1365/29,991 = 4.55%
This is lower than the cost of equity for the firm, which is 8.76%, and the average return
on equity for European banks, which is 11.26%. In the four quarters ended in March
2004, Deutsche Bank paid out dividends per share of 1.50 Euros on earnings per share of
4.33 Euros. The resulting retention ratio is 65.36%.
Retention Ratio = 1 “ Dividends per share/ Earnings per share = 1 “ 1.50/4.33 = 65.36%

6 One of the most famous studies of growth was titled “Higgledy Piggledy Growth” (Little, I.M.D., 1962,
Higgledy Piggledy Growth, Institute of Statistics, Oxford.) precisely because earnings growth was so
difficult to predict based upon history.


If Deutsche maintains its existing return on equity and retention ratio for the long term,
its expected growth rate will be anemic.
Expected Growth Rate Existing Fundamentals = Retention Ratio * ROE = .6536*.0455 = 2.97%
For the next five years, we will assume that the return on equity will improve to the
industry average of 11.26% while the retention ratio will stay unchanged at 65.36%. The
expected growth in earnings per share is 7.36%.
Expected Growth Rate Modified Fundamentals = .6536 * .1126 = .0736

c. Cost of Equity
The dividends and terminal price should be discounted back at a rate that reflects
the risk in the investment to stockholders to arrive at the current value. In chapter 4, we
argued that the only risk that diversified investors see in a stock is market risk and that
this risk can be measured with a beta (in the capital asset pricing model) or multiple betas
(in the arbitrage pricing or multi-factor models). The same reasoning applies here. In fact,
the costs of equity that we estimated for Disney, Deutsche Bank and Aracruz in chapter 4
will be the costs of equity that will be used if we were valuing stock in these companies
using a dividend discount model. The only point that relates specifically to valuation is
that high-growth firms tend to have higher betas than do low-growth firms. Building on
this premise, it is important that, as we change growth rates over time, we also adjust risk
accordingly. Thus, when a firm goes from high growth to low growth, its beta should be
moved towards one to reflect the lower growth.

d. Terminal Value
The last component of the model is the value that you attach to the equity at the end
of a period of high growth. This value is estimated from expected dividends in the first
time period following the high-growth period, the cost of equity in the stable phase, and
the expected stable growth rate in dividends as follows:
Expected Dividendsn +1
Value of Equity in year n =
rn " g n
where rn is the cost of equity in the stable growth period and gn is the expected growth rate in
dividends beyond year n (forever).
Before you estimate terminal value, you need to map out a path for the earnings
growth during the high growth phase to move towards the stable growth rate. The


simplest assumption to make is that your earnings growth rate is constant for the high
growth period, after which the growth rate drops to the stable level, as shown in Figure
Figure 12.2: Two-Stage Growth Model



High Growth Period Stable Growth Period
This is a two-stage model, and its limitation is obvious. It assumes that the growth rate is
high during the initial period and is transformed overnight to a lower, stable rate at the
end of the period. While these sudden transformations in growth can happen, it is much
more realistic to assume that the shift from high growth to stable growth happens
gradually over time. The assumption that the growth rate drops precipitously from its
level in the initial phase to a stable rate also implies that this model is more appropriate
for firms with modest growth rates in the initial phase. For instance, it is more reasonable
to assume that a firm growing at 12% in the high growth period will see its growth rate
drop to 4%, than it is for a firm growing at 40% in the high-growth period. If we assume
that the growth rate and payout ratio are fixed for the high growth period, the present
value of the dividends during the high growth period can be estimated as follows:7
" (1 + g) n %
Dividends0 * (1 + g) * $1- '
(1 + r) n &
PV of High - growth dividends0 =

7 Unlike the stable growth model equation, this one can be used even if the expected growth rate exceeds
the discount rate. While this makes the denominator negative, it will also result in a negative numerator,
and the net effect will be positive. The only condition when it will not work if g=r, but the PV of dividends
in that case will just be the product of the number of years of growth and dividends today since the growth
and the discounting effects each year will cancel out.


A more general formulation would allow for growth during the high growth period,
followed by a gradual reduction to stable growth over a transition period, as illustrated in
figure 12.3:

Figure 12.3: High Growth followed by transition



High Growth Transition Stable Growth Period
High Growth Period
This model allows for growth rates and payout ratios to change gradually during the transition
Whatever path you devise to get your firm to stable growth, it is not just the growth
rate that should change in stable growth. The other characteristics of the firm should also change
to reflect the stable growth rates.
The cost of equity should be more reflective of that of a mature firm. If it is being

estimated using a beta, that beta should be closer to one in stable growth even though it
can take on very high or very low values in high growth.
The dividend payout ratio, which is usually low or zero for high growth firms, should

increase as the firm becomes a stable growth firm. In fact, drawing on the fundamental
growth equation from the last section, we can estimate the payout ratio in stable growth:
Dividend Payout Ratio = 1 “ Retention Ratio = 1 “ Expected growth rate/ ROE
If we expect the stable growth rate to be 4% and the return on equity in stable growth to
be 12%, the payout ratio in stable growth will be 66.67% (1- 4/12).
The return on equity in stable growth, if used to estimate the payout ratio, should be also

reflective of a stable growth firm. The most conservative estimate to make in stable


growth is that the return on equity will be equal to the cost of equity, thus denying the
firm the possibility of excess returns in perpetuity. If this is too rigid a framework, you
can assume that the return on equity will converge on an industry average in the stable
growth phase.
If there is a transition period for growth, as in figure 12.3, the betas and payout ratios should
adjust in the transition period, as the growth rate changes.

12.5. ˜: Terminal Value and Present Value
The bulk of the present value in most discounted cash flow valuations comes from the
terminal value. Therefore, it is reasonable to conclude that the assumptions about growth
during the high growth period do not affect value as much as assumptions about the
terminal value.
a. True
b. False

Closing Thoughts on the Dividend Discount Model
Many analysts view the dividend discount model as outmoded but it is a useful
starting point in valuing all companies and may be the only choice in valuing companies
where estimating cashflows is not feasible. As we noted in chapter 11, estimating free
cashflows for financial service companies is often difficult to do both because the line
between operating and capital expenses is a fuzzy one and because working capital,
defined broadly, could include just about all of the balance sheet. While we can arrive at
approximations of cashflows by making assumptions about capital expenditures, we are
often left in the uncomfortable position of assuming that dividends represent free
cashflows to equity for these firms. Even for firms where we can estimate free cash
flows to equity with reasonable precision, the dividend discount model allows us to
estimate a “floor value” in most cases since firms tend to pay out less in dividends than
they have available in free cashflows to equity.
It is often argued that the dividend discount model cannot be used to value high
growth companies that pay little in dividends. That is true only if we use the inflexible
version of the model where future dividends are estimated by growing current dividends.


In our more flexible version, where both payout ratios and earnings growth can change
over time, the dividend discount model can be extended to cover all types of firms.
There is one final point worth making in this section. We can estimate the value
of equity on a per share basis by using dividends per share or we can obtain the aggregate
value of equity using total dividends paid. The two approaches will yield the same
results, if there are no management options, warrants or convertible bonds outstanding. If
there are equity options, issued by the firm, that are outstanding, it is safest to value the
equity on an aggregate basis. We will consider how best to deal with equity options in
arriving at a value per share later in this chapter.

12.6. ˜: Payout Ratios and Expected Growth
The dividend discount model cannot be used to value stock in a company with high
growth, which does not pay dividends.
a. True
b. False

This file on the web contains, by sector, the industry averages for returns on
capital, retention ratios, debt equity ratios and interest rates.

Illustration 12.3: Valuing equity using the Dividend Discount Model: Deutsche Bank
In illustration 12.2, we estimated the annual growth rate for the next 5 years at
Deutsche Bank to be 7.36%, based upon an estimated ROE of 11.26% and a retention
ratio of 65.36%. In 2003, the earnings per share at Deutsche Bank were 4.33 Euros, and
the dividend per share was 1.50 Euros. Our earlier analysis of the risk at Deutsche Bank
provided us with an estimate of beta of 0.98, which used in conjunction with the Euro
riskfree rate of 4.05% and a risk premium of 4.82%, yielded a cost of equity of 8.76%
(see illustration 4.11).


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