## 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 4.55%:

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:

XT , . Expected Dividendsn+, Value oi Equity in year n = —--—

r - g n 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 12.2.

Figure 12.2: Two-Stage Growth Model gn

### 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

PV of High - growth dividends0 =

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