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3 17 the last 4 years, we arrive at a modified payout ratio of 66.32%. If we substitute this payout ratio into the valuation in Illustration 13.3, the expected growth rate over the next 5 years drops to 8.42%: Expected growth rate = (1- Modified payout ratio) ROE = ( )(0.25) = 8.42% We will still assume a five year high growth period and that the parameters in stable growth remain unchanged. The value per share can be estimated. \$3.00 = (1.0842) 1- (1.0880) ( )( ) \$ (1.0880) P = \$56.75 Note that the drop in growth rate in earnings during the high growth period reduces earnings in the terminal year, and the terminal value per share drops to \$ This value is lower than that obtained in Illustration 13.3 and it reflects our expectation that P&G does not have as many new profitable new investments (earning a return on equity of 25%). Valuing an entire market using the dividend discount model All our examples of the dividend discount model so far have involved individual companies, but there is no reason why we cannot apply the same model to value a sector or even the entire market. The market price of the stock would be replaced by the cumulative market value of all of the stocks in the sector or market. The expected dividends would be the cumulated dividends of all these stocks and could be expanded to include stock buybacks by all firms. The expected growth rate would be the growth rate in cumulated earnings of the index. There would be no need for a beta or betas, since you are looking at the entire market (which should have a beta of 1) and you could add the risk premium (or premiums) to the riskfree rate to estimate a cost of equity. You could use a two-stage model, where this growth rate is greater than the growth rate of the economy, but you should be cautious about setting the growth rate too high or the growth period too long because it will be difficult for cumulated earnings growth of all firms in an economy to run ahead of the growth rate in the economy for extended periods. Consider a simple example. Assume that you have an index trading at 700 and that the average dividend yield of stocks in the index is 5%. Earnings and dividends can be expected to grow at 4% a year forever and the riskless rate is 5.4%. If you use a market risk premium of 4%, the value of the index can be estimated. Cost of equity = Riskless rate + Risk premium = 5.4% + 4% = 9.4%

4 18 Expected dividends next year = (Dividend yield * Value of the index)(1+ expected growth rate) = (0.05*700) (1.04) = 36.4 Expected dividends next year 36.4 Value of the index = = = 674 Cost of equity - Expected growth rate At its existing level of 700, the market is slightly over priced. Illustration 13.5: Valuing the S&P 500 using a dividend discount model: January 1, 2001 On January 1, 2001, the S&P 500 index was trading at The dividend yield on the index was only 1.43%, but including stock buybacks increases the modified dividend yield to 2.50%. Analysts were estimating that the earnings of the stocks in the index would increase 7.5% a year for the next 5 years. Beyond year 5, the expected growth rate is expected to be 5%, the nominal growth rate in the economy. The treasury bond rate was 5.1% and we will use a market risk premium of 4%, leading to a cost of equity of 9.1%: Cost of equity = 5.1% + 4% = 9.1% The expected dividends (and stock buybacks) on the index for the next 5 years can be estimated from the current dividends and expected growth of 7.50%. Current dividends = 2.50% of 1320 = Expected Dividends = \$35.48 \$38.14 \$41.00 \$44.07 \$47.38 Present Value = \$32.52 \$32.04 \$31.57 \$31.11 \$30.65 The present value is computed by discounting back the dividends at 9.1%. To estimate the terminal value, we estimate dividends in year 6 on the index: Expected dividends in year 6 = \$47.38 (1.05) = \$49.74 Expected Dividends6 \$49.74 Terminal value of the index = = = \$1213 r - g \$1213 Present value of Terminal value = \$ = The value of the index can now be computed: Value of index = Present value of dividends during high growth + Present value of terminal value = \$ \$ \$ \$785 = \$943 Based upon this, we would have concluded that the index was over valued at The Value of Growth

5 Investors pay a price premium when they acquire companies with high growth potential. This premium takes the form of higher price-earnings or price-book value ratios. While no one will contest the proposition that growth is valuable, it is possible to pay too much for growth. In fact, empirical studies that show low price-earnings ratio stocks earning return premiums over high price-earnings ratio stocks in the long term supports the notion that investors overpay for growth. This section uses the two-stage dividend discount model to examine the value of growth and it provides a benchmark that can be used to compare the actual prices paid for growth. Estimating the value of growth The value of the equity in any firm can be written in terms of three components: n (1+ g) DPS * (1+ g) * 1-0 n (1+ ke,hg ) P0 = + ke,hg - g (k where e,st DPSn +1 - g )(1+ k n e,hg ) n DPS1 - (k - g DPS + 1 (k e,st -g n ) Extraordinary Growth - DPS 0 k e,st Stable Growth + DPS 0 k e,st Assets in place DPS t = Expected dividends per share in year t k e = Required rate of return P n = Price at the end of year n g = Growth rate during high growth stage g n = Growth rate forever after year n Value of extraordinary growth = Value of the firm with extraordinary growth in first n e,st n ) years - Value of the firm as a stable growth firm 3 Value of stable growth = Value of the firm as a stable growth firm - Value of firm with no growth 19 3 The payout ratio used to calculate the value of the firm as a stable firm can be either the current payout ratio, if it is reasonable, or the new payout ratio calculated using the fundamental growth formula.

6 20 Assets in place = Value of firm with no growth In making these estimates, though, we have to remain consistent. For instance, to value assets in place, you would have to assume that the entire earnings could be paid out in dividends, while the payout ratio used to value stable growth should be a stable period payout ratio. Illustration 13.6: The Value of Growth: P&G in May 2001 In illustration 13.3, we valued P&G using a 2-stage dividend discount model at \$ We first value the assets in place using current earnings (\$3.00) and assume that all earnings are paid out as dividends. We also use the stable growth cost of equity as the discount rates. Current EPS \$3 Value of the assets in place = = = \$ k e,st To estimate the value of stable growth, we assume that the expected growth rate will be 5% and that the payout ratio is the stable period payout ratio of 66.67%: Value of stable growth = ( Current EPS)( Stable Payout Ratio)( 1+ g ) ( \$3.00)( )( 1.05) k e, st g n \$31.91 = \$15.81 n \$31.91 Value of extraordinary growth = \$ \$ \$15.81 = \$19.26 The Determinants of the Value of Growth 1. Growth rate during extraordinary period: The higher the growth rate in the extraordinary period, the higher the estimated value of growth will be. If the growth rate in the extraordinary growth period had been raised to 20% for the Procter & Gamble valuation, the value of extraordinary growth would have increased from \$19.26 to \$ Conversely, the value of high growth companies can drop precipitously if the expected growth rate is reduced, either because of disappointing earnings news from the firm or as a consequence of external events. 2. Length of the extraordinary growth period: The longer the extraordinary growth period, the greater the value of growth will be. At an intuitive level, this is fairly simple to illustrate. The value of \$19.26 obtained for extraordinary growth is predicated on the assumption that high growth will last for five years. If this is revised to last ten years, the value of extraordinary growth will increase to \$ Profitability of projects: The profitability of projects determines both the growth rate in the initial phase and the terminal value. As projects become more

7 21 profitable, they increase both growth rates and growth period, and the resulting value from extraordinary growth will be greater. 4. Riskiness of the firm/equity The riskiness of a firm determines the discount rate at which cashflows in the initial phase are discounted. Since the discount rate increases as risk increases, the present value of the extraordinary growth will decrease. III. The H Model for valuing Growth The H model is a two-stage model for growth, but unlike the classical two-stage model, the growth rate in the initial growth phase is not constant but declines linearly over time to reach the stable growth rate in steady stage. This model was presented in Fuller and Hsia (1984). The Model The model is based upon the assumption that the earnings growth rate starts at a high initial rate (g a ) and declines linearly over the extraordinary growth period (which is assumed to last 2H periods) to a stable growth rate (g n ). It also assumes that the dividend payout and cost of equity are constant over time and are not affected by the shifting growth rates. Figure 13.4 graphs the expected growth over time in the H Model. Figure 13.4: Expected Growth in the H Model g a g n Extraordinary growth phase: 2H years Infinite growth phase The value of expected dividends in the H Model can be written as: P 0 = DPS 0 * (1+g n ) (k e -g n ) + DPS 0 *H*(g a -g n ) (k e -g n )

8 22 where, Stable growth Extraordinary growth P 0 = Value of the firm now per share, DPS t = DPS in year t k e = Cost of equity g a = Growth rate initially g n = Growth rate at end of 2H years, applies forever afterwards Limitations This model avoids the problems associated with the growth rate dropping precipitously from the high growth to the stable growth phase, but it does so at a cost. First, the decline in the growth rate is expected to follow the strict structure laid out in the model -- it drops in linear increments each year based upon the initial growth rate, the stable growth rate and the length of the extraordinary growth period. While small deviations from this assumption do not affect the value significantly, large deviations can cause problems. Second, the assumption that the payout ratio is constant through both phases of growth exposes the analyst to an inconsistency -- as growth rates decline the payout ratio usually increases. Works best for: The allowance for a gradual decrease in growth rates over time may make this a useful model for firms which are growing rapidly right now, but where the growth is expected to decline gradually over time as the firms get larger and the differential advantage they have over their competitors declines. The assumption that the payout ratio is constant, however, makes this an inappropriate model to use for any firm that has low or no dividends currently. Thus, the model, by requiring a combination of high growth and high payout, may be quite limited 4 in its applicability. Illustration 13.7: Valuing with the H model: Alcatel Alcatel is a French telecommunications firm, paid dividends per share of 0.72 Ffr on earnings per share of 1.25 Ffr in The firm s earnings per share had grown at 12% over the prior 5 years but the growth rate is expected to decline linearly over the next 10 years to 5%, while the payout ratio remains unchanged. The beta for the stock is 0.8, the riskfree rate is 5.1% and the market risk premium is 4%. 4 Proponents of the model would argue that using a steady state payout ratio for firms which pay little or no dividends is likely to cause only small errors in the valuation.

9 23 Cost of equity = 5.1% + 0.8*4% = 8.30% The stock can be valued using the H model: Value of stable growth = ( )( ) = \$ / Value of extraordinary growth = ( )( )( ) = 7.64 Value of stock = = The stock was trading at Ffr in May IV. Three-stage Dividend Discount Model The three-stage dividend discount model combines the features of the two-stage model and the H-model. It allows for an initial period of high growth, a transitional period where growth declines and a final stable growth phase. It is the most general of the models because it does not impose any restrictions on the payout ratio. The Model This model assumes an initial period of stable high growth, a second period of declining growth and a third period of stable low growth that lasts forever. Figure 13.5 graphs the expected growth over the three time periods. Figure 13.5: Expected Growth in the Three-Stage DDM

10 24 EARNINGS GROWTH RATES g a g n High Stable growth Declining growth Infinite Stable growth DIVIDEND PAYOUTS High payout ratio Low Payout ratio Increasing payout ratio The value of the stock is then the present value of expected dividends during the high growth and the transitional periods and of the terminal price at the start of the final stable growth phase. P 0 = t=n1 EPS 0 *(1+g a ) t *Π a + (1+k e,hg ) t t=1 t=n2 DPS t + EPS *(1+g )*Π n2 n n (1+k e,t ) t (k e,st -g n )(1+r) n t=n1+1 High growth phase Transition Stable growth phase where, EPS t = Earnings per share in year t DPS t = Dividends per share in year t g a = Growth rate in high growth phase (lasts n1 periods) g n = Growth rate in stable phase Π a = Payout ratio in high growth phase Π n = Payout ratio in stable growth phase k e = Cost of equity in high growth (hg), transition (t) and stable growth (st) Assumptions

11 25 This model removes many of the constraints imposed by other versions of the dividend discount model. In return, however, it requires a much larger number of inputs - year-specific payout ratios, growth rates and betas. For firms where there is substantial noise in the estimation process, the errors in these inputs can overwhelm any benefits that accrue from the additional flexibility in the model. Works best for: This model's flexibility makes it a useful model for any firm, which in addition to changing growth over time is expected to change on other dimensions as well - in particular, payout policies and risk. It is best suited for firms which are growing at an extraordinary rate now and are expected to maintain this rate for an initial period, after which the differential advantage of the firm is expected to deplete leading to gradual declines in the growth rate to a stable growth rate. Practically speaking, this may be the more appropriate model to use for a firm whose earnings are growing at very high rates 5, are expected to continue growing at those rates for an initial period, but are expected to start declining gradually towards a stable rate as the firm become larger and loses its competitive advantages. Illustration 13.8: Valuing with the Three-stage DDM model: Coca Cola Coca Cola, the owner of the most valuable brand name in the world according to Interbrand, was able to increase its market value ten-fold in the 1980s and 1990s. While growth has leveled off in the last few years, the firm is still expanding both into other products and other markets. A Rationale for using the Three-Stage Dividend Discount Model Why three-stage? Coca Cola is still in high growth, but its size and dominant market share will cause growth to slide in the second phase of the high growth period. The high growth period is expected to last 5 years and the transition period is expected to last an additional 5 years. Why dividends? The firm has had a track record of paying out large dividends to its stockholders, and these dividends tend to mirror free cash flows to equity. The financial leverage is stable. Background Information Current Earnings / Dividends Earnings per share in 2000 = \$ The definition of a 'very high' growth rate is largely subjective. As a rule of thumb, growth rates over 25% would qualify as very high when the stable growth rate is 6-8%.

12 26 Dividends per share in 2000 = \$0.69 Payout ratio in 2000 = 44.23% Return on Equity = 23.37% Estimate a. Cost of Equity We will begin by estimating the cost of equity during the high growth phase, expected. We use a bottom-up levered beta of 0.80 and a riskfree rate of 5.4%. We use a risk premium of 5.6%, significantly higher than the mature market premium of 4%, which we have used in the valuation so far, to reflect Coca Cola s exposure in Latin America, Eastern Europe and Asia. The cost of equity can then be estimated for the high growth period. Cost of equity high growth = 5.4% (5.6%) = 9.88% In stable growth, we assume that the beta will remain 0.80, but reduce the risk premium to 5% to reflect the expected maturing of many emerging markets. Cost of equity stable growth = 5.4% (5.0%) = 9.40% During the transition period, the cost of equity will linearly decline from 9.88% in year 5 to 9.40% in year 10. b. Expected Growth and Payout Ratios The expected growth rate during the high growth phase is estimated using the current return on equity of 23.37% and payout ratio of 44.23%. Expected growth rate = Retention ratio * Return on equity = ( )(0.2337) = 13.03% During the transition phase, the expected growth rate declines linearly from 13.03% to a stable growth rate of 5.5%. To estimate the payout ratio in stable growth, we assume a return on equity of 20% for the firm: g 5.5% Stable period payout ratio = 1 - = 1- = 72.5% ROE 20% During the transition phase, the payout ratio adjusts upwards from 44.23% to 72.5% in linear increments. Estimating the Value These inputs are used to estimate expected earnings per share, dividends per share and costs of equity for the high growth, transition and stable periods. The present values are also shown in the last column table Table 13.3: Expected EPS, DPS and Present Value: Coca Cola Year Expected Growth EPS Payout ratio DPS Cost of Equity Present Value

13 27 High Growth Stage % \$ % \$ % \$ % \$ % \$ % \$ % \$ % \$ % \$ % \$ % \$ % \$ % \$ % \$ % \$0.79 Transition Stage % \$ % \$ % \$ % \$ % \$ % \$ % \$ % \$ % \$ % \$ % \$ % \$ % \$ % \$ % \$1.24 (Note: Since the costs of equity change each year, the present value has to be calculated using the cumulated cost of equity. Thus, in year 7, the present value of dividends is: \$1.96 PV of year 7 dividend = = \$ (1.0988) (1.0978) (1.0969) The terminal price at the end of year 10 can be calculated based upon the earnings per share in year 11, the stable growth rate of 5%, a cost of equity of 9.40% and the payout ratio of 72.5% - Terminal price = ( )( 0.725) \$ = \$84.83 The components of value are as follows: Present Value of dividends in high growth phase:\$ 3.76 Present Value of dividends in transition phase:\$ 5.46 Present Value of terminal price at end of transition:\$ Value of Coca Cola Stock :\$ Coca Cola was trading at \$46.29 in May 21, DDM3st.xlss: This spreadsheet allows you to value a firm with a period of high growth followed by a transition period where growth declines to a stable growth rate.

14 28

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