Maintaining Consistency in Multistage Valuation Models

Transcription

1 Maintaining Consistency in Multistage Valuation Models by Larry C. Holland, PhD CFA University of Arkansas at Little Rock Little Rock, AR Telephone: (501) March 6,

2 Maintaining Consistency in Multistage Valuation Models Abstract Valuation analysis which uses the present value of future cash flows often requires a multistage valuation model with a terminal value based on a low continuous growth rate. The analysis in this paper illustrates the importance of maintaining consistent accounting relationships when forecasting future cash flows in a multistage valuation model. This is particularly important when there is a predicted reduction in growth rate underlying a terminal value and the terminal value results in a significant portion of the valuation of the current stock price. Failure to maintain a consistent accounting balance can result in a significant difference in valuation when using multistage valuation models. 2

3 Maintaining Consistency in Multistage Valuation Models Analysts routinely use multistage valuation models that calculate the present value of estimated future cash flows to recommend a fair price for a stock today. A very common approach is to use a higher growth rate for a five to ten-year period, and then a smaller long-term growth thereafter as the firm reaches a more mature phase. However, a change to a smaller growth rate frequently increases the cash flows from a company. For example, the life cycle theory of the firm suggests that a young firm typically has a high growth rate which declines over time to a lower, mature growth rate. The higher growth rate for a young firm typically leads to a significant need for cash to support that growth, while the lower growth for a mature firm often results in significant excess cash. Therefore, when the growth rate of a company begins to slow down as it approaches maturity, the retention rate will likely decrease and the level of dividends will increase. As a result, a mature firm is often characterized as a cash cow and is expected to pay a higher level of dividends. In terms of a valuation model, the financial variables in a forecast such as the retention rate, the capital structure, the capital intensity, and the gross profit margin need to remain in a consistent accounting relationship as a company transitions from a normal growth rate to a slower growth at maturity. Otherwise, a multistage valuation model will be misspecified if an accounting balance is not maintained in the financial factors used as input data to the model. This paper is divided into six sections. The first section sets the stage with a basic introduction and background for the simplest multi-stage valuation model. The second section focuses on the calculation of an appropriate growth rate to drive the valuation analysis. Then a series of pro forma financial statements for an example company provides a foundation for 3

4 calculating the future cash flows for a valuation analysis. The next section utilizes a pro forma forecast over multiple years to illustrate the concept of finding more value in multistage valuation models. The fifth section includes a valuation of Proctor and Gamble to demonstrate the basic idea of this paper with a simple valuation of a real company. Finally, the last section is a summary of the paper. Background At the most fundamental level, the cash flows to the holder of a stock are the future dividends. Thus, the value of a stock would be the present value of all expected future dividends, or D t V 0 = (1 + R E ) t t=1 (1) where V0 = the value at time zero, Dt RE = the dividend at time t, and = the required rate of return for equity cash flows. As a practical matter, estimating future dividends over an extended period of time can be difficult. Therefore, models have been developed that simplify the present value of future dividends. The most widely recognized of these models is the constant dividend growth model, mentioned by Williams (1938) and then popularized by Gordon and Shapiro (1956) and Gordon (1962). V 0 = D 1 R E g n (2) where gn = a constant long-term growth rate. 4

5 An estimate of the long-term growth rate in this model must be less than the required return on equity and normally should be no larger than the long-term growth in the overall economy. This restriction limits the direct application of this model because the current growth rate is frequently larger than the estimated long-term growth in the economy, and often larger than the required return on equity. Thus, a direct application of the constant dividend model for valuation is generally restricted to a very few mature companies with stable cash flows and low growth. More commonly, the constant dividend growth model is applied in a two-stage model. In this case, dividends are estimated for a finite number of years (T) using a faster growth rate. Then a terminal value (VT) estimates the remaining dividends with the constant dividend growth model using a low long-term growth rate (gn) suitable for a mature company, as follows: T D t V 0 = (1 + R E ) t + V T (1 + R E ) T t=1 (3) where D T+t V T = (1 + R E ) t = D T+1 (R E g n ) t=1 (4) As an example, one might assume that the current growth rate would continue for 10 years, and then a terminal value would reflect a constant 2% growth thereafter. Note that a low growth rate for the terminal value is not really a precise forecast that the growth rate will drop suddenly after year 10. Instead, the low growth rate is an artifact of the models we use to calculate the present value of future cash flows the terminal value is more of a placeholder for all the remaining cash flows, which acknowledges that only a low growth rate assumption is 5

6 appropriate for long-term growth in a mature firm. The sudden drop in growth rate is simply a rough (and probably conservative) way to show that the current growth rate will eventually be much lower in the future when the firm is mature, but would not necessarily designate a particular path to the future. In real life, the growth rate of a firm will more likely decline gradually over time. Likewise, in this paper, we are also not forecasting a precise sudden increase in cash flows at the point where the terminal value is applied. Rather, we are providing a method for calculating a level of cash flows consistent with an accounting balance and the assumptions for future growth. One particular issue is estimating the dividend at time T+1 (i.e., DT+1), which is often not straightforward. Although it might seem appropriate, a generally incorrect method of calculating DT+1 when there is a decrease in growth rate to gn is to assume that D T+1 = (1 + g n ) D T (5) Such an assumption is often not consistent with other assumptions. In many cases, a decrease in growth may increase the level of cash flows more than the lower growth rate would imply. For example, Damodaran in his excellent book on valuation illustrates a two stage valuation of Proctor and Gamble as of 2010 (see page 332 of Damodaran, 2012). Year 5 of his example shows a step function increase of the dividend from 3.08 per share to 4.75 per share when the growth rate in earnings decreases from 10% to 3% per year. In this paper, a simple pro forma analysis will show why the dividend (or other cash flow) must be calculated in a manner consistent with an accounting balance at a point of transition where there is a change in growth rates rather than simply growing the cash flows at an assumed lower growth rate. 6

7 Defining the Future Cash Flows and Growth Rates The future cash flows used in the constant dividend growth model are the future dividends, which are the only direct cash flows to equity shareholders. Capital gains from an increase in stock prices would then reflect an increased capacity to pay larger dividends in the future. From this viewpoint, anticipated future dividends are the most direct cash flows for applying valuation to equity shares. Of course, the constant growth model could be applied to other cash flows as well. When applied to other cash flows, the appropriate growth rate must correspond to the particular cash flows identified in the numerator of the model, and the required rate of return must also reflect the riskiness of the cash flows in the numerator. In addition to dividends, the most common cash flows used in valuation analysis are the free cash flow to the firm (FCFF), the free cash flow to equity (FCFE). These three cash flows are defined as FCFF = EBIT (1-t) + Depreciation - Capital Expenditures - NWC (6) or FCFF = EBIT (1-t) - IC (7) where EBIT = Earnings Before Interest and Taxes, t = the tax rate, ΔNWC = the change in net working capital, ΔIC = the change in invested capital, Net Working Capital = Current Assets - Current Liabilities, and Invested Capital = Assets Current Liabilities = NWC + Net Fixed Assets = Debt + Equity 7

8 The free cash flow to equity (FCFE) and dividends are defined as FCFE = FCFF Interest Expense (1 t) + Debt (8) Dividends = Net Income Additions to Retained Earnings (9) The FCFF, FCFE, and dividends are all a function of other variables in the valuation analysis, such as the profitability of the firm, asset efficiency, the assumed level of debt, and the earnings retention rate (b, or the fraction of net income retained). Changes in these other variables complicate the determination of the future growth rates to be used in the valuation analysis. For example, the growth rates for the three variables of earnings before interest and taxes (EBIT), net income, and dividends can be significantly different from each other. In addition to differing growth rates, the absolute levels of net income and dividends over time could also be affected by any assumed changes in the future, especially in the level of debt and the earnings retention rate. The approach used in this paper is to focus on the growth in sales, and then determine the level and growth rate of the resulting cash flows after assuming the profitability of the firm, the asset intensity, and an appropriate level of debt and earnings retention. This approach relies on a given pricing policy, product quality, and the availability of the products of a firm. Then the demand for the products (and thus the level of sales) would be exogenously determined by the customers in the marketplace. In response to this demand, a firm would establish a cost structure (such as the EBIT/Sales and IC/Sales ratios), a capital structure (Debt/IC ratio), and a dividend payout (i.e., the earnings retention, or b). 1 1 The reverse logic of first choosing an earnings retention and level of debt is awkward. In this case, the additions to retained earnings plus any increase in debt would result in an increase in invested capital, which in turn would lead to a given level of increased production. Then the price would need to be altered in order to sell the amount produced, which in turn affects the level of EBIT, net income, and dividends. Beginning with a growth in sales is more convenient and tractable. 8

9 An advantage of focusing on the growth in sales is that it is often useful to observe actual historical growth rates when estimating future growth rates. In this case, calculating the historical compounded growth in sales is very straightforward compared to calculating the growth rate in the resulting cash flows, such as dividends, FCFF, and FCFE. For example, sales are always positive. Therefore, a geometric growth rate or a log regression mean growth rate 2 can easily be determined. In contrast, EBIT and net income can occasionally be negative. Thus an appropriate compounded growth rate can be difficult and sometimes impossible to calculate for these variables. Even if normalized positive values are somehow assumed or generated, the result would then be a less objective measure of the growth rate. As a result, the approach in this paper is to use the growth in sales to drive the determination of resulting cash flows in the valuation analysis. In valuation analysis, it is important to maintain a consistent relationship between the growth rates and the re-investment rates. Often this is accomplished by assuming a relationship between an assumed growth rate, the retention rate (b), and the return on equity (ROE). For example, Bodie, Kane, Marcus (2015, p. xxx) and Damodaran (2012, p. 313) document a reinvestment rate with a version of the equation 3 where g t = b t ROE t (10) ROE t = NI t Equity t 1 (11) 2 A log regression mean growth rate is determined by regressing the log of a variable against the year. Then the resulting compounded growth rate is equal to EXP[Slope Coefficient] 1. This calculation requires all the dependent variable values to be positive (i.e., the log of a negative number is undefined). 3 The asterisk in ROE t* is a reminder that the denominator for this calculation is the Equity from the beginning of the year. The Return on Equity (ROE) is normally calculated from the most recent income statement and balance sheet using the Equity at the end of the year, as ROE t = NI t/equity t. Assuming clean surplus accounting, the sustainable growth rate (SGR) would then be calculated as SGR = (b t ROE t)/(1 b t ROE t). 9

10 This is an expression of the Sustainable Growth Rate (SGR) popularized by Robert Higgins (19XX). The SGR represents the growth in sales that will result in a financial balance that will maintain the same relative capital structure (i.e., using internally generated equity funds plus some additional debt up to the same Debt/Equity ratio). In a more general sense, the ROE can be further described with an extended DuPont equation, ROE t = [ EBIT t (1 t) Sales t Sales t ] [ (1 Interest t ) IC t EBIT t IC t Equity t 1 ] (12) ROE t = [ROIC t ] [ (1 Interest t EBIT t ) IC t Equity t 1 ] From this expression, it can be shown that an assumption of constant (EBIT/Sales) and (Sales/IC) will result in a constant relative capital structure (i.e., IC/Equity ratio) when the ROE is constant. This expression also shows that the ROE is a function of profitability (EBIT/Sales), asset intensity (Sales/IC), and capital structure (IC/Equity). As a result, choosing values for these three variables will establish a value for the ROE. And choosing values for the three more primitive variables allows calculation of any step function change in the level of dividends. Choosing the ROE first can result in ignoring a potential step function change in the level of dividends. Therefore, it is important to maintain the growth rate, capital structure, profitability, asset intensity, and retention rate in a consistent relationship for a valuation analysis to be valid. Illustration with an Example Company An example company is used to illustrate a method for maintaining consistency among assumptions when estimating future cash flows. In the process, several tables of pro forma 10

11 financial statements will be developed. Table 1 provides the initial financial statements for the example company and illustrates a pro forma estimate for the first year of an analysis assuming four possible growth rates in sales. For simplification purposes, each of the following variables will be assumed to remain constant 4 : EBIT Sales = = 0.15 Invested Capital Sales = IC Sales = = 0.70 Interest Expense = 10% of Interest Bearing Debt Tax Rate = 40% This leaves two remaining variables the relative level of debt in the capital structure (Debt/IC ratio) and the retention rate, b (or the percent of net income retained and reinvested within the company). When the retention rate is specified, then the relative level of debt is a result. Likewise, if the relative level of debt is specified, then the retention rate is a result. Appendix 1 shows a derivation of the Debt/IC ratio in terms of any future retention rate, b, or the retention rate for any future Debt/IC ratio, as follows: ( Debt IC ) = t+1 ( Debt IC ) + g t+1 t 1 + g ( EBIT Sales ) t+1 ( IC Sales ) (1 t) b t+1 (1 b t+1 i (1 t)) 4 One could also simply specify a future value for each of these variables rather than maintaining them constant. Holding these variables constant makes it easier to demonstrate the point of maintaining consistency in assumptions when modeling future cash flows. Also, holding constant the EBIT/Sales ratio, the IC/Sales ratio, and the tax rate will result in also holding constant the return on invested capital (ROIC), where ROIC = EBIT (1-t)/IC. A lower future ROIC can be assumed by reducing the gross margin (EBIT/Sales) and/or increasing the capital intensity (IC/Sales). 11

12 b t+1 = ( Debt IC ) t + g t+1 (1 + g t+1 ) ( Debt IC ) t+1 ( EBIT Sales ) Debt ( IC (1 t) i (1 t) ( Sales ) IC ) t+1 Calculating the sustainable growth rate (SGR) and the internal growth rate (IGR) can be helpful in estimating future cash flows and growth rates. The IGR is the growth rate in sales that would result in a financial balance when the absolute level of debt and the earnings retention rate (b) are maintained constant. The SGR is the growth rate in sales that would result in a financial balance when the level of debt is allowed to increase up to the same debt to equity ratio that the firm currently maintains. As mentioned earlier, an assumption that the EBIT margin (EBIT/Sales) and the Capital Intensity (IC/Sales) remain constant will result in a constant capital structure (Debt/Equity or Debt/IC) when sales grow at the SGR. Using the method of calculation in Holland (2012) and assuming a constant retention rate (b), the SGR and IGR for the example company are: SGR = 15 b ROE 1 b ROE = 0.80 ( 90 ) ( 15 = % 90 ) IGR = 15 b ROC 1 b ROIC = 0.80 ( 140 ) 30 (1 0.4) = 9.554% where ROE = Net Income = 15 = 16.67%, Total Equity 90 ROC = Net Income Invested Capital = = 10.71% 12

13 ROIC = EBIT (1 t) Invested Capital = 30 (1 0.4) 140 = 12.86%, b = retention rate or percent net income retained = 0.8, t = tax rate = 40%, and Invested Capital = Interest bearing debt plus total equity = (40+10) + (10+80) = 140. Given these assumptions and the calculations of the SGR and IGR, an important step in a valuation analysis is to determine a consistent forecast of the first year of future cash flows. Table 1 shows pro forma estimates for the example company for the first year of the forecast at four possible growth rates in sales, including growth rates at the IGR of 9.554%, 12%, the SGR of %, and 18%. Note from Table 1 that the internal growth rate is the growth rate that results in no new financing (only internally generated funds), and the level of debt and interest expense remains constant. With a constant retention rate, additional funding is required for growth rates higher than the IGR. In other words, when the growth rate in sales is above the IGR, additional funding (assumed to be debt) is required beyond internally generated funds in order to increase the invested capital, which in turn is required to support the higher growth rate in sales. Note that when sales grow at the SGR rate, the additional debt will be the amount that results in a constant debt to invested capital ratio, which is the definition of the SGR. With a constant retention rate, the debt to invested capital ratio also increases when sales grow more than the sustainable growth rate, and decreases when sales grow at less than the sustainable growth rate. The IGR and SGR thus provide a structure for estimating a consistent first year of future cash flows (such as dividends or FCFF). 13

14 Table 2 shows pro forma estimates for the first year of the analysis at the same four growth rates, but in this case the debt to invested capital ratio (Debt/IC) is held constant and the retention rate is allowed to vary. Note again that the retention rate remains the same as the initial actual data when sales grow at the SGR rate. However, with a constant Debt/IC ratio, the retention rate must decrease when the growth rate is lower than the SGR, and increase when the growth rate is higher than the SGR in order to be consistent with the assumptions. Tables 1 and 2 demonstrate that with the given assumptions, one can either specify the relative level of debt (Debt/IC ratio) or the retention rate (b), and the other variable will then be a result. Thus, only one particular combination of specifying both the debt to invested capital ratio and the retention rate would result in a consistent calculation (as shown earlier, and derived in Appendix 1). This sets the foundation that one should not arbitrarily choose both a future relative level of debt and a retention rate, because it would be easy to choose a combination inconsistent with what could actually occur. This principle becomes more apparent when reviewing the forecasts in Tables 3 and 4, which show a four year forecast for the example company. In this case, an assumption is made that sales grow at 12% for the next two years with the growth rate decreasing to a more stable growth of 2% per year thereafter 5. The idea in this illustration is to focus on the cash flows that occur during the first year of the lower long-term, stable growth rate. 5 This example utilizes a dramatic change in growth rate from 12% to 2% in one year primarily to illustrate in an exaggerated way that cash flows generally increase when the growth rate decreases. In an actual valuation analysis, a multi-stage model would more likely be utilized with a slower decline in growth rates over time. However, the same principle applies but with a less exaggerated effect over time. 14

15 A Pro Forma Forecast Over Multiple Years Table 3 assumes that the relative level of debt remains constant in the future at 35.71% of the invested capital, which automatically determines a retention rate (b) for a given growth rate. For example, note that the assumed growth rate in sales of 12% per year for two years is below the SGR. Therefore, the retention rate in the first year of the pro forma statements must decrease to in year one to maintain consistency with a constant 35.71% debt to capital ratio. This in turn forces the dividend to double from 3.0 to 6.0 in year 1 in order to be consistent with the assumed growth rate. Year 2 shows a lower 12% increase in dividends to 6.72, which is the same as the growth in sales. Then in year 3, the growth rate in sales decreases to 2% per year. The retention rate must again decrease to to be consistent with a 2% growth rate in sales and a constant 35.71% debt to capital ratio. This also means that the dividend in year 3 must nearly triple to to maintain consistency. The reason for this step function increase is the decrease in the change in equity. Note that the annual change in equity from year zero is 6.000, 6.720, and for the first three years. The decrease in the change in equity in the third year means that a lower level of additions to retained earnings is required to be subtracted from net income 6. This is the underlying reason that the retention rate decreases from to from year 2 to year 3 and the dividend nearly triples. After the step function increase in dividends in year 3 because of the decline in the growth rate in sales, the dividend will then continue with a constant 2% growth rate thereafter. This example clearly points out that the dividend in any particular year does not always equal the dividend in the previous year multiplied by (1+g) when there is a significant change in 6 This follows directly from the lower increase in invested capital required to support the lower sales increase, and the constant Debt/IC ratio. 15

16 growth rate. The example also demonstrates that maintaining consistency among assumptions is very important. In this case, there is a significant increase in cash flows at the first year of the forecast as well as the point in the future when the growth rate declines. Thus any change in growth rate should normally be accompanied by a step function change in the level of the cash flows. In a more general sense, the level of dividends is a direct function of the change in the level of equity required to support the growth in the firm, given a constant Debt/IC ratio. This means that the dividend will change in a 1-year step function manner when the equity changes because of a change in growth rate. Appendix 2 shows that the change in dividends when the growth rate changes from gt to gt+1 can be calculated as DIV t+1 = DIV t (1 + g t+1 ) + [g t (1 + g t+1) (1 + g t ) g t+1] EQ t Note that the second term represents a 1-year step function change in dividends when there is a change in growth rates. Thus, when there is a decrease in the growth rate, there is a corresponding step function increase in the level of dividends. When the growth rate remains constant from year to year, the step function is zero and DIVt+1 = DIVt (1+gt+1). Appendix 2 also shows that a decrease in profitability (EBIT/Sales) or an increase in the asset intensity (IC/Sales) will reduce the size of the step function increase in dividends. It also indicates that the instantaneous growth in dividends may not always equal the instantaneous growth in sales. 16

17 In a similar manner, the level of free cash flow to the firm (FCFF) is a direct function of the change in the level of invested capital required to support the growth in the firm, given a constant Debt/IC ratio. The annual change in the level of invested capital from year zero is , , and for the first three years. Again note the decrease in the change in invested capital for the third year. Recall that the formula for calculating the FCFF is EBIT (1-t) minus the change in invested capital. This means that the FCFF will increase in a 1-year step function manner when the change in invested capital decreases because of a change in the growth rate of sales. Appendix 3 shows that the change in FCFF when the growth rate changes from gt to gt+1 can be calculated as FCFF t+1 = FCFF t (1 + g t+1 ) + [g t (1 + g t+1) (1 + g t ) g t+1] IC 1 Note that the second term represents a 1-year step function change in FCFF when there is a change in the growth rate of sales, and that this step function goes to zero when the growth remains constant. The effect on valuation can be significant. Using the dividends in Table 3 (as Billion $) and assuming 2.75 billion shares outstanding and a 15% required return on equity, the value per share of the equity would be V 2 = D 3 R g = = V 0 = D 1 (1 + R) + D 2 (1 + R) 2 + V 2 (1 + R) 2 17

18 V 0 = 6.00 (1.15) (1.15) = = (1.15) 2 V 0 (per share) = = If the step function increase in dividends is not counted, then the value per share would be much lower. In this case, the dividend in year 3 would be D3 = D2 (1+gn). Then the value per share would be V 2 = D 3 = D 2 (1 + g n ) = R g n R g n 6.72 (1.02) = = V 0 = D 1 (1 + R) + D 2 (1 + R) 2 + V 2 (1 + R) 2 V 0 = 6.00 (1.15) (1.15) = = (1.15) 2 V 0 (per share) = = In this example, the value of the equity per share more than doubles from $18.26 to $39.59 when the step function increase in dividends is included. Thus it is clear that there is a potential for finding significantly more value in multistage valuation models if consistent assumptions are maintained. A more dramatic effect occurs if the free cash flow to the firm (FCFF) is used instead of dividends for the future cash flows. In this case, the enterprise value (EV) of the firm is first calculated as the present value of the future free cash flows to the firm. Then the current level of debt is subtracted to yield the current value of equity. Dividing by the shares outstanding yields 18

19 the current value of the equity per share. Following this process and given a weighted average cost of capital (WACC) of 12.16%, the value per share of equity would be EV 2 = FCFF 3 WACC g = = EV 0 = FCFF 1 (1 + WACC) + FCFF 2 (1 + WACC) 2 + EV 2 (1 + WACC) 2 EV 0 = 3.36 (1.1216) (1.1216) = = (1.1216) 2 V 0 = EV 0 Debt 0 = = EV 0 (per share) = = If the step function increase in FCFF is not counted, then the value per share would be much lower. In this case, the FCFF3 = FCFF2 (1+g). Then the value per share of equity would be EV 2 = FCFF 3 WACC g = FCFF 2 (1 + g) WACC g = (1.02) = = EV 0 = FCFF 1 (1 + WACC) + FCFF 2 (1 + WACC) 2 + EV 2 (1 + WACC) 2 EV 0 = 3.36 (1.1216) (1.1216) = = (1.1216) 2 V 0 = EV 0 Debt 0 = = V 0 (per share) = =

20 In this case, the value of the equity per share turns out to be negative, which is not an appropriate result. Again, this illustrates that the step function increase in the FCFF needs to be included in a valuation analysis when the growth rate decreases for the terminal value calculation. Table 4 shows a different perspective by assuming that the retention rate is maintained constant through a change in growth rates, which results in a given relative level of debt. In this case, the 12% growth rate for two years is slightly under the SGR, so the relative level of debt decreases slightly for two years. However, when the growth rate decreases from 12% to 2% per year, the debt to capital ratio declines rapidly from 32.29% to 16.49% within two years. This is the same effect as before, only in this case the amount of new debt is decreased rather than increasing the dividend. A continuation of this trend would show that all debt would be paid off within four years after the decrease in growth. Thereafter, the cash balance would continue to increase dramatically as internally generated cash flows are beyond what is necessary to support the growth of the company. Clearly in this case, maintaining a constant retention with a decreasing level of debt is not typical of what normally occurs. Indeed, mature companies normally have a higher level of debt and pay a larger dividend than growing companies. However, such a non-typical forecast is not at all obvious when estimating a 12% growth rate declining to 2% while holding the retention rate constant. A more reasonable forecast would be to decrease the retention rate (pay higher dividends) when the growth rate declines rather than pay off debt. 7 For example, a reduction of the retention rate from 64.29% to 11.76% would cause the debt to capital ratio to remain constant at that point, as shown in Table 3. A retention 7 Instead of decreasing the retention rate (which increases the dividend), one could repurchase stock with the excess funds rather than pay off debt. The net effect is similar to increasing dividends. Analysts often count the sum of cash dividends plus net stock repurchases (defined as augmented dividends) to take this effect into account. Here, we assume that dividends are directly affected by the retention rate, rather than introducing the complication of stock repurchases from excess cash. 20

21 rate below 11.76% would cause the debt to capital ratio to increase because additional debt would be required to replace the lower additions to retained earnings. And a retention rate above 11.76% would cause the debt to capital ratio to decrease. This example again demonstrates that a future retention rate and relative level of debt should not be chosen arbitrarily. An Example Valuation of Proctor and Gamble as of 2012 Another example using an actual company provides a further illustration on finding additional value in the application of a multistage valuation model. Table 5 shows summarized financial statements for Proctor and Gamble for the year A log regression of the last 11 years of sales indicates a compounded growth of 7.93% per year. This leads to a forecast growth in sales of 8% per year for the next ten years. This growth rate is slightly higher than the SGR of 7.11%. Therefore, at a constant retention rate, the debt to capital ratio is expected to increase slightly over the next ten years from a level of 35.17% in 2012 to about 40% in ten years. Then the growth rate in sales is assumed to drop to a mature growth rate of 2% per year thereafter, similar to a long-term growth rate of the economy. The retention rate for 2012 is 0.445, which is expected to continue for the next 10 years. At that point, the retention rate is assumed to decrease to a level of With a lower retention level, the debt to invested capital ratio will then increase very modestly from 40% to a terminal level of 53% in the long run (i.e., 300 years). Other assumptions for this analysis include the following: 1. Interest-bearing debt remains 71.12% of total debt (2012 level including other liabilities), 21

22 2. Interest expense (net of interest income) remains at the 2012 level of 2.58% of interestbearing debt 3. EBIT margin constant at EBIT/Sales = 19% (2012 level), 4. Tax rate constant at t = 27% (2012 level), 5. Invested Capital to Sales constant at IC/Sales = 1.387, 6. Required rate of return on equity = 8%, 7. Special items, non-operating income, minority, and discontinued operations ignored. The most important thing to note is that the resulting dividend from these assumptions increases markedly in a step function manner for year 11 of the forecast because of the decline in growth rate to 2% per year. The dividend in year 10 is and the dividend in year 11 is Thus it is clear that the dividend in year 11 is not simply the dividend in year 10 times (1.02), which would have been Instead, the temporary growth in dividends in year 11 is over 60% higher because of other factors in the analysis, primarily the decrease in the retention rate. Thereafter, the dividend would continue to grow at 2% per year, equivalent to the growth in sales. The effect on valuation in this analysis is also significant. The present value of all dividends with the assumptions identified yields a value of Billion, divided by Billion shares outstanding, yields a value of $82.74 per share. A simple assumption that the dividend grows 12% for 10 years and 2% thereafter yields a value of only $60.28 per share. Thus, it is illustrated once again the importance of maintaining an accounting balance for all forecasted data when using a multistage valuation model. 22

23 Summary This paper demonstrates that an analyst must be very careful in applying a multistage valuation model in order to maintain consistency in the assumptions of the analysis. Specifically, there frequently may be an ignored value when a growth rate at a terminal value is assumed to decrease to a level near the long-term growth of the economy. A simple example is used to illustrate this effect. Given a consistent set of assumptions, it is shown that the level of future cash flows when the growth rate declines could be significantly higher than a simple projection of growth rates would indicate. Thus, care must be taken in estimating the retention rate and the relative level of debt. When one is specified, the other is a result. Therefore, when an analyst only specifies future growth rates in the cash flows of an analysis, there frequently may be additional value that is not counted when the growth rate declines. Finally, a simple valuation of Proctor and Gamble as of 2012 further demonstrates how easy it is to miss-specify a valuation model when primarily future growth rates are assumed. In this valuation, the dividend in 10 years was 60% higher than a simple projection of growth rates might indicate, and the resulting present value of all future cash flows was 37% higher. Therefore, the example valuations illustrate the importance of maintaining a consistent accounting balance for all forecasted data when using a multistage valuation model. 23

24 References Bodie, Kane, Marcus (2015), Essentials of Investments, New York: McGraw Hill. Damodaran, Aswath (2012), Investment Valuation (University Edition), 3 rd Edition, Hoboken, New Jersey: John Wiley & Sons. Gordon, Myron (1962), The Investment, Financing, and Valuation of the Corporation, IL: Richard D. Irwin. Gordon, Myron and Eli Shapiro (1956), Capital Equipment Analysis: The Required Rate of Profit, Management Science, 3 (1), October 1956, p Holland, Larry (2012), Improved Forecasting Under Alternative Growth Assumptions, Journal of Economics and Finance Education, Volume 11, No. 2, p Williams, John Burr (1938), The Theory of Investment Value, 1997 reprint, Fraser Publishing, c1938, Cambridge: Harvard University Press. 24

25 Table 1: Example Pro Forma Statements at Different Growth Rates: b Constant Pro Forma Pro Forma Pro Forma Pro Forma Actual IGR = 9.554% g = 12% SGR = % g = 18% Income Statement Sales CGS SG&A Depreciation EBIT Interest Expense EBT % Net Income Dividends Additions to RE FCFF Retention Rate (b) Balance Sheet Cash Accounts Receivable Inventory Current Assets Net Fixed Assets Total Assets Accounts Payable Notes Payable Current Liabilities LT Debt Common Stock Retained Earnings Total L&OE Invested Capital Notes Payable LT Debt Interest Bearing Debt Common Stock Retained Earnings Total Equity Invested Capital Debt/IC Ratio 35.71% 32.60% 33.95% 35.71% 37.01% Return on Equity 16.67% 16.17% 16.39% 16.67% 16.88% 25

26 Table 2: Example Pro Forma Statements at Different Growth Rates: Debt/IC Constant Pro Forma Pro Forma Pro Forma Pro Forma Actual IGR = 9.554% g = 12% SGR = % g = 18% Income Statement Sales CGS SG&A Depreciation EBIT Interest Expense EBT % Net Income Dividends Additions to RE FCFF Retention Rate (b) Balance Sheet Cash Accounts Receivable Inventory Current Assets Net Fixed Assets Total Assets Accounts Payable Notes Payable Current Liabilities LT Debt Common Stock Retained Earnings Total L&OE Invested Capital Notes Payable LT Debt Interest Bearing Debt Common Stock Retained Earnings Total Equity Invested Capital Debt/IC Ratio 35.71% 35.71% 35.71% 35.71% 35.71% Return on Equity 16.67% 16.66% 16.67% 16.67% 16.67% 26

27 Table 3: Four-Year Forecast for Example Financial Statements: Debt/IC Constant Year 1 Year 2 Year 3 Year 4 Actual g = 12% g = 12% g = 2% g = 2% Income Statement Sales CGS SG&A Depreciation EBIT Interest Expense EBT % Net Income Dividends Additions to RE FCFF Retention Rate (b) Balance Sheet Cash Accounts Receivable Inventory Current Assets Net Fixed Assets Total Assets Accounts Payable Notes Payable Current Liabilities LT Debt Common Stock Retained Earnings Total L&OE Invested Capital Notes Payable LT Debt Interest Bearing Debt Common Stock Retained Earnings Total Equity Invested Capital Debt/IC Ratio 35.71% 35.71% 35.71% 35.71% 35.71% Return on Equity 16.67% 16.67% 16.67% 16.67% 16.67% 27

28 Table 4: Four-Year Forecast for Example Financial Statements: b Constant Year 1 Year 2 Year 3 Year 4 Actual g = 12% g = 12% g = 2% g = 2% Income Statement Sales CGS SG&A Depreciation EBIT Interest Expense EBT % Net Income Dividends Additions to RE FCFF Retention Rate (b) Balance Sheet Cash Accounts Receivable Inventory Current Assets Net Fixed Assets Total Assets Accounts Payable Notes Payable Current Liabilities LT Debt Common Stock Retained Earnings Total L&OE Invested Capital Notes Payable LT Debt Interest Bearing Debt Common Stock Retained Earnings Total Equity Invested Capital Debt/IC Ratio 35.71% 33.95% 32.29% 24.51% 16.49% Return on Equity 16.67% 16.38% 16.13% 15.09% 14.21% 28

29 Table 5: Summarized Financial Statements for Proctor and Gamble Key Assumptions: Sales growth 8% for 10 Years, 2% thereafter Retention Rate 44.5% for 10 Years, 10% thereafter EBIT Margin = 19% Tax Rate = 27% Billion Shares Outstanding Interest = 2.58% of Interest Bearing Debt Invested Capital/Sales = Required Return Equity = 8% Present Value of Expected Future Dividends: Present Value without Year 11 extra value: per share per share 2012 Pro Forma Pro Forma Pro Forma Pro Forma Actual Year 9 Year 10 Year 11 Year 12 Sales CGS SG&A Depreciation EBIT Interest Expense EBT % Net Income x b x x x x 0.10 x 0.10 Additions to RE FCFF Dividends (pro forma) Current Assets Net Fixed Assets Total Assets Less Current Liabilities Invested Capital Interest Bearing Debt Other Liabilities Total Debt C/S, Surplus-Treasury Retained Earnings Invested Capital Debt/IC Ratio 36.08% 39.50% 39.76% 40.01% 40.24% Sustainable Growth 7.11% 7.50% 7.53% 1.59% 1.60% Return on Equity 14.91% 15.68% 15.74% 15.67% 15.72% 29

30 Appendix 1 Derive (Debt/IC)t+1 given bt+1, or bt+1 given (Debt/IC)t+1 Assume i = Interest Expense/Debt, and Tax Rate (t) are constant, and Sales from time t to time t+1 grow at the rate of gt+1, or Sales t+1 = Sales t (1 + g t+1 ) (1) Assume that (EBIT/Sales) and (IC/Sales) change by a factor of febit and fic respectively from time t to time t+1, then f EBIT ( EBIT t Sales t ) = ( EBIT t+1 Sales t+1 ) (1a) and IC t f IC ( ) = ( IC t+1 ) Sales t Sales t+1 (1b) Multiplying the left side of each equation by (1+gt+1)/(1+gt+1) and factoring out Sales yields EBIT t+1 = f EBIT EBIT t (1 + g t+1 ) (2) IC t+1 = f IC IC t (1 + g t+1 ) (3) The definition of Invested Capital (IC) is the sum of Debt plus Equity. Therefore Debt as a proportion Invested Capital is ( Debt IC ) t+1 = 1 Equity t+1 IC t+1 (4) Substituting Equation 3 into Equation 4 yields ( Debt IC ) t+1 = 1 Equity t+1 f IC IC t (1 + g t+1 ) (5) Assuming no issuance of new equity or repurchases and clean surplus accounting, an increase in Equity is solely due to the retention of Addition to Retained Earnings (ARE), defined as ARE t+1 = b t+1 (Net Income) t+1 (6) 30

31 Therefore, Equityt+1 is equal to Equity t+1 = Equity t + ARE t+1 = Equity t + b t+1 (Net Income) t+1 (7) Substituting Equation 7 into Equation 5 yields ( Debt IC ) t+1 = 1 Equity t + b t+1 (Net Income) t+1 f IC IC t (1 + g t+1 ) (8) Net Income is defined as EBIT less Interest Expense (INT) times one minus the tax rate, or (Net Income) t+1 = (EBIT t+1 INT t+1 ) (1 t) (9) Given a constant i, INT t+1 = i Debt t+1 (10) Substituting Equation 10 into Equation 9 yields (Net Income) t+1 = (EBIT t+1 i Debt t+1 ) (1 t) (11) Substituting Equation 2 into Equation 11 yields (Net Income) t+1 = b t+1 (f EBIT EBIT t (1 + g t+1 ) i Debt t+1 ) (1 t) (12) Substituting Equation 12 into Equation 8 yields ( Debt IC ) t+1 = 1 Equity t + b t+1 (f EBIT EBIT t (1 + g t+1 ) i Debt t+1 ) (1 t) f IC IC t (1 + g t+1 ) (13) Separating the last term in the numerator yields ( Debt IC ) = 1 Equity t + b t+1 (f EBIT EBIT t (1 + g t+1 )) (1 t) t+1 f IC IC t (1 + g t+1 ) b t+1 i (1 t) ( Debt IC ) (14) t+1 31

32 Subtracting the last term from both sides yields ( Debt IC ) + b t+1 i (1 t) ( Debt t+1 IC ) = 1 Equity t + b t+1 (f EBIT EBIT t (1 + g t+1 )) (1 t) t+1 f IC IC t (1 + g t+1 ) (15) Collecting terms of (Debt/IC) yields ( Debt IC ) (1 b t+1 i (1 t)) = 1 Equity t + b t+1 (f EBIT EBIT t (1 + g t+1 )) (1 t) t+1 f IC IC t (1 + g t+1 ) (16) Simplifying the right side yields ( Debt (1 1 ) ( 1 ) ( Debt IC ) f (1 b t+1 i (1 t)) = IC f IC IC ) + g t+1 t t+1 Solving for (Debt/IC)t+1 yields 1 + g t+1 ( f EBIT ) ( EBIT Sales ) f IC ( IC (1 t) b t+1 (17) Sales ) ( Debt IC ) = t+1 (1 1 f IC ) + ( 1 f IC ) ( Debt IC ) t + g t g t+1 ( f EBIT (1 b t+1 i (1 t)) ) ( EBIT Sales ) f IC ( IC Sales ) (1 t) b t+1 (18) Solving for bt+1 yields b t+1 = (1 1 f IC ) + ( 1 f IC ) ( Debt IC ) t + g t+1 (1 + g t+1 ) ( Debt IC ) t+1 ( EBIT Sales ) Debt ( IC (1 t) i (1 t) ( Sales ) IC ) t+1 (19) 32

33 If the (EBIT/Sales) and (IC/Sales) remain constant, then febit and fic will be equal to 1. Under this assumption, the (Debt/IC)t+1 simplifies to ( Debt IC ) = t+1 ( Debt IC ) + g t+1 t 1 + g ( EBIT Sales ) t+1 ( IC Sales ) (1 t) b t+1 (1 b t+1 i (1 t)) (20) Also, if the (EBIT/Sales) and (IC/Sales) remain constant, then bt+1 simplifies to b t+1 = ( Debt IC ) t + g t+1 (1 + g t+1 ) ( Debt IC ) t+1 ( EBIT Sales ) Debt ( IC (1 t) i (1 t) ( Sales ) IC ) t+1 (21) 33

34 Appendix 2 Derive DIVt+1 The dividend paid by a firm is given by the portion of Net Income that is not retained as Addition to Retained Earnings (ARE). This is given by the equation D t+1 = NI t+1 ARE t+1 (1) Assume ICt/Salest, i = Interest Expense/Debt, and Tax Rate (t) are constant. Also assume Sales from time t to time t+1 grow at the rate of gt+1, or Sales t+1 = Sales t (1 + g t+1 ) (2) Assume that (EBIT/Sales) and (IC/Sales) change by a factor of febit and fic respectively from time t to time t+1, then f EBIT ( EBIT t Sales t ) = ( EBIT t+1 Sales t+1 ) (3) and IC t f IC ( ) = ( IC t+1 ) Sales t Sales t+1 (4) Multiplying the left side of each equation by (1+gt+1)/(1+gt+1) and factoring out Sales yields EBIT t+1 = f EBIT EBIT t (1 + g t+1 ) (5) IC t+1 = f IC IC t (1 + g t+1 ) (6) Determine AREt+1 Assume Debtt/ICt is constant. Multiplying both sides of Equation 6 by the Debtt/ICt ratio yields Debt t+1 = f IC Debt t (1 + g t+1 ) (7) The definition of Invested Capital (IC) is the sum of Debt plus Equity. Therefore, subtracting Equation 7 from Equation 6 yields Equity t+1 = f IC Equity t (1 + g t+1 ) (8) 34

35 Subtracting Equityt from both sides of Equation 8 yields Equity t+1 Equity t = f IC Equity t (1 + g t+1 ) Equity t (9) or Equity t+1 = f IC g t+1 Equity t + (f IC 1) Equity t (10) Assuming no issuance of new equity and clean surplus accounting, a change in Equity is solely due to the retention of Addition to Retained Earnings (ARE) from Net Income. Therefore, the ARE is equal to the change in Equity, or ARE t+1 = f IC g t+1 Equity t + (f IC 1) Equity t (11) Adding and subtracting gt+1equityt to the right side of Equation 11 and simplifying yields ARE t+1 = g t+1 Equity t + (f IC 1) (1 + g t+1 ) Equity t (12) Determine NIt+1 Net income can be defined as the after tax difference between EBIT and Interest, or NI t+1 = (EBIT t+1 INT t+1 )(1 t) (13) The Interest expense is assumed to be the debt interest rate (i) times the level of debt. Inserting the relationship of Debtt and Debtt+1 from Equation 7 yields INT t+1 = i Debt t+1 = f IC i Debt t (1 + g t+1 ) (14) Substituting Equation 5 and 14 into Equation 13 yields NI t+1 = (1 + g t+1 ) (f EBIT EBIT t f IC i Debt t )(1 t) (15) Adding and subtracting EBITt and (i Debtt) in the middle term on the right side of Equation 15 and recognizing that (EBITt i Debtt)(1-t) is equal to NIt yields NI t+1 = (1 + g t+1 ) NI t (1 + g t+1 ) [(1 f EBIT ) EBIT t + (f IC 1) i Debt t ) (1 t)] (16) 35