Unformatted text preview: WEB EXTENSION 9A The Required Return Assuming Nonconstant Dividends and Stock Repurchases A s we explained in the chapter, two assumptions underlie the constant dividend growth model: (1) firms do not repurchase any stock, and (2) growth in dividends will be constant. We showed how to incorporate stock repurchases by scaling up dividends by the factor 1 + Rep/Div, where Rep/Div is the number of dollars distributed through repurchases for every dollar paid in dividends. We also assume that the market is in equilibrium, which means that the required return is equal to the expected return. Given these assumptions, the expected return is r rM ^M 1 Rep=Div D1 g P0 (9A1) where g is the longterm growth rate in payouts (which should be the same as the longterm growth rates in sales and earnings). In what follows we explain how to modify the constant growth model to incorporate nonconstant payments. We also apply the constant growth model to the market and to an individual stock. 9.1 ESTIMATING THE EXPECTED MARKET RETURN In the chapter we assumed that the market's ratio of Rep/Div is 1.00, that the market has a projected dividend yield of 3%, and that the market's payouts (dividends and repurchases) have a longterm growth rate of 3.88%. Using Equation 9A1, the expected return on the market is therefore r rM ^M 1 Rep=Div D1 g 1 12:82% 3:88% 9:52% P0 Equation 9A1 (like Equation 97 in the chapter) incorporates the impact of stock repurchases, but it assumes that payouts grow at a constant rate. However, this will be true only if sales and earnings are also growing at a constant rate. This assumption is reasonable for the long term but not for the immediate future. Fortunately, nonconstant shortterm growth is easy to incorporate using Excel. We can use a version of the nonconstant growth approach in Chapter 7 (Equation 76), but instead of solving for the price we solve for the required return. Our objective is to create a time line showing payouts (dividends and repurchases) each year during the nonconstant growth phase. After growth stabilizes at a constant
1 2 Web Extension 9A: The Required Return Assuming Nonconstant Dividends and Stock Repurchases rate, we assume an initial estimate for the required return on the market and then estimate the horizon price using the constant growth formula in Equation 9A1. We find the present value of the payouts and the justestimated horizon price using the initial estimate of the required return on the market. This present value is our initial estimate of the "price" of the market. If this estimated price differs from the actual observed price, we use Excel's Goal Seek feature to change our estimate of the required return until there is a zero difference between estimated price and the actual price. The result is an estimate of the forwardlooking market risk premium that incorporates stock repurchases and nonconstant growth. We now give an example application of this approach. When we wrote this in April 2009, Standard & Poor's was forecasting a $21.97 dividend per share for the S&P 500 for 2009. If companies continue to pay out roughly $1.00 in stock repurchases for every $1.00 of dividends, this suggests that the 2009 total payout of dividends and repurchases will be around $21.97(1 + $1.00/$1.00) = $43.94. Standard & Poor's also estimates that the 2010 growth rate in dividends will be 3.08%. Analysts forecast individual companies' earnings growth rates out to 5 years, so with unlimited time and resources we could aggregate these forecasts into an overall forecast of earnings growth for the market. Because growth in total payouts is usually similar to growth in earnings, we could use the forecast earnings growth rates out to 5 years as estimates of total payout growth during the next 5 years. For illustrative purposes, we take the following shortcut. We have an estimate of longterm growth of 3.88% found in the chapter. We could assume that it would take many years until the constant growth rate is reached, but for illustrative purposes we assume that growth will become constant after Year 5 and beyond. To keep the example simple, we will assume that the growth rate for Year 2 (3.08%) changes by an equal amount each year until it equals the longterm growth rate of 3.88% beyond Year 5. Given these assumptions, the time line of expected growth rates and payouts is shown in Figure 9A1. For example, the payout at Year 1 is (1 + Rep/Div)D1 = (1 + 1.0) ($21.97) = $43.94. The payout at Year 2 is the payout at Year 1 multiplied by the growth rate from Year 1 to Year 2: Payout at Year 2 = $43.94(1 + 0.038) = $44.29. The payouts for Years 3 through 5 are calculated similarly. Next we explain the other calculations shown in Figure 9A1. Note that payouts grow at a constant rate after Year 5. Therefore, we can use the constant growth for^ mula to determine the value of the S&P 500 at Year 5 ( P 5 ) based on the Year5 payout and the longterm constant growth rate: Payout5 1 gL ^ P5 rM  gL (9A2) ^ To find the current price, P 0 , we must calculate the present value of the future expected payouts for Years 1 through 4 and then add to that ^ 5 , the present value P of the price at Year 5. This calculation is shown in Equation 9A3: ^ P0 # ^ Payout1 Payout2 P5 ... Payout5 5 1 2 1 rM 1 rM 5 1 rM 1 rM Payout5 1 gL " # Payout1 Payout2 rM  g ... Payout5 5 1 2 1 rM 1 rM 5 1 rM 1 rM " (9A3) Web Extension 9A: The Required Return Assuming Nonconstant Dividends and Stock Repurchases 3 FIGURE 9A1 Estimating the ForwardLooking Market Risk Premium with an Annual Time Line Key Input/Output: Estimate of rM Estimate of rM = Change the estimate of rM below so that the there is a zero difference between the actual and estimated prices (use Goal Seek). 8.84% Key Output Actual price level of S&P 500 Estimated S&P 500 price from time line below = Target zero differnce between actual and estimate $0.00 Actual price level of S&P 500 = Other Inputs Projected 1year dividend for S&P 500 = Projected repurchase to dividend ratio: Rep/Div = Projected longterm growth in sales, gL = Projected S&P 500 earnings growth from year 1 to 2 = Year Estimated growth rate in payouts = Payout = Dividend + Repurchases = P5 = [Payout5* (1+gL)] / (rMgL) = PV of payouts and PV of P5 (based on rM) = Estimated S&P 500 Price = $843.00 $843.00 = $21.97 1.0 3.88% 3.08% $43.94 3.08% $45.29 3.28% $46.78 3.48% $48.41 3.68% $50.18 $1,009.29 3.88% $843.00 This estimated price will change when we change the input for rM. We know the current price, the expected future payouts, and the longterm constant growth rate. If we assume that the current price is equal to the estimated price, we can substitute these known values into Equation 9A3: " $843 # $50:181 0:0388 rs  0:0388 1 rM 5 (9A3a) $43:94 $45:29 ... $50:18 1 2 1 rM 5 1 rM 1 rM resource
See the worksheet Web 9A in the file Ch09 Tool Kit.xls for an illustration of this approach. We have good news, bad news, and good news. The good news is that rM is the only unknown variable in Equation 9A2a, so we can solve for it. The bad news is that the only way to solve for rM is by an iterative process. But the second piece of good news is that the iterative process is easy to implement with a spreadsheet. Here are the steps we follow to implement this approach. First we assume an initial estimate of the required market return in cell G37. Our initial estimate was 10%. We used rM = 10% to estimate the horizon value at Year 5 using the constant growth formula shown in Figure 9A1. We then found the present value of the horizon value and the annual payouts using Excel's NPV function, which provides an estimate of the S&P 500's current price. With rM = 10%, the estimated S&P 500 price is $688.6; since the actual S&P 500 price is $843, we have a difference of $154.4 between the actual and estimated prices (the difference is shown in cell I43). We then used Goal Seek to set this difference (cell I43) to zero by changing the input for rM in cell G37. The result is rM = 8.84%, as shown in Figure 9A1. Rather than being a full year from Year 0 (now) until Year 1, we wrote this in April, which means the first period is not a full year. We can modify our approach 4 Web Extension 9A: The Required Return Assuming Nonconstant Dividends and Stock Repurchases FIGURE 9A2 Estimating the ForwardLooking Market Risk Premium with Actual Dates Estimate of rM = 8.97% Target zero difference between actual and estimated $0.00 Actual price level of S&P 500 Estimated S&P 500 price from timeline below = Actual price level of S&P 500 = Other Inputs Projected 1year dividend for S&P 500 = Projected repurchase to dividend ratio: Rep/Div = Projected longterm growth in sales, gL = Projected S&P 500 earnings growth from year 1 to 2 = Year Date Estimated growth rate in payouts = Payout = Dividend + Repurchases = P5 = [Payout5+ (1+gL)] / (rM gL) = 0 4/22/09 $843.00 $843.00 = $21.97 1.0 3.88% 3.08% 1 12/31/09 $43.94 2 12/31/10 3.08% $45.29 3 4 12/31/11 12/31/12 3.28% 3.48% $46.78 $48.41 5 6 12/31/13 12/31/14 3.68% 3.88% $50.18 $985.62 P1 = Time 1 PV of payouts 2 through 6 and P5 (based on rM) = Number of days in first period = PV of payout 1 and P1 (based on rM) = Estimated S&P 500 price = 253 $843.00 $853.10 This estimated price will change when rM changes. This estimated price will change when we change the input for rM. by finding the present value at Year 1 of the horizon value and the payouts for Year 2 though Year 5. As shown in Figure 9A2, the value at Year 1 is $853.10. We can then discount this amount back by the actual number of days in the first period. There are 253 days between 4/22/09 and 12/31/09, so the value at the start of the time line is ^ ^ P 1 Payout1 P0 rM 253 1 365 We then set the difference between the estimated price and the actual price to zero. Figure 9A2 illustrates this modified approach with actual dates. The resulting estimate is rM = 8.97%. Because the expected growth rates in the nonconstant period are similar to the longterm growth rate, the adjustment for nonconstant growth doesn't make a big difference: 8.97% versus 9.52% from Equation 9A1. But when growth rates in the nonconstant period are very different from the longterm growth rate, the approach in this Web extension should definitely be used. 9.2 ESTIMATING A STOCK'S EXPECTED RETURN Just as we previously did for the market, we can also estimate the cost of equity for a single company that is expected to have nonconstant growth before settling in to constant growth in the long term. For example, suppose that a company's current dividend Web Extension 9A: The Required Return Assuming Nonconstant Dividends and Stock Repurchases 5 FIGURE 9A3 Estimating a Stock's Cost of Equity Key Input/Output: Estimate of r,
Estimate of r, = Change the estimate of r, below so that the there is a zero difference between the actual and estimated prices (use Goal Seek). Key Output 11.77%
Target zero difference between actual and estimated = Actual stock price Estimated stock price from timeline below = Actual stock price = $32.00  $32.00 $0.00 Other Inputs
Current dividend = Projected repurchase to dividend ratio: Rep/Div = Projected longterm growth in sales, gL = Growth rate from Year 0 to Year 1 = Growth rate from Year 1 to Year 2 = Growth rate from Year 2 to Year 3 = Growth rate from Year 3 to Year 4 = Growth rate from Year 4 to Year 5 = Projected S&P 500 earnings growth from year 1 to 2 = Year Estimated growth rate in payouts = Payout = Dividend + Repurchases = P5 = [Payout5 * (1 + gL)] / (rs gL) = PV of payouts and PV of P5 (based on rs ) = Estimated stock price = $1.65 0.0 5.00% 10.4% 10.4% 10.4% 10.4% 10.4% 3.08% 1 10.4% $1.82 2 10.40% $2.01 3 10.40% $2.22 4 10.40% $2.45 5 10.40% $2.71 $41.95 6 5.00% 0 $32.00 This estimated price will change when we change the input for rs. is $1.65 per share and that the current actual price is $32 per share. Analysts forecast growth of 10.4% each year for the next 5 years and a longterm growth rate of 5% after the fifth year.1 Given these estimates, we can construct a time line with expected future dividend payments. For example, the dividend at Year 1 is D1 = D0(1 + g) = $1.65(1 + 0.104) = $1.82. The estimates of growth and expected future dividends for Years 1 through 5 are shown in Figure 9A3. Note that dividends grow at a constant rate after Year 5. Therefore, we can use the constant dividend growth formula to determine the price at Year 5, based on the Year5 dividend and the longterm constant growth rate: ^ D5 1 gL P5 rs  gL (9A4) To find the current price, P0, we must calculate the present value of the future expected dividend payments for Years 1 through 4 and then add to that P4, the present value of the price at Year 4. This calculation is shown in Equation 9A5: 1 If analysts give separate growth estimates for each year, then apply these as inputs to Figure 9A3. 6 Web Extension 9A: The Required Return Assuming Nonconstant Dividends and Stock Repurchases # D1 D2 P5 ... D5 p0 5 1 2 1 rs 1 rs 5 1 rs 1 rs D5 1 gL " # D1 D2 rs  g ... D5 5 1 2 1 rs 5 1 rs 1 rs 1 rs " (9A5) We know the current price, the expected future dividends, and the longterm constant growth rate. Substituting these known values into Equation 9A5 gives the following equation: " # $2:711 0:05 rs  0:05 1 rs 5 (9A5a) resource
See the worksheet Web 9A in the file Ch09 Tool Kit.xls for an illustration of this approach. 32:00 $1:82 $2:01 $2:71 ... 1 rs 5 1 rs 1 1 rs 2 As we did previously when estimating the market return, we can use the Goal Seek feature of Excel to solve for rs. Using this approach, we find that rs = 11.8%. ...
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