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Text book: Michael C. Ehrhardt (2017). Corporate finance (6th edition). Boston:CengageL.

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When appraising mutually exclusive investments in plant and equipment, financial managers calculate the investments’ equivalent annual costs and rank the investments on this basis. Why is this necessary? Why not just compare the investments’ NPVs? Explain.

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Chapter

Tool Kit Chapter 7 10/27/15
Corporate Valuation and Stock Valuation
7-4 Valuing Common Stocks—Introducing the Free Cash Flow (FCF) Valuation Model
Data for B&B Corporation (Millions)
Constant free cash flow (FCF) = $10
Weighted average cost of capital (WACC) = 10%
Short-term investments = $2
Debt = $28
Preferred stock = $4
Number of shares of common stock = 5
The first step is to estimate the value of operations, which is the present value of all expected free cash flows. Because the FCF’s are expected to be constant, this is a perpetuity. The present value of a perpetuity is the cash flow divided by the cost of capital:
Value of operations (Vop) = FCF/WACC
Value of operations (Vop) = $100.00 million
B&B’s total value is the sum of value of operations and the short-term investments:
Value of operations $100
+ ST investments $2
Estimated total intrinsic value $102
The next step is to estimate the intrinsic value of equity, which is the remaining total value after accounting for the claims of debtholders and preferred stockholders:
Value of operations $100
+ ST investments $2
Estimated total intrinsic value $102
− All debt $28
− Preferred stock $4
Estimated intrinsic value of equity $70
The final step is to estimate the intrinsic common stock price per share, which is the estimated intrinsic value of equity divided by the number of shares of common stock:
Value of operations $100
+ ST investments $2
Estimated total intrinsic value $102
− All debt $28
− Preferred stock $4
Estimated intrinsic value of equity $70
÷ Number of shares 5
Estimated intrinsic stock price = $14.00
The figure below shows a summary of the previous calculations.
Figure 7-2
B&B Corporation’s Sources of Value and Claims on Value (Millions of Dollars except Per Share Data)
Inputs: Valuation Analysis
Constant free cash flow (FCF) = $10 Value of operations $100
Weighted average cost of capital (WACC) = 10% + ST investments $2
Short-term investments = $2 Estimated total intrinsic value $102
Debt = $28 − All debt $28
Preferred stock = $4 − Preferred stock $4
Number of shares of common stock = 5 Estimated intrinsic value of equity $70
÷ Number of shares 5
Estimated intrinsic stock price $14.00
Data for Pie Charts
Short-term investments = $2
Value of operations = $100
Total = $102
Debt = $28
Preferred stock = $4
Estimated equity value = $70
Total = $102
7-5 The Constant Growth Model: Valuation when Expected Free Cash Flow Grows at a Constant Rate
Case 1: The expected free cash flow at t=1 and the expected constant growth rate after t=1 are known.
First expected free cash flow (FCF1) = $105
Weighted average cost of capital (WACC) = 9%
Constant growth rate (gL) = 5%
When free cash flows are expected to grow at a constant rate, the value of operations is:
Value of operations (Vop) = FCF1 / [WACC-gL]
Value of operations (Vop) = $2,625
Case 2: Constant growth is expected to begin immediately.
Most recent free cash flow (FCF0) = $200
Weighted average cost of capital (WACC) = 12%
Constant growth rate (gL) = 7%
When free cash flows are expected to grow at a constant rate, the value of operations is:
Value of operations (Vop) = [FCF0(1+gL)]/[WACC-gL]
Value of operations (Vop) = $4,280
7-6 The Multi-Stage Model: Valuation when Expected Short-Term Free Cash Flow Grows at a Nonconstant Rate
Thurman Corporation’s expected free cash flows are shown below.
Year 0 1 2 3 4
FCF −$20 $80 $100 $110
Growth in FCF 25% 10%
Free cash flows are expected to grow at a 5% rate starting at Year 4 and to continue growing at a 5% rate for the foreseeable future. We know the free cash flow at Year 4 and we know that FCF grows at a constant rate after Year 4. Therefore, we set the horizon date at Year 4.
Free cash flow at beginning of the constant growth phase (FCF4) = $110
Weighted average cost of capital (WACC) = 15%
Constant growth rate (gL) = 5%
HV4 = Vop, at 4 = [FCF4 (1+gL)]/ [WACC-gL]
HV4 = Vop, at 4 = $1,155
Thurman’s time line of expected free cash flows and horizon value is shown below.
Year 0 1 2 3 4
FCF −$20 $80 $100 $110
Horizon value $1,155
Present value of HV4 = $660.375
Present value of free cash flows = $171.745
Total value of operations at Year 0, Vop, at t=0 = $832.120
There is more than one correct way to find the present value of the FCFs and the horizon value. For example, you could find the total cash flows, as shown below, which are equal to the free cash flows except for the last period, when they are equal to the sum of the free cash flow and the horizon value. (It is as though you received the FCF at Year 4 and then “sold” the operations and received cash equal to the horizon value.) You could then find present value of the combined free cash flows and horizon value.
Year 0 1 2 3 4
FCF −$20 $80 $100 $110
Horizon value $1,155
Combined FCF and HV −$20.00 $80.00 $100 $1,265
PV of combined FCF and HV = Total value of operations at Year 0, Vop, at t=0 =
$832.12
Here is a third way to find the present value of the FCFs and the horizon value. The basic idea is to find the value of operations at each date. For the last date, the value of operations is the horizon value. For the previous date, the value of operations is equal to the present value of the sum of the next date’s value of operations and FCF. For example, if you sell the operations immediately after receiving the FCF at Year 3, then the purchaser would receive the FCF at Year 4 plus the value of operations at Year 4 (which is the present value of all cash flows beyond Year 4).
Year 2018 2019 2020 2021 2022
FCF −$20.00 $80.00 $100.00 $110.00
Horizon value $1,155
Vop,t = (FCFt+1 + Vop,t+1)/ (1+WACC)
$832.12 $976.94 $1,043.48 $1,100.00 $1,155.00
Optional Material. You may have noticed that we could have defined the horizon date at Year 3 because we have an estimate of the Year 4 free cash flow, which is expected to grow at a constant rate thereafter. However, we recommend defining the horizon date as the last date in the forecast period even if growth becomes constant at or prior to this date because we have found that this leads to fewer errors. But we illustrate this approach below for the interested reader.
Free cash flow at beginning of the constant growth phase (FCF4) = $110
Weighted average cost of capital (WACC) = 15%
Constant growth rate (gL) = 5%
HV3 = Vop, at 3 = FCF4 / [WACC-gL]
HV3 = Vop, at 3 = $1,100
Thurman’s time line of expected free cash flows and horizon value is shown below.
Year 0 1 2 3
FCF −$20 $80 $100
Horizon value $1,100
Present value of HV4 = $723.268
Present value of free cash flows = $108.852
Total value of operations at Year 0, Vop, at t=0 = $832.120
Following is a summary of the steps used in estimating Thurman Corporation’s value of operations.
Figure 7-3
Thurman Corporation’s Value of Operations (Millions of Dollars)
INPUTS:
gL = 5%
WACC = 15% Projections
Year 0 1 2 3 4
FCF −$20.00 $80.00 $100.00 $110.00 ⟶ ⟶ ⟶ ↴
FCF1 FCF2 FCF3 FCF4 HV = Vop(t=4)
────── ────── ────── ──────
(1+WACC)1 (1+WACC)2 (1+WACC)3 (1+WACC)4 FCF4(1+gL)
─────────
(WACC− gL)
$115.500
PVs of FCFs −$17.391 ⟵ ⟵⤶ 10.00%
$60.491 ⟵⟵⟵⟵ ⟵⤶
$65.752 ⟵⟵⟵⟵ ⟵⟵⟵⟵ ⟵⤶ $1,155.000
$62.893 ⟵⟵⟵⟵ ⟵⟵⟵⟵ ⟵⟵⟵⟵ ⟵⤶
PV of HV $660.375 ⟵⟵⟵⟵ ⟵⟵⟵⟵ ⟵⟵⟵⟵ $1,155.000 ⟵⟵⟵⤶
= ──────
Vop = $832.12 (1+WACC)4
7-7 Application of the FCF Valuation Model to MicroDrive
We begin with MicroDrive’s most recent financial statements and selected additional data.
Figure 7-4
MicroDrive’s Most Recent Financial Statements (Millions, Except for Per Share Data)
INCOME STATEMENTS BALANCE SHEETS
2015 2016 Assets 2015 2016
Net sales $ 4,760 $ 5,000 Cash $ 60 $ 50
COGS (excl. depr.) 3,560 3,800 ST Investments 40 0
Depreciation 170 200 Accounts receivable 380 500
Other operating expenses 480 500 Inventories 820 1,000
EBIT $ 550 $ 500 Total CA $ 1,300 $ 1,550
Interest expense 100 120 Net PP&E 1,700 2,000
Pre-tax earnings $ 450 $ 380 Total assets $ 3,000 $ 3,550
Taxes (40%) 180 152
NI before pref. div. $ 270 $ 228 Liabilities and Equity
Preferred div. 8 8 Accounts payable $ 190 $ 200
Net income $ 262 $ 220 Accruals 280 300
Notes payable 130 280
Other Data Total CL $ 600 $ 780
Common dividends $48 $50 Long-term bonds 1,000 1,200
Addition to RE $214 $170 Total liabilities $ 1,600 $ 1,980
Tax rate 40% 40% Preferred stock 100 100
Shares of common stock 50 50 Common stock 500 500
Price per share $40.00 $27.00 Retained earnings 800 970
Total common equity $ 1,300 $ 1,470
Weighted average cost of capital (WACC) Total liabs. & equity $ 3,000 $ 3,550
10.50% 10.97%
The first step is to calculate the key performance measures that determine free cash flows.
Figure 7-5
Key Performance Measures for MicroDrive (Millions of Dollars)
MicroDrive Industry
2015 2016 2016
Calculating Net Operating Profit after Taxes (NOPAT)
NOPAT = EBIT(1 − T) $330 $300
Calculating Net Operating Working Capital (NOWC)
Operating current assets $1,260 $1,550
− Operating current liabilities $470 $500
NOWC $790 $1,050
Calculating Total Net Operating Capital (OpCap)
NOWC $790 $1,050
+ Net PP&E $1,700 $2,000
OpCap $2,490 $3,050
Investment in operating capital $560
Calculating Free Cash Flow (FCF)
FCF = NOPAT – Investment in operating capital −$260
Calculating Return on Invested Capital (ROIC)
ROIC = NOPAT/Total net operating capital 13.25% 9.84% 15.04%
Calculating the Operating Profitability Ratio (OP)
OP = NOPAT/Sales 6.93% 6.00% 6.92%
Calculating the Capital Requirement Ratio (CR)
CR = (Total net operating capital)/Sales 52.31% 61.00% 46.00%
The next step is to forecast sales, NOPAT, and total net operating capital. We do this by estimating future sales’ growth rates, operating profitability ratios, and capital requirement ratios, as shown in Panel A in the Figure below.
Yearly sales are forecast by letting the previous year’s sales increase by the forecasted sales growth rate. Operating profitability and total net operating capital in a forecasted year are assumed to be proportional to sales in that year.
Figure 7-6
MicroDrive’s Forecast of Operations for the Selected Scenario (Millions of Dollars, Except for Per Share Data)
Status Quo Industry MicroDrive MicroDrive
Panel A: Actual Actual Forecast
Operating Ratios 2016 2015 2016 2017 2018 2019 2020 2021
g = Sales growth rate 15% 5% 10% 8% 7% 5% 5%
OP = NOPAT/Sales 6.92% 6.9% 6% 6% 6% 6% 6% 6%
CR = OpCap/Sales 46.0% 52.3% 61% 61% 61% 61% 61% 61%
Tax rate 40% 40% 40% 40% 40% 40% 40% 40%
Panel B: Actual Forecast
Operating Items 2016 2017 2018 2019 2020 2021
Net sales $5,000 $5,500 $5,940 $6,356 $6,674 $7,007.270
Net operating profit after taxes $300 $330 $356 $381 $400 $420.436
Total net operating capital (OpCap) $3,050 $3,355 $3,623 $3,877 $4,071 $4,274.434
FCF = NOPAT – Investment in OpCap −$260 $25 $88 $128 $207 $216.892
Growth in FCF 252% 45.1% 61.7% 5.0%
ROIC = NOPAT/OpCap 9.84% 9.84% 9.84% 9.84% 9.84% 9.84%
Note: Numbers in the figure are shown as rounded for clarity in reporting. However unrounded values are used for all calculations.
The next step is to estimate the horizon value and the value of operations, beginning with the horizon value.
Free cash flow at beginning of the constant growth phase (FCF2021) = $216.892
Weighted average cost of capital (WACC) = 10.97%
Constant growth rate (gL) = 5%
HV2021 = Vop, 2021 = [FCF2021 (1+gL)]/ [WACC-gL]
HV2021 = Vop, 2021 = $3,814.678
MicroDrive’s time line of expected free cash flows and horizon value is shown below.
Year 2016 2017 2018 2019 2020 2021
FCF $25.000 $88.000 $127.710 $206.564 $216.892
Horizon value $3,814.678
Present value of HV = $2,266.887
Present value of free cash flows = $452.552
Total value of operations at Year 0, Vop, at t=0 = $2,719.439
The figure below shows a summary of these calculations.
Figure 7-7
MicroDrive Inc.’s Value of Operations (Millions of Dollars)
INPUTS: Scenario: Status Quo
gL = 5%
WACC = 10.97% Projections
Year 2016 2017 2018 2019 2020 2021
FCF $25.000 $88.000 $127.710 $206.564 $216.892 ⟶ ↴
FCF2017 FCF2018 FCF2019 FCF2020 FCF2021
────── ────── ────── ────── ────── HV = Vop(2021)
(1+WACC)1 (1+WACC)2 (1+WACC)3 (1+WACC)4 (1+WACC)5
FCF2021(1+gL)
─────────
(WACC− gL)
PVs of FCFs $22.529 ⟵⤶ $227.736
$71.461 ⟵⟵⟵⟵ ⟵⤶ 0.0597
$93.456 ⟵⟵⟵⟵ ⟵⟵⟵⟵ ⟵⤶
$136.217 ⟵⟵⟵⟵ ⟵⟵⟵⟵ ⟵⟵⟵⟵ ⟵⤶ $3,814.678
$128.889 ⟵⟵⟵⟵ ⟵⟵⟵⟵ ⟵⟵⟵⟵ ⟵⟵⟵⟵ ⟵⤶
PV of HV $2,266.887 ⟵⟵⟵⟵ ⟵⟵⟵⟵ ⟵⟵⟵⟵ ⟵⟵⟵⟵ $3,814.678 ⟵⟵⤶
= ──────
Vop = $2,719.44 (1+WACC)5
Note: Numbers in the figure are shown as rounded for clarity in reporting. However unrounded values are used for all calculations.
Estimating MicroDrive’s Intrinsic Stock Price per Share
Value of operations = $2,719.44
Weighted average cost of capital (WACC) = 10.97%
Short-term investments = $0
Short-term debt (notes payable) = $280
Long-term debt (bonds) = $1,200
Preferred stock = $100
Number of shares of common stock = 50
MicroDrive’s total value is the sum of value of operations and the short-term investments:
Value of operations $2,719.44
+ ST investments $0
Estimated total intrinsic value $2,719.44
The next step is to estimate the intrinsic value of equity, which is the remaining total value after accounting for the claims of debtholders and preferred stockholders:
Value of operations $2,719
+ ST investments $0
Estimated total intrinsic value $2,719.44
− All debt $1,480
− Preferred stock $100
Estimated intrinsic value of equity $1,139.44
The final step is to estimate the intrinsic common stock price per share, which is the estimated intrinsic value of equity divided by the number of shares of common stock:
Value of operations $2,719.44
+ ST investments $0
Estimated total intrinsic value $2,719.44
− All debt $1,480
− Preferred stock $100
Estimated intrinsic value of equity $1,139.44
÷ Number of shares 50
Estimated intrinsic stock price = $22.79
The figure below shows a summary of the previous calculations.
Figure 7-8
MicroDrive Inc.’s Intrinsic Stock Price (Millions, Except for Per Share Data)
INPUTS:
gL = 5%
WACC = 10.97%
Year = 2017 2018 2019 2020 2021
Projected FCF = $25 $88.0 $127.71 $206.564 $216.892
Horizon Value: Value of operations $2,719
+ ST investments $0
= $3,815 Estimated total intrinsic value $2,719
− All debt $1,480
Value of Operations: − Preferred stock $100
Present value of HV $2,266.89 Estimated intrinsic value of equity $1,139
+ Present value of FCF $452.55 ÷ Number of shares 50
Value of operations = $2,719.44 Estimated intrinsic stock price = $22.79
Note: Numbers in the figure are shown as rounded for clarity in reporting. However unrounded values are used for all calculations.
7-8 Do Stock Prices Reflect Long-Term or Short-Term Cash Flows?
Managers often claim that stock prices are “short-term” in nature in the sense that they reflect what is happening in the near-term and ignore the long-term. We can use MicroDrive’s results to shed light on this claim.
We previously estimated MicroDrive’s current value of operations. We also estimated MicroDrive’s horizon value at Year 5 and calculated its present value. If we divide the present value of the horizon value, we can estimate how much of MicrDrive’s value is due to cash flows occurring beyond Year 5. In other words, we can determine how much of MicroDrive’s value is due to long-term cash flows and how much is due to short-term cash flows.
Inputs:
Weighted average cost of capital = 10.97%
Horizon year = 5
Horizon value at Year 5 (HV5) = $3,814.678
Value of operations at Year 0 (Vop,0) = $2,719.439
Analysis:
Present value of the horizon value = $2,266.887
Value of operations at Year 0 = $2,719.439
Results:
Percent of current value due to long-term cash flows (i.e., PV of HV5) = 83%
Percent of current value due to short-term cash flows = 17%
For most stocks, the percentage of the current price that is due to long-term cash flows is over 80%.
7-9 Using the Free Cash Flow Valuation Model to Identify Value Drivers
We can use the free cash flow valuation model we developed previously for MicroDrive to determine how the inputs (sales growth, operating profitability, and capital requirements) affect the value of operations and intrinsic stock price. It is very easy to do this in Excel by using the Scenario Manager feature. Following is an explantion of how to use this feature.
The Scenario Manager allows you to specify values for particular cells and then save those values as a “scenario.” If you later change the values in the cells, perhaps to see the impact that the change has on an output, the Scenario Manager allows you to restore the saved scenario without having to re-input the original values. You can create numerous different scenarios, and you can even have the Scenario Manager create a summary that shows the values of the input cells and the values of the output cells for each scenario that you created.
To create a scenario, go to the Data tab in the menu, look in the Data Tools section for What-If Analysis, and then select Scenario Manager. This will open a dialog box that shows seven existing scenarios. If you select the button for “Add…”, you will get another dialog box asking you to give the scenario a name and to specify the “Changing cells.” The “Changing cells” are the cells with values that you want the Scenario Manager to save. For example, we want to save the values for MicroDrive’s estimated sales growth rates, operating profitabiltiy ratio, and capital requirement ratio.
After specifying the “Changing cells”, click “Ok” and you will get a new dialog box asking you to input the values into the changing cells that you want for this scenario. There will already be values shown, which are the values currently in those cells. So if you have already put the values into the cells in the Excel workbook, you won’t have to re-enter them in the dialog box, you can simply click “Ok” and you will have created a new scenario.
The original dialog box gives you several options, including adding a scenario, deleting a scenario, and editing a scenario. It also give you the option to run a “Summary.” If you select the “Summary” button, you get a dialog box asking you to specify some “Results” cells. For example, we specified the cells in this worksheet that have the value of operations, the intrinsic stock price, and the return on invested capital for the last year in the forecast horizon.
After selecting the “Results” cells, you can click “Ok” and the Scenario Manager will create a new worksheet named “Scenario Summary”. This new sheet contains the name of each scenario, the values in the “Changing cells”, and the values in the “Results cells. We copied the information from the “Scenario Summary” into the table below and then formatted the table to make it a bit more reader-friendy.
Figure 7-9
Value Drivers for MicroDrive Inc. (Millions, Except for Per Share Data)
Scenario Additional information not in textbook
(1) Status Quo (2) Higher Sales Growth (Only) (3) Higher Operating Profitability (Only) (4) Better Capital Utilization (Only) (5) Improve Growth and OP (6) Improve Growth and CR (7) Improve Growth, OP, and CR (8) Status Quo but Lower WACC (9) Better OP and CR
Inputs
Sales growth in 1st year 10% 11% 10% 10% 11% 11% 11% 10% 10%
Sales growth in 2nd year 8% 9% 8% 8% 9% 9% 9% 8% 8%
Sales growth in 3rd year 7% 8% 7% 7% 8% 8% 8% 7% 7%
Long-term sales growth (gL) 5% 6% 5% 5% 6% 6% 6% 5% 5%
Operating profitability (OP) 6% 6% 7% 6% 7% 6% 7% 6% 7%
Capital requirement (CR) 61% 61% 61% 52% 61% 52% 52% 61% 52%
Weighted average cost of capital (WACC) 10.97% 10.97% 10.97% 10.97% 10.97% 10.97% 10.97% 9.50% 10.97%
Results
Value of operations $2,719 $2,713 $3,682 $3,576 $3,880 $3,751 $4,918 $3,690 $4,538
Intrinsic stock price $22.79 $22.67 $42.04 $39.91 $46.00 $43.42 $66.76 $42.19 $59.16
Return on invested capital (ROIC) 9.84% 9.84% 11.48% 11.54% 11.48% 11.54% 13.46% 9.84% 13.46%
To better understand why growth doesn’t always add value, we can express the horizon value as:
If the numerator in the fraction in brackes, (1+gL) ROIC − WACC, is negative, then the value of operations will be less than the total net operation capital, OpCapT. If ROIC < WACC/(1+WACC), then growth hurts value. The 2-way data table below show the difference between the value operations and the amount of operating capital for different combinations of growth and ROIC. For input values, we use values similar to those of MicroDrive’s at the horizon.
Input Values (Base Case)
OpCap = $4,274
gL = 5.0%
ROIC = 9.84%
WACC = 10.97%
Output from equation above in yellow.
Vop = $3,815
Vop − OpCap = -$460
A Two-Way Data Table Showing How Combinations of Growth and ROIC Affect the Value of Operations Minus the Value of Operating Capital
Combinations of growth and ROIC that have Vop < OpCap are shown in pink. Notice that for very low values of ROIC, such as the first row with ROIC = 9.7%, growth reduces value (you can see this by looking across the row. For very high values of ROIC, such as the last row with ROIC = 11%, growth adds value. For other combinations, it depends on the relative values of growth, ROIC, and WACC.
gL
−$460 0.0% 2.5% 5.0% 7.5% 9.5%
ROIC 9.70% -$495 -$519 -$562 -$668 -$1,013
9.80% -$456 -$467 -$487 -$536 -$695
9.84% -$442 -$448 -$460 -$488 -$580
9.90% -$417 -$415 -$412 -$403 -$377
10.00% -$378 -$363 -$337 -$271 -$58
10.10% -$339 -$312 -$261 -$139 $260
10.20% -$300 -$260 -$186 -$6 $579
10.30% -$261 -$208 -$111 $126 $897
10.40% -$222 -$156 -$36 $259 $1,215
10.50% -$183 -$105 $39 $391 $1,534
10.60% -$144 -$53 $115 $524 $1,852
10.70% -$105 -$1 $190 $656 $2,171
10.80% -$66 $50 $265 $788 $2,489
10.90% -$27 $102 $340 $921 $2,807
11.00% $12 $154 $415 $1,053 $3,126
7-11 Valuing Common Stocks with the Dividend Growth Model
The Discounted Dividend Approach
The value of any financial asset is the present value of the future cash flows provided by the asset. When an investor buys a share of stock, he or she typically expects to receive cash in the form of dividends and then, eventually, to sell the stock and to receive cash from the sale. However, the price the first investor receives is dependent upon the dividends the next investor expects to earn, and so on for different generations of investors. Thus, the stock’s value ultimately depends on the cash dividends the company is expected to provide and the discount rate used to find the present value of those dividends.
Here is the basic dividend valuation equation:
P0 = D1 + D2 + . . . . DN
( 1 + rs ) ( 1 + rs ) 2 ( 1 + rs ) N
The dividend stream theoretically extends on out forever, i.e., to N = infinity. Obviously, it would not be feasible to deal with an infinite stream of dividends, but fortunately, a relatively simple equation has been developed that can be used to find the PV of the dividend stream, provided it is growing at a constant rate.
Valuing a Constant Growth Stock
In the constant growth model, we assume that the dividend and stock will grow forever at a constant growth rate. Naturally, assuming a constant growth rate for the rest of eternity is a rather bold assumption. However, considering the implications of imperfect information, information asymmetry, and general uncertainty, the assumption of constant growth is often reasonable. It is reasonable to guess that a given stock will experience ups and downs throughout its life. By assuming constant growth, we are trying to find the average of the good times and the bad times, and we assume that we will see both scenarios over the firm’s life. In addition to a constant growth rate, we also need the estimated long-term required return for the stock, and it too must be constant. If these variables are constant, our price equation for common stock simplifies to the following expression:
P0 = D1
( rs – gL )
Generally speaking, the long-run growth rate of a firm is likely to fall between 5% and 8% a year.
Example: Value of a Constant Growth Stock
A firm just paid a $1.15 dividend and its dividend is expected to grow at a constant rate of 8%. What is its stock price, assuming it has a required return of 13.4%?
D0 = $1.15
gL = 8%
rs = 13.4%
P0 = D1 = D0 (1 + gL) = $1.2420
( rs – gL ) ( rs – gL ) 0.0540
P0 = $23.00
Expected Rate of Return on a Constant Growth Stock
Using the constant growth equation introduced earlier, we can re-work the equation to solve for rs. In doing so, we are now solving for an expected return. The expression we are left is:
D1 + gL
P0
This expression tells us that the expected return on a stock comprises two components. First, it consists of the expected dividend yield, which is simply the next expected dividend divided by the current price. The second component of the expected return is the expected capital gains yield. The expected capital gains yield is the expected annual price appreciation of the stock, and is given by gL. This shows us the dual role of gL in the constant growth rate model. Not only does g indicate expected dividend growth, but it is also the expected stock price growth rate.
Example: Expected Rate of Return on a Constant Growth Stock
You buy a stock for $23, and you expect the next annual dividend to be $1.242. Furthermore, you expect the dividend to grow at a constant rate of 8%. What is the expected rate of return on the stock, and what is the dividend yield of the stock?
Inputs:
P0 $23.00
D1 $1.242
gL 8%
13.40%
Dividend yield = 5.40%
What is the expected price of this stock in 1 year?
Application of Constant Growth Model at t=1
P1 = D2
( rs – gL )
D2 = 1.34136
P1 = $24.84
Valuing Nonconstant Growth Stocks
For many companies, it is unreasonable to assume that they grow at a constant growth rate. Hence, valuation for these companies proves a little more complicated. The valuation process, in this case, requires us to estimate the short-run nonconstant growth rate and predict future dividends. Then, we must estimate a constant long-term growth rate at which the firm is expected to grow. Generally, we assume that after a certain point of time, all firms begin to grow at a rather constant rate. Of course, the difficulty in this framework is estimating the short-term growth rate, how long the short-term growth will hold, and the long-term growth rate.
Figure 7-10
Illustrative Dividend Growth at Different Rates Data for figure:
Growth Rates
Year Declining Zero Constant Nonconstant
1 -8% 0% 8% 30%
2 -8% 0% 8% 20%
3 -8% 0% 8% 10%
4 -8% 0% 8% 8%
5 -8% 0% 8% 8%
Dividend
Year Declining Growth: -8% Zero Growth Constant Growth: 8% Long-Term Growth: 8% Year Declining Growth: -8% Zero Growth Constant Growth: 8% Long-Term Growth: 8%
0 $1.15 $1.15 $1.15 $1.15 0 1.1500 1.1500 1.1500 1.1500
1 $1.06 $1.15 $1.24 $1.50 1 1.0580 1.1500 1.2420 1.4950
2 $0.97 $1.15 $1.34 $1.79 2 0.9734 1.1500 1.3414 1.7940
3 $0.90 $1.15 $1.45 $1.97 3 0.8955 1.1500 1.4487 1.9734
4 $0.82 $1.15 $1.56 $2.13 4 0.8239 1.1500 1.5646 2.1313
5 $0.76 $1.15 $1.69 $2.30 5 0.7579 1.1500 1.6897 2.3018
Specifically, we will predict as many future dividends as we can and discount them back to the present. Then we will treat all dividends to be received after the convention of constant growth rate with the Gordon constant growth model described above. The point in time when the dividend begins to grow constantly is called the horizon date. When we calculate the constant growth dividends, we solve for the horizon value (also called a terminal value or a continuing value) as of the horizon date. The horizon value can be summarized as:
HVT = PT = DT+1 = DT (1 + g)
( rs – gL ) ( rs – gL )
This condition holds true, where T is the horizon date. The horizon value can be described as the expected value of the stock at the time period corresponding to the horizon date.
A company’s stock just paid a $1.15 dividend, which is expected to grow at 30% the first year, 20% the second year, and 10% the third year. After three years the dividend is expected to grow constantly at 8% forever. The stock’s required return is 13.4%; what is the price of the stock today?
Figure 7-11
Process for Finding the Value of a Nonconstant Growth Stock
INPUTS:
D0 = $1.15 Last dividend the company paid.
rs = 13.4% Stockholders’ required return.
g0,1 = 30% Growth rate for Year 1 only.
g1,2 = 20% Growth rate for Year 2 only.
g2,3 = 10% Growth rate for Year 3 only.
gL = 8% Constant long-run growth rate for all years after Year 3.
Projections
Year 0 1 2 3 ⟶ ∞
Growth rate 30% 20% 10% 8%
Dividend D0 D0(1+g0,1) D1(1+g1,2) D2(1+g1,2)
Dt $1.15 $1.495 $1.794 $1.973
D1 D2 D3
────── ────── ────── HV3 =
(1+rs)1 (1+rs)2 (1+rs)3
D3(1+gL)
───────
(rs− gL)
$2.131
PVs of Dividends $1.318 ⟵⤶ 5.400%
$1.395 ⟵⟵⟵⟵ ⟵⤶
$1.353 ⟵⟵⟵⟵ ⟵⟵⟵⟵ ⟵⤶ $39.468
PV of $27.065 ⟵⟵⟵⟵ ⟵⟵⟵⟵ $39.468
= ────── ⟵⟵⟵⟵ = ──────
Vop = $31.13 (1+0.134)3 (1+rs)3
Note: Numbers in the figure are shown as rounded for clarity in reporting. However unrounded values are used for all calculations.
7-12 Market Multiple Analysis
Use the following data in the market multiple approach to estimate the stock price per share.
Forecasted earnings per share (EPS) = $7.70
Average peer price/earnings (P/E) ratio = 12
Estimated stock price: $92.40
7-14 Preferred Stock
Consider an issue of preferred stock that pays an $8 dividend and has a required return of 8%. What is the value of this preferred stock?
Vps = Dps ÷ rps
= $8.00 ÷ 8.00%
= $100.00
Some preferred stock has a maturity date. Consider a firm whose preferred stock matures in 50 years, pays a $8 annual dividend, has a par value of $100, and has a required return of 6%. What is the price of this preferred stock?
Years to Maturity (N): 50
Annual Dividend (PMT): $8
Par value (FV): $100
Required return, rd (I/YR): 6%
Vps = $131.52

Harcourt, Inc. items and derived items copyright © 2002 by Harcourt, Inc.

Declining Growth: -8% 0 1 2 3 4 5 1.1499999999999999 1.0580000000000001 0.97336000000000011 0.89549120000000015 0.82385190400000019 0.75794375168000017 Zero GrowthZero Growth

0 1 2 3 4 5 1.1499999999999999 1.1499999999999999 1.1499999999999999 1.1499999999999999 1.1499999999999999 1.1499999999999999 Constant Growth: 8% 0 1 2 3 4 5 1.1499999999999999 1.242 1.3413600000000001 1.4486688000000003 1.5645623040000005 1.6897272883200007 Long-Term Growth: 8% 0 1 2 3 4 5 1.1499999999999999 1.4949999999999999 1.7939999999999998 1.9734 2.1312720000000001 2.3017737600000001Sources of Value

Short-term investments = Value of operations = 2 100

Claims on Value

Debt = Preferred stock = Estimated equity value = Total = 28 4 70 Declining Growth: -8% 0 1 2 3 4 5 1.1499999999999999 1.0580000000000001 0.97336000000000011 0.89549120000000015 0.82385190400000019 0.75794375168000017 Zero Growth

Zero Growth

0 1 2 3 4 5 1.1499999999999999 1.1499999999999999 1.1499999999999999 1.1499999999999999 1.1499999999999999 1.1499999999999999 Constant Growth: 8% 0 1 2 3 4 5 1.1499999999999999 1.242 1.3413600000000001 1.4486688000000003 1.5645623040000005 1.6897272883200007 Long-Term Growth: 8% 0 1 2 3 4 5 1.1499999999999999 1.4949999999999999 1.7939999999999998 1.9734 2.1312720000000001 2.3017737600000001Sources of Value

Short-term investments = Value of operations = 2 100

Claims on Value

Debt = Preferred stock = Estimated equity value = Total = 28 4 70 Declining Growth: -8% 0 1 2 3 4 5 1.1499999999999999 1.0580000000000001 0.97336000000000011 0.89549120000000015 0.82385190400000019 0.75794375168000017 Zero Growth

Zero Growth

0 1 2 3 4 5 1.1499999999999999 1.1499999999999999 1.1499999999999999 1.1499999999999999 1.1499999999999999 1.1499999999999999 Constant Growth: 8% 0 1 2 3 4 5 1.1499999999999999 1.242 1.3413600000000001 1.4486688000000003 1.5645623040000005 1.6897272883200007 Long-Term Growth: 8% 0 1 2 3 4 5 1.1499999999999999 1.4949999999999999 1.7939999999999998 1.9734 2.1312720000000001 2.3017737600000001Sources of Value

Short-term investments = Value of operations = 2 100

Claims on Value

Debt = Preferred stock = Estimated equity value = Total = 28 4 70 Declining Growth: -8% 0 1 2 3 4 5 1.1499999999999999 1.0580000000000001 0.97336000000000011 0.89549120000000015 0.82385190400000019 0.75794375168000017 Zero Growth

Zero Growth

0 1 2 3 4 5 1.1499999999999999 1.1499999999999999 1.1499999999999999 1.1499999999999999 1.1499999999999999 1.1499999999999999 Constant Growth: 8% 0 1 2 3 4 5 1.1499999999999999 1.242 1.3413600000000001 1.4486688000000003 1.5645623040000005 1.6897272883200007 Long-Term Growth: 8% 0 1 2 3 4 5 1.1499999999999999 1.4949999999999999 1.7939999999999998 1.9734 2.1312720000000001 2.3017737600000001Sources of Value

Short-term investments = Value of operations = 2 100

Claims on Value

Debt = Preferred stock = Estimated equity value = Total = 28 4 70 Declining Growth: -8% 0 1 2 3 4 5 1.1499999999999999 1.0580000000000001 0.97336000000000011 0.89549120000000015 0.82385190400000019 0.75794375168000017 Zero Growth

Zero Growth

0 1 2 3 4 5 1.1499999999999999 1.1499999999999999 1.1499999999999999 1.1499999999999999 1.1499999999999999 1.1499999999999999 Constant Growth: 8% 0 1 2 3 4 5 1.1499999999999999 1.242 1.3413600000000001 1.4486688000000003 1.5645623040000005 1.6897272883200007 Long-Term Growth: 8% 0 1 2 3 4 5 1.1499999999999999 1.4949999999999999 1.7939999999999998 1.9734 2.1312720000000001 2.3017737600000001Sources of Value

Short-term investments = Value of operations = 2 100

Claims on Value

Debt = Preferred stock = Estimated equity value = Total = 28 4 70

 

Scenario Summary

Scenario Summary
Current Values: Status Quo Higher Growth (Only) Higher OP (Only) Better CR (Only) Improve Growth and OP Improve Growth and CR Improve Growth, OP, and CR Lower WACC (Only) Better OP and CR
Created by Mike Ehrhardt on 5/22/2014 Modified by Mike Ehrhardt on 5/23/2014 Created by Mike Ehrhardt on 5/22/2014 Modified by Mike Ehrhardt on 5/23/2014 Created by Mike Ehrhardt on 5/22/2014 Modified by Mike Ehrhardt on 5/23/2014 Created by Mike Ehrhardt on 5/22/2014 Modified by Mike Ehrhardt on 5/23/2014 Created by Mike Ehrhardt on 5/22/2014 Modified by Mike Ehrhardt on 5/23/2014 Created by Mike Ehrhardt on 5/22/2014 Modified by Mike Ehrhardt on 5/23/2014 Created by Mike Ehrhardt on 5/22/2014 Modified by Mike Ehrhardt on 5/23/2014 Created by Mike Ehrhardt on 5/23/2014 Modified by Mike Ehrhardt on 5/23/2014 Created by Mike Ehrhardt on 3/21/2015
Changing Cells:
$A$324 Status Quo Status Quo Higher Growth (Only) Higher OP (Only) Better CR (Only) Improve Growth and OP Improve Growth and CR Improve Growth, OP, and CR Lower WACC (Only) Better OP and CR
$F$327 8% 8% 9% 8% 8% 9% 9% 9% 8% 8%
$G$327 7% 7% 8% 7% 7% 8% 8% 8% 7% 7%
$H$327 5% 5% 6% 5% 5% 6% 6% 6% 5% 5%
$E$327 10% 10% 11% 10% 10% 11% 11% 11% 10% 10%
$E$328 6% 6% 6% 7% 6% 7% 6% 7% 6% 7%
$E$329 61% 61% 61% 61% 52% 61% 52% 52% 61% 52%
$C$282 10.97% 10.97% 10.97% 10.97% 10.97% 10.97% 10.97% 10.97% 9.50% 10.97%
Result Cells:
$B$396 $2,719.44 $2,719.44 $2,713.27 $3,681.78 $3,575.63 $3,879.93 $3,751.25 $4,917.91 $3,689.71 $4,537.97
$G$460 $22.79 $22.79 $22.67 $42.04 $39.91 $46.00 $43.42 $66.76 $42.19 $59.16
$I$338 9.84% 9.84% 9.84% 11.48% 11.54% 11.48% 11.54% 13.46% 9.84% 13.46%
Notes: Current Values column represents values of changing cells at
time Scenario Summary Report was created. Changing cells for each
scenario are highlighted in gray.

 

7-4

SECTION 7-4
SOLUTIONS TO SELF-TEST
A company expects a constant FCF of $240 million per year forever. If the WACC is 12%, what is the value of operations?
Expected FCF $240
WACC 12%
Vop = $2,000.00
A company has a current value of operations of $800 million. The company has $100 million in short-term investments. If the company has $400 million in debt and has 10 million shares outstanding, what is the price per share?
Vop $800
ST investments $100
Total value $900
Debt $400
Value of equity $500
Number of shares 10
Price per share $50.00

 

7-5

SECTION 7-5
SOLUTIONS TO SELF-TEST
A company expects to have a FCF in 1 year of $300, which is expected to grow at a constant rate of 3% forever. If the WACC is 11%, what is the value of operations?
Expected FCF1 = $330
Expected gL = 3%
WACC = 11%
Vop = $4,125
A company’s most recent free cash flow was $270. The company expects to have a FCF in 1 year of $300, which is expected to grow at a constant rate of 3% forever. If the WACC is 11%, what is the value of operations?
Expected FCF1 = $300
Expected gL = 3%
WACC = 11%
Notice that the FCF of $270 at t = 0 is irrelevant to the value of operations, because it occurred in the past. The value of operations depends only on the future free cash flows.
Vop = $3,750
A company’s most recent free cash flow was $600 and is expected to grow at a constant rate of 4% forever. If the WACC is 10%, what is the value of operations?
FCF0 = $600
Expected gL = 4%
WACC = 10%
Vop = $10,400

 

7-6

SECTION 7-6
SOLUTIONS TO SELF-TEST
A company expects to have a FCF at Year 10 of $600, which is expected to grow at a constant rate of 8% thereafter. If the WACC is 8%, what is the value of operations at Year 10, HV10?
Expected FCF12 = $600
Expected gL = 4%
WACC = 8%
Vop = $15,600
A company expects a FCF of -$10 million at Year 1 and a FCF of $20 million at Year 2. FCF is expected to grow at a 5% rate after Year 2. If the WACC is 10%, what is the horizon value of operations; i.e., Vop(Year 2)? What is the current value of operations; i.e., Vop(Year 0)?
Long-term growth rate 5%
WACC 10%
Year
1 2
FCF1 FCF2
Expected FCF -$10.00 $20.00
Vop(Year 2) $420.00
PV of expected FCF $7.44
PV of expected Vop(Year 2) $347.11
Vop(Year 0) $354.55

7-7

SECTION 7-7
SOLUTIONS TO SELF-TEST
Cathey Corporation currently has sales of $1,000, which are expected to grow by 10% from Year 0 to Year 1 and by 4% from Year 1 to Year 2. The company currently has and operating profitability (OP) ratio of 7% and a capital requirement (CR) ratio of 50% and expects to maintain these ratios at their current levels. The current level of operating capital is $510. Use these inputs to forecast free cash flow (FCF) for Years 1 and 2. Hint: You must first forecast sales, net operating profit after taxes (NOPAT), and total net operating capital (OpCap) for each year.
Sales0 = $1,000
g0,1 = 10%
g1,2 = 4%
OP = NOPAT/Sales = 7%
CR = OpCap/Sales = 50%
OpCap0 = $510
Year 0 1 2 3
Growth rate in sales 10% 4% 4%
Sales $1,000 $1,100.00 $1,144.00 $1,189.76
NOPAT $77.00 $80.08 $83.28
OpCap $510 $550.00 $572.00 $594.88
Investment in OpCap $40.00 $22.00 $22.88
FCF $37.00 $58.08 $60.40
Growth in FCF 57.0% 4.0%
Cathey Corporation has a 12% weighted average cost of capital. Cathey’s free cash flows, estimated in the previous question, are expected to grow at 4% beginning at Year 2 and continuing for the foreseeable future. What is the horizon value (use Year 2 for the horizon)? What is the current value of operations?
Long-term growth rate 4%
WACC 12%
Year
1 2
FCF1 FCF2
Expected FCF $37.00 $58.08
HV2 = Vop(Year 2) $755.04
PV of expected FCF $79.34
PV of expected HV(Year 2) $601.91
Vop(Year 0) $681.25
Cathey Corporation has $80 in short-term investments, $20 in short-term debt, $140 in long-term debt, $30 in preferred stock, and 10 shares of common stock outstanding. Use the value of operations from the previous question to estimate the intrinsic common stock price per share.
Vop = $681.25
ST investments = $80.00
ST debt = $20.00
Long-term debt = $140.00
Preferred stock = $30.00
Number of shares = 10
Vop $681.25
ST investments $80.00
Total value $761.25
All debt $160.00
Preferred stock $30.00
Value of equity $571.25
Number of shares 10.00
Price per share $57.13

 

7-11

SECTION 7-11
SOLUTIONS TO SELF-TEST
If D1 = $3.00, P0 = $50, and the expected P at t=1 is equal to $52, what are the stock’s expected dividend yield, capital gains yield, and total return for the coming year?
D1 $3.00
P0 $50.00
Expected P1 $52.00
Exp. dividend yield 6.0% =B6/B7
Exp. capital gains yield 4.0% =(B8-B7)/B7
Exp. total return 10.0% =C10+C11
A stock is expected to pay a dividend of $2 at the end of the year. The required rate of return is rs = 12%. What would the stock’s price be if the growth rate were 4%?
D1 $2.00
gL 4%
rs 12%
Stock price $25.00
A stock is expected to pay a dividend of $2 at the end of the year. The required rate of return is rs = 12%. What would the stock’s price be if the growth rate were 0%?
D1 $2.00
gL 0%
rs 12%
Stock price $16.67
If D0 = $4.00, rs = 9%, and g = 5% for a constant growth stock, what are the stock’s expected dividend yield and capital gains yield for the coming year?
D0 $4.00
gL 5%
rs 9%
Expected D1 $4.20
Stock price $105.00
Expected dividend yield 4.00%
Expected capital gains yield 5.00%
Alternatively, you know that the capital gains yield is equal to the growth rate.
Expected capital gains yield = growth rate = 5.00%
Because the total return is rs, the dividend yield is rs minus the capital gains yield:
Expected dividend yield = 4.00%
Suppose D0 = $5.00 and rs = 10%. The expected growth rate from Year 0 to Year 1 (g0 to 1) = 20%, the expected growth rate from Year 1 to Year 2 (g1 to 2) = 10%, and the constant rate beyond Year 2 is gL = 5%. What are the expected dividends for Year 1 and Year 2? What is the expected horizon value price at Year 2? What is the expected price at Time 0?
D0 $5.00
g0 to 1 20%
g1 to 2 10%
gL 5%
rs 10%
Year
1 2
D1 D2
Expected dividends $6.00 $6.60
Expected HVP,2 $138.60
PV of expected dividends $10.91
PV of expected HVP,2 $114.55
Expected price at Time 0 $125.45

 

7-12

SECTION 7-12
SOLUTIONS TO SELF-TEST
Dodd Corporation is a private company that earned $4.00 per share for the most recent year. If the average P/E ratio of a group of comparable public companies is 11, what is an estimate of Dodd’s stock value on a per share basis?
Earnings per share = $4.00
Average comparable P/E ratio = 11.0
Estimated price per share = $44.00
The company in the previous question, Dodd Corporation, has 100,000 shares of common stock owned by its founder. Dodd owes $1,300,000 to its bank. Dodd has 11,400 customers. If the average ratio of total entity value to customers is $500 for a group of comparable public companies, what is Dodd’s estimated total entity value? What is its estimated stock value on a per share basis?
Number of shares = 100,000
Debt = $1,300,000
Number of customers = 11,400
Average comparable ratio of total entity value to number of customers =
$500
Estimated entity value = $5,700,000
− Debt $1,300,000
Intrinsic equity value 4,400,000
÷ Number of shares 100,000
Estimated price per share = $44.00

 

7-14

SECTION 7-14
SOLUTIONS TO SELF-TEST
A preferred stock has an annual dividend of $5. The required return is 8%. What is the Vps?
Dps $5.00
rps 8%
Vps $62.50

 

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