стр. 27 |

difference.

Steps 1вЂ“6 are not repeated. The following steps are iterative.

7. Calculate a relevered beta and equity discount rate using your

initial capital structure and use it to value the п¬Ѓrm.

8. Substitute the п¬Ѓrst calculated fair market value of equity into a

new capital structure and use the new weights to calculate the

next iteration of beta, equity discount rate, and FMV of equity.

9. Keep repeating 7 and 8 until you reach a steady state value for

beta, equity discount rate, and FMV of equity.

LetвЂ™s illustrate this with a couple of examples.

Table 6-1A: The First Iteration

We use a deliberately simple discounted future earnings approach in Ta-

ble 6-1A to illustrate how this process works. Starting with a п¬Ѓrm whose

net income before taxes (NIBT) in 1997, the previous year, was $400,000

(cell D28), we assume a declining growth rate in income: 15% (B7) in

1998, 13% (C7) in 1999, п¬Ѓnishing with 8% (F7) in 2002. We use these

growth rates to forecast income in 1998вЂ“2002. Subtracting 40% for income

taxes, we arrive at net income after taxes (NIAT) of $276,000 in 1998 (B9),

rising to $407,531 in 2002 (F9). The bottom row of the top section is the

present value of NIAT, using the calculated equity discount rate and a

midyear assumption.

The valuation section begins in cell D17 with the sum of the present

value of NIAT for the п¬Ѓrst п¬Ѓve years. The next seven rows are interme-

diate calculations using a Gordon model with an 8% constant growth rate

and the midyear assumption (D17вЂ“D23). Forecast income in 2003 is the

2002 net income times one plus the growth rate [F9 (1 D18) D19

$440,134]. The midyear Gordon model multiple, D20, is equal to SQRT(1

r)/(r g) SQRT(1 D36)/(D36 D18) 8.1456. Multiplying $440,134

8.1456 $3,585,135 (D21), which is the present value of net income after year

2002 as of December 31, 2002. The present value factor for п¬Ѓve years is

0.377146 (D22). Multiplying $3,585,135 0.377146 $1,352,121 (D23), which

is the present value of income after 2002 as of the valuation date, January 1,

1998.

Adding the present value of the п¬Ѓrst п¬Ѓve yearsвЂ™ net income of $1,055,852

(D17) to the present value of the net income after п¬Ѓve years of $1,352,121

(D23), we arrive at our п¬Ѓrst approximation of the FMV of the equity of

$2,407,973 (D24).

Rows 28 through 35 contain the assumptions of the model and the data

necessary to lever and unlever industry average betas and calculate equity

discount rates. The discount rate is in cell D36, though it is calculated in G54

and transferred from there.

Rows 42 through 46 detail the calculation of an unlevered beta of 0.91

(F46) from an average of publicly traded guideline companies. In the capital

structure and iterations section, Row 54 shows the market value of debt and

CHAPTER 6 An Iterative Valuation Approach 181

T A B L E 6-1A

Equity Valuation Approach with Iterations Beginning with Book Equity: Iteration #1

A B C D E F G H

5 1998 1999 2000 2001 2002

6 Net inc before taxes 460,000 519,800 576,978 628,906 679,219

7 Growth rate in NIBT 15% 13% 11% 9% 8%

8 Income taxes (184,000) (207,920) (230,791) (251,562) (271,687)

9 Net inc after taxes 276,000 311,880 346,187 377,344 407,531

10 Present value factor 0.9071 0.7464 0.6141 0.5053 0.4158

11 Pres value NIAT $250,357 $232,777 $212,601 $190,675 $169,441

16 Final Valuation:

17 PV 1998вЂ“2002 net income $1,055,852

18 Constant growth rate in income G 8%

19 Forecast net income-2003 440,134

20 Gordon model mult SQRT(1 R)/(R G) 8.1456

21 Present value-net inc after 2002 as of 12/31/2002 3,585,135

22 Present value factor-5 years 0.377146

23 Present value of net income after 2002 as of 1/1/98 1,352,121

24 FMV of equity-100% interest $2,407,973

27 Assumptions:

28 Net income before tax-1997 400,000

29 Income tax rate 40%

30 Discount rate-debt: pre-tax 10%

31 Discount rate-debt: after-tax 6%

32 Unlevered beta (from F46) 0.91

33 Risk free rate 6%

34 Equity premium 8%

35 Small company premium 3%

36 Equity discount rate R 21.534%

38 Calculation of Equity Discount Rate Using Comparables

40 Equity Unlevered

41 Beta Debt Equity D/E Beta

42 Guideline Company #1 1.15 454,646 874,464 52.0% 0.88

43 Guideline Company #2 1.20 146,464 546,454 26.8% 1.03

44 Guideline Company #3 0.95 46,464 705,464 6.6% 0.91

45 Guideline Company #4 0.85 52,646 846,467 6.2% 0.82

46 Totals or averages 1.04 700,220 2,972,849 23.55% 0.91

49 Capital Structure & Iterations

51 Interest-

52 Bearing Equity Before Relevered Equity FMV

53 t Debt Iteration D/E Beta Disc. Rate Equity

54 FMV debt, eqty at t 1 1 900,000 750,000 1.20 1.5668 21.534% 2,407,973

55 FMV debt, eqty at t 1 2 900,000 2,407,973 0.37 1.1152 17.921% 2,407,973

56 FMV debt, eqty at t 1 3 900,000 2,407,973 0.37 1.1152 17.921% 2,407,973

57 FMV debt, eqty at t 1 4 900,000 2,407,973 0.37 1.1152 17.921% 2,407,973

58 FMV debt, eqty at t 1 5 900,000 2,407,973 0.37 1.1152 17.921% 2,407,973

59 FMV debt, eqty at t 1 6 900,000 2,407,973 0.37 1.1152 17.921% 2,407,973

60 FMV debt, eqty at t 1 7 900,000 2,407,973 0.37 1.1152 17.921% 2,407,973

61 FMV debt, eqty at t 1 8 900,000 2,407,973 0.37 1.1152 17.921% 2,407,973

62 FMV debt, eqty at t 1 9 900,000 2,407,973 0.37 1.1152 17.921% 2,407,973

63 FMV debt, eqty at t 1 10 900,000 2,407,973 0.37 1.1152 17.921% 2,407,973

64 FMV debt, eqty at t 1 11 900,000 2,407,973 0.37 1.1152 17.921% 2,407,973

65 FMV debt, eqty at t 1 12 900,000 2,407,973 0.37 1.1152 17.921% 2,407,973

66 FMV debt, eqty at t 1 13 900,000 2,407,973 0.37 1.1152 17.921% 2,407,973

67 FMV debt, eqty at t 1 14 900,000 2,407,973 0.37 1.1152 17.921% 2,407,973

68 FMV debt, eqty at t 1 15 900,000 2,407,973 0.37 1.1152 17.921% 2,407,973

69 FMV debt, eqty at t 1 16 900,000 2,407,973 0.37 1.1152 17.921% 2,407,973

70 FMV debt, eqty at t 1 17 900,000 2,407,973 0.37 1.1152 17.921% 2,407,973

71 FMV debt, eqty at t 1 18 900,000 2,407,973 0.37 1.1152 17.921% 2,407,973

72 FMV debt, eqty at t 1 19 900,000 2,407,973 0.37 1.1152 17.921% 2,407,973

73 FMV debt, eqty at t 1 20 900,000 2,407,973 0.37 1.1152 17.921% 2,407,973

PART 2 Calculating Discount Rates

182

the book value of equity (our initial guess of market value) as well as the

implied debt/equityratioandreleveredbetaaccordingtoHamadaвЂ™sformula

(Hamada 1972):2

Debt

1 (1 Tax Rate)

levered unlevered

Equity

Cell G54 is the discount rate of 21.534% for the п¬Ѓrst iteration, calculated ac-

cording to the formula

Disc Rate Risk Free Rate ( Equity Premium)

levered

Small Company Premium

We use this discount rate to calculate the п¬Ѓrst iteration of FMV of equity in

cell H54.

Table 6-1B: Subsequent Iterations of the First Scenario

Table 6-1B is identical to Table 6-1A, except that it contains nine iterations

in the capital structure section instead of 1. Also, cell D36 contains the

п¬Ѓnal equity discount rate from Row 62.3 We denote the iteration number

as t, which appears in Column B, Rows 54вЂ“62. When t 1, we obtain

an equity discount rate of 21.534% (G54) and a FMV of the equity of

$2,407,973 (H54), as before. This tells us that our initial guess of the FMV

of the equity, which was the book value of the equity of $750,000 (D54),

is too low.

We substitute the $2,407,973 (H54) п¬Ѓrst iteration of equity into the

new capital structure in D55 to get a debt/equity ratio of 0.37 (E55), as

seen in the second iteration of Table 6-1B. This changes the discount rate

to 17.921% (G55). This results in the second iteration of equity value of

$3,245,701 (H55). We use the new equity as the basis for our third itera-

tion, which we calculate in the same fashion as the previous iteration. We

follow these steps until we reach a steady state, which in this case occurs

in the eighth iteration, with a FMV of $3,404,686 (H61). We must carry

out an additional iteration to know for sure that we have reached a steady

state, which is the purpose of iteration #9.

Table 6-1C: Initial Choice of Equity DoesnвЂ™t Matter

Tables 6-1B and 6-1C demonstrate that the initial choice of equity doesnвЂ™t

matter. Instead of choosing book equity as the starting point, in Table

6-1C we make an arbitrary guess of $5,000,000 (D54) as a starting point.4

2. This equation is most accurate when the п¬ЃrmвЂ™s pretax discount rate for debt is close to the risk-

free rate.

3. Actually, D36 takes on the value calculated in each iteration in G54 through G62, so the discount

rate used in all the calculations changes in each iteration of the spreadsheet.

4. For those who buy the electronic spreadsheet from the author, which is not included with this

book, the steps are: (1) input your initial guess of equity in D54; (2) initialize the

spreadsheet by pressing Control-X; (3) press Control-Z for each iteration. Every time you

press Control-Z, the spreadsheet will calculate one iteration of value, as in Rows 54 to 62.

Repeat pressing Control-Z until you have reached a steady state, i.e., the value in Column

H is the same twice in a row.

CHAPTER 6 An Iterative Valuation Approach 183

T A B L E 6-1B

Equity Valuation Approach with Iterations Beginning with Book Equity

A B C D E F G H

5 1998 1999 2000 2001 2002

6 Net inc before taxes 460,000 519,800 576,978 628,906 679,219

7 Growth rate in NIBT 15% 13% 11% 9% 8%

8 Income taxes (184,000) (207,920) (230,791) (251,562) (271,687)

9 Net inc after taxes 276,000 311,880 346,187 377,344 407,531

10 Present value factor 0.9228 0.7857 0.6690 0.5696 0.4850

11 Pres value NIAT $254,680 $245,045 $231,602 $214,952 $197,669

16 Final Valuation:

17 PV 1998вЂ“2002 net income $1,143,949

18 Constant growth rate in net income G 8%

19 Forecast net income-2003 440,134

20 Gordon model mult SQRT(1 R)/(R G) 11.4763

21 Present value-net inc after 20002 as of 12/31/2002 5,051,106

22 Present value factor-5 years 0.447573

23 Pres value of net income after 2002 as of 1/1/98 2,260,738

24 FMV of equity-100% interest $3,404,686

27 Assumptions:

28 Net income before tax-1997 400,000

29 Income tax rate 40%

30 Discount rate-debt: pre-tax 10%

31 Discount rate-debt: after-tax 6%

32 Unlevered beta (from F46) 0.91

33 Risk free rate 6%

34 Equity premium 8%

35 Small company premium 3%

36 Equity discount rate R 17.443%

38 Calculation of Equity Discount Rate Using Comparables

40 Equity Unlevered

41 Beta Debt Equity D/E Beta

42 Guideline Company #1 1.15 454,646 874,464 52.0% 0.88

43 Guideline Company #2 1.20 146,464 546,454 26.8% 1.03

44 Guideline Company #3 0.95 46,464 705,464 6.6% 0.91

45 Guideline Company #4 0.85 52,646 846,467 6.2% 0.82

46 Totals or averages 1.04 700,220 2,972,849 23.55% 0.91

49 Capital Structure & Iterations

51 Interest-

52 Bearing Equity Before Relevered Equity FMV

53 t Debt Iteration D/E Beta Disc. Rate Equity

54 FMV debt, eqty at t 1 1 900,000 750,000 1.20 1.5668 21.534% 2,407,973

55 FMV debt, eqty at t 1 2 900,000 2,407,973 0.37 1.1152 17.921% 3,245,701

56 FMV debt, eqty at t 1 3 900,000 3,245,701 0.28 1.0625 17.500% 3,385,037

57 FMV debt, eqty at t 1 4 900,000 3,385,037 0.27 1.0562 17.450% 3,402,345

58 FMV debt, eqty at t 1 5 900,000 3,402,345 0.26 1.0555 17.444% 3,404,409

59 FMV debt, eqty at t 1 6 900,000 3,404,409 0.26 1.0554 17.443% 3,404,653

60 FMV debt, eqty at t 1 7 900,000 3,404,653 0.26 1.0554 17.443% 3,404,682

61 FMV debt, eqty at t 1 8 900,000 3,404,682 0.26 1.0554 17.443% 3,404,686

62 FMV debt, eqty at t 1 9 900,000 3,404,686 0.26 1.0554 17.443% 3,404,686

PART 2 Calculating Discount Rates

184

T A B L E 6-1C

Equity Valuation Approach with Iterations Beginning with Arbitrary Equity

A B C D E F G H

5 1998 1999 2000 2001 2002

6 Net inc before taxes 460,000 519,800 576,978 628,906 679,219

7 Growth rate in NIBT 15% 13% 11% 9% 8%

8 Income taxes (184,000) (207,920) (230,791) (251,562) (271,687)

9 Net inc after taxes 276,000 311,880 346,187 377,344 407,531

10 Present value factor 0.9228 0.7857 0.6690 0.5696 0.4850

11 Pres value NIAT $254,680 $245,045 $231,602 $214,952 $197,669

16 Final Valuation:

17 PV 1998вЂ“2002 net income $1,143,949

18 Constant growth rate in net income G 8%

19 Forecast net income-2003 440,134

20 Gordon model mult SQRT(1 R)/(R G) 11.4763

21 Present value-net inc after 20002 as of 12/31/2002 5,051,106

22 Present value factor-5 years 0.447573

23 Pres value of net income after 2002 as of 1/1/98 2,260,738

24 FMV of equity-100% interest $3,404,686

27 Assumptions:

28 Net income before tax-1997 400,000

29 Income tax rate 40%

30 Discount rate-debt: pre-tax 10%

31 Discount rate-debt: after-tax 6%

32 Unlevered beta (from F46) 0.91

33 Risk free rate 6%

34 Equity premium 8%

35 Small company premium 3%

36 Equity discount rate R 17.443%

38 Calculation of Equity Discount Rate Using Comparables

40 Equity Unlevered

41 Beta Debt Equity D/E Beta

42 Guideline Company #1 1.15 454,646 874,464 52.0% 0.88

43 Guideline Company #2 1.20 146,464 546,454 26.8% 1.03

44 Guideline Company #3 0.95 46,464 705,464 6.6% 0.91

45 Guideline Company #4 0.85 52,646 846,467 6.2% 0.82

46 Totals or averages 1.04 700,220 2,972,849 23.55% 0.91

49 Capital Structure & Iterations

51 Interest-

52 Bearing Equity Before Relevered Equity FMV

53 t Debt Iteration D/E Beta Disc. Rate Equity

54 FMV debt, eqty at t 1 1 900,000 5,000,000 1.18 1.0093 17.074% 3,538,676

55 FMV debt, eqty at t 1 2 900,000 3,538,676 0.25 1.0499 17.399% 3,420,038

56 FMV debt, eqty at t 1 3 900,000 3,420,038 0.26 1.0547 17.438% 3,406,499

57 FMV debt, eqty at t 1 4 900,000 3,406,499 0.26 1.0553 17.442% 3,404,901

58 FMV debt, eqty at t 1 5 900,000 3,404,901 0.26 1.0554 17.443% 3,404,712

59 FMV debt, eqty at t 1 6 900,000 3,404,712 0.26 1.0554 17.443% 3,404,689

60 FMV debt, eqty at t 1 7 900,000 3,404,689 0.26 1.0554 17.443% 3,404,687

61 FMV debt, eqty at t 1 8 900,000 3,404,687 0.26 1.0554 17.443% 3,404,686

62 FMV debt, eqty at t 1 9 900,000 3,404,686 0.26 1.0554 17.443% 3,404,686

CHAPTER 6 An Iterative Valuation Approach 185

Table 6-1C is identical to Table 6-1B except in the initial choice of value

of the equity and the intermediate iterations. The п¬Ѓnal FMV is identical.

Note that it does not matter whether your initial guess is too low or too

high: as Table 6-1B is too low and Table 6-1C is too high, but they both

lead to the same FMV.

Convergence of the Equity Valuation Method

While rare, it can happen that the FMV diverges instead of converges. If

the method described above does not converge, an alternative is to take

the average of the resulting FMV of equity and the previously assumed

value as your input into column D when starting the next iteration as

opposed to using just the latest iteration of equity alone. This can be done

by making a small alteration to the spreadsheet.5

INVESTED CAPITAL APPROACH

Tables 6-2A and 6-2B are examples of the invested capital approach. They

are very similar to Table 6-1B for the equity valuation method, with the

following exceptions:

1. We determine earnings before interest but after taxes (EBIBAT)

as the income measure.6 This should be normalized EBIBAT.7

2. We discount EBIBAT using the WACC.

3. We must subtract the market value of debt from the calculated

market value of invested capital to get the market value of

equity.

4. We must calculate a new WACC for every new iteration of FMV

of equity.

5. We do not show the calculation of unlevered beta but will

assume that it has already been calculated to be 1.05.

LetвЂ™s illustrate this with a couple of examples.

Table 6-2A: Iterations Beginning with Book Equity

Earnings before interest and taxes (EBIT) in 1997, the previous year, was

$600,000 (cell D28). We assume a declining growth rate in earnings as

before: 15% (B6) in 1998, 13% (C6) in 1999, п¬Ѓnishing with 8% (F6) in 2002.

We use these growth rates to forecast EBIT in 1998вЂ“2002. Subtracting 40%

for income taxes, we arrive at earnings before interest, but after taxes

(EBIBAT) of $414,000 in 1998 (B8), rising to $611,297 in 2002 (F8). The

growth rates in EBIBAT are identical to those for EBIT because we assume

a constant 40% income tax (D29). The last row of the top section is the

5. Change the formula in D55, which previously was H55, to AVERAGE(D54,H54). Then copy

the formula down Column D.

6. It is better to use cash п¬‚ow (before interest but after taxes), but for simplicity we use EBIBAT.

7. This does not necessarily correspond to the NIBT in Tables 6-1A, 6-1B, and 6-1C, because we are

dealing with a different hypothetical company.

PART 2 Calculating Discount Rates

186

T A B L E 6-2A

WACC Approach with Iterations Beginning with Book Equity

A B C D E F G H I J

4 1998 1999 2000 2001 2002

5 EBIT 690,000 779,700 865,467 943,359 1,018,828

6 Growth rate in EBIT 15% 13% 11% 9% 8%

7 Income taxes (276,000) (311,880) (346,187) (377,344) (407,531)

8 EBIBAT 414,000 467,820 519,280 566,015 611,297

9 Growth rate-EBIBAT 15% 13% 11% 9% 8%

10 Present value factor 0.9308 0.8064 0.6986 0.6052 0.5243

11 Pres value-EBIBAT $385,341 $377,237 $362,767 $342,566 $320,523

14 Final Valuation:

15 PV 1998вЂ“2002 EBIBAT $1,788,434

16 Constant growth rate in EBIBAT 8%

17 Forecast EBIBAT-2003 660,200

18 Gordon model mult SQRT(1 R)/(R G) 14.4646

19 PV-EBIBAT after 2002 as of 1-1-2003 9,549,547

20 Present value factor-5 years 0.488036

21 PV-EBIBAT after 2002 4,660,523

22 Enterprise FMV-100% interest $6,448,957

23 Less FMV of debt (2,000,000)

24 FMV of equity-100% interest $4,448,957

27 Assumptions:

28 EBIT-1997 600,000

29 Income tax rate 40%

30 Discount rate-debt: pre-tax 10%

31 Discount rate-debt: after-tax 6%

32 Unlevered beta 1.05

33 Risk free rate 6%

34 Equity premium 8%

35 Small company premium 3%

36 Wtd avg cost of capital (WACC) 15.428%

38 Capital Structure & Iterations

40 Interest- Interest-

41 Bearing Bearing Equity FMV

42 t Debt Equity Total Debt Equity Disc. Rate WACC Equity

43 FMV debt, eqty at t 1 1 2,000,000 800,000 2,800,000 71.4% 28.6% 30.000% 12.857% 7,776,091

44 FMV debt, eqty at t 1 2 2,000,000 7,776,091 9,776,091 20.5% 79.5% 18.696% 16.099% 3,927,835

45 FMV debt, eqty at t 1 3 2,000,000 3,927,835 5,927,835 33.7% 66.3% 19.966% 15.254% 4,599,240

46 FMV debt, eqty at t 1 4 2,000,000 4,599,240 6,599,240 30.3% 69.7% 19.592% 15.473% 4,411,165

47 FMV debt, eqty at t 1 5 2,000,000 4,411,165 6,411,165 31.2% 68.8% 19.685% 15.416% 4,458,814

48 FMV debt, eqty at t 1 6 2,000,000 4,458,814 6,458,814 31.0% 69.0% 19.661% 15.431% 4,446,410

49 FMV debt, eqty at t 1 7 2,000,000 4,446,410 6,446,410 31.0% 69.0% 19.667% 15.427% 4,449,617

50 FMV debt, eqty at t 1 8 2,000,000 4,449,617 6,449,617 31.0% 69.0% 19.665% 15.428% 4,448,787

51 FMV debt, eqty at t 1 9 2,000,000 4,448,787 6,448,787 31.0% 69.0% 19.666% 15.428% 4,449,002

52 FMV debt, eqty at t 1 10 2,000,000 4,449,002 6,449,002 31.0% 69.0% 19.666% 15.428% 4,448,946

53 FMV debt, eqty at t 1 11 2,000,000 4,448,946 6,448,946 31.0% 69.0% 19.666% 15.428% 4,448,960

54 FMV debt, eqty at t 1 12 2,000,000 4,448,960 6,448,960 31.0% 69.0% 19.666% 15.428% 4,448,957

55 FMV debt, eqty at t 1 13 2,000,000 4,448,957 6,448,957 31.0% 69.0% 19.666% 15.428% 4,448,957

стр. 27 |