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book value of equity. Eventually your initial guess will make no
difference.
Steps 1“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 ¬rm.
8. Substitute the ¬rst 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.
Let™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 ¬rm 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, ¬nishing with 8% (F7) in 2002. We use these
growth rates to forecast income in 1998“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 ¬rst ¬ve years. The next seven rows are interme-
diate calculations using a Gordon model with an 8% constant growth rate
and the midyear assumption (D17“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 ¬ve 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 ¬rst ¬ve years™ net income of $1,055,852
(D17) to the present value of the net income after ¬ve years of $1,352,121
(D23), we arrive at our ¬rst 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 1998“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/equityratioandreleveredbetaaccordingtoHamada™sformula
(Hamada 1972):2
Debt
1 (1 Tax Rate)
levered unlevered
Equity
Cell G54 is the discount rate of 21.534% for the ¬rst 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 ¬rst 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
¬nal equity discount rate from Row 62.3 We denote the iteration number
as t, which appears in Column B, Rows 54“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) ¬rst 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 Doesn™t Matter
Tables 6-1B and 6-1C demonstrate that the initial choice of equity doesn™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 ¬rm™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 1998“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 1998“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 ¬nal 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.
Let™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, ¬nishing with 8% (F6) in 2002.
We use these growth rates to forecast EBIT in 1998“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 ¬‚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 1998“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

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