growth). Notice that the ratio declines continuously as we move down

Column F. The overvaluation of a $10 billion ¬rm using the GM is

12.57%”far less than the overvaluation of the $250,000 ¬rm. The differ-

ences are signi¬cantly greater when using the 72-year log size models, as

including the most volatile years in the regression makes for a greater

difference in the AM versus GM Gordon model multiples. These numer-

ical examples underscore the importance of using the arithmetic mean

when valuing expected future earnings or cash ¬‚ow.

INDRO AND LEE ARTICLE

This article (Indro and Lee 1997) is extremely mathematical, exceedingly

dif¬cult reading. The authors begin by citing (Brealey and Myers 1991),

CHAPTER 5 Arithmetic versus Geometric Means 175

T A B L E 5-4

Comparison of Discount Rates Derived from the Log Size Model Using 60-Year

Arithmetic and Geometric Means

A B C D E F

5 Gordon Model Ratio

Multiples Using

6 Firm Size AM [1] GM [2] AM [3] GM [3] GG / AG [4]

7 $250,000 26.76% 19.12% 5.42 8.32 153.41%

8 $1,000,000 25.09% 18.33% 5.86 8.83 150.61%

9 $25,000,000 21.21% 16.49% 7.24 10.29 142.14%

10 $50,000,000 20.38% 16.10% 7.63 10.67 139.85%

11 $100,000,000 19.54% 15.70% 8.07 11.09 137.34%

12 $500,000,000 17.60% 14.78% 9.35 12.20 130.52%

13 $10,000,000,000 14.00% 13.08% 13.35 15.03 112.57%

Conclusion: The ratio of Gordon Model Multiples decreases with ¬rm size (Column F)

Notes:

[1] Arithmetic Mean (AM) Regression Equation, 60 year model r 41.72% 0.01204 Ln (FMV)

[2] Geometric Mean (GM) Regression Equation, 60 year model. r 26.2% 0.0057 Ln (FMV)

[3] Gordon Model Multiple calculated assuming 6% growth in earnings-midyear assumption. Discount rates are not rounded in these

calculations.

[4] Geometric Gordon Model Multiple / Arithmetic Gordon Model Multiple

who say that if monthly returns are identically and independently dis-

tributed, then the arithmetic average of monthly returns should be used

to estimate the long-run expected return. They then cite empirical evi-

dence that there is signi¬cant negative autocorrelation in long-term equity

returns and that historical monthly returns are not independent draws

from a stationery distribution. This means that high returns in one time

period will tend to mean that on average there will be low returns in the

next period, and vice-versa. Based on this, Copeland, Koller, and Murrin

(1994) argue that the geometric average is a better estimate of the long-

run expected returns.

Indro and Lee show that the arithmetic and geometric means have

upward and downward biases, respectively, and that a horizon-weighted

average of the two is the least biased and most ef¬cient estimator.

If the authors are correct, it would mean that there would no longer

be a single discount rate. Every year would have its own unique

weighted-average discount rate. That would also add complexity to the

use of the Gordon model to calculate a residual value.

Because of the extremely dif¬cult mathematics in the article, it was

necessary to speak to academic sources to evaluate it. Professor Myers,

cited above, did agree that long-term (¬ve-year) returns are negatively

autocorrelated but that there are ˜˜very few data points.™™ He had not fully

read the article, is not sure of its signi¬cance, and did not have an opinion

of it. Ibbotson Associates does not feel the evidence for mean reversion

is that strong, and on that basis is not moved to change its opinion that

the AM is the correct mean. It seems that it will take some time before

this article gets enough academic attention to cause the valuation profes-

sion to make any changes in the way it operates.

PART 2 Calculating Discount Rates

176

BIBLIOGRAPHY

Brealey, R. A., and Stewart C. Myers. 1991. Principles of Corporate Finance. New York:

McGraw-Hill.

Copeland, Tom, Tim Koller, and Jack Murrin. 1994. Valuation: Measuring and Managing

the Value of Companies. John Wiley & Sons, Inc. New York, NY.

Ibbotson Associates. 1998. Stocks, Bills, Bonds and In¬‚ation: 1998 Yearbook. Chicago: The

Associates. 107“08; 153“155.

Indro, Daniel C., and Wayne Y. Lee. 1997. ˜˜Biases in Arithmetic and Geometric Averages

as Estimates of Long-Run Expected Returns and Risk Premia.™™ Financial Management

26, no. 4 (Winter): 81“90.

Joyce, Allyn A. 1995. ˜˜Arithmetic Mean vs. Geometric Mean: The Issue in Rate of Return.™™

Business Valuation Review ( June): 62“68.

CHAPTER 5 Arithmetic versus Geometric Means 177

CHAPTER 6

An Iterative Valuation Approach

INTRODUCTION

EQUITY VALUATION METHOD

Table 6-1A: The First Iteration

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

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

Convergence of the Equity Valuation Method

INVESTED CAPITAL APPROACH

Table 6-2A: Iterations Beginning with Book Equity

Table 6-2B: Initial Choice of Equity Doesn™t Matter

Convergence of the Invested Capital Approach

LOG SIZE

SUMMARY

BIBLIOGRAPHY

179

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your best guess of the FMV of equity or you can use the net

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.