140

T A B L E 4-4A

Discounted Cash Flow Analysis Using 60-Year Model”First Iteration

A B C D E F G

5 Description: 1999 2000 2001 2002 2003 Total

6 Assumptions:

7 Base adjusted cash ¬‚ow $100,000

8 Growth rate in adj cash ¬‚ow 12% 10% 9% 8% 7%

9 Discount rate R 20%

10 Growth rate to perpetuity G 6%

11 Control premium 40%

12 Discount-lack of marketability 35%

14 5 Year Forecasts

16 Forecast cash ¬‚ow $112,000 $123,200 $134,288 $145,031 $155,183

17 Present value factor 0.9129 0.7607 0.6339 0.5283 0.4402

18 PV of cash ¬‚ow $102,242 $93,721 $85,130 $76,617 $68,317 $426,028

20 Calculation of Fair Market Value:

21 Formula

22 Forecast cash ¬‚ow 2003 $164,494 (1 G) * F16

23 Gordon model cap rate 7.8246 SQRT (1 R) / (R G)

24 FMV 2003-in¬nity as of 1/1/2003 $1,287,103 B22 B23

25 Present value factor-5 Yrs 0.4019 1/(1 R) 5 [Where 5 is # yrs from 1/1/98 to 1/1/2003]

26 PV of 2003-in¬nity cash ¬‚ow $517,258 B24 B25

27 Add PV of 1998“2002 cash ¬‚ow 426,028 Total of row 18

28 FMV-marketable minority $943,285 B26 B27

29 Control premium 377,314 B11 B28

30 FMV-marketable control interest 1,320,599 B28 B29

31 Disc-lack of marketability (462,210) B12 B30

32 Fair market illiquid control $858,390 B30 B31

33 Calc of Disc Rate-Regr Eq #2

34 Ln (FMV-marketable minority) 13.7571 Ln(B28)

35 * X coef¬cient of .01039 0.1429 B34 * X coef¬cient-regr #2

36 Constant 0.3750 Constant-regression #2

37 Discount rate (rounded) 23% B35 B36

analysis of a hypothetical ¬rm. The basic assumptions appear in B7“B12.

We assume the ¬rm had $100,000 cash ¬‚ow in 1998. We forecast annual

growth through the year 2003 in B8 through F8 and perpetual growth at

6% thereafter in B10. In B9 we assume a 20% discount rate.

The DCF analysis in B22“B32 is standard and requires little expla-

nation. The present value factors are midyear, and the value in B28 is a

marketable minority interest.15 It is this value ($943,285) that we use to

compare the consistency between the assumed discount rate of 20% (B9)

and the calculated discount rate according to the log size model.

We begin calculating the discount rate using the log size model in

B34, where we compute ln (943,285) 13.7571. This is the natural log of

the marketable minority value of the ¬rm. In B35 we multiply that result

15. See Chapter 7 for explanation of the levels of value and valuation discounts and premiums.

CHAPTER 4 Discount Rates as a Function of Log Size 141

T A B L E 4-4B

Discounted Cash Flow Analysis Using 60-Year Model”Second Iteration

A B C D E F G

5 Description: 1999 2000 2001 2002 2003 Total

6 Assumptions:

7 Base adjusted cash ¬‚ow $100,000

8 Growth rate in adj cash ¬‚ow 12% 10% 9% 8% 7%

9 Disc rate R (Table 4-4A, row 37) 23%

10 Growth rate to perpetuity G 6%

11 Control premium 40%

12 Discount-lack of marketability 35%

14 5 Year Forecasts

16 Forecast cash ¬‚ow $112,000 $123,200 $134,288 $145,031 $155,183

17 Present value factor 0.9017 0.7331 0.5960 0.4845 0.3939

18 PV of cash ¬‚ow $100,987 $90,314 $80,034 $70,274 $61,132 $402,741

20 Calculation of Fair Market Value:

21 Formula

22 Forecast cash ¬‚ow 2003 $164,494 (1 G) * F16

23 Gordon model cap rate 6.5238 SQRT (1 R)/(R G)

24 FMV 2003-in¬nity as of 1/1/2003 $1,073,135 B22 B23

25 Present value factor-5 yrs 0.3552 1/(1 R) 5 [where 5 is # yrs from 1/1/98 to 1/1/2003]

26 PV of 2003-in¬nity cash ¬‚ow $381,179 B24 B25

27 Add PV of 1998-2002 cash ¬‚ow 402,741 Total of row 18

28 FMV-marketable minority $783,919 B26 B27

29 Control premium 313,568 B11 B28

30 FMV-marketable control interest 1,097,487 B28 B29

31 Disc-lack of marketability (384,121) B12 B30

32 Fair market value illiquid control $713,367 B30 B31

33 Calc of Disc Rate-Regr Eq #2

34 Ln (FMV-marketable minority) 13.5721 Ln(B28)

35 * X coef¬cient of .01039 0.1410 B34 * X coef¬cient-regr #2

36 Constant 0.3750 Constant-regression #2

37 Discount rate (rounded) 23% B35 B36

Note: We have achieved consistency in the discount rate assumed (Row 9) and the implied discount rate (Row 37). Also the discount rates match Table 4-3 as we interpolate between

$500k and $750k.

by the x-coef¬cient from the regression, or 0.01039, to come to 0.1429.

We then add that product to the regression constant of 0.3750, which

appears in B36, to obtain an implied discount rate of 23% (rounded, B37).

Comparison of the two discount rates (assumed and calculated) re-

veals that we initially assumed too high a discount rate, meaning that we

undervalued the ¬rm. B29“B31 contain the control premium and discount

for lack of marketability. Because the discount rate is not yet consistent,

ignore these numbers in this table, as they are irrelevant. These topics are

explained in depth in Chapter 7. While the magnitude of the control pre-

mium has been the subject of hot debate, it is merely a parameter in the

spreadsheet and does not affect the logic of the analysis.

PART 2 Calculating Discount Rates

142

T A B L E 4-4C

Discounted Cash Flow Analysis Using 60-Year Model”Final Valuation

A B C D E F G

5 Description: 1999 2000 2001 2002 2003 Total

6 Assumptions:

7 Base adjusted cash ¬‚ow $100,000

8 Growth rate in adj cash ¬‚ow 12% 10% 9% 8% 7%

9 Disc rate R [1] 25%

10 Growth rate to perpetuity G 16%

11 Control premium 40%

12 Discount-lack of marketability 35%

14 5 Year Forecasts

16 Forecast cash ¬‚ow $112,000 $123,200 $134,288 $145,031 $155,183

17 Present value factor 0.8944 0.7155 0.5724 0.4579 0.3664

18 PV of cash ¬‚ows $100,176 $88,155 $76,871 $66,416 $56,853 $388,471

20 Calculation of Fair Market Value:

21 Formula

22 Forecast cash ¬‚ow 2003 $164,494 (1 G) * F16

23 Gordon model cap rate 5.8844 SQRT (1 R)/(R G)

24 FMV 2003-in¬nity as of 1/1/2003 $967,948 B22 B23

25 Present value factor-5 yrs 0.3277 1/(1 R) 5 [where 5 is # yrs from 1/1/98 to 1/1/2003]

26 PV of 2003-in¬nity cash ¬‚ow $317,177 B24 B25

27 Add PV of 1998“2002 cash ¬‚ow 388,471 Total of row 18

28 FMV-marketable minority $705,648 B26 B27

29 Control premium 282,259 B11 B28

30 FMV-marketable control interest 987,907 B28 B29

31 Disc-lack of marketability (345,767) B12 B30

32 Fair market value illiquid control $642,139 B30 B31

[1] Disc Rate 23% (from Table 4-4B, B37) 2% for Speci¬c Company Adjustments 25%

The Second Iteration: Table 4-4B

Having determined that a 20% discount rate is too low, we revise our

assumption to a 23% discount rate (B9) in Table 4-4B. In this case, we

arrive at a marketable minority FMV of $ 783,919 (B28). When we perform

the discount rate calculation with this value (B34“B37), we obtain a

matching discount rate of 23%, indicating that no further iterations are

necessary.

Consistency in Levels of Value

In calculating discount rates, it is important to be consistent in the level

of fair market value that we are using. Since the log size model is based

on returns from the NYSE, the corresponding values generated are on a

marketable minority basis. Consequently, it is this level of value that is

we should use for the discount rate calculations.

Frequently, however, the marketable minority value is not the ulti-

mate level of fair market value that we are calculating. Therefore, it is

crucial to be aware of the differing levels of FMV that occur as a result

CHAPTER 4 Discount Rates as a Function of Log Size 143

of valuation adjustments. For example, if our valuation assignment is to

calculate an illiquid control interest, we will add a control premium and

subtract a discount for lack of marketability from the marketable minority

value. Nevertheless, we use only the marketable minority level of FMV

in iterating to the proper discount rate.

Adding Speci¬c Company Adjustments to the DCF Analysis:

Table 4-4C

The ¬nal step in our DCF analysis is performing speci¬c company ad-

justments. Let™s suppose for illustrative purposes that there is only one

owner of this ¬rm. She is 62 years old and had a heart attack three years

ago. The success of the ¬rm depends to a great extent on her personal

relationships with customers, which may not be easily duplicated by a

new owner. Therefore, we decide to add a 2% speci¬c company adjust-

ment to the discount rate to re¬‚ect this situation.16 If there is no speci¬c

company adjustment, then we would proceed with the calculations in

B22“B32.

Prior to adding a speci¬c company adjustment, it is important to

achieve internal consistency in the ex ante and ex post marketable mi-

nority values, as we did in Table 4-4B. Next, we merely add the 2% to

get a 25% discount rate, which we place in B9. The remainder of the table

is identical to its predecessors, except that we eliminate the ex post cal-

culation of the discount rate in B34“B37, since we have already achieved

consistency.

It is at this point in the valuation process that we make adjustments

for the control premium and discount for lack of marketability, which

appear in B29 and B31. Our ¬nal fair market value of $642,139 (B32) is

on an illiquid control basis.

In a valuation report, it would be unnecessary to show Table 4-4A.

One should show Tables 4-4B and 4-4C only.

Total Return versus Equity Premium

CAPM uses an equity risk premium as one component for calculating

return. The discount rate is calculated by multiplying the equity premium

by beta and adding the risk free rate. In my ¬rst article on the log size

model (Abrams 1994), I also used an equity premium in the calculation

of the discount rate. Similarly, Grabowski and King (1995) used an equity

risk premium in the computation of the discount rate.

16. A different approach would be to take a discount from the ¬nal value, which would be

consistent with key person discount literature appearing in a number of articles in Business

Valuation Review (see the BVR index for cites). Another approach is to lower our estimate of

earnings to re¬‚ect our weighted average estimate of decline in earnings that would follow

from a change in ownership or the decreased capacity of the existing owner, whichever is

more appropriate, depending on the context of the valuation. In this example I have already

assumed that we have done that. There are opinions that one should lower earnings

estimates and not increase the discount rate. It is my opinion that we should de¬nitely

increase the discount rate in such a situation, and we should also decrease the earnings

estimates if that has not already been done.

PART 2 Calculating Discount Rates

144

The equity premium form of the log size model is:

r RF size-based equity premium (4-14)

The size-based equity premium is equal to the return, as calculated by

the log size model, minus the historical average risk-free rate.17

Equity Premium a b ln FMV RF (4-15)

where RF is the historical average risk-free rate. Substituting equation

(4-15) into (4-14), we get:

r RF a b ln FMV RF (4-16)

Rearranging terms, we get:

r a b ln FMV (RF RF) (4-17)

Note that the ¬rst two terms in equation (4-17) are the sole terms

included in the total return version of the log size model. Therefore, the

only difference in calculation of discount rates between the two models

is RF RF, the last two terms appearing in equation [4-17]. Consequently,

the total return of the log size model will exceed the equity premium

version of the model whenever current bond yields exceed historical av-

erage yields and vice versa.

The equity premium term was eliminated in Abrams™ second article

(1997) in favor of total return because of the low correlation between stock

returns and bond yields for the past 60 years. The actual correlation was

6.3%”an amount small enough to ignore.

Bond yields were in the 2“3% range before 1960, under 5% until 1968,

and over 7% from 1975“1993; in 1982 they were as high as 13%. During

the 60-year period from 1939“1998, the low bond yields prevalent in the

1950s and 1960s are balanced by higher subsequent rates, resulting in

little difference in the results obtained using the two models. The 60-year

mean bond yield is 5.64%, as compared with 1998 yields that have ranged

from 5.5% to 6.0%. Thus, current yields are comparable with the 60 year

average yields.

Therefore, it is reasonable to simplify the procedure of calculating

discount rates and eliminate the bifurcation of the discount rate into the

risk-free rate and equity premium components.

17. In CAPM, the latter term is a beta-adjusted equity risk premium, equal to ( equity risk

premium). The equity risk premium (ERP) itself is the arithmetic average of the annual

1998

market returns in excess of the risk-free rate. Mathematically, that is ERP [rmt

t 1926

rFt)/73], where r return and the subscripts m market and F risk-free rate. However,

1998

we can rearrange the equation to ERP [(rmt/73) (rFt/73)] rm rF. This is

t 1926

appropriate for the market as a whole. To calculate a discount rate for a particular ¬rm, in

CAPM we scale the ERP up or down according to the systematic risk as measured by beta.

In log size, we replace the average return on the market with the size-based return for the

¬rm. There is no algebraic scaling, as the log size equation accomplishes the adjustment of

the ERP directly by size.

CHAPTER 4 Discount Rates as a Function of Log Size 145

Adjustments to the Discount Rate

Is Table 4-3 the last word in calculating discount rates? No, but it is the

best starting point based on the available data. Table 4-3 is an extrapo-

lation of NYSE data to privately held ¬rms. While the results appear very

reasonable to me, it would be preferable to perform a similar regression

for NASD data. Unfortunately, the data are not readily obtainable.

Privately held ¬rms are generally owned by people who are not well

diversi¬ed. Table 4-3 was derived from portfolios of stocks that were di-

versi¬ed in every sense except for size, as size itself was the method of

sorting the deciles. In contrast, the owner of the local bar is probably not

well diversi¬ed, nor is the probable buyer. The appraiser may want to

add a speci¬c company adjustment of, say, 2% to 5% to the discount rate

implied by Table 4-3 to account for that. On the other hand, a $100 million

FMV ¬rm is likely to be bought by a well diversi¬ed buyer and may not

merit increasing the discount rate.

Another common adjustment to Table 4-3 discount rates would be

for the depth and breadth of management of the subject company com-

pared to other ¬rms of the same size. In general, Table 4-3 already incor-

porates the size effect. No one expects a $100,000 FMV ¬rm to have three

Harvard MBAs running it, but there is still a difference between a com-

plete one-man show and a ¬rm with two talented people. In general, this

methodology of calculating discount rates will increase the importance of

comparing the subject company to its peers via RMA Associates or similar

data. Differences in leverage between the subject company and its RMA

peers could well be another common adjustment.

Discounted Cash Flow or Net Income?

Since the market returns are based on the cash dividends and the market

price at which one can sell one™s stock, the discount rates obtained with

the log size model should be properly applied to cash ¬‚ow, not to net

income. We appraisers, however, sometimes work with clients who want

a ˜˜quick and dirty valuation,™™ and we often don™t want to bother esti-

mating cash ¬‚ow. I have seen suggestions in Business Valuation Review

(Gilbert 1990, for example) that we can increase the discount rate and

thereby apply it to net income, and that will often lead to reasonable

results. Nevertheless, it is better to make an adjustment from net income

based on judgment to estimate cash ¬‚ow to preserve the accuracy of the

discount rate.

DISCUSSION OF MODELS AND SIZE EFFECTS

The size effects described by Fama and French (1993), Abrams (1994,

1997), and Grabowski and King (1995) strongly suggest that the tradi-

tional one-factor CAPM model is obsolete. As Fama and French (1993, p.

54) say, ˜˜Many continue to use the one-factor Sharpe“Lintner model to

evaluate portfolio performance and to estimate the cost of capital, despite

the lack of evidence that it is relevant. At a minimum, these results here

and in Fama and French (1992) should help to break this common habit.™™

PART 2 Calculating Discount Rates

146

CAPM

Consider the usual way we calculate discount rates using CAPM. We

average the betas of many different ¬rms in the industry, which vary

considerably in size, and apply the resulting beta to a ¬rm that is prob-

ably 0.1% to 1% of the industry average, without correction for size, and

hence risk. Ignoring the size effect corrupts the CAPM results.

This ¬‚aw also applies to the guideline public company method. The

usual approach is to average price earnings multiples (and/or price cash

¬‚ow multiples, etc.) for the various ¬rms in the industry without cor-

recting for size and apply the multiple to a small private ¬rm. A better

method is to perform a regression analysis of market capitalization

(value) as a function of earnings (or cash ¬‚ow) and forecast growth, when

available. I also recommend using another form of the regression with

P/E or P/CF as the dependent variable and market capitalization and

forecast growth as the independent variables.

The beta used in CAPM is usually calculated by running a regression

of the equity premium for an individual company versus the market pre-

mium. As previously discussed, the inability of the resulting beta to ex-

plain the size effect has called into question the validity of CAPM. An

alternative method of calculating beta has been proposed which attempts

to capture the size effect and better correlate with market equity returns,

possibly ameliorating this problem.

Sum Beta

Ibbotson et al. (Peterson, Kaplan, and Ibbotson 1997) postulated that con-

ventional estimates of beta are too low for small stocks due to the higher

degree of auto-correlation in returns exhibited by smaller ¬rms. They cal-

culated a beta using a multiple regression model for both the current and

the prior period, which they call ˜˜sum beta.™™ These adjusted estimates of

beta helped to account for the size effect and showed positive correlation

with future returns.

This improved method of calculating betas will reduce will reduce

some of the downward bias in CAPM discount rates, but it still will not

account for the size effect differences between the large ¬rms in the

NYSE”where even the smallest ¬rms are large”and the smaller pri-

vately held ¬rms that many appraisers are called upon to value. Size

should be an explicit variable in the model to accomplish that.

It may be possible to combine the models. One could use the log size

model to calculate a size premium over the average market return and

add that to a CAPM calculation of the discount rate using Ibbotson™s sum

betas. It will take more research to determine whether than is a worth-

while improvement in methodology.

The Fama“French Cost of Equity Model18

The Fama“French cost of equity model is a multivariable regression

model that uses size (˜˜small minus big™™ premium SMB) and book to

18. The precise method of calculating beta, SMB, and HML using the three-factor model, along

with the regression equation, is more fully explained in Ibbotson Associates™ Beta Book.

CHAPTER 4 Discount Rates as a Function of Log Size 147

market equity (˜˜high minus low™™ premium HML) in addition to beta

as variables that affect market returns. Michael Annin (1997) examined

the model in detail and found that it does appear to correct for size, both

in the long term and short term, over the 30-year time period tested.