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