2.00

0.00

0 5 10 15 20 25 30

FMV ($Millions)

through some general equations where we vary the value of the ¬rm by

a factor of K.

Let r1 the discount rate for Firm #1, whose value FMV1

F I G U R E 4-7

Discount Rates as a Function of FMV

40%

35%

30%

25%

Discount Rate

20%

15%

10%

5%

0%

0 20 40 60 80 100 120

Fair Market Value, Marketable Minority ($ Millions)

For scaling reasons, we eliminate values above $100 million

CHAPTER 4 Discount Rates as a Function of Log Size 137

F I G U R E 4-8

1939“1998 Decile Standard Deviations as a Function of Ln(FMV)

100%

90%

80%

70%

Standard Deviation of Returns

60%

50%

40%

10

30% 9

Std Dev = -3.13% x Ln FMV + 87.77% 8

7 3

R2 = 0.9894

4

6

20%

2

5

3

1

10%

0%

0 5 10 15 20 25 30

Ln(FMV)

Standard deviations of yearly returns are derived from the CRSP Deciles. Data labels are decile numbers. The Y intercept is

the regression intercept, not an actual data point.

r2 the discount rate for Firm #2, whose value FMV2 K

FMV1

r1 a b ln FMV1 (4-6)

regression equation #2 applied to Firm #1

r2 a b ln (K FMV1) (4-7)

regression equation #2 applied to Firm #2

r2 a b [ln K ln FMV1] (4-8)

r2 a b ln FMV1 b ln K (4-9)

r2 r1 b ln K (4-10)

In words, the discount rate of a ¬rm K times larger (smaller) than Firm

#1 is always b ln K smaller (larger) than r1.

Let™s illustrate the nature of this relationship with some speci¬c ex-

amples. First, let™s examine what happens with orders of magnitude of

PART 2 Calculating Discount Rates

138

10. Ln 10 2.302535, so b ln 10 0.01039 2.302585 .02391, or

2.4%. This means that if Firm #2 is 10 times larger (smaller) than Firm

#1, its discount rate should be 2.4% lower (higher) than the Firm #1 dis-

count rate. This result can be seen in Table 4-3. The $10 billion ¬rm has

a discount rate of 13.6%, while the $1 billion ¬rm has a discount rate of

16.0%, which is 2.4% higher. The $100 million ¬rm has a discount rate of

18.4%, which is 2.4% higher than the $1 billion ¬rm. Because of the math-

ematical properties of logarithms, the same percentage change in FMV will

always result in the same absolute change in the discount rate. This phe-

nomenon is also seen in graphs containing log scales. Equal distances on

a log scale are equal percentage changes, not absolute changes.

Let™s try one more useful calculation”an order of magnitude 2. Ln

2 0.6931, so that b ln K 0.01039 0.6931 0.72%. Doubling

(halving) the value of the ¬rm reduces (increases) the discount rate by

0.72%. You can see that in going from a $10 million ¬rm to a $5 million

¬rm, the discount rate has increased from 20.8% to 21.5%, a 0.7% differ-

ence (see Table 4-3).

Now it is possible to construct your own table. All you need to know

is your starting FMV and discount rate. The rest follows easily from the

above formulas. Also, we can easily interpolate the table. Suppose you

wanted to know the discount rate for a $25 million ¬rm. Simply start

with the $50 million ¬rm, where r 19.1%, and add 0.7% 19.8%.

Need for Annual Updating

Tables 4-1 through 4-3 should be updated annually, as the Ibbotson av-

erages change, and new regression equations should be generated. This

becomes more crucial when shorter historical time periods are used, be-

cause changes will have a greater impact on the average values.

Additionally, it is important to be careful to match the regression

equation to the year of the valuation. If the valuation assignment is ret-

roactive and the valuation date is 1994, then one should use a regression

equation for 1939“1994.13

Computation of Discount Rate Is an Iterative Process

In spite of the straightforwardness of these relationships, we have a prob-

lem of circular reasoning when it comes to computing of the discount

rate. We need FMV to obtain the discount rate, which is in turn used to

discount cash ¬‚ows or income to calculate the FMV! Hence, it is necessary

to make sure that our initial estimate of FMV is consistent with the ¬nal

result. If it is not, then we have to use the calculated FMV from the end

of iteration #1 as our new assumed FMV in iteration #2. Using either

equation (4-4) or Table 4-3, that will imply a new discount rate, which

we use to value the ¬rm. We keep repeating the process until the results

are consistent.

It is extremely rare to require more than two iterations to achieve

consistency in the ex ante and ex post values. The reason is that even if

13. Alternatively, one could either use the regression equation in the original article, run one™s own

regression on the Ibbotson data, or contact the author to provide the right equation.

CHAPTER 4 Discount Rates as a Function of Log Size 139

we guess the value of the ¬rm incorrectly by a factor of 10, we will only

be 2.4% off in our discount rate. By the time we come to the second

iteration, we usually are consistent. The reason behind this is that the

discount rate is based on the logarithm of the value. As we saw earlier,

there is not much difference between the log of $10 billion and the log of

$10 million, and multiplying that by the x-coef¬cient of 0.01039 further

reduces the effects of an initial incorrect estimate of value. This is a con-

vergent system 99% of the time with any kind of reasonable initial guess

of value and even most unreasonable guesses.

The need for iteration arises because of the mathematical properties

of the equations we use in valuing a ¬rm. The simplest type of valuation

is that of a ¬rm with constant growth to perpetuity, where we simply

apply the Gordon growth model (˜˜Gordon model™™) to our forecast of cash

¬‚ow for the coming year. For simplicity, we will use the end-year Gordon

model formula, although it is not as accurate as the midyear formula.

We use the following de¬nitions:

CF cash ¬‚ow (available to equity) in year t 1 (the ¬rst

forecast year)

a 0.3750, the regression constant from regression #2

b 0.01039, the x-coef¬cient from regression #2

V fair market value (FMV) of the ¬rm

r the discount rate

Using the Gordon model and ignoring valuation discounts and pre-

miums, the FMV of the ¬rm is:

CF

V (4-11)

r g

Per equation (4-6), our log size equation for the discount rate is:

r a b ln V (4-12)

Substituting (4-12) into (4-11), we get:

CF

V (4-13)

a b ln V g

Equation (4-13) is a transcendental equation with no analytic

solution.14 Therefore, successive approximation is the only method of de-

termining an answer. The simple iterative procedure in Tables 4-4A, 4-4B,

and 4-4C is very easy to use and works in almost all situations.

Practical Illustration of the Log Size Model: Discounted

Cash Flow Valuations

Let™s illustrate how the iterative process works with a speci¬c example.

The assumptions in Tables 4-4A, 4-4B and 4-4C are identical, except for

the discount rate. Table 4-4A is a very simple discounted cash ¬‚ow (DCF)

14. I thank my friend William Scott, Jr., a physicist, for the terminology and the de¬nitive word

that there is no analytic solution.

PART 2 Calculating Discount Rates

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