1. In the second part of the chapter we will explore the valuation impact of appraiser error in

forecasting cash ¬‚ows.

2. While Chapter 4 was updated to include the Ibbotson 1999 SBBI Yearbook results, this chapter

has not. Therefore, this chapter does not contain the 1998 stock market results, which were

very poor for the log size model. As noted in Chapter 4, large ¬rms outperformed small

¬rms. Therefore, the con¬dence intervals calculated in this chapter would be wider if we

were to include the 1998 results, which are reported in the 1999 SBBI Yearbook.

3. Calculation of the log size discount rate is in rows 35“38. The regression equation in these rows

is based on the 1998 SBBI Yearbook and therefore does not match the equation in Table 4-1.

CHAPTER 11 Measuring Valuation Uncertainty and Error 385

#2 in Table 4-1 has 10 observations. The number of degrees freedom is n

k 1, where n is the number of observations and k is the number of

independent variables; thus we have eight degrees of freedom. Using a

t-distribution with eight degrees of freedom, we add and subtract 2.306

standard errors to form a 95% con¬dence interval. The standard error of

the log size equation through SBBI 1998 was 0.76% (B48), which when

multiplied by 2.306 equals 1.75%. The upper bound of the discount rate

calculated by log size is 13% 1.75% 14.75% (B11), and the lower

4

bound is 13% 1.75% 11.25% (B13).

For purposes of comparison, we assume that CAPM also arrives at

a 13% discount rate (B16). We multiply the CAPM standard error of 2.42%

(B49) by 2.306 standard errors, yielding 5.58% for our 95% con¬dence

interval. In cell B15 we add 5.58% to the 12% discount rate, and in cell

B17 we subtract 5.58% from the 12% rate, arriving at upper and lower

bounds of 18.58% and 7.42%, respectively.

Rows 19 to 21 show the calculations of the midyear Gordon model

multiples (GM) (1 r)/(r g). For r 13% 1.75% and g 8%,

GM 21.2603 (B20), which we multiply by the $324 million cash ¬‚ow

(B8) to come to an FMV (ignoring discounts and premiums) of $6.89 bil-

lion (B24).

We repeat the process using 14.75%, the upper bound of the 95%

con¬dence interval for the discount rate (B11) in the GM formula, to come

to a lower bound of the GM of 15.8640 (B19). Similarly, using a discount

rate of 11.25% (the lower bound of the con¬dence interval, B13) the cor-

responding upper bound GM formula is 32.4791 (B21). The FMVs asso-

ciated with the lower and upper bound GMMs are $5.14 billion (B23) and

$10.52 billion (B25), or 74.6% (C23) and 152.8% (C25), respectively, of our

best estimate of $6.89 billion.

Cell C39 shows the average size of the 95% con¬dence interval

around the valuation estimate. It is 39%, which is equal to 1„2 [(1

74.6%) (152.8% 1)]. It is not literally true that the 95% con¬dence

interval is the same above and below the estimate, but it is easier to speak

in terms of a single number.

Row 28 shows the Gordon model multiple using a CAPM discount

rate, which we assume is identical to the log size model discount rate.

Using the CAPM upper and lower bound discount rates in B15 and B17,

the lower and upper bounds of the 95% con¬dence interval for the CAPM

Gordon model are 10.2920 (B27) and 178.5324 (B29), respectively. Ob-

viously, the latter is an explosive, nonsense result, and the average 95%

con¬dence interval is in¬nite in this case.

4. This is an approximation. The exact formula is:

x2

1 0

Y0 ˆ0 t0.025s 1

x2

n i

i

where ˆ 0 is the regression-determined discount rate for our subject company, xi are the

deviations of the natural logarithm of each decile™s market capitalization from the mean log

of the 10 Ibbotson decile average market capitalizations, t0.025 is the two-tailed, 95% t-

statistic, s is the standard error of the y-estimate as calculated by the regression, n 10, the

number of deciles in the regression sample, and x0 is the deviation of the log of the FMV of

the subject company from the mean of the regression sample.

Part 4 Putting It All Together

386

We obtain the same estimate of FMV for CAPM as the log size model

(B32, B24), but look at the lower bound estimate in B31. It is $3.33 billion

(rounded), or 48.4% (C31) of the best estimate, versus 74.6% (C23) for the

same in the log size model. The CAPM standard error being more than

three times larger creates a huge con¬dence interval and often leads to

explosive results for very large ¬rms.

Valuation Error in the Other-Size Firms

The remaining columns in Table 11-1 have the same formulas and logic

as columns B and C. The only difference is that the size of the ¬rm varies,

which implies a different discount rate and therefore different 95% con-

¬dence intervals. In column D we assume the large ¬rm had cash ¬‚ows

of $15 million last year (D5), which will grow at 7% (D7). We see that the

log size model has an average 95% con¬dence interval of 14% (E39)

and CAPM has an average 95% con¬dence interval of 56% (E40).

Columns F and H are successively smaller ¬rms. Note how the min-

imum valuation uncertainty declines with ¬rm size.

The approximate 95% con¬dence intervals for log size are 39%, 14%,

9%, and 7% (row 39) for the huge, large, medium, and small ¬rm, re-

spectively. The CAPM con¬dence intervals also decline with ¬rm size,

but are much larger than the log size con¬dence intervals. For example,

the CAPM small ¬rm 95% con¬dence interval is 23% (I40)”much

larger than the 7% (I39) interval for the Log Size Model.

The Exact 95% Con¬dence Intervals

As mentioned earlier, rows 39 and 40 are a simpli¬ed approximation of

the 95% con¬dence intervals around the discount rates, used to minimize

the complexity of an already intricate series of calculations and related

explanations.

Row 42 contains the exact 95% con¬dence intervals for log size. Note

that the exact 95% con¬dence intervals are larger than their approxima-

tions in Rows 39 to 40. There are no actual 95% con¬dence intervals for

CAPM.5

Aside from the direct effect of size on the calculation of the discount

rate, there is a secondary, indirect effect of size on the con¬dence inter-

vals. All other things being equal, con¬dence intervals are at their mini-

mum at the mean of the data set, which is over $4 billion for the NYSE,

and increase the further we move away from the mean. The huge ¬rm

in column B”and to a lesser extent the large ¬rm in column D”are close

to the mean of the NYSE market capitalization. Therefore, we have two

opposing forces operating on the con¬dence intervals. The mathematics

of the log size equation and Gordon model multiple are such that the

smaller the ¬rm, the smaller the con¬dence interval for the FMV. How-

ever, the smaller ¬rms are far below the mean of the NYSE sample, so

that tends to increase the actual 95% con¬dence interval.

Thus, the direct effect and the indirect effect on the con¬dence inter-

vals work in opposite directions. Jumping ahead of ourselves for a mo-

5. The reason for this is that the CAPM calculations in the SBBI Yearbook are not a pure

regression, because the y-intercept is forced to the risk-free rate.

CHAPTER 11 Measuring Valuation Uncertainty and Error 387

ment, that explains the result in Table 11-2 (which is virtually identical to

Table 11-1 using the 60-year log size regression equation instead of the

72-year equation) that the exact log size con¬dence interval for the small

¬rm is 3%, while it is 2% for the medium ¬rm. If the SBBI Yearbook

compiled similar information for Nasdaq companies, this secondary effect

would be far less, and it is almost certain that the small ¬rm 95% con-

¬dence interval would be smaller than the medium ¬rm con¬dence

interval.

Table 11-2: 60-Year Log Size Model

As mentioned above, Table 11-2 is identical to Table 11-1 except that it

uses the 60-year log size equation instead of the 72-year equation. In this

case we have a much smaller standard error of 0.14% (B35). There is no

comparison to CAPM, because no corresponding data is available. Note

that the actual 95% con¬dence intervals dramatically reduce to 5% of

value for the huge ¬rm (C29) and 2“3% of value for the other size ¬rms

(E29, G29, and I29).

At this point, we remember that there are more sources of uncertainty

than the discount rate, and even with the log size model itself there re-

main questions concerning the underlying data set. I eliminated the ¬rst

12 years of data for reasons that I and others consider valid. Nevertheless,

that adds an additional layer of uncertainty to the results that we cannot

quantify.

Summary of Valuation Implications of Statistical

Uncertainty in the Discount Rate

The 95% con¬dence intervals are very sensitive to our choice of model

and data set. Using the log size model, we see that under the best of

circumstances of using the past 60 years of NYSE data, the huge ¬rms

($5 billion in FMV in our example, corresponding to CRSP Decile #2) have

a 5% (Table 11-2, C29) 95% con¬dence interval arising just from the

statistical uncertainty in calculating the discount rate. All other-size ¬rms

have 95% con¬dence intervals of 2“3% around the estimate (Table

11-2, row 29). If one holds the opinion that using all 72 years of NYSE

data is appropriate”which I do not”then the con¬dence intervals are

wider, with 45% (Table 11-1, C42) for the billion dollar ¬rms and 13%

(G42, I42) to 17% (E42) minimum intervals for small to medium ¬rms.

Actually, the con¬dence intervals around the valuation are not symmetric,

as the assumption of a symmetric t-distribution around the discount rate

results in an asymmetric 95% con¬dence interval around the FMV, with

a larger range of probable error on the high side than the low side.

Huge ¬rms tend to have larger con¬dence intervals because they are

closer to the edge, where the growth rate approaches the discount rate.6

Small to medium ¬rms are farther from the edge and have smaller con-

¬dence intervals. The CAPM con¬dence intervals are much larger than

the log size intervals.

6. Smaller ¬rms with very high expected growth will also be close to the edge, although not as

close as large ¬rms with the same high growth rate.

Part 4 Putting It All Together

388

T A B L E 11-1

95% Con¬dence Intervals

A B C D E F G H I

4 Huge Firm Large Firm Med. Firm Small Firm

5 Cash ¬‚ow-CFt 1 300,000,000 15,000,000 1,000,000 100,000

6 r (assume correct) 13% 19% 24% 28%

7 g constant growth rate 8% 7% 5% 5%

8 Cash Flowt 324,000,000 16,050,000 1,050,000 105,000

9 Discount rate range

10 Log size model

11 Upper bound [2] 14.75% 20.75% 25.75% 29.75%

12 As calculated [1] 13.00% 19.00% 24.00% 28.00%

13 Lower bound [2] 11.25% 17.25% 22.25% 26.25%

14 CAPM

15 Upper bound 18.58% 24.58% 29.58% 33.58%

16 As calculated [1] 13.00% 19.00% 24.00% 28.00%

17 Lower bound 7.42% 13.42% 18.42% 22.42%

18 Gordon model-log size

19 Lower bound [3] 15.8640 7.9903 5.4036 4.6019

20 Gordon-mid [3] 21.2603 9.0906 5.8608 4.9190

21 Upper bound [3] 32.4791 10.5666 6.4105 5.2882

22 FMV-log size model

23 Lower bound [4] 5,139,936,455 74.6% 128,244,770 87.9% 5,673,826 92.2% 483,200 93.6%

24 Gordon-mid [4] 6,888,334,487 100.0% 145,904,025 100.0% 6,153,845 100.0% 516,495 100.0%

25 Upper bound [4] 10,523,225,754 152.8% 169,594,333 116.2% 6,731,077 109.4% 555,257 107.5%

26 Gordon model-CAPM

27 Lower bound 10.2920 6.3488 4.6310 4.0439

28 Gordon-mid 21.2603 9.0906 5.8608 4.9190

29 Upper bound 178.5354 Explodes 16.5899 8.1092 6.3517

30 FMV-CAPM

31 FMV-lower 3,334,607,119 48.4% 101,898,640 69.8% 4,862,595 79.0% 424,611 82.2%

32 FMV-mid 6,888,334,487 100.0% 145,904,025 100.0% 6,153,845 100.0% 516,495 100.0%

33 FMV-upper NA NA 266,268,022 182.5% 8,514,618 138.4% 666,929 129.1%

34 Verify discount rate [5]

35 Add constant 47.62% 47.62% 47.62% 47.62%

36 1.518% * ln (FMV) 34.39% 28.54% 23.73% 19.97%

37 Discount rate 13.23% 19.08% 23.89% 27.65%

38 Rounded 13% 19% 24% 28%

39 Approx 95% conf. int. 39% 14% 9% 7%

log size / [6]

40 Approx 95% conf. int. Explodes 56% 30% 23%

CAPM / [6]

42 Actual 95% conf. int. 45% 17% 13% 13%

log size / [7]

When we add differences in valuation methods and models and all

the other sources of uncertainty and errors in valuation, it is indeed not

at all surprising that professional appraisers can vary widely in their

results.

MEASURING THE EFFECTS OF VALUATION ERROR

Up to now, we have focused on calculating the con¬dence intervals

around the discount rate to measure valuation uncertainty. This uncer-

tainty is generic to all businesses. It was also brie¬‚y mentioned that we

CHAPTER 11 Measuring Valuation Uncertainty and Error 389

T A B L E 11-1 (continued)

95% Con¬dence Intervals

A B C D E F G H I

4 Huge Firm Large Firm Med. Firm Small Firm

44 Assumptions:

46 Log size constant 47.62%

47 Log size X coef¬cient 1.518%

48 Standard error-log size 0.76%

49 Standard error-CAPM 2.42%

Notes:

[1] We assume both the Log Size Model & CAPM arrive at the same discount rate.