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.

[2] The lower and upper bounds of the discount rate are 2.306 standard errors below and above the discount rate estimated by the model. In a t-Distribution with 8 degrees of freedom,

2.306 standard errors approximately yields a 95% con¬dence interval. See footnote [7] for the exact formula.

[3] This is the Gordon Model with a midyear assumption. The multiple SQRT(1 r) / (r g), where r is the discount rate and g is the perpetual growth rate. We use the lower and

upper bounds of r to calculate our ranges. See footnote [7] for the exact calculation of the con¬dence intervals.

[4] FMV Forecast Cash Flow-Next Year CFt 1 Gordon Multiples

[5] Log Size equation uses data through SBBI 1998 and therefore does not match Table 4-1 exactly.

[6] For simplicity of explanation, this is an approximate 95% con¬dence interval and is 2.306 standard errors above and below the forecast discount rate, with its effect on the valuation.

See footnote [7] for the exact con¬dence interval.

[7] These are the actual con¬dence intervals using the exact formula:

x2

1 0

Y0 ˆ0 t0.025s 1,

x2

n i

where the ˆ 0 is the regression-determined discount rate, t0.025 is the two-tailed 95% con¬dence level t-statistic, s is the standard error of the regression (0.76% for Log Size), and xi is the

deviation of ln(mkt cap) of each decile from the mean ln(mkt cap) of the Ibbotson deciles. The actual con¬dence intervals are calculated only for the Log Size Model. CAPM is not a pure

regression, as its y-intercept is forced to the risk-free rate, and therefore the error term is a mixture of random error and systematic error resulting from forcing the y-intercept.

can calculate the 95% con¬dence intervals around our forecast of sales,

cost of sales, and expenses, though that process is unique to each ¬rm.

All of these come under the category of uncertainty. One need not make

errors to remain uncertain about the valuation.

In the second part of this chapter we will consider the impact on the

valuation of the appraiser making various types of errors in the valuation

process. We can make some qualitative and quantitative observations us-

ing comparative static analysis common in economics.

The practical reader in a hurry may wish to skip to the conclusion

section, as the analysis in the remainder of the chapter does not provide

any tools that one may use directly in a valuation. However:

1. The conclusions are important in suggesting how we should

allocate our time in a valuation.

2. The analysis is helpful in understanding the sensitivity of the

valuation conclusion to the different variables (forecast cash

¬‚ow, discount rate, and growth rate) and errors one may make

in forecasting or calculating them.

De¬ning Absolute and Relative Error

We will be considering errors from two different viewpoints:

— By variable”we will consider errors in forecasting cash ¬‚ow,

discount rate, and growth rate.

Part 4 Putting It All Together

390

T A B L E 11-2

95% Con¬dence Intervals”60-Year Log Size Model

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) 15% 19% 23% 26%

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

8 Cash ¬‚owt 324,000,000 16,050,000 1,050,000 105,000

9 Discount rate range

10 Log size model

11 Upper bound [2] 15.32% 19.32% 23.32% 26.32%

12 As calculated [1] 15.00% 19.00% 23.00% 26.00%

13 Lower bound [2] 14.68% 18.68% 22.68% 25.68%

14 Gordon model-log size

15 Lower bound [3] 14.6649 8.8644 6.0608 5.2710

16 Gordon-mid [3] 15.3197 9.0906 6.1614 5.3452

17 Upper bound [3] 16.0379 9.3292 6.2657 5.4217

18 FMV-log size model

19 Lower bound [4] 4,751,416,807 95.7% 142,274,156 97.5% 6,363,826 98.4% 553.459 98.6%

20 Gordon-mid [4] 4,963,589,879 100.0% 145,904,025 100.0% 6,469,480 100.0% 561,249 100.0%

21 Upper bound [4] 5,196,269,792 104.7% 149,734,328 102.6% 6,578,981 101.7% 569,281 101.4%

22 Verify discount rate

23 Log size constant 41.72% 41.72% 41.72% 41.72%

24 1.204% * ln (FMV) 26.88% 22.63% 18.88% 15.94%

25 Discount rate 14.84% 19.09% 22.84% 25.78%

26 Rounded 15% 19% 23% 26%

27 Min 95% conf. int. log 4% 3% 2% 1%

size /

29 Actual 95% conf. int. 5% 3% 2% 3%

log size /

31 Assumptions:

33 Log size constant 41.72%

34 Log size X coef¬cient 1.204%

35 Standard errors-log size 0.14%

— By type of error, i.e., absolute versus relative errors. The

following examples illustrate the differences between the two:

— Forecasting cash ¬‚ow: If the correct cash ¬‚ow forecast should

have been $1 million dollars and the appraiser incorrectly

forecast it as $1.1 million, the absolute error is $100,000 and

the relative error in the forecast is 10%.

— Forecasting discount and growth rates: If the correct forecast of

the discount rate is 20% and the appraiser incorrectly forecast

it as 22%, his absolute forecasting error is 2% and his relative

error is 10%.

We also will measure the valuation effects of the errors in absolute and

relative terms.

— Absolute valuation error: We measure the absolute error of the

valuation in dollars. Even if the absolute error is measured in

CHAPTER 11 Measuring Valuation Uncertainty and Error 391

percentages, e.g., if we forecast growth too high by 2% in

absolute terms, it causes an absolute valuation error that we

measure in dollars. For example, a 2% absolute error in the

discount rate might lead to a $1 million overvaluation of the

¬rm.

— Relative valuation error: The relative valuation error is the

absolute valuation error divided by the correct valuation. This is

measured in percentages. For example, if the value should have

been $5 million and it was incorrectly stated as $6 million, there

is a 16.7% overvaluation.

The Valuation Model

We use the simplest valuation model in equation (11-1), the end-of-year

Gordon model, where V is the value, r is the discount rate, and g is the

constant perpetual growth rate.

CF 1

Gordon model end-of-year assumption7

V CF (11-1)

r g r g

Dollar Effects of Absolute Errors in Forecasting Year 1

Cash Flow

We now assume the appraiser makes an absolute (dollar) error in fore-

casting Year 1 cash ¬‚ows. Instead of forecasting cash ¬‚ows correctly as

CF1, he or she instead forecasts it as CF2. We de¬ne a positive forecast

error as CF2 CF1 CF 0. If the appraiser forecasts cash ¬‚ow too

low, then CF1 CF2, and CF 0.

Assuming there are no errors in calculating the discount rate and

forecasting growth, the valuation error, V, is equal to:

1 1

V CF2 CF1 CF2 CF1

r g r g

1

(CF2 CF1) (11-2)

r g

Substituting CF CF2 CF1 into equation (11-2), we get:

1

V CF (11-3)

r g

valuation error when r and g are correct and CF is incorrect

We see that for each $1 increase (decrease) in cash ¬‚ow, i.e., CF

g).8 Assuming equivalent

1, the value increases (decreases) by 1/(r

growth rates in cash ¬‚ow, large ¬rms will experience a larger increase in

value in absolute dollars than small ¬rms for each additional dollar of

7. For simplicity, for the remainder of this chapter we will stick to this simple equation and ignore

the more proper log size expression for r, the discount rate, where r a b ln V.

8. It would be 1 r / (r g) for the more accurate midyear formula. Other differences when

using the midyear formula appear in subsequent footnotes.

Part 4 Putting It All Together

392

cash ¬‚ow. The reason is that r is smaller for large ¬rms according to the

log size model.9

If we overestimate cash ¬‚ows by $1, where r 0.15, and g 0.09,

then value increases by 1/(0.15 0.09) 1/0.06 $16.67. For a small

¬rm with r 0.27 and g 0.05, 1/(r g) 1/0.22, implying an increase

in value of $4.55. If we overestimate cash ¬‚ows by $100,000, i.e., CF

$100,000, we will overestimate the value of the large ¬rm by $1.67 million

($100,000 16.67) and the small ¬rm by $455,000 ($100,000 4.55). Here

again, we ¬nd that larger ¬rms and high-growth ¬rms will tend to have

larger valuation errors in absolute dollars; however, it turns out that the

opposite is true in relative terms.

Relative Effects of Absolute Errors in Forecasting Year 1

Cash Flow

Let™s look at the relative error in the valuation (˜˜the relative effect™™) due

to the absolute error in the cash ¬‚ow forecast. It is equal to the valuation

error in dollars divided by the correct valuation. If we denote the relative

valuation error as % V, it is equal to:

V

%V relative valuation error (11-4)

V

We calculate equation (11-4) as (11-3) divided by (11-1):

CF/(r g)

V CF

% error (11-5)

V CF/(r g) CF

relative valuation error from absolute error in CF

For any given error in cash ¬‚ow, CF, the relative valuation error is

greater for small ¬rms than large ¬rms, because the numerators are the

same and the denominator in equation (11-5) is smaller for small ¬rms

than large ¬rms.

For example, suppose the cash ¬‚ow should be $100,000 for a small

¬rm and $1 million for a large ¬rm. Instead, the appraiser forecasts cash

¬‚ow $10,000 too high. The valuation error for the small ¬rm is $10,000/

$100,000 10%, whereas it is $10,000/$1,000,000 1% for the large

10

¬rm.

Absolute and Relative Effects of Relative Errors in

Forecasting Year 1 Cash Flow

It is easy to confuse this section with the previous one, where we consid-

ered the valuation effect in relative terms of an absolute error in dollars

in forecasting cash ¬‚ows. In this section, we will consider an across-the-

9. According to CAPM, small beta ¬rms would be more affected than large beta ¬rms. However,

there is a strong correlation between beta and ¬rm size (see Table 4-1, regression #3), which

leads us back to the same result.

10. This formula is identical using the midyear Gordon model, as the 1 r appears in both

numerators in equation (11-5) and cancel out.

CHAPTER 11 Measuring Valuation Uncertainty and Error 393

board relative (percentage) error in forecasting cash ¬‚ows. If we say the

error is 10%, then we incorrectly forecast the small ¬rm™s cash ¬‚ow as

$110,000 and the large ¬rm™s cash ¬‚ow as $11 million. Both errors are 10%

of the correct cash ¬‚ow, so the errors are identical in relative terms, but

in absolute dollars the small ¬rm error is $10,000 and the large ¬rm error

is $1 million. To make the analysis as general as possible, we will use a

variable error of k% in our discussion.

A k% error in forecasting cash ¬‚ows for both a large ¬rm and a small

¬rm increases value in both cases by k%,11 as shown in equations (11-6)

through (11-8) below. Let V1 the correct FMV, which is equation (11-6)

below, and V2 the erroneous FMV, with a k% error in forecasting cash

¬‚ows, which is shown in equation (11-7). The relative (percentage) val-

uation error will be V2/V1 1, which we show in equation (11-8).

1

V1 CF (11-6)

r g

In equation (11-6), V1 is the correct value, which we obtain by mul-

tiplying the correct cash ¬‚ow, CF, by the end-of-year Gordon model mul-

tiple. Equation (11-7) shows the effect of overestimating cash ¬‚ows by k%.

The overvaluation, V2, equals:

1

V2 (1 k)CF (1 k)V1 (11-7)

r g

V2

%V 1 k (11-8)

V1

relative effect of relative error in forecasting cash flow

Equation (11-8) shows that there is a k% error in value resulting from

a k% error in forecasting Year 1 cash ¬‚ow, regardless of the initial ¬rm

size.12 Of course, the error in dollars will differ. If the percentage error is

large, there is a second-order effect in the log size model, as a k% over-

estimate of cash ¬‚ows not only leads to a k% overvaluation, as we just

discussed, but also will cause a decrease in the discount rate, which leads

to additional overvaluation. It is also worth noting that an undervaluation

works the same way. Just change k to 0.9 for a 10% undervaluation instead

of 1.1 for a 10% overvaluation, and the conclusions are the same.