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CHAPTER 10 Empirical Testing of Abrams™ Valuation Theory 373
T A B L E 10-4G

Calculation of Component #1”Delay to Sale”$10 Million Firm [1]


A B C D

4 Coef¬cients Co. Data Discount
5 Intercept 0.1342 NA 13.4%
Revenues2 [2]
6 5.33E 18 2.560E 14 0.1%
7 Value of block-post-discount [2] 4.26E 09 $ 9,489,650 4.0%
8 FMV-marketable minority 100% interest 5.97E 10 $10,000,000 0.6%
9 Earnings stability (assumed) 0.1376 0.4200 5.8%
10 Revenue stability (assumed) 0.1789 0.6900 12.3%
11 Average years to sell 0.1339 1.0000 13.4%
12 Total Discount [4] 5.1%
14 Value of block-pre-discount [5] $10,000,000
16 Selling price $10,000,000
17 Divide by P / E multiple assumed at $ 800,000
12.5 net inc
18 Assumed pre-tax margin 5%
19 Sales $16,000,000
Sales2
20 2.56E 14


[1] Based on Abrams regression of Management Planning, Inc. data-Regression #2, Table 7-10
[2] Equal to Pre-Discount Shares Sold in dollars * (1-Discount). B7 equals B14 only when the discount 0%.
[3] Earnings and Revenue stability are assumed at the averages from Table 7-5, G60 and H60, respectively, for all FMVs. In the
Management Planning data, a correlation analysis revealed that ¬rm size and the stability measures are uncorrelated. Therefore, we
assume the same levels for all FMVs.
[4] Total Discount max(discount, 0), because Disc 0 indicates the model is outside of its range of reasonability.
[5] In our regression of the Management Planning, Inc. data, this was a marketable minority interest value. This is an illiquid control
value and is higher by 12% to 25% than the marketable minority value. The regression coef¬cient relating to market capitalization in
B8 is so small that the difference is immaterial, and it is easier to work with the value available.




T A B L E 10-6A

Calculation of DLOM


A B C D E F G

4 Section 1: Calculation of the Discount For Lack of Marketability
6 1 Col. [C]
7 Pure Discount PV of Perpetual Remaining
8 Component z [1] Discount [2] Value
9 1 0.0% 0.0% 100.0% Delay to sale
10 2 9.0% 9.0% 91.0% Buyer™s monopsony power”thin markets
11 3A 4.9% 5.3% 94.7% Transactions costs”buyers
12 3B 14.3% 1.2% 98.8% Transactions costs”sellers
13 Percent remaining 89.9% Total % remaining components 1 2 3A 3B
14 Final discount 10.1% Discount 1 total % remaining
16 Section 2: Assumptions and Intermediate Calculations:
18 FMV-equity of co. (before discounts) $ 75,000
19 Discount rate r [3] 32.6%
20 Constant growth rate g (Table 10-2, row 24) 2.5%
21 Intermediate calculation: x (1 g)/(1 r) 0.7728
22 Avg # years between sales j 10

[1] Pure Discounts: For Component #1, Table 10-4, cell D12; For Component #2, 9% per Schwert article. For Component #3A and #3B, Table 10-5, cells I8 and I9 2% for public
brokerage costs.
[2] PV of Perpetual Discount Formula: 1 (1 x j)/((1 (1 z)*x j)), per equation [7-9], used for Component #3B.
PV of Perpetual Discount Formula: 1 (1 z)*(1 x j)/((1 (1 z)*x j)), per equation [7-9a], used for Component #3A.
Components #1 and #2 simply transfer the pure discount.
[3] The formula is: 0.5352 (.0186 ln FMV), based on Table 10-1, B34 and B35.




PART 4 Putting It All Together
374
T A B L E 10-6B

Calculation of DLOM


A B C D E F G

4 Section 1: Calculation of the Discount For Lack of Marketability
6 1 Col. [C]
7 Pure Discount PV of Perpetual Remaining
8 Component z [1] Discount [2] Value
9 1 0.0% 0.0% 100.0% Delay to sale
10 2 9.0% 9.0% 91.0% Buyer™s monopsony power”thin markets
11 3A 4.5% 4.9% 95.1% Transactions costs”buyers
12 3B 14.0% 1.3% 98.7% Transactions costs”sellers
13 Percent remaining 89.8% Total % remaining components 1 2 3A 3B
14 Final discount 10.2% Discount 1 total % remaining
16 Section 2: Assumptions and Intermediate Calculations:
18 FMV-equity of co. (before discounts) $ 125,000
19 Discount rate r [3] 31.7%
20 Constant growth rate g (Table 11-2, row 24) 3.0%
21 Intermediate calculation: x (1 g)/(1 r) 0.7822
22 Avg # years between sales j 10

[1] Pure Discounts: For Component #1, Table 10-4, cell D12; For Component #2, 9% per Schwert article. For Component #3A and #3B, Table 10-5, cells I10 and I11 2% for public
brokerage costs.
[2] PV of Perpetual Discount Formula: 1 (1 x j)/((1 (1 z)*x j)), per equation [7-9], used for Component #3B.
PV of Perpetual Discount Formula: 1 (1 z)*(1 x j)/((1 (1 z)*x j)), per equation [7-9a], used for Component #3A.
Components #1 and #2 simply transfer the pure discount.
[3] The formula is: 0.5352 (.0186 ln FMV), based on Table 10-1, B34 and B35.




T A B L E 10-6C

Calculation of DLOM


A B C D E F G

4 Section 1: Calculation of the Discount For Lack of Marketability
6 1 Col. [C]
7 Pure Discount PV of Perpetual Remaining
8 Component z [1] Discount [2] Value
9 1 0.0% 0.0% 100.0% Delay to sale
10 2 9.0% 9.0% 91.0% Buyer™s monopsony power”thin markets
11 3A 4.3% 4.7% 95.3% Transactions costs”buyers
12 3B 13.8% 1.3% 98.7% Transactions costs”sellers
13 Percent remaining 89.8% Total % remaining components 1 2 3A 3B
14 Final discount 10.2% Discount 1 total % remaining
16 Section 2: Assumptions and Intermediate Calculations:
18 FMV-equity of co. (before discounts) $ 175,000
19 Discount rate r [3] 31.0%
20 Constant growth rate g (Table 10-2, row 24) 3.0%
21 Intermediate calculation: x (1 g)/(1 r) 0.7860
22 Avg # years between sales j 10

[1] Pure Discounts: For Component #1, Table 10-4, cell D12; For Component #2, 9% per Schwert article. For Component #3A and #3B, Table 10-5, cells I12 and I13 2% for public
brokerage costs.
[2] PV of Perpetual Discount Formula: 1 (1 x j)/((1 (1 z)*x j)), per equation [7-9], used for Component #3B.
PV of Perpetual Discount Formula: 1 (1 z)*(1 x j)/((1 (1 z)*x j)), per equation [7-9a], used for Component #3A.
Components #1 and #2 simply transfer the pure discount.
[3] The formula is: 0.5352 (.0186 ln FMV), based on Table 10-1, B34 and B35.




CHAPTER 10 Empirical Testing of Abrams™ Valuation Theory 375
T A B L E 10-6D

Calculation of DLOM


A B C D E F G

4 Section 1: Calculation of the Discount For Lack of Marketability
6 1 Col. [C]
7 Pure Discount PV of Perpetual Remaining
8 Component z [1] Discount [2] Value
9 1 0.0% 0.0% 100.0% Delay to sale
10 2 9.0% 9.0% 91.0% Buyer™s monopsony power”thin markets
11 3A 4.1% 4.5% 95.5% Transactions costs”buyers
12 3B 13.6% 1.6% 98.4% Transactions costs”sellers
13 Percent remaining 89.5% Total % remaining components 1 2 3A 3B
14 Final discount 10.5% Discount 1 total % remaining
16 Section 2: Assumptions and Intermediate Calculations:
18 FMV-equity of co. (before discounts) $ 225,000
19 Discount rate r [3] 30.6%
20 Constant growth rate g (Table 10-2, row 24) 4.5%
21 Intermediate calculation: x (1 g)/(1 r) 0.8003
22 Avg # years between sales j 10

[1] Pure Discounts: For Component #1, Table 10-4, cell D12; For Component #2, 9% per Schwert article. For Component #3A and #3B, Table 10-5, cells I4 and I5 2% for public
brokerage costs.
[2] PV of Perpetual Discount Formula: 1 (1 x j)/((1 (1 z)*x j)), per equation [7-9], used for Component #3B.
PV of Perpetual Discount Formula: 1 (1 z)*(1 x j)/((1 (1 z)*x j)), per equation [7-9a], used for Component #3A.
Components #1 and #2 simply transfer the pure discount.
[3] The formula is: 0.5352 (.0186 ln FMV), based on Table 10-1, B34 and B35.




T A B L E 10-6E

Calculation of DLOM


A B C D E F G

4 Section 1: Calculation of the Discount For Lack of Marketability
6 1 Col. [C]
7 Pure Discount PV of Perpetual Remaining
8 Component z [1] Discount [2] Value
9 1 1.9% 1.9% 98.1% Delay to sale
10 2 9.0% 9.0% 91.0% Buyer™s monopsony power”thin markets
11 3A 4.7% 5.3% 94.7% Transactions costs”buyers
12 3B 14.2% 1.9% 98.1% Transactions costs”sellers
13 Percent remaining 87.6% Total % remaining components 1 2 3A 3B
14 Final discount 12.4% Discount 1 total % remaining
16 Section 2: Assumptions and Intermediate Calculations:
18 FMV-equity of co. (before discounts) $ 375,000
19 Discount rate r [3] 29.6%
20 Constant growth rate g (Table 11-2, row 24) 5.0%
21 Intermediate calculation: x (1 g)/(1 r) 0.8100
22 Avg # years between sales j 10

[1] Pure Discounts: For Component #1, Table 10-4, cell D12; For Component #2, 9% per Schwert article. For Component #3A and #3B, Table 10-5, cells I6 and I7 1% for public
brokerage costs.
[2] PV of Perpetual Discount Formula: 1 (1 x j)/((1 (1 z)*x j)), per equation [7-9], used for Component #3B.
PV of Perpetual Discount Formula: 1 (1 z)*(1 x j)/((1 (1 z)*x j)), per equation [7-9a], used for Component #3A.
Components #1 and #2 simply transfer the pure discount.
[3] The formula is: 0.5352 (.0186 ln FMV), based on Table 10-1, B34 and B35.




PART 4 Putting It All Together
376
T A B L E 10-6F

Calculation of DLOM


A B C D E F G

4 Section 1: Calculation of the Discount For Lack of Marketability
6 1 Col. [C]
7 Pure Discount PV of Perpetual Remaining
8 Component z [1] Discount [2] Value
9 1 8.4% 8.4% 91.6% Delay to sale
10 2 9.0% 9.0% 91.0% Buyer™s monopsony power”thin markets
11 3A 4.2% 4.8% 95.2% Transactions costs”buyers
12 3B 13.7% 2.3% 97.7% Transactions costs”sellers
13 Percent remaining 81.4% Total % remaining components 1 2 3A 3B
14 Final discount 18.6% Discount 1 total % remaining
16 Section 2: Assumptions and Intermediate Calculations:
18 FMV-equity of co. (before discounts) $ 750,000
19 Discount rate r [3] 28.3%
20 Constant growth rate g (Table 11-2, row 24) 6.0%
21 Intermediate calculation: x (1 g)/(1 r) 0.8259
22 Avg # years between sales j 10

[1] Pure Discounts: For Component #1, Table 10-4, cell D12; For Component #2, 9% per Schwert article. For Component #3A and #3B, Table 10-5, cells I8 and I9 2% for public
brokerage costs.
[2] PV of Perpetual Discount Formula: 1 (1 x j)/((1 (1 z)*x j)), per equation [7-9], used for Component #3B.
PV of Perpetual Discount Formula: 1 (1 z)*(1 x j)/((1 (1 z)*x j)), per equation [7-9a], used for Component #3A.
Components #1 and #2 simply transfer the pure discount.
[3] The formula is: 0.5352 (.0186 ln FMV), based on Table 10-1, B34 and B35.




T A B L E 10-6G

Calculation of DLOM


A B C D E F G

4 Section 1: Calculation of the Discount For Lack of Marketability
6 1 Col. [C]
7 Pure Discount PV of Perpetual Remaining
8 Component z [1] Discount [2] Value
9 1 5.1% 5.1% 94.9% Delay to sale
10 2 9.0% 9.0% 91.0% Buyer™s monopsony power”thin markets
11 3A 2.7% 3.6% 96.4% Transactions costs”buyers
12 3B 4.4% 1.5% 98.5% Transactions costs”sellers
13 Percent remaining 85.0% Total % remaining components 1 2 3A 3B
14 Final discount 15.0% Discount 1 total % remaining
16 Section 2: Assumptions and Intermediate Calculations:
18 FMV-equity of co. (before discounts) $10,000,000
19 Discount rate r [3] 23.5%
20 Constant growth rate g (Table 10-2, row 24) 8.0%
21 Intermediate calculation: x (1 g)/(1 r) 0.8743
22 Avg # years between sales j 10

[1] Pure Discounts: For Component #1, Table 10-4, cell D12; For Component #2, 9% per Schwert article. For Component #3A and #3B, Table 10-5, cells I20 and I21 2% for public
brokerage costs.
[2] PV of Perpetual Discount Formula: 1 (1 x j)/((1 (1 z)*x j)), per equation [7-9], used for Component #3B.
PV of Perpetual Discount Formula: 1 (1 z)*(1 x j)/((1 (1 z)*x j)), per equation [7-9a], used for Component #3A.
Components #1 and #2 simply transfer the pure discount.
[3] The formula is: 0.5352 (.0186 ln FMV), based on Table 10-1, B34 and B35.




CHAPTER 10 Empirical Testing of Abrams™ Valuation Theory 377
1. As ¬rm size increases, our assumed growth rate, g, increases. By
our analysis of the partial derivatives in the Mathematical
Appendix to Chapter 7, that causes an increase in DLOM.
2. As ¬rm size increases, the log size discount rate, r, decreases. By
our analysis of the partial derivatives in the Mathematical
Appendix to Chapter 7, that also causes an increase in DLOM.
3. As mentioned earlier, for ¬rm sizes under $375,000, we assumed
the delay to sale to be 0.33 years or less, which lead to a zero
discount for component #1. For the $375,000 and $750,000 ¬rms,
we assumed a one-half-year and one-year delay to sale, which
led to a component #1 pure discount of 1.9% (Table 10-6E, B9)
and 8.4% (Table 10-6F, B9), respectively. The latter accounts for
the vast majority of the much higher DLOM for the $750,000
mean selling price ¬rms. Had that been zero, like all of the
others except the $375,000 ¬rm, DLOM for the $750,000 ¬rms
would have been 13.1%”much closer to DLOM for the smaller
¬rms.
4. We assumed a 1% broker™s fee for publicly traded stocks for the
$375,000 and $750,000 ¬rms, while we assumed a 2% fee for the
¬rms under that size. This increased the pure discount for
components #3A and #3B by 1% for those two size categories,
and therefore increased DLOM.
5. Transactions costs decrease as size increases. Buyers™ transactions
costs are 7.7% (Table 10-5, I6) for $25,000 ¬rms and 5.2% for
$750,000 ¬rms (I18), for a difference of 2.5%. Sellers™ transactions
costs are 17.1% (I7) for $25,000 ¬rms and 14.7% (I19) for
$750,000 ¬rms, for a difference of 2.4%.
Items 1 through 4 above cause DLOM to increase with size, while
item 5 causes DLOM to decrease with size. Looking at Table 10-2, it is
clear that the ¬rst four items dominate, which causes DLOM to increase
with size. This is not a result that I would have predicted before. I would
have thought that overall, DLOM decreases with size.
As mentioned earlier in this chapter, had we used a different model,
it would have been possible to assign a pure discount for the delay to
sale of perhaps 3“5% using another model. This would have narrowed
the differences between DLOM for the small ¬rms and the large ones,
but we would still have come to the counterintuitive conclusion that
DLOM increases with ¬rm size.


Calculation of DLOM for Large Firms
The preceding result begs the question of what happens to DLOM beyond
the realm of small ¬rms. To answer this question, we extend our analysis
to Tables 10-4G and 10-6G.
Table 10-4G is otherwise identical to its predecessor, Table 10-4F.
Since we do not have the bene¬t of the IBA data at this size level, we
have to forecast sales in a different fashion. The calculation of component
#1 is still not sensitive at this level to the square of revenues, so we can
afford to be imprecise. Assuming an average P/E multiple of 12.5, we

PART 4 Putting It All Together
378
divide the assumed $10 million selling price by the P/E multiple to arrive
at net income of $800,000. Dividing that by an assumed pretax margin of
1014 (B20, trans-
5% leads to sales of $16 million (B19), which is $2.56
ferred to C6) when squared. That contributes only “0.1% (D6) to the cal-
culation of the pure discount from the delay to sale component (it was
0.0% in Table 10-4F, D6).
The really signi¬cant difference in the calculation comes from cell
D7, which is 4.0% in Table 10-4G and zero in Table 10-4F. The ¬nal
calculation of component #1 is 5.1% (D12) for the $10 million ¬rm, com-
pared to 8.4% for the $750,000 ¬rm. Thus it seems that component #1
rises sharply somewhere between $375,000 and $750,000 ¬rms, but then
begins to decline as the size effect dominates and causes transactions costs
to decline, while not adding any additional time to sell the ¬rm.
Table 10-6G is our calculation of DLOM for the $10 million ¬rm.
Comparing it to Table 10-6F, the DLOM calculation for the $750,000 ¬rm,
the ¬nal result is 15.0% (Table 10-6G, D14) versus 20.4% (Table 10-6F,
D14). Thus, it appears that DLOM should continue to decline with size.
Thus it appears that DLOM rises with size up to about $1 million in
selling price and declines thereafter. Another factor we did not consider
here that also would contribute to a declining DLOM with size is that
the number of interested buyers would tend to increase with larger size,
which should lower component #2”buyer™s monopsony power”below
the 9% from the Schwert article cited in Chapter 7.


INTERPRETATION OF THE ERROR
As mentioned earlier, the magnitude of the error in Table 10-2 is fairly
small. The ¬ve right columns average a 0.4% error (I29) and a 4.2% (I30)
mean absolute error. We can interpret this as a victory for the log size
and economic components models”and I do interpret it that way, to
some degree. However, the many assumptions that we had to make ren-
der our calculations too speculative for us to place much con¬dence in
them. They are evidence that we are probably not way off the mark, but
certainly fall short of proving that we are right.
An assumption not speci¬cally discussed yet is the assumption that
the simple means of Raymond Miles™s categories is the actual mean of
the transactions in each category. Perhaps the mean of transactions in the
$500,000 to $1 million category is really $900,000, not $750,000. Our results
would be inaccurate to that extent and that would be another source of
error in reconciling between the IBA P/E multiples and my P/CF mul-
tiples. It does appear, though, that Table 10-2 provides some evidence of
the reasonableness of the log size and economic components models.
Amihud and Mendelson (1986) show that there is a clientele effect
in investing in publicly held securities. Investors with longer investment
horizons can amortize their transactions costs, which are primarily the
bid“ask spread and secondarily the broker™s fees,9 over a longer period,
thus reducing the transactions cost per period. Investors will thus select


9. Because broker™s fees are relatively insigni¬cant in publicly held securities, we will ignore them
in this analysis. That is not true of business broker™s fees for selling privately held ¬rms.


CHAPTER 10 Empirical Testing of Abrams™ Valuation Theory 379
their investments by their investment horizons, and each security will
have two components to its return: that of a zero bid“ask spread asset
and a component that rewards the investor for the illiquidity that he is
taking on in the form of the bid“ask spread.
Thus, investors with shorter investment horizons will choose secu-
rities with low bid“ask spreads, which also have smaller gross returns,
and investors with longer time horizons will choose securities with larger
bid“ask spreads and larger gross returns. Their net returns will be higher
on average than those of short-term investors because the long-term in-
vestor™s securities choices will have higher gross returns to compensate
them for the high bid“ask spread, which they amortize over a suf¬ciently
long investment horizon to reduce its impact on net returns. A short-term
investment in a high bid“ask spread stock would lose the bene¬t of the
higher gross return by losing the bid“ask spread in the sale with little
time over which to amortize the spread.
Investors in privately held ¬rms usually have a very long time ho-
rizon, and the transactions costs are considerable compared to the bid“
ask spreads of NYSE ¬rms. In the economic components model I assumed
investors in privately held ¬rms have the same estimate of j, the average
time between sales, in addition to the other variables, growth (g), discount
rate, (r), and buyers™ and sellers™ transactions costs, z. There may be size-
based, systematic differences in investor time horizons; if so, that would
be a source of error in Table 10-2.
Suf¬ciently long time horizons may also predispose the buyer to
forgo some of the DLOM he or she is entitled to. If DLOM should be,
say, 25%, what is the likelihood of the buyer caving in and settling for
20% instead? If time horizons are j 10 years, then the buyer amortizes
the 5% ˜˜loss™™ over 10 years, which equals 0.5% per year. If j 20, then
the loss is only 0.25% per year. Thus, long time horizons should tend to
reduce DLOM, and that is not a part of the economic components
model”at least not yet. It would require further research to determine
if there are systematic relationships between ¬rm size and buyers™ time
horizons.



CONCLUSION
It does seem, then, that we are on our way as a profession to developing
a ˜˜uni¬ed valuation theory,™™ one with one or two major principles that
govern all valuation situations. Of course, there are numerous subprin-
ciples and details, but we are moving in the direction of a true science
when we can see the underlying principles that unify all the various
phenomena in our discipline.
Of course, if one asks if valuation is a science or an art, the answer
is valuation is an art that sits on top of a science. A good scientist has to
be a good artist, and valuation art without science is reckless fortune
telling.




PART 4 Putting It All Together
380
BIBLIOGRAPHY
Amihud, Yakov, and Haim Mendelson. 1986. ˜˜Asset Pricing and the Bid“Ask Spread.™™
Journal of Financial Economics 17:223“249.
Miles, Raymond C. 1992. ˜˜Price/Earnings Ratios and Company Size Data for Small Busi-
nesses.™™ Business Valuation Review (September): 135“139.
Pratt, Shannon P. 1993. Valuing Small Businesses and Professional Practices, 2d ed. Burr Ridge,
Ill.: McGraw-Hill.




CHAPTER 10 Empirical Testing of Abrams™ Valuation Theory 381
CHAPTER 11


Measuring Valuation Uncertainty
and Error




INTRODUCTION
Differences Between Uncertainty and Error
Sources of Uncertainty and Error
MEASURING VALUATION UNCERTAINTY
Table 11-1: 95% Con¬dence Intervals
Valuing the Huge Firm
Valuation Errors in the Others Size Firms
The Exact 95% Con¬dence Intervals
Table 11-2: 60-Year Log Size Model
Summary of Valuation Implications of Statistical Uncertainity in the
Discount Rate
MEASURING THE EFFECTS OF VALUATION ERROR
De¬ning Absolute and Relative Error
The Valuation Model
Dollar Effects of Absolute Errors in Forecastng Year 1 Cash Flow
Relative Effects of Absolute Errors in Forecasting Year 1 Cash Flow
Absolute and Relative Effects of Relative Errors in Forecasting Year 1
Cash Flow
Absolute Errors in Forecasting Growth and the Discount Rate
De¬nitions
The Mathematics
Example Using the Error Formula
Relative Effects of Absolute Error in r and g
Example of Relative Valuation Error
Valuation Effects on Large Versus Small Firms
Relative Effect of Relative Error in Forecasting Growth and
Discount Rates
Tables 11-4“12-4b: Examples Showing Effects on Large vs. Small
Firms
Table 11-5: Summary of Effects of Valuation Errors
SUMMARY AND CONCLUSIONS



383




Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
INTRODUCTION
This chapter describes the impact of various sources of valuation uncer-
tainty and error on valuing large and small ¬rms. It will also provide the
reader with a greater understanding of where our analysis is most vul-
nerable to the effects of errors and demonstrate where appraisers need to
focus the majority of their efforts.


Differences between Uncertainty and Error
It is worthwhile to explain the differences between uncertainty and error.
I developed the log size equation in Chapter 4 by regression analysis.
Because the R2 is less than 100%, size does not explain all of the differ-
ences in historical rates of return. Unknown variables and/or random
variation explain the rest. When we calculate a 95% con¬dence interval,
it means that we are 95% sure that the true value of the dependent vari-
able is within the interval and 5% sure it is outside of the interval. That
is the uncertainty. One does not need to make an error to have uncertainty
in the valuation.
Let™s suppose that for a ¬rm of a particular size, the regression-
determined discount rate is 20% and the 95% con¬dence interval is be-
tween 18% and 23%. It may be that the true and unobservable discount
rate is also 20%, in which case we have uncertainty, but not error. On the
other hand, if the true discount rate is anything other than 20%, then we
have both uncertainty and error”even though we have used the model
correctly. Since the true discount rate is unobservable and unknowable
for privately held ¬rms, we will never be certain that our model will
calculate the correct discount rate”even when we use it properly. If one
makes a mistake in using the model, that is what we mean by appraiser
error. For the remainder of this chapter, we will use the simpler term,
error, to mean appraiser-generated error. The ¬rst part of the chapter deals
with valuation uncertainty, and the second part deals with valuation
error.


Sources of Uncertainty and Error
We need only look at the valuation process in order to see the various
sources of valuation uncertainty and error. As mentioned in the Intro-
duction to this book, the overall valuation process is:
— Forecast cash ¬‚ows.
— Discount cash ¬‚ows to present value.
— Calculate valuation premiums and discounts for degree of
control and marketability.
Uncertainty is always present, and error can creep into our results at each
stage of the valuation process.


MEASURING VALUATION UNCERTAINTY
In forecasting cash ¬‚ows, even when regression analysis is a valid tool
for forecasting both sales and costs and expenses, it is common to have

Part 4 Putting It All Together
384
fairly wide 95% con¬dence intervals around our sales forecasts, as we
discovered in Chapter 2. Thus, we usually have a substantial degree of
uncertainty surrounding the sales forecast and a typically smaller, though
material, degree of uncertainty around the forecast of ¬xed and variable
costs. As each company™s results are unique, we will not focus on a quan-
titative measure of uncertainty around our forecast of cash ¬‚ows in this
chapter.1 Instead, we will focus on quantitative measures of uncertainty
around the discount rate, as that is generic.
For illustration, we use a midyear Gordon Model formula,
(1 r)/(r g), as our valuation formula. Although a Gordon model
is appropriate for most ¬rms near or at maturity, this method is inappli-
cable to startups and other high-growth ¬rms, as it presupposes that the
company being valued has constant perpetual growth.


Table 11-1: 95% Con¬dence Intervals
Table 11-1 contains calculations of 95% con¬dence intervals around the
valuation that results from our calculation of discount rate. We use the
72-year regression equation for the log size model. It is the relevant time
frame for comparison with CAPM, since the CAPM results in the SBBI
1998 Yearbook (Ibbotson Associates 1998) are for 72 years.2 Later, in Table
11-2, we examine the 60-year log size model for comparison. For purposes
of this exercise, we will assume the forecast cash ¬‚ows and perpetual
growth rate are correct, so we can isolate the impact of the statistical
uncertainty of the discount rate.
The exact procedure for calculating the 95% con¬dence intervals is
mathematically complex and would strain the patience of most readers.
Therefore, we will use a simpler approximation in our explanation and
merely present the ¬nal results of the exact calculation in row 42.

Valuing the Huge Firm
Because the log size model produces a mathematical relationship between
return and size, our exploration of 95% con¬dence intervals around a
valuation result necessitates separate calculations for different-size ¬rms.
We begin with the largest ¬rms and work our way down.
In Table 11-1, cell B5 we show last year™s cash ¬‚ow as $300 million.
Using the log size model, the discount rate is 13%3 (B6), and we assume
a perpetual growth rate of 8% (B7). We apply the perpetual growth rate
to calculate cash ¬‚ows for the ¬rst forecast year. Thus, forecast cash ¬‚ow
$300 million 1.08 $324 million (B8).
In B12 we repeat the 13% discount rate. Next we form a 95% con¬-
dence interval around the 13% rate in the following manner. Regression


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.
[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.


Absolute Errors in Forecasting Growth and the
Discount Rate
A fundamental difference between these two variables and cash ¬‚ow is
that value is nonlinear in r and g, whereas it is linear in cash ¬‚ow. We
will develop a formula to quantify the valuation error for any absolute


11. Strictly speaking, the error is really k, not k%. However, the description ¬‚ows better using the
percent sign after the k.
12. Again, this formula is the same with the midyear Gordon model, as the square root term
cancels out.




Part 4 Putting It All Together
394
error in calculating the discount rate or the growth rate, assuming cash
¬‚ow is forecast correctly.

De¬nitions
First we begin with some de¬nitions. Let:
V1 the correct value
V2 the erroneous value
r1 the correct discount rate
r2 the erroneous discount rate
g1 the correct growth rate
the erroneous growth rate13
g2
CF cash ¬‚ow, which we will assume to be correct in this section
the change in any value, which in our context means the error
We will consider a positive error to be when the erroneous value, discount
rate, or growth rate is higher than the correct value. For example, if g1
5% and g2 6%, then g g2 g1 1%; if g1 6% and g2 5%, then
g 1%.

The Mathematics
The correct valuation, according to the end-of-year Gordon model, is:
CF
V1 the correct value (11-9)
r1 g1
The erroneous value is:
CF
V2 the erroneous value (11-10)
r2 g2
The error, V V2 V1, equals:
CF CF 1 1
V CF (11-11)
r2 g2 r1 g1 r2 g2 r1 g1
In order to have a common denominator, we multiply the ¬rst term in
round brackets by (r1 g1)/(r1 g1) and we multiply the second term
in round brackets by (r2 g2)/(r2 g2).
(r1 g1) (r2 g2)
V CF (11-12)
(r1 g1)(r2 g2)
Rearranging the terms in the numerator, we get:
(r1 r2) (g1 g2)
V CF (11-13)
(r1 g1)(r2 g2)
Changing signs in the numerator:


13. Actually, only one of the two variables”r2 or g2”need be erroneous. The other one can be
correct, which would make it equal to its r1 or g1 counterpart.




CHAPTER 11 Measuring Valuation Uncertainty and Error 395
(r2 r1) (g2 g1)
V CF (11-14)
(r1 g1)(r2 g2)
which simpli¬es to:
r g
V CF (11-15)
(r1 g1)(r2 g2)
absolute effect of absolute error in r or g14

Example Using the Error Formula
Let™s use an example to demonstrate the error formula. Suppose cash ¬‚ow
is forecast next year at $100,000 and that the correct discount and growth
rate are 20% and 5%, respectively. The Gordon model multiple is 1/(0.25
0.05) 5, which leads to a valuation before discounts of $500,000.
Instead, the appraiser makes an error and uses a zero growth rate. His
erroneous Gordon model multiple will be 1/(0.25 0) 4, leading to a
$400,000 valuation. The appraiser™s error is an undervaluation of $400,000
$500,000 $100,000.
Using equation (11-15),
0 0.05 0.05
V $100,000 100,000
(0.25 0.05)(0.25 0) 0.2 0.25
0.05
100,000 $100,000
0.05

Relative Effects of Absolute Error in r and g
The relative valuation error, as before, is the valuation error in dollars
divided by the correct valuation, or:
CF( r g)/(r1 g1)(r2 g2)
V
% Error (11-16)
V CF/(r1 g1)
r g
V
% Error (11-17)
V r2 g2
relative effects of absolute error in r and g15


14. When r 0, then the formula using the midyear Gordon model is identical to equation
(11-15), with the addition of the term 1 r after the CF, but before the square brackets.
When there is an error in the discount rate, the error formula using the midyear Gordon
model is
(r1 g1) 1 r2 (r2 g2) 1 r1
CF
(r1 g1)(r2 g2)
The partial derivative for g is similar to the discrete equation for change:
V CF
g)2
g (r
Since it is a partial derivative, we hold r constant, which means r 0, and instead of
having r2 g2, we double up on r1 g1, which we can simplify to r g. Again, these
formulas are correct only when CF is forecast correctly.
15. This formula would be identical using the midyear Gordon model, as the 1 r would
appear in both numerators in equation (11-16) and cancel out.




Part 4 Putting It All Together
396
Example of Relative Valuation Error
From the previous example, the relative valuation error is
$400,000
1 20%
$500,000
a 20% undervaluation. Using equation (11-17), the relative error is
0 0.05 0.05
20%
0.25 0 0.25
which agrees with the previous calculation and demonstrates the accu-
racy of equation (11-17). It is important to be precise with the deltas, as
it is easy to confuse the sign. In equation (11-17) the numerator is r
g. It is easy to think that since there is a plus sign in front of g, we
should use a positive 0.05 instead of 0.05. This is incorrect, as we are
assuming that the appraiser™s error in the growth rate itself is negative,
i.e., the erroneous growth rate minus the correct growth rate, (V2 V1)
0 0.05 0.05.

Valuation Effects on Large Versus Small Firms
Next we look at the question of whether large or small ¬rms are more
affected by identical errors in absolute terms in the discount or growth
rate. The numerator of equation (11-17) will be the same regardless of
size. The denominator, however, will vary with size. Holding g2 constant,
r2 will be smaller for large ¬rms, as will r2 g2. Thus, the relative error,
as quanti¬ed in equation (11-17), will be larger for large ¬rms than small
¬rms, assuming equal growth rates.16
Table 11-3 demonstrates the above conclusion. Columns B through D
show valuation calculations for the huge ¬rm, as in Table 11-1. Historical
cash ¬‚ow was $300 million (B6), and we assume a constant 8% (B7)
growth rate as being correct, which leads to forecast cash ¬‚ow of $324
million (B8). Using the log size model, we get a discount rate of 15% (B9),
as shown in cells B14“B17. In B10, we calculate an end-of-year Gordon
model multiple of 14.2857, which differs from Table 11-1, where we were
using a midyear multiple. Multiplying row 8 by row 10 produces a value
of $4.63 billion (B11).
Column C contains the erroneous valuation, where the appraiser uses
a 9% growth rate (C7) instead of the correct 8% growth rate in B7. That
leads to a valuation of $5.45 billion (C11). The valuation error is $821.4
million (D11), which is C11 B11. Dividing the $821.4 million error by
the correct valuation of $4.63 million, the valuation error is 17.7% (D12).
We repeat the identical procedure with the small ¬rm in columns E“G
using the same growth and discount rate as the huge ¬rm, and the val-
uation error is 6.9% (G12). This demonstrates the accuracy of our conclu-
sion from equation (11-17) that equal absolute errors in the growth rate


16. As before, this is theoretically not true in CAPM, which should be independent of size.
However, in reality, is correlated to size.




CHAPTER 11 Measuring Valuation Uncertainty and Error 397
T A B L E 11-3

Absolute Errors in Forecasting Growth Rates


A B C D E F G

4 Huge Firm Small Firm

5 Correct Erroneous Error Correct Erroneous Error

6 Cash ¬‚ow-CFt 1 300,000,000 300,000,000 100,000 100,000
7 g growth rate 8% 9% 8% 9%
8 Cash ¬‚owt 324,000,000 327,000,000 108,000 109,000
9 Discount rate 15.0% 15.0% 26.0% 26.0%
10 Gordon multiple-end year 14.2857 16.6667 5.5556 5.8824
11 FMV 4,628,571,429 5,450,000,000 821,428,571 600,000 641,176 41,176

12 Percentage error 17.7% 6.9%
13 Verify discount rate

14 0.01204 * ln(FMV) 26.80% 26.99% 16.02% 16.10%
15 Add constant 41.72% 41.72% 41.72% 41.72%
16 Discount rate 14.92% 14.73% 25.70% 25.62%
17 Rounded 15% 15% 26% 26%




or discount rate cause larger relative valuation errors for large ¬rms than
small ¬rms.
Let™s now compare the magnitude of the effects of an error in cal-
culating cash ¬‚ow versus discount or growth rates. From equation (11-8),
a 1% relative error in forecasting cash ¬‚ows leads to a 1% valuation error.
From equation (11-17), a 1% absolute error in forecasting growth leads to
a valuation error of 0.01/(r2 g2). Using typical values for the denomi-
nator, the valuation error will most likely be in the range of 4“20% for
each 1% error in forecasting growth (or error in the discount rate). This
means we need to pay relatively more attention to forecasting growth rates and
discount rates than we do to producing the ¬rst year™s forecast of cash ¬‚ows,
and the larger the ¬rm, the more care we should be taking in the analysis.
Also, it is clear from (11-15) and (11-17) that it is the net error in both
r and g that drives the valuation error, not the error in either one indi-
vidually. Using the end-of-year Gordon model, equal errors in r and g
cancel each other out. With the more accurate midyear formula, errors in
g have slightly more impact on the value than errors in r, as an error in
r has opposite effects in the numerator and denominator.

Relative Effect of Relative Error in Forecasting Growth and
Discount Rates
We can investigate the impact of a k% relative error in estimating g by
restating the Gordon model in equation (11-18) below with the altered
growth rate (1 k)g. We denote the correct value as V1 and the incorrect
value as V2.
CF
V2 (11-18)
r (1 k)g
The ratio of the incorrect to the correct value is V2/V1, or:

Part 4 Putting It All Together
398
r g
V2
(11-19)
V1 r (1 k)g

The relative error in value resulting from a relative error in forecasting
growth will be (V2/V1) 1, or:

r g
% Error 1 (11-20)
r (1 k)g

relative error in value from relative error in growth

Thus, if both a large and small ¬rm have the same growth rate, then
the lower discount rate of the large ¬rm will lead to larger relative val-
uation errors in the large ¬rm than the small ¬rm. Note that for k 0,
(11-20) 0, as it should. When k is negative, which means we forecast
growth too low, the result is the same”the under-valuation is greater for
large ¬rms than small ¬rms.
A relative error in forecasting the discount rate shifts the (1 k) in
front of the r in (11-20) instead of being in front of the g. The formula is:

r g
% Error 1 (11-21)
(1 k)r g

relative error in value from relative error in r

Tables 11-4 through 11-4B: Examples Showing Effects on
Large Versus Small Firms
Table 11-4 shows the calculations for k 10% (B38) relative error in fore-
casting growth. Rows 5“6 contain the discount rate and growth rate for
a huge ¬rm in column B and a small ¬rm in column C, respectively. The
end-of-year Gordon model multiples are 50 (B7) and 5.5556 (C7) for the
huge and the small ¬rm, respectively. Multiplying the Gordon model
multiples by the forecast cash ¬‚ows in row 8 results in the correct values,
V1, in row 9 of $15 billion and $555,556, respectively.
Now let™s see what happens if we forecast growth too high by 10%
for each ¬rm. Row 10 shows the erroneously high growth rate of 9.9%.
Row 11 contains the new Gordon model multiples, and row 12 shows V2,
the incorrect values we obtain with the high growth rates. Row 13 shows
the ratio of the incorrect to the correct valuation, i.e., V2/V1, and Row 14
shows the relative error, (V2/V1) 1 81.82% for the huge ¬rm and
5.26% for the small ¬rm.
Rows 20“36 are a sensitivity analysis that show the relative valuation
errors for various combinations of r and g using equation (11-20), with
k 10%. Note that the bolded cells in F20 and F36 match the results in
row 14, con¬rming the accuracy of the error formula. This veri¬es our
observation from analysis of equation (11-20) that equal relative errors in
forecasting growth will create much larger relative valuation errors for
large ¬rms than small ¬rms, holding growth constant. All we need do is
notice that the relative errors in the sensitivity analysis decline as we
move down each column, and as small ¬rms have higher discount rates,
the lower cells represent the smaller ¬rms.

CHAPTER 11 Measuring Valuation Uncertainty and Error 399
T A B L E 11-4

Percent Valuation Error for 10% Relative Error in Growth


A B C D E F G

4 Description Huge Firm Small Firm
5 r 11% 27%
6 g 9% 9%
7 Gordon model 50.0000 5.5556
8 Cash ¬‚ow 300,000,000 100,000
9 V1 15,000,000,000 555,556
10 (1 PctError)*g 9.90% 9.90%
11 Gordon model 2 90.9091 5.8480
12 V2 27,272,727,273 584,795
13 V2/ V1 1.8182 1.0526
14 (V2/ V1) 1 81.82% 5.26%
16 Sensitivity Analysis: Valuation Error for Combinations of r and g
18 Growth rate g

19 Discount Rate r 5% 6% 7% 8% 9% 10%

20 11% 9.09% 13.64% 21.21% 36.36% 81.82% NA
21 12% 7.69% 11.11% 16.28% 25.00% 42.86% 100.00%
22 13% 6.67% 9.38% 13.21% 19.05% 29.03% 50.00%
23 14% 5.88% 8.11% 11.11% 15.38% 21.95% 33.33%
24 15% 5.26% 7.14% 9.59% 12.90% 17.65% 25.00%
25 16% 4.76% 6.38% 8.43% 11.11% 14.75% 20.00%
26 17% 4.35% 5.77% 7.53% 9.76% 12.68% 16.67%
27 18% 4.00% 5.26% 6.80% 8.70% 11.11% 14.29%
28 19% 3.70% 4.84% 6.19% 7.84% 9.89% 12.50%
29 20% 3.45% 4.48% 5.69% 7.14% 8.91% 11.11%
30 21% 3.23% 4.17% 5.26% 6.56% 8.11% 10.00%
31 22% 3.03% 3.90% 4.90% 6.06% 7.44% 9.09%
32 23% 2.86% 3.66% 4.58% 5.63% 6.87% 8.33%
33 24% 2.70% 3.45% 4.29% 5.26% 6.38% 7.69%
34 25% 2.56% 3.26% 4.05% 4.94% 5.96% 7.14%
35 26% 2.44% 3.09% 3.83% 4.65% 5.59% 6.67%
36 27% 2.33% 2.94% 3.63% 4.40% 5.26% 6.25%
38 Relative Error in g 10%

Formula in B20: (which copies to the other cells in the sensitivity analysis) (($A20 B$19)/($A20 ((1 $PctError)*B$19))) 1




Table 11-4A is identical to Table 11-4, with the one exception that the
growth rate is a negative 10% instead of a positive 10%. Table 11-4A
demonstrates that, assuming identical real growth rates, forecasting
growth too low also affects large ¬rms more than small ¬rms.
Table 11-4B is also identical to Table 11-4, except that it measures the
relative valuation error arising from relative errors in calculating the dis-
count rate. Table 11-4B uses equation (11-21) instead of equation (11-20)
to calculate the error. It demonstrates that relative errors in forecasting
the discount rate affect the valuation of large ¬rms more than the valu-
ation of small ¬rms, assuming identical real growth rates.

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