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




1. The formula for the valuation error.
2. The equation number containing the error formula.
3. Whether the error is larger for large ¬rms, small ¬rms, or there
is no difference.
The upper half of the table shows the valuation effects of absolute
errors in forecasting the variables (cash ¬‚ow, discount rate, and growth
rate), and the lower half of the table shows the valuation effects of relative
errors in forecasting the variables.
In 10 of the 12 cells in the table that contain error formulas, the
valuation errors are greater for large ¬rms than for small ¬rms. Only
equation (11-5), which is the relative valuation error resulting from a dol-
lar error in forecasting cash ¬‚ows, affects small ¬rms more than large
¬rms. Equation (11-8), the relative valuation error resulting from a relative
error in forecasting cash ¬‚ows, affects both small and large ¬rms alike. It

CHAPTER 11 Measuring Valuation Uncertainty and Error 401
T A B L E 11-4B

Percent Valuation Error for 10% Relative Error in Discount Rate


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 Flow 300,000,000 100,000
9 V1 15,000,000,000 555,556
10 (1 PctError)*g 12.10% 29.70%
11 Gordon model 2 32.2581 4.8309
12 V2 9,677,419,355 483,092
13 V2/ V1 0.6452 0.8696
14 (V2/ V1) 1 35.48% 13.04%
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% 15.49% 18.03% 21.57% 26.83% 35.48% 52.38%
21 12% 14.63% 16.67% 19.35% 23.08% 28.57% 37.50%
22 13% 13.98% 15.66% 17.81% 20.63% 24.53% 30.23%
23 14% 13.46% 14.89% 16.67% 18.92% 21.88% 25.93%
24 15% 13.04% 14.29% 15.79% 17.65% 20.00% 23.08%
25 16% 12.70% 13.79% 15.09% 16.67% 18.60% 21.05%
26 17% 12.41% 13.39% 14.53% 15.89% 17.53% 19.54%
27 18% 12.16% 13.04% 14.06% 15.25% 16.67% 18.37%
28 19% 11.95% 12.75% 13.67% 14.73% 15.97% 17.43%
29 20% 11.76% 12.50% 13.33% 14.29% 15.38% 16.67%
30 21% 11.60% 12.28% 13.04% 13.91% 14.89% 16.03%
31 22% 11.46% 12.09% 12.79% 13.58% 14.47% 15.49%
32 23% 11.33% 11.92% 12.57% 13.29% 14.11% 15.03%
33 24% 11.21% 11.76% 12.37% 13.04% 13.79% 14.63%
34 25% 11.11% 11.63% 12.20% 12.82% 13.51% 14.29%
35 26% 11.02% 11.50% 12.04% 12.62% 13.27% 13.98%
36 27% 10.93% 11.39% 11.89% 12.44% 13.04% 13.71%
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




is not surprising that the only two exceptions to the greater impact of
valuation errors being on large ¬rms comes from cash ¬‚ows, as value is
linear in cash ¬‚ows. The nonlinear relationship of value to discount rate
and growth rate causes errors in those two variables to impact the val-
uation of large ¬rms far more than small ¬rms and to impact the value
of both more than errors in cash ¬‚ow.
Errors in forecasting growth have the greatest impact on value. Value
is positively related to forecast growth. Errors in forecasting discount
rates are a close second in effect,17 though opposite in sign. Value is neg-
atively related to discount rate. Errors in forecasting the ¬rst year™s cash
¬‚ow by far have the least impact on value.


17. Again, this result comes from using the midyear Gordon model, not the end-of-year formula.




Part 4 Putting It All Together
402
T A B L E 11-5

Summary of Effects of Valuation Errors


Valuation Effects of Absolute Errors in the Variables [1]
Valuation Error Cash Flow Discount Rate r Growth Rate g

Absolute ($) 1 r g r g
V CF V CF V CF
(r g) (r1 g1)(r2 g2) (r1 g1)(r2 g2)
(11-3) (11-15) (11-15)
Large ¬rms Large ¬rms Large ¬rms
Relative (%) V CF r g r g
V V
V CF V (r2 g2) V (r2 g2)
(11-5) (11-17) Note [3] (11-17) Note [3]
Small ¬rms Large ¬rms Large ¬rms
Valuation Effects of Relative Errors in the Variables [1]
Valuation Error Cash Flow Discount Rate r Growth Rate g

Absolute ($) V kV1 Note [4] Note [4]
Note [2] NA NA
Large ¬rms Large ¬rms Large ¬rms
Relative (%) V2 r g r g
1 k %Error 1 %Error 1
V1 r (1 k)g
(1 k)r g
(11-8) (11-21) (11-20)
No difference Large ¬rms Large ¬rms

[1] Each cell shows the formula for the valuation error, the equation number in the chapter for the formula, and whether the valuation error is larger for large ¬rms, small ¬rms, or there is
no difference.
[2] This formula is not explicitly calculated in the chapter. We can calculate it as: V2 V1 [(1 k)V1 V1] kV1.
[3] While there is no difference in the magnitude of valuation errors arising from an error in r or g when we measure value by the end-of-year Gordon model, when we use the midyear
Gordon model, errors in g have slightly more impact than errors in r (and much more impact than errors in cash ¬‚ow).
[4] Omitted because these expressions are complex and add little to understanding the topic.




Another issue in valuation error in using the log size model is that
while an initial error in calculating the discount rate is self-correcting
using an iterative method, an error in calculating cash ¬‚ows or the growth
rate not only causes its own error, but also will distort the calculation of
the discount rate. For example, overestimating growth, g, will cause an
overvaluation, which will lower the discount rate beyond its proper level,
which will in turn cause a second order overvaluation. We did not see
this in our comparative static analysis, because for simplicity we were
working with the Gordon model multiple in the form of equation (11-1).
We allowed r to be an apparently independent variable instead of using
its more proper, but complicated log size form of r a b ln V. Thus,
the proper Gordon model using a log size discount rate is: .

1
V CF
a b ln V g

The secondary valuation error caused by a faulty forecast of cash
¬‚ows or growth rate will be minimal because the discount rate, as cal-
culated using the log size model, is fairly insensitive to the error in the
estimate of value. As mentioned earlier, on the surface, this would not be
a source of error using CAPM, as the discount rate in CAPM does not



CHAPTER 11 Measuring Valuation Uncertainty and Error 403
depend on the magnitude of the subject company™s cash ¬‚ows. However,
that is not really true, as CAPM betas are correlated to size.


SUMMARY AND CONCLUSIONS
We discussed valuation uncertainty in the ¬rst part of this chapter and
valuation error in the second part. Using the past 60 years of NYSE data,
the actual 95% con¬dence intervals around the valuation estimate for our
statistical uncertainty in calculating the discount rate range from 5%
for huge ¬rms down to 2“3% for ¬rms of other sizes, as calculated in
Tables 11-1 and 11-2. Using all 72 years of NYSE data leads to much larger
con¬dence intervals, and using CAPM leads to even much larger con¬-
dence intervals. Additionally, we could calculate the 95% con¬dence in-
tervals around the sales and expense forecast.
Errors in forecasting the growth rate and calculating the discount rate
cause much larger valuation errors than errors in forecasting the ¬rst
year™s cash ¬‚ow. Thus, the bottom line conclusion from our analysis is
that we need to be most careful in forecasting growth and discount rates
because they have the most profound effect on the valuation. Usually we
spend the majority of our efforts forecasting cash ¬‚ows, and it might be
tempting to some appraisers to accord insuf¬cient analytic effort to the
growth forecast and/or the discount rate calculation. Hopefully, the re-
sults in this chapter show that that is a bad idea.
In this chapter we have not speci¬cally addressed uncertainty and
errors in calculating valuation discounts, but one must obviously realize
that they, too, add to the overall uncertainty that we have in rendering
an opinion of value. There is material in Chapter 7 relating to uncertainty
in calculating restricted stock discounts, which forms part of our overall
uncertainty in calculating the discount for lack of marketability.
After analysis of just the uncertainty alone in the valuation”
not even considering the possibility that somewhere we have made an
actual error”a healthy humility about our ¬nal valuation conclusions is
appropriate.


BIBLIOGRAPHY
Ibbotson and Associates. 1998. Stocks, Bonds, Bills and In¬‚ation: 1998 Yearbook. Chicago: The
Associates.




Part 4 Putting It All Together
404
PART FIVE


Special Topics




INTRODUCTION
Part 5, which consists of Chapters 12, 13, and 14, deals with topics that
do not ¬t into any other part of the book. All three are practical ˜˜how-
to™™ chapters.
Chapter 12 concerns valuing startups. The chapter discusses three
topics. The ¬rst is the ˜˜First Chicago™™ approach, which is a weighted
average, multiscenario approach to valuing startups. It has the bene¬t of
breaking down the vast range of possibilities into discrete scenarios that
are more credible than attempting to model all possibilities in a single
scenario. Whereas almost all of this book is my own original work, the
First Chicago Approach and the related section on the venture capital
approach are based on a series of articles by Brad Fowler. It is important
to understand the multiscenario approach, not only for its own sake in
valuing simple start-ups but also as a preparation to understand the de-
cision tree approach in the debt restructuring study.
Chapter 12 also provides an example”again based on Fowler™s
work”of using a venture capital valuation approach. While this is tech-
nically a different valuation approach, we will consider it as essentially
the same topic as the First Chicago approach.
The second topic in Chapter 12 is the presentation of the essential
parts of an actual debt restructuring study I did for a client. It is an
example of using an original adaptation of decision tree logic for incor-
porating the effects of probabilistic milestones into a spreadsheet for the
valuation. In this study the viability of the subject company, the proba-
bility of obtaining venture capital ¬nancing, its ability to survive on its
own without venture capital ¬nancing, and its value depend on the out-
come of four different sales milestones. The logic and structure of this
analysis work well for other types of milestones, such as technological
(e.g., successful development) and administrative (e.g., obtaining Food
and Drug Administration approval).
The third topic in Chapter 12 is presenting an exponentially declining
sales growth model1 to semiautomate the process of modeling different

1
I thank R. K. Hiatt for developing this.




405




Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
sales growth patterns. This is a great time saver in valuing startups using
a top-down approach.2 Typically, sales grow rapidly in the early years,
then more slowly, eventually coming to an expected constant growth rate.
Rather than manually insert every year™s sales growth, the appraiser can
instantly change the entire sales growth pattern over n years by changing
the contents of four spreadsheet cells. Furthermore, it makes extensive
sensitivity analysis, normally a cumbersome procedure, trivial.
ESOP valuation has generated a number of lawsuits. One of the sore
points of ESOP valuation that has led to litigation is the dilution in value
that the ESOP experiences after the sale. Selling stock to an ESOP that
does not have the cash to pay for the stock always causes a dilution in
value to the shareholders the instant the transaction takes place. Of
course, it takes time for the bad news to become known, as usually the
next valuation takes place one year later. Employees may be angry, feeling
that they (through the ESOP) paid too much for the owner™s stock. They
may feel someone has pulled a fast one. This can endanger the life and
health of the business.
In Chapter 13 we develop formulas to calculate the post-transaction
fair market value (FMV) before doing the transaction. This enables the
appraiser to provide accurate information to the ESOP trustee that will
enable both sides to enter the transaction with both eyes open. It also
demysti¬es the dilution in value and provides an accurate benchmark
with which to measure future performance. The chapter also provides
precise formulas with which the appraiser can perform the ¬nancial en-
gineering necessary to enable the owner to reduce his or her transaction
price in order to share some or all of the ESOP™s dilution. While this is
not common, sometimes there are benevolent owners who are suf¬ciently
well off and concerned about their employees to do that.
In general, this is a very mathematical chapter. For those readers who
prefer to minimize the amount of mathematics they must read, we have
included Appendix 13-B, a shortcut chapter.
Chapter 14 is a short, simple chapter that makes use of results in
Chapter 13. When partners or shareholders buy out one another, as a ¬rst
approximation there is no impact to the fair market value per share. This
is certainly true when the buyer has the cash to pay to the seller.
However, when the buyer does not have the cash and the company
itself takes out a loan to ¬nance the purchase, secondary effects occur that
can be signi¬cant. Post-transaction, the ¬rm will be more highly lever-
aged, which increases the discount rate. We use the dilution formulas
from Chapter 13 to provide a benchmark lower limit of fair market value
per share. The appraiser can then employ traditional discounted cash ¬‚ow
analysis to value the ¬rm. The result is likely to be a post-transaction fair
market value per share that is lower than the pre-transaction per share
value.


2
This is in contrast to the bottom-up approach, where the appraiser inserts a series of assumptions
to enable one to forecast sales. This might include line items such as market size, market
share for the subject company, etc.




PART 5 Special Topics
406
CHAPTER 12


Valuing Startups




ISSUES UNIQUE TO STARTUPS
ORGANIZATION OF THE CHAPTER
FIRST CHICAGO APPROACH
Discounting Cash Flow Is Preferable to Net Income
Capital Structure Changes
Venture Capital Rates of Return
Table 12-1: Example of the First Chicago Approach
Advantages of the First Chicago Approach
Discounts for Lack of Marketability and Control
VENTURE CAPITAL VALUATION APPROACH
Venture Capital Rates of Return
Summary of the VC Approach
DEBT RESTRUCTURING STUDY
Backgound
Key Events
Decision Trees and Spreadsheet Calculations
Table 12-3: Statistical Calculation of FMV
Organization
Section 1A: Venture Capital Scenario
Probability of VC Financing after Sale #1
Probability of VC Financing after Sale #2
Generalizing to Probability of VC Financing after Sale #k
Explanation of Table 12-3, Section 1A
Section 1B: The Bootstrap Scenario Assuming Debt Restructuring
with Parent
Section 2: No-Restructure Scenario
Section 3: FMVs per Share under Various Restructure Scenarios
Venture Capital Scenario
No-Restructure Scenario




407




Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
Conclusion
Section 4: Year 2000 Investor Percentage Taken
EXPONENTIALLY DECLINING SALES GROWTH MODEL




PART 5 Special Topics
408
ISSUES UNIQUE TO STARTUPS
A number of issues fairly unique to valuing startups arise chie¬‚y from
the uncertainty associated with new ventures. This uncertainty usually
necessitates a more complex, multiple scenario analysis known as the
First Chicago approach and requires more creativity on the part of the
appraiser than other, more routine assignments.3 In this chapter we also
present a much shorter, easier valuation method for startups, known as
the venture capital pricing approach.
Many new ventures have sequential events (milestones) that may or
may not occur, and the valuation depends upon the probabilities of the
occurrence of these milestones. Often, in order for event n to occur, event
(n 1) must occur”but it may or may not. When valuing such ¬rms,
we often combine the First Chicago approach with decision tree analysis
to arrive at a credible fair market value. This is a much more complex
task than the First Chicago approach by itself. The most common types
of milestones are sales, ¬nancing, technical, and regulatory, the latter two
being universal in the valuation of pharmaceutical and biotechnology
¬rms.
Another issue is that startups typically have a pattern of rapid sales
growth followed by declining sales growth rates, ¬nally reaching some
steady state growth rate. Performing sensitivity analysis can be cumber-
some when the appraiser manually enters sales growth rates under a
number of different scenarios.


ORGANIZATION OF THE CHAPTER
This chapter addresses these issues in three parts. Part 1 consists of the
First Chicago approach of forecasting multiple scenarios, each with its
own discounted cash ¬‚ow analysis. We produce a conditional FMV for
each scenario and then calculate a weighted average FMV based on VC
industry research that speci¬es the probabilities of each scenario coming
to fruition. We also include the venture capital pricing approach in Part
1, as it is short and simple.
Part 2 consists of using a very sophisticated decision tree analysis to
value an early stage ¬rm for the purpose of deciding whether or not to
restructure its debt (the ˜˜debt restructuring study™™). The success or failure
of the ¬rm depends on the outcome of a sequence of four events which
will impact the decision. This came from an actual valuation assignment.
Part 3 consists of a mathematical technique to streamline the process
of forecasting sales for a startup. We call the technique the exponentially
declining sales growth model. This model enables the user to generate a
realistic, exponentially declining sales pattern over the life of the product/
service with ease and greatly simpli¬es and facilitates sensitivity analysis,
as it eliminates or at least greatly reduces the need to manually insert
sales growth percentages in spreadsheets.

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