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.