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pressions that we can calculate in a spreadsheet. This has tremendous
computational advantage, which is not very apparent in a four-milestone
analysis. Increase the number of milestones to 20, and the decision tree
becomes very unwieldy to present, let alone to calculate, while the
spreadsheet is easy. The discussions over the next few pages ultimately
culminate in the development of equations (12-3) through (12-6). The
equations provide the blueprint for the structure of the calculations in
Table 12-3.


Table 12-3: Statistical Calculation of FMV
Table 12-3 is a statistical calculation of the FMV of the common shares of
the Company owned by the existing minority shareholders, based on the
probabilities of the different events occurring and the results of DCF anal-
yses of several different scenario outcomes.

Organization
The table is divided into three sections. In the ¬rst two sections, 1A and
1B, the Company does restructure its debt with the parent. Section 1A is
the calculation of the probability-weighted contribution to the FMV of the
current shareholders™ shares when the Company is successful in obtaining
venture capital. Several possible combinations of events can lead to this
outcome, and we identify the probabilities and payoffs of each combi-
nation in order to calculate the FMV of the common stock owned by the
existing minority shareholders. Section 1B is the probability-weighted
equivalent of section 1A when the Company is not successful in obtaining
venture capital and instead attempts to bootstrap its way to success. The
total of sections 1A and 1B is the FMV of current shareholders™ shares,
assuming the Company restructures the debt.
Section 2 is an analysis of the combination of events in which the
Company does not restructure its debt with the parent. Section 3 is a
summary of the FMVs under the different scenarios and contains calcu-
lations of the per share values. This is the bottom line of the valuation
assignment.

Section 1A: Venture Capital Scenario
In section 1A the primary task is to determine the probability of receiving
venture capital funding. Once we have accomplished that, it is simple to
determine the contribution to FMV from the VC scenario.
Figure 12-1 is a diagram of the decision tree for section 1A. We begin
by noting that there is a 75% probability of making sale #1 and a 25%
probability of not making sale #1, in which case the Company fails. We
denote the former as P(1) 75% and the latter as P( 1) 25%. We

CHAPTER 12 Valuing Startups 417
418




T A B L E 12-3

Statistical Calculation of Fair Market Value


A B C D E F G H I J K

4 Section 1A: Weighted Average Values Assuming Venture Capital Scenario & Debt Restructure With Parent

[C] [D] [G] B18
6 Cum. Product [B] [1 [D] Cumulative [Fn 1] 1 VC% [H] [1 Min] [I]
7 Product [E]

8 Event Conditional Cumulative Venture Prob No VC Current Current Current
9 Probability Joint Cap 1 VC Cond. Shareholders Shareholders Shareholders
10 of Sale Probability Conditional Probability Cum. no VC Prob of VC % Own FMV Control FMV Minor
of Sale Probability

11 #1: Company makes sale #1 75.000% 75.000% 50.000% 50.000% 50.000% 37.500% 50.000% $18,750,000 $14,062,500
12 #3: Company makes sale #2 90.000% 67.500% 60.000% 40.000% 20.000% 20.250% 60.000% $12,150,000 $9,112,500
13 #3: Company makes sale #3 60.000% 40.500% 70.000% 30.000% 6.000% 5.670% 70.000% $3,969,000 $2,976,750
14 #4: Company makes sale #4 80.000% 32.400% 100.000% 0.000% 0.000% 1.944% 85.000% $1,652,400 $1,239,300
15 Totals 63.364% $36,521,400 $27,391,050

Assumptions

18 FMV VC scenario $100,000,000
19 Minority interest discount 25%
(assumed)

21 Section 1B: Bootstrap Scenario Assuming Debt Restructuring With Parent

23 Cum. Product [B] 1 [D] Cum. Prod. P[Si i, [C] [F] Note [1] [H] [I] [1 Min] [J]
[E] (i 1)] {1 [Bt 1]}*[G]

Venture Wtd Avg
26 Cumulative Cap Prob No VC Bootstrap FMV Current
27 Conditional Joint Conditional 1 VC Cond. Conditional Prob of Survival/ Conditional FMV Shareholders
28 Event Probability Probability Probability Probability Cum. No VC Probability No VC FMV Control FMV Minor

29 #1: Company makes sale #1 75.000% 75.000% 50.000% 50.000% 50.000% 30.000% 1.125% 15,286,460 $171,973 $128,980
30 #3: Company makes sale #2 90.000% 67.500% 60.000% 40.000% 20.000% 35.000% 1.890% 15,464,845 292,286 219,214
31 #3: Company makes sale #3 60.000% 40.500% 70.000% 30.000% 6.000% 75.000% 0.365% 15,732,422 57,345 43,009
32 #4: Company makes sale #4 80.000% 32.400% 100.000% 0.000% 0.000% 90.000% 0.000% 16,000,000 0 0
33 Totals 3.380% $521,603 $391,202
35 Section 2: No Debt Restructure With Parent

37 #1: Company makes sale #1 75.000% 75.000% 0.000% 100.000% 100.000% 30.000% 2.250% 7,286,460 $163,945 $122,959
38 #3: Company makes sale #2 90.000% 67.500% 0.000% 100.000% 100.000% 35.000% 9.450% 7,464,845 705,428 529,071
39 #3: Company makes sale #3 60.000% 40.500% 0.000% 100.000% 100.000% 75.000% 6.075% 7,732,422 469,745 352,308
40 #4: Company makes sale #4 80.000% 32.400% 0.000% 100.000% 100.000% 90.000% 29.160% 8,000,000 2,332,800 1,749,600
41 Totals 46.935% $3,671,918 $2,753,938

43 No
Assumptions Restructure Restructure

44 Adjusted FMV $16,000,000 $8,000,000
bootstrap
45 Minority interest discount 25.0%
(assumed)

49 Section 3: Calculation of FMV per Share

50 Restructure No
51 Restructure:
Venture Investor %
52 Capital Bootstrap Total 33.33%

53 Sec 1: venture capital $27,391,050 $391,202 $27,782,252 $2,753,938
scenario
54 Calculation of fully diluted
shares:
55 Original shares 1,000,000 1,000,000 1,000,000
56 Options:
57 200,000 @ $0.50 per share 200,000 200,000 200,000
[2]
58 66,667shares @ $0.75 per 66,667 0 0
share
59 100,000 shares @ $1.00 per 100,000 0 0
share
60 Preferred stock conversion 9,624 0 0
[3]
61 Total option shares 376,290 200,000 200,000
62 Original shares plus options 1,376,290 1,200,000 1,200,000
63 Proposed issuance to 1,300,000 1,300,000 0
president
64 Shares to outside investors 0 0 600,000
[4]
65 Fully-diluted shares [5] 2,676,290 2,500,000 1,800,000
66 Fully-diluted FMV/share- $10.235 $0.156 $10.391 $1.530
post transaction
419
420




T A B L E 12-3 (continued)

Statistical Calculation of Fair Market Value


A B C D E F G H I J K

68 Section 4: 2000 Investor Percentage Taken

70 Control
FMVs

71 t2000 FMV-40% disc rate” $8,000,000
control basis
72 Less: minority interest 25.0%
discount-% (assumed)
73 Less: minority interest ($2,000,000)
discount-$
74 2000 FMV-40% discount $6,000,000
rate”minority basis
75 Percentage required for $2 33.3%
million investment


Notes:
[1] Column I Calculations: Beginning with FMV for Event #4, we subtract $750,000 for not reaching each of Events #4 and #3 and $500,000 for not reaching Event #2. All previous numbers are tax effected and present valued.
[2] Only the 200,000 shares are applicable in all scenarios. The remaining options apply only to the V.C. Scenario
[3] Assume 4 to 1 Preferred-to-Common conversion ratio, per CFO, as follows:


Preferred stock-stated value $400,000
FMV per share of common $10.391
Multiply by 4 $41.56
Convert to # common shares 9,624


[4] In the Bootstrap-No Restructure Scenario, the Company falls $1 million short of cash and owes $1 million to the parent. We assume it will have to take on $2M investment for 33% of the stock. See Section 4.
[5] Actually, fully-diluted shares will be more, as will FMV when VC shares are included. In Section 1A, Columns H and I, we calculated the FMV of the current shareholders™ shares, which is simpler than using actual FMV and wtd avg shares
F I G U R E 12-1

Decision Tree for Venture Capital Funding


VC Found
P(VC2)=0.2025
P(VC2|2)=0.6

VC Found Make Sale 3
Make Sale 2
P(VC1|1)=0.5 P(3|2)=0.6
P(VC1)=0.375 P(3)=0.081
P(2)=0.3375
P(2|1)=0.9
Make Sale 1
P(1)=0.75 No VC Found
P(-VC2|2)=0.4 P(-VC2)=0.135
START No VC Found
No Sale 3
P(-VC1|1)=0.5 P(-VC1)=0.375 No Sale 2
No Sale 1 P(-3|2)=0.4 P(-3)=0.054
P(-1)=0.25 P(-2|1)=0.1 P(-2)=0.0375
Company Fails




P(VC4|4)=1.0 VC Found
P(VC4)=0.01944
VC Found
P(VC3|3)=0.7
Make Sale 4
P(VC3)=0.0567
P(4)=0.01944
P(4|3)=0.8
No VC Found
Make Sale 3 P(-VC4)=0.0
P(-VC4|4)=0.0
P(3)=0.081
No VC Found
No Sale 4
P(-VC3)=0.0243
P(-4)=0.00486
P(-VC3|3)=0.3 P(-4|3)=0.2


Many of the probabilities in this figure appear in Table 12-3, Section 1A, Columns B, D, and G.
Also P(-VC1|1) is equivalent to [1-P(VC1|1)] in the text and P(-2|1)=[1-P(2|1)] etc.




denote the conditional probabilities of subsequent sales as P( j j 1),
where j is the sale number. For example, P(2 1) is the conditional proba-
bility of making sale #2, given that the Company already made sale #1.
The probability of making sale #2 is the probability of making sale #1
multiplied by the conditional probability of making sale #2, given that
the Company makes sale #1, or: P(2) P(1) P(2 1) 0.75 0.9
0.675. Also note that P(1) is the same as P(1 0) since there is no sale zero.

Probability of VC Financing After Sale #1. If the Company makes
sale #1, there is a 50% conditional probability of receiving VC funding at
that time. We denote that event as VC1, which means receiving VC fund-
ing after sale #1 but before sale #2 is attempted,8 and we denote its con-
ditional probability of occurrence as P(VC1 1), i.e., the probability of VC
funding after sale #1, given that sale #1 occurs. The probability of receiv-
ing VC funding after the ¬rst sale is the conditional probability of the
¬rst sale occurring times the conditional probability of VC funding, given
the sale.9 The statistical statement is: P(VC1) P(1) P(VC1 1), where
P(1) is the probability of making sale #1. Thus P(VC1) 0.75 0.5
0.375.
We denote the conditional probability of failure to obtain VC funding
after sale #1 as P( VC1 1) 1 P(VC1 1) 0.5. Thus the absolute prob-


8
From now on, when we say ˜˜after sale i,™™ we also mean ˜˜but before the Company attempts sale
i 1.™™
9
For the ¬rst sale, the conditional probability and the absolute probabilities are identical.


CHAPTER 12 Valuing Startups 421
ability of not receiving VC ¬nancing after sale #1 is P( VC1) P(1)
P( VC1 1) 0.75 0.5 0.375, which is the same result as P(VC1). This
occurs because the conditional probability of obtaining venture capital,
given that the Company makes the ¬rst sale, is 50%. At any other prob-
ability, P(VC1 1) P( VC1 1). These statements generalize for sale i, i
1, 2, 3, 4.

Probability of VC Financing after Sale #2. Let™s move on to the
next step in our analysis: sale #2 and the probability of VC funding after
it. If the Company receives VC after sale #1, we have already quanti¬ed
that above. Our task in this iteration is to quantify the probability of VC
funding if it did not come after sale #1 but does come after sale #2. Thus,
the chain of events we are quantifying in this round is: sale #1 ’ VC1
’ sale #2 ’ VC2, i.e., the Company makes sale #1, doesn™t receive venture
capital, makes sale #2, then receives venture capital.
The probability of obtaining VC funding after sale #2 is:
P(VC2) P(1) [1 P(VC1 1)] P(2 1) P(VC2 2)
0.75 (1 0.5) 0.9 0.6 0.2025 (12-1)
Note that the conditional probability of VC ¬nancing, given that the Com-
pany makes sale #2, P(VC2 2) 0.6, compared to 0.5 after sale #1. In
general, it makes sense that the conditional probability of receiving VC
¬nancing rises with each new key sale.
We can rearrange equation (12-1) as:
P(VC2) P(1) P(2 1) [1 P(VC1 1)] P(VC2 2) (12-2)

In other words, the probability of obtaining VC ¬nancing after sale #2 is
the cumulative joint probability of making both sale #1 and sale #2 times
the conditional probability of not obtaining VC funding after sale #1 times
the conditional probability of obtaining VC funding after sale #2.

Generalizing to Probability of VC Financing after Sale #k. We can
generalize the probability of obtaining VC funding after sale #k as:10
k k1
P(VCk) P(i i 1) [1 P(VCj j)] P(VCk k) (12-3)
i1 j0


Equation (12-3) states that the probability of obtaining venture capital
¬nancing after sale #k is the cumulative joint probability of sale #k oc-
curring times the cumulative joint probability of having been refused VC
¬nancing through sale #(k 1) times the conditional probability of re-
ceiving VC ¬nancing after sale #k.
Finally, the total probability of obtaining VC ¬nancing is the sum of
equation (12-3) across all n sales, where n 4 in this example:


10
Of course, P(1 0) P(1), as the former has no meaning. Also, in the ¬rst iteration of equation
(12-3), i.e., when j 0, the term P(VCj j ) is the cumulative probability of receiving VC
¬nancing from sale #0, which is a zero probability. Thus 1 P(VCj j ) goes to 1.0, as it
should.


PART 5 Special Topics
422
n k k1
P(VC) P(i i 1) [1 P(VCj j)] P(VCk k) (12-4)
k1 i1 j0


Explanation of Table 12-3, Section 1A. Column A lists the sales
events described above, and column B lists their associated conditional
probabilities in cells B11“B14, i.e., P(1) 75% (B11), P(2 1) 90% (B12),
etc. Column C is the cumulative joint probability, which is just the cu-
mulation of the conditional probabilities. For example, the cumulative
joint probability of making sale #4 is P(1) P(2 1) P(3 2) P(4 3)
75% 90% 60% 80% 32.4% (C14), where the conditional proba-
bilities we multiply by each other are in cells B11“B14. Cells C11“C14
n
represent the term P(i i 1) in equations (12-3) and (12-4).
i1
Column D is the president™s forecast of the conditional probability of
obtaining venture capital ¬nancing. Each conditional probability is
P(VCj j), i.e., the probability of obtaining VC ¬nancing after sale #j, given
that the Company makes sale #j, but before attempting sale #j 1. Every
subsequent sale increases the probability of obtaining venture capital be-
yond the level of the previous event. The conditional probability of VC
¬nancing rises from 50% (D11) after sale #1 to 60%, 70%, and 100% for
sales #2, #3, and #4, respectively (D12“D14).
Column E, the conditional probability of not receiving VC ¬nancing
after each sale, is one minus column D. Column F is the cumulative prod-
k1
uct of column E. It is the [1 P(VCj/j )] in equation (12-3) when we
j0
use the cumulation of the previous sale. For example, the probability of
obtaining VC ¬nancing after the sale to company #4 is the cumulative
joint probability of making sale #4, which is 32.4% (C14) the cumulative
joint probability of not having obtained VC ¬nancing after the ¬rst three
sales, which is 6% (F13) the conditional probability of making sale #4,
which is 100% (D14) 1.944% (G14).
Finally, the probability of obtaining VC ¬nancing, according to equa-
tion (12-4), is 65.364% (G15), the sum of column G. The FMV of the com-
pany, if it obtains VC ¬nancing, is $100 million (B18), which we deter-
mined with a DCF analysis.
Column H is one minus the percentage that Mr. Smith estimates the
venture capital ¬rm would take in the company™s stock. After sale #1, he
estimates the venture capitalist would take 50%, leaving 50% (H11) to the
existing shareholders after the conditional transaction. If the Company
makes the sale to company #2, it will be in a stronger bargaining position,
and Mr. Smith estimates the venture capitalist would take 40% of the
Company, leaving 60% (H12) to existing shareholders after the transac-
tion. If the Company makes the sale to company #3, then he estimates
the venture capitalist would take 30% of the Company, leaving 70% (H13)
to the existing shareholders after the transaction. Finally, if the Company
makes the sale to company #4, then he estimates the venture capitalist
would take 15% of the Company, leaving 85% (H14) to the existing share-
holders after the transaction.
Columns I and J are the FMVs of the current shareholders™ shares on
a control and minority basis resulting from obtaining venture capital ¬-

CHAPTER 12 Valuing Startups 423
nancing. Later on, we will add in the current shareholders™ FMV from
bootstrapping the Company to come to a total current shareholders™ FMV
for the debt restructure option. Column I is the control value FMV and
is obtained by multiplying the probability of obtaining VC ¬nancing in
column G times the $100 million FMV of the Company if it receives VC
¬nancing (B18) times column H, the current shareholder ownership per-
centages. Column J is the FMV on a minority interest basis, which is
column I times one minus the minority interest discount of 25.0% (B19),
the magnitude of which is an arbitrary assumption in this analysis. The
total FMVs of current shareholder shares are $36,521,400 (I15) and
$27,391,050 (J15) on a control and minority basis, respectively.
The ¬nal equation describing the FMV is:11
n k k1
FMV (VC) P(i i 1) [1 P(VCj/j)]
k1 i1 j0



P(VCk k) SH%k $100 million (12-5)

In words, the contribution to FMV from the VC scenario is the sum of
the probabilities of obtaining VC, which we quanti¬ed in equation
(12-4), times the $100 million FMV of the company, assuming it is VC
¬nanced.

Section 1B: The Bootstrap Scenario Assuming Debt
Restructuring with Parent
Bootstrapping occurs when the Company fails to attract venture capital
but still manages to stay in business. The bootstrap scenario includes both
success and failure at its attempts to bootstrap. Figure 12-2 shows the

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