American Journal of Economics and Business Administration 1 (2): 194-205, 2009ISSN 1945-5488© 2009 Science Publications

A Theory of the Risks of Venture Capital Financing
Edmund H. Mantell
Department of Finance, Lubin School of Business, Pace University,
1 Martine Avenue, White Plains, New York, USA, 10606
Abstract: Problem statement: There is no general theory analyzing how the time-varying cash flows
of venture capitalist financing affect the likelihood of success of a new venture. This research
addressed that lacuna in the literature. Approach: Research in the area of venture capital financing
was needed because of the importance of new ventures as germinators of technological innovation.
The research in this study developed a general economic theory quantifying the risk of venture failure
associated with time-varying cash flows of financing. Each occasion when an entrepreneur made an
overture to a venture capitalist to elicit a financing commitment was defined to be a distinct“solicitation event”. The series of financial commitments elicited from venture capitalists wereassumed to have the characteristics of independently distributed random variables. It was assumed thatthe entrepreneur must secure a minimal aggregate commitment in order to ensure development of the
project; failure to secure that amount caused the venture to be aborted. The theory of stochastic
processes was applied to derive the practical implications as regards the risk of abortion. Results: It
was shown that the aggregate financing commitment secured by an entrepreneur in a finite time had
stochastic properties corresponding to those of a statistical renewal process. The research derived
limiting conditions on the probability that entrepreneur’s venture will be aborted because of his failure
to secure the minimal aggregate commitment. The main result was that if the number of solicitations
by the entrepreneur is large and the financial commitments were independently distributed randomvariables with finite means and variances, the probability distribution governing venture survival is theNormal distribution. Conclusion: The study derived four analytical propositions quantifying the tradeoffs between the risk and the expected return associated with venture capital financing. The policy
implications of the results imply the benefits of mitigating information asymmetry. Some of the risk
faced by the entrepreneurs could be attenuated if information about the risk/return preferences ofventure capitalists were known to the entrepreneurs prior to solicitation. Some of the risks faced by theventure capitalists could be attenuated if information about the risk/return characteristics of the
proposed investment project could be accurately and transparently communicated to the venture
capitalist during the solicitation event. If either or both of these information deficits were palliated, themarket for venture capital would operate more efficiently.
Key words: Entrepreneurial solicitations, venture capital financing, stochastic processes, probability ofentrepreneurial insolvency, renewal process
investment activity during most of the period studied,
its impact is remarkable[30]”.
In view of the growing significance of the venture
capital industry in the US, it is odd that so little
attention has been given to the development of a formal
theory of entrepreneurial behavior in the venture capital
solicitation process. There is a large body of literaturedescribing the attitudes and the investment behavior of
the venture capitalists[6-8,18,33]. There are empiricalstudies documenting the realized rates-of-return to theventure capitalists employing different exit strategies[2].
There is an abundant supply of how-to-do-it books andarticles dispensing practical guidance to entrepreneurs

INTRODUCTION
The germination of corporate vitality via venture
capital is growing rapidly. By the end of year 2001,
companies financed with venture capital since the1970s accounted for 5.9% of the jobs in the UnitedStates and 13.1% of US gross domestic product in year
2000 (National Venture Capital Association).
These statistics, as well as others that can be
adduced, evidence the tremendous impact that venture
capital has had on employment and revenue generationin the United States during the past thirty years. “Given
that venture capital was less than one percent of US
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Am. J. of Economics and Business Administration, 1 (2): 194-205, 2009seeking financing (An especially well known how-todo-it book is Pratt)[22]. There are books and articles
describing the financial contracting process (A generalrecent survey is Hart[9]. A much more focuseddescription of financial contracting in the venture
capital industry is Sahlman[27]). I have not found any
general theory purporting to analyze the intertemporal
properties of entrepreneurial funding. The lacuna isconfirmed by a very recently published study wherein
the author commented “...though entrepreneurial firms
are a pivotal source of new employment in Europe, theentrepreneur’s perspective on capital acquisition israrely discussed in the literature”[26]. That author
conducted a case study, consisting mainly if interviews
with successful entrepreneurs. However, without arigorous theory of the solicitation process, the statistics
yielded by the data collected might “yield only a few
banalities”[19].
This study establishes a theory of the venture
capital funding by focusing attention on the activities ofthe entrepreneur at the incipiency of the investment
process: Namely the solicitation event.
A “solicitation event” is a series of activities
wherein the entrepreneur (or the entrepreneurial group)
searches for a venture capitalist to solicit, proposes aninvestment to a venture capitalist and elicits a financial
commitment or, more frequently, a rejection. A seriesof such events constitutes the solicitation process. Thisway of describing a solicitation event makes it possible
to characterize it as a binary-valued variable; i.e., either
the entrepreneur secures a financial commitment, or hedoes not.
This study describes the series of solicitation
events as the realization of random variables. The
successes and the failures of the entrepreneurial groupare shown to have characteristics conforming to thoseof a stochastic process. I exploit some of the properties
of such processes to derive behavioral implications aswell as inferences pertaining to the probability
distributions governing the success of the funding
solicitations.

intermediaries, the market for venture capital would
tend to dry up. This is because relatively poorly
informed investors who were drawn into failinginvestments would decline to provide venture capitalfinance. The investment allocation problems associatedwith risk and uncertainty are assigned to those persons
who are willing and competent to manage them.
Venture capitalists are sufficiently specialized andexperienced in high-risk investments to cope withproblems of information asymmetry. This managementof information asymmetry tends to lead to the so-celled
problem of adverse selection. Venture capitalists havebeen characterized as institutions that resolve
information asymmetries[1,24].
Venture capital firms often finance projects which,at the time of funding, have neither revenues nor even a
product in existence[14]. However, an entrepreneur (oran organized group) will often approach venture
capitalists (hereafter “VCs”) for funding at different
stages in the gestation of the nascent venture. The
multiple stages of the finance solicitation activities havebeen adumbrated by Kozmetsky et al.[15]:
•
•
•

•

•

•

MATERIALS AND METHODS
Synopsis of the venture capital funding process:From the perspective of the microeconomic theory offinancial institutions, the venture capitalist is a kind of
financial intermediary between entrepreneurs and theinvesting public[3]. The venture capitalist manages
funds for downstream investors (i.e., the buyers of thestock when and if, the business goes public) who are
not interested in direct investment in high risk/highreturninvestments.
Without
suchfinancial

Seed financing-capital provided to an entrepreneur
to prove a concept. It may include productdevelopment but does not involve initial marketing
Start-up financing-financing used in productdevelopment and initial marketingFirst-stage financing-financing provided tocompanies that have expended their initial capital(often in developing a prototype) and require funds
to initiate commercial manufacturing and sales
Second-stage financing-working capital used forthe initial expansion of a company that is
producing and shipping a product and has growing
accounts receivable and perhaps inventories
Third-stage financing-funds providing for major
expansion of a company whose sales volume isincreasing and that is breaking even or starting toshow a profit
Fourth-stage, mezzanine, or bridge financingcapital funds invested in a company expected to go
public within six months to a year

When a VC makes a funding commitment to an
entrepreneur it almost always entails staged financing.
That is, the VC makes a firm commitment to offer afraction of the funds needed, with the understandingthat future funding is contingent on firm performance.
This is financially equivalent to taking a sequence of
call options on the entrepreneur’s project. The VC thenevaluates whether the project has reached its
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Am. J. of Economics and Business Administration, 1 (2): 194-205, 2009
performance targets and decides whether to proceedwith additional funding. Consequently, the initialfunding commitment is seldom for 100% of theentrepreneur’s needs.
To put the financial significance of these stages
into practical perspective, in year 1999 “roughly 30%
apiece was invested (by VCs) in first and second roundsof financing and close to 40% was invested in third andlater rounds, with the remainder (apparently close tozero) devoted to seed rounds”[31]. The multi-stage
character of solicitation activities is embodied in thetime series of solicitation events.

trying to educate (the entrepreneur) about the realworld.... The terms of the (venture capitalist’s)
counteroffer will likely call for staged infusions ofcapital over time.... Remember that the venture isscheduled to run out of capital periodically; if itcannot raise capital at the second or third rounds,
then it goes out of business and the entrepreneur isout of a job”)
If an affirmative funding decision is made by a VC
in the interval ∆t i , that decision is manifested as amonetary commitment in the same interval. The
commitment may constitute a capital funding
appropriation for a single period, or it may
constitute serial commitments for funding theventure in specified amounts at predeterminedpoints in future time. Whatever form it may take,the commitment secured in interval ∆t i issymbolized by I( ∆t i ) . It is assumed that theentrepreneurial group regards the sequence ofcommitments {I( ∆t i ) | i = 1, 2,...T} as a stochastic
process whose elements are independently
distributed on the non-negative real line (there issome evidence of collaboration among venture
capital firms in the form of syndication of theirinvestments. For example, a 1994 study by
Lerner[16] examines three rationales for thesyndication of venture capital investments. He useda sample of 271 private biotechnology firms.
Lerner found that syndication is commonplace,even in the first round investments. Syndication
often ensures that the ownership stake of theventure capitalist stays constant in later venturerounds. For the purposes of the analyses in thispaper a syndicate of VCs is defined as one VC
acting independently of other VCs who are notmembers of the syndicate) Historically, only oneout of 50-100 investment opportunities reviewed
by any given VC has actually been financed(Tankersley[28] and Zider[33] “Even though thestructure of venture capital deals seems to put theentrepreneurs at a steep disadvantage, they
continue to submit far more plans than actually get
funded, typically by a ratio of more than ten toone”. This percentage has some empirical validity,
as found by Maier and Walker[17]. Venture impact,2007 page 10 stated: “For every 100 business plansthat come to a venture capital firm for funding,usually only 10 or so get a serious look and only
one ends up being funded”). To the extent that thishistorical statistic is stationary, most of theelements of the series {I(∆t i )} will be equal to zero
(One commentator has suggested that if an

Assumptions characterizing the solicitation process:The solicitation process that is the subject of this study
is characterized by a set of simplifying assumptions.
These are enumerated below:
1.

2.

3.

4.

The entire time period during which the solicitation
activities are carried out is called the funding
solicitation period. The point in time that marks the
inception of the solicitation activities is symbolizedby t0. The funding solicitation period is assumed toconsist of T intervals of equal length: ∆t i = t i − t i −1
for i = 1,...., T. In this study it is assumed that T isa large number
It is assumed that in each interval theentrepreneurial group solicits at least one source ofventure capital funding. A solicitation event
consists of three (or possibly four) distinct phases:(a) identify suitable VCs to be solicited, (b) makean overture to a suitable VC to determine whether
he will entertain a full-blown proposal, (c) present
a business plan to the VC who agrees to entertain
the proposal and (d) carry out negotiations forfunding and control. In order to simplify the model
it is assumed that solicitation activities that arecommenced in each time interval are completed in
the same time interval
It is assumed that the funding decisions of the VCs
who are solicited in each of the T time intervals are
made during the same in the interval in which they
are solicited
The length of the funding solicitation period isTmeasured as ∑ ∆t i
i =1

5.

It is assumed that the terminal point of thesolicitation period for this venture is immutable(The consequences of an immutable terminal pointin the solicitation period has been vividly describedby Sahlman[27]. “After picking himself up off thefloor, the venture capitalist will begin a process of196

Am. J. of Economics and Business Administration, 1 (2): 194-205, 20096.

7.

investment opportunity is rejected by a number offirms, it may get an “overshopped” reputation
because VCs freely trade information and aturndown by one firm may influence others. See
Tankersley[28]. Assumption (5) expressly precludes
this behavior). In addition, the entrepreneurialgroup will recognize that they must expect tosolicit many VCs in order to secure adequate
funding. This translates to mean that T is likely to
be a large integer
Another characteristic of the venture capitalindustry that affects the validity of Assumption (5)is that VC firms also protect themselves from risk
by coinvesting with other VCs. It is the exception,
not the rule, for one VC to finance an individualcompany entirely. Rather, venture firms prefer tohave two or three VCs involved in most stages offinancing[33]. This is a form of syndication (supra).
A reason for this is that agency costs of investing
are reduced when other reputable venture
capitalists are syndicating a deal and the network
allows an efficient transfer of information about
competing firms and technologies. To the extent
that this preference is ubiquitous in the communityof VCs, it will tend to induce an increase in thenumber of VCs solicited by an entrepreneurialgroup. This attitude will also tend to cause T to belarge
It is assumed that the members of theentrepreneurial group formulate homogeneous
expectations about the parameters of the
distributions governing the elements in thesequence {I(∆t i )} . The expected values of theaggregate commitments in each interval aresymbolized by E[I(∆t i )] = µi and the variances of
the aggregate commitments in each interval aresymbolized by Variance of I( ∆t i ) = vari
It is assumed that the entrepreneurial group
allocates a budget of B dollars to underwrite theexpenses of carrying on solicitation activities
during the entire funding solicitation period. In
each of the T intervals the group will spend a
constant portion of the aggregate amount budgeted.
The budget allocation in each solicitation interval
is symbolized by b ≤8.

success of the venture, but it is a sine qua non forsuccess
9. It is assumed that if the group does not secureadequate funding on or before the expiration of thefunding solicitation period the venture will beaborted10. It is assumed that if the entrepreneurial groupsecures commitments equal to or exceeding K
dollars prior to the expiration of the funding
solicitation period, the solicitation activities willcontinue to the end of the funding solicitation
period. To protect their investment or tocompensate for the lack of available working
capital from traditional sources, VCs provide laterstage financing to their successful companies whootherwise would be unable to expand to a
profitable maturity stage
Properties of the commitment process: Consider thefinancial status of the entrepreneurial group when it hascompleted solicitations in t of the intervals, that is, there
are T-t intervals remaining in which the entrepreneurialgroup will continue its solicitation activities.
Let St represent the aggregate capital funding thathas been committed to the group at the end of interval t.
This is calculated as:t

St = ∑ I(∆t i )i =1

The dollar magnitude of St can be described inprobabilistic language; the entrepreneur has made asequence of draws from t independent probability
distributions. Each of those draws results in a nonnegative random variable, i.e., the dollar commitment.
The sequence of dollar commitments sums to St. Thatsum consists of the realizations of t random variables
drawn from the process {I( ∆t i ) | i = 1, 2,...t} . Cox[4]describes this system as “an (ordinary) renewal
process” (The analytical approach to renewal theory
adopted by Cox casts it in terms of a population of“components” with “failure times” governed by acontinuous random variable. He states: “It is again to be
stressed that the terms component and failure-time canbe given many different interpretations”. In this paper
the analogue of the “nth failure-time” (i.e., the nthrenewal) consists of the investment commitment in thenth solicitation interval).
Periodically (A recurring event ξ is called periodic
if there exists an integer λ f 1 such that ξ can occur only
at trials number λ,2λ,3λ,… The greatest λ with thisproperty is called the period of ξ. In this paper theBT

It is assumed that the entrepreneurial group hasdetermined the minimum amount of financing
required for a successful continuation of theirproject at the expiration of the funding solicitation
period. That minimum is symbolized by K dollars.
Funding at the minimal level does not guarantee
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Am. J. of Economics and Business Administration, 1 (2): 194-205, 2009
periodicity of the process {St} is a consequence ofAssumption (1)), the sum St is “renewed” by the
random variable I( ∆t i ) such that in interval ∆t i the firstdifference of the sum is Si − Si −1 = I(∆t i ) . A brief
description of the application of renewal processes tofinance can be found in Ziemba and Vickson[32]. A
general treatment can be found in the monograph byCox cited above.
Considered ex ante at the end of interval t, the net
aggregate investment capital that will be available at theend of the funding solicitation period is a randomvariable. It can be expressed as Eq. 1:

Proof: Pursuant to Assumption (9) above, theprobability that the venture will be aborted is equal tothe probability that the venture will be undercapitalized
at the end of the solicitation period. The probability
distribution is symbolized as P( ). The probability ofundercapitalization can be written as Eq. 3: T
P(undercapitalization) = P(ST p K) = P  ∑ I(∆t i ) p K *t  (3)
i
=
t
+
1



Pursuant to Assumptions (1) and (5), the number of
independently distributed random variables summed isT-t. If that number is large, then the sum satisfies a
condition sufficient to invoke the Central Limit
Theorem (i.e., CLT). The application of the CLT in thiscontext states that the probability distribution governing

T

ST = ∑ I( ∆t i ) + St + B − bt

(1)

i − t +1

T

For any fixed value of t<T, ex post the partial sum
St is a known value. Thus we can condense the threeconstant terms on the right hand side of Eq. 1 anddefine:K*t = (K − St ) − (B − bt)therenewalequation ∑ I(∆t i )isapproximatelyi = t +1

Normal. The formal expression of the CLT entails someTT

i = t +1

i = t +1

additional notation. Let I t = ∑ µi and let σ 2t = ∑ vari .

(2)

The former is the sum of the commitments expected in
the remaining T-t solicitation events and the latter is thevariance of that sum. Let the variable Zt be defined as:

The quantity K*t defined in Eq. 2 measures themarginal required commitment; that is, the aggregate
commitment that must be secured by theentrepreneurial group in the future T-t intervals. The
first parenthetical term on the right side of Eq. 2
represents the deficit remaining between (a) the
minimum capitalization required for a launch of theventure and (b) the gross commitments to time t. The
second parenthetical term represents the difference
between the aggregate amount budgeted to underwrite
the solicitation activities and the portion of it that will
be expended by date t.
Expression (1) can now be adapted to the express
the probability that the aggregate investment capitalcommitted at the end of the funding solicitation time
horizon will be inadequate to launch the venture, given
the aggregate commitments to time t. That is to say, theconditional probability of undercapitalization calculated
ex ante at time t with ex post knowledge ofcommitments to that time. The probability is expressed
as Proposition 1.

T∑ I(∆t ) − I

Z t = i = t +1

i

t

σ 2t

The CLT states:
lim P(Z t ≤ X) ≅ F(X)

T − t →∞

where, F(.) symbolizes the cdf of the Normal
distribution with a mean of 0 and a variance equal to 1.
This theorem allows us to derive an expression that is
an approximate measure of the probability identified in
Eq. 3. This is accomplished using simple algebra:
 T
P(undercapitaliztion) = P  ∑ I(∆t i ) p K*t  i = t +1

= P  I + σ t Z t p K *t K* − I = P  Zt p t t σt Proposition 1: If the number of solicitations is large
and the capital commitments by the VCs are
independently distributed random variables with finite
means and variances, the probability that the venture
will be aborted is a normally distributed function of theminimal aggregate capital commitment.

 K* − I ≅ F t t  σt 

This completes the proof of Proposition 1.
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Am. J. of Economics and Business Administration, 1 (2): 194-205, 2009
It is a property of all cumulative probability
distributions that they are monotone non-decreasing
functions of their argument (s). Thus, in this context it
is obvious that any increase in the argument
X t = (K *t − It ) σ t will lessen the likelihood that theventure will secure its capitalization requirements.
Perhaps less obvious is the effect on the probabilityof undercapitalization of a change in the variance of therandomly distributed commitments in the remainder ofthe solicitation period, ceteris paribus. This issue bears
on the behavior of the entrepreneurial group in carryingout its solicitation activities subsequent to time t. The
implications of the CLT can be summarized in
Proposition 2.

c.

∂X t ∂σ t = 0

Pursuant to the strictly monotonic property of theNormal cdf, an increase in Xt will increase theprobability of undercapitalization. Likewise, a decreasein Xt will lessen the probability of undercapitalization.
The direct implication of monotonicity is that if thevariance of the aggregate future commitments
increases, ceteris paribus, the probability ofundercapitalization will be increased when the sum of
their expected values exceeds the target funding.
The probability of undercapitalization will decrease
when the variance of future commitments increases,
ceteris paribus, if target funding exceeds the sum of the
expectations. This completes the proof of Proposition 2.

Proposition 2:
a.

b.

c.

If the expected aggregate commitment exceeds themarginal required commitment, an increase in thevariance of the aggregate commitment will
increase the probability of undercapitalization
If the expected aggregate commitment is less thanthe marginal required commitment, an increase in
the variance of the aggregate commitments willlessen the probability of undercapitalization
If the expected aggregate commitment equals themarginal required commitment, a change in thevariance of the aggregate the future commitment
will have no effect on the probability ofundercapitalization

Behavioral implications of proposition 2: Thebehavioral implications of proposition 2 reflect thecollective attitude towards risk manifested by theentrepreneurial group. It is well beyond the scope of
this study to undertake a comprehensive discussion of
the definition and the feasibility of establishing acollective attitude towards risk. This study assumes thatthe members of the group agree to act as if they shared
a common attitude.
At the threshold of the behavioral analysis it will
be useful to prescind the different manifestations of risk
faced by the entrepreneurial group. One apparent risk is
the risk that the group will fail to secure aggregate
commitments that are required to launch the venture;that is, the risk that St p K . That risk can be called theterminal risk.
The other kind of risk is the uncertainty manifested
in the variation of the interim cash flow commitments
antedating the terminal point of the funding solicitation
period. That risk is measured by the elements of theseries {σ 2t | t = 1,2,....T} .
The reason for distinguishing between these two
manifestations of risk is to establish a group utilityordering relation between them. As between these two
manifestations of risk, it is assumed that theentrepreneurial group assigns primacy to the terminal
risk. The collective group attitude regarding theterminal risk can be expressed in terms of the behavior
aimed at attaining the objective of the group: The group
adopts a policy of managing and carrying out thesolicitation process in order to minimize the probability
that the venture will fail to meet its capitalization
requirements at the terminal point of the funding
solicitation period. All other objectives of the group are

Proof: The proof of Proposition 2 proceeds byanalyzing the partial derivative of the argument of theNormal cdf with respect to σt, that is, taking the partial
derivative of Xt w.r.t. σt, we have:∂X t It − K*t
=
∂σ t
σ2t

It is obvious that sign [ ∂X t ∂σ t ] = sign[ It − K*t ].
Recall that It represents the expected value of the sumof the randomly distributed commitments in the future
T-t periods. Also recall that K*t represents the marginal
required commitment. The following inferences are
immediate:a.

If the aggregate expected commitment at time t
exceeds the marginal required commitment, then
∂X t ∂σ t f 0b.

If the aggregate expected commitment at time t isequal to the marginal required commitment, thenIf the aggregate expected commitment at time t isless than the marginal required commitment, then
∂X t ∂σ t p 0

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Am. J. of Economics and Business Administration, 1 (2): 194-205, 2009
subordinated to that primary objective. The group may
manifest risk-aversive or risk-seeking attitudes only tothe extent that those attitudes do not motivate behavior
that is inconsistent with attainment of the group’s
primary objective.

riskiness of the cash flow of the future commitments,
ceteris paribus.
Given this construction, Proposition 1 implies thefollowing behavior: If the expected aggregate
commitment
exceeds
themarginal
requiredcommitment, a risk-seeking group will carry on its
solicitation activities in such a way as to lessen theriskiness of the cash flow of future commitments. Thiscounterintuitive result is a direct implication of part (a)of Proposition 1.

Behavioral implications of a group risk-aversive
attitude: Assume that the collective group attitude isone of risk aversion. This is construed to mean that thegroup will avoid any purposeful action that causes anincrease in the riskiness of the cash flow of the futurecommitments, ceteris paribus. This presumed attitudereflects recent research in the area of managerial
attitudes and practices for managing risk. It embodies
the psychological approach developed by, interalia[12,5,10,29]. All these papers are case studies adducing
empirical evidence respecting managerial attitudes
towards defining, measuring and managing risk). The
fundamental notion is one of “loss aversion” ---notably the avoidance of the maximum conceivable
loss. The maximum loss, in the context of this study,
would be the failure by the entrepreneur to elicit
aggregate commitments sufficient to meet its
capitalization requirements at the terminal point of thefunding solicitation period. A failure of that magnitude
results in the death of the investment project. From theperspective of the entrepreneur, nothing could be
worse. That attitude has been empirically studied inseveral recent publications.
Given this construction, Proposition 1 implies thefollowing: A risk-aversive group will manage its future
solicitation activities in such a way as to cause anincrease in the riskiness of the future commitments if
the expected value of those aggregate commitments
exceeds the marginal required commitment. This is an
instance in which a risk-aversive group will seek out
additional risk in its interim cash flow in order to lessen
the terminal risk of failing to meet its targeted
objective.
This counterintuitive result is a direct implicationof part (b) of Proposition 1. The riskiness of theaggregate future commitments is measured by thevariance of the sum of those randomly distributed
commitments. Proposition 1(b) establishes that anincrease in that variance will lessen the probability ofundercapitalization, given the inequality specified inthat part of the proposition.

Behavioral implications of group risk-neutrality: A
collective attitude of risk-neutrality means that thegroup regards the changes in the variability of the
future commitments as irrelevant to its solicitation
activities, ceteris paribus.
Proposition 1(c) implies that if the expected
aggregate commitment is equal to the marginal
required commitment, the group will not alter its
solicitation behavior in response to changes in theriskiness of the future commitments unless thosechanges produce an inequality of the two capitalamounts.
Mean-variance analysis when the solicitationintervals are independent random variables:
Assumption 1 defined the solicitation intervals as fixedand of equal length. This study relaxes that assumption
to enhance the verisimilitude of the theory.
The Synopsis of this study suggested that there areat least three and perhaps four distinct phases of thesolicitation process in each interval. The amount oftime allocated to phase (a) is entirely controlled by theentrepreneurial group (one well-known entrepreneurwith a track record of successful experiences in raisingventure capital delivered a speech in which he revealed
to incipient entrepreneurs (and others) some of thefund-raising practices that worked well for him.. Firston his list of “tips” was an exhortation to “... carefullytarget four or five venture firms that have investment
strategies consistent with your company’s needs”.
Hoffstein[11]); the amount of time allocated to phases(b), (c) and (d) is only partially controlled by the group.
The search activity in phase (a), to the extent that it iscarried out carefully and with due attention to the kindof information to be discovered, will often represent themost time consuming portion of the total solicitation
activities (An example of the “due attention” to theinformation discovered by the search is the avoidanceof approaching a later stage investor if theentrepreneurial group is looking for seed money).

Behavioral implications of a group risk-seeking
attitude: A group risk-seeking attitude is construed tomean that the group will conduct its solicitation
activities in such as way as to cause an increase in the200

Am. J. of Economics and Business Administration, 1 (2): 194-205, 2009{I(∆t i )} arei.i.d.

where

E[I(∆t i )] = µandVariance[I(∆t i )] = σ2 .

Thecounterpart

elements

oftheprocesses{∆t i | i = 1,2,...} and {I( ∆t i ) | i = 1, 2,...t} are not, in general,

independent. The correlation between corresponding
elements is symbolized by:ρ=Cov[∆t i , I(∆t i )]σ 2∆t σ 2I assume Cov[∆t i ,I(∆t j )] = 0 for i ≠ j .
Define the stochastic process {N(Ti ) | i = 1,2,...} as acounting process such that N(Ti) represents the total
number of solicitation events completed up to time Ti.
Formally, N(Ti) has the following properties:

Fig. 1: Distribution of the sources of entrepreneurial
funding in the US in 2003[30]The systematic component of the solicitationinterval reflects the deliberations and the search
activities carried out by the entrepreneurial group to
identify VCs to approach. As a general matter, there areat least six distinct sources of funding, each with its
own special interests and attitudes
towards risk.
Figure 1 displays the relative distribution of the sourcesof entrepreneurial funding in year 2003. The
entrepreneur must allocate sufficient time to identify
the optimal VCs to approach.
To the extent that the VC participants in theprocess exercise control over the tempo of thesolicitation activities in phases (b-d), the time required
to complete the solicitation in each interval cannot beknown by the entrepreneurs with certainty ex ante. The
entrepreneurial group regards each solicitation interval
as consisting of the sum of a systematic part and arandomly distributed part.
Hereafterthesolicitationintervals{∆t i | i = 1,2,...} are assumed to be i.i.d. random variables(i) N(Ti) > 0 for all i(ii) N(Ti) is integer valued(iii) If j < i, then N(Tj) < N(Ti)(iv) For j < i, N(Ti) - N(Tj) equals the number ofsolicitation events that have occurred in theinterval Ti-Tj
These properties suffice to establish N(Ti)i = 1,
2,..} as a renewal process[25]. Now we can define astochastic process that measures the aggregate capitalcommitments that have been secured from the total
number of VCs in all the solicitation events up to time
T n.
The notation can be simplified by omitting thesubscript appended to Tn. Hereafter, N(T) will signify
the total number of VCs who have been solicited in thearbitrary interval [t0, T]. The aggregate commitments inthat interval is:with E[∆t i ] = ∆ t and Variance [ ∆t i ] = σ 2∆t . I assume
the entrepreneurial group knows (or can estimate) thenumerical values of these parameters.
Define the random interval Tn as the cumulative
time elapsed between the inception of solicitation
activity (at time t0) and the conclusion of thesolicitation event in the interval ∆tn. That cumulative
elapsed time interval is calculated as:

ST = I(∆t1 ) + I(∆t 2 ) + ... + I(∆t N(T) )(4)

The stochastic process {ST} is called a renewal
reward process by Ross[25] and a cumulative renewal
process by Cox[4] and Karlin[13]. All these researchers
analyze parametric and other characteristics of thisprocess. Cox displays the limiting properties of themean and variance of {N(T)} for large T as follows:n

Tn = ∑ ∆t ii =1

E[N(T)] ≈Assumption 5 is also amended. Hereafter theelements of the process of the capital commitments
201

T
∆t

and Var[N(T)] ≈ T

σ 2∆t

(∆t )

3

(5)

Am. J. of Economics and Business Administration, 1 (2): 194-205, 2009
The counterpart parameters of {ST} for T→∞ are
given by Cox[4] as follows:
E[ST ] = T

µ
∆t

 σ2

µ2
µ
Var[ST ] = T  ∆t + σ2∆t
− 2σσ∆tρ32
 ∆ t(∆t )( ∆ t ) 

(6)

(7)

Equation 6 can be substituted into Eq. 7 to express
the variance of {ST} as a function of its expected value.
The expression is written in Eq. 8:
Var[ST ] =1  σ 2∆t22 [E(ST )] − 2σσ ∆t E(ST )ρ + Tσ ∆t  T

Fig. 2: Level curves displaying the risk/return tradeoff
for different correlations between the randomly
distributed solicitation interval and therandomly distributed finance commitment in
the interval(8)Equation 8 establishes that the variance of {ST} is aquadratic function of its expected value, inter alia. For
arbitrary numerical values of the parameters ∆ t , σ∆tand σ, the shape of the locus of Eq. 9 is determined by
the algebraic sign of the correlation between thesolicitation intervals and the commitments secured inthose intervals. Figure 3 illustrates the relationshipbetween risk and expected return.
Figure 2 displays three members of a family of locithat differ from each other only with respect to a single
parameter; namely the algebraic sign of the correlation
between the randomly distributed solicitation interval[i.e., ∆ti] and the randomly distributed capitalcommitment secured in that interval [i.e., I(∆ti)]. The
Fig. 3 shows how the variance of the aggregate
commitments at time T is functionally related to the
expected value of that aggregate at the same time,
assuming different algebraic signs of the correlations.
Generally, the loci are shaped as one would expect;
namely sloping upward to the right. This is merely agraphic confirmation of the proposition that as the
expected value of the aggregate commitments
increases, its riskiness likewise increases. There isnothing remarkable in that result. However, theordering of the curves implies a less obvious result;namely that as the correlation decreases, ceteris paribus,
the riskiness increases. Compare, for example, a casewhere the capital commitment is statisticallyindependent of the solicitation interval with the casewhere the two are positively correlated. Then for any
arbitrary expected aggregate, the riskiness will be larger
if the correlation is independent than if the correlationis positive. The loci displayed in Fig. 2 suggest
inferences summarized in Proposition 3.

Fig. 3: Level curves of the variance showing thecertainty equivalent varying with the correlation
Proposition 3: Given arbitrary values for threeparameters: The mean solicitation interval, the varianceof the solicitation interval and the variance of therandomly distributed capital commitments:
a.

b.

202

If
the
randomly
distributed
investmentcommitments are not positively correlated withtime allocated by the entrepreneur to solicitation of
those commitments, then the variance of theaggregate capital commitments is a monotone
increasing function expected value of the aggregate
capital commitment
Iftherandomly
distributed
investmentcommitments are positively correlated with time
allocated by the entrepreneur to solicitation of
those commitments, then variance of the aggregate
capital commitments is initially a decreasing
function and then an increasing function of the
expected value of the aggregate capitalcommitments

Am. J. of Economics and Business Administration, 1 (2): 194-205, 2009
symbolized by θ (considered from a behavioral
perspective, constancy of the coefficient of absoluterisk aversion is construed to mean that the risk premiumthat an entrepreneur would pay to eliminate a given
amount of variance does not depend on the expectedlevel of the aggregate commitment). If the entrepreneur
assigns a fixed value K* as a certainty equivalent, thenEq. 9 graphs as a straight line that can be superimposedon the loci displayed on Fig. 1. The result of thatsuperposition appears in Fig. 3.
On Fig. 3 the dashed straight line is the graphic
representation of the entrepreneur’s certainly equivalent
symbolized by Eq. 9. For any fixed value of capital, say
K*, the dashed line shows the combination of the
expected aggregate commitment/risk of commitmentthat the entrepreneur identifies as the certainty
equivalent of K*. That is to say, at any point on thedashed line the entrepreneur would be indifferent
between the fixed capital commitment K* and arandomly determined aggregate commitment with an
expected value/standard deviation given by the coordinates of the point on the line.
The certainty equivalent locus intersects each of
the level curves associated with different correlations.
For a fixed value of the certainty equivalent K*, it canbe shown that the slope of the certainty equivalent locus

As a practical matter, one would expect that if anentrepreneurial group allocates more time and effort tothe solicitation process, ceteris paribus, the individual
capital commitments that the group can expect to secure
will likewise increase, i.e., ρ>0. The counterintuitive
implication of part (b) of Proposition 3 is that for somerange of increases in the expected aggregate
commitments, the riskiness of the aggregate capitalcommitment decreases.
Solicitation behavior for constant absolute risk
aversion: Define a collective utility function for theentrepreneurial group as U(X) where X is a monetaryamount. Let U be a three-times continuously
differentiable function that assigns to its argument acollective utility for the entrepreneurial group. I assume
that the function U has the properties U′>0 and U′′>0.
These properties of the aggregate utility function
suffice to define the group’s collective attitude towards
risk.
If X is a random variable with an expected value
symbolized by X , then the so-called “coefficient of
absolute
risk
aversion”
isdefinedasθ(X) = − U(X) / U′(X) . The usefulness of the coefficientof absolute risk aversion in this context is that it can beapplied to identify the certainty equivalent of theuncertain outcome of the solicitation process.
Milgrom and Roberts[20] explain that the certainty
equivalent “…is the amount of income, payable forcertain, that the (decision maker) regards as equivalent
in value to the original, random income”.
Let ST symbolize the certainty equivalent of ST.
That certainty equivalent can be calculated as (Milgrom
and Roberts[20] for a derivation of Eq. 9:

is equal tocertainty equivalent locus will decrease as thecoefficient of absolute risk aversion increases, ceterisparibus. The behavioral implications of the geometry
can be summarized in the following Proposition 4.
Proposition 4: For a fixed certainty equivalent of theaggregate capital commitment, the mean and thevariance of the aggregate capital commitment display
the following properties:

1
ST = E[ST ] − θ(E[ST ])Var(ST )2a.

Where:
θ(E[ST ]) = −U′′(E[ST ])U′(E[ST ])2. This implies that the slope of the
θ

(9)
b.

The certainty equivalent can be used to identify themean and the variance of the aggregate capitalcommitment in the solicitation process implied by thecorrelation between the randomly distributed
solicitation intervals and the random commitments
secured in those intervals. In order to achieve this it isnecessary to assume that the coefficient of absolute risk
aversion is a non-negative constant hereafter

If the correlation between the solicitation intervals
and the capital commitments secured in thoseintervals is not positive, an increase in theentrepreneur’s coefficient of absolute risk aversion
will be associated with a larger mean and variance
If the correlation between the solicitation intervals
and the capital commitments secured in thoseintervals is positive, an increase in theentrepreneur’s coefficient of absolute risk aversion
will be associated with a larger mean and avariance that first decreases and then increases

Part (a) of Proposition 4 has a counterintuitive
sense; it suggests that as the entrepreneur’s aversion to203

Am. J. of Economics and Business Administration, 1 (2): 194-205, 2009
information about the risk/return characteristics of the
proposed investment project could be accurately andtransparently communicated to the VC during thesolicitation event. If either or both of these information
deficits were palliated, the market for venture capitalwould operate more efficiently. How might thepalliation be effected?
To the extent that the venture capital industry
has the characteristics of a public good, governmental
intervention might be justified as a means of mitigatingsome of the problems posed by the information
asymmetry. The establishment of an information
depository, operated by a disinterested governmental
authority, could help to disseminate information among
investors as well as entrepreneurs.

risk increases, ceteris paribus, he will carry out thesolicitation activities in such a way as to result in an
increase in the expectation and the uncertainty of theaggregate capital commitments. But this inference isstrictly true only in the (arguably) unrealistic cases
where the correlations between the time allocated to thesolicitation events is not positively correlated with thecapital commitments secured by those events.
In the most plausible scenario where the correlationis positive, namely Part (b) of the Proposition, Fig. 3
demonstrates that an increase in absolute risk aversion
is associated with a larger expected value and a smaller
variance for a range of different certainty equivalents.
RESULTS AND DISCUSSION
This study examines the risk-return relationships ina scenario wherein an entrepreneur solicits financial
commitments from a series of venture capitalists. The
four propositions derived in this study are based on thestochastic properties of the sequence of solicitation
events as well as on the risk-attitudinal characteristicsof the entrepreneur. The study applies ergodic theorems
to derive statistical and behavioral inferences bearing
on the parameters of the stochastic processes
characterizing the solicitation events.
A main result of this study is Proposition 1; it
derives a limiting probability distribution governing the
likelihood that an entrepreneur will succeed insoliciting from venture capitalists the financial
commitments that he needs to launch his venture.
Propositions 2 and 3 exploit the properties of thestochastic process of solicitation events to deriveinferences bearing on the relationship between thevariation in the series of capital commitments and theentrepreneur’s expectation of aggregate financing.
Proposition 4 construes the ways in which thestochastic properties of the solicitation events influence
the solicitation behavior of the entrepreneur. That
proposition applies the theory of certainty equivalents
and the coefficient of absolute risk aversion to showthat changes in either of these attitudinal parameters
will affect the tradeoff between the risk and the
expected return in the solicitation process.

CONCLUSION
The study derived four analytical propositions
quantifying the trade-offs between the risk and the
expected return associated with venture capitalfinancing. The policy implications of the results imply
the benefits of mitigating information asymmetry.
Some of the risk faced by the entrepreneurs could be
attenuated if information about the risk/returnpreferences of venture capitalists were known to theentrepreneurs prior to solicitation. Some of the risks
faced by the venture capitalists could be attenuated if
information about the risk/return characteristics of the
proposed investment project could be accurately andtransparently communicated to the venture capitalist
during the solicitation event. If either or both of theseinformation deficits were palliated, the market for
venture capital would operate more efficiently.
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attenuated if information about the risk/returnpreferences of VCs were known to the entrepreneurs
prior to solicitation. On the other hand, some of therisks faced by the VCs could be attenuated if

3.

4.

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