The Optimal Allocation of the
Investment Capital for R&D Projects
at the Commercial Stage with the
Kelly Criterion
Kim, Gyutai
Department of Industrial Engineering
Chosun University
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The Table of Contents
1.Introduction
2.A Short Survey on Real Options Valuing
Theories and The Kelly Criterion
 The survey on the ROV theories will be talked focusing on R&D projects.
 In the discussion of the Kelly criterion, its basic concept will be introduced.
3. Real Options Valuing Theories and the Kelly
Criterion
 The development of the optimal capital allocation of the investment fund
for the R&D project will be addressed.
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….Continued
4. A Numerical Example


The methodology developed in the paper will be demonstrated
with a numerical example which was already publish in the
journal.
It was modified to fit to the purpose of the paper.
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1. Introduction
 It is a main purpose of an investment project that an investor
makes an effort to attain a maximum level of profit with an
optimal allocation of investment capital within a minimum
time horizon.
 To the end, many researchers including Thorpe and Ziemba
have done much work on the subject which is related with so
called the Kelly criterion.
 The effectiveness of the criterion has been proven by many
researchers such that it now becomes so popular in the
financial investing and gambling fields.
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 Unfortunately no work has been done on the real
investment project with the application of the Kelly
criterion. In this paper, we will show one small case of a
real investment project for which the Kelly criterion may
be applicable.
 The methodology will be discussed for which the cash flows
generated at the commercial stage of an R&D investment
project are assumed to follow a normal probability
distribution function.
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2. A Short Survey on Real Options Valuing
Theories and The Kelly Criterion
I.
ROV Theories

Myers first introduced a financial option pricing theory developed by
Black and Scholes used to evaluate a real investment project in 197
and coined it as a real options valuing theory (ROV).
Kester defined a R&D project as a call option and maintained that
since its value was greatly affected by an interest rate, the deferral of
an investing timing, and uncertainty inherent in the project, the
option pricing theory played better role in economically evaluating
the R&D project than traditional discounted cash flow (DCF)
techniques such as a net present value (NPV) or an internal rate of
return (IRR).
Nicholas described a real case study of a real options valuing theory
at which Merck pharmaceutical company performed an economic
appraisal of the development of a new drug.


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II. The Kelly Criterion

The Kelly criterion provides us with the investment tool to
maximize the geometric mean of return on investment played
over the long period of time under the following assumptions:

to get the positive mathematical expectation,
to be an independent trials process,
to reinvest the returns,
to play the games over the long run and
to allow the fractional betting.




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 Kelly approached the criterion in two different aspects.

One is the criterion widely spoken and used to calculate the optimal
fixed fraction
The other is one which maximizes the rate of return on
investment using the Bayesian decision theory.
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민감도 분석예제
예제 9.1]
조건] 초기 투자비용: 125,000(천원), 잔존가액: 40,000(천원)-초기투자비용의 32%, 내용연수: 8년,
감가상각방법: 정액법, 연간 생산량: 2,000개, 단위당 가격: 50(천원), 단위당 변동비용: 15(천원),
연간 고정비용: 10,000(천원), 프로젝트 수명: 5년, 법인세율: 25%, MARR: 15%
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%)  59
,680(천원)  0Templates
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Most of the Kelly criterion starts deriving the optimal fixed fractional
bet size with the following equation .
G  lim
N 
 N  ln  X N X0 
1
where,
G: an exponentially growth rate of the investment return.
N: a total number of investing or a period of time of investing.
X0: a total amount of the initial investment fund.
 Equation (1) becomes a variety of forms depending on the structure
of an odd system.
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 The simplest form of Equation (1) is formulated with one that the
odd system is of a 1 to 1 relationship, which is mathematically
expressed in the following equations:
g ( f )  p ln 1  f   q ln 1  f 
g ( f *)  p ln p  q ln q  ln 2
where,
f*: 2p-1.
f*: an optimal betting ratio.
f: a betting ratio.
g(f): a maximum geometric return when invested with f*.
p: a winning probability.
q: a losing probability.
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An Illustrative Example of the Kelly Criterion
Fig. 2. The value of g(f) as f changes at p=0.7
 It was found that f*=0.4 and g(f*)=8.2%.
 It implies that an investor may maximize his geometric return of 8.2% per
unit time if he invested only 40% of his total investment capital over a long
period to time.
 The graph shows that as f→1-, g(f)→.
 Thus we can find that g(f)=0 when f=0.7165. It means that if the investor
invested 71.65% of his total investment capital over a long period of time, he
will certainly ruin sometime Free
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3. Real Options Valuing Theories and the Kelly
Criterion
It is required to explain the implication of Vt,j at the first time to
understand a binomial lattice model.
Fig. 3. The change in the value of the project under a
binomial lattice model.
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Now we will describe the process of developing the
methodology to determine an optimal allocation of the
investment capital with the Kelly criterion under which the real
options value follow a binomial lattice model.
 If the value of the project has increased j times by time=t, then Vt,j
may be expressed in the following equation.
j
t j 

Vt , j  V0 1  f u  1
1  f 1  d 



 Applying the Kelly criterion for the equation above, we obtain a
geometric return on the project in the following equation.
 Vt , j
Gt , j ( f )  ln 



 j  ln 
1  f
V0 

 t  j   ln 1 
f
 u  d 
1  d 
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 Since the main purpose of the Kelly criterion is to find the
value of f which maximizes the expected logarithmic return on
the project, we need to derive an expected value of the last
equation in the previous slide.
 The value of the project, Vt,j, is determined by a number of
increases of the project value in a variable of j in the last
equation in the previous slide, its expected value may be
demonstrated in the following equation .

E Gt , j ( f )
  E  j   ln 1  f u  1
+E  t  j  ln 1  f (1  d ) 
ln 1  f (u  1)

=t  p  
1  f (1  d )

+t  ln 1  f (1  d ) 
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 Since there is a high likelihood that an investor may be fully ruined as he
increases his betting ratio of f, it is better to measure its risk with a traditional
wisdom of variance.
(7)
 The mathematical expression of variance
(8) for the case is given in the following
equation .
Var Gt , j

 f 

 t  p  1  p 
ln 1  f (u  1) 


1  f (1  d )


2
The following mathematical expressions have widely been used as inputs to
model a binomial lattice model by researchers .

1

exp   t 
u  exp
 t  2 


p


u  d 
1


d  exp
=
1
u



 t 
2


1  RB   d 
=
where,
t: T/N.
: an expected return
the project.
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p:a risk-neutral probability.
u  d 
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4. A Numerical Example
The case study discussed in this section was taken from the Shockely, et.al.
paper and was modified to fit for the purpose of this paper.
Wahoo (a disguised name) company wanted to perform an economic analysis
for the investment project planned to develop a new drug to cure a virusrelated animal disease.
The company already got admitted to receive a financial support from a
Federal and State government for the development of the preclinical stage of
the project.
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 Table 1 was constructed by considering the following conditions,
which plays a vital role in an economic evaluation of the R&D
investment project.
 The expected annual revenue was projected to be $85 million for the sale of the
new drug, but not to be any change in the projected revenue.
 It was estimated that the company would take up 3% and 50% of the market in
2007 and 2017, respectively. It was assumed that the annual market growth rate
was of a linear function.
 The company projected that a cost of goods sold, sales and general administrative
expenses, and working capital would be 30%, 25%, 10% of total revenue,
respectively. The company assumed that a corporate tax rate would be 12.1%.
 The company assessed that building a facility to produce the drug and a logistics
structure to sell it in 2007 would require $10 million, which were planned to be
depreciated with a straight-line method.
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The company guesstimated that the market share of the drug would
decrease by 20% annually after 2017. It assumed that a minimum
attractive rate of return and a risk-free discount rate would be 11% and
5%, respectively.
It was assumed that an investment capital would be always constant.
It was also assumed that only one-fourth of the total investment capital
for commercialization would be invested in the first commercialization
stage in 2007, and
depending on the result of the first
commercialization stage, the rest of the investment capital would be
invested in the following stage.
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A binomial lattice model for the project of Wahoo
Company
 400.4  1.63379
d   224.9
 0.56169
400.4 
u  654.7
In the first time, we derived the optimal betting ratio
considering a general binomial lattice model with the
1
u  (u  1)  (1.63511  1)  0.63511 following findings and assumptions.
First, a risk-free interest rate for 3-year Koran government
1
d  (1  d )  (1  0.56169)  0.48310
bonds came up with around 4%.
It was assumed that a probability that the value of the project
increased was 60%. Then, we could find the values of u and d
as followings:
Using the values above, we could determine the values of u1
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Now, we need to calculate f* using the following equation whose
explanation was omitted due to the limited page. It is noted that when putting
the value of d1 into the following equation, we need to have a minus sign for
it.
1  R  R  pu1  qd1 
B
B

f* 
 R  u1 R  d1 
B
 B






1  0.04 0.04  (0.6)(0.63511)  (0.4)( 0.56169)

(0.04  0.63511)(0.04  0.56169)
=0.33806
 The term of f*=0.33806 implies that the company should invest only
33.8% of the total investment capital in the commercialization stage.
 If the company invests such portion, then it would obtain
WN 
Gn ( f *  0.33806)  ln 
  0.0582717
W

0
This number provides a rate of return of 1.2% per unit time.
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Fig. 5 shows change in the value of G(f) as the probability varies.
Looking the graph in Fig. 5, we can detect that the value of G(f) rapidly
decreases and finally approach negative infinity as f→1” and p→0.
On the other around, G(f) goes up to a maximum value.
In this case, if the company invests 87.7% of its total investment capital, it
would absolutely be ruined.
Fig. 5. The value of G(f) with changes in the value of f and p
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 Now we would analyze the betting ratio applying the Kelly criterion to the
binomial lattice model referring to the discussion in Section III.
 To perform the analysis, we need to calculate E[Gt,j(f)] and Var[Gt,j(f)] and
the expected value and standard deviation of the value of the project.
 Then, those values were estimated to be as followings:
  16%,   60%
u  1.822, d  0.549, p  0.500
We selected G5,j(f) to calculate E[G5,j(f)] and Var[G5,j(f), which are as followings:
 (1  0.822 f ) 

 (1  0.45 f ) 
E G5, j  f   5  (0.5)  ln 


Var G5, j  f   5  0.5  0.5

+5  ln 1  0.451 f 
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
 1  0.822 f  
 ln

1

0.45
f




2
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Fig. 6. The expected value and variance of the project
Fig. 6 shows the sensitivity analysis on the value of E[G5,j(f)]
and Var[G5,j(f) with changes in the values of f and p.
Investigating the graph in Fig. 6, we might find the optimal
betting ratio of 93%, which was derived by MATHEMATICA.
This optimal betting ratio yields a rate of return on the project
of 13% per unit time.
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Fig. 7. The sensitivity analysis on the expected
value with change in the value of f
 Looking at the graph in Fig. 6 and 7, it can be
easily recognized that a variance of the project
geometrically increases at which f >0.4 and the
expected value rapidly decreases at which f >0.93.
 If the company borrows more money from the
outside and invests it, then we can see that it would
completely be ruined at f =1.7
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 Most of research on the Kelly criterion has been focused on a financial
investment project solely and it has been reported that the Kelly criterion
plays a key role in a successful financial investing activity.
However, not much work has been done on a real investment project. So
this paper was concerned with doing a little research on the application of
the Kelly criterion to an economic analysis of a real investment project
And we found some interesting results from this research, but we believe
that we need further research on this topic to refine the work and
investigate the reason why the rates of return found in this paper are so
much different.
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투자프로젝트의 NPV(6%)의 분포
단위: 천원
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의사결정나무분석법
- 의사결정나무분석법(Decision Tree Analysis: DTA)
 불확실한 미래에 일어날 일을 도식화하여 보여주는 대표적인 방법
 의사결정마디(decision node) : 투자프로젝트의 대안들을 표시
 선택마디(chance node) : 의사결정마디의 대안들에 영향을 미치는 임의의 사건에 관한 확률
 청산 값 : 화폐가치로 표시된 숫자
 역방향으로 계산
 청산 값인 화폐가치를 극대화하는 대안을 최선의 대안으로 선정
예제 9.9] DTA를 이용한 투자프로젝트의 위험성 분석
조건] 3가지 대안: (1) 이 투자프로젝트를 지금 당장 포기하는 것- 2,000만원 처분비용 발생,
(2) 현재 발생하고 있는 비용수준에서 현 투자프로젝트를 지속적으로 실행하는 것
- 5,000만원 비용 발생,
(3) 현재 실행 중인 투자프로젝트의 성공 가능성을 향상시키기 위하여 현재의 발생
비용을 증가시키는 것 - 1억원 비용 발생
1년 후 수익: 3억원, 현 수준 노력에 의한 투자프로젝트의 성공/실패 확률: 40%/60%,
현 수준의 노력보다 두 배의 노력을 할 경우 투자프로젝트의 성공/실패 확률: 80%/20%
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의사결정나무분석법
대안종류
풀이] DTA기법을 이용한 투자의사결정
- 평균수익
 포기대안 = -20,0001.0 = -20,000(천원)
 유지대안 = 250,0000.4+(-50,000)  0.6
=70,000(천원)
 증가대안 = 200,0000.8+(-100,000)  0.2
=140,000(천원)
- 증가대안의 청산 값이 가장 크므로 증가대안 선택
확률정보
p=1
청산 값
-20,000
포기대안
p=40%
250,000
유지대안
증가대안
p=유지대안 성공확률
q=유지대안 실패확률
P=증가대안 성공확률
Q=증가대안 실패확률
q=60%
-50,000
P=80%
200,000
Q=20%
-100,000
투자프로젝트에 관한 의사결정나무
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실물옵션가치론
- 옵션(Option)
 특정의 투자자산을 특정한 기일 내에 특정한 가격으로 매입하든지 혹은 처분할 수 있는 권리
 콜옵션(call option), 풋옵션(put option)
 유럽피언옵션(European option), 아메리칸옵션(American option)
- 실물옵션(real option)
 실물투자자산에 관한 투자의사결정문제에서 미래에 발생할 더 바람직한 선택을 할 수 있는 권리
- 실물옵션의 대표적 사례
 실물투자프로젝트의 투자시점을 연기할 수 있는 유연성
 현재 실행 중인 투자프로젝트를 축소하거나 증가하거나 혹은 포기할 수 있는 유연성
- DCF기법과의 차이점
 DCF 기법
실물옵션의 사례들을 투자평가 과정에 적절하게 반영하지 못함
 투자프로젝트에 내재된 불확실성으로부터 초래되는 투자프로젝트의 가치를 향상시킬 수 있는 기
회를 투자평가과정에서 무시
 실물옵션 기법
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 투자프로젝트의 투자시점을 다양한
측면에서
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이항격자옵션가치모델
(binomial lattice option pricing model : BLOPM)
- 1단계 BLOPM으로 주식옵션(stock option) 가치 구하기
조건]
- “SS”회사 주식 현재가치: 43만원
- 행사가격(exercise price): 45만원
- 주가가 48만원이 될 확률: 60%
- 주가가 38만원이 될 확률: 40%
- 무위험 이자율: 5%
- 유럽피언 옵션
상승확률=60%
주가: 43만원
옵션가치: ?
주가: 38만원
옵션가치: 0
하락확률=40%
풀이] 주식옵션의 가치 계산
- 주식과 옵션으로 구성된 복제포트폴리오
(replicating portfolio)구성
- 옵션은 공매도
- 무위험차익거래는 없음
- 불확실성을 완전히 소멸시키도록 주식
매입 수()를 결정
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주가: 48만원
옵션가치: 3만원
t=0
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t=1
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주식옵션의 가치를 계산하는 과정
- 1단계: 주가가 상승할 경우의 복제포트폴리오의 가치
48  3
- 2단계: 주가가 하락할 경우의 복제포트폴리오의 가치
38
- 3단계: 1단계와 2단계의 복제포트폴리오 가치가 같도록 연립방정식 구성하여  구하기
48  3  38
  0. 3
- 4단계:  값을 이용하여 복제포트폴리오의 가치를 계산(상승 또는 하락의 경우 모두 결과 동일)
48  3  48(0.3)  3  14.1(만원)
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주식옵션의 가치를 계산하는 과정
- 5단계: 복제포트폴리오의 현재가치 구하기(무위험 이자율=5%)
11.4( P / F ,5%,1)  10.857(만원)
- 6단계: 5단계에서 구한 복제포트폴리오 가치는 현재의 포트폴리오 가치와 동일
43  f  10.857
43(0.3)  f  10.857
f  2.043(만원)
- “SS”회사의 주식 콜옵션의 현재가치는 20,430원
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실물옵션가치 결정
예제 9.10] 실물옵션가치 결정하기
조건] 초기투자자본=2,000,000(천원), 투자프로젝트 현재가치=1,900,000(천원), 상승가치=2,834,450(천원),
하락가치= 1,273,610(천원), 무위험할인율=8%, 실물옵션만기=1년
38
V1u: 28억3,445만원
상승확률=60%
V0:19억원
옵션가치: ?
하락확률=40%
t=0
V1d: 12억7,361만원
t=1
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실물옵션가치 결정
풀이]
(1) 전통적인 DCF을 토대로 한 투자의사결정
투자프로젝트의 NPV  PV  초기투자자본
 19  20  1(억원)
- 이 투자프로젝트의 NPV= -1억원이기 때문에 DCF 의사결정 기법에 의해서는 투자프로젝트 기각
(2) 실물옵션가치론을 토대로 한 투자의사결정
- 1단계: 투자프로젝트가 상승할 경우의 복제포트폴리오의 가치
28.3445   8.3445
 상승할 경우의 1년 후 옵션의 가치 = 투자프로젝트 – 초기투자자본
= 28억 3,445만원 – 20억원 = 8억3,445만원
- 2단계: 투자프로젝트가 하락할 경우의 복제포트폴리오의 가치
12.7361
 하락할 경우의 1년 후 옵션의 가치 = 0
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실물옵션가치 결정
- 3단계: 1단계와 2단계의 복제포트폴리오 가치가 같도록 연립방정식 구성하여  구하기
28.3445  8.3445 12.7361
  0.5346
- 4단계:  값을 이용하여 복제포트폴리오의 가치를 계산
28.3445  8.3445 28.3445(0.5346)  8.3445
 6.809(억원)
-5단계: 복제포트폴리오의 현재가치 구하기(무위험 이자율=8%)
6.809( P / F ,8%,1)  6.305(억원)
- 6단계: 5단계에서 구한 복제포트폴리오 가치는 현재의 포트폴리오 가치와 동일
19  f  6.305
19(0.5346)  f  6.305
f Powerpoint
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실물옵션가치 결정
 SNPV  NPV  ROP
3억8,530만원  1억원  ROP
ROP  4억8,530만원
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