African Journal of Business Management Vol. 5(12), pp. 49164933, 18 June, 2011
Available online at http://www.academicjournals.org/AJBM
DOI: 10.5897/AJBM10.1598
ISSN 19938233 ©2011 Academic Journals
Full Length Research Paper
Valuation of investment on a firm with trade credit
under uncertainty: A real options approach
PoYuan Chen
1,2
*, HorngJinh Chang
1
and IMing Jiang
2
1
Graduate Institute of Management Sciences, Tamkang University, 151 Yingchuan Road, Tamsui,
Taipei County, Taiwan.
2
Department of Financial and Tax Planning, Jinwen University of Science and Technology, 99 Anchung Road, Sindian
District, New Taipei City, Taiwan, R. O. C.
2
Faculty of Finance, School of Management, Yuan Ze University, 135 YuanTung Road, Chungli, Taoyuan, Taiwan.
Accepted 18 January, 2011
This paper intends to propose a corporate valuation framework by incorporating both the stochastic
product price and the stochastic interest rate in a delay payment context. By using ItoIsometry
theorem, we derived the analytical solution for corporate value, based on which sensitivity analyses
and further simulations for the real option value are performed. Some critical factors were considered in
the sensitivity analysis of corporate value: the drift and the volatility in the price and in the interest rate
processes, the price elasticity of demand, the cost rate, the market share, as well as the length of time
period for delay payments. To valuate an opportunity of investment on a firm with trade credit under
uncertainty, a real options model was employed. The simulation results for the real option value
indicated that increasing demand elasticity, market share, number of time periods for delay payments,
interest rate drift, price volatility, and interest rate volatility all contribute to increasing real option value,
whereas the increasing cost rate and price drift lead to the decreasing real option value as expected.
Key words: Trade credit, stochastic price, stochastic interest rate, real option.
INTRODUCTION
The mechanism of trade credits is frequently adopted in
businesstobusiness (B2B) transactions to provide
buyers with more flexibility in their working capital
management and thereby enhance their financial liquidity.
The payment terms in B2B transactions are also
employed to facilitate the billing system and attract the
buyers to maintain their loyalty. Such mechanism permits
buyers to arrange a more flexible payment scheduling.
From the perspective of buyers, the payment terms of
accounts payable are considered as trade credits, which
allow them to defer the payable amounts until the
maturity of the credit period and hence increase their
financial ability in management of net working capitals.
Besides, the delayed cash outflows in the procurement
activities could also be invested in financial instruments
with guaranteed returns and safety over the credit period
between the invoice date of merchandise purchases and
*Corresponding author. Email: mikecpy@ms7.hinet.net Tel:
+886282122000 ext. 6375. Fax: +886282122399.
the payment date of account payable settlements. Robb
and Silver (2006) classify the supplier trade credits into
two categories: date term and day term. Dateterm entails
buyer’s payments on a specific date (on 20
th
or 30
th
in
each month), while dayterm permits the buyer to
postpone its payments without incurring any interest
charges for a specified period of time (20 or 30 days).
Based on a dayterm credit policy, the optimal payment
period in a cost minimization model is derived by Jamal,
Sarker, and Wang (2000), who divide the payment period
into two sub periods: the credit period incurring no
interest charge and the overdue period incurring some
interest charges. They conclude that buyer’s payment
decisions should depend on the multiplier of unit price
relative to unit cost. The smaller the multiplier, the earlier
the buyer should pay. However, the optimal payment date
should not be earlier than the maturity of a credit period
in terms of inventory cost minimization. Bhaba et al.
(1994) investigate the optimal order quantity and
maximum allowable shortage in terms of inflation and
time value of money. They conclude that the decisions on
order quantity and allowable shortage depend evidently
Chen et al. 4917
Figure1. The monthly price and quantity processes for the LCD product in a 10year period.
on the difference between the inflation and discount rate.
Dave and Patel (1981) optimize the total costs to obtain
the optimal number of replenishments with a time linearly
dependent demand over a finite time horizon. Kim et al.
(1995) propose a profit maximization model from the
perspective of the supplier to obtain the optimal length of
credit period by using decreasing demand functions of
the retail price. In their model, the supplier and the buyer
should jointly determine the retail price and order size.
Petersen and Rajan (1997) conduct an empirical study on
the importance of trade credit for 3404 small firms in the
USA from the dataset of the National Survey of Small
Business Finances (NSSBF). They find out that larger
firms borrow and extend more trade credit, which serving
as a strategic tool to increase business scale. NG et al.
(1999) empirically investigate determinants of trade credit
from a survey of credit managers from 2538 firms, drawn
from COMPUSTAT files. They classify trade credit into
two categories: net term (net 30) and twopart term (10/2
net 30). Factors impacting the choice of different term are
discussed. Fatoki and Odeyemi (2010) conduct a
questionnaire survey and conclude that 331 SMEs
(small or mediumsized enterprise) out of total 417
respondents in South Africa apply for trade credits from
their suppliers (creditors), but only 71 respondents are
approved. The result reveals that most SMEs consider
trade credit as a better financing channel than ordinary
bank loan. However, the suppliers choose the tradecredit
receivers in a strict and secure way.
Once a profitmaximizing firm receives trade credits, it
should actively invest the delayed dollars in instruments
with higher and guaranteed returns to increase the
corporate value even only for a short period. In view of
this, we intend to formulate a stochastic model to valuate
such a firm facing price and interest rate risks. The
approaches of corporate valuation fall into two
categories: marketbased and DCFbased (Discounted
Cash Flow) models. The marketbased model dates back
to the work of Merton (1974), who proposes the
stochastic asset value in a call option, whose exercise
price is the contingent claim on debt. Based on his option
framework, many related models have been developed
by Black and Cox (1976), and Landes and Loistl (1992),
etc. On the other hand, the frameworks employing the
DCF to calculate the corporate value can be found in the
works by Rappaport (1986), and Copeland et al. (1996).
The main theme of these works lies in the discounted
present value of all future cash inflows. Casey (2001)
computes the discounted value from a series of uncertain
future payments until the date of insolvency and then
obtains the present value distribution.
In this paper, we propose a model for corporate
valuation from the perspective of a buyer with dayterm
trade credits under the stochastic interest rates, based on
the short rate of the Merton model (1973). According to
the time series data acquired from the Taiwan Economic
Journal (TEJ) in the period of 2001 to 2010, we found
that the monthly price for the LCD component evolves
stochastically as seen in Figure 1. The interest rate of 3
month Tbill also evolves stochastically as shown in
4918 Afr. J. Bus. Manage.
Figure 2. The monthly interest rate process for 3month Tbill in a 10year period.
Figure 2. The quantity demanded demonstrates a
negative correlation with the price. Therefore, the
inversed demand function is adopted in our model. The
corporate value can then be derived by discounting all
the future profits, which also evolve as a stochastic
process due to the randomness in the interest rate and in
the price. The sensitivity of corporate value was analyzed
to provide readers with more insights into the interactions
among the following factors: the drift and the volatility for
the stochastic interest rate and for the price, the price
elasticity of demand, the market share, the cost rate, the
length of each period, and the number of delay periods.
METHODOLOGY
Assumption and notation
In this model, we calculate the expected discounted value of future
cash flows for a buyer. In the context of trade credits offered by the
supplier, the buyer’s cash outflows can be delayed for
k
periods
with the length of each period denoted by
δ
. The buyer has an
opportunity to invest the deferred amounts payable in financial
instruments with safety and guaranteed returns until the maturity
date of credit period, e.g. bank savings in LIBOR market, Floating
Rate Note, etc. We consider the rate of return as a stochastic
process and adopt the short rate structure proposed by Merton
(1973). The short rate in his framework evolves stochastically on
the continuous time basis. The product price also evolves
stochastically. Because of the stochastic effects, the uncertainties
are propagated into the future cash flows and then into the
discounted present value, whose expected value represents the
corporate value in our model. According to the Merton model of
short rate, we denote the instantaneous interest rate at time t as
t
r
,
which stochastically evolves as
t r r t
dr dt dW
α σ= +
(1)
or equivalently,
0 0 0
( ), 0
t r r t
r r t W W W
α σ
= + + − =
(2)
where
r
α
1
is a constant drift,
r
σ
is a constant volatility, and
t
W
is a standard Brownian motion with a filtration
{
}
( );0
t t
≥
F,
t
dW
is the normally distributed increment with the expected value
0 and the variance 0
= − =
dt t t
.
The product price also evolves stochastically as:
t
p p t
t
d P
d t d Z
P
α σ= +
(3)
where
p
α
is a constant drift,
p
σ
is a constant volatility, and
t
Z
is
1
Tuckman (2002) divides
α
into two components: the expected growth and
the risk premium.
a standard Brownian motion with a filtration
{
}
( );0
t t
≥
F,
t
dZ
is
the normally distributed increment with the expected value 0 and
the variance 0
= − =
dt t t
. It is assumed that
t
dW
and
t
dZ
are
independent.
In order to calculate the time value of money, we denote a
continuous compounding money market account as
,
t k
β
, and
define it as follows:
,
t k
s
t
r d s
t k
e
δ
β
+
=
. (4)
where
δ
is the constant length of each period between two
consecutive transactions.
We assume that the market demands are price sensitive and can
be defined by an inversed demand function of Carruth et al. (2001)
as follows:
1
−
=
t t
P Q
ε
. (5)
The buyer’s marginal cost is assumed proportional to its revenue
according to the assumption made by Schwartz and Moon (2001),
who divide the cost structure into two components: a fixed term and
a variable term, which is proportional to the revenue. We also
assume that the buyer could control its cost rate at a fixed level
relative to the revenues through its managerial activities such as its
bargaining power to negotiate with the supplier. In the context of
trade credits offered by the supplier, the payments for marginal
costs incurred at time
t
can be delayed for
k
periods. We denote
the costs incurred at time
t
, but actually paid at time
t k
δ
+
as
,
t k
C
, which is defined as follows:
,
t k t
C P
γ
= ⋅
, (6)
where
γ
is the constant cost rate,
0,1,2,
k
=
.
Framework
It is assumed that the market share of the firm in question is
constant and denoted by
s
. The firm’s revenue at time t, denoted
by
t
R
, is therefore as follows:
1
−
= ⋅ ⋅ =
t t t t
R P s Q sP
ε
. (7)
Because the firm can defer the payments of accounts payable
for
k
periods and invest the delayed amounts in interestbearing
financial instruments, the buyer’s cash profits at time t, denoted
by
t
π
, can be expressed as:
,,
1
,
( )
t t k t k k t k t k k
t k t k k
R s Q C
sP
δ δ δ δ
ε
δ δ
π β
β γ
− − − −
−
− −
= ⋅ − ⋅ ⋅
= −
(8)
Chen et al. 4919
We assume that the transactions occur at the time when it is the
multiple of the fixed time length
δ
. That is, the buyer periodically
purchases merchandises from the supplier at time
1
δ
,
2
δ
,
and
3
δ
, etc, while the supplier allows the buyer to postpone its
amounts payable for
k
periods. The buyer can simultaneously sell
and deliver the goods it received to the final customers on the cash
basis. Therefore, no inventoryrelated costs are considered in this
model. We denote
t
V
as the corporate value at time
t
, and define it
over an infinite time horizon as follows:
n
n
T
t t t T
n k
V e
ρ
π
∞
−
+
=
=
E
(9)
where
, 0,1,2,.
= =
n
T n nδ and
{
}
t
⋅
E is the conditional
expectation given the information set at time
t
. The discount rate is
assumed exogenous and denoted by
ρ
.
Substitute Equations (2), (4), and (8) into (9), we have (proved in
Appendix A):
1
1 2
( ) ( )
t t
V s P I I
ε
γ
−
= ⋅ − ⋅
where
2
(1 )( )
2
1
=
t T
n p
s n p n k
t T
n k
r ds T T
t
n k
I e
σ
ρ ε α
+
−
+
−
∞
− + − −
=
E
, (10)
and
2
(1 )( )
2
2
=
p
n p n k
T T
t
n k
I e
σ
ρ ε α
−
∞
− + − −
=
E
. (11)
Equations (10) and (11) can be simplified as follows (proved in
Appendix B): .
2 2 3
2
2 2 2 3
1
( ) ( ) ( )
2 6
1
1
(1 )( ) ( )
2 2
1
r
t r
p
r p r
r k k k
k k
e
I
e
α
ρ δ δ σ δ
σ
ρδ α δ ε α δ σ δ
− + +
− + + − − +
=
−
(12)
22
((1 )( ) )
2
1
p
p
k
e
I
e
ρδ
σ
ε α ρ δ
−
− − −
=
−
(13)
The following constraint should also be satisfied due to the
convergence condition:
2
2 2 2
1
( (1 )( ) )
2 2
p
r p r
k k
σ
ρ α δ ε α σ δ
> + − − +.
4920 Afr. J. Bus. Manage.
Figure 3. The sensitivity of corporate value to drift and volatility of interest rate under parameters.
p p
t t
0.01,0.001,s 0.2,1.2,1/52,
k 4,0.2,r 0.065,P 0.008,0.12
α = σ = = ε = δ =
= γ = = = ρ =
By combining Equations (12) and (13), we can finally derive the analytical solution for corporate value as follows:
2 2 3
2 2
2 2 2 3
1
( ) ( ) ( )
2 6
1
1
(1 )( ) ( ) ((1 )( ) )
2 2 2
( ) ( )
1 1
− + +
−
−
− + + − − + − − −
= ⋅ − ⋅
− −
r
t r
p p
r p r p
r k k k
k
t t
k k
e e
V s P
e e
α
ρ δ δ σ δ
ρδ
ε
σ σ
ρδ α δ ε α δ σ δ ε α ρ δ
γ
(14)
where parameters should satisfy
2
2 2 2
1
( (1 )( ) )
2 2
p
r p r
k k
σ
ρ α δ ε α σ δ
> + − − +
.
According to equation (14), corporate value at time t (
t
V
) is a
function of stochastic price (
t
P
) and interest rate (
t
r
) at time t. In
our formulation, an opportunity of investment on a firm with trade
credit under uncertainty is taken into account. A real option model
for such an investment opportunity is adopted. Subsequently, we
can simulate the process of corporate value and then obtain the
numerical solution for the real option value as follows:
(
)
(
)
0 0 0
(;,)
−
= −
t
t t
t
F V P r Max E V I e
ρ
, (15)
where
F
denote the real option value, defined by Pindyck (1991),
and
I
denotes the constant sunk cost for the investment
opportunity.
Analytical solution
The sensitivity of corporate value based on equation (14) is
analyzed and illustrated in Figures 3 through 8. It can be seen in
Figure 3 that the corporate value increases with increasing interest
rate drift and volatility. However, in Figure 4 the price drift has a
negative impact on corporate value, while the price volatility
demonstrates a positive impact on corporate value, even though
such an impact is not so significant.
Figure 5 illustrates the relationship among the cost rate, the
demand elasticity, and the corporate value, while Figure 6
demonstrates the relationship among the cost rate, the market
share and the corporate value. They confirm the business practice
that higher cost reduces the cash inflows generated from
operational profits and hence lowers corporate value. Nonetheless,
the cost rate affects corporate value insignificantly when demand
elasticity is as low as 1.1 and the market share approaches 0.1.
The impact of the cost rate becomes more significant with the
increase in price elasticity and market share. On the other hand, the
impacts of the demand elasticity and the market share on corporate
value become more significant with the decreases in the cost rate.
However, corporate value is a convex function of price elasticity in
Figure 5, whereas it is a concave function of the market share in
Chen et al. 4921
Figure 4. The sensitivity of corporate value to drift and volatility of price under parameters:
r r
t t
0.01,0.001,s 0.2,1.2,1/52,
k 4,0.2,r 0.065,P 0.008,0.12
α = σ = = ε = δ =
= γ = = = ρ =
Figure 5. The sensitivity of corporate value to cost rate and to demand elasticity under parameters:
r r p p
t t
0.01,0.001,0.01,0.001,s 0.2,
1/52,k 4,r 0.065,P 0.008,0.12
α = σ = α = σ = =
δ = = = = ρ =
Figure 6. It can be inferred that small change in price causes large
change in quantities when the price elasticity is larger. As a result,
more revenue and more corporate value are anticipated in Figure 5.
In Figure 7, the firm obtains more value when it can defer its
payments for the purchased inventory as late as possible due to the
trade credits it receives form the supplier. It can be shown that
corporate value increases with longer time to maturity of payment,
indicated by the increasing number of delay periods (
k
) in our
framework. However, the length of each time period (
δ
)
represents the time interval between two consecutive transactions.
The shorter length of each period implies that transactions occur
4922 Afr. J. Bus. Manage.
Figure 6. The sensitivity of corporate value to cost rate and to market share under parameters:
r r p p
t t
0.01,0.001,0.01,0.001,1.2,
1/52,k 4,r 0.065,P 0.008,0.12
α = σ = α = σ = ε =
δ = = = = ρ =
Figure 7. The sensitivity of corporate value to time length and to number of delay periods under
r r p p
t t
0.01,0.001,0.01,0.001,0.2,
1.2,S 0.2,r 0.065,P 0.008,0.12
α = σ = α = σ = γ =
ε = = = = ρ =
more frequently and hence more cash flows are generated. As a
result, a corporation with higher transaction frequency, indicated by
shorter length of each period in our framework, results in higher
corporate value. That accounts for the reason why corporate value
is negatively correlated with the length of each period. That is, the
shorter the length of each period, the higher the corporate value.
Chen et al. 4923
Figure 8. The sensitivity of corporate value to initial interest rate and price under parameters:
r r p p
0.01,0.001,0.01,0.001,0.2,
1.2,S 0.2,k 4,1/52,0.12
α = σ = α = σ = γ =
ε = = = δ = ρ =
Because corporate value is a convex function of the time period
length, it becomes smoother when the length of each period is
larger. In Figure 8, the initial price demonstrates more significant
impacts on corporate value than the initial interest rate. It can be
inferred that the lower price induces more market demand, and thus
generates more revenues and more corporate value. As a result,
the impact of the price on corporate value is negative. Contrarily,
the impact of the initial interest rate is slightly positive as expected.
RESULTS
We considered the 3month Tbill rate as a proxy for the
short rate of the Merton model and approximated the
random fluctuations of the short rate ranging from 0.00 to
0.0625 in the period of 2000 to 2010 as shown in Figure
2. We therefore adopted 0.065 as initial interest rate for
the simulations of the interest rate process. Moreover,
Chan et al. (1992) estimate the drift and volatility for the
interest rate process to be 0.005 and 0.02 respectively.
We therefore chose 0.005 and 0.015 to be benchmark
parameters for interest rate drift and volatility respectively.
As seen in Figure 1, the parameters of drift and volatility
for LCD price were estimated to be 0.039 and 0.350,
respectively. Based on such parameter estimations, we
set the benchmark parameters for the stochastic price
process in our simulations as follows: ±0.01, ±0.05, ±0.09
for the price drift and price volatility. In order to investigate
the impacts of different parameters on real option value,
we simulated by using different values of those
parameters, including the number of delay periods, the
cost rate, the market share, and the price elasticity.
Simulation results
Similar to the stochastic dominance methodology in the
study of new product development by Kleczyk (2008)
who adopts Monte Carlo simulation, the stochastic
programming by simulation approach was adopted by
this work to obtain numerical solution for equation (15) in
a 10year period with 10,000 paths for each replication.
For each sample path, we generated two independent
and random standard normal values for
t
dW
and
t
dZ
for
equations (1) and (3). Based on those two random
values, the price and the interest rate processes were
then simulated for the computations of the corporate
value and those of the real option value in equations (14)
and (15). The simulated sample paths for the product
price, the interest rate, and the corporate value, as well
as the discounted investment value are illustrated in
Figures 9 through 12. The results calculated from the
10,000 paths are presented in Tables 1 through 4. The
numbers in each cell are the mean and the standard
deviation (in parentheses) from the 10,000 sample paths
under different parameter combinations. It can be seen in
4924 Afr. J. Bus. Manage.
Figure 9. Simulated sample paths for the stochastic interest rate process.
Figure 10 that the price drift is negative, causing the price
to have a downtrend. However, such a downtrend in price
may induce more demand and thus generates more
value for the firm as shown in Figure 11. However, the
discounted value for such an investment opportunity is
shown in Figure 12 as decreasing with time passage.
The numerical results can be found in the following
tables. In Table 1, the real option value increases with
increasing number of delay periods. Even though the
price drift is positive in Table 2, the real option value still
increases with increasing number of delay periods. It
implies that when the interest rate process has an
uptrend, the firm should seek for the trade credit
opportunity to obtain the higher option value. Besides, it
can be shown that the real option value increases with
the increasing price elasticity of demand whatever the
drifts for the interest rate and for the price are. Similar to
the effects of volatility in financial options, the impact of
volatility for both the interest rate and the price on real
option value are also positive. The cost rate is not
surprisingly a negative factor to the option value as seen
in the third column of those tables, while the market share
has a positive impact on the option value as we can see
in the last column. It can be seen that Tables 3 and 4
almost demonstrate the same results as Figures 1 and 2
do. However, two major differences can be found: the
effect of interest rate drift becomes positive and the effect
of interest rate volatility becomes insignificant.
Conclusion
This paper integrated the stochastic interest rate model
proposed by Merton (1973) and a stochastic price
process into corporate valuation to investigate the
impacts of the drift and the volatility for the interest rate
and for the price, the cost rate, time to maturity of delay
payments, the price elasticity of demand, and the market
Chen et al. 4925
Figure10. Simulated sample paths for the stochastic price process.
Figure 11. Simulated sample paths for the derived stochastic corporate value.
4926 Afr. J. Bus. Manage.
Figure 12. Simulated sample paths for the discounted values of (VtI).
flows were then discounted and summed up as expected
corporate value. An analytical solution for corporate value
was obtained and further sensitivity analyses were
performed. Based on the analytical results, we concluded
that corporate value increases with the increasing interest
rate drift and volatility, but decreases with the increasing
price drift. Besides, the price elasticity of demand has a
positive impact on corporate value. Not surprisingly, the
cost rate, a percentage of the selling price, for purchased
inventory has a negative impact on corporate value.
However, corporate value is less sensitive to the cost rate
when price elasticity is smaller. Similarly, the market
share has less significant impacts on corporate value
when the cost rate approaches 1.0. It suggests that the
profitmaximizing firm in our setting should give first
priority to entering a market with higher price elasticity
and reinvesting its delayed payments in a financial
instrument with higher growth rate. Higher volatility of the
interest rate is taken into account as second priority while
the profit margins for the goods sold are lastly considered
if the price elasticity and the market share are smaller.
Based on the analytical solution for corporate value,
real option values were obtained through further
simulations. The numerical results, presented in Tables 1
through 4, demonstrate that the drift of the interest rate is
quite critical in the derivations of the real option value.
The real option value increases with decreasing price drift
and increasing interest rate drift. Regarding the volatility
for both the interest rate and the price, their impact on
real option value is consistently positive. The price
elasticity of demand and the market share have positive
impacts on real option value, while the cost rate has
expectedly negative impacts.
The framework discussed in this paper has its
implications: the reinvestment opportunities due to the
delay payments of trade credit offered by the supplier
allow the firm to obtain higher corporate value, once it
correctly chooses a niche market to enter, such as the
product market with higher demand elasticity and higher
market share, as well as the financial market with positive
growth and higher volatility for returns. The analytical
solution obtained in this paper could be also employed as
the basis for the strategic investment decisions, such as
market entry and exit policies in further studies. Other risk
sources such as foreign exchange rate in the financial
market could be also taken into account in the extension
price. The interest rate was applied to the cash
reinvestments due to the delay payments. An inversed
demand function proposed by Carruth et al. was adopted
in our formulation.
To construct the value dynamics for the firm in ques
tion, we derived a stochastic profit process. The profit
share on corporate value and on the real option value in a
context where a firm receives trade credits from its
supplier. In our framework, the uncertainty of the profits
comes from the stochastic interest rate and the stochastic
of this model to explore the effects of other
uncertainties.
Chen et al. 4927
Table 1. Sensitivity of real option value obtained from simulated 10,000 sample paths under parameters: = 1 week, = 0.12, P0= 0.008, r0= 0.065,r = 0.005,r r= 0.005.
Option value (F)
Delay periods () Price elasticity () Cost rate () Market share (S)
4 8 12 1.2 1.6 2.0 0.2 0.5 0.8 0.2 0.5 0.8
Price drift (p) 0.09
121.9
(0.004)
129.3
(0.020)
136.7
(0.05)
121.9
(0.004)
2290.1
(0.470)
37541.2
(65.35)
759.4
(0.010)
440.6
(0.010)
121.9
(0.004)
9.8
(0.170)
38.7
(0.030)
121.9
(0.004)
0.05
105.5
(0.002)
112.1
(0.020)
118.6
(0.04)
105.5
(0.002)
1639.6
(0.170)
154368.4
(9.03)
696.7
(0.010)
401.1
(0.010)
105.5
(0.002)
12.7
(0.150)
28.4
(0.030)
105.5
(0.002)
0.01
91.4
(0.001)
97.3
(0.010)
103.2
(0.03)
91.4
(0.001)
1267.5
(0.070)
9688.1
(2.56)
642.5
(0.010)
366.9
(0.010)
91.4
(0.001)
15.1
(0.130)
19.6
(0.020)
91.4
(0.001)
Price volatility
(p)
0.01
91.4
(0.001)
97.3
(0.010)
103.2
(0.03)
91.4
(0.001)
1267.5
(0.070)
9688.1
(2.56)
642.5
(0.010)
366.9
(0.010)
91.4
(0.001)
15.1
(0.130)
19.6
(0.020)
91.4
(0.001)
0.05
91.9
(0.150)
97.8
(0.150)
103.8
(0.20)
91.9
(0.150)
1285.1
(5.76)
9992.7
(115.2)
644.4
(0.200)
368.1
(0.150)
91.9
(0.150)
15.1
(0.490)
20.2
(0.340)
91.9
(0.150)
0.09
93.3
(0.560)
99.3
(0.580)
105.3
(0.62)
93.3
(0.560)
1331.6
(20.27)
10831.7
(447.8)
649.0
(0.970)
371.1
(0.700)
93.3
(0.560)
14.9
(0.880)
21.5
(0.970)
93.3
(0.560)
Table 2. Sensitivity of real option value obtained from simulated 10,000 sample paths under parameters: = 1 week, = 0.12, P0= 0.008, r0= 0.065,r= 0.005,r= 0.005.
Option value (F)
Delay periods () Price elasticity () Cost rate () Market share (S)
4 8 12 1.2 1.6 2.0 0.2 0.5 0.8 0.2 0.5 0.8
Price drift
(p)
+0.01
85.0
(0.010)
90.6
(0.010)
96.2
(0.030)
85.0
(0.010)
1135.5
(0.040)
8160.8
(1.300)
618.1
(0.020)
351.5
(0.010)
85.0
(0.010)
16.2
(0.120)
15.6
(0.020)
85.0
(0.010)
+0.05
73.5
(0.010)
78.6
(0.010)
83.7
(0.020)
73.5
(0.010)
935.5
(0.010)
6196.8
(0.480)
573.8
(0.010)
323.6
(0.010)
73.5
(0.010)
18.1
(0.100)
8.4
(0.020)
73.5
(0.010)
+0.09
63.3
(0.010)
68.0
(0.010)
72.6
(0.020)
63.3
(0.010)
768.4
(0.080)
4769.5
(0.580)
534.6
(0.010)
299.5
(0.010)
63.3
(0.010)
19.7
(0.090)
2.5
(0.010)
63.3
(0.010)
Price volatility
(p)
0.01
85.0
(0.010)
90.6
(0.010)
96.2
(0.030)
85.0
(0.010)
1135.5
(0.040)
8160.8
(1.300)
618.1
(0.020)
351.5
(0.010)
85.0
(0.010)
16.2
(0.120)
15.6
(0.020)
85.0
(0.010)
0.05
85.5
(0.140)
91.1
(0.160)
96.8
(0.190)
85.5
(0.140)
1149.6
(4.530)
8373.3
(82.90)
619.8
(0.180)
352.6
(0.140)
85.5
(0.140)
16.2
(0.450)
16.2
(0.320)
85.5
(0.140)
0.09
86.8
(0.500)
92.5
(0.560)
98.2
(0.580)
86.8
(0.500)
1186.5
(16.30)
8948.7
(301.6)
624.2
(0.910)
355.4
(0.650)
86.8
(0.500)
15.9
(0.821)
17.4
(0.910)
86.8
(0.500)
4928 Afr. J. Bus. Manage.
Table 3. Sensitivity of real option value obtained from simulated 10,000 sample paths under parameters: = 1 week= 0.12, P0= 0.008, r0= 0.065, p =0.01, p= 0.01.
Option value (F)
Delay periods () Price elasticity () Cost rate () Market share (S)
4 8 12 1.2 1.6 2.0 0.2 0.5 0.8 0.2 0.5 0.8
Interest rate drift (r)
+0.005
91.4
(0.01)
97.3
(0.01)
103.2
(0.04)
91.4
(0.01)
1267.6
(0.05)
9688.1
(2.39)
642.5
(0.01)
366.9
(0.01)
91.4
(0.01)
15.2
(0.13)
19.6
(0.02)
91.4
(0.01)
+0.010
94.4
(0.01)
103.4
(0.01)
112.5
(0.03)
94.4
(0.01)
1290.1
(0.06)
9855.5
(2.52)
645.6
(0.02)
370.0
(0.01)
94.4
(0.01)
14.7
(0.13)
21.5
(0.02)
94.4
(0.01)
+0.015
97.5
(0.01)
109.7
(0.01)
122.0
(0.04)
97.5
(0.01)
1312.8
(0.07)
10024.0
(2.82)
648.2
(0.01)
373.1
(0.01)
97.5
(0.01)
14.2
(0.14)
23.4
(0.02)
97.5
(0.01)
Interest rate volatility
(r)
0.005
91.4
(0.01)
97.3
(0.01)
103.2
(0.04)
91.4
(0.01)
1267.6
(0.05)
9688.1
(2.39)
642.5
(0.01)
366.9
(0.01)
91.4
(0.01)
15.2
(0.13)
19.6
(0.02)
91.4
(0.01)
0.010
91.4
(0.01)
97.3
(0.07)
103.2
(0.19)
91.4
(0.01)
1267.6
(0.11)
9688.3
(2.91)
642.5
(0.01)
366.9
(0.01)
91.4
(0.01)
15.2
(0.20)
19.6
(0.06)
91.4
(0.01)
0.015
91.4
(0.04)
97.3
(0.19)
103.2
(0.43)
91.4
(0.04)
1267.6
(0.19)
9688.7
(3.42)
642.5
(0.01)
366.9
(0.01)
91.4
(0.04)
15.2
(0.28)
19.7
(0.12)
91.4
(0.04)
Table 4. Sensitivity of real option value obtained from simulated 10,000 sample paths under parameters:= 1 week= 0.12, P0
= 0.008, r
0= 0.065,p = 0.01,p = 0.01.
Option Value (F)
Delay Periods () Price Elasticity () Cost Rate () Market Share (S)
4 8 12 1.2 1.6 2.0 0.2 0.5 0.8 0.2 0.5 0.8
Interest rate
drift (r)
+0.005
85.0
(0.01)
90.6
(0.01)
96.2
(0.03)
85.0
(0.01)
1135.5
(0.02)
8160.8
(1.30)
618.1
(0.01)
351.5
(0.01)
85.0
(0.01)
16.2
(0.12)
15.6
(0.02)
85.0
(0.01)
+0.010
87.8
(0.01)
96.4
(0.01)
104.9
(0.03)
87.8
(0.01)
1153.9
(0.04)
8280.5
(1.50)
620.6
(0.02)
354.4
(0.01)
87.8
(0.01)
15.8
(0.12)
17.4
(0.02)
87.8
(0.01)
+0.015
90.7
(0.01)
102.2
(0.01)
113.8
(0.03)
90.7
(0.01)
1172.5
(0.04)
8400.9
(1.52)
623.8
(0.01)
357.3
(0.01)
90.7
(0.01)
15.3
(0.12)
19.2
(0.02)
90.7
(0.01)
Interest rate
volatility (r)
0.005
85.0
(0.01)
90.6
(0.01)
96.2
(0.03)
85.0
(0.01)
1135.5
(0.02)
8160.8
(1.30)
618.1
(0.01)
351.5
(0.01)
85.0
(0.01)
16.2
(0.12)
15.6
(0.02)
85.0
(0.01)
0.010
85.0
(0.01)
90.6
(0.06)
96.3
(0.07)
85.0
(0.01)
1135.5
(0.06)
8160.9
(1.76)
618.1
(0.01)
351.5
(0.01)
85.0
(0.01)
16.2
(0.18)
15.7
(0.06)
85.0
(0.01)
0.015
85.0
(0.04)
90.7
(0.18)
96.5
(0.40)
85.0
(0.04)
1135.5
(0.13)
8161.2
(2.16)
618.1
(0.01)
351.5
(0.01)
85.0
(0.04)
16.2
(0.26)
15.7
(0.11)
85.0
(0.04)
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4930 Afr. J. Bus. Manage.
APPENDIX
Appendix A
Substitute Equations (2), (4), and (8) into (9), we have:
2
1
,
1 1
(1 )( )
1 1
2
( )
( )
+
+
−
−
+
−
+
−
∞
−−
+ − + −
=
∞ ∞
−
−− −
+ + −
= =
− + − −
− +
− −
= −
−
−
n
n n
t T
n
s n
t T
n k n
n k n
t T
n p
s n p n k
n
t T
n k
T
t t t T k t T k k
n k
r ds T
T
t t T t t T k
n k n k
r ds T T
T
t t t t
V sP e
sP e sP e
s P e sP e
ρε
δ δ
ρ
ρε ε
δ
σ
ρ ε α
ρ
ε ε
β γ
γ
γ
E
E E
E E
2
2 2
(1 )( )
2
(1 )( )
(1 )( )
1 1
2
2
1 1
1 2
1
1 2
( )
( ) ( )
( ) ( )
−
+
−
−
+
−
∞ ∞
− −
= =
∞ ∞
− + − −
− + − −
− −
= =
− −
−
−
= ⋅ − ⋅
= ⋅ − ⋅
s
p
p n k
t T
n
p p
s n p n k
n p n k
t T
n k
T
n k n k
r ds T T
T T
t t t t
n k n k
t t
t
P e sP e
s P I s P I
s P I I
σ
ε α
σ σ
ρ ε α
ρ ε α
ε ε
ε ε
ε
γ
γ
γ
E E
where
2
(1 )( )
2
1
=
t T
n p
s n p n k
t T
n k
r ds T T
t
n k
I e
σ
ρ ε α
+
−
+
−
∞
− + − −
=
E,
and
2
(1 )( )
2
2
=
p
n p n k
T T
t
n k
I e
σ
ρ ε α
−
∞
− + − −
=
E.
Appendix B
To simplify Equations (10) and (11), we firstly calculate the expected exponential function with an Ito integral in the
power. According to Stochastic Fubini Theorem, we can rewrite
t T s
n
r u
t T t T
n k n k
dW ds
t
e
σ
+
+ +
− −
E as
t T t T
n n
r u
t T u
n k
dsdW
t
e
σ
+ +
+
−
E. Then
integrating the inner part in the exponential function gives the result as follows:
( )
t T
n
r n u
t T
n k
t T u dW
t
e
σ
+
+
−
+ −
E
. (B1)
We define an Ito integral as
0
( ) ( )= Δ
t
u
I t u dW
where ( )
Δ = + −
n
u t T u
is a nonrandom function. Because
( )
Δ
u
is
( )
u
F measurable for each time index
0
≥
u and it satisfies
2
0
( )
Δ < ∞
t
E u du, Equation (12) can be rewritten in the
form of an Ito integral as follows:
Chen et al. 4931
( )
( ) ( )
r n n k
I t T I t T
t
e
σ
−
+ − +
E
.
It can be shown that for each
0
≥
t,
( )
I t
is a normally distributed random variable with the expected value 0 and the
variance
2
0
( )
Δ
t
u du
.Subsequently, we calculate the variance for the Ito integral as follows.
( ) ( )
( )
2
2
2
( ( ) ( ))
[ ( ) ( ) ] [ ( ) ( ) ]
[ ( ) ( ) ]
−
− −
−
+ − +
= + − + − + − +
= + − +
n n k
t n n k t n n k
t n n k
Var I t T I t T
I t T I t T I t T I t T
I t T I t T
E E
E
By applying the ItoIsometry theorem, we have
(
)
2
2
2
[ ( ) ( ) ]
[ ( ) ]
( )
n
n k
n
n k
t n n k
t T
t
t T
t T
n
t T
I t T I t T
u du
t T u du
−
−
−
+
+
+
+
+ − +
= Δ
= + −
E
E
2 2
3
2 2
3 3
[( ) 2( ) ]
[( ) ( ) ]
3
.
3
n
n k
nn k
t T
n n
t T
t T
n n t T
t T t T u u du
u
t T u t T u
k δ
−
−
+
+
++
= + − + +
= + − + +
=
Therefore,
( ( ) ( ))
−
+ − +
n n k
I t T I t T is a normal distribution with the expected value 0 and the variance
3 3
3
k
δ
. Similarly,
( ( ) ( ))
r n n k
I t T I t Tσ
−
+ − + is also a normal distribution with the expected value 0 and the variance
2 3 3
3
r
k
σ δ
when
r
σ
is a constant volatility.
The following lemma (Shreve, 2000) is employed in our derivation for the corporate value.
Lemma 1.
Let the random variable
ξ
has the Gaussian law
2
(,)
Normal
µσ
. Then the random variable
=
e
ξ
ς has the expected
value
2
1
2
+
e
µ σ
and the variance
2 2
2
( 1)
+
−
e e
µ σ σ
.
Based on Lemma 1, Equation (B1) can be simplified as follows:
( )
2 3 3
2 3 3
1 1
2 3 6
0( ) ( )
t T s
n
k
r
r u
t T t T r
r n n kn k n k
dW ds
kI t T I t T
t t
e e e e
σ δσ
σ δ
σ
+
+ +
−− −
+ ⋅+ − +
= = =
E E. (B2)
By substituting Equation (B2) into Equation (10), we can derive the analytical solution for
1
I
as follows:
4932 Afr. J. Bus. Manage.
( )
2
2
2
(1 )( )
2
1
( ( )) ( (1 )( )
2
(1 )( ) ( ( ))
2
t T
n p
s n p n k
t T
n k
t T
n
p
t r n k r s t T n p n k
T n k
n k
t T
n k
p
n p n k t T r n k
n k
r ds T T
t
n k
r a s t T W W ds T T
t
n k
T T r ds a s t T d
t
I e
ee
σ
ρ ε α
σ
σ ρ ε α
σ
ρ ε α
+
−
+
−
+
+ − + −
−
−
+
−
− + −
−
∞
− + − −
=
∞
+ − + + − − + − −
=
− + − − + + − +
=
=
=
E =
EE
( )
t T t T t T
n n n
r s t T
n k
t T t T t T
n k n k n k
s W W ds
n k
σ
+ + +
+
−
+ + +
− − −
∞
+ −
=
2
2 2
2
2 2
(1 )( ) ( ) ( )
2 2
( (1 )( )( )
2 2
t T
n
p
r
n p n k t T n n k r s t T
n k n k
t T
n k
t T s
n
p
r
r u
p t T
n k
t T t T
n k n k
T T r T T k W W ds
t
n k
dW ds
n n k r k k
t t
n k
e
e e
σ
α
ρ ε α δ σ
σ
α
σ
ρ δ ε α δ δ δ
+
− + − +
− −
+
−
+
+
−
+ +
− −
∞
− + − − + − + + −
=
∞
− + − − − + +
=
=
= ⋅
E
E E
2
2 2 2 3
1
( (1 )( )( ) ( )
2 2 6
p
r
p r
t T
n k
n n k k k
r k
t
n k
e e
σ
α
ρ δ ε α δ δ σ δ
δ
+
−
∞
− + − − − + +
=
= ⋅
E. (B3)
According to the definition of the stochastic short rate process in Equation (1), we have
( )
( ).
t T
n k
t r n k r t T
n k
t r n k r t T
n k
r k
r T dW k
r T k k dW
δ
α σ δ
α δ σ δ
+
−
− +
−
− +
−
= + +
= + ⋅ + ⋅
Because
+
−
t T
n k
dW is the increment for the standard Brownian motion and is normally distributed, we can obtain another
normal distribution with the mean and the variance as follows:
(
)
( )
2 2 2 2 2 2
( ) ( ( ) )
( )
t T t r n k t r
n k
t T r n k r
n k
r k r T k r n k k
t
r k k T k n k
t
E
Var
δ α δ α δ δ
δ δ σ δ σ δ
+ −
−
+ −
−
+ ⋅ + − ⋅
−
= =
= =
Again, according to Lemma 1, we have
2 2 2
1
( ( ) ) ( )
2
( ).
t r r
t T
n k
r n k k k n k
r k
t
E e e
α δ δ δ σ δ
δ
+
−
+ − ⋅ + −
=
Equation (B3) then becomes
2
2 2 3 2 2 2 3
1 1
( ( ) ) ( ) ( (1 )( )( ) ( )
2 2 2 6
p
r
t r r p r
r n k k n k k n n k k k
n k
e
σ
α
α δ δ σ δ ρ δ ε α δ δ σ δ
∞
+ − + − + − + − − − + +
=
. (B4)
Simplify Equation (B4), we can derive the analytical solution for
1
I
as follows.
Chen et al. 4933
2
2 2 3 2 2 2 3
2
2 2 3 2 2 2 3
1 1
( ( ) ) ( ) ( (1 )( )( ) ( )
2 2 2 6
1
1 1
( (0) ) (0)( ) ( ( ) (1 )( )(0) ( )
2 2 2 6
1
( (1) ) (1)(
2
p
r
t r r p r
p
r
t r r p r
t r
r n k k n k k n n k k k
n k
r k k k k k
r k
I e
e
e
σ
α
α δ δ σ δ ρ δ ε α δ δ σ δ
σ
α
α δ δ σ δ ρ δ ε α δ δ σ δ
α δ δ
∞
+ − + − + − + − − − + +
=
+ + + − + − − + +
+ +
=
=
+
2
2 2 3 2 2 2 3
2
2 2 3 2 2 2 3
2 2 3
2
2
1
) ( ( 1) (1 )( )(1) ( )
2 2 6
1 1
( (2) ) (2)( ) ( ( 2) (1 )( )(2) ( )
2 2 2 6
1
( ) ( ) ( )
2 6
1
(1 )( )
2 2
1
p
r
r p r
p
r
t r r p r
r
t r
p
r p
k k k k
r k k k k k
r k k k
k
e
e
e
σ
α
σ δ ρ δ ε α δ δ σ δ
σ
α
α δ δ σ δ ρ δ ε α δ δ σ δ
α
ρ δ δ σ δ
σ
ρδ α δ ε α δ
+ − + + − − + +
+ + + − + + − − + +
− + +
− + + − − +
++
=
−
2 2 3
( )
r
kσ δ
In order to obtain the converged solution for the above infinite series, parameters in our model should satisfy the
following constraint:
2
2 2 2
1
( (1 )( ) )
2 2
p
r p r
k k
σ
ρ α δ ε α σ δ
> + − − +.
Similarly, simplifying equation (11) can solve for
2
I
as follows.
2
2 2 2 2
0 1 1 2 2 3 3
2
(1 )( )
2
2
(1 )( ) (1 )( ) (1 )( ) (1 )( )
2 2 2 2
( ) (1 )( )(0 ) ( 1) (1
2
...
p
n p n k
p p p p
k p k p k p k p
p
p
T T
t
n k
T T T T T T T T
t
k k
t
I e
e e e e
e e
σ
ρ ε α
σ σ σ σ
ρ ε α ρ ε α ρ ε α ρ ε α
σ
ρ δ ε α δ ρ δ
−
+ + +
∞
− + − −
=
− + − − − + − − − + − − − + − −
− ⋅ + − − − ⋅ + +
=
= + + + +
= +
E
E
E
2 2 2
2
)( )(1 ) ( 2) (1 )( )(2 ) ( 3) (1 )( )(3 )
2 2 2
((1 )( ) )
2
...
1
p p p
p p p
p
p
k k
k
e e
e
e
σ σ σ
ε α δ ρ δ ε α δ ρ δ ε α δ
ρδ
σ
ε α ρ δ
− − − ⋅ + + − − − ⋅ + + − −
−
− − −
+ + +
=
−
The following constraint should also be satisfied due to the convergence condition:
2
(1 )( )
2
p
p
σ
ε α ρ
− − <
.
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