Abstract This study attempts to apply real options and expand the model designed by Linand Huang [Lin, T.T., Huang, Y.T.: J. Technol. Manage. 8(3), 59–78 (2003)], which helpsventure capital (VC) companies to optimize project exit decisions. The expected discountedfactor and a jump-diffusion process combine to assess the value of a start-up company, anddetermine the threshold of the exit timing of liquidation or convertibility for establishing theoptimal disinvestment evaluation model for VC companies. When the project value is belowV ∗
L , the VC company carries out liquidation, but when the project value exceeds V ∗C , the
VC company performs convertibility. The project value is ranging between(V ∗
L , V ∗C
), and
the best choice is holding the decision and waiting to carry out the rights of liquidation andconvertibility next time. Besides, this work attempts to identify the expected discounted timein terms of the investment time for VC companies.
Keywords Venture capital · Jump-diffusion process · Discounted factor · Liquidation ·Convertibility
1 Introduction
According to the Taiwan Venture Capital Yearbook (Year 2003), the global stock marketentered a bear market after 2000 and the capital market deteriorated at the same time. Theglobal economy faced a recession during this period, and began to turn around after 2002.
T. T. Lin (B)Department of International Business, National Dong Hwa University,No. 1, Sec. 2, Da Hsueh Rd., Shou-Feng, Hualien, 974 Taiwane-mail: [email protected]
C.-C. KoInstitute of Business & Management, National Chao Tung University, Taipei, Taiwan
C.-W. ChangGraduate School of Management, Ming Chuan University, Taipei, Taiwan
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818 T. T. Lin et al.
However, the initial public offering (IPO) market in Taiwan remained strong, despite thedrop of almost 20% in the Taiwan stock index. 203 IPOs occurred during 2002, representinga 33.6% increase compared to 152 IPOs during 2001. This phenomenon contrasted withmajor global capital markets, including the New York Stock Exchange, the NASDAQ, andthe London Stock Exchange, which all showed a decrease in IPOs during this period.
Generally, Taiwanese Venture Capital (VC) companies are highly active in IPOs and playimportant roles in creating new entrepreneurs. One third of new IPOs involve VC companies,which means that Taiwanese VC companies have much more chances to succeed and exitchannels in the investments of start-up companies. According to the Taiwan Venture CapitalYearbook (Year 2003), 27.62% of VC was invested during the entrepreneur stage, 47.2% dur-ing the growth stage, and 20.33% during the mature stage. In addition to capital investments,VC companies will assist companies at the seed and early stages in business managementand development by leveraging their professional experiences, and help companies to launchIPOs as early as possible. Pandey and Angela (1996) also noted that Taiwanese investmentcompanies only put 10% of their capitals into VC Funds. According to Andreas and Uwe(2001), a couple of exit channels of start-up companies exist for VC companies, including (1)Liquidation, (2) Buy-Back: Invested companies buying back outstanding shares held by VCcompanies, (3) Secondary Purchase: Selling shares to institutional investors, (4) Trade Sale:Selling the shareholdings of the invested companies to other companies, and (5) IPO: themost valuable method among those exit channels of start-up companies. However, accordingto Mason and Harrison (2002), Trade Sale of start-up companies is the most widely usedchannel by business angels for realizing capital gains. According to Andres and Uwe (2001),a correlation exists between exit decision of start-up companies and contract design for VCcompanies. VC companies must give up the (exit) shareholdings of the invested enterprisesduring a certain time frame. Sometimes, VC companies and entrepreneurs have differentopinions regarding exit plans (e.g. IPO or Trade Sale). Outside investors agree to the optimalexit plan of issuing convertible warranties. Thus, VC companies have frequently used con-vertible warranties in their investments. During the expansion period, VC companies appearto exit the current investment projects to attract new investors and increase total profits.
By applying the real options approach (ROA), reconsidering the discounted factor andfollowing the jump-diffusion process in cash flows of the company evaluation model, thisassay extends the model of Lin and Huang (2003) to establish the exit strategy and modelfor VC companies in start-up investment projects. This assay is also used to determine theoptimal exit point (liquidation or conversion) and establish the evaluation model for the opti-mal exit plan of start-up companies. Furthermore, Lin et al. (2006) explored exit decision offinancial institutions in duopolistic loan market with game options approach and analyzedhow uncertainty influences loan decisions for the financial institution.
The remainder of this paper is organized as follows: Section 2 deals with model estab-lishment, including probing the optimal liquidation and conversion model based on the con-tinuous basic model and jump-diffusion process. Section 3 then conducts the relative valueanalysis and sensitivity analysis. Finally, Section 4 presents conclusions.
2 Proposed model
This assay assesses the optimal VC exit plan using ROA. By applying the new discountedfactor concept and considering Poisson’s jump-diffusion process, this study also discusseshow VC companies obtain the maximum investment value by deciding to liquidate or convertto preferred stocks upon expiration of the investment contract.
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Applying jump-diffusion processes to liquidate and convert venture capital 819
2.1 Continuous model
Assuming project value of VC investment in a start-up company is defined as V (t), thevalue variation growth over time is described by the Geometric Brownian Motion (GBM) asfollows:
dV (t) = αV (t)dt + σ V (t)dW (t). (1)
Among those factors, α: drift over time, σ : volatility over time, dW (t) is the increment ofstandard Winner process W (t) of zero mean and unit standard deviation
√dt .
The project value of VC investment in the start-up company FI (V ) after dt , the incrementof project value d FI (V ) is calculated according to Itô’s Lemma(Itô 1951),
d FI (V ) = FI V (V )dV + 1
2FI V V (V )dV 2. (2)
Assuming VC companies do not have any risk preference, following incremental time dt ,they will expect the project value to equal the risk-free interest r f multiplied by FI (V )dt ,which is
r f × FI (V )dt = E [d FI (V )] ={αV FI V (V ) + 1
2σ 2V 2 FI V V (V )
}dt. (3)
By slightly manipulating the above equation, we obtain:
1
2σ 2V 2 FI V V (V ) + αV FI V (V ) − r f × FI (V ) = 0. (4)
Equation 4 is a second order homogeneous differential function, and its general solutionform has the format FI (V ) = AV u . Applying the solution to the above equation to obtainthe following quadratic equation:
1
2σ 2u2 +
(α − 1
2σ 2
)u − r f = 0. (5)
The two roots of the above equation are:
u1 =( 1
2σ 2 − α) +
√( 12σ 2 − α
)2 + 2σ 2r f
σ 2 > 1; (6)
u2 =( 1
2σ 2 − α) −
√( 12σ 2 − α
)2 + 2σ 2r f
σ 2 < 0. (7)
The project value upon liquidation is:
FI 1(V ) = a1V u1 + b1V u2 . (8)
The liquidation value is zero when the project value is zero, VC companies do not need topay an additional liquidation fee, that is, FI 1(0) = 0. b1 in Eq. 8 must be zero. Thus, thecontract value upon liquidation is FI 1(V ) = a1V u1 .
With the same inference as the liquidation value, the contract value upon conversion isFI 2(V ) = a2V u1 + b2V u2 . Among those exit plans, if VC companies choose to liquidate,they will incur a liquidation cost L0; if VC companies choose to give up the right to liquidateand wait for the right of conversion, they will incur two costs, the conversion cost C0 and the
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820 T. T. Lin et al.
liquidation cost upon conversion L0 × (V ∗C/V ∗
L )β11 , where (V ∗L /V ∗
C )β11 is the present valuefactor. See Appendix 1 for more detailed inferences.
From the above inference, the contract value is conducted upon both liquidation FI (V ∗L )
and conversion FI (V ∗C ) to calculate the optimal exit threshold for VC companies to select
liquidation or conversion. Refer to Dixit and Pindyck (1994) for finding the threshold andproject value using value-matching and smooth-pasting conditions as follows:
According to the value matching condition (VMC), assuming the optimal liquidationthreshold is V ∗
L , the liquidation threshold upon liquidation V ∗L minus the liquidation cost L0
should equal the project value upon liquidation FI 1(V ∗L ). According to the smooth pasting
condition (SPC), the incremental project value should match the first order of the differentialfunction when the project value at the threshold point produces the same marginal profit forexit plan. The above two conditions are shown as follows:
{V MC : V ∗
L − L0 = FI 1(V ∗L );
S PC : d[V ∗L −L0]
dV ∗L
= d FI 1(V ∗L )
dV ∗L
.(9)
Based on Eq. 9 and after some manipulation, the optimal liquidation threshold V ∗L and the
project value coefficient a1 upon liquidation are conducted as follows:
V ∗L = u1
u1 − 1L0; (10)
a1 = 1
u1(V ∗
L )1−u1 . (11)
Assuming the optimal conversion threshold is V ∗C , the project value is evaluated by V ∗
C uponconversion minus the liquidation cost C0 and L0 × (V ∗
C/V ∗L )β11 should equal the project
value upon conversion FI 2(V ∗C ). The above result is the VMC.
Moreover, according to SPC, the incremental project value should match the first orderdifferential function. The two conditions are shown as below:
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
V MC : V ∗C −
[C0 + L0
(V ∗
CV ∗
L
)β11]
= FI 2(V ∗C );
S PC :d
[
V ∗C −
(
C0+L0
(V ∗
CV ∗
L
)β11)]
dV ∗C
= FI 2(V ∗C )
dV ∗C
.
(12)
Furthermore, upon conversion, the project value minus the liquidation cost and the presentvalue of the conversion cost is assumed to equal the liquidation contract value FI 2(V ∗
L ) =a2
(V ∗
L
)u1 + b2(V ∗
L
)u2 ; restated,
V ∗L −
[
L0 + C0
(V ∗
L
V c∗
)β11]
= FI 2(V ∗L ). (13)
Rearrange the equations upon liquidation and conversion as follows:
{V ∗
L − a1(V ∗
L
)u1 − L0 = 0;1 − a1u1
(V ∗
L
)u1−1 = 0,(14)
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Applying jump-diffusion processes to liquidate and convert venture capital 821
and⎧⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎩
V ∗C −
[C0 + L0
(V ∗
CV ∗
L
)β11]
− [a2
(V ∗
C
)u1 + b2(V ∗
C
)u2] = 0;
1 − L0β11
[(V ∗
C)β11−1
(V ∗L )
β11
]− [
a2u1(V ∗
C
)u1−1 + b2u2(V ∗
C
)u2−1] = 0;
V ∗L −
[L0 + C0
(V ∗
LV ∗
C
)β11]
− [a2
(V ∗
L
)u1 + b2(V ∗
L
)u2] = 0.
(15)
Since no closed form solution exists for the optimal conversion threshold V ∗C and the conver-
sion project value coefficient a2, b2 upon conversion, the solution is inferred via the valueanalysis method. When the project value is below V ∗
L , VC companies will choose to liqui-date the investment in the start-up company; when the project value exceeds V ∗
C , the VCcompanies will choose to convert and exercise the option. However, when the project valueis between V ∗
L and V ∗C , the optimal strategy is to maintain the current situation and continue
evaluating the investment in the start-up company to determine the optimal solution.
2.2 Jump-diffusion model
Assuming project value of VC investment in the start-up company is defined as V (t), whichis the same definition as the continuous process model mentioned above. The value variationgrowth over time is determined by GBM for the continuous process and by Poisson processfor the discreet process based on the jump-diffusion model as follows:
dV (t) = αV (t)dt + σ V (t)dW (t) − θV (t)dq(t). (16)
Among those factors of the project value,α: drift over time, σ : volatility over time, and dW (t):the increment of standard Winner process W (t) of zero mean and unit standard deviation√
dt . The jump-process follows Poisson process with an arrival rate of λ1:
dq (t) ={
1, with prob. λ1dt;0, with prob. 1 − λ1dt,
θ1: the magnitude of influence for the jump size in the jump-process.Given the project value of the investment FI (V ) after dt , the increment of the project
value is calculated according to Itôs Lemma (Itô 1951)
d FI (V ) = FI V (V )dV + 1
2FI V V (V )dV 2 − λ1 [FI (V ) − FI (θ1V )] dt. (17)
Assuming VC companies have no risk preference, following incremental time dt , they expectthe project value to equal the risk-free interest r f multiplied by FI (V )dt , namely:
r f × FI (V )dt = E [d FI (V )]
={αV FI V (V ) + 1
2σ 2V 2 FI V V (V ) − λ1 FI (V ) + λ1 FI (θ1V )
}dt.(18)
With some manipulation, we obtain:
1
2σ 2V 2 FI V V (V ) + αV FI V (V ) − (λ1 + r f )FI (V ) + λ1 FI (θ1V ) = 0. (19)
The general solution to Eq. 19 can be expressed as FI (V ) = AV u . Applying the general solu-tion form to the above equation with some manipulation, we obtain the following quadraticequation:
123
822 T. T. Lin et al.
1
2σ 2u2 +
(α − 1
2σ 2
)u − (λ1 + r f ) + λ1θ
u1 = 0. (20)
The project value upon liquidation is:
FI 1(V ) = a1V u1 + b1V u2 . (21)
The liquidation value is zero when the project value is zero, and VC companies do not need topay an additional fee for liquidation (as seen in Section 2.1). To ensure the consistency withthe contract value being converted into zero, b1 in Eq. 21 must be zero. Thus, the contractvalue upon liquidation is FI 1(V ) = a1V u1 . With the same inference of the liquidation value,the contract value upon conversion is FI 2(V ) = a2V u1 + b2V u2 . Among those exit plans,if VC companies choose to liquidate, a liquidation cost of L0 will occur; if VC companieschoose to give up the right to liquidate and wait for the conversion right, two costs will occur,the conversion cost C0 and the liquidation cost upon conversion L0 × (V ∗
C/V ∗L )β11 . This
study conducts the contract value upon both liquidation FI (V ∗L ) and conversion FI (V ∗
C ) todetermine the optimal exit threshold for VC companies to select liquidation or conversion.
According to VMC, assuming the optimal liquidation threshold is V ∗L , the liquidation
threshold upon liquidation V ∗L minus the liquidation cost L0 should equal the project value
upon liquidation FI 1(V ∗L ). According to SPC, the incremental project value should be equal
to the first order of differential function, which matches the same marginal profit in thresholdpoint. The above two conditions are demonstrated as follows:
⎧⎨
⎩
V MC : V ∗L − L0 = FI 1(V ∗
L ),
S PC : d[V ∗L −L0]
dV ∗L
= d FI 1(V ∗L )
dV ∗L
.(22)
Based on the above description, this study conducts the optimal liquidation threshold V ∗L and
the project value coefficient a1 upon liquidation as follows:
V ∗L = u1
u1 − 1L0; (23)
and
a1 = 1
u1(V ∗
L )1−u1 . (24)
Based on the same inference as previously mentioned, assuming the optimal conversionthreshold is V ∗
C , the project value V ∗C upon conversion minus the liquidation cost C0 and
L0 × (V ∗C/V ∗
L )β11 should equal the project value upon conversion FI 2(V ∗C ). The incremental
project value should be the same as the first order of the differential function, which canmatch the marginal profit at decision point. The analytical result is as below:
⎧⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
V MC : V ∗C −
[C0 + L0
(V ∗
CV ∗
L
)β11]
= FI 2(V ∗C ),
S PC :d
[
V ∗C −
(
C0+L0
(V ∗
CV ∗
L
)β11)]
dV ∗C
= FI 2(V ∗C )
dV ∗C
.
(25)
Furthermore, upon conversion, assuming the project value minus the liquidation cost andthe present value of the conversion cost equals the liquidation contract value FI 2(V ∗
L ) =a2
(V ∗
L
)u1 + b2(V ∗
L
)u2 , it will become:
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Applying jump-diffusion processes to liquidate and convert venture capital 823
V ∗L −
[
L0 + C0
(V ∗
L
V ∗C
)β11]
= FI 2(V ∗L ). (26)
Rearrange the liquidation and conversion equations as follows:
{V ∗
L − a1(V ∗
L
)u1 − L0 = 0,
1 − a1u1(V ∗
L
)u1−1 = 0,(27)
and⎧⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎩
V ∗C −
[C0 + L0
(V ∗
CV ∗
L
)β11]
− [a2
(V ∗
C
)u1 + b2(V ∗
C
)u2] = 0,
1 − L0β11
[(V ∗
C)β11−1
(V ∗L )
β11
]− [
a2u1(V ∗
C
)u1−1 + b2u2(V ∗
C
)u2−1] = 0,
V ∗L −
[L0 + C0
(V ∗
LV ∗
C
)β11]
− [a2
(V ∗
L
)u1 + b2(V ∗
L
)u2] = 0.
(28)
Since no closed form solution exists for the optimal conversion threshold V ∗C and the conver-
sion project value coefficient a2, b2 upon conversion, the solution is inferred using the valueanalysis method. When the project value is below V ∗
L , VC companies choose liquidation;when the project value exceeds V ∗
C , they choose conversion and exercise the option. However,when the project value is between V ∗
L and V ∗C , the optimal strategy is to maintain the current
status and keep evaluating the project value to determine the optimal solution.
2.3 Expected discount time
First, a discounted time E[T ∗
1
]interval is assumed to exist in
[V ∗
L , V ∗C
]. Referring to Appen-
dix 1, the expected discounted factor is defined as E[e−rT ∗1 ] = (V ∗
L /V ∗C )β11 . The change in
the logarithm using the rule of Jensen’s inequality (E(−rT ∗1 ) ≥ log E(e−rT ∗
1 )) is derived asfollows: −r × E
[T ∗
1
] ≥ β11 × log(V ∗
L /V ∗C
). After rearranging the equation, the expected
discounted time can be derived as:
E[T ∗
1
] ≥ β11
rlog
(V ∗
C
V ∗L
)(29)
Secondly, an expected discounted time E[T ∗
2
]is assumed to exist in interval
[V0, V ∗
L
].
From Appendix 2, the expected discounted factor is calculated by E[e−rT ∗2 ] = (V0/V ∗
L )β21 .Rearranging the equation based on the same inference, the expected discounted time E
[T ∗
2
]
is greater than or equals:
β21
rlog
(V ∗
L
V0
)(30)
2.4 Model comparison
Lin and Huang (2003) discuss the exit plans for VC companies using ROA based on Cossinet al. (2002). This proposed model compares exercising liquidation with exercising conver-sion to obtain the optimal exit evaluation model by establishing two exit thresholds. Thisassay continues the research of Lin and Huang (2003), based on mutual preference of riskby revising the discounted factor and applying the Jump process to identify the threshold
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824 T. T. Lin et al.
Table 1 Model comparison
Andreas andUwe (2001)
Lin and Huang (2003) Proposed model
Approach Utility function ROA ROAOptimal threshold None V ∗
R : Optimalredemptionthreshold
V ∗L : Optimal liquidation andconvertibility threshold
Discounted Factor None (V ∗
CV ∗
L)
rα : discounted factor r
α (V ∗
CV ∗
L)β11 : discounted factor β11
Project of value path According togeometric Brownianmotion dVt =αVt dt + σ Vt dW (t)
upon liquidation and conversion. In addition, the expected time required to reach the deci-sion threshold is derived by applying the expected discounted factor to the expected discountpoint. The model comparison is listed as Table 1.
3 Numerical and sensitivity analysis
Since it is difficult to obtain data from single case for value analysis, this study applies therelative variance and coefficients from the Taiwan Venture Capital Yearbook, published bythe Taiwan Venture Capital Association (Year 2003). The numerical solution is also obtainedusing the non-liner function solution from the software polyamth5.1.
3.1 Continuity model
Now we apply continuity model to turn to numerical solutions and sensitivity analysis toverify these intuition.
3.1.1 Numerical analysis
First, the parameters β11 and u1, u2 of the discounted factor u1 > 1, u2 < 0 are calculated.The simulated parameter data are as follows: volatility σ = 0.790; drift rate α = 3.150E-04;risk-free interest rate r f = 0.020; discount interest rate r = 0.220; project value V0 = 21(million NT dollars). Meanwhile, the simulation results are β11 = 1.476, u1 = 1.059,and u2 = −0.060. Other variables by hypothesis and parameter date are liquidation costL0 = 3.25 million NT dollars and conversion cost C0 = 4.2 million NT dollars. The resultsare obtained as: V ∗
L = 57.9 million NT dollars; V ∗C = 133.9 million NT dollars; a1 = 0.975;
a2 = 0.816; b2 = 0.074. Thus, if the invested project value is lower than the optimal liquida-tion threshold V ∗
L = 57.9 million NT dollars, the optimum exit plan for VC companies is toliquidate the project; if the invested project value exceeds the optimal conversion thresholdV ∗
C = 133.9 million NT dollars, the optimum exit plan for VC companies is to exercise the
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Applying jump-diffusion processes to liquidate and convert venture capital 825
Fig. 1 Drift rate α to the optimalthreshold
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.000 0.001 0.005 0.010
V Liquidation Threshold Conversion Threshold
α
Fig. 2 Discounted interest ratesr to the optimal threshold
0.0
0.5
1.0
1.5
0.170 0.220 0.270 0.320 0.370 0.420 0.470 0.520 r
V Liquidation Threshold Conversion Threshold
conversion option. Moreover, the contract value upon the liquidation threshold is:
F1(V ∗L ) = a1 × (
V ∗L
)u1 = 0.975 × (0.579)1.059 = 0.547 (million NT dollars).
The contract value upon the conversion threshold is:
This section performs a sensitivity analysis of the simulated variances in this assay, including:drift rate α; discounted interest rate r ; liquidation cost L0; conversion cost C0; parametersu1, u2, and β11. Under the hypothesis that the other parameter’s variances are consistent, thevariable is varied to identify the changes in optimal thresholds V ∗
L and V ∗C for both magnitude
and direction.From Fig. 1, both the optimal liquidation threshold V ∗
L and the optimal conversion thresh-old V ∗
C increase with increasing α, and the incremental also increases with increasing α.Figure 2 shows that the optimal liquidation threshold V ∗
L is not influenced by r when theoptimal conversion threshold V ∗
C decreases with increasing r .Figure 3 displays that both the optimal liquidation threshold V ∗
L and the optimal conversionthreshold V ∗
C increase with increasing L0.Figure 4 illustrates that the optimal liquidation threshold V ∗
L is not influenced by C0 sincethe optimal conversion threshold V ∗
C increases with increasing C0.Figure 5 demonstrates that both the optimal liquidation threshold V ∗
L and the optimalconversion threshold V ∗
C decrease with increasing u1 (but in less magnitude than u1).Figure 6 shows that the optimal liquidation threshold V ∗
L is not influenced by u2 sincethe optimal conversion threshold V ∗
C increases with increasing u2. However, the increase isminimal.
Figure 7 illustrates that the optimal liquidation threshold V ∗L is not influenced by β11 when
the optimal conversion threshold V ∗C decreases with increasing β11 (but in less magnitude
than β11).
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826 T. T. Lin et al.
Fig. 3 Liquidation cost L0 to theoptimal threshold
Now we apply jump-diffusion model to turn to numerical solutions and sensitivity analysisto verify these intuition.
3.2.1 Numerical analysis
First, this study calculates the parameters β11 and u1, u2 of the discounted factor withu1 > 1, u2 < 0. The simulated parameter data are as follows: volatility σ = 0.790; driftrate α = 3.150E-04; risk-free interest rate r f = 0.020; discounted interest rate r = 0.220;
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Applying jump-diffusion processes to liquidate and convert venture capital 827
magnitude θ1 = 1.200E-03; arrival rate λ1 = 0.006; project value V0 = 21 (million NT dol-lars). Moreover, the simulation results are β11 = 1.486, u1 = 1.076, and u2 = −0.053. Othervariables by hypothesis and parameter date include liquidation cost L0 = 3.25 million NTdollars and conversion cost C0 = 4.2 million NT dollars. The numerical results are obtainedas: V ∗
L = 45.8 million NT dollars; V ∗C = 106.3 million NT dollars; a1 = 0.986; a2 = 0.781;
b2 = 0.073. Thus, if the invested project value is below the optimal liquidation thresholdV ∗
L = 45.8 million NT dollars, the optimum exit plan for VC companies is liquidation; ifthe invested project value exceeds the optimal conversion threshold V ∗
C = 106.3 million NTdollars, the best exit plan for VC companies is conversion. In addition, the contract valueupon the liquidation threshold is:
F1(V ∗L ) = a1 × (
V ∗L
)u1 = 0.986 × (0.458)1.076 = 0.426 (Million NT dollars).
The contract value upon the conversion threshold is:
This subsection performs a sensitivity analysis of the simulated variances in this assay, includ-ing: drift rate α; discounted interest rate r ; project value effect intensity θ1; arrival rate λ1;liquidation cost L0; conversion cost C0; parameters u1, u2 and β11. Assuming the other vari-ances are consistent, the variable is varied to identify the changes in the optimal thresholdsV ∗
L and V ∗C for both magnitude and direction.
Figure 8 shows that both the optimal liquidation threshold V ∗L and the optimal conversion
threshold V ∗C increase with increasing α, but the magnitude of the increase is minimal.
Figure 9 reveals that the optimal liquidation threshold V ∗L is not influenced by r as the
optimal conversion threshold V ∗C decreases with increasing r (but in less magnitude than r ).
Figure 10 indicates that both the optimal liquidation threshold V ∗L and the optimal con-
version threshold V ∗C decrease with increasing λ1 (but in less magnitude than λ1).
Figure 11 illustrates that both the optimal liquidation threshold V ∗L and the optimal con-
version threshold V ∗C increase with increasing θ1, but the magnitude of the increase is very
minimal.Figure 12 illustrates that both the optimal liquidation threshold V ∗
L and the optimal con-version threshold V ∗
C increase with increasing L0.Figure 13 illustrates that the optimal liquidation threshold V ∗
L is not influenced by C0 asthe optimal conversion threshold V ∗
C increases with increasing C0.Figure 14 illustrates that both the optimal liquidation threshold V ∗
L and the optimal con-version threshold V ∗
C decrease with increasing u1 (but in less magnitude than u1).Figure 15 reveals that the optimal liquidation threshold V ∗
L is not influenced by u2 as theoptimal conversion threshold V ∗
C increases with increasing u2, but the magnitude of increaseis minimal.
Figure 16 shows that the optimal liquidation threshold V ∗L is not influenced by β11 since
the optimal conversion threshold V ∗C decreases with increasing β11 (but in less magnitude
than β11).
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828 T. T. Lin et al.
Fig. 8 Drift rate α to the optimalthreshold
0.0
0.5
1.0
1.5
-0.001 0.000 0.001 0.005
V Liquidation Threshold Conversion Threshold
α
Fig. 9 Discounted interest rate rto the optimal threshold
0.0
0.5
1.0
1.5
0.020 0.070 0.120 0.170 0.220 0.270 0.320 0.370 r
V Liquidation Threshold Conversion Threshold
Fig. 10 Average arrival rate λ1to the optimal threshold
Now we turn to numerical solutions and sensitivity analysis to verify these intuitions.
3.3.1 Numerical analysis
The simulated parameter data include the following: λ1 = 0.006, θ1 = 1.200E-03,λ2 = 0.04, and θ2 = 0.008. According to the numerical analysis from the previous model,the variables are found to have the following values (Shown as Table 2).
This study also computes the expected discounted time on both the original model andthe Jump-diffusion model (Shown as Table 3)
3.3.2 Sensitivity analysis
This section conducts a sensitivity analysis of the simulated variances in this assay, including:r, β11, β21, V0, V ∗
L , and V ∗C . Assuming that the other variances are consistent, the variable is
adjusted to identify the changes of the expected discounted time E[T ∗
1
]and E
[T ∗
2
].
123
830 T. T. Lin et al.
Table 3 Expected discounted time
Continuity model Jump-diffusion model
Interval[V0, V ∗
L
]E
[T ∗
2] ≥ β21
r log
(V ∗
LV0
)=
2.955 (years)
E[T ∗
2] ≥ β21
r log
(V ∗
LV0
)=
2.371 (years)
Interval[V ∗
L , V ∗C
]E
[T ∗
1] ≥ β11
r log
(V ∗
CV ∗
L
)=
2.443 (years)
E[T ∗
1] ≥ β11
r log
(V ∗
CV ∗
L
)=
2.470 (years)
Table 4 Summary of sensitivityanalysis for expected discounttime
+: Positive correlation;−: Negative correlation;×: Zero correlation
Parameters and variables E[T ∗
1]
E[T ∗
2]
r − −β11 + ×β21 × +
V0 × −V ∗
L − +
V ∗C + ×
First, the discounted point in interval[V ∗
L , V ∗C
]: E
[T ∗
1
] ≥ (β11/r) log(V ∗
C/V ∗L
). Sep-
arately taking the first order of differential to β11, r, V ∗L , and V ∗
C yields the following: (1)∂ E
[T ∗
1
]/∂r ≥ −(β11/r2)
(V ∗
C/V ∗L
); (2) ∂ E
[T ∗
1
]/∂β11 ≥ (1/r) log
(V ∗
C/V ∗L
)> 0; (3)
∂ E[T ∗
1
]/∂V ∗
L ≥ −(β11/r)(1/V ∗
L
); (4) ∂ E
[T ∗
1
]/∂V ∗
C ≥ (β11/r)(1/V ∗
C
)> 0.
Secondly, the discounted point in interval[V0, V ∗
L
]E
[T ∗
2
] ≥ (β21/r) log(V ∗
L /V0).
Separately taking the first order of differential to β21, r, V0, and V ∗L yields the following:
(1) ∂ E[T ∗
2
]/∂r ≥ −(β21/r2)
(V ∗
L /V0); (2) ∂ E
[T ∗
2
]/∂β21 ≥ (1/r) log
(V ∗
L /V0)
> 0;(3) ∂ E
[T ∗
2
]/∂V ∗
0 ≥ −(β21/r)(1/V ∗
0
); (4) ∂ E
[T ∗
2
]/∂V ∗
L ≥ (β21/r)(1/V ∗
L
)> 0. The
sensitivity analysis for the expected discounted point to all variables in both intervals issummarized as below (Table 4).
4 Conclusion
By applying ROA, reconsidering the present value interest factor, and applying the jump-diffusion process company evaluation model, the proposed assay extends the research of themodel of Lin and Huang (2003) to establish an exit strategy and model for VC companieswho invest in the start-up companies. This assay can also determine the optimal exit threshold(liquidation or conversion) and build up an evaluation model for the optimal exit plan. Whenthe project value is below the liquidation threshold V ∗
L , VC companies should decide toliquidate the start-up companies; meanwhile, when the project value exceeds the conversionthreshold V ∗
C , they choose to convert by converting their shares in an open market. However,when the project value is between V ∗
L and V ∗C , the best strategy is to maintain the current
status and continue assessing the optimal solution.In the example involving the numerical analysis of the threshold and coefficients, the
liquidation model coefficient a1 in the continuity-model is lower than the liquidation modelcoefficient a1 in the jump-diffusion model. Other values including liquidation threshold V ∗
L ,conversion threshold V ∗
C , and project value coefficients, a2 and b2, are all higher in thecontinuity-model than the jump-diffusion model. In the numerical analysis example for theexpected discounted point, in interval
[V0, V ∗
L
], the expected time in the continuity-model
123
Applying jump-diffusion processes to liquidate and convert venture capital 831
exceeds that in the jump-diffusion model, while in interval[V ∗
L , V ∗C
], the expected time in the
jump-diffusion model exceeds that in the continuity-model. The above explains that the deci-sions are more strongly influenced by a sudden change in the period between liquidation andconversion than before liquidation; in the sensitivity analysis, the expected discounted pointis positively correlated to all the parameters in the denominator and negatively correlated toall the parameters in the numerator.
Acknowledgements The first author would like to thank the National Science Council of the Republic ofChina for financially supporting this research under Contract No. NSC95-2416-H-259 -015-.
Appendix 1
Assuming the expected discounted factor in the continuous period[V ∗
Since f1(v) equals zero upon v = 0, the coefficient A12 in Eq. A7 must be zero. Thus, thevalue function is f1(v) = A11v
β11 . T ∗1 is very minimal when V ∗
L → V ∗C and the discounted
factor f1(v) ≈ 1; thus f1(V ∗L ) = 1. In other words,
A11V ∗β11L = 1 ⇒ A11 =
(1
V ∗L
)β11
(A8)
The value function of f1(v) = (v/V ∗C )β11 can be obtained.
123
832 T. T. Lin et al.
Upon the point of V ∗C turning back to the point of V ∗
L , the value function is f1(V0) =(V ∗
L /V ∗C )β11 . Thus the discounted factor is E[e−rT ∗
1 ] = (V ∗L /V ∗
C )β11 .
Appendix 2
Assuming the expected discounted factor in the continuous period[V0, V ∗
L
]is E[e−rT ∗
2 ] withthe same inference as Appendix 1, based on Eq. A6, the equation of parameter β2 can bederived as:
1
2σ 2β2
21 +(
α − 1
2σ 2
)β21 − (λ2 + r) + λ2θ
β212 = 0, (A9)
and the value function is f2(v) = (v/V ∗L )β21 .
Upon the point of V ∗L turning back to the point of V0, the value function is f2(V0) =
(V0/V ∗L )β21 . Thus the discounted factor is E[e−rT ∗
2 ] = (V0/V ∗L )β21 .
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