Timing options for a startup with early termination and competition risks

Proposition 3.1.

After the competitor’s arrival, the firm’s value function defined in ( 2.2 ) is given by V ˜ ( x ) (3.3) = − α k 1 ( ρ − μ ) ( a ˜ ∗ ) 1 − k 1 x k 1 + α ρ − μ x − β ρ if x > a ˜ ∗ , 0 if 0 < x ⩽ a ˜ ∗ , where k 1 = h 1 ( 0 ) < 0 (see ( 3.2 )), and the optimal threshold is explicitly given by (3.4) a ˜ ∗ = ρ − μ ρ k 1 k 1 − 1 β α > 0 .

The firm’s optimal timing to abandon is given by the first time the value function V ˜ ( X t ) reaches zero. This occurs when X t = a ˜ ∗ , so we have (3.5) τ ˜ a ∗ = inf { t ⩾ 0 : X t ⩽ a ˜ ∗ } . In particular, the optimal abandonment threshold a ˜ ∗ is slightly smaller than the ratio β / α , but note that g ( x ) < 0 for x < β / α . That means that, even if the firm is currently incurring a loss, it does not immediately abandon but will wait for the profit to improve in the future. When the price falls further to the lower level a ˜ ∗ , then the firm will decide to abandon. The explicit expression of a ˜ ∗ is amenable for sensitivity analysis, which we summarize as follows.

Corollary 3.2.

The optimal threshold a ˜ ∗ is increasing in ρ, β, and decreasing in μ, σ, α.

From Proposition 3.1, we see that V ˜ ( x ) is increasing and becomes nearly linear when x is large since the term x k 1 ( k 1 < 0 ) diminishes to zero. In fact, it’s dominated by the linear part, even if x is close to a ˜ ∗ (see the proof of Corollary 3.3 in the Appendix).

Corollary 3.3.

V ˜ ( x ) is increasing convex in x. In addition, when x is sufficient large, V ˜ ( x ) is increasing in μ, α, and decreasing in ρ, β.

Next we turn to the optimal abandonment problem V ( x ) . The explicit solution of V ˜ ( x ) will become the input to the problem for V ( x ) . We derive the optimal threshold-type strategy. Let a ∗ denote the abandonment price level before new competition arrives. New competition will bring about a reduction in revenue for the firm, described by the fraction α. If the revenue impact is low, then the firm will choose to stay longer in the market and exit in a lower threshold, and thus we expect a ∗ > a ˜ ∗ . In contrast, when the firm’s revenue is significantly reduced, then it is more likely that the firm will experience a negative cash flow, and may be forced to leave the market earlier. This means that the post-competition threshold a ˜ ∗ is higher than the pre-competition level a ∗ . As is intuitive, the aforementioned first and second cases correspond to, respectively, the large and small values of α. As we show below, the two cases are separated by the critical value of α 0 , defined by α 0 = ( 1 − ρ p 1 ( k 1 − 1 ) ( ρ + λ 2 − μ ) k 1 ( p 1 − 1 ) ( ρ − μ ) ( ρ + λ 2 ) (3.6) × ( 1 − K β ) ) − 1 , where p 1 = h 1 ( λ 2 ) as in (3.2).

Proposition 3.4.

The firm’s pre-competition optimal abandonment problem ( 2.3 ) is solved as follows:

(I) If K ⩾ β and α 0 ⩽ α ⩽ 1 , then there exists a unique a ∗ ∈ [ a ˜ ∗ , + ∞ ) such that α ( k 1 − p 1 ) k 1 ( ρ − μ ) ( a ˜ ∗ ) 1 − k 1 ( a ∗ ) k 1 + ρ + α λ 2 − μ ρ + λ 2 − μ p 1 − 1 ρ − μ a ∗ (3.7) − p 1 ( β λ 2 + ρ K ) ρ ( ρ + λ 2 ) = 0 . The value function V ( x ) is given by V ( x ) (3.8) = C 1 x p 1 − α k 1 ( ρ − μ ) ( a ˜ ∗ ) 1 − k 1 x k 1 + ρ + α λ 2 − μ ρ + λ 2 − μ 1 ρ − μ x − β λ 2 + ρ K ρ ( ρ + λ 2 ) if x > a ∗ , 0 if 0 < x ⩽ a ∗ .

(II) If K < β , or K ⩾ β and 0 ⩽ α < α 0 , then there exists a unique a ∗ ∈ ( 0 , a ˜ ∗ ) such that ( p 1 − k 1 ) ρ ( ρ + λ 2 − μ ) + k 1 μ λ 2 ( 1 − p 1 ) ( 1 − k 1 ) ρ ( ρ + λ 2 ) ( ρ + λ 2 − μ ) × β ( a ˜ ∗ ) − p 2 ( a ∗ ) p 2 + p 1 − 1 ρ + λ 2 − μ a ∗ (3.9) − p 1 K ρ + λ 2 = 0 . The value function V ( x ) is given by V ( x ) (3.10) = C 2 x p 1 − α k 1 ( ρ − μ ) ( a ˜ ∗ ) 1 − k 1 x k 1 + ρ + α λ 2 − μ ρ + λ 2 − μ 1 ρ − μ x − β λ 2 + ρ K ρ ( ρ + λ 2 ) if x > a ˜ ∗ , C 3 x p 1 + C 4 x p 2 + 1 ρ + λ 2 − μ x − K ρ + λ 2 if a ∗ < x ⩽ a ˜ ∗ , 0 if 0 < x ⩽ a ∗ . In both cases (I) and (II), the firm’s optimal abandonment time without competition is given by (3.11) τ a ∗ = inf { t ⩾ 0 : X t ⩽ a ∗ } .

The coefficients in above expressions are given by (3.12) C 1 = − v 1 ( a ∗ ) ( a ∗ ) p 1 , C 2 = ( p 1 − k 1 ) ρ ( ρ + λ 2 − μ ) + k 1 μ λ 2 ( 1 − p 1 ) ( k 1 − 1 ) ρ ( ρ + λ 2 ) ( ρ + λ 2 − μ ) C 2 = × ( ( a ˜ ∗ ) p 2 − p 1 − ( a ∗ ) p 2 − p 1 ) β ( p 2 − p 1 ) ( a ˜ ∗ ) p 1 (3.13) C 2 = + v 2 ( a ˜ ∗ ) − v 1 ( a ˜ ∗ ) ( a ˜ ∗ ) p 1 − v 2 ( a ∗ ) ( a ∗ ) p 1 , C 3 = 1 1 − ( a ∗ ) p 1 − p 2 ( a ˜ ∗ ) p 2 − p 1 C 3 = × ( C 2 + ( a ˜ ∗ ) p 2 − p 1 ( a ∗ ) p 2 v 2 ( a ∗ ) (3.14) C 3 = − v 2 ( a ˜ ∗ ) − v 1 ( a ˜ ∗ ) ( a ˜ ∗ ) p 1 ) , (3.15) C 4 = − C 3 ( a ∗ ) p 1 + v 2 ( a ∗ ) ( a ∗ ) p 2 , v 1 ( x ) = − α k 1 ( ρ − μ ) ( a ˜ ∗ ) 1 − k 1 x k 1 v 1 ( x ) = + ρ + α λ 2 − μ ρ + λ 2 − μ 1 ρ − μ x (3.16) v 1 ( x ) = − β λ 2 + ρ K ρ ( ρ + λ 2 ) , (3.17) v 2 ( x ) = 1 ρ + λ 2 − μ x − K ρ + λ 2 , and p 2 = h 2 ( λ 2 ) .

Proposition 3.4 gives the exact conditions to determine the ordering of the thresholds a ∗ and a ˜ ∗ . As an example, let β = α K so that the post-competition profit stream g ( x ) be proportional to f ( x ) , i.e. g ( x ) = α ( x − K ) . Proposition 3.4 indicates that α > α 0 , and thus we have a ∗ > a ˜ ∗ in case (II). In this example, the firm will have a smaller profit stream after the competitor’s arrival. To compensate the reduction in profit, the firm opts to stay longer in the market with the competitor and exit at a lower threshold.

To determine which case applies, the simple first step is to check whether K is larger than β or not. If K < β , it falls into case (II). Otherwise, we need to check the values of the parameters α and α 0 (see (3.6)). Note if K = β , then α 0 = 1 , which can be seen directly from (3.6). Next, Corollary 3.5 describes the behavior of α 0 and its sensitivity in the arrival rate of competition λ 2 .

Corollary 3.5.

Suppose K > β , then we have 0 < α 0 < 1 . Moreover, α 0 is strictly decreasing with respect to λ 2 , and it admits a limit α 0 ∞ : = lim λ 2 → + ∞ α 0 = ( 1 + ρ ( 1 − k 1 ) k 1 ( ρ − μ ) ( 1 − K β ) ) − 1 (3.18) ∈ ( 0 , 1 ) .

This also leads to the curious question: under what conditions does the abandonment level stay unchanged after the competitor’s entry? From the proof of Proposition 3.4 in the Appendix, we see that a ∗ = a ˜ ∗ if and only if K ⩾ β and α = α 0 . As a concrete example, let β = K and α = 1 , then a ∗ = a ˜ ∗ . This is intuitive since in this case the competitor’s entry does not affect the firm’s profit stream, and thus its abandonment timing.

4. Optimal entry timing problem

We now analyze the firm’s optimal entry timing problem ψ ( x ) defined in (2.4). Suggested by Fig. 1(a), the firm can choose to cancel the project, enter the market, or wait at anytime during the incubation period. This leads us to determine the upper and lower thresholds, e ∗ and c ∗ , for entry and cancellation by the firm.

Proposition 4.1.

The firm’s optimal entry timing problem ( 2.4 ) is given by (4.1) ψ ( x ) = V ( x ) if x ⩾ e ∗ , J ( x ) if c ∗ < x < e ∗ , 0 if 0 < x ⩽ c ∗ , where J ( x ) = D 1 x q 1 + D 2 x q 2 (4.2) − a ρ + λ 1 − μ x − b ρ + λ 1 , and q i = h i ( λ 1 ) as in ( 3.2 ). The constants { e ∗ , c ∗ , D 1 , D 2 } are found from the system of equations: D 1 ( c ∗ ) q 1 + D 2 ( c ∗ ) q 2 (4.3) − a c ∗ ρ + λ 1 − μ − b ρ + λ 1 = 0 , D 1 q 1 ( c ∗ ) q 1 − 1 + D 2 q 2 ( c ∗ ) q 2 − 1 (4.4) − a ρ + λ 1 − μ = 0 , D 1 ( e ∗ ) q 1 + D 2 ( e ∗ ) q 2 (4.5) − a e ∗ ρ + λ 1 − μ − b ρ + λ 1 = V ( e ∗ ) , D 1 q 1 ( e ∗ ) q 1 − 1 + D 2 q 2 ( e ∗ ) q 2 − 1 (4.6) − a ρ + λ 1 − μ = d d x V ( x ) | x = e ∗ . The corresponding optimal entry and cancellation times are, respectively, (4.7) τ e ∗ = inf { t ⩾ 0 : X t ⩾ e ∗ } and τ c ∗ = inf { t ⩾ 0 : X t ⩽ c ∗ } .

Remark 4.2.

To our best knowledge, the above system of nonlinear equations do not admit explicit solutions. Even in the simpler perpetual American installment call/put option problem studied in Ciurlia and Caperdoni [2], Kimura [6], similar systems of equations arise and no explicit solutions are given. Nevertheless, these equations can be solved efficiently by standard root-finding methods, such as the Newton–Raphson method, available in many computational softwares.

The value function ψ ( x ) depends on the early termination risk parameter λ 1 as we can see from J ( x ) in (4.2) and the coefficients in (4.3)–(4.6). We observe from (4.1) that ψ ( x ) equals to V ( x ) for x ⩾ e ∗ . Therefore, the competition risk, which exists after the firm’s entry, has an indirect effect on the firm’s decisions prior to entering the market since V ( x ) , and thus e ∗ and c ∗ , depend on λ 2 . Moreover, noticing that V ( x ) appears in (4.3)–(4.6), a ∗ enters the system of equations for e ∗ and c ∗ indirectly through the value function V ( x ) that is tied to a ∗ . As a result, both e ∗ and c ∗ depend on a ∗ .

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Fig. 2.

The value functions ψ ( x ) and V ( x ) in cases (I) and (II). Left: c ∗ = 1.21 , a ˜ ∗ = 3.03 , a ∗ = 3.22 and e ∗ = 6.66 . Right: c ∗ = 2.55 , a ˜ ∗ = 6.06 , a ∗ = 5.16 and e ∗ = 10.81 . Common parameters: μ = 0.03 , σ = 0.2 , ρ = 0.05 , a = 0.1 , b = 0.1 , K = 10 , λ 1 = 0.1 , λ 2 = 0.2 , β = 7 , and α = 0.6 (left), 0.3 (right).

Figure 2 shows the value functions ψ ( x ) and V ( x ) , along with four thresholds c ∗ , a ˜ ∗ , a ∗ , and e ∗ . From both panels, we see that ψ ( x ) and V ( x ) are both increasing in x as a higher expected net present value is attained at a higher price level. The value function ψ ( x ) dominates V ( x ) due to the timing option (to enter the market) embedded in ψ ( x ) . If the firm chooses to enter immediately at some x, then we have ψ ( x ) = V ( x ) . This occurs for x ⩾ e ∗ . By checking the condition in Proposition 3.4, we have K = 10 > β = 7 > 0 and α 0 = 0.47 , and in turn, α = 0.6 > 0.47 = α 0 in Fig. 2(a) while α = 0.3 < 0.47 = α 0 in Fig. 2(b). In other words, the left and right panels represent case (I) with a ∗ ⩾ a ˜ ∗ and case (II) with a ˜ ∗ > a ∗ in Proposition 3.4 respectively.

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Fig. 3.

Sensitivity of the firm’s strategies with respect to early termination rate λ 1 .

Also, we observe that the cancellation threshold c ∗ is smaller than both post-entry abandonment thresholds a ∗ and a ˜ ∗ . During the incubation period, the option to enter induces the firm to be willing to incur a loss in order to wait for the opportunity to enter the market. However, once the firm has entered the market, the entry option vanishes, and the firm demands more profit to stay in the market, and will exit at a level higher than c ∗ .

5. Sensitivity analysis

In this section, we examine the effects of the early termination risk and the competitor’s arrival on the firm’s optimal strategies. Common parameters are of the same values as in Fig. 2.

5.1. Early termination

First, we conduct a sensitivity analysis of the thresholds with respect to the early termination rate λ 1 . As is seen in Fig. 3, the pre/post-competition abandonment levels do not change with λ 1 since the firm has already entered the market. For the entry and cancellation thresholds, we see that e ∗ decreases but c ∗ increases as λ 1 increases. In other words, a high early termination risk induces the firm to enter the market early, even if that means capturing a lower profit stream upon entry. Moreover, the firm may also cancel the project earlier because the expected profitability is reduced by the higher termination risk. In Fig. 3, we also observe that the threshold e ∗ is higher than a ∗ while c ∗ is lower than a ∗ .

Figures 3(a) and 3(b) depict the two cases a ˜ ∗ ⩽ a ∗ and a ˜ ∗ > a ∗ respectively. The impact of early termination risk is different in these two scenarios. If the competitor’s entry causes a significant shrinkage in the firm’s profit as in case (II), then the post-competition abandonment level a ˜ ∗ will be higher than the pre-competition one a ∗ . Also, the firm’s entry level e ∗ is decreasing in λ 1 . In Fig. 3(b), we see that e ∗ will eventually fall below a ˜ ∗ but stay above a ∗ . In this case, whenever the competitor enters, the firm switches its abandonment level to the higher level a ˜ ∗ . This forces the firm to abandon immediately involuntarily if the current value of X is below a ˜ ∗ .

5.2. Competition risk

The risk of competition will influence the firm’s pre-competition exit strategy, described by a ∗ , in a non-trivial way. We discuss the four scenarios of different sensitivities in Fig. 4. In the following discussion, we keep in mind that a ˜ ∗ doesn’t depend on λ 2 and α 0 is decreasing in λ 2 by Corollary 3.5.

Figure 4(a) illustrates the first scenario with K < β . This corresponds to case (II) in Proposition 3.4, where we have shown that a ∗ < a ˜ ∗ for all λ 2 > 0 . In addition, a ∗ is increasing in λ 2 , which means that the firm will choose to exit early under the threat of competitor’s entry.

For the remaining three scenarios in Figs 4(b)–4(d), we have K ⩾ β . By Corollary 3.5, we have 1 > α 0 ( 0 ) > α 0 ( λ 2 ) > α 0 ∞ > 0 for all λ 2 > 0 . In Fig. 4(b), a ∗ is again increasing in λ 2 . In this figure, we have α ⩽ α 0 ∞ , which belongs to case (II) in Proposition 3.4, and thus a ∗ < a ˜ ∗ .

Figure 4(c) shows that a ∗ is first increasing then decreasing in λ 2 . This is because the constant α 0 is first larger than α but then smaller than α as λ 2 increases. Therefore, as λ 2 increases, it changes from case (II) to case (I) in Proposition 3.4. That explains why a ∗ is smaller than a ˜ ∗ at first, and the relation reverses for large λ 2 .

Lastly, in Fig. 4(d) we have α ⩾ α 0 ( 0 ) , which corresponds to case (I) in Proposition 3.4. We have shown that a ∗ ⩾ a ˜ ∗ for all λ 2 . In contrast to the other scenarios, the pre-competition threshold a ∗ is decreasing and approaches a ˜ ∗ as λ 2 increases. In this scenario, the impact of the competitor’s entry on the firm’s profit is much less than in other scenarios, as indicated by the higher value of α.

In summary, scenarios (a), (b), and the first part of scenario (c) belong to case (II), in which the competitor’s potential entry will lead to a significant profit reduction. Therefore, the firm exits early by selecting a higher pre-competition abandonment threshold a ∗ as the competitor’s arrival rate increases. The opposite happens to scenario (d) and the second part (large λ 2 ) of scenario (c). As λ 2 increases, which means high risk of facing competition in the future, the firm tries to stay longer before the competitor’s entry to make more profits in case of the potential significant loss afterwards. This results in a decreasing pre-competition abandonment threshold a ∗ .

Despite different behaviors of a ∗ , there is a common phenomenon that a ∗ approaches a ˜ ∗ as λ 2 increases. The intuitive explanation is as follows. When the competition risk is very high, the firm will adjust its abandonment level a ∗ in order not to be affected too much when the competitor eventually enters. That means that the firm’s abandonment level a ∗ before the competitor’s entry will be close to the post-competition abandonment level a ˜ ∗ .

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Fig. 4.

Sensitivity of the abandonment thresholds, a ∗ (pre-competition) and a ˜ ∗ (post-competition), with respect to the competitor’s arrival rate λ 2 .

In addition to a ∗ , the entry and cancellation thresholds e ∗ and c ∗ are also affected by the competitor’s arrival rate λ 2 after the firm’s market entry. In two scenarios, Fig. 5(a) and Fig. 5(b) respectively take the same parameters as Fig. 4(b) and Fig. 4(d), so they represent respectively case (II) and (I) in Proposition 3.4. Figure 5(a) shows that both e ∗ and c ∗ are increasing in λ 2 under case (II). In particular, the entry threshold e ∗ is seen to be lower than the abandonment threshold a ˜ ∗ when λ 2 is small, but then surpasses it for large λ 2 . In this case, the competitor’s arrival will lead to a significant post-entry profit reduction. Therefore, as is intuitive, the firm requires a higher entry level to ensure a higher post-entry profit so as to make the entry worthwhile. Nevertheless, under case (I), Fig. 5(b) shows that both e ∗ and c ∗ are decreasing in λ 2 . In other words, when the post-entry competition arrival is not expected to have a significant impact on profit, the firm will opt to enter the market earlier at a lower entry threshold.

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Fig. 5.

Sensitivity of the entry and cancellation thresholds, e ∗ and c ∗ , with respect to the competitor’s arrival rate λ 2 .