A Sensitivity Analysis of the eal Option Model Junichi Imai and Takahiro Watanabe November 25, 2003 Abstract In this paper, we analyze the value of the flexibility of the project under the demand uncertainty. In the real option analysis the project could be deferred until the next period. This possibility of waiting the investment can be regarded as a real option to defer that is one of the typicalrealoptionsinthecorporatefinance. Itisshowninthispaperthat the optimal decision can be classified with respect to the investment size and the value of the real option takes the maximum value when the net present value is equal to zero. This paper also examines the sensitivity of the real option value to the volatility. The model could be extended to the real option analysis under competition. 1 Introduction This paper develops a simple discrete model to analyze the project value in the corporatefinance. Traditionally, the project value is measured by the net present value (NPV) which is consistent with the finance theory that the company should maximize the shareholders value. It has been pointed out, however, that the net present value can not capture the flexibility, and accordingly, the project manager often underestimates the project value if the NPV method is used. The real option analysis (OA) is proposed to overcome this problem. In the real option analysis the flexibility of the project is taken as a real option which is exercised in the future only when the uncertainty turns out to be favorable for the exercise. Black and Scholes(1973) and Merton(1973) propose the financial option pricing theory. Any financial option can be evaluated by applying no-arbitrage principle. For the application of the no-arbitrage principle to the real option analysis it is necessary to assume that the underlying market is complete. It is This research is supported by Grant-in-Aid for Scientific esearch in Japan Society for the Promotion of Science. Faculty of Policy Studies in Iwate Prefectural University (IPU). Faculty of Economics in Tokyo Metropolitan Univeristy. 1
difficult to accept this assumption in many real option analyses since the underlying assets can be hardly observed. Therefore, in this paper we apply the equilibrium approach that is now widely accepted in the real option analysis 1. In this paper, we analyze the value of the flexibility of the project under the demand uncertainty. In the real option analysis the project could be deferred until the next period. This possibility of waiting the investment can be regarded as a real option to defer that is one of the typical real options in the corporate finance. It is shown in this paper that the optimal decision can be classified with respect to the investment size. Our analysis reveals the following implication. When the investment cost becomes too large or too small there exists no real option values. The real option value is not monotonous with respect to the investment size. The value of the real option takes the maximum value when the net present value is equal to zero. The real option to defer includes two types. In the first type the company puts off the decision to entry while the investment is immediately carried out in the NPV analysis. The second option represents the option not to give up the opportunities while in the NPV analysis the company gives up investing. In addition to this sensitivity analysis to the investment, this paper also examine the sensitivity of the real option to the volatility of the underlying asset, which is one of the most important issues in the real option analysis. It is proven that There exists a boundary volatility where the real option value becomes positive. Once the volatility is large enough the real option values monotonously increases with regard to the volatility, which is consistent with the results of the standard option analyses. This paper is organized as follows. In section 2, we set up a simple binomial model that can analyze the value of the real option. Section 3 defines the real option value as well as the traditional net present value and completes the sensitivity analysis. Concluding remarks are in section 4. 2 Model Description In this section we set up a simple valuation model of a real option. Consider a company which has an opportunity to invest a project. This project produces 1 It can be proven that the equilibrium approach is consistent with the no-arbitrage principle under some conditions. See Duffie(1996), for example. 2
some merchandise under the demand uncertainty. We assume that the demand of the merchandise follows one-period binomial model. Let Y 0 denote the initial demand. After one period the demand of the merchandise, denoted by Y 1 could either move up to Y 1 = uy 0, or move down to Y 1 = dy 0 where u and d are rate of the demand in one period, respectively. We assume that u = 1 d > 1 is satisfied for simplicity. Under this demand uncertainty the project starts at time zero and ends at time one. The model is really simple but it enables us to complete the full analysis for valuing the project. At time zero the company can either start the project or defer the project; i.e., the company has a real option to defer. The cash flow that the company obtains at each time depends on whether the project is active or not. Assume that the cash flow at time zero can be defined as D i Y 0 (i =0, 1) where D 1 represents cash flow per one unit of demand when the project has begun while D 0 represents cash flow per unit of demand when the project does not start at time zero. Similarly, the cash flows at time one are defined as D i uy 0 (i =0, 1) and D i dy 0 (i =0, 1), respectively. We assume that D 1 >D 0 is satisfied. If the underlying asset of the real option can be traded in the complete market we can apply the no-arbitrage principle to value the real option 2. It is difficult, however, to apply the principle to our model since the demand of the merchandise cannot be observed in the market. In this paper, we take an equilibrium approach that is a typical assumption for the real option analyses 3. Especially, the demand risk in this paper can be considered private risk or unsystematic risk that is independent of the market risk. Since an investor pays no risk premium with respect to the unsystematic risk in equilibrium, we can assume that the investors are risk neutral in the valuation model. Let p denote a probability of the demand to move up and q denote a probability to move down that is equal to 1 p. Letr be the risk free rate for one period and let define =1+r. The expected demand at time one is E [Y 1 ]=puy 0 + qdy 0 µy 0 (1) Note that we define µ = pu + qd. To examine the impact of a real option, we consider two cases of valuation. The first approach is based on a net present value analysis in which the company does not have an option to defer the project. Thus, it must decide whether the project should be started at time zero. The second approach is based on the real option analysis which includes a value of the real option to defer the project until time one. Thus, the difference of these two values represents the net value of the real option. Accordingly, we can examine the impact of the real option by comparing these two values. 2 Mason and Merton(1985) insist that the value of real options can be evaluated with noarbitrage principle if the underlying securities are observed in the markets which has the same risk profile as the real options. 3 For example, Cox and oss, Constantinides, and McDonald and Siegel propose the equilibrium approach for the real option pricing. 3
3 A Sensitivity Analysis This section examines the impact on the real option values when the parameter values are changed. The impact of the real option can be measured by comparing the net present value and the value of the real option with flexibility. 3.1 A Sensitivity to the Investment Let V0 NPV denote the net present value of the project, which does not take any flexibility into consideration. The value of the project when the company decides not to entry, denoted by v out,is v out = D 0 Y 0 + 1 E [D 0Y 1 ]=D 0 Y 0 1+ µ. where D 0 Y 0 is the cash flow at time zero and 1 E [D 0Y 1 ] is the discounted value of the expected cash flow at time one. The value of the project when the company decide to entry by paying the investment I at time zero, which is denoted by v in, becomes v in = D 1 Y 0 I + 1 E [D 1Y 1 ]=D 1 Y 0 1+ µ I The optimal decision can be solved by comparing these two values, which is described by the following proposition. Proposition 1 The company should entry the project at time zero if I I, and should not entry if I>I where I represents the boundary investment that is I =(D 1 D 0 ) 1+ µ Y 0. (2) On the other hand, the real option approach takes a possibility to defer the project at time zero. It can be derived by backward reduction. In the real option framework we first compute the project value at time one on condition that the company does not entry the project at time zero. First, consider the optimal strategy at time one. Let V 1 (1) denote the project value at time one when the company entries at time one, and V 0 (1) denote the project value at time one when the company does not entry the project, on condition that it does not entry at time zero. Then, V 1 (1) = (D 1 Y 1 I) V 0 (1) = D 0 Y 1 The company makes a decision so that the project value is maximized. Then, the optimal decision depends on the boundary investment costs, I u and I d where 4
I u =(D 1 D 0 ) uy 0 (3) I d =(D 1 D 0 ) dy 0 (4) The optimal decision at time one can be described as follows. Proposition 2 When the company does not entry at time zero the optimal decision at time one depends on the investment cost; Namely, If I I d the company should entry regardless of the demand at time one. if I d I I u the company should entry when the demand moves up to uy 0 and should not entry when the demand moves down to dy 0. if I>I u the company should not entry regardless of the demand at time one. According to Proposition 2 we consider the optimal decision at time zero when the company has the real option to defer the project. The additional boundary investment I Q is defined as I Q 1+ q =(D 1 D 0 ) Y d 0 1 p. (5) The relations among the boundary investment costs, I u,i d,i and I Q are ordered to the following proposition. Proposition 3 The following orders are satisfied. 1. if 1+ µ >uthen Id <I u <I <I Q. 2. if 1+ µ u then Id <I Q <I <I u. Proof. First we show the following equivalent relations. It is shown I u <I u<1+ µ Iu <I Q I u <I (D 1 D 0 ) uy 0 < (D 1 D 0 ) 1+ µ Y 0 u<1+ µ pu + qd u<1+ u pu < 1+qd u 1 p < 1+ qd I u <I Q. 5
Next, we show that the sign of I Q I and that of I Q I u is identical. I Q I + qd =(D 1 D 0 ) Y 0 p (D 1 D 0 ) 1+ µ Y 0 1 =(D 1 D 0 ) Y 0 { ( + qd) ( p)( + µ)} ( p) 1 =(D 1 D 0 ) Y 0 {( + qd) u ( p)} ( p) = p µ + qd (D 1 D 0 ) Y 0 p u = p I Q I u Finally, since d<1 < +qd p and IQ >I d always holds. Consequently, we have shown that these inequalities are satisfied. Using the above proposition we can solve the optimal decision at time zero when the company has the real option to defer the project. Let V 1 (0) denote the project value at time zero when the company invests at time zero, and V 0 (0) denote the project value at time zero when the company does not invest at time zero, but will make an optimal decision at time one. Note that V 1 (0) = V1 NPV. The decision can be divided into two cases. Case 1:1+ µ >u Proposition 4 In case 1 the optimal decision for the company which has a real option to defer is equivalent to the decision which is based on the net present value analysis; Namely, the company entries at time zero when I I and it never entries if I>I. Proof. According to proposition 3 the inequality I>I leads to I>I u,which means that if the company does not entry at time zero it should not invest at time one either regardless of the demand. Therefore, its decision is identical to the decision of the NPV method. Since I>I means that the company should not invest at time zero it never entries in this case. When I I proposition 1 indicates V 0 (0) = D 0 Y 0 + 1 (D 1µY 0 I) 0pt0pt I I d D 0 Y 0 + 1 {p (D 1µY 0 I)+qD 0 dy 0 } I d I I u D 0 Y 0 + 1 D 0µY 0 I u <I I By comparing V 0 (0) with V 0 (1) we can conclude whether the company should entry at time zero. If I I d then µ V 1 (0) V 0 (0) = (D 1 D 0 ) Y 0 1 1 I >I d µ 1 1 I>I d I 0. 6
Next, if I d I I u then V 1 (0) V 0 (0) = (D 1 D 0 ) 1+ q d Y 0 1 p I = 1 p I Q I Since I Q >I u accordingtoproposition3andi I u Finally, if I u I I then V 1 (0) V 0 (0) > 1 p (I u I) 0. V 1 (0) V 0 (0) = I I>0 Accordingly, it is shown that the company should invest at time zero when I I holds. In the first case where 1 + µ >uholds the company never exercises its real option to defer the project and to entry at time one. This means that the holding the real option is useless in this case. The case indicates that the volatility of the demand is not large enough so that the real option is valuable. Case 2: 1+ µ u Proposition 5 In case 2, the optimal decision for the company that has the real option can be described as follows. 1. Invest at time zero if I I Q 2. Defer the investment at time zero if I Q I I u. At time one invest when the demand moves up, and do not invest otherwise. 3. DonotinvestifI>I u. Proof. Suppose I>I u. According to proposition 2 the optimal decision at time one on condition that the company does not invest at time zero is not to invest regardless of the demand at time one. Therefore, the optimal decision is equivalent to the net present value analysis. Thus, proposition 1 indicates that the company should not entry the investment in this case. Next, when I d < I I u is considered proposition 2 indicates that the optimal decision for the company on condition that it does not entry at time zero is to invest when the demand moves up at time one, and not to invest when the demand moves down. Then, V 1 (0) V 0 (0) = 1 p I Q I, which indicates that the company should invest if I d < I I Q, defer the investment at time zero and make an optimal decision at time one depending on the demand if I Q I I u. This proves the proposition. 7
Investment I I I Q I Q <I I I <I I u I u <I NPV invest at time zero do not invest donotinvestattime0 (2) invest if Y 1 = uy 0 do not invest if Y 1 = dy 0 eal option(o) (1) invest at time zero O-NPV 0 (3)do not invest 1 p I I Q p (Iu I) 0 Table 1: Investment and the optimal Decision in Case 2 Table 1 shows the summary of the optimal decision for the company which depends on the size of the initial investment I. In case 2, because the volatility of the demand is large there could exist a difference between a real option analysis and the net present value analysis. Note that when the investment is too large or too small there is no real option value because the real option to defer the project is useless. In short, there exists the real option value only when the investment I Q <I<I u is satisfied. Figure 1 illustrate the values the net present value and the real option.both of the net present value and the real option value decrease as the investment cost increases. But when the investment cost equals to I Q the optimal decision for the real option analysis is changed. While the company invests at time zero in the NPV analysis the company exercises the real option and defer the project until time one in the real option analysis. At the point of I = I the company gives up investing in the NPV analysis while the company still pursues the possibility of investment the project at time one. The net value between the NPV and the real option records a maximum value when the investment is equal to I. 3.2 A Sensitivity to the Volatility of the Demand It is interesting for the real option analysis to examine a sensitivity of the real option value to the volatility of the demand. In the simple binomial model the volatility σ is computed as σ 2 = p (u µ) 2 + q (d µ) 2. Assuming that d = 1 / u leads to µ σ 2 =(u µ) µ 1. (7) u Equation (7) indicates that the size of the volatility can be controlled by the parameter u assuming that the expected rate of demand µ is constant. It is important to note that the net present value of the project is not affected 8
Figure 1: The project value base on the NPV and the OA the change of the volatility since in essence the NPV analysis does not take the uncertainty into account. Since I is constant regardless of the volatility, we consider two cases, which are I<I and I I. Consider first when I<I, which is illustrated in Figure 2 In this case the company always invests at time zero in the NPV analysis. The net present value is D 1 1+ µ Y0 I. In the real option analysis when σ is small the company invests at time zero. The real option value is constant because it is assumed that the expected rate of demand is constant. When σ = σ 1 which represents that I = I Q the optimal decision is changed. The company waits to invest until time one and makes the optimal decision that depends on the demand Y 1, which corresponds to the area (2) in Table 1. After that the real option value increases as the volatility increases. This implies that the value of real option is increasing after the value becomes positive. Next, we consider the case when I I. It is illustrated in Figure 3 In this case the company never entries the project in the NPV analysis. Thus, the valueinthiscaseisd 0 1+ µ Y0. When the volatility is small the company never entries the project in the real option analysis as well. When σ = σ 2 that represents I = I u the company could entry the project at time one when the demand moves up. This increases the value of real option to defer. 9
Figure 2: The project values when I<I Figure 3: The project values when I I 10
4 Concluding emarks This paper investigates the flexibility of the project under the future demand uncertainty. Wetakethisflexibility as a real option to defer the project and evaluates it using equilibrium approach. We analyze the impact of the investment and the volatility on the net value of the real option. The model developed in this paper is too simple for the company to compute the project value in practice. However, it can be used to understand the relation between the traditional net present value method and the real option analysis, impact of the investment cost on the real option value, and effect of the volatility. Furthermore, it can be extended to the model in which the competition is involved. The model developed here becomes a good foundation for the analysis the real option under the competitive environment, which will be a future research. 11