The investment game in incomplete markets

Transcription

1 The investment game in incomplete markets M. R. Grasselli Mathematics and Statistics McMaster University Pisa, May 23, 2008

2 Strategic decision making We are interested in assigning monetary values to strategic decisions. Traditional, these include the decision to: create a new firm;

3 Strategic decision making We are interested in assigning monetary values to strategic decisions. Traditional, these include the decision to: create a new firm; invest in a new project;

4 Strategic decision making We are interested in assigning monetary values to strategic decisions. Traditional, these include the decision to: create a new firm; invest in a new project; start a real estate development;

5 Strategic decision making We are interested in assigning monetary values to strategic decisions. Traditional, these include the decision to: create a new firm; invest in a new project; start a real estate development; finance R&D;

6 Strategic decision making We are interested in assigning monetary values to strategic decisions. Traditional, these include the decision to: create a new firm; invest in a new project; start a real estate development; finance R&D; abandon a non-profitable project;

7 Strategic decision making We are interested in assigning monetary values to strategic decisions. Traditional, these include the decision to: create a new firm; invest in a new project; start a real estate development; finance R&D; abandon a non-profitable project; temporarily suspend operations under adverse conditions.

8 Many other non-financial decisions can be addressed in the same framework. For instance, the decision to: enroll in an MBA program;

9 Many other non-financial decisions can be addressed in the same framework. For instance, the decision to: enroll in an MBA program; get married;

10 Many other non-financial decisions can be addressed in the same framework. For instance, the decision to: enroll in an MBA program; get married; change the constitution of a country;

11 Many other non-financial decisions can be addressed in the same framework. For instance, the decision to: enroll in an MBA program; get married; change the constitution of a country; introduce environmental laws;

12 Many other non-financial decisions can be addressed in the same framework. For instance, the decision to: enroll in an MBA program; get married; change the constitution of a country; introduce environmental laws; develop a controversial highway;

13 Many other non-financial decisions can be addressed in the same framework. For instance, the decision to: enroll in an MBA program; get married; change the constitution of a country; introduce environmental laws; develop a controversial highway; commit suicide!

14 Elements of Valuation In all of the previous problems, we can identify the following common elements: uncertainty about the future; To account for these elements, we are going to base our decisions on values obtained using the following theoretical tools:

15 Elements of Valuation In all of the previous problems, we can identify the following common elements: uncertainty about the future; some degree of irreversibility; To account for these elements, we are going to base our decisions on values obtained using the following theoretical tools:

16 Elements of Valuation In all of the previous problems, we can identify the following common elements: uncertainty about the future; some degree of irreversibility; timing flexibility; To account for these elements, we are going to base our decisions on values obtained using the following theoretical tools:

17 Elements of Valuation In all of the previous problems, we can identify the following common elements: uncertainty about the future; some degree of irreversibility; timing flexibility; interaction with other people s decisions. To account for these elements, we are going to base our decisions on values obtained using the following theoretical tools:

18 Elements of Valuation In all of the previous problems, we can identify the following common elements: uncertainty about the future; some degree of irreversibility; timing flexibility; interaction with other people s decisions. To account for these elements, we are going to base our decisions on values obtained using the following theoretical tools: Net Present Value: intrinsic advantages

19 Elements of Valuation In all of the previous problems, we can identify the following common elements: uncertainty about the future; some degree of irreversibility; timing flexibility; interaction with other people s decisions. To account for these elements, we are going to base our decisions on values obtained using the following theoretical tools: Net Present Value: intrinsic advantages Real Options: value of waiting

20 Elements of Valuation In all of the previous problems, we can identify the following common elements: uncertainty about the future; some degree of irreversibility; timing flexibility; interaction with other people s decisions. To account for these elements, we are going to base our decisions on values obtained using the following theoretical tools: Net Present Value: intrinsic advantages Real Options: value of waiting Game Theory: erosion of creation of value due to competition

21 Successes and Limitations of Real Options Real options accurately describe the value of flexibility in decision making under uncertainty.

22 Successes and Limitations of Real Options Real options accurately describe the value of flexibility in decision making under uncertainty. According to a recent survey, 26% of CFOs in North America always or almost always consider the value of real options in projects.

23 Successes and Limitations of Real Options Real options accurately describe the value of flexibility in decision making under uncertainty. According to a recent survey, 26% of CFOs in North America always or almost always consider the value of real options in projects. This is due to familiarity with the option valuation paradigm in financial markets and its lessons.

24 Successes and Limitations of Real Options Real options accurately describe the value of flexibility in decision making under uncertainty. According to a recent survey, 26% of CFOs in North America always or almost always consider the value of real options in projects. This is due to familiarity with the option valuation paradigm in financial markets and its lessons. But most of the literature in Real Options is based on different combinations of the following unrealistic assumptions: (1) infinite time horizon, (2) perfectly correlated spanning asset, (3) absence of competition.

25 Successes and Limitations of Real Options Real options accurately describe the value of flexibility in decision making under uncertainty. According to a recent survey, 26% of CFOs in North America always or almost always consider the value of real options in projects. This is due to familiarity with the option valuation paradigm in financial markets and its lessons. But most of the literature in Real Options is based on different combinations of the following unrealistic assumptions: (1) infinite time horizon, (2) perfectly correlated spanning asset, (3) absence of competition. Though some problems have long time horizons (30 years or more), most strategic decisions involve much shorter times.

26 Successes and Limitations of Real Options Real options accurately describe the value of flexibility in decision making under uncertainty. According to a recent survey, 26% of CFOs in North America always or almost always consider the value of real options in projects. This is due to familiarity with the option valuation paradigm in financial markets and its lessons. But most of the literature in Real Options is based on different combinations of the following unrealistic assumptions: (1) infinite time horizon, (2) perfectly correlated spanning asset, (3) absence of competition. Though some problems have long time horizons (30 years or more), most strategic decisions involve much shorter times. The vast majority of underlying projects are not perfectly correlated to any asset traded in financial markets.

27 Successes and Limitations of Real Options Real options accurately describe the value of flexibility in decision making under uncertainty. According to a recent survey, 26% of CFOs in North America always or almost always consider the value of real options in projects. This is due to familiarity with the option valuation paradigm in financial markets and its lessons. But most of the literature in Real Options is based on different combinations of the following unrealistic assumptions: (1) infinite time horizon, (2) perfectly correlated spanning asset, (3) absence of competition. Though some problems have long time horizons (30 years or more), most strategic decisions involve much shorter times. The vast majority of underlying projects are not perfectly correlated to any asset traded in financial markets. In general, competition erodes the value of flexibility.

28 Alternatives The use of well known numerical methods (e.g finite differences) allows for finite time horizons.

29 Alternatives The use of well known numerical methods (e.g finite differences) allows for finite time horizons. As for the spanning asset assumption, the absence of perfect correlation with a financial asset leads to an incomplete market.

30 Alternatives The use of well known numerical methods (e.g finite differences) allows for finite time horizons. As for the spanning asset assumption, the absence of perfect correlation with a financial asset leads to an incomplete market. Replication arguments can no longer be applied to value managerial opportunities.

31 Alternatives The use of well known numerical methods (e.g finite differences) allows for finite time horizons. As for the spanning asset assumption, the absence of perfect correlation with a financial asset leads to an incomplete market. Replication arguments can no longer be applied to value managerial opportunities. The most widespread alternative to replication in the decision-making literature is to introduce a risk-adjusted rate of return, which replaces the risk free rate, and use dynamic programming.

32 Alternatives The use of well known numerical methods (e.g finite differences) allows for finite time horizons. As for the spanning asset assumption, the absence of perfect correlation with a financial asset leads to an incomplete market. Replication arguments can no longer be applied to value managerial opportunities. The most widespread alternative to replication in the decision-making literature is to introduce a risk-adjusted rate of return, which replaces the risk free rate, and use dynamic programming. This approach lacks the intuitive understanding of opportunities as options.

33 Alternatives The use of well known numerical methods (e.g finite differences) allows for finite time horizons. As for the spanning asset assumption, the absence of perfect correlation with a financial asset leads to an incomplete market. Replication arguments can no longer be applied to value managerial opportunities. The most widespread alternative to replication in the decision-making literature is to introduce a risk-adjusted rate of return, which replaces the risk free rate, and use dynamic programming. This approach lacks the intuitive understanding of opportunities as options. Finally, competition is generally introduced using game theory.

34 Alternatives The use of well known numerical methods (e.g finite differences) allows for finite time horizons. As for the spanning asset assumption, the absence of perfect correlation with a financial asset leads to an incomplete market. Replication arguments can no longer be applied to value managerial opportunities. The most widespread alternative to replication in the decision-making literature is to introduce a risk-adjusted rate of return, which replaces the risk free rate, and use dynamic programming. This approach lacks the intuitive understanding of opportunities as options. Finally, competition is generally introduced using game theory. Surprisingly, game theory is almost exclusively combined with real options under the hypothesis of risk-neutrality!

35 A one period investment model Consider a two factor market where the discounted prices for the project V and a correlated traded asset S follow: (us 0, hv 0 ) with probability p 1, (us (S T, V T ) = 0, lv 0 ) with probability p 2, (1) (ds 0, hv 0 ) with probability p 3, (ds 0, lv 0 ) with probability p 4, where 0 < d < 1 < u and 0 < l < 1 < h, for positive initial values S 0, V 0 and historical probabilities p 1, p 2, p 3, p 4.

36 A one period investment model Consider a two factor market where the discounted prices for the project V and a correlated traded asset S follow: (us 0, hv 0 ) with probability p 1, (us (S T, V T ) = 0, lv 0 ) with probability p 2, (1) (ds 0, hv 0 ) with probability p 3, (ds 0, lv 0 ) with probability p 4, where 0 < d < 1 < u and 0 < l < 1 < h, for positive initial values S 0, V 0 and historical probabilities p 1, p 2, p 3, p 4. Let the risk preferences be specified through an exponential utility U(x) = e γx.

37 A one period investment model Consider a two factor market where the discounted prices for the project V and a correlated traded asset S follow: (us 0, hv 0 ) with probability p 1, (us (S T, V T ) = 0, lv 0 ) with probability p 2, (1) (ds 0, hv 0 ) with probability p 3, (ds 0, lv 0 ) with probability p 4, where 0 < d < 1 < u and 0 < l < 1 < h, for positive initial values S 0, V 0 and historical probabilities p 1, p 2, p 3, p 4. Let the risk preferences be specified through an exponential utility U(x) = e γx. An investment opportunity is model as an option with discounted payoff C t = (V e rt I ) +, for t = 0, T.

38 European Indifference Price The indifference price for the option to invest in the final period as the amount π that solves the equation max E[U(x +H(S T S 0 )] = max E[U(x π+h(s T S 0 )+C T ] H H

39 European Indifference Price The indifference price for the option to invest in the final period as the amount π that solves the equation max E[U(x +H(S T S 0 )] = max E[U(x π+h(s T S 0 )+C T ] H H Denoting the two possible pay-offs at the terminal time by C h and C l, the European indifference price is explicitly given by π = g(c h, C l ) (2) where, for fixed parameters (u, d, p 1, p 2, p 3, p 4 ) the function g : R R R is defined as g(x 1, x 2 ) = q ( ) γ log p 1 + p 2 p 1 e γx 1 + p2 e γx (3) q ( ) p 3 + p 4 log γ p 3 e γx 1 + p4 e γx, 2 with q = 1 d u d.

40 Early exercise When investment at time t = 0 is allowed, it is clear that immediate exercise of this option will occur whenever its exercise value (V 0 I ) + is larger than its continuation value π C.

41 Early exercise When investment at time t = 0 is allowed, it is clear that immediate exercise of this option will occur whenever its exercise value (V 0 I ) + is larger than its continuation value π C. That is, from the point of view of this agent, the value at time zero for the opportunity to invest in the project either at t = 0 or t = T is given by C 0 = max{(v 0 I ) +, g((hv 0 e rt I ) +, (lv 0 e rt I ) + )}.

42 A multi period model Consider now a continuous-time two factor market of the form ds t = (µ 1 r)s t dt + σ 1 S t dw dv t = (µ 2 r)v t dt + σ 2 V t (ρdw + 1 ρ 2 dz).

43 A multi period model Consider now a continuous-time two factor market of the form ds t = (µ 1 r)s t dt + σ 1 S t dw dv t = (µ 2 r)v t dt + σ 2 V t (ρdw + 1 ρ 2 dz). We want to approximate this market by a discrete time processes (S n, V n ) following the one period dynamics (1).

44 A multi period model Consider now a continuous-time two factor market of the form ds t = (µ 1 r)s t dt + σ 1 S t dw dv t = (µ 2 r)v t dt + σ 2 V t (ρdw + 1 ρ 2 dz). We want to approximate this market by a discrete time processes (S n, V n ) following the one period dynamics (1). This leads to the following choice of parameters: u = e σ 1 t, h = e σ 2 t, d = e σ 1 t, l = e σ 2 t, p 1 + p 2 = e(µ 1 r) t d, p 1 + p 3 = e(µ 2 r) t l u d h l ρσ 1 σ 2 t = (u d)(h l)[p 1 p 4 p 2 p 3 ], supplemented by the condition p 1 + p 2 + p 3 + p 4 = 1.

45 Numerical Experiments We now investigate how the exercise threshold varies with the different model parameters.

46 Numerical Experiments We now investigate how the exercise threshold varies with the different model parameters. The fixed parameters are I = 1, r = 0.04, T = 10 µ 1 = 0.115, σ 1 = 0.25, S 0 = 1 σ 2 = 0.2, V 0 = 1

47 Numerical Experiments We now investigate how the exercise threshold varies with the different model parameters. The fixed parameters are I = 1, r = 0.04, T = 10 µ 1 = 0.115, σ 1 = 0.25, S 0 = 1 σ 2 = 0.2, V 0 = 1 Given these parameters, the CAPM equilibrium expected rate of return on the project for a given correlation ρ is ( ) µ1 r µ 2 = r + ρ σ 2. (4) σ 1

48 Numerical Experiments We now investigate how the exercise threshold varies with the different model parameters. The fixed parameters are I = 1, r = 0.04, T = 10 µ 1 = 0.115, σ 1 = 0.25, S 0 = 1 σ 2 = 0.2, V 0 = 1 Given these parameters, the CAPM equilibrium expected rate of return on the project for a given correlation ρ is ( ) µ1 r µ 2 = r + ρ σ 2. (4) The difference δ = µ 2 µ 2 is the below equilibrium rate of return shortfall and plays the role of a dividend rate paid by the project, which we fix at δ = σ 1

49 Known Thresholds In the limit ρ ±1 (complete market), the closed form expression for the investment threshold obtained in the case T = gives V DP = 2.

50 Known Thresholds In the limit ρ ±1 (complete market), the closed form expression for the investment threshold obtained in the case T = gives V DP = 2. This should be contrasted with the NPV criterion (that is, invest whenever the net present value for the project is positive) which in this case gives V NPV = 1.

51 Known Thresholds In the limit ρ ±1 (complete market), the closed form expression for the investment threshold obtained in the case T = gives V DP = 2. This should be contrasted with the NPV criterion (that is, invest whenever the net present value for the project is positive) which in this case gives V NPV = 1. The limit γ 0 in our model corresponds to the McDonald and Siegel (1986) threshold, obtained by assuming that investors are averse to market risk but neutral towards idiosyncratic risk.

52 Known Thresholds In the limit ρ ±1 (complete market), the closed form expression for the investment threshold obtained in the case T = gives V DP = 2. This should be contrasted with the NPV criterion (that is, invest whenever the net present value for the project is positive) which in this case gives V NPV = 1. The limit γ 0 in our model corresponds to the McDonald and Siegel (1986) threshold, obtained by assuming that investors are averse to market risk but neutral towards idiosyncratic risk. For our parameters, the adjustment to market risks is accounted by CAPM and this threshold coincides with V DP = 2

53 Dependence with Correlation and Risk Aversion !=0.5!=2!= "=0 "=0.6 "= threshold 1.5 threshold !1! correlation risk aversion

54 Dependence with Volatility !=0!=0.6!= !=0!=0.6!= !=0!=0.6!= threshold threshold threshold volatility volatility volatility

55 Dependence with Dividend Rate threshold 8 6 "=0 "=0.6 "= ! !

56 Dependence with Time to Maturity 1.5 Low risk aversion!= Higher risk aversion!= "=0 "=0.6 "= threshold "=0 "=0.6 "=0.9 threshold time to maturity time to maturity

57 Values for the option to invest !=0!=0.99 (V!I) + option value V Figure: Option value as a function of underlying project value. The threshold for ρ = 0 is and the one for ρ = 0.99 is

58 Combining options and games For a systematic application of both real options and game theory in strategic decisions, we consider the following rules:

59 Combining options and games For a systematic application of both real options and game theory in strategic decisions, we consider the following rules: 1. Outcomes of a given game that involve a wait and see strategy should be calculated by option value arguments.

60 Combining options and games For a systematic application of both real options and game theory in strategic decisions, we consider the following rules: 1. Outcomes of a given game that involve a wait and see strategy should be calculated by option value arguments. 2. Once the solution for a given game is found on a decision node, its value becomes the pay-off for an option at that node.

61 Combining options and games For a systematic application of both real options and game theory in strategic decisions, we consider the following rules: 1. Outcomes of a given game that involve a wait and see strategy should be calculated by option value arguments. 2. Once the solution for a given game is found on a decision node, its value becomes the pay-off for an option at that node. In this way, option valuation and game theoretical equilibrium become dynamically related in a decision tree.

62 One period expansion option under monopoly Suppose now that a firm faces the decision to expand capacity for a product with uncertain demand: { hy0 with probability p Y 1 = ly 0 with probability 1 p, (5) correlated with a traded asset

63 One period expansion option under monopoly Suppose now that a firm faces the decision to expand capacity for a product with uncertain demand: { hy0 with probability p Y 1 = ly 0 with probability 1 p, (5) correlated with a traded asset The expansion requires a sunk cost I.

64 One period expansion option under monopoly Suppose now that a firm faces the decision to expand capacity for a product with uncertain demand: { hy0 with probability p Y 1 = ly 0 with probability 1 p, (5) correlated with a traded asset The expansion requires a sunk cost I. The state of the firm after the investment decision at time t k is { 1 if the firm invests at time tk x(k) = (6) 0 if the does not invest at time t k

65 One period expansion option under monopoly Suppose now that a firm faces the decision to expand capacity for a product with uncertain demand: { hy0 with probability p Y 1 = ly 0 with probability 1 p, (5) correlated with a traded asset The expansion requires a sunk cost I. The state of the firm after the investment decision at time t k is { 1 if the firm invests at time tk x(k) = (6) 0 if the does not invest at time t k The cash flow per unit demand for the firm is denoted by D x(k).

66 The NPV solution If no expansion occurs at time t 0, then the value of the project at time t 0 is v out = D 0 Y 0 + g(d 0 hy 0, D 0 ly 0 ) = D 0 Y 0 + π 0 (D 0 Y 1 ).

67 The NPV solution If no expansion occurs at time t 0, then the value of the project at time t 0 is v out = D 0 Y 0 + g(d 0 hy 0, D 0 ly 0 ) = D 0 Y 0 + π 0 (D 0 Y 1 ). If expansion occurs, then the value of the project at time t 0 is v in = (D 1 Y 0 I ) + g(d 1 hy 0, D 1 ly 0 ) = D 1 Y 0 + π 0 (D 1 Y 1 ).

68 The NPV solution If no expansion occurs at time t 0, then the value of the project at time t 0 is v out = D 0 Y 0 + g(d 0 hy 0, D 0 ly 0 ) = D 0 Y 0 + π 0 (D 0 Y 1 ). If expansion occurs, then the value of the project at time t 0 is v in = (D 1 Y 0 I ) + g(d 1 hy 0, D 1 ly 0 ) = D 1 Y 0 + π 0 (D 1 Y 1 ). textcolorredif the decision needs to be taken at time t 0, then according to NPV the firm should expand provided v in v out, that is, if the sunk cost I is smaller then I NPV = (D 1 D 0 )Y 0 + (π 0 (D 1 Y 1 ) π 0 (D 0 Y 1 )). (7)

69 The RO solution By contrast, if the decision to invest can be postponed until time t 1, then the value of the project when no investment occurs at time t 0 is v wait = D 0 Y 0 + π 0 (C 1 ), where C 1 denotes the random variable C 1 = C 1 (Y 1 ) = max{d 0 Y 1, D 1 Y 1 I } D 0 Y 1.

70 The RO solution By contrast, if the decision to invest can be postponed until time t 1, then the value of the project when no investment occurs at time t 0 is v wait = D 0 Y 0 + π 0 (C 1 ), where C 1 denotes the random variable C 1 = C 1 (Y 1 ) = max{d 0 Y 1, D 1 Y 1 I } D 0 Y 1. Accordingly, the firm should invest at time t 0 provided v in v wait, that is, if the sunk cost is smaller than I RO = (D 1 D 0 )Y 0 + (π 0 (D 1 Y 1 ) π 0 (C 1 )). (8)

71 The RO solution By contrast, if the decision to invest can be postponed until time t 1, then the value of the project when no investment occurs at time t 0 is v wait = D 0 Y 0 + π 0 (C 1 ), where C 1 denotes the random variable C 1 = C 1 (Y 1 ) = max{d 0 Y 1, D 1 Y 1 I } D 0 Y 1. Accordingly, the firm should invest at time t 0 provided v in v wait, that is, if the sunk cost is smaller than I RO = (D 1 D 0 )Y 0 + (π 0 (D 1 Y 1 ) π 0 (C 1 )). (8) Since the function g is non-decreasing in each of its arguments, I NPV I RO = π 0 (C 1 ) π 0 (D 0 Y 1 ) 0. (9)

72 The RO solution By contrast, if the decision to invest can be postponed until time t 1, then the value of the project when no investment occurs at time t 0 is v wait = D 0 Y 0 + π 0 (C 1 ), where C 1 denotes the random variable C 1 = C 1 (Y 1 ) = max{d 0 Y 1, D 1 Y 1 I } D 0 Y 1. Accordingly, the firm should invest at time t 0 provided v in v wait, that is, if the sunk cost is smaller than I RO = (D 1 D 0 )Y 0 + (π 0 (D 1 Y 1 ) π 0 (C 1 )). (8) Since the function g is non-decreasing in each of its arguments, I NPV I RO = π 0 (C 1 ) π 0 (D 0 Y 1 ) 0. (9) That is, according to RO, the firm is less likely to expand at time t 0.

73 One period duopoly Consider two firms L and F each operating a project with an option to re-invest at cost I and increase cash flow according to the uncertain demand Y.

74 One period duopoly Consider two firms L and F each operating a project with an option to re-invest at cost I and increase cash flow according to the uncertain demand Y. Let D xi (t m)x j (t m) denote the cash flow per unit of demand of firm i.

75 One period duopoly Consider two firms L and F each operating a project with an option to re-invest at cost I and increase cash flow according to the uncertain demand Y. Let D xi (t m)x j (t m) denote the cash flow per unit of demand of firm i. Assume that D 10 > D 11 > D 00 > D 01.

76 One period duopoly Consider two firms L and F each operating a project with an option to re-invest at cost I and increase cash flow according to the uncertain demand Y. Let D xi (t m)x j (t m) denote the cash flow per unit of demand of firm i. Assume that D 10 > D 11 > D 00 > D 01. We say that there is FMA is (D 10 D 00 ) > (D 11 D 01 ) and that there is SMA otherwise.

77 Equilibrium strategies Lemma Under FMA: 1. If I < IF h, then the equilibrium strategy is (1, 1) for high and low demand. 2. If IF l < I < I F h and I < I L l, then the equilibrium strategy is (1, 1) for high demand and (1, 0) for low demand. 3. If IF h < I < I L l, then the equilibrium strategy is (1, 0) for high and low demand. 4. If IF l < I < I F h and I L l < I, then the equilibrium strategy is (1, 1) for high demand and (0, 0) for low demand. 5. If IL l < I < I L h and I F h < I, then the equilibrium strategy is (1, 0) for high demand and (0, 0) for low demand. 6. If I > I h F, then the equilibrium strategy is (0, 0) for high and low demand.

78 A multi-period investment game Consider two firms L and F each operating a project with an option to re-invest at cost I and increase cash flow according to an uncertain demand dy t = µ(t, Y t )dt + σ(t, Y t )dw.

79 A multi-period investment game Consider two firms L and F each operating a project with an option to re-invest at cost I and increase cash flow according to an uncertain demand dy t = µ(t, Y t )dt + σ(t, Y t )dw. Suppose that the option to re-invest has maturity T, let t m, m = 0,..., M be a partition of the interval [0, T ] and denote by (x L (t m ), x F (t m ) {(0, 0), (0, 1), (1, 0), (1, 1)} the possible states of the firms after a decision has been at time t m.

80 A multi-period investment game Consider two firms L and F each operating a project with an option to re-invest at cost I and increase cash flow according to an uncertain demand dy t = µ(t, Y t )dt + σ(t, Y t )dw. Suppose that the option to re-invest has maturity T, let t m, m = 0,..., M be a partition of the interval [0, T ] and denote by (x L (t m ), x F (t m ) {(0, 0), (0, 1), (1, 0), (1, 1)} the possible states of the firms after a decision has been at time t m. Let D xi (t m)x j (t m) denote the cash flow per unit of demand of firm i.

81 A multi-period investment game Consider two firms L and F each operating a project with an option to re-invest at cost I and increase cash flow according to an uncertain demand dy t = µ(t, Y t )dt + σ(t, Y t )dw. Suppose that the option to re-invest has maturity T, let t m, m = 0,..., M be a partition of the interval [0, T ] and denote by (x L (t m ), x F (t m ) {(0, 0), (0, 1), (1, 0), (1, 1)} the possible states of the firms after a decision has been at time t m. Let D xi (t m)x j (t m) denote the cash flow per unit of demand of firm i. Assume that D 10 > D 11 > D 00 > D 01.

82 A multi-period investment game Consider two firms L and F each operating a project with an option to re-invest at cost I and increase cash flow according to an uncertain demand dy t = µ(t, Y t )dt + σ(t, Y t )dw. Suppose that the option to re-invest has maturity T, let t m, m = 0,..., M be a partition of the interval [0, T ] and denote by (x L (t m ), x F (t m ) {(0, 0), (0, 1), (1, 0), (1, 1)} the possible states of the firms after a decision has been at time t m. Let D xi (t m)x j (t m) denote the cash flow per unit of demand of firm i. Assume that D 10 > D 11 > D 00 > D 01. We say that there is FMA is (D 10 D 00 ) > (D 11 D 01 ) and that there is SMA otherwise.

83 Derivation of project values (1) Let V (x i (t m 1 ),x j (t m 1 )) i (t m, y) denote the project value for firm i at time t m and demand level y.

84 Derivation of project values (1) Let V (x i (t m 1 ),x j (t m 1 )) i (t m, y) denote the project value for firm i at time t m and demand level y. Denote by v (x i (t m),x j (t m)) i (t m, y) the continuation values: v (1,1) (1,1) g(v i (t m+1, y u ), (V (1,1) i (t m+1, y d )) i (t m, y) = D 11 y t + e r t v (1,0) (1,0) g(v L (t m+1, y u ), (V (1,0) L (t m+1, y d )) L (t m, y) = D 10 y t + e r t v (0,1) (0,1) g(v L (t m+1, y u ), (V (0,1) L (t m+1, y d )) L (t m, y) = D 01 y t + e r t v (1,0) (1,0) g(v F (t m+1, y u ), (V (1,0) F (t m+1, y d )) F (t m, y) = D 01 y t + e r t v (0,1) (0,1) g(v F (t m+1, y u ), (V (0,1) F (t m+1, y d )) F (t m, y) = D 10 y t + e r t v (0,0) (0,0) g(v i (t m+1, y u ), (V (0,0) i (t m+1, y d )) i (t m, y) = D 00 y t + e r t

85 Derivation of project values (2) For fully invested firms, the project values are simply given by V (1,1) i (t m, y) = v (1,1) i (t m, y).

86 Derivation of project values (2) For fully invested firms, the project values are simply given by V (1,1) i (t m, y) = v (1,1) i (t m, y). Now consider the project value for firm F when L has already invested and F hasn t: V (1,0) F (t m, y) = max{v (1,1) F (t m, y) I, v (1,0) F (t m, y)}.

87 Derivation of project values (2) For fully invested firms, the project values are simply given by V (1,1) i (t m, y) = v (1,1) i (t m, y). Now consider the project value for firm F when L has already invested and F hasn t: V (1,0) F (t m, y) = max{v (1,1) F (t m, y) I, v (1,0) F (t m, y)}. Similarly, the project value for L when F has invested and L hasn t is V (0,1) L (t m, y) = max{v (1,1) L (t m, y) I, v (0,1) L (t m, y)}.

88 Derivation of project values (3) Next consider the project value for L when it has already invest and F hasn t: { V (1,0) v (1,1) L (t m, y) = L (t m, y) if v (1,1) F (t m, y) I > v (1,0) F (t m, y), (t m, y) otherwise. v (1,0) L

89 Derivation of project values (3) Next consider the project value for L when it has already invest and F hasn t: { V (1,0) v (1,1) L (t m, y) = L (t m, y) if v (1,1) F (t m, y) I > v (1,0) F (t m, y), (t m, y) otherwise. v (1,0) L Similarly, the project value for F when it has already invest and L hasn t is { V (0,1) v (1,1) F (t m, y) = F (t m, y) if v (1,1) L (t m, y) I > v (0,1) L (t m, y), (t m, y) otherwise. v (0,0) F

90 Derivation of project values (4) Finally, the project values V (0,0) i are obtained as a Nash equilibrium, since both firms still have the option to invest.

91 Derivation of project values (4) Finally, the project values V (0,0) i are obtained as a Nash equilibrium, since both firms still have the option to invest. The pay-off matrix for the game is Firm F Invest Wait Invest (v (1,1) Firm L L I, v (1,1) F I ) (v (1,0) L I, v (1,0) F ) Wait (v (0,1) L, v (0,1) F I ) (v (0,0) L, v (0,0) F )

92 FMA: dependence on risk aversion. 200 Gamma = 0.01 Rho = Gamma = 0.1 Rho = Project Value 50 0!50!100!150 V00F V00L V11L!I V10L!I V01L Project Value 50 0!50!100!150 V00F V00L V11L!I V10L!I V01L! Demand! Demand 200 Gamma = 1 Rho = Gamma = 2 Rho = Project Value 50 0!50!100!150 V00F V00L V11L!I V10L!I V01L Project Value 50 0!50!100!150 V00F V00L V11L!I V10L!I V01L! Demand! Demand Figure: Project values in FMA case for different risk aversions.

93 FMA: dependence on correlation. 90 VL Gamma = 1 VF Gamma=1 Rho = VL Gamma = 1 VF Gamma = 2 Rho = Project Value Project Value V00F V00L 20!0.8!0.6!0.4! Correlation 30 V00F V00L 20!0.8!0.6!0.4! Correlation 90 VL Gamma = 2 VF Gamma = 1 Rho = Project Value V00F V00L 20!0.8!0.6!0.4! Correlation Figure: Project values in FMA case as function of correlation.

94 SMA: dependence on risk aversion 200 SMA Gamma= SMA Gamma= Project Value 50 0!50!100!150 V00F V00L V11L!I V10L!I V01L Project Value 50 0!50!100!150 V00F V00L V11L!I V10L!I V01L! Demand! Demand 200 SMA Gamma=1 200 SMA Gamma = Project Value 50 0!50!100!150 V00F V00L V11L!I V10L!I V01L! Demand Project Value 50 0!50!100!150 V00F V00L V11L!I V10L!I V01L! Demand Figure: Project values in SMA case for different risk aversions.

95 SMA: dependence on correlation. 200 VL Gamma = 1 VF Gamma = 1 Y0= VL Gamma = 1 VF Gamma = 2 Y0= Project Value 100 Project Value V00F V00L!0.8!0.6!0.4! Correlation 0 V00F V00L!0.8!0.6!0.4! Correlation 200 VL Gamma = 2 VF Gamma = 1 Y0= Project Value V00F V00L!0.8!0.6!0.4! Correlation Figure: Project values in SMA case as function of correlation.

96 SMA x FMA 200 FMA/SMA Gamma= FMA/SMA Gamma= Project Value 50 0!50!100!150 V00F FMA V00L FMA V11L!I V01L V00L SMA V00F SMA Project Value 50 0!50!100!150 V00F FMA V00L FMA V11L!I V01L V00L SMA V00F SMA! Demand! Demand 200 FMA/SMA Gamma=1 200 FMA/SMA Gamma= Project Value 50 0!50!100!150 V00F FMA V00L FMA V11L!I V01L V00L SMA V00F SMA Project Value 50 0!50!100!150 V00F FMA V00L FMA V11L!I V01L V00L SMA V00F SMA! Demand! Demand Figure: Project values for FMA and SMA.