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Lecture: Continuous Time Models with Investment Applications Simon Gilchrist Boston Univerity and NBER EC 745 Fall, 2013 Brownian Motion Brownian motion (Wiener process): Continous time stochastic process with three properties: Markov process: probability distribution for all future values depends only on its current value. Independent increments: probability distribution for the change in the process is independent of any other non-overlapping time interval. Changes in the process over any finite interval are normally distributed witha variance that increases linearly with the time interval. Formal Definition If z(t) is a wiener process then any change in z, ∆z corresponding to a time interval ∆t satisfies the following conditions: √ ∆z = εt ∆t εt ∼ N (0, 1) E(εt εt+s ) = 0 f or t 6= s Intution: Change in z(t) over a finite interval T . Divide T into n = T /∆t : ∆z = z(s + T ) − z(s) = n X √ εi ∆t i=1 E(∆z) = 0 V (∆z) = n∆t = T Brownian motion with drift Brownian motion with drift: dx = αdt + σdz where dz is a Wiener process. Over any finite interval ∆t, ∆x is distributed normal with E(∆x) = α∆t, V (∆x) = σ 2 ∆t. Random walk representation of Brownian motion: Show that dx is the limit of a discrete time random walk with drift. Suppose ∆x = ∆h = −∆h with prob p with prob q = 1 − p then E∆x = (p − q)∆h and V (x) = E(∆x2 ) − E(∆x)2 = (1 − (p − q)2 )∆h2 = 4pq∆h2 Binomial distribution Let a time interval t have n = t/∆t discrete steps, then xt − xo is a serise of n independent trials with ∆h a success occurring with prob p and −∆h a failure, occurring with prob (1 − p) = q. So xt − xo has a binomial distribution with: E(xt − xo ) = n(p − q)∆h = t(p − q)∆h/∆t and V (xt − xo ) = n ∗ ((1 − (p − q)2 )∆h2 = 4pqt∆h2 /∆t Random walk representation of Brownian motion: Choose ∆h, p, q so that mean and variance of xt − xo depends only on t and not on step-size ∆t or jump ∆h : √ ∆h = σ ∆t p= α√ 1 α√ 1 1+ ∆t //q = 1− ∆t 2 σ 2 σ then p−q = α√ α ∆t = 2 ∆h σ σ This implies α ∆h 2 = αt σ 2 ∆t α 2 ∆t σ 2 V (xt − xo ) = t 1 − σ E(xt − xo ) = t and so lim V (xt − xo ) = tσ 2 . ∆t→0 Comments: Brownian motion is limit of discrete time random walk where mean and variance are independent of step-size ∆t and jump ∆h. This limiting process has the property that variance grows linearly per unit of time. For any finite interval, total distance travelled is infinite as ∆t → 0 : |∆x| = ∆h so E |∆x| = ∆h and nE |∆x| = t tσ ∆h =√ →∞ ∆t ∆t Brownian motion is not differentiable in the conventional sense: ∆x ∆h = ∆t ∆t → ∞ so dx/dt does not exist and we cannot compute E(dx/dt). We 1 can compute E(dx) and dt E(dx) however. Ito Processes Generalize brownian motion (Ito processes): dx = a(x, t)dt + b(x, t)dz where dz = wiener process and a(x, t), b(x, t) are non-random function of state. E(dx) = a(x, t)dt so a(x, t) is instantaneous rate of drift. Instantaneous variance: V (dx) = E(dx2 ) − E(dx)2 = a(x, t)2 dt2 + 2E((a(x, t)b(x, t)dtdz) + b(x, t)2 var(dz) The first two terms are of order dt2 and dt3/2 so that V (dx) = b(x, t)2 var(dz) = b(x, t)2 dt Example 1: Geometric brownian motion: Let dx = αxdt + σxdz If x is a geometric brownian motion then F (x) = ln(x) is brownian motion with drift: dF = (α − 0.5σ 2 )dt + σdz This implies that ln(xt /xo ) ∼ N α − 0.5σ 2 t, σ 2 t Using properties of the log-normal we have E(xt ) = xo eαt 2 V (xt ) = x2o e2αt (eσ t − 1) Present values Also we have the present value expression: Z ∞ Z ∞ −rt E x(t)e dt = xo e−(r−t) dt o o = xo r−α The drift rate α can be interpreted as the dividend growth rate. Example 2: Continous time AR1 (Ornstein-Uhlenbeck) Let dx = η(µ − x)dt + σdz Then E(xt ) = µ + (xo − µ)e−ηt → µ as t → ∞ V (xt ) = σ2 σ2 (1 − e−2ηt ) → as t → ∞ 2η 2η If η → ∞ x is a constant. Need to adjust both σ, η to vary the degree of mean reversion. Ito’s Lemma: Ito process is continuous but not differentiable. What about functions of x, F (x, t) where dx = a(x, t)dt + b(x, t)dz Consider taylor-series expansion of F (x, t) (ignore higher order derivatives of t): dF = ∂F 1 ∂2F 2 1 ∂3F 3 ∂F dt + dt + dx + dx + .. ∂x ∂t 2 ∂x2 6 ∂x3 We want all terms of order dt : dx is of order dt (dx)2 = b(x, t)2 dt +higher order terms Ito’s Lemma This implies dF dF ∂F ∂F 1 ∂2F 2 dt + dt + dx 2 ∂x ∂t 2 ∂x 2 ∂F 1 ∂ F ∂F + a(x, t) + b(x, t)2 dt = ∂t ∂x 2 ∂x2 ∂F + b(x, t) dz ∂x = Taking expectations we have: 2 ∂F ∂F 1 2 ∂ F + a(x, t) E(dF ) = + b(x, t) dt ∂t ∂x 2 ∂x2 2 Because of uncertainty the term 12 b(x, t)2 ∂∂xF2 is of first-order. I.e. owing to Jensen’s inequality, if the function is concave at x uncertainty lowers the value of dF . Example: Geometric brownian motion Let dx = αxdt + σxdz Let F (x) = ln(x) 2 1 ∂F ∂F 2 ∂ F + b(x, t) dt + b(x) dz dF = a(x) ∂x 2 ∂x2 ∂x −1 1 1 = α + σ 2 x2 dt + σx dz 2 2 x x 1 = α − σ 2 dt + σdz 2 The log of geometric brownian motion is a brownian motion with drift. Dynamic programming in continuous time: Start with discrete time problem: F (x, t) = max{π(x, u, t)∆t + u 1 E F (x0 , t + ∆t)|x, u 1 + ρ∆t where π(x, u, t) is flow profit given state x and policy u. Rearrange to get ρ∆tF (x, t) = max{π(x, u, t)∆t+E F (x0 , t + ∆t) − F (x, t)|x, u u Divide by ∆t and take limit as ∆t → 0 ρF (x, t) = max{π(x, u, t) + u 1 0 0 E F (x , t)|x, u } dt 1 Suppose x follows an Ito process: dx = a(x, u, t)dt + b(x, u, t)dz then up to o(∆t) E F (x0 , t + ∆t) − F (x, t)|x, u = [Ft (x, t) + a(x, u, t)Fx (x, t) 1 2 b (x, u, t)Fxx (x, t)]∆t + 2 We now have that the return equation satisifies: ρF (x, t) = max{π(x, u, t) + Ft (x, t) + a(x, u, t)Fx (x, t) u + 1 2 b (x, u, t)Fxx (x, t)} 2 Hamilton-Jacobi-Bellman equation If there is an infinite horizon and a() and b() don’t depend explicitly on time then the value F (x) satisfies the ordinary differential equation: 1 ρF (x) = max{π(x, u) + a(x, u)F 0 (x) + b2 (x, u)F 00 (x)} u 2 This is the continuous time equivalent of the Bellman equation. Optimal stopping problem: Discrete time Let π(x) denote flow profit of a machine. Let Ω(x) denote the terminal payoff. Assume that π(x) − ρ 1+ρ Ω(x) Φ(x0 |x), is increasing in x. Assume that the distribution function is first-order stochastic dominant (i.e. an increase in x shifts the probability distribution of x0 to the right) – example AR1, Random walk. Optimal policy Value function: 1 E F (x0 |x F (x) = max Ω(x); π(x) + 1+ρ Solution: stop if x < x∗ for some value x∗ to be determined. Optimal stopping problem in continuous time Assume Ito process for x : dx = a(x)dt + b(x)dz Profit relative to flow value of terminal payoff π(x) − ρΩ(x) is increasing in x. Return function: F (x) = max Ω(x); π(x) + 1 E [F (x + dx|x] 1 + ρdt Solution: stop if x < x∗ for some value x∗ to be determined. Value on continuation region If x > x∗ the return function satisfies ρF (x) = π(x) + 1 E(dF ) dt which implies: 1 ρF (x) = π(x) + a(x)F 0 (x) + b2 (x)F 00 (x) f or x > x∗ 2 Because x∗ is endogenous, we need two boundary conditions to solve this differential equation. Optimality conditions Value matching: F (x∗ ) = Ω(x∗ ) Smooth pasting: Fx (x∗ ) = Ωx (x∗ ) Suppose Ω(x) = 0. At boundary: 1 0 = π(x∗ ) + b2 (x∗ )F 00 (x∗ ) 2 The solution implies wait until π(x) ≤ π(x∗ ) < 0 before stopping (t is worthwile to incur some loss before closing down the machine). Heuristic argument for smooth pasting: Suppose Fx < Ωx at x∗ . We have upward kink. Then there exists an x∗∗ > x∗ such that Ω(x∗∗ ) > F (x∗∗ ) and we should stop at x∗∗ . Suppose Fx > Ωx at x∗ . We have downward kink. Then payoff is convex at optimum and there is value to waiting and determining what the realized value of x will be. Smooth pasting – slightly less heuristic Assume a = 1, b = 1, Over the interval ∆t, x rise by ∆h with √ p = 1/2[1 + falls by ∆h with prob q = 1/2[1 − ∆t]. √ ∆t] and Consider the alternative policy of waiting until ∆t to take action. Return from waiting Return is G = π(x∗ )∆t + 1 [pF (x∗ + ∆h) + qΩ(x∗ − ∆h)] 1 + ρ∆t = π(x∗ )∆t + 1 [pF (x∗ ) + Fx (x∗ )∆h) + q (Ω(x∗ ) + Ωx (x∗ )∆h)] 1 + ρ∆t +higherorder terms Let ∆t → 0 but recognize that ∆t is of order ∆h2 and goes to zero faster than ∆h. Apply value matching condition, we get 1 G = F (x∗ ) + [Fx (x∗ ) − Ωx (x∗ )]∆h > F (x∗ ) 2 Option to invest Assume value of a project evolves according to geometric brownian motion (log-normal dividends): dV = αV dt + σV dz Project manager can pay I to exercise an option and get V . Let F (V ) denote the value of the investment opportunity: F (V ) = max E (VT − I)e−ρT T Here Vt denotes the payoff to investing at t. Assume α < ρ (otherwise wait forever). Deterministic case: (σ = 0) Value of payoff, given initial value Vo : V (t) = Vo eαt Value of investing at time T is therefore: F (V ) = (V eαT − I)e−ρT Suppose α < 0. In this case, invest now if V > I. Otherwise, never invest. Suppose 0 < α < ρ. F (V ) > 0 even if V < I since V is growing exponentially. Optimality First-order-condition: dF (V ) = −(ρ − α)V e−(ρ−α)T + ρIe−ρT = 0 dT Invest now (set T = 0) if V >V∗ = Suppose ρ I>I ρ−α ρ I>V >I ρ−α Project has positive net-present value at T = 0 but you should still wait to invest. Intuition: cost of investing discounted at higher rate (ρ) than benefit, discounted at ρ − α. Solution Solution is therefore 1 ρI T = max ln ,0 α (ρ − α) V ∗ and (ρ − α) V ρ/α αI f or V < V ∗ F (V ) = ρ−α ρI = V − I f or V > V ∗ Stochastic case:(σ > 0). When is it optimal to invest I in return for an asset worth V ? Assume V follows geometric brownian motion: dV = αV dt + σV dz Investment rule: optimal cutoff V > V ∗ Continuation region: Value of investment project determined by capital gain. ρF dt = E (dF ) Continuation value Apply Ito’s Lemma: 1 dF = F 0 (V )dV + F 00 (V )(dV )2 2 V is Brownian motion: 1 dF = αV F 0 (V )dt + σV F 0 (V )dz + σ 2 V 2 F 00 (V )dt 2 Take expectations: 1 E(dF ) = αV F 0 (V )dt + σ 2 V 2 F 00 (V )dt 2 Bellman’s equation holds in the continuation region: 1 ρF (V ) = αV F 0 (V ) + σ 2 V 2 F 00 (V ) 2 Solution We are looking for a solution to the differential equation 1 ρF (V ) = αV F 0 (V ) + σ 2 V 2 F 00 (V ) 2 which is satisfied at V > V ∗ . Because V ∗ is endogenous, we have a free-boundary problem. Boundary conditions Value matching: F (V ∗ ) = V ∗ − I Smooth pasting F 0 (V ∗ ) = 1 We also have F (0) = 0 i.e. if V = 0 , with geometric brownian motion it remains at zero. Waiting to invest Rewriting the value matching condition we have V ∗ − F (V ∗ ) = I This implies that manager will wait to invest, even if the project has positive net present value. Reasons: Dividend growth (as in non-stochastic case) Uncertainty – higher uncertainty raises the option value F (V ∗ ) and delays investment. Explicit solution: Guess: F (V ) = AV β1 Value matching implies AV ∗β1 = V − I Smooth pasting implies β1 AV ∗β1 −1 = 1 Combine these we get V∗ = β1 I β1 − 1 A= V∗−I β V∗ 1 and Solving for coefficients We now need to solve for β 1 . Plug guess into differential equation. Let δ = ρ − α. Equation is satisified if β is a root of 1 2 σ β(β − 1) + (ρ − δ) β − ρ = 0 2 There are two roots to this equation. The positive root satisfies s 1 (ρ − δ) (ρ − δ) 1 2 ρ β1 = − + − +2 2 >1 2 2 2 σ σ 2 σ Comparative statics: β1 is decreasing in σ, we have that V ∗ is increasing in σ – as uncertainty increase, we wait longer to invest. (The wedge between V ∗ and I increases.) β1 is increasing in δ, we have that V ∗ is decreasing in δ – as the growth adjusted discount for profits increases, we invest sooner. β1 is decreasing in ρ (holding δ constant). The more we discount costs relative to growth-adjusted benefits, we wait longer to invest. Limiting behavior: As σ → ∞, V ∗ → ∞ As σ → 0, if α > 0 β1 → ρ ρ and V ∗ → I > I ρ−δ δ As σ → 0, if α ≤ 0 β1 → ∞ and V ∗ → I Implications for user cost: Assume profit flow of machine is geometric brownian motion: dπ = απdt + σπdz Value of profit stream is Z Vt = E ∞ t πs e−ρ(s−t) ds = πt ρ−α Investment rule is πt > π ∗ = β (ρ − α) I > (ρ − α) I β−1 Quadratic equation implies β 1 (ρ − α) I = ρ + σ 2 β1 β−1 2 Critical value of profits satisfies: 1 π ∗ = ρ + σ 2 β1 I > ρI 2 Comments: Uncertainty increases the hurdle rate – the effective user cost of capital that should be applied when evaluating a given project. With no uncertainty, the user cost is ρ and does not depend on α. In other words, although without uncertainty, we may still wait to invest, our decision is based on standard user cost arguments – i.e. waiting for the flow value of profits to exceed the flow cost of the investment. Abel and Eberly: Generalized adjustment cost framework to include fixed costs and irreversibility. Operating profit: π(kt , εt ) where kt is current capital and εt is a random shock to profits. Assume πk > 0, πkk ≤ 0. Shock process: dεt = µ(εt )dt + σ(εt )dz where z is brownian motion. Capital accumulation: dkt = (It − δkt )dt Investment costs: Purchase-sale costs Pk− I(I < 0) + Pk+ I(I ≥ 0), Pk+ > Pk− Adjustment costs: continous, strictly convex, twice differentiable. Minimized at I = 0. Fixed costs: – non-negative if I 6= 0 Augmented adjustment cost function: Express the augmented adjustment cost function as vC(I, K) where v = 1 if I 6= 0 = 0 if I = 0 Limits: lim C(I, K) = lim C(I, K) = C(0, K) I→0− I→0+ where C(0, K) is fixed cost. Also: CI (0, K)+ ≥ 0 and CI (0, K)+ ≥ CI (0, K)− Value of the firm: Hamilton-Jacobi-Bellman equation: 1 rV (K, ε) = max π(k, ε) − vc(I, k) + E(dV ) I,v dt Taylor expansion: 1 1 dV = Vk dk + Vε dε + Vkk (dk)2 + Vεε (dε)2 + Vkε dkdε + .. 2 2 Note that dk = (I − δK)dt so that dk 2 = o(dt) Also (dε)2 = σ 2 (ε) dt + o(dt) Expected firm value This implies 1 dV = Vk (I − δK)dt + Vε (µ(ε)dt + σ(ε)dz) + Vεε σ 2 (ε)dt 2 Taking expectations: 1 2 EdV = Vk (I − δK) + Vε µ(ε) + Vεε σ (ε) dt 2 Bellman’s equation Let q = Vk then 1 2 rV = max π(k, ε) − vc(I, k) + q(I − δk) + µ(ε)Vε + σ (ε)Vεε I,v 2 Bellman’s equation says that we can choose I, v to solve max [qI − vC(I, k)] I,v Optimal investment First consider v = 1. Let Ψ(q, k) = max [qI − c(I, k)] I Let I ∗ (q, k) satisfy: CI (I ∗ (q, k), k) = q f or q < CI (0, k)− or q > CI (0, k)+ I ∗ (q, k) = 0 f or CI (0, k)− < q < CI (0, k)+ Comments: CII > 0 implies that I ∗ (q, k) is strictly increasing in q over range of action. If C(I, k) differentiable at I = 0, CI (I ∗ (q, k), k) = 0 f or all q If C(I, k) non-differentiable at zero, we have a range of inaction. I ∗ (q, k) < 0 if q < CI (0, k)− = 0 if CI (0, k)− < q < CI (0, k)+ > 0 otherwise These results imply that I ∗ (q, k) is non-decreasing in q. Optimal choice of v If v = 0, I = 0, and qI − v(CI, k) = 0. If v = 1, Ψ(q, k) = qI ∗ (q, k) − C(I ∗ (q, k), k). Look at shape of Ψ(q, k) : Ψq (q, k) = I ∗ (q, k) < 0 if q < CI (0, k)− = 0 if CI (0, k)− < q < CI (0, k)+ = I ∗ (q, k) > 0 otherwise Also Ψqq (q, k) = Iq∗ (q, k) > 0 in action range Result: Ψ(q, k) is convex in q and attains minimum on interval CI (0, k)− < q < CI (0, k)+ . Optimal policy: Let q1 ≤ q2 be roots of Ψ(q, k). Optimal policy is then: b k) = I ∗ (q, k) < 0 if q < q1 I(q, b k) = 0 if q1 < q < q2 I(q, b k) = I ∗ (q, k) > 0 if q > q2 I(q, Possible cases: Unique root: occurs with no fixed costs and differentiability. Implies no range of inaction. Exactly two roots: only occurs with fixed cost. Implies range of inaction. Continuum of roots: occurs if there is no fixed cost but non-differentiability. Implies Range of inaction. Comments: Range of inaction depends on adjustment costs not π function or ε process. If there are fixed costs or non-differentiability of C(0, k) we have a non-degenerate range of inaction. b k) will have a With fixed costs the optimal policy I(q, discontinuity. Solving for q: Differentiate Bellman equation w.r. to k : 1 2 b k)−δq+qk (I−δk)+µ(ε)V b rVk = πk (k, ε)−b v Ck (I, ε,k + σ (ε)Vεεk 2 Get E(dq) using Ito’s Lemma: 1 E(dq) = qk (Ib − δk)dt + µ(ε)qε dt + σ 2 (ε)qεε dt 2 Here we use fact that q = Vk so qε = Vkε , qεε = Vεεk . From Bellman equation we now have: b k) + (r + δ)q = πk (k, ε) − vbCk (I, 1 E(dq) dt i.e. required return on the marginal unit of capital equals marginal product plus expected capital gain. Solution Lemma: Suppose xt is a diffusion process and a > 0 then Z ∞ xt = Et gt+s e−as ds 0 is a solution to 1 Et dx − axt + gt = 0 dt This implies Z ∞h i πk (kt+s , εt+s ) − vbt+s Ck (Ibt+s , kt+s ) e−(r+δ)s ds qt = Et 0 So qt is the expected present discounted value of the marginal return to capital. Relating q to observables: If π, vC(I, k) are linearly homogenous then qo = Vo /ko To show this, consider 1 dk 1 E(d(qk)) = E(dq)k + q dt dt dt = b k) k + q(I − δk) (r + δ)q − πk + vbCk (I, Apply linear homogeneity to get 1 b k) + q Ib − vbCI Ib E(d(qk)) = rq − (π − vbC(I, dt If I > 0, v = 1, q = CI . If I = 0, v = 0. so last term is zero. 1 b k) E(d(qk)) = rq − (π − vbC(I, dt Again apply the lemma Z ∞h i qo ko = Et π(kt+s , εt+s ) − vbt+s C(Ibt+s , kt+s ) e−rs ds = Vo 0 Example: Linear homogeneity and fixed cost = bK. Assume C(I, k) = kc(I/k, 1) = kG(I/k) Then I/k = G0−1 (q) < 0 if q ≤ q1 = 0 if q1 < q < q2 = G0−1 (q) > 0 ιf q ≥ q2 Example of q : π(k, p) = max pLα K 1−α − wL = hpθ k L where h = (1 − α)αα/1−α w−α/(1−α) > 0, θ = then Z qt = hEt 0 ∞ pθt+s e−(r+δ)s ds 1 >1 1−α Specific example Suppose dp = σpdz which implies ln(pt+s /pt ) ∼ N (−0.5σ 2 s, σ 2 s) and Et (pθt+s ) = pθt exp(1/2(θ (θ − 1) σ 2 s so that qt = hpθt . r + δ − 0.5θ (θ − 1) σ 2 In this case, as σ increases, Tobin’s Q will increase. So will investment. Intuition Profit functions are convex in prices. Although the total profit function is linearly homogenous, given a quasi-fixed factor capital, the flexibility of labor implies convexity with respect to prices. Thus, a mean preserving spread to the price implies higher variable profits and therefore more investment. Comment: this is true even if there are irreversibilties and fixed costs to investing. Why? In this model, fixed costs are flow fixed costs, i.e. whenever investment is non-zero, you must pay the cost. There isn’t really an option to invest aspect to the model. (Hence range of inaction does not depend explicitly on the stochastic process ε). This model does nicely illustrate the point made by Hartman, Abel and others that investment may increase with uncertainty owing to the fact that profit functions are convex in prices however. References: Dixit, A. and Pindcyk, Investment under uncertainty, Princeton Press. 1994. Chapters 2-5. Abel, Andrew and Janice Eberly, “A unified model of investment under uncertainty”, AER, 1994. Pindyck, Robert, “Irreversibility, Uncertainty and Investment”, Journal of Economic Literature, Vol XXIX, 1991, 1110-1148. Sodal, Sigbjorn, “A simplified exposition of smooth pasting”, Economic Letters, 58, 1998, 217-223.