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BOOMS, RECESSIONS AND FINANCIAL TURMOIL: A FRESH LOOK AT INVESTMENT DECISIONS UNDER CYCLICAL UNCERTAINTY Yu-Fu Chen Economic Studies School of Social & Environmental Sciences University of Dundee Michael Funke Department of Economics Hamburg University June 2009 Hamburg University 1 Department of Economics Michael Funke 1. Introduction Referring to the current financial crisis, Blanchard (2009) offers the following observation. “Crisis feeds uncertainty. And uncertainty affects behaviour, which feeds the crisis. Were a magic wand to remove uncertainty, the next few quarters would still be tough (…), but the crisis would largely go away”. Hamburg University 2 Department of Economics Michael Funke 2. A Model of (Partially) Irreversible Investment The Cobb-Douglas production function is given by (1) Yt K t N 1, 0 1, where K is the capital stock, N is the constant employment level, and is a parameter determining the shares between capital and labour in production. Hamburg University 3 Department of Economics Michael Funke It is assumed that the firm faces a stochastic isoelastic demand function (2) p Yt 1 Z t , 1, where p denotes the price, Z denotes the random demand shock, and is an elasticity parameter that takes its minimum value of 1 in a perfectly competitive environment. Hamburg University 4 Department of Economics Michael Funke Therefore, current profits, measured in units of output, are defined as (3) Z t K t1 N 2 Cost I t xK t wN , where 1 and 2 1 , I t is gross investment, x denotes constant service expenses for capital, w is the constant real wage, and Cost() are the total investment expenditures denoted by the following functions: 1 2 p K I t 2 I t for I t 0 (4) Cost I t 0 for I t 0 1 p K I t I t2 for I t 0. 2 Hamburg University 5 Department of Economics Michael Funke Purchase (resale) costs are the costs of buying (selling) capital. We assume that p K p K 0 . Adjustment costs, I t2 2 , are continuous and strictly convex in I, and is a positive parameter. The adjustment of capital over time is denoted by dKt (5) I t K t , dt where represents the constant depreciation rate. Hamburg University 6 Department of Economics Michael Funke To capture probabilistic state transitions over time, Markov-switching models popularized by Hamilton (1989, 1990) provide an attractive analytical framework. A key step in the modelling stage is the specification of the number of regimes. Before embarking on the modelling exercise, we first try to detect different regimes. In order to determine the number of regimes, we use the Chicago Board Options Exchange (CBOE) volatility indices. Hamburg University 7 Department of Economics Michael Funke Figure 1: CBOE Volatility Indices VIX and VBO 100.00 90.00 80.00 70.00 volatility 60.00 50.00 40.00 30.00 20.00 10.00 0.00 17/12/1996 17/12/1997 17/12/1998 17/12/1999 17/12/2000 17/12/2001 17/12/2002 17/12/2003 17/12/2004 17/12/2005 17/12/2006 17/12/2007 17/12/2008 Hamburg University 8 Department of Economics Michael Funke (1) Although there is no perfect correspondence between cyclical phases and regimes, apparently two regimes (booms vs. recessions) have existed between 1996 and 2007. (2) The depth of the current recession suggests the existence of a third regime indicating episodes of financial turmoil and sharp contractions. Given the evidence in Figure 1, we propose a nonlinear analytical framework with three regimes (two recessionary states and one expansionary state). Hamburg University 9 Department of Economics Michael Funke We assume that the demand process follows the continuous-time stochastic (geometrical Brownian motion) Markov switching processes (6) dZ t i Z t dt i Z t dWt , for i = 0, 1, 2 Where dWt t dt denotes the increments of a standard Wiener process, t is an i.i.d. sequence with mean zero and a standard deviation of unity, . i is the drift parameter, and i2 the variance parameter. It is assumed that if the boom (state 2) occurs, the drift and the variance parameters 2 are and 2 respectively; 2 Hamburg University 10 Department of Economics Michael Funke if the recession state (state 1) emerges, they are 1 and 1 , respectively; 2 and if the financial turmoil state (state 0) occurs, the parameters are given by 0 and 0 , respectively. It is expected that the value of the drift (growth 2 of demand) of the state 2 is higher than the one of the states 1 and 0, i.e. 2 1 0 holds. The corresponding volatility parameters are in the opposite order 0 1 2 . The next aim is to describe the connections between the phases of cycles. We propose the following 3×3 transition matrix providing information on how business cycle phases are related. 1 t 0 t (7) 0 1 t t , 0 t 1t 1 0 t 1t Hamburg University 11 Department of Economics Michael Funke where ( ) denotes the probability of changing from boom state to 0 1 financial turmoil state (recession state). Correspondingly, ( ) represents the transition probability from recession state (financial turmoil state) to boom state. It is assumed that there are no transitions between recession state and the financial turmoil state. This constraint based on simple economic considerations simplifies the analysis significantly. Thus, the firm’s profit-maximisation problem is denoted by: (8) V max E Z t K t N xK t wN Cost I t e dt Z Z 0 , K K 0 1 2 rt It 0 where r is the discount rate. Hamburg University 12 Department of Economics Michael Funke Applying Ito’s Lemma, the stochastic nature of this optimization problem requires the solution to the following Bellman equations for the states 0, 1 and 2: rV0 max ZK 1 N 2 xK wN Cost I V0 K I K I (9) 0 ZV0 Z 0 Z 2V0 ZZ 2 V2 V0 , 2 rV1 max ZK 1 N 2 xK wN Cost I V1K I K I (10) 1 ZV1Z 12 Z 2V1ZZ 2 V2 V1 , Hamburg University 13 Department of Economics Michael Funke (11) rV2 max ZK 1 N 2 xK wN Cost I V2 K I K I 2 ZV2 Z 2 Z 2V2 ZZ 0 V0 V2 1 V1 V2 1 2 2 where V0, V1, and V2 represent the value of the firm in the states 0, 1, and 2 respectively. The nature of the solution of this problem is now intuitive. The investment policy that maximizes profits has a simple and intuitive form: the q-type investment function for I for the states 0, 1 and 2 is denoted by (12) p K I qi , where for i = 0, 1, 2. Hamburg University 14 Department of Economics Michael Funke The solutions for q 0, q1 and, q 2 all consist of particular solutions and gene- ral solutions so that q 0 q 0 q 0 , q1 q1P q1 , q 2 q 2 q 2 . P G G P G 1 1 2 x (13) P q0 a0 ZK N , r x (14) q1P a1 ZK 1 1 N 2 , r x (15) q 2 a3 ZK 1 1 N 2 P , r Hamburg University 15 Department of Economics Michael Funke The general solutions for q 0, q1, and q 2 represent the net value of options and are (16) G q0 Ai ZK i 1 3 1 1 i 6 Ai ZK i 4 1 1 i (17) G q1 Bi ZK i 1 3 1 1 i 6 Bi ZK i 4 1 1 i (18) G q2 3 Ci ZK i 1 1 1 i 6 Ci ZK i 4 1 1 i where 1 2 3 0 4 5 6 and they are the six charac- teristic roots of the following equation for ß. Hamburg University 16 Department of Economics Michael Funke It can be noted that the positive ß terms of (16)-(18) are generally related to investment options and the negative ß terms are linked to disinvestment terms. r 1 1 2 2 1 0 1 r 1 1 0 0 1 1 2 1 2 2 2 (19) r 1 1 1 1 12 1 0 r 1 1 1 1 12 1 2 2 1 r 1 1 0 0 1 . 1 2 2 Hamburg University 17 Department of Economics Michael Funke Note that the relationships between Ai, Bi, and Ci are according to the following equations: r 1 1 0 0 1 1 2 (20) Ai C , i 1,..., 6, 2 i r 1 1 1 12 1 1 (21) B 2 C , i 1,...,6. i i Hamburg University 18 Department of Economics Michael Funke The set of boundary conditions that applies to this optimal stopping problem is composed by the value matching conditions (22) q Z , A , A , A , A , A , A q P Z q G Z , A , A , A , A , A , A p , 0 0 1 2 3 4 5 6 0 0 0 0 1 2 3 4 5 6 K (23) q Z 0 0 , A , A , A , A , A , A q Z q Z , A , A , A , A , A , A p , 1 2 3 4 5 6 P 0 0 G 0 0 1 2 3 4 5 6 K (24) q Z 1 1 , B , B , B , B , B , B q Z q Z , B , B , B , B , B , B p , 1 2 3 4 5 6 P 1 1 G 1 1 1 2 3 4 5 6 K (25) q Z 1 1 , B , B , B , B , B , B q Z q Z , B , B , B , B , B , B p , 1 2 3 4 5 6 P 1 1 G 1 1 1 2 3 4 5 6 K (26) q Z , C , C , C , C , C , C q Z q Z , C , C , C , C , C , C p , P G 2 2 1 2 3 4 5 6 2 21 2 2 1 2 3 4 5 6 K (27) q Z 2 2 , C , C , C , C , C , C q Z q Z , C , C , C , C , C , C p , 1 2 3 4 5 6 P 1 2 G 1 2 1 2 3 4 5 6 K Hamburg University 19 Department of Economics Michael Funke and the corresponding smooth-pasting conditions (28) q 0 Z 0 , A1 , A2 , A3 , A4 , A5 , A6 0, Z 0 q 0 Z 0 , A1 , A2 , A3 , A4 , A5 , A6 0 (29) , Z 0 (30) q1 Z 1 , B1 , B 2 , B3 , B 4 , B5 , B 6 0 , Z 1 (31) q1 Z 1 , B1 , B 2 , B3 , B 4 , B5 , B6 0 , Z 1 (32) q 2 Z 2 , C1 , C 2 , C 3 , C 4 , C 5 , C 6 0 , Z 2 (33) q 2 Z 2 , C1 , C 2 , C 3 , C 4 , C 5 , C 6 0 . Z 2 Making use of the value-matching and smooth-pasting conditions, we get the boundary values that separate the space into two regions: one where it is optimal to exercise the investment option and another where it is not. Hamburg University 20 Department of Economics Michael Funke 3. Model Simulations The unit time length corresponds to one year. Our base parameters which were chosen for realism are 0 = 0.35, 1 = 0.25, 2 = 0.15, 0 = -0.04, . 1 = 0.01, 2 = 0.025, = 0.07, = 1.5, = 0.3, = 0.33, = 0.15, p K 1.0, p K 0.4 , r = 0.05, x = 0.1, K0 = N0 = 1.0, and = 1.50. We set the baseline standard switching probabilities = 0.25, = 0.4, 0 0.02 and 0 0.1, respectively. Hence, the expected duration for booms is (1- 0 - 1 )/( 0+ 1 ) = 0.88/0.12 = 7.3 years. The expected duration of a recession is (1- )/ = 0.6/0.4 = 1.5 years, and the expected duration of a period of financial turmoil is (1- )/ = 0.75/0.25 = 3.0 years. Hamburg University 21 Department of Economics Michael Funke Figure 2: The Impact of Upon the Z Thresholds 2 0.8 Financial turmoil Dis-investment thresholds 1.9 0.7 Invesment tthresholds 1.8 0.6 1.7 0.5 Boom 1.6 0.4 Recession Recession 0.3 Financial turmoil 1.5 1.4 0.2 Boom 0.1 1.3 1.2 0 0.15 0.175 0.2 0.225 0.25 0.15 0.175 0.2 0.225 0.25 Hamburg University 22 Department of Economics Michael Funke Figure 3: The Impact of Upon the Z Thresholds 1.9 0.8 Financial turmoil Dis-investment thresholds 1.8 0.7 Invesment tthresholds 1.7 0.6 1.6 0.5 Boom 1.5 0.4 Recession Financial turmoil 1.4 Recession 0.3 1.3 0.2 1.2 0.1 Boom 1.1 0 0.25 0.3 0.35 0.4 0.45 0.25 0.3 0.35 0.4 0.45 Hamburg University 23 Department of Economics Michael Funke Figure 4: The Impact of 0 Upon the Z Thresholds 1.9 0.8 Dis-investment thresholds 1.8 Financial turmoil 0.7 Invesment tthresholds 1.7 0.6 1.6 0.5 Recession Boom Recession 1.5 0.4 Financial turmoil 1.4 0.3 1.3 Boom 0.2 1.2 0.1 0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2 0 0 Hamburg University 24 Department of Economics Michael Funke Figure 5: The Impact of 1 Upon the Z Thresholds 1.9 0.8 Financial turmoil Dis-investment thresholds 1.8 0.7 Invesment tthresholds 1.7 0.6 1.6 0.5 Boom Recession 1.5 0.4 Recession Financial turmoil 1.4 0.3 1.3 Boom 0.2 1.2 0.1 0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2 1 1 Hamburg University 25 Department of Economics Michael Funke Figure 6: The Impact of 0 Upon the Z Thresholds 2.2 1 0.9 Dis-investment thresholds 2 Financial turmoil 0.8 Invesment tthresholds 0.7 1.8 0.6 0.5 Boom 1.6 0.4 Recession Financial turmoil Recession 0.3 1.4 0.2 Boom 0.1 1.2 0 0.28 0.305 0.33 0.355 0.38 0.28 0.305 0.33 0.355 0.38 0 0 Hamburg University 26 Department of Economics Michael Funke Figure 7: The Impact of 1 Upon the Z Thresholds 2 0.8 1.9 Dis-investment thresholds 0.7 Invesment tthresholds 1.8 Financial turmoil 0.6 1.7 0.5 Boom 1.6 0.4 Financial turmoil Recession 1.5 0.3 Recession 1.4 0.2 Boom 1.3 0.1 1.2 0 0.18 0.205 0.23 0.255 0.28 0.18 0.205 0.23 0.255 0.28 1 1 Hamburg University 27 Department of Economics Michael Funke Figure 8: The Impact of 2 Upon the Z Thresholds 2 0.8 Dis-investment thresholds 1.9 0.7 Financial turmoil Invesment tthresholds 1.8 0.6 1.7 0.5 Boom 1.6 0.4 Recession Recession 0.3 Financial Turmoil 1.5 1.4 0.2 Boom 0.1 1.3 1.2 0 0.125 0.15 0.175 0.2 0.225 0.125 0.15 0.175 0.2 0.225 2 2 Hamburg University 28 Department of Economics Michael Funke Figure 9: The Impact of 2 Upon the Z Thresholds 2 0.8 Financial turmoil Dis-investment thresholds 1.9 0.7 Invesment tthresholds 1.8 0.6 1.7 0.5 Boom 1.6 0.4 Recession 1.5 Recession 0.3 Financial turmoil 1.4 0.2 1.3 Boom 0.1 1.2 0 0.02 0.026 0.032 0.038 0.044 0.02 0.026 0.032 0.038 0.044 2 2 Hamburg University 29 Department of Economics Michael Funke Since the focus of the paper is investment, we next present a translation from thresholds to investment and the capital stock and assess the impact of the three regimes upon investment. In other words, we “reverse engineer” time series for investment and the capital stock from our setup. In this validation stage, we also test the ability of our model to replicate some business cycle characteristics by using numerical simulations of the dynamic system. We specify a sequential iterations method that allows us to generate discrete realizations of the nonlinear dynamical system and investigate the oscillations, given the chosen levels of parameters. Hamburg University 30 Department of Economics Michael Funke Equation (6) is proxied by the following discrete stochastic differential equation – the Euler scheme, (34) Z t t Z t i Z t t i t Z t t , t ~ N 0,1 for i = 0, 1, 2 where the normal random variables, t , are generated via the central limit theorem. As the time passes, the term Z t fluctuates according to the corres- ponding stochastic processes and K will depreciate as long as Z t is staying within the no-action area. Hamburg University 31 Department of Economics Michael Funke If Z t hits the threshold Z 1 in state 1 or the threshold Z 0 in state 0, the firm will invest according to (35) It q i Z t p K qi Z t q1 Z 1 for i = 0, 1, 2 After the level of investment is determined, the corresponding capital stock is computed using the capital accumulation constraint (36) K t 1 K t I t K t , which become the initial value of K for the time t+1, by which the new thresholds are recomputed accordingly for the time t+1. Hamburg University 32 Department of Economics Michael Funke Up to now, we have interpreted the model as applying to a single firm. Suppose that we re-interpret the model at the macroeconomic level, i.e. K and I now represent economy-wide gross investment and the capital stock, respectively, and the interpretation of q is likewise altered. Unlike micro- economic data, aggregate investment series look smoother since micro- economic adjustments are far from being perfectly synchronized. The question arises as to whether aggregation eliminates all traces of infrequent lumpy microeconomic adjustment. We again focus on investment (I), and we model aggregate investment in terms of average investment of a large number of individual firms indexed by i [1,2000]. Hamburg University 33 Department of Economics Michael Funke Finally, we employ a “hybrid” model that endogenises the business cycle turning points. Suppose that at each turning point, the proportion of firms experiencing a peak (trough) at time t is assumed to be drawn at random from a standard normal distribution around the predefined turning points in Figure 10. Hamburg University 34 Department of Economics Michael Funke Figure 10: A Sample Path of the Demand Shock (Z), the Z-Thresholds, Installed Capital (K), and Optimal Investment 2 Z and Z thresholds 1.5 Z0+ Z1+ 1 Z2+ Zt 0.5 Z2 Z0 Z1 0 0 6 12 18 24 30 time 2.5 2.4 2.2 2 2 1.8 Investment, I 1.6 1.5 1.4 1.2 K 1 1 0.8 0.6 0.5 0.4 0.2 0 0 0 6 12 18 24 30 0 6 12 18 24 30 time time Hamburg University 35 Department of Economics Michael Funke Figure 11: Aggregate Dynamics for Perfectly Synchronised Business Cycle Turning Points (a) Investment Dynamics for = 1 0.5 0.4 0.3 I 0.2 0.1 0 0 6 12 18 24 30 Time (b) Investment Dynamics for = 2 (c) Investment Dynamics for = 4 0.5 0.5 0.4 0.4 0.3 0.3 I I 0.2 0.2 0.1 0.1 0 0 0 6 12 18 24 30 0 6 12 18 24 30 Time Time Hamburg University 36 Department of Economics Michael Funke Figure 12: Aggregate Dynamics for Heterogeneous Business Cycles Turning Points (a) Investment Dynamics for = 1 and N(0, 0.00) around state-switching points 0.5 0.4 0.3 I 0.2 0.1 0 0 6 12 18 24 30 Time (b) Investment Dynamics for = 1 and N(0, 0.05) (c) Investment Dynamics for = 1 and N(0, 0.1) around state-switching points around state-switching points 0.5 0.5 0.4 0.4 0.3 0.3 I I 0.2 0.2 0.1 0.1 0 0 0 6 12 18 24 30 0 6 12 18 24 30 Time Time Hamburg University 37 Department of Economics Michael Funke 4. Summary and Conclusions The focus of this article has been the incorporation of jump dynamics into real options models in order to improve understanding of cyclical investment behaviour, especially in the most volatile era. The Markov-switching modelling approach allows the derivation of analytical and numerical results on option pricing, taking into account that firms not only either observe or infer the current state of the system but also make predictions about future regime switches. The chief implication of the model is that recessions and financial turmoil periods are important catalysts for waiting. Hamburg University 38 Department of Economics Michael Funke