Using discounted flexibility values to solve for decision costs in sequential investment policies.

Transcription

1 Using discounted flexibility values to solve for decision costs in sequential investment policies. Steinar Ekern, NHH, 5045 Bergen, Norway Mark B. Shackleton, LUMS, Lancaster, LA1 4YX, UK Sigbjørn Sødal, UiA, 4604 Kristiansand, Norway April 2016

2 Policy implied decision values and costs 1 1 Overview Real options; maximizing project/firm NPV with uncertainty but flexibility; developed alongside financial options but less advanced theory and empirics Black Scholes (73) [1], Merton (73) [2], Cox Ross Rubinstein (76) [3] Operations research links; using market consistent (risk neutral) discount rates Myers (77) [4], Brennan Schwartz (85) [5], Dixit Pindyck (94) [6], Trigeorgis (96) [7]. Gamba Fusari (09) [8] motivate and value the real options of design modularity (splt, subs, augm, excl, invt, port). Interactions are more important for real options (than traded), especially for multi stage and network investments Links to costly (frictional) reversibility (Abel and Eberly 96, [9]), Q theory and marginal cost of capital (Hayashi 82 [10], Abel et al. 96 [11]) Discount factor approach; Dixit, Pindyck, Sødal (99) [12], Sødal (06) [13]

3 Policy implied decision values and costs Flexibility modelling to date, + additional PV investment costs.. (e.g. ± ) treated as inputs in a search for matching revenue value thresholds.. (e.g. ), i.e. X to P Choose diffusion, obtain ode, solve functions ( ) ( ) (often backward from an end point) s.t. boundary conditions at No analytic solution from X to P with numerics only; hard to customise and can t tackle large or closed systems with cycles + We graphically unpick flexibility sequence and values using discounts ( ) + Diffusion (ode) choice occurs after the flexibility modelling + Using augmented smooth pasting or weighted beta first order conditions give... + Matrix/vector solutions for X V i.e. input policy thresholds P output costs X and values V + Additional insights for flexibility valuation in more complex and closed systems.

4 Policy implied decision values and costs 3 2 Canonical hysteresis Dixit (89) [14] and others study a firm that can switch between two operational modes, off (idle ) oron(full ) earning (with present value = ) ( ) ( )+ ( + ) at = (1) ( ) ( )+ ( ) at = Flexibility is described as a call or put on firm revenues with costs as strikes; these have dynamic values ( ) ( ) functions of but not time itself (homogeneous). Assume switch on at = and off at = (fixed policy points which are hit at random stopping times). Work through consequences for (six vars) option values at policy points and present value costs and frictions.

5 Policy implied decision values and costs Value matching and smooth pasting As well as tesing for matching of option and operational value at exercise, the first order condition for optimality requires smoothness of transition in ( ) ( )+ ( ± ) (2) ( ) ( ) + at = and = The beta of an option (or present value) is ( )= ( ) so the smooth pasting condition premultied by is equivalent to a weighted beta or rate of return matching condition ( ± has zero beta). ( ) ( )= ( )+1 at = and = (3)

6 Policy implied decision values and costs Discounting between beginning and end option values The value of an option at the beginning of its life (on the left in vector U) isa fraction of its value at the end (on the right in vector W) " U # = " D # " W ( ) 0 1 = 4 ( ) # (4) These option have been evaluated at fixed policy points, on the top row and on the bottom. Arbitrary (for now) fractions or discounts 1 4 and 1 16 were chosen (these can also be interpreted as growth factors of 16 & 4).

7 Policy implied decision values and costs Value matching in vector form The same vectors can be used to represent value matching at these thresholds. W # + Z # = " U # + Y # X ( + = ) + " ( ) " 0 " 0 " # (5) Extra vectors Z Y areusedtorepresentthepresentvalueofcashflows before and after option conversion, whilst X carries the fixed cost of operations and frictional switching costs, i.e. = + = Treating and discounts as inputs, there are four equations for six variables, two more equations needed from smooth pasting.

8 Policy implied decision values and costs Smooth pasting using betas in matrices Assume (will show how) the betas of the call and put are =2and = 1 (the call varies 2:1 with but the put -1:1). β #" W # + " β Z # = " β #" U # + β Y ( ) ( + = ) " " 0 (6) To capture smooth pasting, these go into matrices β β that premultiply W U (β β = I the identity matrix here). #

9 Policy implied decision values and costs Matrix solution Two equations do not involve costs so solve for U W first then X W = [β W β U D] 1 (β Y β Z) (7) U = D [β W β U D] 1 (β Y β Z) X = U W + Y Z The option values are like a perpetuity (with factor β β D)onacashflow β Y β Z It is possible to solve for W U X from thresholds, discounts and betas(but not the other way round this is the root finding problem).

10 Policy implied decision values and costs Example with geometric Brownian motion = =( ) + (8) Apply Ito s lemma to [ ( )] = ( ) an expected change over interval the Bellman is solved by ( ) ( ) with fixed betas shown 0 = ( ) 2 ( ) 2 +( ) ( ) ( ) (9) = ( ) = s µ = ± E.g. annual parameters =4% 4% 20% give =2 1.

11 Policy implied decision values and costs Discount functions Carry functional form of each option in a (dynamic) discount applied to its end value ( ) = ³ ( ) (10) ( ) = ³ ( ) Discounts are expectations of next stopping time that give the functional form (& betas) of options value within thresholds as well as their separation ³ = ³ = h ( ) = i = Ã h ( ) = i = Ã! (11)!

12 Policy implied decision values and costs 11 6 Simple hysteresis between P4=4 and P1=1 5 4 P+Vf V P Vi X4= X1= Underlying project value Figure 1: GBM =2 1 with =4 1 gives = , i.e. =1 962 and =0 321

13 Policy implied decision values and costs 12 VM DI SP Conditions = + = ( ) q q 1 4 ( ) =1 + =1 + Solutions (bold), inputs = = q q = = Table 1: Given = and discounts of a quarter and a sixteenth, these six equations are used to solve for. VM stands for value matching, DI for discounting and SP for smooth pasting (or dollar beta matching).

14 Policy implied decision values and costs 13 RN diffusion Call/put discount functions quad= 0for ( ) + + ( ) + ³ = ³ : 6 ³ = ³ : > ³ = ( ) : 6 ³ = ( ) : > µ = µ ³ ³ ( 1) + ( ) ( 1) Table 2: Discounts for Geometric, Arithmetic and Mean Reverting processes. 2.8 Discount functions for other processes

15 Policy implied decision values and costs 14 3 Switching with a third, gamma mode ( ) Z [ ] = b b = b = b where b = ( ) ( 1) The perpetuity value of cashflow power ( ) arises from a constant yield b (given in last line, nb implies b is positive). The present value of power flow b reverts to for =1or a constant 1 for =0 The beta of this present value with respect to the driver (but not the option )is = =. Under GBM for = and both ( ( )) and ( ) have iso-betas b and = b that givethesamescale,i.e. b = b = so that ( ) ( ) b.

16 Policy implied decision values and costs Three threshold matching & discounting W + Z = U + Y X + 0 = + 0 (12) U = D W 0 0 ( ) = ( ) ( ) 0 (13)

17 Policy implied decision values and costs 16 full flow + + power flow idle no flow policy threshold option used ( ) ( ) ( ) option gained ( ) ( ) ( ) Table 3: Investment network graph at three thresholds (horizontal; put, call, call) for three state system (vertical; idle, power, full). Value matching at investment occurs vertically, diffusion and discounting horizontally.

18 Policy implied decision values and costs 17 Condition =+ =+ = + = ( ) = ( ) = ( ) = + = + = Solutions (bold) inputs = = = = = = = = = Table 4: and 9equations

19 Policy implied decision values and costs 18 Two call stages and a put Total value incl. flex Underlying revenue value P P+Vf (full inc put) P (full only) P^0.5 + Vg (power + flex) P^0.5 (power only) Vi (idle inc call) X1= X2=1.061 X4=0.742 Figure 2: Investments across three mode example with =4 2 1 for GBM and power values = Table 3 shows the investment graph whilst this plot shows the idle, power and full flex values with their investment quantities (value matching occurs vertically at =4 2 1).

20 Policy implied decision values and costs Smooth pasting at three thresholds With two calls and one put the beta matrices carry extended entries, is the beta of ( ) β W = β U + β Y β Z β = = β = (14)

21 Policy implied decision values and costs 20 4 Two way discounting Now consider splitting the gamma call into a call (up) and a put down (to idle), the put from full returns the system to the gamma state at W + Z = U + Y X + 0 = + 0 (15)

22 Policy implied decision values and costs Table 5: Investment graph with four thresholds (horizontal) and three states (vertical). Value matching at investment occurs vertically, diffusion and discounting occurs horizontally. Note the two way switching from within the power state.

23 Policy implied decision values and costs Two way discount factors in the gamma state Flexibility is a linear combination of call and put and discounts which can knock out ( )= ( )+ ( ) (16) ( ) = h i ( ) = min( ) (17) = ( ) ( ) ( ) (18) ( ) = h i ( ) = max( ) = ( ) ( ) ( ) where =1 ( ) ( )

24 Policy implied decision values and costs Two way discount matrix D These discount factors are put into the matrix D that generates U = DW U = D = = D 0 ( ) 0 0 ( ) 0 0 ( ) ( ) 0 0 ( ) 0 0 ( ) 0 (19)

25 Policy implied decision values and costs Two way growth matrix G The inverse of D is G a matrix that says how option values grow from fixed past values W = GU W = G = = G 0 ( ) ( ) 0 ( ) ( ) 0 ( ) ( ) 0 (20)

26 Policy implied decision values and costs Dynamic values U ( ) from static W To draw out the beta sensitivity of each row, this operation must be applied line by line. (U ( )) 0 = ( ) ( ) ( ) ( ) 0 = ( ) ( ) ( ) ( ) = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) =(D ( ) W) 0

27 Policy implied decision values and costs Evaluating the beta matrix β for U and β for W This can be done with a dynamic D ( ) (in appendix of paper) β ( ) U ( ) = D 0 ( ) W = D 0 ( ) GU (21) β ( ) = D 0 ( ) G β = D 0 G (W( )) 0 = (G( )U) 0 =(G( )) 0 U (22) β W = G 0 U = G 0 DW β = G 0 D

28 Policy implied decision values and costs Smooth pasting the four threshold system Starting with value matching, free up the dynamics in at the threshold, differentiate and scale by then work backwards to produce matrices β β GU = DW + Y Z X (23) G 0 U = D 0 W +[Y Z] 0 G 0 DW = D 0 GU +[Y Z] 0 β W = β U + β Y β Z

29 Policy implied decision values and costs Solutions for four threshold, three state system β W = = β U +(β Y β Z) =

30 Policy implied decision values and costs 29 Type VM DI SP Condition: Values (sols in bold) = = = = = = = = = = = = Table 6: Eight option values, four frictions and 12 equations.

31 Policy implied decision values and costs 30 6 Investment ladder idle, power and full switching 5 ility 4 x ib fle g in d 3 clu in e a lu 2 V 1 P+Vf (full inc put) P (full only) P^0.5 + Vp (power inc switch) P^0.5 (power only) Vi (idle inc call) X1 X2 X3 X Underlying (full) project value P Figure 3: GBM ladder with = with idle, power and full values vertically; (dynamic) values between policy anchors filled in.

32 Policy implied decision values and costs 31 5 Restrictions for maxima Starting from U each option later smooth pastes into other(s) in W one each of W U at the same threshold are paired maxima (or minima) But smooth pasting (rate of return: Shackleton and Sødal [15]) conditions are not sufficient for maxima Re-evaluate/perturb from local solution to test for second order condition W = U + Y Z X max W = [I D] 1 [Y Z X] If recovered investment costs not equal to target X (i.e. P output required), iterate on thresholds P until X(P) =X

33 Policy implied decision values and costs 32 6 Summary Sødal et al. (99, 06) [12], [13] developed discount factors. For any given process these determine the relative value of static policy values and specify the dynamics of the option betas, i.e. i) Separate problem constants e.g. from (W from U) and ii) Produce betas required at policy boundaries Can separate the flexibility modelling from the diffusion/pde choice and Cashflows can be varied flexibly within a circular network (modular design) and solved using matrices Investment is a bipartite and directed graph (see Wilson [16]) with solutions W U X Adds intuition to optimal smooth pasting using discount, growth matrices and their scaled partials Allows more investment problems to be tackled with a wider range of diffusions.

34 Policy implied decision values and costs 33 References [1] Fischer Black and Myron Scholes. The pricing of options and corporate liabilities. Journal of Political Economy, 81(May June): , [2] Robert C. Merton. The theory of rational option pricing. Bell Journal of Economics, 4(1): , [3] John C. Cox, Stephen A. Ross, and Mark Rubinstein. Option pricing: a simplified approach. Journal of Financial Economics, 7: , [4] Stewart C. Myers. Determinants of corporate borrowing. Journal of Financial Economics, 5: , [5] Michael John Brennan and Eduardo S. Schwartz. Evaluating natural resource investments. Journal of Business, 58(2): , [6] Avinash K. Dixit and Robert S. Pindyck. Investment under Uncertainty. Princeton University Press, 1994.

35 Policy implied decision values and costs 34 [7] Lenos Trigeorgis. Real options: Managerial flexibility and strategy in resource allocation. MIT Press, [8] Andrea Gamba and Nicola Fusari. Valuing modularity as a Real Option. Management Science, 55(11): , [9] Andrew B. Abel and Janice C. Eberly. Optimal investment with costly reversibility. Review of Economic Studies, 63: , [10] Fumio Hayashi. Tobin s marginal and average : A neoclassical interpretation. Econometrica, 50(1): , [11] Andrew B. Abel, Avinash K. Dixit, Janice C. Eberly, and Robert S. Pindyck. Options, the value of capital, and investment. The Quarterly Journal of Economics, 111(3): , [12] Avinash K. Dixit, Robert S. Pindyck, and Sigbjørn Sødal. A markup interpretation of optimal investment rules. The Economic Journal, 109(455): , 1999.

36 Policy implied decision values and costs 35 [13] Sigbjørn Sødal. Entry and exit decisions based on a discount factor approach. Journal of Economic Dynamics and Control, 30(11): , [14] Avinash K. Dixit. Entry and exit decisions under uncertainty. Journal of Political Economy, 97: , [15] Mark B. Shackleton and Sigbjørn Sødal. Smooth pasting as rate of return equalization. Economics Letters, 89(2 November): , [16] Robin J. Wilson. Introduction to Graph Theory. Longman, third edition, 1985.