essays on regime switching models in finance

We …rst consider an irreversible investment timing decision by adding a hidden Markov process to model the state of the economy in continuous time.. David and Veronesi2000, and Bu¢ ngton

Essays on Regime Switching Models in Finance

by

Hong Miao

A DISSERTATION SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE

DEGREE OF DOCTOR OF PHILOSOPHY

HASKAYNE SCHOOL OF BUSINESS

CALGARY, ALBERTA

April 2008

© Hong Miao 2008

          ISBN: 978-0-494-41142-1

The undersigned certify that they have read, and recommend to the Faculty of Graduate Studies for acceptance, a dissertation entitled “Essays on Regime Switching Models in Finance” submitted by Hong Miao in partial fulfillment of the requirements for the degree of Doctor of Philosophy

Supervisor, Dr Robert J Elliott, Haskayne School of Business

Dr Anatoliy Swishchuk, Department of Mathematics & Statistics

Dr Alfred Lehar, Haskayne School of Business

Dr Tony Ware, Department of Mathematics & Statistics

_ External Examiner, Dr Aditya Kaul, University of Alberta

Date

ii

This dissertation consists of four essays focusing on applications of regime switching models in finance

The first essay discusses the investment timing problem in a regime switching framework

We consider a firm facing a future real investment opportunity whose investment cost depends on economic situations Our approach considers the investment timing as a perpetual American option when the strike price switches between two possible values depending on the economic situations It gives better optimal investment policy than the widely used standard real option method

The second essay extends the general equilibrium pricing model into an economy with two “states” Based on assumptions of a CRRA utility function, we have derived a partial differential equation satisfied by the representative agent's cost function A form of the solution of the partial differential equation has been given in general equilibrium with intermediate consumption In the case when the representative agent does not have intermediate consumption, we have found an explicit solution of the cost function A closed-form expression for the riskless rate has been derived We have also provided a partial differential equation satisfied by any contingent claim written on any risky asset in the market The stochastic discount factor has been investigated in our framework Based

on the stochastic discount factor, we have suggested an explanation for the equity premium puzzle

The third essay studies a stochastic volatility model in a regime switching world We have derived closed form expressions for all the parameters by using the filtering

iii

shortfall in a regime switching economy Based on the Student-t distribution assumption,

we have suggested an approach to evaluate value at risk and expected shortfall We use the Student-t distribution to capture the fat-tail phenomenon and regime switching to model the volatility clustering Closed form expressions for computing value at risk and expected shortfall for both a single asset and a portfolio are proposed The approach is easy to apply

iv

Shanshan Hong

My parents, Huanlian Dong and Shuhua Miao

My supervisor, Dr Robert J Elliott and his wife, Ann Elliott

v

I would like to thank my supervisor, Dr Robert J Elliott, for his unceasing support and guidance I also thank the other two professors on the supervisory committee, Dr Anatoliy Swishchuk and Dr Alfred Lehar, for their comments and suggestions which have improved this dissertation Thanks also go to Jin Yu at Vienna School of Finance for his cooperation

vi

Approval Page……… ii

Abstract……… iii

Dedication……… v

Acknowledgements……… vi

Table of Contents……… vii

CHAPTER ONE: INTRODUCTION……… 1

CHAPTER TWO: ESSAY 1: INVESTMENT TIMING UNDER REGIME SWITCHING……… 4

2.1 Introduction……… 4

2.2 Optimal Investment Models……… 7

2.2.1 Model Setting……… ….……… ……… 10

2.2.2 The Benchmark Case……… ….….……… 12

2.2.3 Standard Real-options Model……… ….……… 17

2.2.4 Full Information Case……… …….……… 18

2.3 A Numerical Example ……… ……… ……… 22

2.3.1 Expected Actual Payoff in the Benchmark Case … ….……… 24

2.3.2 Expected Actual Payoffs in the Standard Real-Option Case………

25 2.4 A Numerical Example……….……… 26

2.5 Conclusions……….……… 29

CHAPTER THREE: ESSAY 2: GENERAL EQULIBRIUM MODEL WITH REGIME SWITCHING……… ………

30 3.1 Introduction……….……… 30

vii

3.3.1 General Equilibrium with Intermediate Consumption……… 36

3.3.2 General Equilibrium without Intermediate Consumption……… ……… 44

3.4 A Partial Differential Equation for Contingent Claims ……… 47

3.5 Stochastic Discount Factor and Market Price of Risk ……… 51

3.6 Concluding Remarks……… 55

CHAPTER FOUR: ESSAY 3: STOCHASTIC VOLATILITY MODEL WITH FILTEING……… …………

57 4.1 Introduction……… 57

4.2 Assumptions………… ……….……… 58

4.3 Change of Measure ……… 61

4.4 Parameter Estimates……… ……… 69

4.4.1 Expectation Maximization……… 70

4.4.2 Specific Processes ……… 70

4.4.3 Estimate of p ji……….……… 80

4.4.4 Estimate of α, β, θ and μ….……… ……… 84

4.5 Conclusions……… ……… 86

CHAPTER FIVE: ESSAY 4: VaR and EXPECTED SHORTFALL: A NON-NORMAL REGIME SWICHING FRAMEWORK……… …………

87 5.1 Introduction……… 87

5.2 Definitions and Model Setup……….……… 90

5.2.1 Definitions and Features of VaR ……… 90

5.2.2 Definitions and Features of ES ……… 91

viii

5.5 Parameter Estimation ……… ……… 97

5.5.1 Change of Measure…… ……… 97

5.5.2 Parameter Estimation….……… 102

5.6 Closed-form VaR and ES for a Single Asset……….………… ……… 105

5.7 Closed-form VaR and ES for a Portfolio……… 106

5.8 Conclusion……… 108

CHAPTER SIX: SUMMARIES AND FUTURE RESEARCH……… ………… 109

REFERENCES……….……… 111

APPENDIX ……… 118

ix

Hong Miao

This dissertation consists of four essays focusing on applications of regime switchingmodels in …nance The …rst essay addresses the investment timing problem in a regimeswitching framework The second essay discusses the general equilibrium pricing modelwhen the economy has two states The third essay studies a stochastic model using

…ltering The fourth essay presents a computational method for value at risk and expectedshortfall when the returns of …nancial assets follow a Student-t distribution with twodi¤erent regimes

1 Introduction

Regime switching models were introduced into …nance by Hamilton (1988) Since then,studies have been done in various areas of …nance, for instance, stock market price move-ment, exchange rate modelling, and interest rate modelling Empirically, economic sit-uations impact …nancial decisions and price movements of …nancial instruments Thepurpose of this study is to extend the applications of regime switching models to invest-ment timing, general equilibrium pricing models, volatility models and risk management.This dissertation includes four essays with consolidated references at the end

The …rst essay elaborates regime switching and optimal investment timing in a realoption framework We …rst consider an irreversible investment timing decision by adding

a hidden Markov process to model the state of the economy in continuous time The cost

of the investment is driven the Markov chain Therefore, the investment cost is either K1

or K2 depending on whether the economy is in a ‘low cost’or ‘high cost’state K1and K2

can be considered as strike prices of a perpetual American option By introducing thisspeci…c structure of stochastic investment costs, the essay presents a di¤erent optimalexercise policy for the …rm It can be optimal for the …rm to exercise much earlier thanotherwise suggested by a standard real option model Moreover, a range of exercisingtrigger prices is determined Of course, the value of the growth option that the …rm faces

is also higher than the one derived by a conventional real option model Thus, we showthat an optimal timing policy suggested by the conventional real option model mightruin the investment opportunities

In the second essay, we have developed a continuous time general equilibrium model

in an economy which has two states, a ‘good’ state and a ‘bad’ state There are twotypes of shocks in the economy: small shocks and large shocks The small shocks whichonly a¤ect the individual price movements are modeled by Brownian motions The largeshocks, the states of the economy, are modeled by a continuous time Markov chain There

are n basic risky assets, one riskless assets and contingent claims written on the riskyassets in the market The states of the economy a¤ect the expected returns and thevariances of the assets We investigate the asset pricing problem in general equilibriumwith a representative agent who maximizes a cost function Based on the assumption

of a CRRA utility function, we have derived a partial di¤erential equation satis…ed bythe representative agent’s cost function A form of the solution of the partial di¤erentialequation is given in general equilibrium with intermediate consumption In the case whenthe representative agent does not have intermediate consumption, we …nd an explicitsolution of the cost function A closed-form expression for the riskless interest rate hasbeen derived We also give a partial di¤erential equation satis…ed by any contingent claimwritten on basic risky asset The stochastic discount factor is de…ned and computed inour framework Based on the stochastic discount factor, we suggest an explanation forthe equity premium puzzle

In the third essay, we generalize the Stochastic Volatility model by allowing the ity to follow di¤erent dynamics in di¤erent states of the world The dynamics of the

volatil-"states of the world" are represented by a Markov chain We estimate all the ters by using Filtering and the EM algorithm Closed form estimates for all parametersare derived in this paper These estimates can be updated using new information as itarrives

parame-In the third essay, we have developed a regime switching framework to compute theValue at Risk and Expected Shortfall measures Although Value at Risk as a risk measurehas been criticized by some researchers for lack of subadditivity, it is still a central tool inbanking regulations and internal risk management in the …nance industry In contrast,Expected Shortfall (ES) is coherent and convex, so it is a better measure of risk thanValue at Risk Expected Shortfall is widely used in the insurance industry and has thepotential to replace Value at Risk as a standard risk measure in the near future We have

proposed regime switching models to measure value at risk and expected shortfall for asingle …nancial asset as well as …nancial portfolios Our models capture the volatilityclustering phenomenon and variance independent variation in the higher moments byassuming the returns follow Student’s t-distributions.

The dissertation is organized as follows The …rst essay on investment timing ispresented in Chapter 2, the second essay on general equilibrium model in Chapter 3,the third essay on stochastic volatility in Chapter 4, and the fourth essay on value atrisk and expected shortfall in Chapter 5 Chapter 6 concludes the dissertation with asummary and directions for future research, followed by the consolidated references forall the chapters Some Proofs are in the appendix

2 Essay 1: Investment Timing under Regime Switching

2.1 Introduction

The formulation of optimal investment strategies is always of interest for …rms This

is also one of the most important topics in capital budgeting Literature in this areausually divides into two components: the capital allocation and the investment timingdecisions The standard capital budgeting approach of capital allocation is to computethe Net Present Value of the possible projects and invest in those with positive N P V s.The investment timing problem is usually solved by the standard real option techniquewhich is addressed by papers including Titman (1985), Brennan and Schwartz (1985),McDonald and Siegel (1986), and Dixit (1989) The books by Dixit and Pindyck (1994),and Trigeorgis (1996) o¤er a classical treatment Schwartz and Trigeorgis (2004) collectclassical readings and recent contributions in real options and investment under uncer-tainty, as the title of the book indicates On the other hand, Grenadier (2000) studies

…rms’optimal option exercise policies in a continuous time Nash Cournot Equilibrium.The option value under strategic competition deteriorates quickly

Grenadier and Wang (2005) derive an optimal contract that best aligns the incentives

of owners and managers Their method is di¤erent to the standard real option approach

in which only one strike price exists for the project They assume that the projectgenerates two sources of value, one of which can be observed only by the manager Thissource of value can take two di¤erent values and the manager can increase the possibility

of the realization of the high value by exerting e¤ort However, in reality, the managerhave limited ability to increase the probability of the high value

We argue that it might be macroeconomic variables, rather than the manager’s e¤ort,which trigger the change of the parameters used in the model We model the state of theeconomy by a Markov chain with a …nite state space Markov regime switching modelshave been used in electrical engineering since the 1960s Hamilton (1989) proposed the a

regime switching model to study postwar US real GNP associated with business cycle SeeHamilton (2005) for a recent survey The modelling of asset price processes with regimeshifts has also been adopted by others Veronesi (1999) derives a rational expectationsequilibrium model of asset values in the present of regime shifts David and Veronesi(2000), and Bu¢ ngton and Elliott (2002) value contingent claims on an underlying withuncertain parameters.

This paper is intended to elaborate regime switching and optimal investment timing

in a real option framework The paper di¤ers from the existing literature in a signi…cantway In this paper we …rst consider an irreversible investment timing decision by adding

a hidden Markov process to model the state of the economy in continuous time Thecost of the investment is driven by the Markov chain Therefore, the investment cost

is either K1 or K2 depending on whether the economy is in a ‘low cost’ or ‘high cost’state K1 and K2 can be considered as strike prices of a perpetual American option.This is reasonable as people do observe business cycles of macroeconomic variables thatdetermine the investment costs of the project By introducing this speci…c structure ofstochastic investment costs, the paper presents a di¤erent optimal exercising policy forthe …rm We have shown that it is optimal for the …rm to exercise much earlier thanotherwise suggested by a standard real option model Moreover, a range of exercisingtrigger prices is determined It will be shown later that the value of the growth optionthat the …rm faces is also higher than the one derived by a conventional real optionmodel

There are a number of papers dealing with stochastic investment costs for real optionvaluation, or equivalently stochastic strike prices for the pricing of a …nancial contingentclaim Fischer (1978) and Margrabe (1978) derive explicit solutions to an exchange optionthat gives the holder the right to exchange one risky asset to another one McDonaldand Siegel (1986) study a similar irreversible investment timing policy when the value of

the project and the required investment cost follow two correlated geometric Brownianmotions Since the quotient of these two geometric Brownian motions is still a geometricBrownian motion, their valuation problem reduces to …nding a threshold value for theratio of the value and the cost of the project This type of problem could be solved

by using either of above processes as numeraire Like the optimal exercising intervalderived in our paper, their approach shows the optimal trigger value of the project couldtake values from zero to in…nity, though the critical ratio of the value and the cost isuniquely determined This is to be expected because in their framework investment costfollows a di¤usion model that could reach any value between zero and in…nity in the nexttime instant However, McDonald and Siegel (1986) points out that for the analysis it isnecessary to assume value F and V are geometric Brownian motions This assumption

is reasonable for the project value V , but may be less so for the investment cost F McDonald and Siegel (1986) also suggest a mean reverting process for prices which arenot present values Since Markov switching models can model business cycles easily, theycould be a better candidate for modeling nonpresent value prices

Pindyck (1993) isolates the uncertainty embedded in the investment cost and ines a …rm’s irreversible investment decision when the cost is exposed to both technicaland input risks The project value in his model is constant Blenman and Clark (2005)derive closed-form solutions to European options with stochastic strike prices, which pre-serve constant elasticity of the strikes with respect to the price They apply the technique

exam-to value an IPO decision of a privately owned …rm

The modeling of stochastic investment costs by geometric Brownian motions has, atleast, two drawbacks First, the investment cost could go to anywhere in the interval ofzero to in…nity at the next second This is obviously not realistic of the uncertainty costs.Second, as pointed out by McDonald and Siegel (1986), the prices of physical assets have

to converge to their equilibrium levels, and not grow exponentially as suggested by a

geometric Brownian motion.

The main contribution of our paper is to overcome the limitations discussed above Weassume, instead, that the investment costs follow a Markov chain with a …nite state space,though for convenience of exposition we restrict our analysis to a two state Markov chain.This model con…nes the uncertainty of the investment cost to a controllable range instead

of di¤usively moving from zero to in…nity Moreover, by calibrating the parameters inour model, that is the elements in the Q-matrix, it is possible to have our stochasticinvestment cost return to its equilibrium level in the long run Hence, our assumption ofregime switching investment cost is less misspeci…ed than ones with constant or stochasticcosts driven by Brownian motions

The paper is organized as follows In Section II, real option models are developed toinvestigate the investment timing problem Three cases with di¤erent information avail-ability are considered Two thresholds for exercising the projects with full informationare derived In Section III, the actual payo¤s of the the …rm are derived and discussed inthe presence of a hidden Markov chain, and the advantages of the full information model

is presented by a numerical example Section IV concludes the paper

2.2 Optimal Investment Models

Consider a …rm which has an investment opportunity If the …rm invests in the project,

it will produce a product The output of the …rm at time t is given by Qt, which isassumed to be a constant The demand for the product, and hence the equilibrium price,

is stochastic and speci…ed by an isolastic inverse demand curve:

Pt= tQt 1:

Here t is an industry wide shock and 0 < < 1 is a given parameter

Further, we assume the industry shock follows a geometric Brownian motion

Suppose the …rm only faces zero …xed costs and no variable production costs, at time

t the pro…t, or the net cash ‡ow from the investment is given by

states We restrict our study to two states for convenience of exposition.

We then consider this project as a perpetual American call option It is optimal forthe …rm to launch the project if certain values of the project pass some thresholds In

a standard real option framework, the strike price is constant However, in our model,

we suppose the project has two strike prices, or di¤erent investment costs, depending onthe states of the economy, or the states of the company Grenadier and Wang (2005)suppose the investment is a constant K, but the project generates a value componentwhich takes two values 1and 2 Only the manager can observe 1and 2 In our model

we consider the strike price as the investment cost minus the value component :

To introduce our model formally, we make the following assumptions

Assumption 2.1 We consider a real option model on a …ltered probability space ( ; F; Q)where F is to be determined for di¤erent cases The capital market is free of arbitrageand complete

Assumption 2.2 In the market there exists a riskless asset that is essentially a bankaccount Let the continuously compounded bank rate be r Then if Bt denotes the value

in the bank account at time t of $1 at time 0;

Bt = ert:

We suppose that < r in order to obtain a …nite value of the project

Proposition 2.1 The value of the entire project can be calculated by computing the tional expectation of the investors as

Now model the states of the world by a continuous time Markov chain X = fXt; t 0g :

We suppose the economy has only two states: ‘low cost’and ‘high cost’ The state pace

of the chain X can be taken to be the two unit vectors e1 = (1; 0)0; e2 = (0; 1)0 in R2:The state of the economy is known to the …rm The value of the project is then thepayo¤ of a perpetual American call option, so we shall discuss the value of a perpetualcall on S for which the strike price can take one of the two values K1 or K2: The strikeprice at time t is determined by the Markov chain X = fXt; t 0g which takes twovalues corresponding to K1 or K2:If K denotes the vector (K1; K2)and h ; i is the scalarproduct in R2 then the strike price in e¤ect for the call at time t is

@ a1 a2

a1 a2

1C

A : The semi-martingale

representation of the Markov chain (see Elliott (1993)) is

Xt= X0+Rt

0AXsds + Mt; (2.4)

where M = fMt; t 0g is an R2-valued martingale process

Assumption 2.4 We assume the independence of the Brownian motion and the Markovchain Accordingly the internal …ltrations generated by respective Brownian motion, Fw

t =(ws; 0 s t) and Markov chain FX

to the decision maker

2.2.2 The Benchmark Case

We shall decide the optimal investment policy for the …rm under the regime switchingframework For comparision, we …rst consider a simple case with a rational investmentcost based on the state of the economy when the project is intitiated Therefore, the

…rm faces an optimal exercise problem with a constant investment cost identical to theratinal expectation of the investment cost at time zero.

The rational investment cost should be the expectation of hK; Xti : We …rst estimatethe investment costs K1 and K2 by computing the expectation of hK; Xti based onobserving the original state of the chain Thus, we shall …rst calculate the expectation

of the Markov chain at time t, i.e E[Xt]:

Solving the above SDE by multiplying the integrating factor, eAt, and assuming theinitial value of (Xt)t 0 is a known value X0, we have,

Xt= eAt X0+

Z t 0

0B

@ a2 a2

a1 a1

1C

A e

ct

where, c = a1+ a2:

Proof See Appendix A

Then the investment costs are:

1 If the initial value of X0 = e1;, i.e the economy at time zero turns out to be at alow cost state

K1 = a2+ a1e ct K1

c + 1 e

ct a1K2

2 If the initial value of X0 = e2;, i.e the economy starts at a high cost state

K1 = E [Kt] = E [hK; Xti]

=hK; E [Xt]i = hK; e1P(Xt = e1) + e2P(Xt = e2)i

= K; (q11t; q12t) = K1q11t+ K2q12t:

Assumption 2.5 The …rm’s rational expectation of the investment cost is its long runexpectation By long run, we mean the limit as t ! 1.

Under Assumption 2.5, we have

Finally, we obtain the time homogeneity

The problem now becomes like a standard real option one We wish to computethe investment timing, or critical values, indicating when to start the project and theconsequent present value of the project Applying the standard approach as in Dixitand Pindyck (1994) ; the value of the project, Di S; Ki , i = 1; or 2, must satisfy thefollowing di¤erential equation:

12

2S2@

2Di

@S2 + (r )S@Di

@S rDi = 0: (2.10)

We are considering a perpetual option, so Di does not depend on time Further Di

satis…es the following boundary conditions

Solving equation (2:10) ; with boundary conditions (2:11) (2:13) ; we have:

Proposition 2.3 Under Assumption 2.5, if the economy is at the cost state i, i = 1; 2,

at time zero (when the project is initiated), that is, X0 = ei; i = 1; 2; the present value

of the option at time zero and the critical value Si are

and the common critical value is S = S1 = S2:Here the value of K1 = K2 is determined

by equation (2.8)

In this case the investment costs are rationally chosen and the existence of the regimechanges are known We call this case the benchmark and we shall show that the optimalinvestment policy improves on the benchmark case That is, using the full informationdoes increase the value of the project for the …rm We also notice that the optimaltiming policies for the two di¤erent initial states of economy coincide because the longrun expectation of the investment costs are identical

2.2.3 Standard Real-options Model

The second case is one in which the …rm does not know the timing of the regime changes,but at time zero, the …rm does observe the state of the economy, that is it knows X0 The

…ltration, or Information set, is Fw

t If the economy is in the ‘low cost’state, the …rmchooses the investment cost K1 and maintain that during the whole life of the project;otherwise, it chooses K2:This is then a standard real-option situation where the …rm candecide either to invest in the project or wait The objective of the …rm is to maximizethe value of the project The strike prices of the option are Ki , i = 1; 2: Similarly to theprevious case, applying the standard approach as in Dixit and Pindyck (1994) ; we have:Proposition 2.4 If the economy is in state i, i = 1; 2; at time zero, the present value ofthe project and the critical value ^Si are:

Here the values of the project should be less than those in the benchmark case sincethe decision is not rational Although the economy has regime changes, the …rm doesnot observe the changes, or does not use this information.

Corollary 2.1 When compared to the benchmark case, the misspeci…ed investment costmodel implies an investment strategy which is too early when the low cost state economyoccurs at time zero, and which is too late when the economy starts in the high cost state

Proof We must show

^

S1 S1 = S2 S^2: (2.17)

However, this follows trivially from K1 K1 = K2 K2

2.2.4 Full Information Case

In this situation, the …rm does observe the existence of the regime changes, and willinvest depending on the state of the world Its’information set is Ftw;X The investmenttiming problem now becomes a perpetual American Options with two strike prices, K1

and K2: The investment timing depends on the payo¤ of the project

Recall that we suppose K1 K2; the case when K1 K2 is just the opposite of thefollowing discussion Analogously to the case when K1 = K2; we suppose there are twocritical prices S1 and S2:When Xt = ei; i = 1; 2;it is better to hold the option if St Siand to exercise if St > Si: Again the value process has the dynamics

Here, X0 can take either the value e1 or the value e2:

of the project if it is launched

Proposition 2.5 For 0 < S S1; the …rm does not invest, and under the two regimesthe project has the values

For S2 S; the …rm launches the project in both states of the economy, and the values

of the project is given by:

D1 = S K1;

and

D2 = S K2:

Proof See Appendix B

Proposition 2.6 The critical values S1;and S2 for launching the project are determined

by the following two equations, which must be solved numerically:

a2 3K1 a2 3K2 r 3K2

r + a2+a2S1(1 3)

a2 4K1 a2 4K2 r 4K2

r + a2+a2S1(1 4)

+ a2 +

a2 4K1

r + a2 :

Proof See Appendix C

The tractability is unfortunately lost due to the highly non-linear system giving S1and S2 We resort to numerical analysis to compare the payo¤s of di¤erent cases in asubsequent section

The recent advances in asset pricing theory use a …rm’s real investment activities toexplain the conditional dynamics in expected asset returns We are also interested in therisk implications of a …rm’s growth opportunity given that investment cost is stochastic

To focus on the added uncertainty of investment cost, we con…ne our analysis to the casethat a …rm has no assets in place and has only one growth opportunity There is no …xedoperating cost, either Following Carlson, Fisher, and Giammarino (2004), we calculatethe …rm’s systematic risk sensitivity, , as follows,

Dif-ferentiating equation (2.24) with respect to S and using the de…nition of by equation(2.25), we have

because 2 > 1 > 1: We interpret the derived as follows Firstly, our result is identical

to the one derived in Carlson, Fisher, and Giammarino (2004) after recognizing VG

i = Vi(because value of asset in place is zero) and ViF = 01 Secondly, our model suggests ahigher risk premium for a growth stock because of the additional economy-wide shock

of the investment cost Thirdly, our nests the in Carlson, Fisher, and Giammarino(2004) by letting a1 = a2 = 0 (hence 1 = 2) Obviously, assuming a1 = a2 = 0, we goback to an economy with constant investment cost

2.3 Hidden Losses

In this section, we investigate hidden losses due to not knowing the hidden Markov chainfor a set of base parameters In other words, the ‘optimal’option values derived abovefor the benchmark and standard Real Option cases are di¤erent from the expected actualpayo¤s that the …rm receives This is because the investment costs are in fact stochasticbut are assumed to be constant in these two cases

To proceed, we introduce de…nitions of …rst passage times of a geometric Brownianmotion and present a lemma for …rst passage times

1 For de…nititons of V G ; V i ; and, Vf, see Carlson, Fisher, and Giammarino (2004).

De…nition 2.3 We de…ne the …rst passage times of a geometric Brownian motion asfollows:

Lemma 2.2 For the …rst passage time of a geometric Brownian motion,

Solving the di¤erential equation and …nding the appropriate coe¢ cient that matchesboundary condition completes the proof.

In the subsequent sections, we take # be either r or r + c:

2.3.1 Expected Actual Payo¤ in the Benchmark Case

In the previous section, we have computed the present values of the project subject todi¤erent assumptions In the benchmark case, we simpli…ed the problem by assumingthe strike price to be the owner’s rational expectation of the investment cost In thereal world, when the project is launched, even the manager does not observe the state ofthe economy, the investment cost will still be Ki depending on the state of the economy

at that time Thus, the actual payo¤ will be di¤erent from the values in Proposition2:3: Therefore, to compare the value of the project in the three cases we compute theexpected actual payo¤ of the project

Recall that in the benchmark case the optimal timing policy is independent of theinitial state of the economy Hence, we only present, without loss of generality, the casewhen the economy is in the ‘low cost’state at time zero

Proposition 2.7 The expected actual payo¤ to the …rm in the benchmark case is strictlygreater than the one derived by the ‘optimal’ timing policy derived in Proposition 2.3

We denote the actual payo¤ by P1 Explicitly we have

Proof In the following, all expectations are taken under Q and conditional on F0.

The equality in the second line follows from the de…nition of K and Lemma 2.2, with

# = r + c: Setting = 2 gives the last equality

Proposition 2.7 can be interpreted as saying that the expected actual payo¤ of the

…rm is higher than the option value we have derived in Proposition 2.3 Therefore, theexercise strategy in Proposition 2.3 is optimal only with respect to the information set

Fw

t That is, if K = 0 and/or a1 = 0, the di¤erence vanishes and, in fact, we have ahidden trivial single state Markov chain

2.3.2 Expected Actual Payo¤s in a Standard Real Option Case

Similarly to Proposition 2.7, we have the following analogous results

Proposition 2.8 If the economy starts in the ‘low cost’state, the expected actual payo¤

of the …rm is strictly less than the option value we derived in Proposition 2.4 Otherwise,the expected actual payo¤ is strictly greater Denote the expected payo¤s by ^P1 and ^P2,respectively Then,

Proof Similar to the proof of Proposition 2.7.

Again, Proposition 2.8 shows that the expected payo¤s to the …rm di¤er from theoption value determined ‘optimally’in Proposition 2.4 Although we do observe the de-viation of the expected payo¤ from the option value, we have not said anything aboutwhich model is superior Intuitively, one expects the full information model would im-prove on the other two models with consider only partially revealed information Now,

we resort to a numerical example

2.4 A Numerical Example

Suppose we have a possible project which …ts into the previous framework We shallnumerically show how the optimal policy works and why knowing the full informationabout the project creates value for the …rm

We assume the Q-matrix takes the value:

A =

0B

@ 0:5 0:30:5 0:3

1C

A :

That is we set a1 = 0:5;and a2 = 0:3:The other parameter values are taken to be r = 3%;

= 2:5%; = 10%; S0 = 100; K1 = 105; and K2 = 110: Thus, applying Proposition 2.3

to Proposition 2.8, we have the following results:

1 For the benchmark case, K1 = K2 = 108:125; 1 = 2:45; the present value of theproject, D1 S0; K = 17:0399, and the critical values S = 182:72 The actualexpected payo¤ is P1 = 17:0412 which is very close to the present value

2 For the standard real option case: If the economy is in the ‘low cost’state at timezero, the present value of the project, the critical value, and the actual expectedpayo¤ are ^D1(S0; K1) = 17:78, ^S1 = 177:44 and ^P1 = 17:0148; respectively If the

economy is in the ‘high cost’state at time zero, the values are: ^D2(S0; K2) = 16:62,

^

S2 = 188:89, and ^P2 = 17:0305 respectively

3 For the full information case, we have: S1 = 169:49; S2 = 220:71;and D1 = 17:4076;

D2 = 17:4056 which are the values of the project depending on the state of theeconomy Notice both of these values are greater than the values provided by thebenchmark case and the standard real option case Therefore, when St reaches169:49; and the economy is in the ‘low cost’ state, the …rm should launch theproject according to the full information case optimal investment policy If St

reaches 220:7; and during the period of time when St is in the interval [169:49;220:71) the economy stays in the ‘high cost’ state, the …rm launches the projectimmediately, and the state of the economy does not matter anymore

Comparing the results from the three cases, we have the following remarks:

If the economy is in the ‘low cost’ state at time zero, the standard real optionapproach suggests an earlier exercise than the benchmark case since ^S1(177:44) <

S (182:72):If the economy is in the ‘high cost’state at time zero, the model implies

a later exercise than the benchmark case, since ^S2 > S : These results perfectly …tthe prediction of Corollary 2.1

The standard real option approach decreases the value of the project since both ^P1;and ^P2 are smaller than P1: This is also expected because in this case investmentcosts used to determine the optimal timing policy are far way from the actualcosts incurred when the project is launched The …rm is penalized by adopting amisspeci…ed exercising strategy

The full information case optimal strategy implies a lowest trigger value S1 =169:49 with the highest present value, of the project’s payo¤ to the …rm, 17:4076:

The values of the project dominates the values from the other two cases This isremarkable, since the result indicates an earlier exercise and a higher payo¤ thanboth the benchmark and the standard real option cases On the other hand, wealso notice that S2 = 220:71, but it is unlikely to happen that the project is going

to be launched at this threshold As we mentioned above, this can only happen ifthe economy is in the ‘high cost’ state before the …rst trigger value S1 = 169:49;and it maintains in the state until the value process St reaches the second triggervalue S2 = 220:71:For the continuous Markov chain Xt, the probability of having

no transition in some time s is

This simple example says that if there are di¤erent capital costs in di¤erent states

of the economy, which is common in the real world, the regime switching approach is abetter method of investigating investment timing problems Although the standard realoption is broadly used, it does often ruin investment opportunities