optimal timing of relocation

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Optimal Timing of Relocation

José Azevedo-PereiraDepartment of Management, and CIEF, ISEG, Portugal

[email protected]

Gualter CoutoDepartment of Economics and Management, and CEEAplA,

Universidade dos Açores, [email protected]

Cláudia NunesMathematics Department, and CEMAT, IST, Portugal

[email protected]

IST, UTL, Lisboa, 26 de Outubro de 2007

mailto:[email protected]



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Structure• Context

• Problem relevance

• Stochastic framework

• The decision problem

• Optimal timings of relocation and corresponding volatilities

– Truncated-exponential distribution

– Gamma distribution

• Comparative statics illustrations

• Traditional analysis

• Concluding remarks

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Context• Between 1970 and 1991 the net inflows of foreign direct

investment, in percentage of GDP, approximately doubled worldwide;

• Between 1991 and 2000, the speed of this process increased dramatically (six fold);

• Globalization tends to be understood as one of the most important questions faced by developed economies;– A consequence has been a significant wave of

production relocation;

• Cheap and readily available information imposes the need to make swift decisions and simultaneously reduce uncertainty, creating an environment prone to relocation.

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Problem relevance

• Focus on relocation has not been much explored in finance;

• Focus in a significant segment of the literature is not on the rationale for the decision to move from one location to the other, but on the corresponding consequences in terms of risk and return;

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Problem relevance (cont.)

• Flexibility regarding the place where to locate production creates value;

• Flexibility is conceptually regarded as an option, and valued using simple real options frameworks;

• Focus is on the perceived increase in efficiency needed to justify the decision to relocate from one place to another.

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Stochastic framework• McDonald and Siegel (1986) and Dixit and Pindyck

(1994): the decision maker needs to consider not only the advantages associated to the increased efficiency provided by newly available locations, but also the costs associated to the loss of the option to relocate;

• Locations identified and distinguished in accordance to their efficiency;

• Increased levels of efficiency will materialize in the ability to generate more output with same level of input or to use lower levels of input to generate the same output;

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Stochastic framework (cont.)• Arrival of new information regarding the availability of

new locations with improved levels of efficiency, modelled as a Poisson arrival process, in a stochastic environment;

• Risk-neutral firm, with a constant discount factor, r, like in Dixit and Pindyck (1994), were (.) denotes the corresponding cash flow;

• Dynamic model with an infinite planning horizon, assuming that when the firm invests in a new location, it incurs a sunk cost, I;

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Stochastic framework (cont.)• Two levels of uncertainty:

– Regarding the moment in which a new location becomes available;

– Regarding timing of subsequent increases in efficiency;

• Given the uncertainty inherent to the diverse factors, it is reasonable to admit that we are not dealing with a learning process with embedded continuous leaps in terms of information arrival. Therefore, it makes sense the use of an exponential distribution to model the time that elapses between the moments in which two consecutive efficiency optimizing locations become available;

• New locations become available independently of the firm (Poisson process, ).

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Stochastic framework (cont.)

• We let:

(t) – efficiency of the firm;

(t) – efficiency of the best location available at time t, with 0≤(t)≤(t);

(t) – is a random variable;

(t), t+, corresponding stochastic process;

For any t, best efficiency can be written as follows:

01

N t

jj

t U

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The decision problem

• The decision to relocate can be stated as a capital budgeting decision problem;

• Each time a new (and more efficient) location becomes available, the firm has to decide if it stays in the same place or if it changes to the new location;

• At each time, the firm has to decide between continuing in the present location or stop, and move to a new location;

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The decision problem (cont.)• There exists a critical value * such that, for every t:

– If (t)>* the firm decides to invest (stopping action) in this new location;

– If (t)≤* , the optimal decision is to continue in its current site, and wait for other locations to become available;

• We call * the optimal switching level (Dixit and Pindyck, 1994), found using the so-called value matching condition;

• T* is a stopping time, for the Poisson process N (Ross, 1996).

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The decision problem (cont.)

• If the firm decides to change its current location, than for all (t)>* :

(Termination payoff)

• If the firm does not change its current location, i.e., if (t) ≤* , the corresponding value is given by:

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Optimal timings of relocation and corresponding volatilities• Values of firms facing relocation decisions will depend on:

– Levels of efficiency increase;

– Dynamics inherent to the process that governs efficiency evolution, modelled using different density distribution functions;

• For each one we derive:

– Value of the firm, F(.);

– Optimal switching level, *;

– First two order moments for the time of adoption of the new location, E[T*] and E[(T*)2], respectively.

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Optimal timings of relocation and corresponding volatilities• Truncated-exponential distribution

– If we modelled increases in efficiency by an exponential distribution, then we would be implicitly assuming an increase in the efficiency that could be arbitrarily large, with non-zero probability;

– Given the historical background at this level, this possibility does not seem reasonable;

– We consider a bounded (M) increase in the efficiency.

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– The optimal switching level, * , is the solution of the following equation:

with:

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– Value of the firm is given by F():

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Optimal timings of relocation and corresponding volatilities• Gamma distribution

– Investment decisions are subject to several, eventually independent, risk factors;

– It makes sense to introduce distribution functions that enable the analytical treatment of more than one state variable, in order to approximate the modelling exercise to corporate reality.

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– The optimal switching level, * , is the solution of the following equation:

with:

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– Value of the firm is given by F():

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Comparative statics illustrations

• Parameters of the random variables were chosen so that they have the same expected value (but different volatility), i.e., every time a new location becomes available, the expected increase in efficiency will be the same in both cases;

• In particular, we consider the following instances:

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Comparative statics illustrations• We consider a Cobb-Douglas production function;

• We assume values for the output price, p; input price, w; discount rate, r; sunk cost of investment, I; initial efficiency, 0; and the rate at which new locations become available, ;

• Numerical results for these situations:

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0

10

20

30

40

50

60

70

0,1 0,3 0,5 0,7 0,9 1,1 1,3 1,5

Expected value of U

Opt

imal

sw

itchi

ng le

vel

Figure 1: Behaviour of * for the truncated-exponential distribution.


0

10

20

30

40

50

60

70

0,1 0,3 0,5 0,7 0,9 1,1 1,3 1,5

Expected value of UO

ptim

al s

witc

hing

leve

l

Figure 2: Behaviour of * for the gamma distribution.

The optimal policy seems robust in terms of the particular distribution used, as values of * are nearly the same for both distributions.

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2,3

2,4

2,4

2,5

2,5

2,6

2,6

2,7

2,7

2,8

2,8

0 50 100 150 200 250 300 350 400 450

Input Price

Op

tim

al s

wit

chin

g le

vel

ExpT

Gamma

Figure 3: Behaviour of * for the truncated-exponential distribution and for the gamma distribution as a function of the input price.


Higher levels of input related costs lead naturally to increases in the value of the option to delay an investment. Therefore, increases in input prices lead to increases in optimal switching levels.

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2,2

2,3

2,4

2,5

2,6

2,7

2,8

2,9

0 5.000 10.000 15.000 20.000

Investment Cost

Op

tim

al s

wit

chin

g le

vel

ExpT

Gamma

Figure 4: Behaviour of * for the truncated-exponential and gamma distributions as a function of the investment cost.


Higher levels of investment costs lead naturally to increases in the value of the option to delay an investment. Therefore, increases in investment costs lead to increases in optimal switching levels.

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2,0

2,5

3,0

3,5

4,0

4,5

5,0

0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09

Discount Rate

Op

tim

al s

wit

chin

g le

vel

ExpT

Gamma

Figure 5: Behaviour of * for the truncated-exponential distribution and for the gamma distribution as a function of the discount rate.


Higher levels of discount rate mean higher time value of money and, consequently, a higher potential loss for postponing the decision to relocate.

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Traditional analysis• The traditional capital budgeting decision suggests that an

investment should be implemented when the value of the corresponding operating net cash inflows exceeds the present value of the inherent investment costs;

• Assumes implicitly that the investment is reversible or, if irreversible, that it corresponds to a “now or never” type of opportunity;

• In ROA investment opportunities are seen as options;

• Traditional NPV tends to lead to premature investment, i.e., to non-optimal solutions that are significantly different from those based in a ROA framework.

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Concluding remarks• In a period of globalization and market integration, the

problem of relocation is especially relevant, given its socio-economic implications;

• Using a dynamic framework, we have analysed the problem of the optimal timing for the relocation of a firm that faces a constant and irreversible level of investment expenditure to move its production site and reached a new set of closed form solutions;

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Concluding remarks (cont.)• According to our research, the optimal timing of relocation

is significantly affected by the uncertainties related to both the expected rhythm that characterizes the arrival of information regarding the availability of new, more efficient locations sites, and the degree in efficiency improvement from one to the other;

• Our simulation results suggests the results of our model are in accordance with economic rationale;

• Our framework suggests that the timing of relocation is significantly longer than that inherent to the traditional NPV analysis.

optimal timing of relocation

Documents

levels of uncertainty

arrival of new information

available information

improved levels of efficiency

increased efficiency

terms of information

available locations

decision maker