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Investment Lags
A Numerical Approach
M. Al-Foraih, P. Johnson and P. Duck
School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, U.K.
Keywords:
Real Option, Stochastic Models, Time-Varying Demand, Investment Lag.
Abstract:
In this paper we use a mixture of numerical methods including finite difference and body fitted co-ordinates
to form a robust stable numerical scheme to solve the investment lag model presented in the paper by Bar-Ilan
and Strange (1996). This allows us to apply our methodology to models with different stochastic processes
that does not have analytic solutions.
1 INTRODUCTION
Most investment projects take a long time to become
operational so there are often periods where a firm
will incur losses before the project starts generating
income. Such a period might be referred to as the
“construction lag”, “time to build ” or “Investment
Lag” (Costeniuc et al., 2008). These investment lags
can be quite lengthy which can result in a serious cost
for the investor, an example of which is described by
(MacRae, 1989) where it could take up to 10 years
to see the positive income when investing in a power
generating plant – similar situations can be found in
investment projects on natural resources. For exam-
ple, when an oil company buys a license from a gov-
ernment, it takes time to search fields and estimate
the fields’ reserve quantity before the beginning of oil
production. Thus, when evaluating a project such as
this the “lag” should be taken into consideration. If
the sale price of a firm’s product is modelled by a
stochastic process, then the lag brings added risk to
the project since the price may rise or fall during this
lag, resulting in a negative cash flow. This situation
and its effect on an investment has been studied by
(Gauthier and Morellec, 2000) and they implied that
it a has significant consequences on investment deci-
sions.
The use of option theory to value and assess in-
vestment decisions has a long history going back to
(Myers, 1977), but it was (Brennan and Schwartz,
1985) that first allowed the project to be mothballed
rather than abandoned so that it could be reopened at a
later date. They showed that if there was a fixed cost
to move between the states, the decision to start the
project would happen at a price higher than the deci-
sion to mothball. Following on from this, (Bar-Ilan
and Strange, 1996) applied investment lags on irre-
versible investments and they found that a lag can re-
duce the effects of uncertainty in an investment, since
the investor has more time to act on an unexpected
fall in the price or changes in the investment. In order
to generate the results for their model, they present an
analytic technique,(see Brekke and Øksendal, 1994,
for more details. The method as described by (Bar-
Ilan and Strange, 1996) is flawed in that it relies on the
particular form of the process, so they can only solve
the problem with a simple geometric Brownian mo-
tion. The contribution of this paper is to apply a more
generic numerical approach which can be extended to
many classes of stochastic processes. We present a
robust numerical technique for solving generic prob-
lems of this type.
2 MODEL FRAMEWORK
We follow the general framework as laid down by
(Bar-Ilan and Strange, 1996) in valuing a firm that
can pay (on delivery) k ≥ 0 units to exercise an ir-
reversible option to produce and sell 1 unit of product
per unit time forever. The marginal cost of production
is ω per unit, and both the future revenues and costs
are discounted at the rate of ρ. The project can later be
abandoned at a cost of l ≥0. The price of the product
P
t
follows a standard geometric Brownian motion
dP
P
= µdt + σdz. (1)
282
Al-Foraih M., Johnson P. and Duck P..
Investment Lags - A Numerical Approach.
DOI: 10.5220/0004920702820287
In Proceedings of the 3rd International Conference on Operations Research and Enterprise Systems (ICORES-2014), pages 282-287
ISBN: 978-989-758-017-8
Copyright
c
2014 SCITEPRESS (Science and Technology Publications, Lda.)