THE COSTS AND BENEFITS
OF PROJECT-LEVEL COMMERCIAL RESEARCH
David G. Laughton
Department of Finance and Management Science
Department of Mining, Metallurgical and Petroleum Engineering
University of Alberta
Edmonton, Alberta CANADA T6G 2R6
S. Frimpong and J. M. Whiting
Department of Mining, Metallurgical and Petroleum Engineering
University of Alberta
Edmonton, Alberta CANADA T6G 2G6
3 January 1993
Rev. 16 February 1994
This research has been supported (for DGL) by the Natural Science and Engineering
Research Council of Canada, Imperial Oil University Research Grants, Interprovincial
Pipeline Co., Saskoil, Exxon Corp., the MIT Center for Energy and Environmental Policy
Research, and the Social Science and Humanities Research Council of Canada, and by
the Central Research Fund, a Nova Faculty Fellowship, the Muir Research Fund and the
Institute for Financial Research of the University of Alberta, and (for SF) by a Canadian
International Development Agency-University of Alberta Doctoral Fellowship.
ABSTRACT
The management of an option to undertake some project-level research may
depend jointly on the endogenous uncertainty to be resolved by the research activity and
any exogenous macroeconomic uncertainty that influences the project cash-flows. A
systematic examination, using modern asset pricing methods, of a class of simple
research options reveals a complex taxonomy of management patterns based on the
benefits to be supplied by the research, the duration of the research programme, and its
cost. The value of the project at its initiation is presumed to be, in any endogenous state,
a linear function of a single exogenous uncertain macroeconomic variable. The benefits
of the optional research programme are classified in a 2x2 matrix by whether the effect is
to reduce uncertainty about the slope of the project value function or its intercept, or to
increase the project value function by increasing the slope or decreasing the intercept.
This taxonomy is useful in a more general examination of project-level research
management.
THE COSTS AND BENEFITS
OF PROJECT-LEVEL COMMERCIAL RESEARCH
1.
INTRODUCTION
Project-level commercial research is an activity undertaken by an organisation
primarily to resolve some uncertainty relevant for the management of a single project. It
includes most natural resource exploration, market research, project feasibility studies,
and product design. It also includes some forms of process research. Understanding
when and how to perform project-level research is an important function for many
organisations in much of the economy. The management issues that arise in any
particular situation may depend jointly on the endogenous uncertainty to be resolved by
the research itself, and on any exogenous uncertainty that influences the project cashflows. We have found that the tools of modern asset pricing theory can be useful in
attempts to understand these issues.
In this paper, we analyse a class of simple research options, which nevertheless
reveals a complex underlying taxonomy of optimal management patterns. The purpose
of the paper is to outline the taxonomy that has been found. The results are useful in the
analysis of more complex situations.
To be specific, we examine research programmes where there is only one possible
activity that, once begun, will take a given time, have a given cost, and resolve some
given time-independent project uncertainty. There is no uncertainty about the research
itself nor is there any research dynamics. [insert text xxx]
We look at projects where the amount of each project cash-flow, after the project
is begun, may depend on the project state, the choice of project design, and the time since
the project start. The amount of each project cash-flow is also a non-decreasing linear
function of the contemporaneous realisation of a single exogenous economic time-series
variable.
For ease of exposition, but without loss of analytical generality, we take this
exogenous variable to be the project output price. The evolution of the price and its
valuation are both modelled to be based on by a time-independent scalar lognormal
random-walk diffusion process. Under these circumstances, the exogenous economic
state at any time is determined by the price at that time.
The management options include an initial perpetual timing option on the choice
between doing the research (the "research" choice) or beginning the project without doing
the research (the "uncertain-start" choice). If the research choice is made, there is a
subsequent perpetual timing option on the project alone. The project design choice is
made when the project is begun. The set of design choices available may depend on the
project state at that time. Finally, the uncertainty resolved by the research is resolved by
starting the project, if the research is not done.
In our analysis, the "project value function" is a useful construct. This is the
value of the project at the time the project is begun considered as a function of the project
state (which may be the "uncertain" state before the project uncertainty is resolved, or
one of the "known" states after), the contemporaneous output price (which determines the
exogenous economic state), and the chosen project design. In the class of models just
outlined, the project value is a time-independent non-decreasing linear function of the
price. Therefore, the project value function may be characterised by its slope with
respect to price, which we call "the effective output" of the project, and its price intercept,
which we call "the effective average cost".
The benefits of a research programme can be classified by the type of decision
that would be influenced by the information gathered. In the situations we consider,
these are the timing of the project, and the project design. We have found it useful to
begin by examining those situations where timing is the only issue. If the research
reduces the uncertainty in the project cash-flows, then the project manager may make
finer distinctions about the project value function. This permits a more finely-tuned
management of the project timing option in response to exogenous economic movements,
even if there is only one project design possible (so that considerations of design choice
are moot). If different project designs are possible, information gathered by undertaking
the research may allow the manager to make better design choices, and thus obtain, with
any given timing decision, more project value in some states.
In this paper, we model uncertainty reduction in the cash-flows and its effect on
timing explicitly. However, we do not provide a specific optimisation model for the
design choices. Instead, we treat the effects of such choices in reduced form by
specifying directly, for any given state, the value from the best of the possible design
choices in that state. Moreover, we presume, again for simplicity, that the resulting
optimised project value function (which, by abuse of terminology, we also call the project
value function) is also linear in the price.
Another way of classifying the benefits of research is to determine the aspects of
the potential project value that are affected by the research. In our examples, the project
value depends on the two parameters of the project value function: the effective output
and the effective average cost.
Based on these two classifications, we have constructed a two-by-two matrix of
research benefits to consider. We examine situations where the research will refine
knowledge about the project value by resolving some uncertainty about the effective
output or the effective average cost of the project, or increase the project value in a
known way by allowing a better design optimisation that increases the effective output or
decreases the effective average cost. We examine each of these four types of benefits
separately.
The set of situations that we consider is laid out in detail in Section 2. There is a
common model for the output price and its valuation, and, for each situation, a project
value function and a model of the possible research activity. In an Appendix, we show
how a linear project value function arises from a linear project cash-flow model, and how
the effective output and effective average cost of the project value function may be
related, in a particular class of project models, to the stream of actual project outputs and
costs.
Sections 3 and 4 consider situations where the only benefit of the research would
be to reduce the uncertainty in the project cash-flows, given a project timing policy. In
Section 3, we show that research should not be undertaken if the reduction or elimination
of uncertainty in the effective output is the only benefit. In Section 4, we examine
uncertainty in the effective average cost, and begin to show the importance of research
duration as a costly feature of research.
Sections 5 and 6 focus on research benefits that stem solely from an allowed
improvement in the project cash-flow pattern, given a project timing policy, without any
reduction in the uncertainty in those cash-flows. In Section 5, we consider the benefit of
higher effective output. One interesting result is that there are situations where the
research should be done if the output price is either high or low enough, but not in the
middle. This can happen if there is a moderate amount of price uncertainty and normal
backwardation in the output forward prices, and if the research takes a moderate amount
of time and costs a moderate amount. Finally, in Section 6, we consider the benefit of
lower effective average cost.
In Section 7, we summarise, conclude and describe potential extensions.
2.
2.1.
THE MODEL
The Output Price and Valuation Model
The model for the output price is specified by an initial price and the pattern of
possible price changes at subsequent times. The price change over any small time
interval is modelled to be a normal random variable. The expected price change, over a
small time interval of duration dt beginning at a time t, is modelled to be a known
proportion, adt, of the price at that time, Pt. For simplicity of presentation, we take a to
be time-independent. The variance of the price change is modelled to be proportional to
the length of the time interval as well, and to the square of the price at the beginning of
the interval. The constant of proportionality is called the square of the volatility, s2,
which is modelled to be known and, for simplicity, is also taken to be time-independent.
This price model is called a lognormal random-walk diffusion model. The
dynamics may be represented by the following equation for the uncertain change in price
from time t to time t+dt:
(1)
where dzt is a normal random variable with variance dt, and independent of previous
price changes.
When this equation is integrated over time, we find that the prices are, in any state
at any time s, a set of lognormal variables with expectations and associated covariances:
(2)
where Es(Pt | Ps = P) is the expectation in a state at time s of the price at time t, if the
price in the state at time s is P. Notice that the associated covariances at time s do not
depend on the price at that time, and are known with certainty at any time.
If the price behaves in this way, there is at least one simple consistent model for
the valuation of claims to future prices. The expected proportional rate of change at any
time in the value of such a claim (also called its expected rate of return) is the sum of the
risk-free rate of return, r, and a risk premium proportional to the amount of uncertainty in
the price as measured by the volatility of the relevant price expectation. The
proportionality constant in the risk premium is called the price of risk for the uncertainty
in the output price. The volatility in any price expectation can be shown, for this model,
to be the volatility of the price itself, s. For this model to be consistent, future risk-free
rates of return and prices of risk must be modelled not to be uncertain. For simplicity of
presentation, we take them also to be time-independent.
The difference at any time between the expected rate of return for the claim to the
next price and the expected proportional rate of change in the price itself is called the
rate-of-return shortfall, c, in the price. It parameterises the term structure for the value of
claims to future prices:
(3)
where Vs(Pt | Ps = P) is the value at time s of the claim to the price at time t, if the price
at time s is P.
This price and valuation model is essentially the same as that used by Brennan
and Schwartz (1985) in their analysis of an operating mine, and has been used elsewhere
in the "real options" literature.
Derivative asset valuation methods may be used to find the value of the claim to
any future cash-flow, the amount of which is determined by the price at the time of the
cash-flow (Jacoby and Laughton 1991, 1992, Cox et al. 1985). This will be of use in our
analysis. The resulting formulation is:
Vs(CF | Ps = P) = ∫dms (Pt | Ps = P) XCF(Pt) ,(4)
where Vs(CF | Ps = P) is the value at time s of the claim to the cash-flow, CF, if the price
at time s is P; t is the time of the cash-flow; XCF shows the functional dependence of the
cash-flow amount on the contemporaneous price, Pt ; and dms(Pt| Ps = P) is the state
price measure at time s for the output price states at time t, if the price at time s is P. For
this price model, the state price measure is given by:
dms (Pt | Ps = P) = exp (-r (t - s)) dLN [Pt ; P exp ((r - c)(t - s)), s2 (t - s)] ,(5)
where dLN[.;E,var] is a lognormal measure with expectation, E, and associated variance,
var.
Equation 4 states that the first moment of a cash-flow amount, with respect to the
output price state price measure, is the value of the claim to the cash-flow. By definition,
the expectation of the cash-flow amount is its first moment with respect to the output
price probability measure. Therefore, the discounting for time and risk that is inherent in
the valuation may be seen by comparing how these two measures affect first moments.
The probability distribution at time s for the output price Pt, if the price at time s
is P, is:
dLN [Pt ; P exp (a (t - s)), s2 (t - s)] .(6)
How does the state price measure differ from this?
Notice that the state price measure incorporates a time discount factor:
exp (-r (t - s)) .(7)
This directly provides the source of time discounting in the valuation.
The source of risk discounting is more subtle. Notice that there is a difference in
the expectations that respectively parameterise the probability measure and the lognormal
measure that is imbedded in the state price measure. The true price expectation is used in
the probability measure, while the risk-discounted price expectation is used in the state
price measure. This makes the expectation of the lognormal measure imbedded in the
state price measure less than that for the probability measure. More relative weight is
given in the state price measure to states with a low commodity price. If commodity
prices are correlated with the health of the economy, then these are the states in which
most investors would have a greater preference for receiving extra cash on the margin.
Therefore, in the formation of their state prices, this upward risk-adjustment to the state
probability makes some sense. Moreover, it also lowers the first moment of any cashflow amount that increases with the output price. This is the source of risk discounting.
Notice that valuation depends on the risk-free rate of return, r (which is the only
parameter relevant in the valuation of risk-free cash-flows), the rate-of-return shortfall, c,
of the output price (which is the only parameter relevant in the valuation of pure output
price claims), and the volatility in the price, s (which, together with the other two
parameters, determines the valuation of claims to payoffs that have a nonlinear
dependence on the price).
The integral formulation of the valuation problem in Equation 4 is equivalent to
the following fixed boundary problem, which is of the type first formulated by Black and
Scholes (1973) and refined by Merton (1973):
(8)
where V(P,s) is defined to be Vs(CF | Ps = P).
This differential equation results from setting the risk-free rate of change in value
equal to the actual change in value of an instantaneously risk-free portfolio made up from
the claim to the payoff under consideration and an appropriate amount of the claim to the
underlying price. The terminal boundary condition states simply that the value of the
claim to the cash-flow at the time of the cash-flow is the amount of the cash-flow. The
lower boundary condition comes about because the price is guaranteed to remain zero, if
at any time it is zero. Therefore, the value in such a state is the value of the risk-free
claim to the cash-flow that will occur if the price is zero. The upper boundary condition
results from an asymptotic analysis of the integral in Equation 4 . The particular form of
the condition in Equation 8 is valid if the amount of the cash-flow being valued is a
smooth function of the price, and increases linearly, or less than linearly, with the price at
high prices.
If the cash-flow amount to be claimed depends on the contemporaneous price, but
the timing of the cash-flow is at the discretion of the claim-holder, then the valuation
problem becomes:
,(9)
where P* is the set of prices where the claim-holder claims the cash-flow, if it has not
already been claimed.
The time derivative drops out of the differential equation, because the value of the
opportunity does not depend directly on the time of the valuation. The terminal condition
is replaced by an unstated transversality condition at infinite time. The boundary
conditions are supplemented by two other conditions, which are used to specify the states
in which an outstanding claim is made. By the first, the claim value in any state in the set
of states where the claim is made, P*, is set equal to the cash-flow amount claimed. By
the second, the value in any state where the claim is not made is greater than the cashflow amount that would otherwise be received.
In the particular parameterisation of the price and valuation model that we use in
our examples, the risk-free rate of return, r, is 3% per year, the rate-of-return shortfall, c,
is 5% per year, and the volatility, s, is 20% on an annual basis. Figure 1 shows, for this
model, the term structure of 10th and 90th percentiles of the price (the solid lines) if the
term structure of medians is flat and scaled to be 1 (the central dotted line). It also shows
the term structure of values of claims to the price (the dashed line).
Our results would differ with changes in these parameters, and, in particular,
would exhibit qualitative differences in some instances if the risk-free rate of return were
more than the rate-of-return shortfall. In these circumstances, the term structure of output
forward prices, given by:
P exp ( (r - c) t) ,(10)
would increase with the term rather than decrease. We do not address the comparative
statics of the price and valuation model parameters in this paper, although we note the
results that may depend on having a decreasing term structure of forward prices (i.e., on
the existence of normal backwardation).
2.2.
The Project Model
In the examples considered in this paper, the value of the project when begun is a
non-decreasing linear function of the output price at that time. (An example, which is
described below, is given by the dashed line in Figure 2.) As is shown in the Appendix,
this linear dependence occurs, for example, if the project cash-flows at any time after the
project is begun are themselves linear in the contemporaneous price. As was noted in the
Introduction, the slope of the linear project value function is called the "effective output"
of the project and the price intercept is called the "effective average cost". If all the
production and sales for the project were to occur instantaneously, and if the project cashflow were a revenue (the product of price and output) less a price-independent cost, the
effective output and average cost would be the actual output and average cost. Hence the
names.
For any project examined in this paper, we specify the effective output and
effective average cost directly, not by specifying any underlying project parameters. We
work with a basic situation where the effective output is 30M units and the effective cost
is $20/unit. These parameters merely set the price and value scales to be considered in
this analysis, and can be changed, under appropriate scaling of the results, without loss of
generality.
The value of the option to undertake the basic project is shown in Figure 2 as a
function of price. The dashed line shows the value if the project must be started
immediately or not at all (the "now-or-never" option). The is given by the positive
portion of the project value function. In this circumstance, the project should be begun at
any price above the effective average cost of $20/unit. If there is an option to wait to
begin the project (the timing option), the value is given by the solid line. This option is
equivalent to an American call option on the output price, and is found by using Equation
9, where the amount of the cash-flow to be claimed is given by the project value function.
There is a region at low price where waiting is optimal and the differential equation
holds, and a region at high price where the project should be started immediately. The
boundary between these two price regions, which is called the critical price of the option,
is proportional to the effective average cost. For the price and valuation model used in
this paper, the proportionality factor is 1.63. This means that, in the basic situation, the
project should be begun at prices above $32.65/unit (to the closest $0.05/unit), where, as
Figure 2 shows, the now-or-never value and the timing-option value are the same.
Similar calculations may be done to find the value and management of each post-research
project timing option within the situations considered in this paper.
2.3.
The Research Programme Model
The model of the research programme is parameterised by the benefit provided by
the research, the duration of the research, and the research cost. We presume that the cost
is incurred at the time that the research is begun.
In our analysis of each type of research benefit, we first use Equation 9 in the
manner described above to evaluate the states where the information determined by the
research is known (the known states), and a situation where no research can be done (the
"no-research" situation). We then evaluate the situation where the research can be done
instantaneously at zero cost (the "free-research" situation) by finding the expectation,
across the post-research project states, of the value of the timing option for the project
then. The no-research and free-research situations are the polar cases of no information
and free information. If the project opportunity were to have the same value in these two
polar cases, it would never be optimal to undertake a research programme that would take
any time, or involve any direct cost, and the analysis would be over.
Otherwise, we examine a set of situations, where, in each situation, one research
programme is possible. The research programmes differ by duration and direct cost.
Each has a duration of zero, 1 or 2 years, and a cost of zero (if the duration is not zero),
$1M, $2M, $5M or $10M. The value of the research choice at the time that choice is
made, gross of the research cost, may be calculated using Equation 8 over the duration (if
any) of the research programme. The amount of the cash-flow to be claimed is again
given by the expectation, across the post-research project states, of the value of the timing
option for the project then. The value and management of the initial timing option is
determined using Equation 9 where the amount of the cash-flow to be claimed is the
larger of the optimised project value at an uncertain project start and the research value
net of the research cost. We explore this initial timing option value by examining the
incremental value of the option to undertake the research compared to the no-research
situation, and the value lost compared with the free-research situation.
The benefits of the research in each analysis are specified as part of the
description of that analysis.
3.
NO BENEFIT: REDUCING UNCERTAINTY IN THE EFFECTIVE OUTPUT
Figure 3 shows the value of the option to undertake a project, as a function of the
contemporaneous price of the project output, under various conditions where a
programme of research can be used to reduce some uncertainty about the effective output,
while not affecting the prior distribution of this parameter or revealing any new
information about the effective average cost. In the situations considered, there are, for
simplicity, two equiprobable known project states, one with a "high" and one with a
"low" effective output. This uncertainty in the effective output is ±4M units of output
around an expectation of 30M units. The effective average cost is $20/unit in each
known state.
The graphs again come in pairs. The dashed line in each pair shows the value in a
given project state, if the project must be begun immediately or not at all (the "now-ornever" value), while the solid line shows the value if there is an option to begin the
project at any time.
The two outer pairs of lines are the values in the two possible known project
states. If the timing of the project is optional, it will be begun in either state if and when
the price is above $32.65/unit. This critical price is the given multiple,1.63, of the
effective average cost, $20.00/unit, in each state. (Note again that the timing-option and
the now-or-never value are the same above the critical price.)
The central pair of lines shows the value of the project option if no research is
possible. The manager must manage the option without knowing which of the two
known project states will be realised. Notice that the effective average cost, by
construction, is $20.00/unit, just as it is for the two possible known project states.
Therefore, the critical price for project initiation is again $32.65/unit.
Because the timing of the project, which is the only decision that arises in this
situation, is the same whether the research is undertaken or not, the information from the
research, while it refines the value of the project timing option, is of no value itself. This
is confirmed by the observation that the "no-research" value is equal to the expectation of
the value in the known states, which is also the value of the project option if the research
could be done instantaneously and with no cost. If the option to undertake free research
does not add any value, then the option to undertake costly research, or research that
takes time, can only detract from value if a plan is introduced to exercise that option.
Therefore, costly research or research that takes time should never be contemplated.
4.
BETTER TIMING: REDUCING UNCERTAINTY IN THE EFFECTIVE
AVERAGE COST
In the following set of examples, a research programme, if undertaken, would
resolve an equiprobable uncertainty of ±$4/unit about an expected average cost of
$20/unit, and accomplish nothing else. The effective output is 30M units in each known
state.
The scope of the problem is shown in Figure 4.1, where the value in various
situations of the option to undertake the project is graphed against the output price. The
dashed lines again show the value in now-or-never situations, while the solid lines show
the value for timing options. The two outer pairs of graphs are again the values in the
two known project states, with a low effective average cost of $16/unit and a high
effective average cost of $24/unit respectively. The central dashed line and the lower of
the two central solid lines are the values for the no-research situation. The other central
solid line shows the values for the timing option if instantaneous research is possible at
no cost. This free-research value is the expectation at each output price of the knownstate timing-option value.
If waiting is allowed, the project would be begun in the low-cost known state at
any price above $26.10/unit, in the high-cost known state at any price above $39.20/unit,
and in the no-research situation at any price above $32.65/unit. Therefore, without the
information to be given by the research, the project might be undertaken at lower prices
(between $32.65/unit and $39.20/unit), if the cost is high, or delayed at higher prices
(between $26.10/unit and $32.65/unit), if the cost is low, than would be desirable if the
information given by the research were available. Therefore, the option to undertake the
research may have value. Indeed, the free-research value is greater than the no-research
value at prices below $39.20/unit.
The spread between the no-research and free-research timing-option graphs shows
the maximum incremental value that can be obtained from the option to do the research,
or, equivalently, the maximum value that is lost because the research involves a cost or a
delay rather than being instantaneous and without cost.
Notice that this difference goes to zero at zero price, because all values are zero if
the price is zero. In Figure 4.2a (which is described in detail below), we show that the
difference is positive at low positive prices where the project manager should wait to act.
This incremental value is positive because the research option enhances the value of the
payoff for which the investor is waiting.
The maximum research option value is also zero at prices above $39.20/unit,
which is the critical price for immediate initiation of the project in the high cost project
state. Immediately starting the project is optimal at those prices regardless of the
research results, which makes the option to undertake the research worthless.
We now examine situations where there is an option to undertake research that is
costly or takes time, beginning with examples where the research is costly but takes no
time.
4.1.
Instantaneous Research
Figure 4.2a shows the incremental value of the project, if there is an option to
undertake instantaneous research, over the value in a situation where no research is
possible. The top curve is for a research programme that has no cost (the free-research
situation). It is the difference between the free-research and no-research lines (the two
central timing-option graphs) in Figure 4.1. The rest of the curves are for options to
undertake research that costs $1M, $2M, $5M, and $10M. The value decreases with the
research cost.
Notice, as mentioned above, that the value is positive at low prices, and goes to
zero at zero price. It is also zero at high prices. In this high-price region, the project is
begun immediately without doing any research. If the research is essentially free, this
happens at prices at or above the critical price of the high cost project, $39.20/unit.
(Table 1 lists, for all situations discussed in Section 4, the lower boundary of each price
region where a different action is optimal, except for a low-price waiting region if one
exists.) As the research cost increases, there is a shift away doing the research (the
"research" choice) toward beginning the project without doing the research (the
"uncertain-start" choice), and the lower boundary of the high-price "uncertain-start"
region decreases .
The curve for the free-research situation shows that the most value that can be
added by having a research option is just over $20M, and it occurs at a price of just under
$28.50/unit. If the research were to cost more than this maximum value, it should not be
done under any circumstance. Finally, the price where each research option has its
maximum incremental value is the same for all the situations in this series.
Figure 4.2b shows the research option value from the opposite vantage point by
graphing the value lost from not being able to determine the project state instantaneously
and at no cost. The top curve is the value lost if no research can be done. It is again the
difference between the free-research value and the no-research value, and is the same as
the top curve in Figure 4.2a. The remaining curves are the value lost relative to the freeresearch situation in situations where the research costs $10M, $5M, $2M, and $1M. The
value lost, when compared to the free-research situation, increases with the cost.
Notice that the value lost has a maximal plateau at the research cost, if that cost is
less than the maximum cost for which the research might be done. These regions of
constant value lost (bounded, for the $10M research option, by the points labelled "a" and
"b" in the figure) are the price regions where the research choice should be made (the
"research" region). They occur around the price, $28.50/unit, where the most value is lost
in the no-research situation, and between $26.10/unit and $39.20/unit, the critical prices
for beginning the project in the two possible known cost states.
At prices above and below each plateau in Figure 4.2b, the value lost decreases to
zero. The region below the plateau extends in each case to zero price. If the price is in
this region, the project manager should wait to find out if the price will increase
sufficiently to make doing anything worthwhile. This called the "waiting" region. If the
research is essentially (but not quite) costless, waiting is optimal below $26.10/unit, the
lowest price for which the project would be begun if the research showed that the project
had a low effective average cost. As we consider situations with larger and larger
research costs, there is the shift away from the research choice toward waiting, and the
boundary between the waiting region and research regions increases.
The region of declining value above each plateau extends in each case to the
critical price, $39.20/unit, for beginning the high cost project, which is marked "d" on the
figure. In this region of prices, the project manager should do one of two things.
At lower prices within this region (between "b and "c" for the situation with a
$10M research option), the project manager should "dither" between the possibility of
beginning the project without doing any research, which it should do if the price goes up
enough (above "c" for the $10M research option), and doing the research, which it should
do if the price goes down enough (below "b" for the $10M research option). This
dithering region is reflected, on the one hand, by a level of value lost less than the
research cost, because it is possible that the research will not be undertaken if the price
increases enough, and, on the other hand, by a level of value lost that is less than the
value lost in the no-research situation, because it is still possible that the research will be
undertaken if the price decreases enough.
At higher prices (between "c" and "d" for the $10M research option), the value
lost is the same as would be lost in a situation in which no research is possible. At these
prices, the project manager should decide to begin the project immediately without the
research. Below the critical price for high-cost project, there is still value lost relative to
the value of a free-research situation because, if the effective average cost of the project
turns out to be high, a project start would be premature when considered ex post facto.
This is costly when compared to what would be done if the free research option were
available.
The upper and lower boundaries of the dithering region are lower in price, the
more the cost of the research: There is a shift away from the research choice as its cost
increases. Moreover, the dithering region expands in size if the research cost is larger.
For research costs above the maximal value of the free research option, the research
region does not exist and the dithering region is subsumed by the low-price waiting
region.
4.2.
Research Duration as a Cost
The time that a programme of research takes is also a costly feature of that
research. This is shown in Figures 4.3a-c for research programmes that take one year to
complete. Figures 4.3a and 4.3b are analogous to Figure 4.2a and 4.2b.
Figure 4.3a shows the incremental value of the free (no cost, zero duration)
research option over the no-research situation. It also shows the incremental value of the
research options where the research takes a year and costs $0, $1M, $2M, $5M, or $10M.
The free-research value is once again the top curve, and is the same as in Figure
4.2a. The value of each of the 1-year research options is less and decreases with the
research cost. Comparison with Figure 4.2a shows that the "uncertain-start" region
(where the project is begun without doing the research) extends to lower prices, and the
maximum values are less and occur at lower prices than if the research had the same cost
and took no time.
Any research programme that might be undertaken must cost no more than about
$12M, in contrast to the maximum of over $20M if the research were to take no time. All
of this reflects the cost of the research duration, and the shift, with increased research
time, away from the research choice toward either waiting or an uncertain project start.
Finally, the price where the maximum value of the research option occurs decreases with
the cost, from over $26/unit to about $25/unit.
Figure 4.3b shows that the value lost for these options relative to the free-research
situation again increases with research cost toward the value lost if there can be no
research. The value lost in the research region is not constant, but increases with the
price: The higher the price, the greater is the value of the potential project cash-flow, and
the more costly is any delay in receiving those cash-flows, including a delay caused by
research.
Because value lost increases with price in the research region, it is difficult to
determine, from Figure 4.3b, the location of the boundaries between the waiting, research
and dithering regions. These are shown in Table 1, but they are also illustrated in Figure
4.3c, much as there are in Figure 4.2b for the instantaneous research options. Figure 4.3c
exhibits graphs of the incremental value lost for the costly 1-year research options, not
respect to the free-research situation, but over the no-cost 1-year option.
The value lost, when it is defined with the effects of research duration cancelled
out in this way, has a flat maximum at the direct cost of the research, if that cost is below
the maximum for which a 1-year research option would be undertaken at some price.
Figure 4.3c shows, in the same manner as does Figure 4.2b, the research, waiting and
dithering regions.
An examination of Table 1 shows that the research region generally is smaller, if
the research takes a year, than it would be if it were instantaneous, and it occurs at lower
prices. The smaller research regions reflect the increased indirect cost of a 1-year
research option. The lower research regions reflect the desirability of preparing for the
possibility that the output price at the time the research is completed will be in the region
where an immediate project start would be desired if the low cost project exists. If the
term structure of output price medians is flat, and if the price is at the lower boundary of
the research region when the research is begun, the probability that the price might be in
this "project-start" region at the end of the research period ranges from 13% for the $1M
research programme to 22% for the $10M research programme. With higher research
costs, the project manager is less willing to purchase insurance against the eventuality of
what would be considered ex post facto delays in the project. This increases the
probability at the marginal trade-off that such delays might happen.
The dithering region is lower and larger for the 1-year than for the instantaneous
research options. In this respect, increased research time has the same qualitative effect
as increased research cost. The region is larger, because there is more at stake in
committing to a research. It is lower, because the indirect cost of the research duration
shifts the trade-off at the margin in favour of an immediate project start.
4.3.
Longer Research Durations
If the research time is made longer, the effects noted in this analysis persist and
become more pronounced. The 2-year no-cost option value has a maximum of about
$8M, down from $12M for the 1-year no-cost option. This means, among other things,
that the opportunity to undertake a $10M 2-year research programme would never be
exercised.
The maximum research option value occurs in the price range of $22/unit to
$24/unit, down from $25/unit to $27/unit. The value lost increases more quickly with
price, because the longer delay in production has more of an effect for a given price
change.
The lower boundary of the research region, if it exists, is consistently shifted
down by about $2.5/unit from where it is for the 1-year research options. The upper
boundary is shifted down by about $4/unit. The decrease in the lower boundary of the
uncertain-start region is about $0.8/unit for the no-cost option ranging down to $0.25/unit
for the $5M option. These boundary shifts mean that, because of the costliness of the
extra research duration, the waiting region and the research regions are further
compressed, and the dithering and uncertain-start regions are further expanded.
5.
MORE PROJECT VALUE: MORE EFFECTIVE OUTPUT
In the following set of examples, we examine the effects of research options, with
the same costs and durations as those examined in Section 4, in situations where the
research does not resolve any uncertainty about the cash-flow, but does allow the project
manager to avoid losing 2M units of effective output from the 30M units that is otherwise
available. The effective average cost is $20/unit.
Figure 5.1 shows the value of the venture with no research option and with free
research available. Notice that the free-research value is always strictly greater than the
no-research value, and that this difference increases with the output price.
5.1.
Instantaneous Research
Figure 5.2a shows the incremental value (over the no-research situation) of the
instantaneous research options. (We show this value out to a price of $70/unit because
there is some important structure for 1-year research options at these high prices, and we
wish to present a consistent set of figures.) The value of the research options increases
with the output price, and decreases with the cost of the research. The value increase
continues into the high price regime. This is due to the increasing value of the output that
is not lost if the research is undertaken.
Figure 5.2b shows the value lost relative to the free-research situation. For any
costly research option, this graph again has a flat maximum at the research cost. Costly
research is not undertaken before it is needed, and so, in the region below the critical
price for initiating the post-research project, $32.65/unit, the project manager waits. The
critical price for undertaking costly research increases with the cost of the research, from
$32.65/unit, as there is a shift away from doing the research. (These boundaries are
shown, along with other boundaries discussed in Section 5, in Table 2). An uncertain
project start is never an optimal alternative.
5.2.
Research Duration as a Cost
The pattern of optimal behaviour is again more complex if the research takes
time. This is shown in Figures 5.3a-c for research programmes that take 1 year to
complete. These are analogous to Figures 4.3a-c.
Figure 5.3a shows the incremental value over the no-research situation of the free
research option (for which the research has no cost and takes no time), and of the 1-year
research options. The free-research value is again the top curve, and is the same as in
Figure 5.2a.
We may begin our examination of Figure 5.3a by looking at the research option
values at high output prices. At these prices, the research (if any is possible) and then the
project would be undertaken as soon as possible. Waiting would not be optimal.
Undertaking the year of research delays the receipt of the project value. This delay has
two effects.
First, the delay causes some extra discounting of the revenues from the project.
When viewed from the time the research is begun, as opposed to when the project is
begun, this is equivalent to an effective loss of output. This loss brings the effective
output for the research choice, viewed from the commencement of the research, down to
28.5M units at high prices. Therefore, the net gain at high prices from doing the research
decreases from 2M units of effective output for the instantaneous research options to
0.5M units fro the 1-year options. This is the slope of the high price part of the graphs in
Figure 5.3a of the incremental research option values.
Second, the price intercept of the asymptote to the high-price part of the value
function for the research choice, viewed from the commencement of the research, of a 1year option is different from the effective average cost of the uncertain-start choice. As
an example, consider the no-cost research options. The research delay effects less
discounting in the value of the costs of the project than in the value of the revenues,
because of the normal backwardation in the forward prices. The smaller decrease in the
value of the costs than in the effective output increases the effective average cost of the
asymptotic value function for the research choice up to $20.40/unit from $20/unit, the
level with no research delay.
Thus, we have a situation, where, with no waiting allowed, the uncertain-start
value function has a lower slope than research value function, 28M units as opposed to
28.5M units, and also a lower price intercept, $20/unit as opposed to $20.40/unit. The
two value functions intersect at a price of $41.48/unit. Above this price, if waiting is not
allowed, doing the research gives the higher value, while below it, immediately starting
the project is better. Below a price of $20/unit, abandoning the project dominates both.
Below the critical price between the research and uncertain-start regions, the research
option provides no incremental value, if waiting is not allowed, and the graph for each
situation in the equivalent of Figure 5.3a would show zero incremental value below the
critical price for that situation. The critical price increases with the cost of the research, as
there is a shift away from doing the research. Note that this is the behaviour of the price
intercept of the asymptote to the high-price part of the graphs in Figure 5.3a.
The value of the actual 1-year research options in Figure 5.3a exhibits a more
complex structure at low to medium prices. This is because of the timing options built
into each situation.
First, recall the optimised value function constructed above in circumstances
where no waiting is allowed. Research, if possible, is optimal only at high prices.
However, if the research allows the project manager subsequently to wait to begin the
project, then, because of the value in that option to wait, there may be a low price
research region as well. Indeed, if the research is costless, the research choice has
positive value at all positive prices. Therefore, it dominates abandonment at all prices
and an uncertain project start at low enough prices. At a low enough research cost, part
of this research region still exists. However, abandonment would again dominate at low
prices. Therefore the optimised value function, if research with subsequent waiting is
allowed as an option, would be constructed once again from a low price abandonment
region and a high price research region. However, there may also be an intermediate
uncertain-start region. If an uncertain-start region does exist, there may also another
research region below the uncertain-start region and above the low-price abandonment
region
Given this structure for an optimised value function, the timing option to claim
this value is best managed by having a waiting region at low prices and a research region
at high prices. However, there may be an intermediate structure, if the optimised value
function has any intermediate structure itself. If an intermediate uncertain-start region
exists in the value function, it may persist in the structure with the timing option, buffered
from the high-price research region by a dithering region. Similarly, if the lower-price
research region exists in the value function, it may persist as well, buffered from the
uncertain-start region above it by a dithering region as well.
We may now return to Figure 5.3a. An increase in the value function for the
payoff of a timing option (in this case, for the initial timing option, caused by the
presence of the option to do the research) always increases the value of the timing option
when it is far enough "out of the money". At zero price, the value of the option to
undertake the project, with or without the additional research option, is zero. Therefore
the incremental value of the research option increases from zero at zero price. However,
as the price enters the region in which the project manager would want to begin the
project if no research were possible, the effect of the delay caused by the research starts
to decrease the incremental value of the research option. This continues until the price
approaches the high-price regime, where, as is shown above, the option to undertake the
research increases in value relative to the no-research value.
Note that, if the cost of the research is low enough, the incremental value of the
research option remains positive and there is no intermediate structure where an uncertain
project start is optimal. However, if the cost of the research is high enough in this set of
situations, the incremental value of the research option does go to zero and there is a
region where an uncertain project start is best. It remains to be seen, whether, for these
research costs, there is also a lower-price research region below the uncertain-start
region.
Figure 5.3b shows the value lost relative to the free-research option. Value lost
increases steadily, because either output is lost or time is lost. It is difficult again to
determine from Figure 5.3b the location of the boundaries between the waiting, research
and dithering regions. These are shown in Table 2, but they are also illustrated by Figure
5.3c, which, again, graphs the value lost in the no-research situation, and the costly 1year research options, compared to the no-cost 1-year option. Figure 5.3c also shows that
there is an uncertain-start region at intermediate prices if the research cost is more than a
little less than $2M (given by the local minimum in the "no research" graph at just above
a price of $40/unit). It also shows that, for research costs from there up to just below
$15M (the local maximum in the same graph at just below $30/unit), there is lower-price
research region.
For each option, the research region (or lower research region, if there are two
research regions) once again has a lower price boundary than for the equivalent
instantaneous option. This reflects again the desirability of preparing for the possibility
that the output price, at the time the research is completed, will be in the region where an
immediate project start would be desired with the information provided of the research at
hand. If the term structure of output price medians is flat, the probability that the price
might be in this post-research project-start region at the end of the research period, given
that the price is at the lower boundary of the research region when the research is begun,
ranges from 21% for the $1M research programme to 36% for the $10M research
programme. Again, a higher cost for the research makes the project manager less willing
to purchase insurance against the ex post facto eventuality of a delay in the project.
The rest of the structure has already been discussed. The uncertain-start region
and the small dithering regions between the uncertain-start region and each of the
research regions also increase in size with the cost of the research.
5.3.
Longer Research Durations
If the research duration is long enough, the effective loss of output at high prices
due to the project delay is enough to overwhelm the benefits of the research. This is
shown in Figure 5.4a, which graphs the research values for the free research and the 2year research options. The effective output at high price, if and when the research is
undertaken, is about 27M units, which is less than the 28M units that would be obtained
from an uncertain project start. Therefore, the 2-year research options, in contrast to the
1-year options, have no value at high prices.
Each may still have some value at lower prices, because, at prices in a region
below the critical price for initiating the no-research project ($32.65/unit), the time taken
to do the research would otherwise be spent, at least to some extent, in waiting for the
price to increase to a level where it would be best to begin the project. Note that the
maximum value of the no-cost 2-year research option is less than $10M, which means
that the $10M 2-year research option does not have any incremental value at any price.
The rest of the analysis is qualitatively very similar to the research duration
comparisons in Section 4.3.
6.
MORE PROJECT VALUE: LOWER EFFECTIVE AVERAGE COST
In this set of examples, we examine the effects of research options that allow the
project manager to avoid an increase in effective average cost of $2/unit above the basic
level of $20/unit. The effective output is 30M units.
Figure 6.1 shows the value of the venture with no research option and with free
research available. It is similar to the situation where the research permits the project
manager to avoid an effective loss of output, except that the incremental value of the freeresearch option is constant at high prices.
6.1.
Instantaneous Research
Figure 6.2a shows the incremental value (over the no-research situation) of the
instantaneous research options. (Again, we show this value out to a price of $70/unit
because there is some important structure at these prices for the options to undertake
research programmes that take time.) The value of the research option increases with the
output price up to the price region where the research is undertaken and the cost saving is
realised. Unlike the situations where the research results in an improvement in the
effective output, the research option value does not increase further at higher prices.
Nevertheless, as in the last set of instantaneous research examples, an uncertain project
start is not optimal at any price.
Figure 6.2b shows the value lost relative to the free research option. For each
research option, this graph has a flat maximum at the research cost. Costly research is
not undertaken before it is needed, and so the project manager waits if the price is less
than the critical price, $32.65/unit, for developing the post-research project. The critical
price for undertaking a costly research programme increases again with the research cost,
from this critical initiation price of $32.65/unit, as there is a shift away from doing the
research. Indeed, because the post-research project is the same as the post-research
project in the last section, the trade-offs between waiting and expending the research cost
are the same, and thus the boundaries are the same as well. (These boundaries are shown,
along with other boundaries discussed in Section 6, in Table 3).
6.2.
Research Duration as a Cost
The pattern of optimal behaviour is more complex again if the research takes
time. This is shown in Figures 6.3a-c for research that takes a year to complete.
Figure 6.3a shows the incremental value over the no-research situation of the free
research option (for which the research has no cost and takes no time), and of the 1-year
research options. The free-research value is again the top curve and is the same as in
Figure 6.2a. Notice that the 1-year options would not be undertaken at high prices. The
loss of effective output at the time the research is undertaken, caused by the project delay
due to the research, swamps, at high enough prices, the price-independent cost benefit of
the research.
Figure 6.3b shows the value lost relative to the free-research option. Value lost
increases steadily until the maximum cost increase is swallowed. It is difficult once
again to determine from this figure the location of the boundaries between the waiting,
research and dithering regions. These are shown in Table 3, but they are also illustrated
by Figure 6.3c, which again graphs the value lost in the no-research situation and for the
costly 1-year research options relative to the no-cost 1-year option. An examination of
Table 3 shows that, for each research cost, the research region begins again at lower
prices than for the equivalent instantaneous option, reflecting the desirability of preparing
for the possibility that the output price at the time the research is completed will be in the
post-research project-start region. In fact, as with the instantaneous options, the tradeoffs between waiting and doing the research are essentially the same for these examples
as with the examples in the last section, and the boundaries are the same. Therefore, if
the term structure of output price medians is flat, the probability that the price might be in
the post-research project-start region at the end of the research period, given that the price
is at the lower boundary of the research region when the research is begun, ranges once
again from 21% for the $1M research programme to 36% for the $10M research
programme. Again, a higher cost for the research makes the project manager less willing
to purchase this insurance against project delay.
Most of the rest of the structure is similar to previous examples where an
uncertain project start is optimal at high prices. The lower boundary of this uncertainstart region decreases with an increase in the cost of the research. The small dithering
region between the uncertain-start region and the research region increases in size with
the cost of the research.
6.3.
Longer Research Durations
If the research duration is made longer, the effective loss of output at the initiation
of the research increases and overwhelms more readily at high prices the benefits of the
research. The rest of the analysis of boundary movements is qualitatively very similar to
the other research duration comparisons.
7.
7.1.
CONCLUSIONS
General
We have shown how modern asset pricing theory can be used to analyse projectlevel research management and value. In doing so, we have restricted ourselves to a set
of simple situations. The environment is stationary. There is one potential research
programme consisting of one activity with a known cost, duration and information
content. There is an initial perpetual timing option for the choice between the research
and an uncertain project start, followed, if the research is chosen, by a perpetual timing
option for the project. Finally, there is an exogenous lognormal time-series variable
(taken, for ease of exposition and without loss of analytical generality, to be the project
output price) upon which the project cash-flows have an increasing linear dependence.
This work begins to integrate the "real options" literature of corporate finance and the
decision analysis literature on research management.
We have classified the potential benefits of research programmes in this class of
situations into a two-by-two matrix. One dimension considers the aspect of the future
project decision that is affected by the research: better project timing, which results from
more refined information about the project value, and a higher project value function,
which results from better project design. The other dimension considers the parameter of
the project value function that is affected by the research: the effective output, which
measures the degree of dependence of the project value on the contemporaneous output
price, and the effective average cost, which is break-even price for the project considered
on a now-or-never basis.
We have examined each type of benefit separately for research programmes with
known cost, duration, and information potential. We have shown how different types of
benefits cause different patterns of research management and value. The first step in our
analysis of a given type of benefit is to graph the incremental value of the free-research
option over the situation where no research is possible. If this is anywhere positive, we
can then analyse other research options, first, by research duration, and then, for each
duration, by research cost. The results for each type of benefit are as follows.
7.2.
No Benefit: Reducing Uncertainty in the Effective Project Output
A reduction in uncertainty about the effective output alone would not influence
the project management policy, and the option to undertake research to achieve such a
reduction by itself has no value under any circumstances.
7.3.
Better Timing: Reducing Uncertainty in the Effective Average Project Cost
A reduction in uncertainty about the effective average cost can have value. If
there is an upper limit on the possible effective average cost, the value of the freeresearch option is zero above a critical price, which marks the lower boundary of the
price region where the project would be begun, even if the results of the research were
known.
This critical price, the research option value at prices below it, and the location of
the price region where the research is undertaken, all decrease with duration for the nocost research options. We conjecture that the critical price is positive for any finite
duration.
The maximal value of the no-cost research option, for a given duration, sets the
upper limit on the cost of research, with the same duration, that can give any positive
value. For research of a given duration, the critical price for an uncertain project start,
the research option value below that price, and the size of the research region decrease
with increasing cost to zero at the upper-limit cost just mentioned.
Finally, we conjecture that any no-cost research option has positive incremental
value in a price region of positive size for any positive amount of uncertainty to be
resolved. Moreover, at levels of uncertainty where a given research option is valuable at
some prices, the extent of that price region, and the value there, increases with the
amount of uncertainty to be resolved.
7.4.
More Project Value: More Effective Output
If a research programme prevents an effective loss of output, then, if it is free, the
value of the option to undertake it increases indefinitely with the price. This is also true
of costly instantaneous research options. However, there is a critical research duration
where the value of the effective output not lost by undertaking the research is effectively
lost at high prices because of the delay induced in undertaking the project. An option to
undertake a no-cost research programme with a duration above this critical duration has a
critical price above which an uncertain project start is optimal. We conjecture that this
critical price decreases with duration. The analysis of costly research with these high
durations is similar to that for research where the benefit is a reduction in cost
uncertainty.
Research options, for programmes with positive durations below the critical
duration, have a more complex structure. For low enough research cost, there is a region
of increasing value at low prices, which is followed, at intermediate prices, by decreasing
value before a high price regime is reached, where value increases again. If the cost is
high enough, there is an intermediate price region where an uncertain project start is
optimal, and the research option has no value. In our example, the options where an
uncertain start is optimal at intermediate prices also have a research region at lower
prices. This low-price research region is conjectured to disappear at high enough
research costs, leading to an incremental value pattern that is zero in the uncertain-start
region and below.
The value of the free research increases with the effective output that would be
lost if the research were not done. The critical duration also increases with the amount of
the potential loss. For any given research duration and cost, the extent of the region
where an uncertain project start would be optimal, if such a region exists, would decrease
as the loss increases.
7.5.
More Project Value: Lower Effective Average Cost
If research allows the project manager to avoid an increase in the effective
average cost of the project, then, if it is free, the option to undertake it has an increasing
incremental value up to the post-research critical price for project initiation. Above this
price, the value is constant at the total value of the costs avoided.
If the research is costly (with a cost less than the total cost increase that the
research would allow the project manager to avoid) and instantaneous, there is a similar
pattern of value. The level of the high-price value plateau is at the difference between
avoided post-research cost increase and the research cost. It is reached at a price that
increases with the research cost. If the research cost is greater than the avoided postresearch cost increase, then the research option has zero incremental value, and is
equivalent to having no research option at all.
The options to undertake no-cost research of positive duration have zero value at
high enough prices, because the avoided post-research cost increase is dominated at high
enough prices by the value lost due to project delay. The lower boundary of the highprice regime, where the research option value is zero, decreases with the duration of the
research, as does the value of the research option below that price. We conjecture that,
for any finite duration, the boundary is positive.
The addition of a direct cost further decreases this price boundary, and the value
below it, until a duration-dependent upper limit is reached where the research value is
uniformly zero. The actual research region also decreases in extent with the cost of the
research up to this upper limit. The upper limit decreases with research duration, but, we
conjecture, is positive if the duration is finite.
Finally, the incremental value of a research option, if it has any value, increases
with the increase in effective average cost that would occur if the research were not done.
If the research is not free, the region where the research would be undertaken also
increases in extent, and if the research has positive duration, the critical price for an
uncertain project start increases.
7.6.
Further Work
While the qualitative structure of the comparative statics with respect to the level
of research benefits seems fairly clear, a quantitative exploration is in order. Moreover,
as we mentioned in the Introduction, any real example of a research option would present
a more complex pattern of benefits. With the structure of the basic patterns of benefits
understood, more complex patterns should be systematically examined. Lastly, we have
not examined in this paper the comparative statics of research value and management
with respect to the parameters of the economic model: the volatility and the rate-ofreturn shortfall (which is measure of the amount of risk discounting) in the exogenous
economic variable, and the amount of time discounting. Although the general structure
of the effects of these parameters on an investment timing option overall is known
(Merton 1973, Laughton 1993a), the detailed incremental effects, when these parameters
are considered in interaction with the project and research parameters, has yet to be
explored.
A more basic exploration of the project design dimension of the project decision,
and how it is influenced by a research option, is also in order. The research model itself
should be expanded, so that there may be internal structure within a given research
programme, including sequential programmes that would generalise the work of Pindyck
(1992), as well as a choice, at any stage of a research programme, of possible research
activities. The interaction of the research process, the project initiation decision, and
subsequent flexibility in the operation of the project is an area that is worthy of
exploration. There are also tax and regulation issues that may be addressed by
aggregating from project-level analyses such as this. These can include not only the
effect of different tax structures on venture management, but also the effect, for example,
of environmental or safety regulation, which can require background research that can
take long, uncertain periods of time to complete before a project is allowed to begin.
APPENDIX
AN UNDERLYING PROJECT MODEL
If the output of a project at a given time is independent of the price, the
expectation, at the time the project is begun, of any future revenue, when conditioned on
the output price at the time of the revenue, is the product of the price-independent
expected output for that time and the conditioning output price. The expectation of each
cost does not depend on the price. Thus, any expected net cash-flow, if it is the
difference of the expected revenue and the expected cost, is linear in the price at the time
of the cash-flow.
According to Equation 3, the value, at any given time, of a claim to any then
future output price is proportional to the then current output price. Moreover, any
uncertainty in the output profile is not correlated with the uncertainty in the price or any
macroeconomic risk factor. Therefore, at the time the project is begun, the value of any
future revenue is the product of the then expected output at the time of the revenue and
the value of the claim to the output price at that time. This product is proportional to the
price at the time the project is begun.
Finally, because any uncertainty in the costs is not correlated with the uncertainty
in the price or any macroeconomic risk factor, the value of any cost is the expectation of
that cost discounted at the risk-free rate of return. This cost value is independent of the
output price.
The value of the project when begun is the sum of the value of these cash-flows:
(a-1)
where s is the time the project is begun, qt is the expected output in the time interval at
time t, and costt is the expected cost in that interval. This value is linear in the price at
the time the project is begun. (See the dashed line in Figure 2 for an example.)
The slope of the value function in Equation a-1 is the first sum in that equation.
We call this sum the "effective output" because, as can be seen from the sum itself, it
would be the actual output if the output were all produced and sold instantaneously at the
time the project is begun. The price intercept, which is the ratio of the value of the costs
(the second sum in Equation a-1) to the effective output (the first sum), may be called the
"effective average cost" because, as can be seen from Equation a-1, it would be the actual
average cost if all the project output were produced and sold instantaneously.
If the project does not occur instantaneously, how do the effective output and the
effective average cost vary with the underlying project parameters? First, the effective
output increases with the actual total output. Second, it also increases with earlier timing
of the output. Third, the effective average cost increases with the cost levels. Fourth, if
the there is normal backwardation, then the effective average cost increases if the output
is pushed back in time because the effective average output decreases more than the value
of the costs. Otherwise, it decreases. Finally, an increase in the total output decreases the
actual average cost at any given time, if it increases output at that time and if there are
returns to scale. It also stretches out the project, if there are capacity constraints. If there
is normal backwardation, the longer project life will tend, by itself, to increase the
effective average cost as already noted. Therefore, any scale effects must be large
enough compared to the effects of a longer project life to make the effective average cost
decrease with an increase in the actual total output. If there is no normal backwardation,
an increase in the total output unambiguously decreases the effective average cost if there
are returns to scale or capacity constraints.
REFERENCES
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58 (1985) 135-157
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Dykstra, C. and D.G. Laughton "When is Enough Enough: Proving Reserves for a
Frontier Pipeline" Unpublished (1993)
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Journal of
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Table 1
$4/Unit Uncertainty in Effective Average Cost
Lower Boundaries of Action Regions
Time Cost
(years) ($M)
Research
($/unit)
Dither
($/unit)
Uncertain Start
($/unit)
No research
-----
-----
32.65
0
0
1
2
5
10
0.00
26.25
26.35
26.65
27.20
----36.80
35.80
33.80
31.60
39.20
37.80
37.25
36.05
34.70
1
0
1
2
5
10
0.00
19.45
20.70
22.65
24.50
28.50
28.25
28.00
27.20
25.80
34.10
33.95
33.80
33.45
32.90
2
0
1
2
5
10
0.00
16.95
18.35
20.65
-----
24.25
24.00
23.80
23.05
-----
33.30
33.20
33.15
32.90
32.65