Mobile Edition
Print
Get the Mag
Weekly UpdatesINTRODUCTION
The simplest form of real options is the deferral or delay option. The mathematics used are based on the financial call option. For that financial case, if the purchase of a stock is delayed and an option is purchased instead, then any dividend is lost since the stock is not owned. For real options, project delays may carry a benefit but most delays also involve a cost. The dividend model has been used to determine the option value of both financial and real options. However, real options must consider a much broader array of possibilities.
Let us begin with three examples of why real options may be useful in evaluating potential delays in projects that engineering economists must evaluate. First, potentially higher future market prices for the output of a mining or petroleum project may well justify continuing to make lease payments for a tract whose development is currently uneconomic. Second, possible advances in uncertain technological capabilities may justify continuing R&D for a project that is "on hold" rather than being terminated. Third, a pharmaceutical firm might be evaluating whether to proceed with plant construction while waiting for regulatory approval.
The common thread in these examples and all real delay options is that something may change the project economics during the delay period, so that the project becomes worthwhile. European-style real options include projects that are waiting for an event to happen, such as regulatory approval. American-style real options tend to be situations where the firm is waiting for conditions to improve, such as higher oil prices making previously uneconomical oil fields become economically attractive. Given that expected value decision-making is often used, the change may simply be a shift in the probability distribution for key project elements.
While many authors point out the possible advantages of delaying a decision, we have found few that point out that there are potential costs associated with that delay. Proceeding immediately starts the cash flow or benefit stream as soon as possible, while delay in the decisions means those cash flows are lost or delayed.
There are a number of possible models for the costs of a delay associated with a real deferral option. These are summarized with a title that suggests the basis for the model. These include:
* No cost associated with delay (or cost is ignored)
* Lost dividend model
* Limited market window (cash flows during delay are lost)
* Limited technology-generation life (cash flows are delayed but not lost)
* Loss of cash flows during delay but project life is uncertain and considered infinite
* Loss of cash flows during delay and permanent loss of market share (reduced future inflows)
* Potential changes to all project cash flows.
In this article we numerically address all but the last two of these models, which are described conceptually.
In Eschenbach et al. (2007), we assert that for many real engineering projects, delay is often better characterized as terminating a project that might be reevaluated and restarted later. These engineering projects are executed by teams of individuals, and when a project is delayed the team is disbanded and assigned to other projects--often with other departments, divisions, or organizations. Not only is the intellectual capital of the project dispersed, but by the time restart is considered, change is virtually inevitable: new technology, tighter regulations, changed market conditions for inputs and outputs, etc. Thus, at the project restart, virtually everything must be revisited and redesigned. These delay costs for redoing the initial work impact the cost to exercise the real option--which is an additional issue that is not addressed in this article.
Unfortunately, as detailed in the next section, far too often the model chosen is the first unrealistic option of ignoring the delay costs. Real projects rarely have a zero cost of delay and we have not identified one. More realistic models must consider changes in technology, regulations, market conditions, and market share. As a starting point, we use models of uniform cash flows where some are lost or delayed.
The article is structured as follows. First, the literature is reviewed to determine the waiting cost models that have been used in the past. Next, the lost dividend model that is used in financial and real options is described. The next section describes three uniform cash flow models that are explicitly based on lost or delayed cash flows. The final sections describe nonuniform cash flow models and close with conclusions and recommendations.
DELAY COST MODELING IN THE LITERATURE
In the arena of financial options, a zero delay cost can be appropriate. Time periods may be measured in days or weeks, and there may be no significant financial cost for waiting. For pricing stock options, delays may involve missing the payment of a dividend. The initial Black-Scholes (1973) model did not include this waiting cost, but the authors specifically stated that the model applied where there were no dividends. One of the first modifications of the model was to include the impact of dividends.
However, for real options in the context of engineering economy, delays may be measured in years and the losses are linked to changes in the cash flows rather than to lost dividends. Delaying projects means that the benefit stream of revenues or reduced costs is delayed. Thus, we suggest that the omission of waiting costs should be considered a near fatal flaw in most engineering economic analyses that include real deferral options.
Instead, because of the delays and changes in the cash flows, we assert that it is essential to match the model chosen to the reality of the project under analysis. This is in contrast to many articles and analyses that ignore the cost of waiting and assume that the value of the project is stable except for the changes measured by the volatility coefficient. The omission of waiting costs can certainly be justified for introductory presentations such as Park (2007), but we believe that it is less justifiable in analyses intended to support real-world decision making.
We have examined articles in The Engineering Economist from 1999 through 2007 as a representative sample on how waiting costs have been treated. Table 1 includes references in order of publication for the quarter of the 28 articles published that included some cost of waiting.
Nembhard et al. (2000) point out that the Black-Scholes equation needs to be modified where there is a loss in value of the underlying asset during a delay. Cottrell and Sick (2002) point out that there is an opportunity cost of waiting, such as the loss of sales of a new product, or a benefit foregone from not having a project in operation. They also point out that first-mover benefits may be lost to the competition due to a delay. Others before us have properly included waiting costs that were matched to their situation, but many more have not.
Unfortunately, the cost of waiting is usually dealt with in detail only in those books that treat real options in detail. Articles and book chapters may simply mention it or pass it by altogether. Perhaps the first detailed look at the cost of waiting was by Amram and Kulatilaka (1999), where the cost of waiting was recognized as a "leakage in value." Copeland and Antikarov (2003) basically avoid the issue. Mun (2006) demonstrates several models, largely dealing with the treatment of dividends.
In reality, the cost of waiting can nullify the value of a real option. In one reality-based example (Lewis et al. 2007), the value of the option was reduced from $29 million to $8 million by including waiting costs. Obviously, ignoring waiting costs can dramatically overstate the value of an option. In the cited case, ignoring the cost of waiting resulted in a recommendation of delaying the project. Including the cost of waiting changed the recommendation to proceed immediately.
LOSS OF DIVIDENDS TO MODEL DELAY COSTS
One of the ways to evaluate real options is the use of the Black-Scholes model, which in a financial context uses the loss of dividends at the rate D to model the cost of waiting. The financial variables are translated to the present worth of the future cash flows ([S.sub.0]), the cost (X), project volatility ([sigma]), interest rate (r), time of delay (T), and the cost of waiting rate (D). When the cost of waiting, W, is explicitly calculated, the [S.sub.0][e.sup.-DT] term can be replaced with ([S.sub.0] - W). The value of the option (C) is expressed as Equation (1). Note that while they appear different, Equations (2a) and (2b) are equivalent ways to include the waiting cost in the calculation of [d.sub.1].
C = [S.sub.0][e.sup.-DT] [PHI]([d.sub.1]) - [Xe.sup.-rt] [PHI]([d.sub.2]) (1)
where [PHI]([d.sub.x]) is the cumulative standard normal distribution of the variable [d.sub.x] [e.sup.-DT] is the discounted cost of waiting, represented by a fraction
[d.sub.1] = ln([S.sub.0][e.sup.-DT] / X) + (r + [[sigma].sup.2] / 2)T / [sigma][square root of T] [d.sub.2] = [d.sub.1] - [sigma][square root of T] (2a)
[d.sub.1] = ln([S.sub.0] / X) + (r - D + [[sigma].sup.2] / 2)T / [sigma][square root of T] [d.sub.2] = [d.sub.1] - [sigma][square root of T] (2b)
If we contrast the value of So with and without the delay cost subtracted, we can define Equation (3) as the cost of waiting. At later points in the article we will express this as the percentage of the value So that is lost. As noted above, there are two equivalent ways of expressing the delay cost:
[S.sub.0][e.sup.-DT] = ([S.sub.0] - W)
or
[W.sub.D] = [S.sub.0](1 - [e.sup.-DT]) (3)
FEED