Reversion, Timing Options, and Long-Term Decision-Making

• Many observers of managerial processes have come to the conclusion that discounted cash flow methods lead to a damaging neglect of long-term and strategic investments (e.g., Hayes and Garvin [5], MacCallum [9], and Dertouzos et al [4]). Some critics have argued for putting less weight on financial analysis and more on managerial intuition. Myers [10] has countered that the problem is the inappropriate application of financial analysis rather than the use of financial analysis in general. He suggests that improper accounting for risk in future cash flows frequently leads to the use of discount rates that are too high.

This results in relative undervaluation of typical long-term decision alternatives. He also suggests that organizations may underestimate, or neglect altogether, the value of options stemming from managerial decisions. Because the creation and exercise of future options is of the essence of strategic decision-making, and since there tend to be more options imbedded in longer-term investments, the under-valuation of future options would induce a bias against strategic or long-term decision alternatives.

In this paper, we use modern asset pricing methods to examine one possible reason for excessive risk discounting. If the cash flows being discounted have an increasing dependence on an uncertain variable that tends to revert to a long-term equilibrium path in the face of short-term shocks and this reversion is ignored, then the uncertainty in the cash flows will be overestimated. If this uncertainty leads to excessive, systematic risk discounting, then the project will be undervalued.

We show how to classify the effects of such reversion on asset value, as well as the implications of ignoring it. Our example include both "now-or-never" decisions about a production project and choices that involve a project timing option. The reverting variable in these examples is the project output price. For some examples, the measure of "long-term versus short-term" is the operating duration of the project; for others, it is the length of the timing option. Throughout, we use a set of valuation models designed for relative ease of calculation and usefulness for managers.1

All of the situation that we examine satisfy three conditions. First, the investing organization is a price-taker in the output market, so that the price is an underlying exogenous variable. Second, uncertainty in future output prices is the only uncertainty underlying the decision to be made, and this uncertainty result in positive risk discounting in the valuation of claims to any fixed future output. Third, the structure of the potential production opportunity (i.e., the profiles of production and sales, and project costs) is independent of when the project is undertaken. The first two conditions allow us to focus on a simple specific model. The third condition is imposed so that the effects of reversion can be isolated from those due to any direct dependence of the project cash flows on time.)

Output price reversion has a straightforward effect on "now-or-never" decisions about project alternatives, provided there are no operating options to be considered. The stronger the reversion, the lower the uncertainty in long-term revenues, which, in turn, may require less risk discounting. Thus, any neglect or underestimation of reversion may bias against project alternatives with more long-term revenues, other things being constant. Moreover, the use of a single discount rate to value (on a now-or-never basis) project alternatives with different operating lives may introduce a bias against long-term investments when there is reversion in the project output price.

If options are imbedded in the project alternatives being considered, the effects of output price reversion are more complex. As noted, because reversion tends to decrease long-term price uncertainty, if may reduce the risk premium or discount factor and raise the value of the underlying asset claim (here, a claim to a cash flow proportional to the long-term output price). This may increase the value of claims to cash flows that increase with long-term prices, such as call options, and decrease the value of claims to cash flows that decrease with prices, such as put options. This phenomenon may be referred to as the "risk-discounting" effect.)

Less uncertainty also tends to reduce directly the value of long-term options of any type. This may be called the option "variance" effect, which reinforces the risk-discounting effect for put options and mitigates, if not overwhelms, it for call options.]

Finally, the reversion of future term structure for central tendencies of the price can have direct effects on asset values. These may be called "future-reversion" effects. They exist for American options, for which the timing of the option exercise is discretionary, and may exist for options whose payoffs occur over a period of time. The details of these effects on an option can depend on whether or not the option is in-the-money now, and whether the reversion is to prices where the option would be in -or out-of-the money in the future.]

In Section I, we introduce the class of price models to be examined. We restrict the analysis to price processes that have a lognormal structure. This condition allows us to present an easily integrated form of the conditional distribution (in any future state) of the term structure of prices, and permits us to apply a nonstochastic discounting framework to the valuation of related price claims. We also wish to facilitate the valuation of project options, such as the initial timing option considered below. Therefore, we examine price models that result in a simple state space (i.e., a one-dimensional state space indexed by the contemporaneous output price).

The output price models we use are each specified using a process for the expectation of prices where the key feature is an exponentially decaying term structure of expectation volatilities. New information has a greater impact on expectations for the prices that will occur a year or two in the future than on expectation for prices that will occur in ten or 20 years. Moreover, the proportional drift in the resulting process for the price itself has a term that is logarithmic in the price, illustrating the reversion forces at work in the price itself. Finally, the pattern of future conditional term structure of price medians shows the reversion directly. 2

In Section II, we show the effects of different levels of reversion on the valuation of options on projects where production and sales occur instantaneously. These timing options are equivalent to American call options on the output price, and we examine options that are currently at-the-money. These simple examples allow us to focus on the effects of reversion without the cash flow complexity of multiperiod projects. The comparisons are based on the presumption that, while managers have some knowledge of the output price uncertainties over the medium term, and of the appropriate valuation of these risks, they may misspecify them over the long-term. We find that counteracting risk-discounting and variance effects give conflicting results for the European options. However, the early exercise premium for an American option is also influenced by a future-reversion effect. For the at-the-money options examined, the early exercise premium increases with the degree of reversion, with the net result that neglecting reversion gives a bias against longer-term investment timing options.

Section III demonstrates how reversion can affect the evaluation of options on projects of different operating length. We show flow a bias against long-term alternatives might occur, in a now-or-never analysis of the projects, through the use in the valuation of the longer project of a corporate discount rate based on the valuation of the shorter project. The results provide a clear example of the bias inherent in single-rate discounting methods. We also show what happens if reversion is neglected in an otherwise correct valuation of these projects in the presence of an initial timing option. In the examples that we have examined, the bias remains against the project with a longer operating duration, even if that bias is mitigated somewhat by the timing option. However, when considering the same operating project, longer timing options tend to be relatively overvalued if reversion is neglected, in contrast to the results in Section II for options on single-period projects. We show how the two effects are related for the set of examples considered.

In Section IV, we conclude and suggest areas of future work. Derivations of the output price process and comparative-statics conditions are set forth in the appendices.

I. The Price Model and Valuation Method A. The Price Model

The output price model can be formulated in terms of a process for the evolution over time of the price expectations. The process is based on an approximation that the information needed to determine the revision of future expectations is imbedded in the most recent unanticipated revision in the expectation of current prices.

For any given period, s to s + ds, the revision of expectations for all times at or after s + ds is determined by a single normal random variable, dzs, which is normalized to have zero mean and a variance equal to the length of the time period, ds.[3] This variable represents information coming from the output market during the period just after time s, in the form of the final movement in the expectation for the price at the end of that period. It is independent of the other dz's, because each dz represents new information at a different time.

The revision of each price expectation is modelled to be proportional to the expectation of that price at the beginning of the period and to the normalized information for that period. Thus, given the expectation at the beginning of the period s for the price that will occur at time t, Es(Pt), the change over that period in the expectation of that price is taken to be of the form

dsEs(Pt) = Es(Pt) sigmas, t dzs , (1)

where the proportionality constant, sigmas,t, is the volatility of the expectation of the price at time t seen from the period beginning at times.

We can think of the pattern of volatilities in any given period as an influence function which reflects the relative effect of information arriving during that period on expectations of prices at different times. If, for a given times, sigmas, tis constant for all t, then a shock at s has the same proportional influence on expectations for a price far in the future as for one near at hand. If sigmas,t declines in the term, t - s, of the price, then the influence of new information arriving at s is "decaying" as one looks farther into the future.

It is plausible that, in many aspects of the economy, a development now is less and less relevant to the state of the economy the farther out in the future we look. In effect, information becomes stale-dated. This will happen, for example, for prices in markets that are influenced by long-term forces of supply and demand, which limit the length of time that an exceptionally "low "or "high" price can be sustained. After a short shock, the price tends to revert to some "normal" long-term equilibrium path,[4] perhaps determined by the long-run marginal cost or (in the case of a cartelized commodity) the long-ran, profit-maximizing price sought by cartel managers. The greater is this reversion tendency, the greater is the decay in the effect of new information on future prices.

In the subsequent applications, two restrictions are placed on the form of the volatility in a future price expectation. First, the volatility at any future time, s, must be modelled as known with certainty at the time of the analysis (i.e., it may not vary according to the state of the economy at the time s). This restriction means that the probability distribution in any current or future state for the then future prices is a multivariate lognormal distribution. It also means that a simple nonstochastic discounting model, described in Section I.B., can be appropriate for valuation (Jacoby and Laughton [6]).

Second, the decay in the volatility term-structure can be expressed by an exponential form,

sigmas, t = sigmas exp[-gamma(t - s)], (2)

where gamma is the rate of decay. The amount of reversion may also be measured by the half-life of the decay process, which is related to the decay rate by

H = In(2)/gamma. (3)

In Appendix A, we show that a one-dimensional state space, indexed by the contemporaneous price, occurs only for price models from a slightly more general class of processes with multivariate lognormal probability distributions.

Finally, for ease of presentation, the short-term volatility, sigma s, is held to be constant for all s, and denoted as o. Therefore,

dsEs(Pt) = Es(Pt) sigma exp[-gamma(t - s)]dzs. (4)

The form of the price model commonly used in "real options" work (e.g., Pindyck [11], and Brennan and Schwartz [2]) is a process for the price itself rather than the price expectations. The price process for the model given by Equation (4) is derived in Appendix A to be

[multiple line equation(s) cannot be represented in ASCII text]

where Mo(Pt) is the current median of the price at t, and betat is the growth rate at term t in the current term structure of price medians:

[multiple line equation(s) cannot be represented in ASCII text] (6)

The current price medians are related to the current expectations by

[multiple line equation(s) cannot be represented in ASCII text] (7)

where varo,t is the current variance of the logarithm of that price, [multiple line equation(s) cannot be represented in ASCII text] (8)

As can be seen in this formulation of the price model, the contemporaneous price is, as required, a sufficient state variable for the price dynamics. The price reversion is evident in the logarithmic term in Equation (5).

There is an integrated formulation of the model (also derived in Appendix A) given by the conditional term structure of price medians in each future price state and the covariances at each time of the price logarithms. The median at time s (if the price at time s is P) of the price at time t is given by

[Multiple line equation(s) cannot be represented in ASCII text] (9)

and the covariance at time s (in any state at that time) of the prices at times tl and t2 is given by

[multiple line equation(s) cannot be represented in ASCII text] (10)

Notice that the second factor in Equation (9), which gives the effect on the conditional medians of the information arriving between times zero and s, shows the reversion in the price model. If there is reversion (gamma > 0), the power in this shock factor approaches zero as the term (t - s) becomes larger, so that this factor approaches one, and the conditional future medians revert to the original medians. If there is no reversion (gamma = 0), the power is one and the proportional changes to medians are the same for all terms.

B. The Mechanics of Derivative Asset Valuation[5]

The underlying assets of the derivative asset valuation are the set of output price claims. The current term structure of their values is calculated as

[Multiple line equation(s) cannot be represented in ASCII text] (11)

where Vo(Pt) is the current value of the claim maturing at time t. The expected rate of return at time s for the claim maturing at time t, mus,t is the sum of the risk-free rate, r (taken to be constant for convenience), and a risk premium that is proportional to the amount of volatility at time s in the expectation of the price at time t, sigmas,t. The proportionality constant or price of risk,phi, is also assumed constant over time for ease in presentation.[6] It is presumed to be positive, so that there is risk discounting in the valuation of the output price claims. Using the expression for sigmas, t in Equation (2) and remembering that sigmas has been modelled to be a constant s, the expected return at time s for the claim maturing at time t is given by

mus, t = r + phi sigma exp[-gamma (t - s)]. (12)

The structure of forward prices implicit in these price claim values is the term structure of expectations of the prices with respect to their risk-adjusted distribution (Cox, Ingersoll, and Ross [3]). In our lognormal price model, the risk-adjusted distribution is also lognormal and the covariances of the price logarithms with respect to the risk-adjusted distribution are the same as those with respect to the true distribution (Jacoby and Laughton [6]).

This risk-adjusted distribution may be labelled by the state at which it is defined, where each state is determined by its time s and the realized level of the price at that time, Ps (which we denote by P). We denote the risk-adjusted distribution of price scenarios for the state (s, P) by dms(P>/= s I Ps = P), and use it in the computation at time s of the value of any asset with cash flows that depend only on the then future prices. The value of an asset thus is the sum of the value of the claims to the individual cash flows (indexed by CF) that are a part of the asset:

Multiple line equation(s) cannot be represented in ASCII text (13)

and the value of the claim to each individual cash flow may be determined with state pricing methods by taking the risk-adjusted expectation of the cash flow amount and then discounting for time:

Vs(CF | Ps = P) = exp[-r(tcF - s)] integral of dms(P>/=s | Ps = P)Xcp(P>/=s), (14)

where tCF is the time of the cash flow CF and XCF captures the functional dependence of the amount for the cash flow CF on the time series of prices. Value calculations can also be based on a Black-Scholes-Merton boundary problem that corresponds to the integral in Equation (14) (Cox, Ingersoll, and Ross [3], and Laughton and Jacoby [8]).[7]

C. Comparative Statics

In constructing the comparative statics for the study of alternative assumptions about reversion, we presume that managers think about the uncertainty and riskiness within the time span of the principal revenue flows of typical new investments. For example, a manager who is familiar with projects that produce most of their output within a ten-year horizon after the start of the project will tend to think about price conditions over this particular horizon. We summarize this medium-term view by the conditions prevailing around the center of the project's productive life, i.e., five years into the future. We thus presume that managers focus on a medium-term "reference time" of five years, which we denote tref = 5, and hold constant their impression of the variance and discounting for the price at this time under alternative models of reversion. This presumption about managerial behavior makes it more difficult to show bias against the long-term than would a very short-term reference time (e.g., zero).

Under this framework, our comparisons need to give equivalent risk treatment to cash flows at tref by keeping fixed the current probability distribution of the price at that time, and the risk. discounting in the price claim with that maturity. To achieve the desired probability distribution, different degrees of reversion are accompanied by adjustments to the short-term volatility, s. We also preserve the current risk-discounting of the price claim maturing at tref by an adjustment in the price of risk phi. For example, if sigma = 0.1 and phi = 0.4 in annual terms in the absence of reversion, then for tref = 5 years and H = 3 years, the corresponding short-term volatility and price of risk are 0.160 and 0.421, respectively. The procedure for making these adjustments is presented in Appendix B.

Exhibit I shows the current term structure of the current price distributions under the two models if the current term structure of price medians is flat at $20. The medians are shown by the central dotted line. The other lines show 0.9 and 0.1 fractiles. The effect of the change from a model with no reversion (H = oo), shown by the solid lines, to a model with reversion (H < infinity), shown by the dashed lines, is to pull in the distant tails of the probability distribution of the output price, and to fatten the fractiles for t < tref. The "knowledge" of the long-term distribution embodied in the reversion process keeps the price closer to the current term structure of medians, and indeed causes the total amount of uncertainty to approach a constant in the long-term.

A similar figure for the fractiles of any conditional distribution in the future would show (reflecting the results in Equation (9) for the conditional medians) that the fractiles in the no-reversion model are shifted up or down by the ratio of the realized price at the conditioning time to the original median of that price. In the model with reversion, on the other hand, the fractiles of prices at long terms into the future would tend to revert to the current fractiles.

II. Evaluation of Projects With One Operating Time

Because many factors come into play if cash flows occur over several years, we begin with examples of projects where production and sales take place instantaneously. The timing options for these projects are simply American call options on the output price. We compare price models where the current term structure of medians of the price is held constant. This is like maintaining the same "base case" in different scenario analyses.

To reveal the mechanisms by which reversion influences project value, we decompose the American option value into three component parts. We first express the American value as the value of a European call option of equivalent length plus a premium for the possibility of early exercise. Using put-call parity, we then decompose the value of the European call into two additive parts: the current value of the "call obligation" (which is the current value of price claim less the current value of a risk-freeclaim to the exercise price) and the value of the corresponding European put option.

In Exhibits 2 through 6, we show the effects of different degrees of reversion on these components of the value of American call options of different maturities. In these comparisons, the risk-free rate, r, is 0.03 per year, and the current price medians are all $20, as is the exercise price. The short-term volatility and the price of risk in the no-reversion price model are 0.1 and 0.4 in annual terms, respectively.

Exhibit 2 shows the value of the call obligation. The value of the obligation with a term of tref= 5 years is, by construction, independent of the degree of reversion. The obligation value is negatively (positively) related to the degree of reversion for maturities less (more) than tref. This is simply due to the risk-discounting effect, which reflects the greater (lesser) overall uncertainty, given higher degrees of reversion, for prices at times less (more) than t ref. 8 tref.

A complementary pattern is also observable in the value of the European put options, shown in Exhibit 3. Two influences are at work here. First, we may define a put obligation as a short position in the call obligation. Reversion, through the risk-discounting effect, increases the short-term (up to tref) put obligation values and decreases the long-term values. This tends to increase (decrease) the short-term (long-term) put option values. Because reversion increases (decreases) the price variance for terms shorter (longer) than tref, it also tends to increase (decrease) short-term (long-term) put values directly through the variance effect. The two effects reinforce each other for puts, resulting in a relative overvaluation of long-term puts if reversion is ignored.

For the European call option, the risk-discounting effect and the variance effect counteract each other, and no clear pattern exists. The call obligation value decreases (increases) with greater reversion for short (long) terms, while the put option value does the opposite. At a maturity of tref, the values for different degrees of reversion are equal by construction. Exhibit 4 shows the results of combining these two values to form the European call option values for our set of examples.

The effect of reversion on early exercise is shown in Exhibit 5. In contrast to the classic Black-Scholes case (where the current price claim values are constant over all maturities), the rate-of-return shortfall is nonzero, even when H = infinity. Early exercise commands a premium in all the examples because there is always a positive expected shortfall. Moreover, the premium rises with the degree of reversion. This occurs because positive fluctuations in the potential option payoff are more likely to be short-lived, which increases the value of the right to exercise early in the face of any given positive fluctuation. This effect also tends to lower the exercise boundary, which increases the probability of early exercise. These are future-reversion effects. In the long-term, however, reversion decreases the probability that any given positive fluctuation will occur, counteracting the effects of these fluctuations. This is the variance effect. The future-reversion effect dominates for the parameters examined. We have not explored the limits of the parameter range over which this dominance holds.

Combining the European call option value and the early exercise premium gives the overall value of the American call, shown in Exhibit 6. The future-reversion and risk-discounting effects dominate the variance effect so that the value increases with reversion, the more so the longer the term of the option.

Recall that the call obligations and call options are examples of very simple projects that involve production and sales at a single time. Our examples are constructed so that each project has zero value if the production and sales occur now. Price reversion affects the value of these projects if production and sales must or can occur in the future. In particular, a neglect of reversion results in a relative bias (because of a risk-discounting effect) against those projects where production and sales must be undertaken at some time in the far future. Because of a complex combination of risk-discounting, variance and future-reversion effects, there also is a bias against long-term timing options on such projects.

Ill. Projects With Many. Periods of Operation

We next consider the effects of reversion on projects where production and sales occur at more than one time. We use the same class of price models, taking the half-life of the reversion to be H = 3 years. We first consider the "now-or-never" evaluation of a mutually exclusive pair of projects, and show the errors that can be introduced if price reversion is not properly incorporated into the analysis. We then examine the effects of reversion on the evaluation of mutually exclusive options to undertake these projects at any point within specified period of time.

The salient difference between these projects is that one has an operating life of L = 10 years, while the other has an operating life of L = 20 years. Initial project costs are taken to be known with certainty: $100M for the ten-year project and $150M for the 20-year project. Each has a known, constant stream of annual operating costs of $11.75M per year. For each project, the annual output (which begins one year after initial investment) is constant at 1.405M units. With no reversion, the ten-year project, if undertaken immediately, would currently have zero value, while the 20-year project would have negative value.

A. Now-or-Never Projects

The results of the analysis of these projects on a now-or-never basis are shown in Exhibit 7. For correct valuation, the analysis must account properly for the difference in project operating life and for the degree of reversion in prices. Failure to do so will introduce bias into the comparison between the long-lived and the short-lived project. The exhibit shows the correct valuation, and the result of two such errors. The left-hand panel of the exhibit (first line) shows the true valuation of the two projects if H = 3 years. The value of the ten-year project is $3.62M, much less than the value of $17.02M for the longer alternative. While both are worthwhile, the long-term alternative is clearly preferred to the short-term.

An appropriate discount rate for each project may be defined as the single constant annual discount rate which, if used in discounting the cash flows in the median price scenario, yields the correct value of the project.9 The project discount rates thus defined are presented on the second line. The annual discount rate for the ten-year project is 0.093. The discount rate of the 20-year project is lower, at 0.075, because reversion in the prices tends to lower the volatility in revenues for the out years and thus the risk-discounting in the overall revenue stream. This effect can also be seen in the pattern of discount rates for the valuation of the revenue stream in each project considered in isolation: 0.063 for the shorter project and only 0.053 for the longer.

Now consider what happens if the difference in discounting is not taken into account and the longer project is discounted at the rate appropriate for the shorter project. This result is shown on the fourth line of the exhibit. The value of the longer project would then be set to -$3.76M, or $20.78M less than its true value. Such faulty analysis in this case would incorrectly suggest that the shorter project should be chosen.

In this situation, however, the bias would not occur if there were no reversion. As the right-hand panel of the exhibit shows, without reversion (H = infinity), the short-term project (which would have zero value under these circumstances) would be preferred to the long-term project (which would be worth -$19.67M). Moreover, the use of a short-term discount rate would bias the value of the long-term project upwards, to -$11.83M. 10 It is only with price reversion that any bias is introduced against the long-term when the same discount rate is used for long-term as for short-term projects.

Finally, note that neglect of reversion would introduce bias against the long-term in an otherwise correct valuation of this pair of projects. With reversion, taking the 20-year project is the best alternative, while without reversion, it is the worst.

B. Project Timing Options

We now consider the relative value of the projects if, after one is chosen, it may be begun in any year up to a specified maturity (or relinquishment), time T, where T is allowed to vary from zero (the now-or-never case) to ten years. As stated, we presume that the time of project start does not affect the profile of production and sales or the project costs relative to that start time.11 The current project value is denoted as V0(L,T). Once again, the analysis of the effects of reversion assumes that the correct half-life is H = 3 years. The correct results are compared to the results of an analysis that incorrectly assumes that there is no reversion (H = infinity).

This valuation applies a two-method approach to the valuation of the investment timing option described in Laughton and Jacoby [8]. The first stage of the analysis is to use Equations (11) to (14) to calculate, for each project under each price model, the value of the project in each possible starting state. Because both the cash flow model and the underlying output price model are stationary with respect to the starting time, this value is independent of the starting time, but it does depend on the particular output price at the starting time, which we call the starting price. Exhibit 8 presents "value functions" which show the value of each of the two projects at the time it is initiated as a function of the starting price, for price models with and without reversion. [12]

The value functions for each of the two projects differ under the two price models. First, for each project, the dependence of the value function on the starting price is weaker with reversion. This difference is greater for the long-term project. Second, the starting price at which each project has a "now-or-never" value of zero (the starting price intercept of the value function) is lower with reversion. Again, the difference is greater for the long-term project.

The differences in the sensitivity of the value to the starting price are the result of counterbalancing forces, of which there are three. The first is the future reversion effect. Without reversion, shocks to the price have permanent effects, and higher or lower prices are likely to be maintained. With reversion, the term structure of price distributions after a shock will tend, in the long-term, toward the original distributions. This decreases the dependence of statistic of the conditional price distribution (such as the value function) on the conditioning price. The longer the term and the higher the degree of reversion, the greater this future-reversion effect tends to be. It thus tends to decrease the dependence of the value function on the starting price for a project under reversion, and the effect is stronger for longer-term projects.

The second and third forces are due to risk-discounting effects. Greater risk discounting of short-term revenues under the reversion model also tends to decrease the dependence of the value function on the starting price, simply by decreasing the value at all starting prices. However, unlike the first force, it does so more for shorter-term projects. Decreased risk discounting for longer-term revenues increases the starting price dependence of the value functions under the reversion model by increasing the value for all starting prices. It does so more for the longer project than for the shorter. The first of the three forces dominates in this example.

The differences in the starting price intercept of the value function also result from a combination of effects. At prices below the current $20 price, the revenues are more valuable if there is reversion, both because reversion decreases risk discounting in the valuation of the revenues and because, in states defined by price levels below the current price, reversion increases the term structure of the conditional price medians. The risk-discounting effect occurs also in future states at the current (and long-term median) price, and it is sufficiently large so that, if there is reversion, both projects are in-the-money, more so for the long-term project. Recall that, without reversion, the short-term project is at-the-money and the long-term project is out-of-the-money.

In the second step of the procedure, the value function of each project is used as an input to a Black-Scholes-Merton formulation of the American option for starting that project. [13] The result of this calculation is the value and the critical starting price exercise boundary, P*s, at each possible starting time s for this timing option.[14] Exhibit 9 shows this boundary for each project under each price model, given an option of length T = 5 years. At year five, the option offers only a now-or-never choice, and the project is started at prices where its then current value is positive. For each project, the critical starting price, P*s, at s = T = 5 years is the starting price intercept of the value function in Exhibit 8. For the years before s = T = 5 years, the price must be higher than the now-or-never break-even price to justify starting instead of waiting, because the option to wait has value. The value of the option to wait is greater, and the critical starting price is higher, the longer the length of the option. Note that, because there is greater chance in the model without reversion of still higher prices when the contemporaneous price is above its original median, the option to wait is worth more in any future high price state and a higher price is required to justify starting either project. This effect is larger for the longer-term project.

Exhibit 10 shows the value, Vo(L,T), of options on each project under both price models for option lengths ranging from T = 0 (now-or-never) to a maximum of T = 10 years. (The results for T = 0 are the same as those shown for "now" projects in Exhibit 7, with negative values set to zero.) As one would expect, the option value is a nonde-creasing function of the option length, T, regardless of the project or price model.

If the proper specification of price reversion is H = 3 years, for all option lengths from T= 0 to T= 10 years, the 20-year project would be started now. Also, the option to undertake it is more valuable than the option on the ten-year project, although by decreasing amounts for longer options as the timing option on the ten-year project increases in value. However, without reversion, the option on the ten-year project would be more valuable. Therefore, the neglect of reversion in an otherwise correct valuation would result in a bias against the project with a long operating duration if there is management flexibility about when to start the project, just as it would in a setting in which management is faced with a now-or-never decision. Although the option to wait reduces somewhat the effect of the bias introduced by misspecification of the degree of reversion, the effect of the option value in this case does not overcome the much greater difference in now-or-never values for the longer project.

For either project, neglecting reversion would provide a relative bias in favor of longer-term timing options. This is the opposite of the result for the projects with one time of production and sales, as shown in Section II. The difference is that the value functions for the ten-year and 20-year projects depend on the amount of reversion, while the value function for the instantaneous project does not. For the ten-year project, the decreased slope of the value function under reversion makes the major difference. For the 20-year project, the options are out-of-the-money without reversion, while they are so far in-the-money under reversion that the option to wait has no incremental value for any option length.

IV. Conclusions and Extensions

Reversion in the cash flows of different investment alternatives can have a substantial impact on their relative evaluation, particularly if some of the alternatives are short-term, while others have long-term implications. We have analyzed the effects of such reversion for situations in which the decision cash flows increase with a variable that reverts and have shown examples where this reverting variable is the output price. For simplicity, we have restricted ourselves to a case where the output price claims may be valued within a nonstochastic discounting framework, and to a pattern of reversion that allows the economic state at any time to be parameterized by the contemporaneous output price. The resulting model exhibits a decaying term structure of price expectation volatilities, thus linking the notion of the decay over time of the effect of economic shocks with reversion to a long-term equilibrium.

We found that reversion can affect value through three channels. First, by lowering long-term uncertainty, it may decrease the amount of risk discounting and increase underlying asset values (the risk-discounting or value effect). Second, by decreasing long-term uncertainty, it directly decreases option values (the variance effect). Finally, the future reversion of the term structure of the cash flow determinants can have direct effects on value (future-reversion effects).

The analysis of the choice between projects with differing operating lives shows that reversion may justify the use of lower discount rates in the valuation, on a now-or-never basis, of long-term as compared to short-term projects. If output prices do exhibit reversion, there may be a bias against projects with long operating durations if all alternatives are evaluated using the same single (short-term) discount rate.

In otherwise correct valuations that neglect reversion by modelling the price as following a random walk, there may also be a bias for or against long-term decision alternatives depending on the specifics of the situation. A bias against the long-term is typically caused by over-discounting of long-term cash flows. A bias in favor of the long-term occurs when the overdiscounting is dominated by an overestimate of the value of options imbedded in the decision alternative being considered, which is typically caused by an overestimation of the variance of the long-term cash flows. The effects of reversion in the future term structure of the cash flows can result in different biases, depending on the situation.

In our timing option examples, the neglect of reversion results in a bias against options for projects with long operating duration. However, for the same project with at least a moderate operating duration, neglecting price reversion may introduce a relative bias in favor of longer options.

This work can be extended in several directions. First, random-walk models, which are the mainstay of the "real options" literature, do not account for reversion. Our results suggest that it may be useful to revisit earlier work based on random-walk models and modify the analysis for situations where reversion may be a consideration.

Second, a more complete study should be done of the effects on project values of the interaction of cash flow reversion with different measures of project duration, as well as with other aspects of asset structure such as the degree of operating leverage, the relative size of early sunk costs, and different types of operating flexibility. In addition to the initial timing option considered in this paper, other types of flexibility to be addressed include initial capacity and technology choices, and post-startup options such as scale adjustments, temporary shut-downs, and abandonment. More difficult tasks would include finding the effects of other types of term-dependent uncertainty (such as cyclical behavior) in a simple project setting such as that used in this paper, or categorizing the effects of this or other types of term-dependent behavior in more complex settings where the project being considered is not stationary with respect to the time of its initiation.

This research has been supported by the Natural Science and Engineering Research Council of Canada, Imperial Oil University Research Grants, Interprovincial Pipeline Co., Saskoil, Exxon Corp., the Social Science and Humanities Research Council of Canada; and the Central Research Fund, a Nova Faculty Fellowship, the Muir Research Fund, and the Institute for Financial Research of the University of Alberta; and by the MIT Center for Energy and Environmental Policy Research.

1 Jacoby and Laughton [6] provide a description of the approach that does not require an extensive background in financial economics. Laughton and Jacoby [8] show an application to managerial flexibility.

2 Our specification is close to one outlined by Treynor and Black [13] in an early paper applying modern asset pricing concepts to project evaluation. A similar specification for the uncertainty in short-term, risk-free interest rates has been used in the valuation of derivative securities of treasury bonds (Turnbull and Milne [14]).

Battacharya [1] also values a reverting cash-flow stream. His model is not simply integrable, however. His valuation method is based on a continuous-time capital asset pricing model with a single risk factor and a nonstochastic price of risk for the cash flow uncertainty. Kulatilaka and Marcus [7] have used a similar model to describe commodity price reversion.

• 3 The specialization to one piece of information in each period is made for simplicity. The model can be generalized to include two types of information, incorporating both long-term decay and cyclical effects of new information. Such a model would result in a state space for analysis of future options that is much more complex than the one considered here.
• 4 Pindyck and Rubinfeld [12, pp. 462-465] have found possible instances of this type of behavior. They reject non-reversion models for oil and copper prices on the basis of Dickey-Fuller uit root tests using more than 100 years of data.
• 5 For a more extensive treatment of this valuation model and the compromises between operational feasibility and "best" valuation theory, see Jacoby and Laughton[6].
• 6 Valuation with known expected returns having this structure may occur, for example, in Cox-Ingersoll-Ross[3] economies where there is a representative agent with known impatience and logarithmic risk aversion, and where the structure of the production opportunity set is state-independent. The valuation of projects in this paper may be viewed as partial equilibrium exercises in an economy that has these characteristics, at least to a good approximation for purposes of financial market price determination.
• 7 The crucial aspect of our price mode, when used in Black-Scholes-Merton boundary problem, is the rate-of-return shortfall on the short-term price claim. This is the difference between the expected rate of return on the short-term price claim and the expected rate of price appreciation. For our model, the rate-of-return shortfall at time t, if the price at time t is P, is:

[Multiple line equation(s) cannot be represented in ASCII text]

The first two terms in this formula are the rate of return on the claim to the short-term price at time t (Equation (12) with s=t), while the last three terms are the negative of the expected rate of price appreciation at time t(Equation (5)).

If the price were for a storable commodity, the ct would be the convenience yield for that commodity (Brennan and Schwartz [2]). This is the ratio of its convenience value to its price, where the convenience value is the value of the services received, net of the costs incurred, from holding physical inventories of the commodity. Notice that the convenience yield in Equation (15) is an increasing linear function of the logarithm of the price.

• 8 This effect is accompanied by a decrease in the price expectations with more reversion. More reversion reduces the variance of the logarithm of the price, which decreases the factor that relates the price expectation to the price median (Equation (7)). For the given level of the price of risk and the volatility of short-term price expectations, the risk-discounting effect always dominates. When the risk-discounting or "value" effect is mentioned, the reader may assume that this small influence on the expectations is taken into account.
• 9 We use this discount rate to compare with a standard scenario-based discounted cash flow analysis based on the median price scenario.
• 10 The discount rate for the long-term project is higher because there is no reversion to counteract the effects of the greater operating leverage that exists for the longer-term cash flows.
• 11 While this is done primarily so that the effects of different assumptions about output price reversion are not confounded with effects of a direct time-dependence of the cash flows, many potential nonrenewable resource developments do, at least approximately, satisfy this condition.
• 12 For this valuation, the cash flows are modelled to occur at annual intervals.
• 13 In these calculations, it is presumed that an option may be exercised only at times occurring at annual intervals. This corresponds to the annual cumulation of the project cash flows used in the calculation of the project value functions. If the possible exercise times were modelled to be continuously distributed and cash flows to flow continuously, the qualitative results would not differ.
• 14 For study of the details of the optimal solution, and for representation of the results in a simulation context, the values may then be recomputed conditional on P* (Laughton and Jacoby [8]).

GRAPH: Exhibit 1. Current Output Distributions, With 0.5 Fractile or Median (Dotted) and 0.9 and 0.1 Fractiles (No Reversion (Solid) and Reversion With H = 3 years (Dashed))

GRAPH: Exhibit 2. Call Obligation Value vs. Maturity Time for Half-Life = 1 year, 3 years, 5 years, infinity

GRAPH: Exhibit 3. European Put Value vs. Maturity Time for Half-Life = 1 year, 3 years, 5 years, infinity

GRAPH: Exhibit 4. European Call Value vs. Maturity Time for Half-Life = 1 year, 3 years, 5 years, infinity

GRAPH: Exhibit 5. Call Early Exercise Premium vs. Maturity Time for Half-Life = 1 year, 3 years, 5 years, infinity

GRAPH: Exhibit 6. American Call Value vs. Maturity Time for Half-Life = 1 year, 3 years, 5 years, infinity

Exhibit 7 Bias in Now-or-Never Project Valuation From Mishandling of Reversion Legend for Chart A - Project Length B - Reversion (H = 3 years)--10 years C - Reversion (H = 3 years)--20 years D - No Reversion (H = infinity)--10 years E - No Reversion (H = infinity)--20 years A B C D E Value 3.62 17.02 0.000 -19.67 Discount rate, project 0.093 0.075 0.101 0.110 Discount rate, revenue 0.063 0.053 0.067 0.067 Value (10-year rate) 3.62 -3.76 0.000 -11.83

GRAPH: Exhibit 8. Now-or-Never Project Value vs. Starting Price for Project Length = 10 Years and 20 Years, for Reversion (H = 3 years) and No Reversion (H = infinity)

GRAPH: Exhibit 9. Five-Year Timing Options: Critical Starting Price vs. Starting Time for Project Length = 10 Years and 20 Years, Reversion (H = 3 years) and No Reversion (H = infinity)

GRAPH: Exhibit 10. Project Timing Option Value vs. Maturity Time, Project Length = 10 Years and 20 Years, Reversion (H = 3 years) and No Reversion (H = infinity)

References

1. S. Battacharya, "Project Valuation with Mean-Reverting Cash-Flow Streams," Journal of Finance (December 1978), pp. 1317-1331.

• 2. M.J. Brennan and E.S. Schwartz, "Evaluating Natural Resource Investments," Journal of Business (April 1985), pp. 135-157.
• 3. J.C. Cox, J.E. Ingersoll, Jr., and S.A. Ross, "An Intertemporal General Equilibrium Model of Asset Prices," Econometrica (March 1985), pp. 363-384.
• 4. M.L. Dertouzos, R.K. Lester, R.M. Solow, and the M1T Commission on Industrial Productivity, Made in America: Regaining the Productive Edge, Cambridge, MA, MIT Press, 1989.
• 5. R.H. Hayes and D.A. Garvin, "Managing as if Tomorrow Mattered," Harvard Business Review (May-June 1982), pp. 71-79.
• 6. H.D. Jacoby and D.G. Laughton, "Project Evaluation: A Practical Asset Pricing Method," The Energy Journal (Vol. 13, No. 2, 1992), pp. 19-47.
• 7. N. Kulatilaka and A. Marcus, "A Real Options Primer," Working Paper 93-10, Boston University, School of Management, 1993.
• 8. D.G. Laughton and H.D. Jacoby, "A Two-Method Solution to the Investment Timing Problem," in Advances in Futures and Options Research 5, F. Fabozzi (ed.), Greenwich, CT, JAI Press, 1991.
• 9. J.S. MacCallum, "The Net Present Value Method: Part of Our Investment Problem," Business Quarterly (Fall 1987), pp. 7-9.
• 10. S.C. Myers, "Finance Theory and Financial Strategy," Interfaces (January-February 1984), pp. 126-137.
• 11. R.S. Pindyck, "Irreversibility, Uncertainty, and Investment," Journal of Economic Literature (September 1991), pp. 1110-1152.
• 12. R.S. Pindyck and D. Rubinfeld, Econometric Models and Economic Forecasts, New York, NY, McGraw-Hill, 1991.
• 13. J.L. Treynor and F. Black, "Corporate Investment Decisions," in Modern Developments in Financial Management, S.C. Myers (ed.), New York, NY, Praeger, 1976.
• 14. S.M. Turnbull and F. Milne, "A Simple Approach to Interest-Rate Option-Pricing," Review of Financial Studies (Vol. 4, No. 1, 1991), pp. 87-120.

Appendix A. Derivation of the Output Price Process

We begin here with a price model formulated as an initial value problem for the price expectations. The dynamics follow a scalar lognormal process as in Equation (1). From this, we find a sufficient constraint (which is a slight generalization of Equation (2)) such that the conditioning state variable of the model at any time is the contemporaneous price. Then, we present the model as an initial value problem for the price itself, as in Equation (5) with an initial condition for the price, and show an explicit representation of all the conditional price scenario distributions (Equations (9) and (10)).

If we define the expectation at time u of the price at time t to be Eu,t congruent Eu(Pt), the formulation of the model as an initial value problem for the price expectations is (if the initial time is s):

duEu,t = Sigma[sub u,t Eu,t dzu

Es, t given. (AI)

The process for the logarithm of the expectation of the price at time t, constructed from Equation (A1) using Ito's lemma, may be integrated from time s to time s', and then exponentiated, to show that

[multiple line equation(s) cannot be represented in ASCII text] (A2)

If the time of the final expectation, s', is set to the time of the price, t, then this becomes an equation of the price, which may be rewritten in terms of its median, Ms,t congfruent Ms(Pt), as

[multiple line equation(s) cannot be represented in ASCII text] (A3)

Taking the logarithm of both sides, and defining the log-price as Rhot approximately equals log(Rhot), and the logarithm of the price median as ms, t -- log(Ms, t), we have

[multiple line equation(s) cannot be represented in ASCII text] (A4)

The differential of the log-price is then

[multiple line equation(s) cannot be represented in ASCII text] (A5)

If the price is to be a sufficient state variable for the model, then the drift term in Equation (A5) must depend only on the price. This will be the case only if the stochastic part of the drift term is determined (path by path) by the stochastic part of the price. This requires that the inte-grands of the integrals in Equation (A5) and the log-price equation (Equation (A4)) be proportional to each other. If this is so, then the term structure of expectation volatilities satisfies an initial value problem of the form

DeltatSigmau,t = -atSigmau,t

Sigmau,u given. (A6)

Therefore, if we define the short-term volatility at any time to be Sigmat approximately equals Sigmatt,thprice differential is

dtpt=(Deltat ms,t + atms,t -at pt)dt + Sigmatdzt (A7)

This is independent of the conditioning time, s, only if the term structure of medians satisfies an initial value problem of the form

Deltatms,t + at ms,t = alphat

ms, s = P[sub s (A8)

The differential of the log-price is then of the form

[multiple line equation(s) cannot be represented in ASCII text] (A9)

and the price, using Ito's lemma, satisfies an initial value problem with differential equation

[multiple line equation(s) cannot be represented in ASCII text]

The conditional price scenario distributions of this model are joint lognormal as can be seen from Equation (A3) for the price. The covariances of the associated normal distribution are determined by the expectation volatilities

[multiple line equation(s) cannot be represented in ASCII text] (A11)

so that the solution of the initial value problems for the medians and for the volatilities determines the medians and associated covariances and thus the complete structure of the distribution of the price scenarios:

[multiple line equation(s) cannot be represented in ASCII text]

[multiple line equation(s) cannot be represented in ASCII text]

where

[multiple line equation(s) cannot be represented in ASCII text] (A13)

The equation for the medians may be rewritten as an evolution equation for the medians at some later time s' in terms of the price at time s' and the medians at an earlier time s:

[multiple line equation(s) cannot be represented in ASCII text] (A14)

The class of models that we use in this paper is specialized so that at is a constant o and at is a constant Gamma.

Appendix B Comparative Statics With Finite Reference Times

For H = infinity, the current associated variance of the price at time tref has the form Sigma2tref. We calculate the magnitude of the appropriate adjusted volatility for a finite half-life relative to the volatility for H = infinity, which, in these examples, is maintained at 0. 1 in annual terms. Therefore, the s appropriate for a finite half-life is a function of this short-term volatility for H = infinity, as well as the reference time and the half-life. It is denoted Sigma(tref, Eta, Sigma). The current associated variance of the price at tref, given a finite half-life H, is

[multiple line equation(s) cannot be represented in ASCII text]

where g approximately equals to ln(2)/H. Equating this to the variance for H = infinity gives an expression for the adjusted volatility:

[multiple line equation(s) cannot be represented in ASCII text] (B2)

For example, the adjusted volatility for tref= 5 years, H = 3 years and Sigma = 0.1 is Sigma(5, 3, 0.1) = 0.160.

Similarly, we want to preserve the current risk discounting of the price claim maturing at tref, which can be accomplished by an adjustment in the price of risk Phi. It also becomes a function of Sigma, H and tref as well as the price of risk for H = infinity which is set at 0.4 in annual terms for all of these examples. The risk discount factor with H = infinity has the form exp(Phi Sigma tref). If the adjusted price of risk is denoted Phi(tref, Eta, Sigma, Phi), the risk discount factorforfinite His

[multiple line equation(s) cannot be represented in ASCII text]

Combining Equations (B2) and (B3) gives the following expression for the adjusted price of risk:

[multiple line equation(s) cannot be represented in ASCII text]

For Phi = 0.4, the adjusted price of risk is Phi(5, 3, 0.1, 0.4) = 0.421.

By David G. Laughton and Henry D. Jacoby

David G. Laughton is Adjunct Professor in the Faculties of Engineering and Business, University of Alberta, Edmonton, Alberta, and Henry D. Jacoby is the William F. Pounds Professor of Management, Sloan School of Management, Massachusetts Institute of Technology, Cambridge, Massachusetts.