Real Asset Valuation: A Back-to-basics Approach

VOLUME 20 | NUMBER 2 | spRiNg 2008

APPLIED CORPORATE FINANCEJournal of

A M O R G A N S T A N L E Y P U B L I C A T I O N

In This Issue: Valuation and Corporate portfolio Management

Corporate portfolio Management RoundtablePresented by Ernst & Young

8 Panelists: Robert Bruner, University of Virginia; Robert Pozen,

MFS Investment Management; Anne Madden, Honeywell

International; Aileen Stockburger, Johnson & Johnson;

Forbes Alexander, Jabil Circuit; Steve Munger and Don Chew,

Morgan Stanley. Moderated by Jeff Greene, Ernst & Young

Liquidity, the Value of the Firm, and Corporate Finance 32 Yakov Amihud, New York University, and

Haim Mendelson, Stanford University

Real Asset Valuation: A Back-to-Basics Approach 46 David Laughton, University of Alberta; Raul Guerrero,

Asymmetric Strategy LLC; and Donald Lessard, MIT Sloan

School of Management

Expected Inflation and the Constant-Growth Valuation Model 66 Michael Bradley, Duke University, and

Gregg Jarrell, University of Rochester

Single vs. Multiple Discount Rates: How to Limit “Influence Costs” in the Capital Allocation process

79 John Martin, Baylor University, and Sheridan Titman,

University of Texas at Austin

The Era of Cross-Border M&A: How Current Market Dynamics are Changing the M&A Landscape

84 Marc Zenner, Matt Matthews, Jeff Marks, and

Nishant Mago, J.P. Morgan Chase & Co.

Transfer pricing for Corporate Treasury in the Multinational Enterprise 97 Stephen L. Curtis, Ernst & Young

The Equity Market Risk premium and Valuation of Overseas investments 113 Luc Soenen,Universidad Catolica del Peru, and

Robert Johnson, University of San Diego

stock Option Expensing: The Role of Corporate governance 122 Sanjay Deshmukh, Keith M. Howe, and

Carl Luft, DePaul University

Real Options Valuation: A Case study of an E-commerce Company 129 Rocío Sáenz-Diez, Universidad Pontificia Comillas

de Madrid, Ricardo Gimeno, Banco de España, and

Carlos de Abajo, Morgan Stanley

Bne of the most important responsibilities of corporate managers is to evaluate and choose among major investment projects. The role of analysis in this decision-making is to help iden-

tify the alternatives that managers should consider and to support high-quality conversations, using information from throughout the organization, that lead to the best choices possible. This is true whether decisions are made centrally or by the business units. One requirement of asset valuation in this context is that it judge competing decision alternatives on a “level playing field.” A second requirement is that it support an effective division of labor between the front-line manag-ers who possess the most relevant technical and commercial information about the business or project in question, and the corporate center charged with providing the inputs for economy-level variables and ensuring the consistency and integrity of the entire process.

One reason why the prevailing standard for asset valuation, discounted cash-flow (DCF) analysis, has persisted despite widespread recognition of its limitations is that it is believed to meet these two criteria at least as well as other alterna-tives. With DCF, all proposals face the same corporate hurdle rate, creating the perception of a level playing field. Project or business unit-level considerations are “rolled up” into the projected cash flows, while corporate-wide risk adjustment and financing considerations are built into the discount rate, thus achieving the desired division of labor. But DCF, at least as commonly practiced, has two limitations that introduce a number of biases and violate the two criteria.

First, most DCF analyses are based on a “static” view in which future decisions are assumed to depend only on information available now and not on additional informa-tion that would be available when the decision is made. For this reason, static DCF does not allow front-line managers to model future flexibility explicitly and makes it difficult to value—and indeed tends to undervalue—those investments for which flexibility is an important source of value.

Second, most DCF implementations use a single discount rate in the analysis of all decisions or, at best, establish different discount rates for just a few large classes of decisions. This one-size-fits-all approach to the valuation of risk may be adequate for understanding the value of the “average” asset, but it fails to reflect the variety of asset designs that can feature very different

types and levels of uncertainty. For example, the use of a single discount rate, chosen with an average asset in mind, typically undervalues proposals to spend in the near term to reduce longer-term costs or increase longer-term revenues. Moreover, it systematically misvalues investments with so-called “non-linear” payoffs—not only investments with built-in flexibility (which has been the focus of the real options literature) but, as we discuss below, those presenting the challenges posed by complex interaction of multiple uncertainties, certain types of tax or other contractual obligations, and real asset design features like constraints on production capacity.

To be sure, managers can and do attempt to improve DCF valuation by adjusting discount rates to reflect differences in risk among business units or types of decisions, or changes over time in risk or other circumstances. But in the absence of clear methods for making such adjustments, the process is likely to be overly influenced by political considerations, undermining the level-playing-field role of valuation. Many managers are aware, at least at an intuitive level, of some of the biases of DCF, and some managers attempt to circumvent them using ad hoc approaches such as assigning high “strategic” value to particular decisions. But such practices undermine the ability of senior management to enforce common standards of analy-sis across business units, asset classes, and types of decisions.

In this paper, we show that overcoming the limitations of conventional “static, single-rate” DCF valuation requires two separate types of tools: a disciplined process for adjusting values for risk that goes beyond the single rate; and a method of modeling future decisions that makes valuation “dynamic” rather than static. We present an approach for adjusting values for risk that rigorously applies the principles of time and risk discounting to each potential cash flow rather than, as in the case of DCF, to an aggregated measure of the cash flows. We show that adjusting for risk on a “state-by-state” basis ensures consistency with external markets and internal consistency in the treatment of risk. We call this “market-based valuation,” or “MBV,” for reasons made clear shortly.

To model future flexibility, we propose an approach based on decision tree analysis (DTA). Some organizations already routinely use a limited form of DTA to analyze a few select types of flexibility. Our proposal is to broaden the application of DTA significantly. To distinguish this from the relatively narrow use of DTA in current practice, we call our proposal

by David Laughton, University of Alberta, Raul Guerrero, Asymmetric Strategy LLC, and Donald Lessard, MIT Sloan School of Management

O

46 Journal of Applied Corporate Finance • Volume 20 Number 2 A Morgan Stanley Publication • Spring 2008

Real Asset Valuation:A Back-to-basics Approach

complete decision tree analysis, or “CDTA.”In sum, our approach provides two important techni-

cal improvements over DCF: consistency in accounting for risk through MBV and capture of flexibility in decision-making through CDTA. By improving their valuation approaches in these two ways, companies can satisfy two often competing organizational requirements. They can enable unbiased, decision-oriented modeling of specific commercial situations. At the same time, they can strengthen senior management’s ability to articulate clearly, and consis-tently, the shared rules of engagement for decision evaluation throughout the organization.

In discussing potential bias in valuation approaches, it is useful to start by setting up a benchmark. In a corporation funded with publicly traded securities, or one mandated to act as if it were, this benchmark is the financial market value of the assets involved. DCF is intended to estimate such values when they are not directly observable. Many corporate analysts are aware of at least some of the ways in which static, single-rate DCF fails to estimate financial market value properly. Never-theless, many of these same people also assume that whether the DCF valuation result is “right” in an absolute sense or not, the corporate-wide use of a single discount rate has the virtue of penalizing all decision alternatives equally, and so preserving the relative ranking of the alternatives. But in fact, this usually will not be the case. Different valuation tools will return different relative rankings of managerial choices, not just different absolute value estimates.1 Toward the end of the paper, we show an example of this in a case study of a carbon capture and storage (CCS) opportunity. This oppor-tunity can be managed in different ways, each requiring a total investment of about $100M. In a version of the case, if managers execute the management policy suggested by DCF methods, they destroy about $13M in value by turning an asset worth more than $8M into one with a negative value of close to $5M.

The first part of our challenge, then, is to identify the source of the biases in current methods and propose ways of eliminating them. The second part is to demonstrate how our suggested technical changes can help management improve the process by which a company uses valuation estimates and other information to make decisions. The crux of our argument is straightforward: a corporate-wide discount rate policy is simply too blunt an instrument to control the huge variety of corpo-rate initiatives that must be evaluated. In fact, as we show later, a single discount rate cannot even be relied on when evaluat-ing different alternatives for developing a single asset if the alternatives differ substantially in terms of variables like capital intensity, operating leverage, or expected life. And even with

a varying discount rate, static DCF technology itself is too limited and inflexible to reflect the economics of most corpo-rate decisions, particularly those that are strategic in nature.

In the corporate valuation process that we propose, local teams focus on the characteristics of the project that are unique, while the corporate center specifies the rules of the game and controls the modeling of pervasive, economy-level risks. The proposed division of labor is as follows:

(1) Asset teams or business unit managers develop the set of decision alternatives to be formally analyzed and “own” the modeling of asset-level uncertainties, such as the amount of oil in a field or the likelihood of developing a new technol-ogy. Local teams have a clear comparative advantage over the corporate center in these aspects of the valuation process. A suitable peer review can serve as a check on this activity.

(2) Central finance staff specifies a standardized treatment of economy-level risks, such as commodity price risk. This function is analogous to a central group’s responsibility for, say, determining and disseminating a corporate-wide hurdle rate. But note that, in our approach, the group does not focus on modeling the risk of typical company projects, but rather on the risk associated with fundamental economic drivers such as oil prices or the growth in GDP. A specialized central unit usually has a comparative advantage in understanding how to model such risks, which must be modeled uniformly across the corporation to ensure consistency in the valuation process.

(3) Senior management centrally specifies a valuation approach that is general enough to handle most problems involving different types of decisions and the interaction of different types of uncertainty,2 thereby minimizing the need for ad hoc “excursions” outside of the framework.

In this paper, we do not focus on how an organization might manage the process of implementing the changes we are proposing, but we do want to make one comment in that vein here. Many firms that use DCF have something like this division of labor already. For those firms, the approach we advocate is designed to maximize the gain in analytical under-standing while minimizing the amount and cost of change. For those that are not organized in this way, our approach makes it easier to achieve this alignment if desired.

We recognize that it is not typical to address asset pricing techniques and organizational concerns within the same paper. But as we will demonstrate, the technical and organizational sides are closely linked. Our aim in the pages that follow is to present a broad enough picture of the practical advantages and applicability of our methods suitable for managers, while at the same time providing enough detail to enable valuation specialists to begin to apply our recommendations to their internal processes.

1 Note that selling the asset, or even the entire company, is frequently an alterna-tive, so if nothing else, an incorrect valuation changes the order of this monetization alternative.

2. From a computational point of view, this is similar to what Borison (2005a) calls “the integrated approach,” but, as we show in the Appendix, the basis we provide for it is very different, and in ways that have important practical implications for its use.

Journal of Applied Corporate Finance • Volume 20 Number 2 A Morgan Stanley Publication • Spring 2008 47

The Four Linked Aspects of Asset ValuationAt the level of fundamental theory, we are proposing noth-ing that goes beyond the foundations laid by Fischer Black, Myron Scholes, and Robert Merton in their pioneering work on option pricing in the early 1970s.3 The academic exposi-tion of asset pricing requires analysts, first of all, to account for differences in risk among assets. The theory also instructs analysts to account for changes in the risk of a given asset over time, and to reflect the value of “contingency” in decision-making—that is, the ability of management to respond to changes in key variables over time.4 Our suggestions for imple-menting these foundational concepts are an elaboration of the approach to real asset valuation that was first clarified by Michael Brennan and Eduardo Schwartz over 20 years ago,5 and that has been further developed and refined by many since then. But even with such a long gestation period, the ongoing debate in the real options literature suggests that the logic behind this approach is not well understood and needs to be revisited in a systematic way.6

With this goal in mind, in this section we break asset evaluation down into the construction and use of four linked models and discuss the characteristics of each, note potential problems with current approaches, and present our proposed improvements. The four models are these:

(1) a decision model, which contains an explicit statement of the alternatives the corporation is considering;

(2) a cash-flow model, which shows the relationships required to determine the cash flows;

(3) an uncertainty model, which provides the probability distribution of the uncertain inputs to the cash-flow model; and

(4) a valuation model, which shows how value is deter-mined from the streams of possible cash flows created by the other models.

Expanding the decision model. It may seem unneces-sary to ask analysts to state explicitly what is actually being decided at the start of an evaluation, but in fact the way a team frames the problem at hand—that is, the act of articulating its decision model—can have a significant effect on the actions that are eventually taken. Our main concern is that companies often do not model “contingency”—again, the possibility of management response to changes in key variables—during the decision-modeling stage, even when they are aware of important sources of flexibility in the future management of the asset.

For example, a car company may frame its decision model for an opportunity it is considering as follows: “When in the next five years, if at all, should we launch a battery-powered model using our patented but not completely developed battery technology?” Putting the question this way requires a

choice among alternatives such as “launch in 2009,” “launch in 2010,” and so forth, or “launch never.” This rules out the possibility that management will wait for still more informa-tion about the likely success of the project before making a decision. The actions taken today, based on that decision model, are effectively determined within the context of a future launch that is pre-committed.

For most situations, however, pre-commitment is a poor model of how a company actually intends to manage its assets. For the auto company, the asset is the opportunity to launch a new green model. The decision rule the car company actually intends to follow with respect to the launch is probably something along the lines of: IF gasoline prices remain high, car buyers remain interested in green products, our produc-tion costs are low enough to make the car marketable, and the overall demand for cars is high enough, and IF we have done the work needed to be able to launch, THEN we will launch; ELSE we will continue to monitor developments. That rule readily takes the form of a decision-tree structure. There are multiple sequential decisions—for example, how different pre-launch choices might constrain the possible times at which we can launch, or how such choices could influence the demand or cost conditions at any time we might wish to launch. There are also many sources of uncertainty, including external, economy-level uncertainties like overall demand for cars and the price of gasoline, as well as internal, asset-level uncertainties, such as how well our particular patented battery technology will work and when it will be available given different patterns of research effort. You have to put this type of structure into your evaluation process if you wish to consider contingent strategies explicitly and thoroughly. If you have not put it in, you are effectively modeling pre-committed decisions.

We believe that asset realities, such as the contingencies we have just discussed, should inform the decision model, which in turn should determine the cash flow and uncertainty models. In our own experience with companies, a different causal chain tends to prevail—one in which the existing uncer-tainty and valuation models effectively determine the way decisions get framed. If “deterministic” base-case cash flows are used in the DCF model, there tends to be little model-ing of uncertainty, and decision modeling remains outside of the process. On the other hand, if probability-weighted expected cash flows are used, as they should be, distribu-tions of outcomes are usually simulated, but often without adequate modeling of the decisions involved. We cannot fault analysts or asset champions for choosing to spend their time on the static DCF analyses that are understood and expected by senior management. But, as we have already suggested, there is a major downside to this kind of decision-making: a

3. Black and Scholes (1973) and Merton (1974).4. Even in introductory books such as Brealey, Myers and Allen (2006), contingent

decisions are discussed and real options is proposed as a modeling tool.

5. Brennan and Schwartz (1985). 6. Some of these debates have occurred in this journal. See, for example, the fol-

lowing articles on the “Georgetown Debate”: Borison (2005a), Copeland and Antikarov (2005) and Borison (2005b).


systematic failure to make the most of the wealth of informa-tion, both inside and outside of the organization, about the best way of increasing its long-run value. To help address this problem, management must develop and support an analytical technology that is capable of handling contingent as well as pre-committed decisions.

Those of our readers who are familiar with the real options (RO) literature will filter our discussion of the decision model through that lens. The essence of RO is that flexibility has value, and we certainly agree with that concept. We do have concerns, however, about how the concept is implemented in specific real options valuation models, and we urge readers to apply the same criteria when judging RO models as when judging DCF models. Most important, does the technol-ogy encourage and enable analysts to make the most of the information that the organization has about a particular asset? Forcing an asset to match a financial-market “look alike,” which is the essence of many RO approaches7, will only rarely capture the economics of actual decisions involving real assets. And requiring the flexibility to be modeled as an “add on” to an asset without any flexibility—which is the essence of many other approaches8—is also very restrictive.

The cash-flow model. With a decision model in place, the analyst next develops a cash-flow model. This is the arena where different types of inputs are gathered together to deter-mine the stream of cash flows that will occur in any possible future scenario. The cash-flow model takes the form of a set of relationships such as:

cash flowt = revenue

t - cost

t - tax

t.

In this case, there would also be sub-models for revenue, costs, and taxes. For example, in the battery car example, the revenue could be modeled as the price of the car multiplied by the quan-tity sold. There would then be a sales model that relates the quantity sold to the price set and to drivers of demand like the overall consumption or GDP index and the price of gasoline.

The cash-flow model has two types of inputs. First there are the variables, such as the time of launch in our battery car example, that are used to specify which decision alternative is being considered. These control variables are the link between the decision model and the cash-flow model. Second are the uncertain inputs, such as the GDP index and the price of gasoline in our example, that drive the overall uncertainty in the cash flows. These inputs are important enough that they have their own model: the uncertainty model.

The uncertainty model. In the example we are consider-ing, the uncertainty model would include a joint probability

distribution of the GDP level and gasoline prices.9 On the cost side there might be a technical success variable that specifies the uncertainty that the remaining R&D being done on the battery technology will make the battery more or less expen-sive to make.

There are three points to be made about the uncertainty model. First, it must be quantitative, even if this is difficult to do. In the case presented near the end of this paper, we model the future price of CO

2 emissions based on limited informa-

tion, but what is the alternative? Second, if the uncertainty model is to support the analysis of future decisions made with information available only in the future, we must have some idea of how that information will arrive over time (e.g. will the company learn about a variable a little at a time or all at once.) Therefore it will need to include both the probability distribution of the uncertain inputs into the cash model and any variable that describes that information flow. Third, it is important from a valuation point of view, as well as from an organizational one, to distinguish between those variables that are correlated with uncertainty in the overall economy, such as the price of gasoline or the GDP index, and those that are uncorrelated with the economy, as the technical success variable may be.10

The valuation model. The role of the fourth and final model, the valuation model, is to account for the two characteristics of cash flows that are of primary concern to corporate investors: timing and risk. Most methods of asset valuation treat time by discounting with a risk-free discount rate. Where methods tend to differ is in their treatment of risk. How this is done, and in particular at what level of aggregation, is central to the ability of a valuation approach to account for risk consistently. The current best practice in this regard is known as “state-pricing.”11 As discussed in more detail below, state pricing is a more general version of the single risk-adjusted discount rate approach taught in introductory MBA courses—and it can be traced back, in the form we use it, to the Black-Scholes and Merton approach to asset valuation and its descendants.

In the next few sections, we present some of the potential problems with single-rate discounting and how they can be managed, at least in relatively simple situations, by a shift to an MBV method called “forward pricing.” We will also see, however, that forward pricing can take us only so far, and that it breaks down in cases involving non-linear cash flows, which arise with managerial flexibility and for other reasons. Once our description of forward pricing is done, we will have the vocabulary to present formal definitions of MBV and CDTA.

7. An example is much of the RealOptions ToolKit described in Mun (2003).8. The most widely known example is Copeland and Antikarov (2002).9. Note that, in so-called “deterministic” DCF analyses, the analyst does not work

with entire probability distributions, but rather a single forecast or a small number of forecasts. In our terminology, these single point estimates of gasoline price, GDP, etc. are still an uncertainty model, but a limited one.

10. The uncertainty about the success of the technology will not be correlated with uncertainty in the overall economy if, on the one hand, it is measured by physical, sci-entific criteria—not influenced by the economy and if, on the other hand, our project is

small enough not to influence the economy by itself. It will satisfy this second condition if, for example, it is but one of many such projects, at least one of which is bound to succeed.

11. MBV and state pricing are closely related but not synonymous. MBV refers to a general approach for carrying out valuations by using financial market data as much as possible in a consistent manner. State pricing is a specific technique to value cash-flows by considering them on a state-by-state basis. We shall see that the MBV approach can be implemented by using state pricing or, in certain circumstances, by using a less general technique called “forward pricing.”


We then return to the more complex situations characterized by non-linear cash flows, where we will find that we must turn to the state pricing implementation of MBV.

DCF and MBV for Simple Cash-flow ModelsThe formula you are probably most familiar with for DCF valuation looks something like:

V = Σt expected cash flow

t / (1 + k) t

where V stands for value, Σt means “sum over times labeled by

t” and k is the DCF discount rate. We refer to 1 / (1+k)t as the discount factor for the time t. One point before we continue, the continuously compounded version of the formula for the discount factor is e-k*t, where k is the continuously compounded discount rate. This is the formula most commonly used in financial markets and we shall use continuous compounding henceforth in this paper.

We will find it useful to express the discount factor as a product of two factors—one to compensate for the time value of money and the other to compensate for risk—so that: Discount Factor = Risk Discount Factor * Time Discount Factor. The time discount factor is generated by the risk-free rate, and it accounts for the time value of riskless money. The risk discount factor accounts for risk only and the rate that produces it is frequently called the risk premium.

Let’s explore a very simple cash flow model: Cash flow t =

Revenue t - Costs

t. When we apply a discount factor to this

cash flow, we are implicitly applying the same factor to both the revenues and the costs. Almost always, the uncertainty of each component of cash flow will be different, with revenues typically more uncertain than costs. A single discount factor may still give a good estimate of value, but only if it is a kind of weighted average of the “true” discount factors for the cash flow components. The problem, however, is that the correct average discount factor will change whenever the “true” discount factors for revenues or costs change, or the weights of revenues and costs in the overall cash flow change. For example, in a growing company, the ratio of revenues to costs typically changes over time. To remain consistent with such a change in the asset structure, the single discount factor would have to change, even if the riskiness of revenues and costs themselves remained constant across projects and time.12

What happens if we abandon the single corporate rate version of DCF and try to adjust the discount rate to the situation we face? The technical problem is that, in most situations, analysts are likely to have less intuition about what the overall discount rate should be than about the discount rates for the components of the cash flow.13 Consider, for example, the case where an analyst models the revenues of

the potential battery car business unit as a linear function of the GDP index and the price of gasoline, while the costs are assumed to be known with certainty. An example would be the following model for the cash flow in 2015 (assuming we launch before then):

100M gallons * price of gasoline + $2M * US GDP index - $150M, where the GDP index for this year is 100.In this case, the appropriate discount rate for the risk-free

component of the cash flow—the negative $150M—is simply the risk-free rate.

As for the gasoline price component, recall that the forward price of a commodity is the given known amount of cash that one can contract for today to buy a commodity at a defined future time. Therefore, the forward price is simply the expected commodity price discounted for risk but not time. The component of the cash flow related to the gasoline price is equivalent to a claim to 100M gallons of gasoline in 2015. If we have the forward price for gasoline, we can value this by discounting the quantity (100M gallons * gasoline forward price) for time only.

There are forward prices for gasoline that run to maturi-ties of three years. In cases where there are reliable market forward prices for the relevant maturities, they should be used. In their absence, forward prices can generally be estimated using a forward market model based on both past and current forward prices.14

What about the component of overall cash flow that is proportional to the GDP index? One can generalize the concept of forward price to refer to any cash-flow amount, not just a commodity price. Indeed, there are the equivalent of forward contracts on stock market indices that are traded in futures markets in volumes much larger than the trading in the stock markets themselves. Thus one could imagine a forward price for the GDP index. If we can estimate it, the value of the claim to the overall cash flow being considered would be:

(100M gallons * gasoline forward price + $2M * GDP index forward price - $150M) * time discount factor.The forward price of the GDP index is just the amount of

the cash flow in question discounted for risk but not time. A number of studies have shown that the discount for risk in a broad-based economic index like GDP or the S&P 500 equals, in annual terms, about half the annualized standard deviation of the forecast return of that index.15 Using this result, the risk discount factor built into the forward price for the GDP index could be roughly estimated initially as follows:

exp (-0.5*cumulative annualized uncertainty in the GDP Index forecast)

12. These points are well recognized, but often swept under the rug. For an early exposition see Robichek and Myers (1966), Myers (1974) made this type of argument in introducing the APV approach and Lessard (1979) extended it in international applica-tions involving multiple uncertainties, tax regimes, etc.

13. This technical problem is on top of the organizational problems associated with this procedure.

14. Such models are in common use in financial markets. See Geman (2005) for a recent monograph on this topic.

15. Brealey, Myers and Allen (2006) refer to this in their discussion of the capital asset pricing model (CAPM). An example is Ibbotson and Sinquefield (1976).


The organization could also choose to put resources into a more accurate determination using a more detailed model of this risk discounting.

Non-linearity and ValuationThe corporation that is willing to move away from a single discount rate for all cash flows and cash-flow components can get much more insight into the effects of risk on value. We refer to this “multiple-rate” DCF technology broadly as forward pricing because of the central role played in it by forward prices.16

Unfortunately, forward pricing works only when the cash flow can be divided into additive components, one for each driver with a forward price observed or otherwise deter-mined—or, in other words, when the cash-flow model is linear in those drivers. Managerial flexibility and other sources of “non-linearity” violate this condition.

Flexibility introduces non-linearity because different decisions made in the future will change the relationship between inputs and cash flows in different ways. For example, suppose that the car company we have been discussing faces a choice at a single time in the future to launch the battery-powered product or not, and we have estimated the value at launch as follows:

500M gallons * gasoline price + $10M * GDP index -$3000MIf we want to model the realistic assumption that the

company will launch only if the value is positive, this contin-gency translates to a value at startup of:max(500M gallons * gasoline price + $10M * GDP index - $3000M,0)This is not a linear function of the gasoline price and GDP index.17

Non-linearity can arise for other reasons as well. When there are capacity constraints, for example, production and thus sales cannot rise beyond a certain level. The car company can always build another factory; but while it is doing so, the cash flows will not move in a linear fashion with the drivers. Contractual terms are another common source of non-linear-ity. The car company may have a previous owner that requires an additional payment (a clawback or earnout) if profitability exceeds a certain level. Tax regimes are another potential source of non-linearity through, for example, tax loss carryforwards or income limits on the use of tax credits.

In the DCF analysis of the linear cash flow model we presented above, the expected cash flows are determined solely in terms of the expected forecasts of the cash flow drivers. But this approach breaks down in the face of non-linearity because we then need to know the entire distribution of the driver, not just its expectation, to calculate the expected cash flow.18 This requires a shift in mindset—one that can be facilitated by simulating the value drivers, for example, and letting the spreadsheet or program calculate the expected cash flow.

Forward pricing breaks down for essentially the same reason. From a computational point of view, the correct expected cash flow can be calculated using simulation, as we have just suggested. But this approach leaves open the question of how to treat the effect of risk on value when the dependence of the cash flows on their underlying determinants is compli-cated by non-linearity. To treat risk properly, we turn to state pricing, which we shall introduce shortly.

MBV-CDTAOur walk-through of the four models in asset evaluation (as yet incomplete for the valuation model) has already suggested what we would like to improve. First, we would like to be able to model decisions that present themselves as complex deci-sion trees. Second, we would like a method for treating risk more consistently. These two dimensions are a very general way to distinguish between valuation approaches.

Our method improves on current practice along both dimensions. It consists of two approaches—Complete Decision Tree analysis (CDTA) and Market-based Valua-tion (MBV)— that can be used independently but are most powerful when combined. Along the risk valuation dimen-sion, MBV uses techniques, and as much as possible data, from financial markets to ensure an internally and exter-nally consistent treatment of risk. The focus on financial market data is why we call it “market-based” valuation.19 It accounts for risk at its sources, and maintains the entire distribution of cash flows until the final step of the valua-tion.20 Along the decision modeling dimension, CDTA uses decision trees to model decisions throughout the asset life cycle made in response to the resolution of all types of dynamic uncertainty.

In the next section (and the Appendix) we provide a more detailed account of how MBV works in general. Before we do, however, a few more words about what we mean by “complete”

16. This approach is also sometimes called certainty equivalent valuation. Certainty equivalent is just a more general term for forward price.

17. For example, if the GDP index is 100 and the gasoline price is anything less than $4 per gallon, the launch does not take place and the value turns out to be zero. But at a GDP index level of 100 and above $4 per gallon, the start-up value of the project increases by $50M for each 10¢ per gallon increase in price. A different relationship between inputs and value over different ranges for inputs is the defining characteristic of non-linearity.

18. With linear cash flow models, we can get away with knowing only expectations of the inputs because of the rule in statistics that “the mean of the sum is the sum of the means”. This rule cannot be applied if the cash-flow is not a sum of the uncertain input

variables, i.e. is not linear.19. The term “MBV” was initiated in 2003 by the participants at a Society of Pe-

troleum Engineers workshop on asset valuation in the upstream petroleum industry to replace an older term—“modern asset pricing” (MAP)—since this notion of “modern” was then over 30 years old.

20. We explain each of these choices in the course of the paper, but here is the quick explanation for the reader already familiar with some of this material. We adjust for market-priced risk in fundamental drivers using state pricing, model the interaction of the risk-adjusted variables with other parameters, accounting for non-linear payoffs, and then discount the risk-adjusted (or as some others refer to it the “risk-neutral”) cash-flow expectations for time.


decision tree analysis. Most companies that use DTA do so only to examine information-gathering decisions made early in the asset life cycle, such as R&D or, in the case of mining and oil and gas, exploration and appraisal. When they do so, moreover, they usually look at the impact of changes in only the asset-level variables while ignoring the effect of uncertain changes over time in the external commercial environment. CDTA is “complete” both in the sense of modeling flexibility throughout the entire asset life cycle (for example, by allow-ing the eventual closing costs of a mine to influence decisions today) and modeling within a single framework the impact on decision-making of changing uncertainty in economy-level as well as asset-level variables.21

State Pricing and MBVState pricing is the technical engine of MBV for general cash-flow models and the basis of the current standard for asset valuation in financial markets. A “state” is a condition at a given time of the variables in the uncertainty model.22 For example, one state in an uncertainty model for the battery car opportunity for the year 2015 might be specified by the gaso-line price at $3.15 per gallon, the GDP index at 101.5, and the technical success of the engineering design of an engine that delivers 100 horsepower. Another state might be gas at $2.80 per gallon, the GDP index at 102.1, and the technical failure of the engineering design. In this example, the gasoline price, GDP, and engineering outcome are all “state variables.”23

To gain an intuitive understanding of state pricing, consider a situation where there is only one state variable, the GDP index for next year.24 If you had the opportunity to buy a bond (which we call a high-GDP bond) that paid $10,000 a year from now if the GDP index were between its 50th and 100th percentile (say, 103–106), and an opportunity to buy a second bond (the low-GDP bond) that paid $10,000 a year from now if the GDP index were between its 0th and its 50th percentile (say 96 to 103), what price would you pay for each? In particular, would these prices be the same?

If the economy is doing well and growing at perhaps significantly greater than 3% next year, your job and other aspects of your finances are more likely to be going well, and the extra $10,000 would be nice to have: perhaps it represents an upgrade on which new car you buy. If the economy is doing poorly, your overall finances are likely to be going less well, and the extra $10,000 may mean the difference between

staying in your current apartment or moving to a cheaper neighborhood. Most people would agree the low GDP bond is worth more to them. This is simply a manifestation of risk aversion: We prefer to smooth our income so that the lows are not too low; and to achieve this goal, we are willing to give up the possibility that the highs are very high.

We could repeat this exercise for increasingly finer equal probability partitions of the state variable, until we had a different bond for each state, which we could call a state-bond. Each bond would have a different price, but since all the bonds have the same expected payoff and mature at the same time, this must mean that a different adjustment for risk would apply to each state. This is the result that provides the general principle underlying our pricing algorithm—namely, that different states may have different risk adjustments.

We can use this result to understand better how DCF works and where it might have problems. With DCF, we first deter-mine an average measure of the cash flow at each time—that is, we estimate the expected cash flow. This average is actually (at least supposed to be) a probability —weighted sum over the cash flow that occurs in each state. Then we discount the expected cash flow using the risk discount factor (RDF) and the time discount factor (TDF). We can “open up” the expected cash flow calculation to see what is actually going on:

E (CF) = sum over all states ‘s’ (cash flow in each state * probability of each state occurring)The formula for value is:V = E(CF)*RDF*TDF

In this equation, the RDF and the TDF are constants, so we can write:

V = sum over all states ‘s’ (cash flow in ‘s’ *probability of ‘s’ * RDF * TDF)

DCF accounts for risk with the same risk discount factor for each state at any given time.

The corresponding equation for state pricing in MBV is very similar, except we use a different risk discount factor for each state:

V = sum over all states ‘s’ (cash flow in ‘s’*probability of ‘s’ * RDF

s * TDF)

This makes clear the difference between DCF and state pric-ing. DCF uses an average risk discount factor and applies it to all states; state pricing uses individual risk factors.25

The linear/non-linear distinction that drives all of this can seem quite mathematical, but we can offer a good heuristic.

21. Some readers will be aware that it is difficult to model the appropriate discount rate to use within a decision tree, if DCF methods are used to value payoffs on the tree and if an attempt is made to find a discount rate appropriate for the situation as opposed to using a corporate hurdle rate. We can sidestep this problem by using MBV to adjust for risk first, then discounting these risk-adjusted payoffs with the risk-free rate within the decision tree.

22. Recall that these are the uncertain inputs into the cash flow model and any other variables needed to determine the dynamic behavior of information about those variables.

23. When we introduce state pricing, some students and clients express concern about the large, possibly limitless, number of states. The intuition is right, but this is the reality of the potential futures for a typical real asset, and, as we shall show in the

example below, modern desk-top computer power can be harnessed to manage this large number of states to our advantage. A large number of states can actually be used to increase our understanding of a situation, as we shall see in the Gascom case.

24. This is more or less implicit assumption when you use the CAPM and a beta calculated on the S&P 500 Index to derive a discount rate.

25. These risk discount factors are also called “risk adjustments” in the literature. The risk adjustments multiplied by the probabilities are called “risk-adjusted probabilities” or “risk-neutral probabilities”. The risk-adjusted probabilities multiplied by the relevant time discount factor are called “state prices” The state prices are the fundamental economic concepts, which is why the whole technique is called “state pricing” We discuss all these terms further in the appendix.


If you naturally think about a situation in terms of overall averages, the cash flow model in your head is likely to be linear. If instead you think of the situation in separate pieces (separate states) your cash flow model is probably non-linear, and you may need state-pricing. For example, almost no one speaks of the overall average performance of start-up firms. We speak of the average performance of firms that last for more than three years, and put those that fail early on in a separate category. This is a non-linear idea: we have a piece of the data, firms that collapse early, that gets treated one way (perhaps even ignored), and another piece of the data, firms that survive three years, that gets treated a different way. We also don’t naturally think of the overall average revenues of new projects such as the battery-powered car.26 We speak and think in terms of a 30% chance that the car will not be launched at all, and a 70% of achieving $10M a year revenue, conditional on launch. Do you see how you are segregating your thinking into a launch and a non-launch state?

State-by-state discount factors are determined collectively using information we observe directly in financial markets, such as forward prices. or they can be calculated from finan-cial market structures and data. They are conceptually the same as the risk discount factors we considered for the state bonds above. This allows our risk discounting to reflect both the structure of the underlying uncertainty that actually exists in the uncertainty model for each situation we analyze and the risk premia the financial markets assign to that uncer-tainty.27

Although we show how this works out in the case study that follows, we have two comments for you to keep in mind:

First, all cash flows, whether linear or otherwise, can be properly priced using this state-by-state approach. Second, if we can observe or otherwise determine the forward price of each of the state variables that define the uncertainty model, we are a long way toward being able to determine what we need to do our state pricing valuations. We provide more detail on this final point in the Appendix.

Gascom Case Study28

We now provide an example showing the application of MBV-CDTA to a realistic commercial situation with several sequential decisions of different types that must be made in response to changes in both economy-wide and asset-level uncertainties.

Let us go back to late 2004.29 Gascom has made the

decision to develop an already well-defined offshore natural gas field, but has to make another decision. Production from this field would produce CO

2 as a by-product, and any CO

2

emissions to the atmosphere would be subjected to penalty. One alternative for managing the emissions is to build a carbon capture and storage (CCS) facility that will compress the CO

2, pipe it to an otherwise useless underground reservoir,

and inject it there. The alternative to building a CCS facility is to emit the CO

2 to the atmosphere and pay whatever charges

are then in force.30 The cost of the facility is expected to vary little and is

modeled as known with certainty. If the CCS facility is built, the major operating cost will be the natural gas from the field that is used as the energy source for the compression, trans-port, and injection of the CO

2. There is some uncertainty

about the amount of energy that will be needed for injec-tion. This uncertainty will be resolved once injection begins, but can be resolved earlier by a $5M well test, which would also allow Gascom to tailor the design of the injection sites and save 10% of the total gas needed for the CCS facility. There are also two possible injection schemes—“basic” and “enhanced”—that involve a tradeoff between injection drill-ing costs ($5M more for enhanced) and the amount of gas required for injection (10% less for enhanced, in addition to the 10% saving if the well test is done).

We carry out a separate analysis for each of two possible regulatory regimes for CO

2 emissions. Under “cap and trade”

regulation, the price of CO2 emissions will be set by a market

for permits and is uncertain. With carbon excise tax regulation, every tonne of CO

2 emitted will incur a charge that is equal

to the current expected price for that year under the cap and trade system. Since the quantity of CO

2 emitted is modeled

as known and the cost per tonne is fixed, the costs of emitting the CO

2 under a carbon tax are known with certainty.31

Gascom has an immediate decision to make. It can build space into its production platform now, at a cost of $10M, for the compression aspects of its CCS facility. If this is done, the cost of the whole CCS facility in the future will be $90M. Otherwise, it can do nothing now; but if Gascom decides to build the CCS facility later, it will have to retrofit the produc-tion platform at an extra cost of $30M.

Setting up the Four Models. With this background infor-mation, we can construct the decision model and the cash-flow model for this opportunity. We show one part of each of these models to illustrate what is involved.

The decision model consists of a series of statements like

26. Of course, people can and do calculate overall averages for analysis, but our point is that you should do so with caution if you tend to describe the situation in a “on one hand this, but on the other hand that” basis

27. Note that forward pricing does just this in cases of linear cash flows.28. The full Gascom example is available in Laughton (2008).29. The first version of this case was presented in 2002 (Laughton et al 2002). The

version presented here uses economy-level uncertainty and valuation models that were updated in late 2004. All monetary amounts are in 2005 US$. The asset-level uncer-

tainty model, decision model and cash-flow model have been expanded for this paper.30. Note that, for Gascom, the “revenues” from building the CCS facility are the

avoided emissions charges.31. If a carbon tax were the regulatory instrument, in reality there would likely be

some uncertainty about its level. For simplicity, we have presumed that the government can commit to it with certainty. The thrust of our results will go through so long as the tax is more certain than the price would be under cap and trade.


the following: “in 2006 and later, if nothing has been done other than the possible pre-investment in 2005, Gascom can choose to do nothing, to conduct the well test, or to build the basic or the enhanced CCS facility.”

The cash-flow model consists of equations like the follow-ing for the net cash flow at time t, assuming the CCS facility has been built and is operating:

1M tonne * CO2 _price

t – $1M - basic_no_test_gas_

amount*(1-0.1*(well_test + enhanced)) * gas_pricet

if t>startup_time, operatingt=1

In this relationship, the economy-level uncertain inputs are the CO

2 and gas prices at the time of the cash flow. The asset-

level uncertain input is the amount of gas needed if no test is done and the basic injection scheme is used: “basic_no_test_gas_amount.” The other variables—“well_test,” “enhanced,” “startup_time,” and “operating

t”—are all so-called “control

variables” that are used to define the decisions that affect the cash flow. More specifically, “well_test,” “enhanced,” and “operating

t” are yes/no switches where the “yes” choice for

doing the well test, building the enhanced injection scheme or operating the CCS facility at time t, respectively, is 1 and the “no” choice is 0.

We next turn to the uncertainty model. Gas and CO2

prices, as noted, are economy-level uncertain variables. While there is a lot of data about the past and current behavior of the gas market that can be used to inform the gas price model,32 there is very little data about CO

2 price behavior. Expert

opinion, informed by the use of a large-scale integrated assess-ment tool,33 about the supply of CO

2 to the atmosphere, the

political demand for emissions reductions, and the marginal cost of meeting those reductions, was sought and incorporated into the CO

2 model.34

Figure 1 shows the expected CO2 prices used in the analy-

sis,35 as well as the median prices and the 80% confidence interval around the medians. There are long-term equilib-rium forces in the CO

2 markets that tend to bring the prices

down when they get too high and drive them up when they get too low. As a result, the amount of uncertainty about the

Figure 1 CO2 Price Model

0

10

20

30

40

50

60

70

2005 2015 2025 2035 2045 2055

CO2 Price Model

Time (years)

Price

($

/tonn

e )

median

expect

forward price

p10

p90

32. In this version of the analysis, we used a first “rough-cut” model of gas prices. It was not based on any econometric analysis of past gas forward prices. In a corporation with a fully operational MBV system, the econometrics would be done.

33. In this case the model of the MIT Joint Program on the Science and Policy of Glob-al Change was used. Details may be found at http://web.mit.edu/globalchange/www/

34. The numbers used for this and the gas price model were intended to give the best assessment possible of the true probability distribution. Frequently in DCF “valuations” done now, “conservative” assumptions about the uncertainty variables are used to gener-ate information about what “value” could look like if things go bad. This is stress testing, not valuation. Another common practice is to generate distributions of DCF “values”, one for each possible realization of the future, and statistics of this distribution are fed into the decision-making process. This may be useful simulation, but it is also not value estima-

tion. Value is a price and, as such, is a number not a distribution. Our contention is that in addition to this and other types of simulation, decision-making that should be based on value maximization should have access to the best unbiased value estimates possible.

35. These are obtained from a dynamic probability model. For the reader interested in the details, it is a correlated two-factor (one for gas and one for CO2) geometric diffusion model in the expected price forecasts with volatility that declines exponentially with the term of the forecast and that is time-dependent in the short-term. There are detailed eco-nomic, political and physical stories behind the uncertainty and valuation models (see Laughton 2008). These details are not important for the purposes of this paper, but it is important that they exist and could provoke necessary discussions within a corporation about their validity and limitations.


prices grows at a decreasing rate the farther out we look into the future. Thus, the per-period uncertainty about long-term prices is less than the per-period uncertainty about short-term prices. Natural gas prices exhibit similar long-term equilibrium behavior. We shall return to this point when we get to the valuation model. Uncertainty about the two prices is positively correlated over the time horizon we are considering.

The gas amount is an asset-level variable whose uncer-tainty is resolved solely through managerial choice: either well test or startup. It is not correlated with the economy-level inputs of the gas or CO

2 price.36

Finally we have the valuation model. For our DCF analy-sis it is simply a discount rate, which has been specified to be 10% per year. For MBV we have time discounting and state risk discount factors. The time discounting is determined by a constant real risk-free rate of 3% per year. The pattern of state risk discount factors is driven primarily by the risk discounting built into the CO

2 and gas forward prices.37 Note

in Figure 1 that, just as the proportional amount of uncer-tainty in the CO

2 prices approaches a constant over longer

periods of time, so does the discounting of risk.38 The same is true of gas forward prices. Therefore, as shown in Figure 2, the per-period average discounting for risk in the prices goes down as we look over longer periods of time. As we show in the Appendix, state variables that are not correlated with overall economic uncertainty, like the basic no-test amount of gas needed, do not contribute to the risk discounting in a state-pricing valuation.

Static DCF vs MBV. We begin with a decision model where Gascom must commit now to its future courses of action. It can perform the well test now if it wishes and use the resulting information before committing. This provides a comparison between single-rate DCF and a slightly enhanced form of forward pricing MBV without the complications caused by future flexibility.39 It is the type of analysis that would result if Gascom framed its decision model as “in what year should we build the CCS facility?” Later, we will add back flexibility using a CDTA analysis.

Table 1 shows the value obtained for various choices that could be made with respect to pre-investment, the well test, the injection scheme, and the timing of startup. If the well test is done, there are three start times, one for each of the

possible amounts of the gas needed to run the facility. If there is a choice to be made between the basic (B) and enhanced (E) injection schemes, the choice is indicated. Each possible combination of these control variables is called a “policy” and is valued separately. The single value assigned to the CSS facil-ity is the maximum found after a search over all policies. Table 2 shows the value of the CCS opportunity and the optimal policy in all of the cases we shall consider: DCF vs. MBV, cap and trade vs tax, pre-committed policies vs. full flexibility.

Consider first the DCF and MBV results under cap and trade with pre-committed policies. The first decision that Gascom must make is whether to pre-invest in the CCS facility. This decision involves a tradeoff between spending a certain $10M today and saving a possible $30M in the future. The DCF analysis discounts the $30M savings at a 10% corporate-wide discount rate, whereas MBV discounts it at the economically consistent discount rate, which for this known cost is the risk-free rate of 3%.40 As shown in Table 2, the optimal startup time is 2016 under MBV and 2021 under DCF, making the actual tradeoffs as follows:

(1) under DCF: spend $10M today to save a value of $30M * e-10%*16, or $ 6.1M.

(2) under MBV: spend $10M today to save a value of $30M * e-3%*11, or $ 21.6M.

This comparison reflects the bias of DCF against spending now to save later—a bias that can be attributed primarily to DCF’s discounting the future savings at an inappropriately high rate.

This bias appears again in the choice of injection design: basic with DCF and enhanced with MBV. DCF suggests that you not spend money on the injection design to save future gas costs. MBV suggests you should. The main cause of this differ-ence is that the long-term savings in gas cost are undervalued by the 7% discount rate for risk used in the DCF analysis.41 In Figure 2, we can see that the appropriate average risk discount rates for gas falls from 7% per year for very short-term gas to 1.5% per year for 20-year gas, as compared to the constant 7% per year rate applied by DCF. The overdiscounting of long-term cash flows in markets with long-term equilibrium forces is a pervasive bias in single-rate DCF valuation.

The decision not to do the well test is the same in DCF

36. The amount of gas needed for any given asset design will be determined by the structure of the storage reservoir, and will not be affected by the economy. The uncer-tainty in the amount of gas needed for this particular activity is too small to influence the economy as a whole.

37. As we show in the Appendix, some basic market consistency conditions constrain the state risk discount factors so that the relevant forward prices are at the “center” of the distribution of state prices in the same way that expectations are at the center of a prob-ability distribution. In the model we use, the risk discount factors are such that the state prices have the same proportional spread around the forward prices for gas and CO2 as the probability distribution for the actual prices around their expectations.

38. Recall that a commodity forward price is the price discounted for risk (forward price = expected price * risk discount factor). Therefore, flipping this relationship around, the risk discount factor is the ratio of the forward price to the expected price. It is this ratio that approaches a constant (roughly 0.56) in Figure 1.

39. We actually use a slightly subtle combination of general state pricing and forward pricing to do the valuations here. The cash flow model is non-linear even in this decision model with pre-commitment, because the gas cost once the facility is operating is the uncertain gas price multiplied by the uncertain gas amount. However, in any given state about the amount of gas needed (so that the amount of gas becomes fixed), the cash flows are linear in the CO2 and gas prices and we can use forward pricing methods to determine the value contingent on that asset-level state occurring. We then use state pricing methods to take the probability–weighted sum over the gas amount states of these values to get the overall value. As noted above, because the uncertainty in the gas needed is not correlated with overall economic uncertainty, there is no further risk discounting in this probability-weighted sum.

40. A model with uncertain startup costs would not change the essential insights, if the uncertainty were not too large.

41. The 7% DCF risk premium arises from a total DCF discount rate of 10% less the 3% risk-free rate for time.


and MBV, so the last difference to discuss is the difference in the optimal timing of the startup: 2016 with MBV, 2021 with DCF. The tradeoff here is the value of the operating cash flow you gain by building the facility vs. the time value of delaying the expenditure of the startup costs. DCF under-values long-term CO

2 for the same reasons it undervalues

long-term gas. And the net effect is that DCF undervalues the long-term potential operating cash flows of the CCS facility, which biases DCF toward a later startup. DCF also overvalues the time value of delaying the startup costs by using a 10% discount rate instead of the appropriate rate of 3%, creating a further bias toward a later startup date. Finally, DCF suggests that you not pre-invest, which makes the startup cost higher than with MBV. And this in turn has the effect of increasing the value of delay even more.

Is the difference between the suggested actions with DCF

and MBV important? Using the DCF policy, the MBV value would be a negative $4.7M, which is $13M less than the value using the MBV policy.

What happens if CO2 regulation is accomplished through

the carbon excise tax? By design, the expected cash flows of the tax are the same as under cap and trade regulation. The DCF values depend only on the expected cash flows and not on their uncertainty, and hence they do not change from the cap and trade results. On the other hand, MBV assigns a higher value to the revenues for the CCS facility under tax regulation because the revenues are certain and hence risk-free. And for this reason, startup occurs earlier under MBV, and the well test gets done because more gas is used (and can be saved by doing the test) with the earlier start.42

In this analysis, we have several examples of a general result that we cannot emphasize enough. The change in

Table 1 Results of DCF and MBV Valuation with no Future Flexibility

DCF - Cap and Trade or Carbon Tax MBV - Cap and Trade MBV - Carbon Tax

Value if Invest Best Year Value if Invest Best Year Value if Invest Best Year

Build basic facility $ 6.65 M 2021 - $ 3.09 M 2018 127.09 2013

Pre-Invest+basic $ 3.50 M 2018 $ 7.86 M 2016 141.05 2012

Well Test+basic $ 2.47 M 2020, 2021, 2022 - $ 2.36 M 2016, 2017, never 127.27 2012, 2012, 2013

Pre-invest + Well Test+basic

- $ 0.32 M 2017, 2018, 2019 $ 7.14 M 2014, 2016, 2017 141.54 2011, 2012, 2013

Build enhanced faciliy $ 6.41 M 2021 - $ 2.92 M 2017 128.13 2013

Pre-invest + Enhanced

$ 3.25 M 2018 $ 8.31 M 2016 142.33 2012

Well Test + Basic or Enhanced

$ 2.47 M B2020, B2021, B2022 - $2.21 M B2016, E2017, never 128.48 B2012, E2012, E2013

Pre-invest + Well Test + Basic or Enhanced

- $ 0.32 M B2017, B2018, B2019 $ 7.23 M B2014, E2015, E2016 142.84 E2011, E2011, E2012

Best Policy Build basic facility in 2021 with no pre-investment, no well-test

Build enhanced facility in 2016 with pre-investment, no well test

Build enhanced facility in 2011, 2011, 2012 with pre-investement and the well test

Approach and Regime: No Contingency Full Flexibility

Value Policy Value Modal Policy

DCF - Cap and Trade $6.65 Build basic 2021 $10.00 Build basic 2014-2030

DCF - Carbon Tax $6.65 Build basic 2021 $6.75 Build basic 2019-2024

MBV- Cap and Trade $8.31 Preinvest, build enhanced 2016 $13.59Preinvest, well test, build enhanced 2012-2017

MBV - Carbon Tax $142.84Preinvest, well test, build enchanced in 2011-2012

$143.80Preinvest, well test, build enhanced 2010-2015

Table 2 Values and policies without and with Future Flexibility


valuation approach from DCF to MBV changes not just the value estimates in spreadsheets, but the decisions that should be taken to realize the most value possible.

Adding in flexibility. Table 2 also shows how the values and policies change with DCF and MBV under the two types of regulation when we move to the fully flexible decision model. As we discussed earlier, the non-linearity arising from flexibility requires the analyst to consider the entire distri-bution of drivers and not just their expectations or forward prices. To estimate the distributions, we simulate 100,000 to 800,000 CO

2 and gas price path realizations.43 The discrete

gas amount probabilities are handled directly.Figures 3, 4, and 5 show the pattern, for the first 10,000

price path realizations, of the DCF and MBV choices in 2013 and the MBV choices in 2021, assuming cap and trade regula-tion, pre-investment in 2005, and no other action by the time of the choice.44 The different regions on the graph represent different combinations of CO

2 and gas prices for which a

particular policy is optimal. For example, in the region colored by the dark gray dots, it is optimal to continue waiting.

The analysis produces this data for each year for all possible states of the CCS asset45 and therefore allows us to calculate the probability of being in any given state and taking any given action in any given year. Figure 6 shows the probability for the timing of the set of actions leading to startup that are most likely for the DCF and MBV analy-ses under cap and trade and tax regulation. For example, the black solid and dashed lines show the probability in any given year, using the optimal DCF policy under cap and trade and tax regulation, respectively, of building the basic facility without doing the well test (the only option with significant probability under DCF). The equivalent green lines are the probability under MBV for doing the well test, which is the most likely action under MBV (76% under cap and trade, close to 100% under tax regulation). The gray lines show the probability of following up the test under MBV with building the enhanced facility, which is again the most likely choice (66% under cap and trade, 94% under tax regulation).

We can see from this figure that, with DCF-CDTA, the value of flexibility stems almost entirely from the responsive-

42. Because the well test is done, there are three startup designs and times, one for each possible gas needs state. The enhanced design is recommended in all states. The timing is 2011 in the low-needs state and 2012 in the others.

43. A look at a given single price path in this sample would reveal that because the prices are built up probabilistically over time, these paths are not smooth. They do mimic the sorts of continuous yet jagged patterns you see in most past time series of economic data.

44. The optimization techniques used to find these choices are based on the work of Longstaff and Schwartz (2001). The algorithms and the presentation methods used were developed with Mike Paduada and Mike Samis while David Laughton Consulting Ltd. had an alliance agreement with the Mining and Metals Group of Amec Americas Ltd.

45. The plots are best viewed as a time-lapsed animation to give an idea of what might happen with the asset.

Figure 2 Discount Rates of Risk: DCF versus Forward pricing

Discount Rates for Risk

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

2005 2010 2015 2020 2025 2030 2035 2040Term

Rat

e (1

/yea

r)

CO2

gas

DCF


0 5 10 15 20 25 30 35 40 45 50

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ural

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x($

/mcf

)

Gray = Wait Black= Test Light Gray = Enhanced StartGreen = Basic Start

CO2 Price ($/tonne)

16

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4

2

ness of the start-up timing to the natural gas and CO2 prices.

This responsiveness produces a spread in the start-up time probability distribution around the 2021 start for the static decision model. This spread is relatively narrow in the case of tax regulation, where gas prices are the only uncertainty in play. It is broader for the cap and trade regulation. Indeed, with cap and trade, there is a 15% probability under the DCF recommended policy that the CCS facility is never built.

The MBV-CDTA story is more complex. Recall that the optimal policy to follow under the static decision model is to pre-invest and then build the enhanced facility in 2016 without doing the well test. With flexibility, the most likely action is to do the test earlier on and then build the enhanced facility. However there are significant probabilities of other sequences of actions leading to startup.

Let us now return to the plots of optimal action in terms of the prices (Figures 3, 4, and 5). As can be seen in all the plots, given low CO

2 prices, it is optimal to wait to

see whether it will be worthwhile in the future to build the CCS facility. At high prices it pays to build the facility and begin storing CO

2. A region of prices where testing is optimal

occurs only with MBV and only in the early years, in states where CO

2 prices are moderate (and the costs of delaying

the availability of storage are low) and gas prices are high, making gas savings from testing more valuable. The CCS design pattern makes sense as well: early on in the asset life and with high gas prices, it is best to invest in the enhanced facility to save gas costs.

Figure 3 MBV-CDTA under Cap and Trade Choices in 2013 if Preinvestment Made/No Other Action Yet

Figure 5 DCF-CDTA under Cap and Trade Choices in 2013 if Preinvestment Made/No Other Action Yet

Figure 4 MBV-CDTA under Cap and Trade Choices in 2021 if Preinvestment Made/No Other Action Yet

0 5 10 15 20 25 30 35 40 45 50

Nat

ural

Gas

x($

/mcf

)


CO2 Price ($/tonne)

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Note finally that the DCF biases against spending now to save or enhance later (no well testing and smaller enhanced facility regions) remain very evident in the flexible analysis, as they were in the pre-committed analysis.

Conclusions from the Case. This case shows the power of MBV-CDTA analysis to support decision-making about asset design, selection, and management. It also makes clear the biases inherent in the single-rate DCF approach to valua-tion and the importance of getting all the models used in the valuation right. The importance of a formal treatment of flexibility is shown by the change in the recommended management policy from no testing to a high probability of testing. The importance of getting the uncertainty model right is shown by the very large differences between the cap and trade and the tax analyses. Finally, this still relatively simple example shows the limitations of the most commonly proposed real options approaches. There is no way that this opportunity could be shoehorned into an analysis based on a typical financial instrument analogy, and the flexibil-ity is much too complex to be treatable as an add-on to a non-flexible version of the asset, even if the right version of the non-flexible asset could be identified.

Organizational Benefits of MBV-CDTAIn order for valuation to provide a level playing field, it must be done in a logical and consistent fashion rather than tailored

to particular features of a given project or investment. MBV-CDTA supports this requirement, as well as the desired division of labor between headquarters and operating manag-ers, by offering the technical method to pursue a transparent, divide-and-conquer approach to project evaluation.

MBV is consistent with the principle of “value additivity,” which underpins the ability to consider assets one-by-one as well as attributing different components of assets to one or another strategic source of value. It also allows the corporation to separate the analysis of economy-level drivers from the analy-sis of local asset-level technical and commercial drivers that are influenced by asset managers. And of course, MBV shows how to combine the separate analyses into a single result.

Additionally, breaking up asset evaluation into four distinct tasks—decision modeling, cash-flow modeling, uncertainty modeling, and valuation—clarifies where the opportunities are for different groups to offer their input. We would expect a company’s modeling approaches—and its modeling of economy-wide uncertainty in particular—to be managed by the corporate center. A centrally supported valuation platform could include a common set of probability distributions and state prices for an array of economy -level drivers that are relevant to a particular business. One can imagine in the not too distant future a shift from PC-based spreadsheet models with risk plug-ins to a web-based system in which the probability distributions and state prices for the core

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uncertainties facing a given company are provided centrally.At the same time, the separation supported by

MBV-CDTA would allow the local operating managers to focus on their technical and business expertise instead of trying to master advanced financial concepts. This stands in marked contrast to many real options methods, which are not amenable to separation into an economy-wide versus asset-level split. In our hypothetical web-based platform, local analysts could model technical and commercial aspects of the projects “on top of” the centrally provided information.

In particular, local operating managers are responsible for information about asset-level variables. As we show in the Appendix, unless one of these variables is correlated with economy-wide developments, it makes no direct contribution to risk discounting. Therefore, local operating managers do not need an expert understanding of how financial markets price risk and how to estimate risk discount factors. It is only the central finance unit that owns the economy-level uncertainty model for the corporation that must develop this expertise.

This division of labor quite closely matches the division implicit in DCF where risk adjustments are rolled into a centrally determined discount rate and cash flows are modeled locally, ideally with central oversight or peer review. It also is consistent with practice in firms that have adopted a more formal decision-theoretic perspective, but it integrates much better with the core finance function than is often the case with such approaches. As a result, for the majority of firms it preserves the existing division of labor and work flow and thereby minimizes difficult political and cultural elements of change. For firms whose process is either totally centralized or operates completely on a case by case method, it provides a roadmap toward a more desirable approach, one that allows asset teams or business unit managers to develop the set of decision alternatives to be formally analyzed and “own” the modeling of asset-level uncertainties, central finance staff to specify a standard treatment of economy-level risks, and senior management to specify a common valuation approach to ensure consistency and facilitate governance. Yet it is general enough to avoid myriad special cases.

Beyond its role in fostering a practical division of labor, MBV-CDTA further gives analysts the flexibility to model assets in a way that matches and illuminates the actual decisions at hand. It provides a powerful decision model, clearly showing which alternatives become optimal under which circumstances. This emphasis on matching the struc-ture of the decision is most clearly visible in Figures 3, 4, and 5 of the Gascom case study, which show the “zones” of future states where different actions and investment alterna-tives become viable. MBV-CDTA supports clear thinking and open dialogue by preserving distributions of outcomes as well as expected values, emphasizing decision points, and calling attention to which uncertainties are most critical

when making current and future choices. One can easily imagine revisiting these analyses periodically as the underly-ing drivers evolve to see which alternatives have become more or less attractive. This provides a forward-looking window into future business decisions that helps managers retain the organizational readiness required to exploit real options and, when warranted, to close down future options that no longer appear viable.

Finally, our approach increases the visibility and transpar-ency of key assumptions, which is critical to communication and organizational buy-in. It also brings the valuation of risk to the foreground, forcing much more explicit consideration of which drivers are priced and which are not and highlight-ing the interaction between local and global variables.

While the method we have presented is applicable to any set of asset decisions, the required investment in the develop-ment of understanding and tools is most easily justified for projects involving: (1) enduring assets (particularly those with different lives across alternatives); (2) future decisions that depend on the evolution of key variables; and (3) technical, contractual, and fiscal non-linearities. Although these features are prominent in virtually all valuations and choices in oil and gas and other natural resource investments, they are also common in tangible and intangible “platform” investments in manufacturing and service industries.

Some readers may be left wondering: Do we have to do all of this at once or can we gradually wade into this approach to valuation? As we mentioned early on, the management of change is not the focus of this paper. Nevertheless, we do want to say in answer to this question that the break-down of the changes into those relevant for implementing MBV and CDTA separately mean that an organization can implement them sequentially in either order if that suits its circumstances. Moreover, there is no need for a discrete shift from current practice to a complete implementation of either MBV or CDTA. There is scope for gradual movement, both technically and organizationally, on either front toward what we propose.

Conclusion We have presented a general alternative to single-rate DCF called MBV that adjusts for risk at the level of the individual cash flow realizations instead of aggregate cash flows. MBV is an implementation of the time-state pricing basics that under-pin virtually all modern valuation approaches. It provides more accurate valuations for all but the simplest assets.

We have also shown that we can complement MBV with a more complete decision tree framework called CDTA that increases the “richness” or realism of the modeling environ-ment. Recent improvements in computational methods (and computing power) make this approach feasible for assets with a realistic degree of complexity, as we have shown in the Gascom example. As a valuation tool, the combination


of MBV and CDTA dominates those real options approaches that model assets restrictively as analogies to traded finan-cial instruments or that treat limited forms of flexibility as add-ons to a static asset model.

The key advantages of the method from a valuation perspective are numerous. Most important, by going back to basics in the pricing for risk, it allows market-based risk adjustments for all types of uncertainty on all elements of the cash flows. MBV-CDTA can also readily accommodate non-linearities in the cash flows, thus overcoming the well known biases in DCF.

Like many back-to-basics approaches, MBV-CDTA tends to transform implicit assumptions into explicit ones. We appreciate that the increased transparency comes at a cost of increased complexity. We would argue that the complexity is “out there” in the business environment, and the valuation approach merely reflects it. A major advantage of MBV-CDTA, however, is that its structure encourages, indeed requires, a divide-and-conquer mentality. This provides more levers for corporate managers to pull, and requires more consistency on the part of project champi-ons. The CDTA component takes into account the value of flexibility that these champions sometimes try to reflect in nebulous claims of “strategic value.” Overall, MBV-CDTA actually allows the corporate center to increase its influence over the evaluation of asset decisions while at the same time increasing the use of asset-level expertise.

david laughton is the Principal at David Laughton Consulting Ltd,

and an Adjunct Professor in the School of Business at the University of

Alberta.

raul guerrero is Managing Partner at Asymmetric Strategy LLC.

donald lessard is Epoch Foundation Professor of International

Management at the MIT Sloan School.

Appendix: Some Further Details on State PricingIn the body of the paper, we considered the implementation issues with MBV-CDTA, providing a minimum of vocabulary and technical detail. In this Appendix, we delve deeper into two issues: first, the frictionless (or perfect) financial markets approximation and its implications for the role of valuation in decision-making; and, second, the basics of state prices. This is done with an eye towards helping readers answer some of their own, and their colleagues’, “yes, but…” questions. Other important issues must await treatment elsewhere. These include many details of how to specify and param-eterize usable “industrial strength” decision, uncertainty, and valuation models, and how to perform, as required by CDTA, the complex searches for a best decision.

The bulk of asset pricing theory is built on the approxi-mation that financial markets are frictionless. This is not because anyone believes that real markets lack frictions such as transactions costs, information asymmetries, and barri-ers to the creation of markets or the quick equilibration of prices. Rather, it is because there is currently no complete approach for handling these frictions, and the conjecture is that their effect on valuation is small compared to other types of approximation errors, such as those in the cash flow and uncertainty models.

We believe the current best practical solution to account for market frictions is to split the analysis of value into a strict valuation portion based on the perfect market approximation and an “enterprise risk management and other considerations” portion. As we expand on below, on the strict valuation side, frictionless markets lead to value additivity, and projects, defined properly, can be considered on a stand-alone basis. On the enterprise risk management side, new projects must be considered in light of the company’s existing and poten-tial portfolio of assets, and at least some attempt is made to account for the effects of frictions on value.

As we discussed in the body of the paper, valuation produces a ranking of projects and actions within projects, not just numbers. It is possible the company may choose to overrule this ranking given the concerns that stem from finan-cial market frictions. For example, a common complaint of senior managers against valuation is that it can suggest taking the project that risks $100M in order to produce a net value of $12M over the project that risks only $50M to produce a net value of $9M. The validity of this complaint, with which we have some sympathy, is based entirely on the consideration of market imperfections. In this particular case, managers may be worried about difficulties in raising capital, which would not be an issue if there were no financial market frictions. Or they might be worried about the risk of a costly bankruptcy, again an issue only with imperfections. Hence, their intuition to choose the smaller, less valuable project might be right if the larger project would crowd out other valuable opportunities by starving them for capital, or if a $100M loss could bankrupt


additivity is the formal principle that allows an analyst to value a bond by separately valuing each cash flow, for example. Although this separation by time is the most common applica-tion, there is no reason why cash flows cannot be segregated into components and valued separately, as suggested by Myers with the adjusted present value approach. And of course, value additivity also allows us to value cash flows state-by-state.

A good way to think about the law of one price is to remember that cash flows inside the company are the same as cash flows outside of the company. Did you just immediately disagree with that statement in your mind? If you did, you probably jumped to a scenario where the company “needs the cash,” but this is an enterprise risk-management/market imperfection consideration which valuation always explicitly sets aside. For valuation, the cash flows from a real asset are a commodity that can be diversified with cash flows from financial assets. If this can be done in frictionless markets, technical risks that are not affected by, and do not affect, the overall economy can be completely diversified away. That is, we can combine the cash flows from the purely technical project with financial assets until the variance of the resulting portfolio is negligibly small. On the other hand, pervasive, market-wide risks, such as oil and gas prices, interest rates, or stock market performance cannot be diversified away.2 Most people are risk-adverse, and hence the variance associated with these economy-level risks must be compensated via some sort of risk premium.

Corporate decision-makers often struggle with the prescription to value purely asset-level risks with no risk premium—equivalently, to discount them with the risk-free rate. The reaction is understandable, since we seem to be equating the variability in those technical outcomes that have purely local asset-level effects—for some companies the aspect of their business they spend the most time understanding and attempting to influence—with a sure thing such as the payoff for a government bond. The variability in asset-level outcomes does “matter.” A 52% chance of technical success is always better and more valuable than a 51% chance of success. But the effect of this variability on value is completely described by the probability of outcomes for the technical uncertainty and requires no further adjustment for risk.

To understand this result, consider a state variable defined by the amount of oil in place in a small field. The states for this variable are determined purely in physical terms.3 What is happening in the economy has no influence on it, as it was determined long ago, and the outcome of the variable is too small to have an influence back onto the overall economy.

the company (or give customers, suppliers, or shareholders reason to doubt its future solvency). However, the discussion would be moot if the value of the $100M alternative were either $9.01M or $200M, and not $12M. Issues arising from market imperfections for a $100M investment would likely be worth more than $0.01M, and while they might be worth as much as $3M, they are almost surely not worth as much as $191M. Valuation provides a screen to allow the organi-zation to focus on tough risk management calls, where the risk management must be done with real assets as opposed to financial engineering.1

We believe it is important not to blur the line between valuation and enterprise risk management. We would recom-mend management conduct “clean” valuations of both projects, unfettered by ad hoc devices such as artificially high discount rates to penalize the capital intensive project. Once the valuations are complete, additional considerations can be weighed, including the implications of asset decisions for enterprise risk management, the costs of transmitting appro-priate information about “private risks” to financial markets (and the implications of those costs for the availability of capital), and the management issues of information sharing, incentive alignment, and control. As there are no comprehen-sive, useable quantitative models to deal with these issues at this time, we believe they are best handled using qualitative judgments about the use of the value estimates, or at best by semi-quantitative adjustments to them such as scorecards. In particular, attempts to quantify the impact of market imper-fections by using a corporate utility function (Smith and Nau 1995, Borison 2005a) as a metric for decision-making are ill advised. Corporate utility is an ad hoc device without theoretical foundation that is not connected with the real economic and organizational issues that arise from financial market frictions. Its use tends to obscure the complex issues that are involved.

The job of valuation, then, whether it be DCF, EVA, real options, or MBV-CDTA, is to estimate the value of project cash flows under the assumption of frictionless finan-cial markets. Under this assumption, trading takes place to enforce the “Law of One Price”: assets with the same cash flow consequences have the same price. A corollary of the Law of One Price is the Principle of Value Additivity: the value of a collection of cash-flow claims is the sum of the values of the individual claims. This principle allows us to partition the cash flows associated with a particular complex asset into different parts associated with simpler assets, value each of the simpler assets separately, and then sum the resulting values. Value

1. Resource constraints other than capital, particularly human resource constraints, may be another important reason for overruling the choice dictated by the stand-alone value calculation. We would argue this is a case of incomplete modeling. The portfolio of projects should be selected subject to the constraints the company faces. It is almost always the case that the company cannot staff up immediately to meet all opportunities. Hence, it is appropriate to consider project value relative to the size and capability of the project management team. But our main point stands—the valuation should be carried out first, including the opportunity cost (or shadow price) of constrained resources to the

extent these can be estimated (a topic we shall not address here), and other consider-ations should be weighed outside of the valuation framework.

2. Specific groups of assets can be hedged against each other to yield cash flows with no variance. Given the positive correlation between assets, this is usually achieved by going long some assets and short others. On an aggregate basis, however, all assets that exist are “net long” and the logic of hedging cannot be applied economy-wide.

3. Oil in place is different from reserves, which is a mixed physical-economic con-struct, and, as such, less useful for analysis.


To simplify the following discussion, we limit ourselves to a single time period. The simplest claim to value in this single time period is one that pays a constant, say $5, in every state, such that:

V = $5 * time discount factor * Σ over all s (probability of state s * risk adjustment for state s)However, we also know that the value of a claim that pays

$5 with certainty is just $5 * time discount factor. Therefore, it must be that:

1 = Σ over all s (probability of state s * risk adjustment for state s)The product of the true probability of state s occurring

and the risk adjustment for each state s is a positive number and these numbers sum to one. These are the defining proper-ties of a probability distribution. Therefore we call this product the risk-adjusted probability of the state. It is useful terminol-ogy to call any statistic, such as an expectation, derived using this distribution a risk-adjusted statistic.

What other constraints do we have on the risk-adjusted probabilities? If we consider a general cash-flow, CF, which varies by state and occurs at time t, the state pricing equation tells us that the value of a claim to CF is:

V = Σ

s (CF in state s * risk adjusted probability of s

occurring) * time discount factor for time t = risk-adjusted expectation of CF * time discount factor for time tBut recall that this value is also the forward price for the

cash-flow discounted for time. Combining these two relation-ships, we find that the forward rice for any cash-flow is just its risk-adjusted expectation. Because of this, any forward price directly observable from market data (like the short-term gasoline forward prices in the battery car example) or calcu-lated in some other way (like the GDP index forward prices or the long-term gasoline forward prices to which we referred in the body of the paper) puts another constraint on the risk-adjusted probability distribution. For many uncertainty and valuation models in current use, the whole of the risk-adjusted probability distribution can be estimated once we know the real distribution and the risk-adjusted expectations of all of the state variables.

We show how this all works out in a simple little example of the natural gas market over one time period. In our model of the market there are three states: one state now and two equi-probable states a year from now. In one state a year from now, the price is higher than expected. In the other it is lower. The prices are as follows:

Price in One Year$14.50 with 50% probability, or$9.50 with 50% probability

How would an investor price a bond that pays $1000 a year from now if the amount of oil is high (the “success bond”) and another similar bond that pays off if it is low (the “failure bond”), where high and low are defined so that there is a 50-50 probability of either happening? Under these circumstances, both bonds are economically identical to an arm’s length inves-tor, with each offering an expected payoff one year from now of $500 in situations where there is no difference in overall well being. Therefore they should have the same price.

Two big misunderstandings arise when a manager oversee-ing the particular field in question considers these bonds. First, the manager may object that the outcome of these bonds is perfectly correlated with his personal well being. That may be true, but of course he is charged with running the company for the benefit of its owners, not for his personal gain.4 The shareholder owners are diversified in their holdings and there-fore break the correlation between the field outcome and their personal wealth. Second, the asset-level risk under consider-ation may potentially bankrupt the company. In that case, we are back to our original suggestion: value the oil-in-place bonds using consistent principles—in this case, as we shall see shortly, using no risk premium to account for the amount of oil in the field. Then, adjust for the market imperfection of costly bankruptcy.

For the arm’s-length investor, we have determined that the two oil-in-place bonds are identical in effect, and their value must be equal. Since together the two bonds give a risk-free payoff one year from now, the sum of their value must be $1000 discounted for time alone. Hence, the value of each bond must be $500, or 1000 times the probability of success or failure respectively, discounted for time. This is the formal way to show that there is no additional risk adjustment in the valuation due to technical uncertainty. This sort of argument can be generalized to situations where there are many mutually exclusive and exhaustive states with different probabilities. It can be further generalized to look at risk adjustment in states characterized by both asset-level and economy-level variables. In that case, the risk adjustment for the combined state is equal to the risk adjustment for the economy-level variables by themselves.

We now outline how the risk adjustments for economy-level variables can be estimated using observed market variables.

The general state pricing formula is:V

= Σ overall times t of interest

(Σ over all states ‘s’ at time t (cash flow in state s * probability of state s * risk adjustment for state s) * time discount factor for time t )

4. There is clearly a management control issue here. Board members and senior executives must design appropriate incentives for the manager to bring his actions in line with what shareholders would like.


V = 0.3325 up state price X $14.50 gas in up + 0.6175 down state price X $9.50 gas in down = $10.69 = $11.25 * 0.95As we mention in the body of the paper, if forward prices

are not available, we can use an asset pricing model such as the CAPM to estimate a forward price, then proceed as above.

So now we know what to do in one period with one simple type of uncertainty with an approximation of two end-of-period states. What about multiple states, multiple variables, and multiple time periods?

Dealing with multiple states and multiple types of uncer-tainty does not require any new basic concepts, just more mathematical and modeling detail.

Dealing with multiple periods requires one more idea, which we only sketch here. The basis of compound interest is the observation that, in a world without uncertainty, holding a multi-period bond is the same as holding a series of one-period bonds where the proceeds of each one-period bond are “rolled over” into buying the next. The law of one price then requires that the price of the multi-period bond be the product of the one period prices.5 There is a generalization to this in a world of uncertainty represented by multi-period generalizations of the one-period tree we just discussed. In such a tree, any state price for a state s several periods into the future is just the product of the one period state prices for the states on the path in the tree that lead to the state s. If we can model the one period state prices as we just did above, we can therefore determine all of the state prices on the tree.

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Most of the literature deals in state prices rather than risk adjustments, but these are a final small step away. The value today of a claim to $1 in the up state equals the probability of getting the $1 multiplied by the total discount factor, so: 50% * $1 * 1/(1+50.38%) = $0.3325. The corresponding state price for the down state is $0.6175. We can double check these results by recalling that the state prices should sum to the time discount factor, which they do:

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Real Asset Valuation: A Back-to-basics Approach

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Transcript of Real Asset Valuation: A Back-to-basics Approach