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Regulated prices and real options Graeme Guthrie n School of Economics and Finance, Victoria University of Wellington, PO Box 600, Wellington, New Zealand

a r t i c l e in f o

abstract

Available online 2 July 2012

This paper shows how the cash ﬂows received by an unregulated ﬁrm operating in a workably competitive market can be replicated for a regulated ﬁrm. The only change to standard regulatory practice is that each time the regulated ﬁrm invests, the amount added to its rate base is the product of its capital expenditure and a multiplier, greater than one, that captures the reduction in value of the unregulated ﬁrm’s growth options that occurs whenever the ﬁrm invests. The regulated ﬁrm is allowed to earn a rate of return equal to its weighted-average cost of capital, applied to this rate base. Four possible approaches to estimating the size of the multiplier are presented, each based on an established real-options investment model. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Regulation Workable competition Real options Rate base

1. Introduction The infrastructure that plays an increasingly important role in determining overall economic performance – airﬁelds, electricity, gas, ports, rail, roading, telecommunications, and water networks – has natural monopoly characteristics that mean infrastructure providers are typically subject to some form of regulation. The emergence of new technologies will require signiﬁcant investment in many industries, such as the next generation networks needed in the telecommunications sector. In many developed countries, much existing infrastructure is nearing the end of its economic life, so that substantial investment will be required. Infrastructure investment is just as important in developing countries. It is therefore essential that regulators consider the impact of the prices they set on ﬁrms’ investment incentives.1 Modern investment theory recognizes that ﬁrms often have ﬂexibility regarding when and how they invest, and it has sought to incorporate that ﬂexibility into decision making using real options analysis. See, for example, textbook treatments by Dixit and Pindyck (1994) and Guthrie (2009). An important source of real option value derives from ﬁrms’ ﬂexibility regarding the timing of investment, since this allows them to wait and see how future uncertain economic conditions evolve, for example, before committing to large irreversible investments. A key insight is that the total economic cost of an individual project is not just the capital expenditure involved, but also includes the reduction in value of the ﬁrm’s growth options due to investment. This manifests itself in decision making in two ways: (i) investment is optimal only when the value of the completed project exceeds the required capital expenditure by at least the amount of the reduction in growth-option value and (ii) for a project with a unique internal rate of return, investment is optimal only when that internal rate of return exceeds the project’s weighted-average cost of capital (WACC) by some strictly positive premium. Real options analysis is especially important in the telecommunications sector, due to the high capital intensity, the rapid rate of technological change and rapidly growing demand making ongoing investment essential, and the high levels

n

Tel.: þ 64 4 4635763; fax: þ64 4 4635014. E-mail address: [email protected] 1 See Cambini and Jiang (2009) and Guthrie (2006) for recent surveys of the regulation–investment literature.

0308-5961/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.telpol.2012.04.013

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of uncertainty surrounding future demand and capital costs. Applications span the whole range of goods and services provided by ﬁrms participating in this sector. Recent examples of the application of real options analysis include the problem of when to adopt new technology in the wireless industry (Harmantzis & Tanguturi, 2007; Sadowski, Verheijen, & Nucciarelli, 2008), deciding when to deploy broadband infrastructure (Angelou & Economides, 2008, 2009; Krychowski, 2008), and the deployment of point-of-sale debit services for electronic banking (Schwartz & Zozaya-Gorostiza, 2003). Many applications of real-option investment theory to regulatory economics focus on the access-pricing problem and use straightforward modiﬁcations of standard investment-timing models. For example, Hausman (1997) and Pindyck (2007) calculate regulated access prices that allow the investing ﬁrm to recover its cost of capital, Hori and Mizuno (2006) analyze the timing of investment by two competing ﬁrms when the follower is entitled to regulated access to the leader’s asset, and Guthrie, Small, and Wright (2006) compare the welfare performance of access-pricing rules that depend on the historical cost or the replacement cost of the asset being accessed. Another strand of the literature focuses on the effect of price caps on investment. For example, Dixit (1991) considers a perfectly competitive industry, Dobbs (2004) considers a monopolistic one, and Roques and Savva (2009) consider an oligopoly that includes the models of Dixit (1991) and Dobbs (2004) as limiting cases. Evans and Guthrie (forthcoming) investigate welfare-maximizing price-cap regulation in the presence of scale economies. This literature has established that overall welfare can be enhanced if regulated prices appropriately incorporate the value of ﬁrms’ investment options. Regulators are slowly coming around to the view that, at least for some industries (or, more precisely, some services in some industries), it may be appropriate to recognize real options when setting regulated prices. For example, the European Commission (2010) has recommended that when regulating access to next generation networks, regulators should add a premium to ﬁrms’ WACCs reﬂecting various sources of investment risk, including uncertainty regarding future demand and technological change. Previously the telecommunications regulator in the United Kingdom, Ofcom, ‘‘propose[d] to analyze the case for the application of real options to individual wholesale products on a case-by-case basis’’, but warned that ‘‘it would need to determine that the appropriate adjustmentywas relevant in a competitive or contestable environment’’ (Ofcom, 2005, p. 101). Ofcom subsequently rejected the use of real options analysis, although it did claim to be ‘‘open to arguments about the best way of reﬂecting uncertainty, and its effect on investment incentives, in access pricing methodologies’’ (Alleman, 2008, p. 138). Ofcom seems to recognize that ﬁrms use sophisticated techniques when evaluating investments, but sees no place for such techniques in price setting, arguing that ‘‘it is not the regulator’s job to conduct investment appraisals on behalf of ﬁrms. If the regulator gets its calculation right, then ﬁrms – using whatever methodologies they choose to evaluate their options, possibly including a real options approach – should be in a position to make efﬁcient investment decisions’’ (Alleman, 2008, p. 139). The regulatory economics literature has demonstrated that regulated prices need to incorporate some adjustment for real options if investment is to maximize overall welfare. It seems that, for the foreseeable future, regulators will not be comfortable incorporating real options in regulated prices unless it is possible to do so using simple, straightforward, techniques. This paper proposes one technique that regulators could use. Setting regulated prices can be thought of as a two-step process, especially as it relates to the recovery of the costs associated with investment in physical capital. The ﬁrst step determines what capital costs can be recovered by the regulated ﬁrm. The second step determines how to recover those costs, and requires speciﬁcation of the depreciation proﬁle and the allowed rate of return, as well as other characteristics of the regulatory regime such as the rules determining the timing of future regulatory hearings and how prices will be adjusted at such hearings.2 This paper concentrates on the ﬁrst step and describes a methodology for determining the level of costs to be recovered that is compatible with the wide range of approaches to the second step that we observe in practice. For example, this paper’s approach can be used in jurisdictions where regulators use accounting-based depreciation proﬁles (such as straight-line and annuity-based depreciation), and also where regulators use various forms of depreciation that are intended to mimic outcomes in workably competitive markets (such as tilted annuities). It can also be used when the timing of future regulatory hearings is ﬂexible (and typically chosen by the regulated ﬁrm requesting price increases) or ﬁxed as part of the regulatory regime. This paper proposes making an adjustment to the regulated ﬁrm’s rate base, in such a way that each time the regulated ﬁrm invests its rate base is increased by an amount equal to the market value the new assets would have at the time of investment if they were owned by an unregulated ﬁrm operating in a workably competitive market; the ﬁrm’s allowed rate of return would equal its WACC.3 This can be achieved by adding the sum of capital expenditure and the reduction in value of the ﬁrm’s growth options to the ﬁrm’s rate base each time it invests. In many established real option models, the loss of growth-option value is a constant multiple of capital expenditure. Therefore, this paper proposes applying a multiplier to capital expenditure when it is added to the regulated ﬁrm’s rate base. The multiplier can be estimated from

2 Section 2 of Guthrie (2006) describes the variety of approaches regulators have adopted to implement the second step. Evans and Guthrie (2005) discuss the relationship between the rule for adjusting the rate base from one year to the next – that is, the depreciation proﬁle – and the allowed rate of return that is needed in order to achieve cost recovery. 3 Many practitioners have argued in favor of incorporating real options in regulated prices by adding a premium to a ﬁrm’s WACC when setting the allowed rate of return. However, this approach confounds the two steps of price setting – determining what costs to recover and how to recover them – in ways that complicate the rules for price setting and have so far discouraged regulators from adopting the approach.

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G. Guthrie / Telecommunications Policy 36 (2012) 650–663

existing models and using industry parameters, with the proviso that one parameter – which measures the degree of competition in a hypothetical workably competitive market – must be speciﬁed by the regulator. Application of the multiplier to capital expenditure is the only change to current practice in many regulated industries, which is to set the regulated ﬁrm’s rate base equal to the (depreciated) historical cost of its assets-in-place, or under some schemes of incentive regulation to the estimated replacement cost of the assets (Evans & Guthrie, 2005). In particular, the ﬁrm’s allowed rate of return equals its WACC, and this is applied to its augmented rate base, and the usual depreciation arrangements are followed, except that they apply to the augmented rate base. The approach proposed in this paper can be implemented along exactly the same lines as the conventional one, except that capital expenditure is scaled up by the appropriate multiplier when it is added to the rate base. Section 2 discusses the theoretical and empirical evidence that competition between ﬁrms does not generally eliminate the value of ﬁrms’ investment timing options. Section 3 explains how the market value of an unregulated ﬁrm naturally decomposes into the sum of the values of its cash holdings, its assets-in-place, and its growth options. This section demonstrates how this decomposition evolves over time as the ﬁrm implements its investment program. Section 4 introduces the real-option-multiplier approach to setting regulated prices and demonstrates its application. The overall approach can be given a solid theoretical grounding using real-option models of investment timing in workably competitive markets to calculate the real-option multiplier, and four possible models are described in Section 5. Some concluding remarks are offered in Section 6. 2. Competition and option value It is often argued by those who oppose incorporating real options in regulated prices that theory proves competition eliminates option value; that is, investment-timing options are valuable only for ﬁrms that have some market power. Indeed, this result holds in an economy in which there is an inﬁnite number of identical ﬁrms with a constant-returns-toscale technology, investment is instantaneous, there is no bound on the rate of investment, and all agents have identical information sets.4 However, subsequent theoretical research has relaxed many of these assumptions and shown that the option value of investment-timing ﬂexibility is not eliminated by competition in many circumstances. For example, Williams (1993) shows that timing options can be valuable if there exists a cap on the rate of development; Childs, Ott, and Riddiough (2002) show that timing options have positive value when the ﬁrm that invests ﬁrst reveals information about the true state of demand; Novy-Marx (2007) shows that they can be valuable when there is cross-sectional variation in ﬁrm size; the results in Guthrie (2010) show that they can be valuable when price-taking ﬁrms have different cost structures. The question of how competition affects timing-option values can therefore be answered only by looking at the empirical evidence. Survey evidence reveals that ﬁrms, many of them operating in highly competitive markets, set hurdle rates substantially in excess of their WACC (Jagannathan, Meier, & Tarhan, 2011; Poterba & Summers, 1995). One explanation of this behavior is that projects must earn a return higher than the ﬁrm’s WACC if the investment payoff is to compensate the ﬁrm’s owners for the reduction in value of growth options resulting from investment. Of course, it is possible that this behavior is caused by factors other than consideration of real options, and that these factors will not be present in the subset of ﬁrms comparable to those operating in regulated industries. For example, mid-level managers may exaggerate the forecast rates of return on proposed projects above the true level when seeking funding from top-level management, in an attempt to increase the likelihood that projects under their control are funded.5 If senior management responds by setting hurdle rates above the WACC in order to ﬁlter out poor projects being advocated by empire-building mid-level managers, then the observed high hurdle rates may be unrelated to the ﬁrm’s real options. The greater transparency surrounding expected rates of return on projects undertaken by regulated ﬁrms means that such an adjustment is unnecessary, so that the hurdle rate (and hence the allowed rate of return) can equal the WACC. The important issue for regulators is therefore whether the higher hurdle rates observed in practice are related to factors that should be reﬂected in regulated prices (such as the effect of investment on the value of ﬁrms’ growth options) or not (such as internal control factors). The key test is whether the actual returns earned by ﬁrms in workably competitive markets exceed their WACC on average: if they do, then regulated prices should be set in such a way that regulated ﬁrms can also earn average returns in excess of their WACC. Empirical research indicates that average rates of return exceed the WACC. Chirinko and Schaller (2009) use data on investment expenditure and actual project outcomes for 16,140 U.S. publicly traded ﬁrms during the period 1980–2001. They ﬁnd expected returns well in excess of estimated costs of capital. Because Chirinko and Schaller use actual project outcomes, the premia they detect cannot be explained as responses to project-manager optimism: project managers can inﬂate a project’s forecast cash ﬂows in their project evaluation, but they cannot inﬂate its actual cash ﬂows. If the only explanation for hurdle rates being higher than the estimated cost of capital was that this is a response to project-manager optimism, then ﬁrms would invest to the point that the marginal unit of installed capital earns – on average – the ﬁrm’s cost of capital and nothing more. However, Chirinko and Schaller show that the marginal unit of installed capital earns 4 5

See, for example, Leahy (1993) and Dixit and Pindyck (1994, chap. 8). See, for example, Berkovitch and Israel (2004).

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considerably more than the cost of capital on average. The premia they ﬁnd are economically signiﬁcant. For example, they estimate that ﬁrms with limited resale markets set hurdle rates 5.1% higher than their cost of capital; ﬁrms with low depreciation rates set them 2.2% above their cost of capital; those with high demand uncertainty set them 7.3% above their cost of capital (Chirinko & Schaller, 2009, p. 391). It follows that ﬁrms operating in a wide range of industries, and often in highly competitive markets, expect to earn returns in excess of their WACC whenever they invest. This, in turn, implies that if regulators want to mimic workable competition then the ﬁrms they regulate should also be allowed to earn returns in excess of their WACC whenever they invest. 3. Investment and the three components of ﬁrm value The regulatory scheme proposed in this paper attempts to replicate aspects of outcomes in workably competitive markets. The value of ﬁrms operating in such markets can be decomposed into the sum of the values of their cash holdings, assets-in-place, and growth options. As this decomposition is the motivation for the regulatory scheme, this section uses a simple numerical example to illustrate how investment affects the decomposition of a ﬁrm’s value in a workably competitive market.6 Implementation of the scheme will be described in Section 4. Consider an unregulated ﬁrm operating in a workably competitive market and suppose that competition has not completely eliminated the value of the ﬁrm’s investment timing options; in particular, the ﬁrm is able to adopt a hurdle rate in excess of its cost of capital. The ﬁrm initially holds 200 dollars in cash and owns options to invest in two projects. Each project requires capital expenditure of 100 dollars and generates a perpetual cash ﬂow of x dollars per annum beginning one year after investment occurs. For simplicity, all cash ﬂows are assumed to have zero systematic risk and the risk-free interest rate is 4% per annum. That is, the ﬁrm’s WACC is 4%. Suppose also that the ﬁrm sets a hurdle rate of 5% per annum. The earliest that the ﬁrst project will have x¼5 is two years from now; the earliest that the second project will reach this threshold is four years from now. To make the analysis as simple as possible, the ﬁrm does not pay any dividends and there are no taxes. The ﬁrm’s cash ﬂows are calculated in the ﬁrst four columns of Table 1. This information leads directly to the market value summarized in the next four columns. The details of the calculations are summarized as follows: 1. Each year the ﬁrm’s cash ﬂow comprises interest earned on its cash balance, plus cash ﬂow from operations, minus any capital expenditure. The ﬁrm’s total cash ﬂow is positive in all years except for years 2 and 4, when it invests in the two projects. 2. Each year, the market value of the ﬁrm is calculated as the present value of all current and future cash ﬂows. The value of assets-in-place equals 125 dollars when only one project has been built and 250 dollars when both projects have been built. Since the ﬁrm pays no dividends, the cash stock increases by the amount of total cash ﬂow each year. The value of the ﬁrm’s growth options equals the value of the ﬁrm minus the sum of the cash stock and the value of assetsin-place. The ﬁrm’s market value is determined by its cash ﬂows, which are determined by its investment decisions and outcomes in the product market. It is sometimes erroneously suggested that the returns from exercising ﬁrms’ real options are already included in the historical rates of return on the market that are used to estimate the WACC.7 However, in efﬁcient ﬁnancial markets shareholders will not earn excess returns on average, no matter what investment policies ﬁrms follow: prices will be bid up (or down) in order to eliminate the prospect of such returns occurring. This argument applies to ﬁrms’ growth options, just as it applies to other assets. The current market value of a ﬁrm’s growth options is determined by the market in such a way that excess returns are not earned on average. This means that the option premium is built into the market value of the owning ﬁrm, not into the rate of return that shareholders receive from trading in the ﬁrm’s shares. In particular, even if a ﬁrm chooses a hurdle rate in excess of its cost of capital, its shareholders will not earn an excess rate of return on their investment in the ﬁrm’s shares on average. Consider the example above. Investors in this ﬁrm earn a rate of return over the ﬁrst year equal to 254:26244:48 ¼ 0:04 244:48 and over the second year they earn a rate of return equal to 264:43254:26 ¼ 0:04: 254:26 6 In order to isolate the key insights, this example assumes that all future outcomes are known with certainty. However, the decomposition of market value into three components – and the effect of investment on the values of the components – also applies in the more realistic setting where future outcomes are uncertain. Indeed, the models used in Section 5 to formalize the ideas here are built upon stochastic processes that describe the random nature of future outcomes. 7 This common error is discussed in Pindyck (2007, footnote 23).

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G. Guthrie / Telecommunications Policy 36 (2012) 650–663

Table 1 Effect of investment on growth options and assets-in-place. Year

0 1 2 3 4 5 6 7 8 9 10 ^

Net cash ﬂow

Market value

Interest

Operations

Capex

Total

Cash

Assets-in-place

Growth options

Firm

8.00 8.32 4.65 5.04 1.44 1.90 2.37 2.87 3.38 3.92 ^

0.00 0.00 5.00 5.00 10.00 10.00 10.00 10.00 10.00 10.00 ^

0.00 100.00 0.00 100.00 0.00 0.00 0.00 0.00 0.00 0.00 ^

8.00 91.68 9.65 89.96 11.44 11.90 12.37 12.87 13.38 13.92 ^

200.00 208.00 116.32 125.97 36.01 47.45 59.35 71.72 84.59 97.98 111.90 ^

0.00 0.00 125.00 125.00 250.00 250.00 250.00 250.00 250.00 250.00 250.00 ^

44.48 46.26 23.11 24.04 0.00 0.00 0.00 0.00 0.00 0.00 0.00 ^

244.48 254.26 264.43 275.01 286.01 297.45 309.35 321.72 334.59 347.98 361.90 ^

Growth options Assets-in-place Cash

350 300 250 200 150 100 50 0

1

2

3

4

5

6

7

8

9

10

Fig. 1. Behavior of the ﬁrm’s market value.

The market values reported in Table 1 for year 3 onwards also imply that investors earn a 4% rate of return in all future years. That is, even though the ﬁrm chooses a hurdle rate of 5%, the market rate of return from investing in the ﬁrm’s shares is only 4%. As this example demonstrates, investor rates of return for individual ﬁrms do not contain an option premium when those rates of return are calculated using the ﬁrm’s market value. The same is true for portfolios of ﬁrms. In particular, the historical rates of return used to estimate the WACC do not include a real option premium. The market value of an unregulated ﬁrm equals the sum of its cash holdings, the market value of its assets-in-place, and the market value of its growth options. The individual components, and the sum, change over time as market conditions change. In practice, and in contrast to the simple numerical example used here, these changes will often be unpredictable. While the ﬁrm is waiting to invest, its growth options may ﬂuctuate in value, falling in value when the prospects for investment worsen and increasing in value when they improve. The value of the growth options today will be set by the market at such a level that the expected capital gain does not allow excess returns on average. Each time a ﬁrm invests, the sum of the changes in the market values of assets-in-place and growth options equals the amount of capital expenditure. Thus, if investment reduces the value of the ﬁrm’s growth options, then its assets-in-place must increase in value by the sum of the amount of capital expenditure and the fall in value of the growth options. Growth options do not generate cash ﬂow until they are exercised. Consequently, if they have a positive market value today then it must be the case that the market expects them to generate a positive net cash ﬂow in the future. This behavior can be seen in the example above. Fig. 1 shows how the value of the ﬁrm and its constituent values of cash, assets-in-place, and growth options evolve over time. The black bars show the value of the ﬁrm’s growth options, the white bars the value of its assets-in-place, and the gray bars its cash holdings. Initially the growth option increases (very slightly) in value, even though it is generating no cash ﬂow for the ﬁrm. The ﬁrst time that the ﬁrm invests, the cash stock falls by the 100 dollars of capital expenditure, the growth option falls in value by 23.15 dollars, and the assets-in-place increase in value by 125 dollars. The net gain, 1.85 dollars, equals the 4% return required on the growth option. The second time that the ﬁrm invests, the cash stock also falls, the value of the growth option falls, and the value of assets-in-place rises. Note that each time the ﬁrm invests the value of the ﬁrm’s assets-in-place increases by more than the amount of capital expenditure. The ﬁrm spends 100 dollars of capital expenditure, but subsequently earns a net cash ﬂow of 5 dollars

G. Guthrie / Telecommunications Policy 36 (2012) 650–663

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from the project each year. This can be thought of either as earning a rate of return of 5% on its capital expenditure (that is, its WACC of 4% and a real option premium of 1%) or earning the ﬁrm’s WACC of 4% on the total cost of 100 þ 25 ¼ 125 dollars, where 100 dollars is the amount of capital expenditure and 25 dollars is the reduction in value of the ﬁrm’s growth options that occurs at the time of investment. The latter interpretation is the key to the regulatory approach proposed in this paper. 4. Compensation via rate-base adjustments Motivated by the discussion in the previous section, the guiding principle behind the regulatory scheme proposed in this paper is that each time a regulated ﬁrm invests, its rate base should increase by the hypothetical market value at the time of investment of the ﬁrm’s additional assets-in-place if the ﬁrm was unregulated and operating in a workably competitive market. The decomposition of the unregulated ﬁrm’s market value into cash, assets-in-place, and growth options can be used to interpret the amount that should be added to the rate base. Immediately before investment the ﬁrm’s market value can be decomposed into V ¼ cash þ ½assets-in-place þ½growth options and immediately afterwards it can be decomposed into V n ¼ cashcapex þ ½assets-in-placen þ½growth optionsn , where the n s indicate post-investment market values. If the investment is anticipated then V ¼ V n , so that ½assets-in-placen ½assets-in-place ¼ capex þ ½growth options½growth optionsn : That is, the unregulated ﬁrm’s assets-in-place increase in value by an amount equal to the sum of the capital expenditure and the reduction in value of the ﬁrm’s growth options. This amount can be interpreted as being the total cost of investment: that is, the ﬁrm’s owners sacriﬁce a combination of cash and reduced growth-option value when they allow the ﬁrm to invest. Under the regulatory scheme proposed in this paper, it is this total cost of investment that is added to the rate base each time the regulated ﬁrm invests. As Section 5 demonstrates, in several established real-options models of workably competitive markets the reduction in the value of a ﬁrm’s growth options when it invests is proportional to its level of capital expenditure. If the constant of proportionality equals M1 ¼

½growth options½growth optionsn capex

then ½assets-in-placen ½assets-in-place ¼ M capex: That is, each time the regulated ﬁrm invests, its rate base increases by the product of its capital expenditure and the realoption multiplier M. The multiplier approach works by augmenting each item of capital expenditure by a ﬁxed percentage at the time it is added to the rate base. This is the only change to widespread practice. In particular, the ﬁrm’s allowed rate of return equals its WACC, and this is applied to its augmented rate base, and the usual depreciation arrangements are followed, except that they apply to the augmented rate base. The capital expenditure is augmented by an amount equal to the reduction in market value of the ﬁrm’s growth options that occurred as a consequence of the investment. This ensures that the ﬁrm’s owners receive compensation only for the ﬁrm’s capital expenditure and the loss in value of growth options associated with that capital expenditure.8 The form that this compensation takes depends on how the remaining aspects of the regulatory scheme are set. If the regulatory regime allows for prices to be adjusted in the future to ensure full cost recovery, then the regulated ﬁrm will eventually be able to recover the sum of its actual investment of funds in the project and the estimated reduction in growth-option value that occurred when the ﬁrm invested. In contrast, if the regulatory scheme does not fully protect the regulated ﬁrm from future demand shocks, then the market value of the ﬁrm’s allowed cash ﬂows, measured at the time of investment, will equal the sum of its actual investment of funds in the project and the estimated reduction in growth-option value, but full ex post cost recovery will not be guaranteed.9 The following continuation of the earlier example demonstrates how the multiplier approach would work in practice. It features the same ﬁrm as in the previous example. However, whereas the ﬁrm was unregulated in that example, in this one it is regulated according to the multiplier approach. As in the previous example, the ﬁrm pays no dividends and there are no taxes.10 The ﬁrm’s cash ﬂows are calculated in columns 2–5 of Table 2. This information leads directly to the market value summarized in the ﬁnal column. The details of the calculations are summarized as follows: 8 In particular, post-investment ﬂuctuations in the value of the ﬁrm’s growth options due to changes in market conditions have no impact on the ﬁrm’s rate base, and therefore no effect on its allowed revenue. 9 How regulatory schemes allocate risk between ﬁrms and customers is described in Section 2 of Guthrie (2006). 10 Consistent with the perpetual nature of the asset, no allowance is made for depreciation. Finite asset lives could be introduced, but doing so complicates the arithmetic without providing any additional insights.

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G. Guthrie / Telecommunications Policy 36 (2012) 650–663

Table 2 Demonstration of the multiplier approach to regulation. Year

0 1 2 3 4 5 6 7 8 9 10 ^

Rate base

0.00 0.00 125.00 125.00 250.00 250.00 250.00 250.00 250.00 250.00 250.00 ^

Net cash ﬂow

Market value

Interest

Operations

Capex

Total

Cash

Firm

8.00 8.32 4.65 5.04 1.44 1.90 2.37 2.87 3.38 3.92 ^

0.00 0.00 5.00 5.00 10.00 10.00 10.00 10.00 10.00 10.00 ^

0.00 100.00 0.00 100.00 0.00 0.00 0.00 0.00 0.00 0.00 ^

8.00 91.68 9.65 89.96 11.44 11.90 12.37 12.87 13.38 13.92 ^

200.00 208.00 116.32 125.97 36.01 47.45 59.35 71.72 84.59 97.98 111.90 ^

244.48 254.26 264.43 275.01 286.01 297.45 309.35 321.72 334.59 347.98 361.90 ^

1. The rate base is initially zero and each time the ﬁrm invests the rate base is increased by the product of the ﬁrm’s capital expenditure and the real option multiplier M ¼1.25. 2. Each year the ﬁrm’s cash ﬂow comprises interest earned on its cash balance, plus earnings from regulated activities calculated so that the ﬁrm earns a rate of return on its (augmented) rate base equal to its WACC of 4%, minus any capital expenditure. 3. Each year, the market value of the ﬁrm is calculated as the present value of all current and future cash ﬂows. In contrast to the unregulated ﬁrm, here the ﬁrm’s cash ﬂows are determined by its investment decisions and the rate of return allowed by the regulator. The resulting cash ﬂows then determine the market values of the ﬁrm as a whole and its various components. Note that the ﬁrm’s cash ﬂows are identical to those in the earlier example and that its rate base tracks the value of the corresponding unregulated ﬁrm’s assets-in-place. 5. Implementation If Ofcom is any guide, regulators are attempting to engineer outcomes similar to those that would eventuate in workably competitive markets. A regulator attempting to achieve this goal using the scheme advocated in this paper must estimate the effect that investment has on the value of the growth options owned by hypothetical unregulated ﬁrms operating in workably competitive markets. If the ﬁrms being regulated were operating in workably competitive markets then there would be no need for regulation, so any method for calculating option values needs to recognize that competition would not work in these particular markets. This means that there is no point in building models of particular ﬁrms that capture all aspects of their circumstances and then asking what competition would look like. For instance, if there are sufﬁciently strong economies of scale in investment then a competitive process would tend to result in a single ﬁrm dominating the market. An option value for such a ﬁrm would be contaminated by the market power that the regulator is trying to offset. It follows that any attempt to estimate workably-competitive option values cannot be tied too closely to the particular circumstances of the ﬁrm being regulated. Some aspects of that ﬁrm’s situation will need to be varied for the hypothetical workable competition to make sense. This paper has a practical focus, aiming to give regulators a mechanism for incorporating real options in regulated prices. This is most likely to happen if the mechanism is simple enough that it can be understood and subjected to scrutiny by a wide range of stakeholders. Such a mechanism will inevitably be based on a highly stylized model, which is why this section draws on four separate approaches that the regulator might consider, each based on a well known real option model that might reasonably be regarded as describing the situation facing an unregulated ﬁrm operating in a workably competitive market. The models’ various parameters are described in Table 3. Most of them can be estimated to match the circumstances of the ﬁrm being regulated, but one parameter in each model – the one determining the nature of competition – cannot be. In each case, if the relevant parameter was set to match the regulated ﬁrm’s circumstances, there would be no competition and the implied option values that the regulator would calculate would include the beneﬁts of market power. Therefore, the regulator needs to decide what workable competition means for the parameter in question. For example: How many ﬁrms operate in a workably competitive market? On average, how long can a ﬁrm operating in a workably competitive market delay investment before it is preempted? The role of the real option model is to translate the regulator’s assumption about workable competition into a real option multiplier that can be used to ensure the regulated ﬁrm receives adequate compensation for the value of growth options lost when investing. The models considered in this section make different assumptions and attempt to capture different aspects of the situations facing real-world ﬁrms. In all cases the models incorporate managers’ ﬂexibility regarding investment decisionmaking and, in particular, how they respond to uncertainty regarding future outcomes. In the ﬁrst case, managers have the

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Table 3 Notation. Common to all models r

m s w

Risk-free interest rate Mean annual growth in risk factor Standard deviation of annual growth in risk factor Risk-adjusted discount rate for risk factor

Section 5.1 1=f

Average number of years until preemption in hypothetical workably competitive market

Section 5.2 C/I 1=f

Opex–capex ratio Average number of years until preemption in hypothetical workably competitive market

Section 5.3 n

eD

Number of ﬁrms in hypothetical workably competitive market Price elasticity of demand

Section 5.4

eS

Price elasticity of supply in hypothetical workably competitive market

ﬂexibility to delay investment until the investment environment improves. In the second case, they have the additional ﬂexibility of suspending production in the future if economic conditions deteriorate. In the third case, multiple rounds of investment are possible, all occurring at the time of managers’ choosing. Finally, in the fourth case, managers choose the timing of investment recognizing that investment in new assets affects the market value of existing assets. 5.1. Preemption when investment destroys a delay option McDonald and Siegel (1986) developed one of the earliest and simplest models of investment timing. In their model a ﬁrm holds a perpetual option to develop a project. The market value of the completed project (including the value of any embedded options, such as the option to contract or expand in the future) and the required amount of capital expenditure evolve randomly over time, and investment is irreversible. Their model is easily modiﬁed so that the development option is not perpetual. Speciﬁcally, the option can be assumed to be extinguished at some unknown future date. This can be thought of as capturing the threat of preemption that might arise in a workably competitive market. Under perfect competition, the ﬁrm might expect to be preempted if it delayed investing – even for an instant – when the asset created by investment is worth more than the capital expenditure required. In this case, the expected time until preemption is zero. At the other extreme, if the ﬁrm is the only one able to invest it could delay indeﬁnitely without any threat of preemption. In this case, the expected time until preemption is inﬁnite. Workable competition will lie somewhere between these two extremes. As shown in Appendix A.1, in the special case where there is no capital-expenditure risk, this leads to the option multiplier11 M ¼ 1þ

1

y1

,

where 1 wrm y¼ þ þ 2 s2

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2ðr þ fÞ 1 wrm 2 þ þ 2 s2 s2

and r is the risk-free interest rate, w is the project’s WACC, m is the average annual growth rate in the value of the completed project, s is the volatility of the annual growth rate in the value of the completed project, and f is the reciprocal of the average number of years until the ﬁrm’s investment is preempted. All parameters except for f can be calibrated to the ﬁrm (or industry) being regulated. Two of them (r and w) are calculated as part of the regulator’s usual WACC estimation procedure. The others (m and s) will require more work, with the approach adopted depending on the nature of the asset and the data available. For example, if historical usage data are available, then m and s can be estimated from the average and standard deviation of actual growth rates.12 In some cases, it may be possible to infer values of m and s from comparison ﬁrms, using a de-levering process analogous to that used to convert equity betas into asset betas (Guthrie, 2009, Section 14.2.3). 11

The general case with stochastic capital costs is not much more complicated and leads to a similar expression for y. It can be calculated by adding

f to the interest rate where it appears in the investment thresholds given in McDonald and Siegel (1986). 12

In the general model, with stochastic capital costs, historical data on capital prices can be used in much the same way. Averages and standard deviations of actual growth rates in usage and capital prices, together with the correlation coefﬁcient between usage growth and capital-price growth, are sufﬁcient to estimate m and s.

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Table 4 Option multipliers when investment destroys a delay option. Average years until preemption Multiplier

1=f M

1 2.00

100 1.86

10 1.46

5 1.33

2 1.21

1 1.15

0 1.00

The market power enjoyed by the regulated ﬁrm means that it may have considerable ability to wait without fear of losing its growth option. Thus, the calibrated level of f would be close to zero. The regulator therefore needs to choose a level of f that it believes reﬂects what workable competition would look like in the industry under consideration. In practice, it would be more natural to choose the average length of time until preemption and then set f equal to the reciprocal of this number. The regulator might consider factors such as the length of time needed to plan and implement investment projects in the industry being regulated. In those industries with long investment lead times, 1=f might reasonably take a relatively high value. Dixit and Pindyck (1994) discuss the special case of no preemption risk. Their baseline example has r¼ 0.04, w¼ 0.04, m ¼ 0, and s ¼ 0:2, which implies an option multiplier of M¼2 (Dixit & Pindyck, 1994, p. 153). Table 4 shows the effect of the average time until preemption for these parameter values. Notice that the only way that the option multiplier equals one (so that no compensation for lost growth-option value is required) is if the ﬁrm would be preempted immediately if it delayed undertaking the investment: this is perfect competition, not workable competition. The principal advantages of this approach are its simplicity, its transparency, the fact that it is not tied too closely to any particular ﬁrm or industry (although the particular parameter values used would be industry-speciﬁc), and the intuitive way in which it captures the effect of workable competition on investment-timing ﬂexibility. 5.2. Preemption when investment creates a suspension option In the example just considered, the ﬁrm owns a single investment option, which is destroyed when it invests. However, investment can also create options, and the example in this subsection demonstrates that the multiplier approach is ﬂexible enough to handle this situation as well. In the case considered, which is a modiﬁcation of the model introduced by McDonald and Siegel (1985) and analyzed by Dixit and Pindyck (1994, chap. 6), investment simultaneously destroys the option to wait and invest at a future date and creates the option to suspend production without cost. The output price, which evolves according to geometric Brownian motion, has an annual growth rate with mean m and standard deviation s. Operating expenditure is C per unit of output and investment requires lump sum capital expenditure of I. The risk-free interest rate equals r and the risk-adjusted discount rate for price risk is w. As in Section 5.1, the average number of years until the ﬁrm’s investment is preempted equals 1=f. Appendix A.2 shows that the real option multiplier M for this model is b 1 x b2 C=I b 2 , M ¼ 1þ 2 b2 y wm b2 y r b2 y where x is deﬁned implicitly by x y C=I b y 1b1 C=I b1 C=I x b2 þ1 þ 2 ¼ þ wm C=I b1 b2 wm b1 b2 r y1 r y1 and 1 wrm b1 ¼ þ þ 2 s2

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2r 1 wrm 2 þ , þ 2 s2 s2

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 wrm 2r 1 wrm 2 þ , b2 ¼ þ þ 2 2 s2 s2 s2 1 wrm y¼ þ þ 2 s2

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2ðr þ fÞ 1 wrm 2 þ : þ 2 s2 s2

Note that the multiplier is a function only of C=I, r, m, s, w, and f. All parameters except for f can be calibrated to the ﬁrm (or industry) being regulated using the approaches described in Section 5.1. The only additional parameter is the ratio of operating expenditure to capital expenditure, which will be speciﬁed as part of any regulatory price-setting process. Dixit and Pindyck (1994) discuss the special case of no preemption risk. Their baseline example has r¼ 0.04, w¼ 0.04, m ¼ 0, s ¼ 0:2, and C=I ¼ 0:1 which implies an option multiplier of M¼3.80 (Dixit & Pindyck, 1994, Section 6.2.B). Table 5 shows the effect of the average time until preemption for these parameter values, using the same values for f as in Table 4. It considers three different levels of the opex–capex ratio, with larger values of C=I corresponding to more valuable suspension options.

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Table 5 Option multipliers when investment creates a suspension option. Average years until preemption Multiplier (C=I ¼ 0:00) Multiplier (C=I ¼ 0:01) Multiplier (C=I ¼ 0:10)

1=f M M M

100 1.86 2.06 3.37

1 2.00 2.23 3.80

10 1.46 1.56 2.14

5 1.33 1.41 1.80

2 1.21 1.26 1.48

1 1.15 1.18 1.32

0 1.00 1.00 1.00

Table 6 Option multipliers when investment creates expansion options. Number of ﬁrms Multiplier (eD ¼ 0:5) Multiplier (eD ¼ 2:0)

n M M

1 n/a 2.00

2 n/a 1.33

3 3.00 1.20

4 2.00 1.14

5 1.67 1.11

10 1.25 1.05

1 1.00 1.00

5.3. Competitive equilibrium when investment creates expansion options The option created by investment in the example in Section 5.2 is embedded in the assets built as part of the ﬁrm’s investment. An alternative possibility, considered here, is that investment creates the option to carry out further investment. The net reduction in the value of the ﬁrm’s growth options – which is what determines the size of the realoption multiplier – equals the value of the options destroyed by investment minus the value of the options it creates. Grenadier (2002) derives equilibrium investment policies in a competitive industry in which the number of ﬁrms is ﬁnite and exogenously speciﬁed, physical capital is inﬁnitely divisible, and each ﬁrm’s equilibrium investment policy involves an ongoing ﬂow of capital expenditure. In the special case of Grenadier’s model where the demand shock follows geometric Brownian motion and the price elasticity of demand is constant, the option multiplier is M ¼ 1þ

1 , neD 1

where n is the number of ﬁrms and eD (which is required to exceed 1=n) is the price elasticity of demand.13 Although the underlying assumptions and the implied investment behavior are unrealistic, this approach has the advantage that it only requires estimation of the price elasticity of demand.14 Standard techniques are available for this task. Note that the regulator will need to specify the number of ﬁrms in its hypothetical workably competitive market. Table 6 shows the effect of the number of ﬁrms on the size of the real option multiplier for two levels of the price elasticity of demand, corresponding to inelastic and elastic demand. 5.4. Competitive equilibrium when investment cannibalizes assets-in-place In the three cases considered above, investment activity by individual ﬁrms has no effect on the value of their existing assets-in-place. A common issue facing telecommunications ﬁrms is that investment in new technology can cannibalize their existing assets-in-place. Novy-Marx (2007) has developed a model of competitive equilibrium in an industry in which ﬁrms replace, rather than augment, their existing capacity whenever they invest. That is, new investment strands all of a ﬁrm’s older assets. In his model an inﬁnite number of atomistic ﬁrms compete in a product market. At any point in time, different ﬁrms potentially have different productive capacities, and this heterogeneity leads to cross-sectional variation in the opportunity cost of investment. There are decreasing returns to scale in investment, which prevents any single ﬁrm from growing until it takes over the entire market. This assumption is not likely to match the realities of regulated ﬁrms but – as discussed earlier – such differences are the price we must pay for measuring option values in workably competitive markets. In Novy-Marx’s model, the capital expenditure required to build an asset with q units of capacity equals qg , for some positive constant g. The price elasticity of supply is shown to equal eS ¼ 1=ðg1Þ. Speciﬁcally, beyond the level of the output price that triggers investment, each one percent increase in the output price is associated with an eS percent increase in the capital stock. As shown in Appendix A.3, the option multiplier is M ¼ 1þ

1

eS

:

13 Grenadier (2002, Eq. (30)) gives the value of one unit of capacity once it is in place. Evaluating this expression when the demand driver is at the investment threshold (that is, when X ¼ X n ðQ Þ), shows that each dollar of capital expenditure increases the value of assets-in-place by M dollars. 14 Alternative speciﬁcations (for example, linear demand) are possible (Grenadier, 2002, Section 6). In many of these speciﬁcations the option multiplier will depend on more parameters than just n and eD . For example, demand volatility will be relevant. However, the additional parameters can be estimated in much the same way as the parameters in the ﬁrst approach. For example, r and w are already available from the WACC calculations, while m and s can be estimated from historical usage data.

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Table 7 Option multipliers when investment cannibalizes assets-in-place. Price elasticity of supply Multiplier

eS M

0.25 5.00

0.50 3.00

1.00 2.00

2.00 1.50

5.00 1.20

1 1.00

Regulated ﬁrms will most likely have lower levels of eS than would occur in a workably competitive market. It is therefore necessary to vary eS from its estimated value in order to capture the effects of workable competition. The regulator needs to specify a level of eS that it believes is appropriate: no other parameters are required to be speciﬁed. Table 7 shows the effect of the price elasticity of supply on the real option multiplier. 6. Concluding remarks This paper has demonstrated how the investment payoffs of an unregulated ﬁrm operating in a workably competitive market can be replicated for a regulated ﬁrm. The only change to standard regulatory practice is that each time the regulated ﬁrm invests, the amount added to its rate base is the product of its capital expenditure and a multiplier, greater than one, that captures the reduction in value of the unregulated ﬁrm’s growth options that occurs whenever the ﬁrm invests. The regulated ﬁrm is allowed to earn a rate of return equal to its WACC, applied to this augmented rate base. Four possible approaches to estimating the size of the real option multiplier were presented, each based on an established model. They each attempt to capture the outcomes that would be observed in a workably competitive market. Each approach requires the regulator to specify a single parameter to operationalize the notion of workable competition, and in each case the remaining parameters can all be calibrated to the industry or ﬁrm being regulated. The multiplier approach described in this paper offers one approach to determining the costs that a regulated ﬁrm should be allowed to recover. The next step in regulatory price-setting is to specify pricing rules that ensure the allowed costs are actually recovered. A crucial part of this calculation is the choice of depreciation proﬁle, as this determines the paths of prices and proﬁts over time (Biglaiser & Riordan, 2000) and the allowed rate of return that the ﬁrm must be allowed to earn (Evans & Guthrie, 2005). The multiplier approach is compatible with any depreciation proﬁle that, in expectation, fully depreciates the asset over its economic lifetime.15 For example, the multiplier approach can be used with the straight-line and annuity-based depreciation rules favored by some regulators, or the more sophisticated proﬁles (such as tilted annuities) that other regulators use to try and capture the effects of changes in technology and changes in competition in hypothetical workably competitive markets. One issue that some stakeholders may raise is whether or not this approach gives the regulated ﬁrm a license to print money. The ﬁrm is allowed to recover M dollars in total for each dollar of capital expenditure. The suggestion will be that, as long as the regulator sets M at a level greater than 1, the ﬁrm has an incentive to over-invest. Such concerns are unfounded, for three reasons. First, when a ﬁrm invests it incurs capital expenditure and destroys valuable real options. The total cost of one dollar of capital expenditure is M dollars. Thus, according to this approach the ﬁrm is being compensated only for the total cost of its investment. Second, the repeated interaction between regulator and ﬁrm means that the scheme need not give the regulated ﬁrm an incentive to over-invest. It is well known that this interaction can lead to equilibria in which the regulator continues to allow agreed cost recovery (due to the ﬁrm’s ability to punish the regulator by ceasing investment if the regulator deviates from the agreed cost-recovery rule) and the ﬁrm continues to undertake agreed investment (due to the regulator’s ability to punish the ﬁrm by restricting cost recovery if it detects over-investment).16 Third, the multiplier applies only to capital expenditure, not operating expenditure. The regulator is able to assess capital expenditure before allowing it into the rate base. This monitoring restricts the ﬁrm’s ability to exploit any undesirable incentives that survive the repeated ﬁrm–regulator interaction. The weak link of the approach proposed here is the need for the regulator to specify one parameter, which cannot be estimated using data from the industry being regulated. Although this parameter has different interpretations in different models, it plays a common role in all cases: measuring the strength of competition in the workably competitive market. The impossibility of estimating this parameter using data from the regulated industry is simply a reﬂection of the fact that the industry in question is not, and cannot be, workably competitive. That is why it is being regulated. However, the models can be used to guide the regulator. Rather than making a subjective assessment of the size of the multiplier, it can make a subjective assessment of some aspect of a workably competitive market. For example, the regulator can specify how long a ﬁrm in a workably competitive market could delay investment before being preempted (the ﬁrst and second approaches), the number of ﬁrms active in a workably competitive market (the third approach), or the price elasticity of supply (the fourth approach). A fruitful topic for future research would be to identify workably competitive industries that 15 Schmalensee (1989) shows that, provided the allowed rate of return equals the ﬁrm’s cost of capital, the net present value from investment equals zero for any depreciation schedule that fully depreciates the ﬁrm’s assets over their service life. The regulator is therefore free to choose any depreciation schedule, as long as the allowed rate of return equals the ﬁrm’s cost of capital. There exists a considerable literature comparing different depreciation schedules, dating back to Baumol (1971). 16 See Salant and Woroch (1992) and Gilbert and Newbery (1994) for formal models of such situations, and Guthrie (2006, Section 6.2) for a general discussion.

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can serve as role models for regulated industries and estimate factors such as the price elasticity of supply, which would be a useful guide to regulators when setting the level of the real-option multiplier.

Acknowledgments The comments of two anonymous referees are gratefully acknowledged. Appendix A A.1. Modifying McDonald and Siegel (1986) Consider the special case where the expenditure required to build the project is constant (and equal to I). The riskneutral process for the market value of the completed project is dV t ¼ ðrw þ mÞV t dt þ sV t dxt : The investment option dies with probability f dt during any interval of time of length dt. The risk-free interest rate equals r. Denote the market value of the project rights by F(V). The (risk-adjusted) expected payoff from waiting dt years and then reconsidering the situation is ð1f dtÞðF þEn ½dFÞ þ f dt 0 ¼ F þ En ½dFfF dt: This expected payoff equals F þ rF dt when ﬁnancial markets are in equilibrium, so that F satisﬁes 0¼

En ½dF 1 ðr þ fÞF ¼ ðrwþ mÞVF 0 ðVÞ þ s2 V 2 F 00 ðVÞðr þ fÞF: dt 2

If the ﬁrm invests as soon as V Z V^ then the project rights are worth y V , FðVÞ ¼ ðV^ IÞ V^ where

y¼

1 wrm þ þ 2 s2

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2ðr þ fÞ 1 wrm 2 þ : þ 2 2 2 s s

The optimal investment threshold, which maximizes F(V), is y 1 I ¼ 1þ I: V^ ¼ y1 y1 That is, the ﬁrm will invest only when the market value of the completed project is greater than or equal to MI, where M ¼ 1þ

1

y1

:

This is the real option multiplier implied by this model. A.2. Modifying McDonald and Siegel (1985) Consider a completed project that generates one unit of output per unit of time, which can be suspended temporarily without cost. Operating expenditure is C per unit of output and the output price evolves according to geometric Brownian motion dPt ¼ ðrwþ mÞP t dt þ sP t dxt : Investment, which requires lump sum capital expenditure of I, can be delayed indeﬁnitely, but the investment option dies with probability f dt during any interval of time of length dt. The risk-free interest rate equals r. This is a modiﬁcation of the project introduced by McDonald and Siegel (1985) and analyzed in Dixit and Pindyck (1994, chap. 6). From Section 6.2.A of Dixit and Pindyck (1994), once investment occurs, the assets-in-place will be worth 8 b1 > 1b2 C b2 C P > > þ if P o C, > < C b1 b2 wm b1 b2 r AðPÞ ¼ b2 > P C 1b1 C b1 C P > > > þ þ if P Z C, : wm r C b1 b2 wm b1 b2 r

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where 1 wrm b1 ¼ þ þ 2 s2

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2r 1 wrm 2 þ þ 40 2 s2 s2

and

b2 ¼

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 wrm 2r 1 wrm 2 þ þ o 0: þ 2 2 2 2 2 s s s

Using the same approach as in Appendix A.1, the value of the ﬁrm prior to investment, F(P), satisﬁes the ordinary differential equation En ½dF 1 ðr þ fÞF ¼ ðrw þ mÞPF 0 ðPÞ þ s2 P2 F 00 ðPÞðr þ fÞF: dt 2 If the ﬁrm invests as soon as P Z P^ then the project rights are worth y P ^ , FðPÞ ¼ ðAðPÞIÞ P^ 0¼

where 1 wrm y¼ þ þ 2 s2

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2ðr þ fÞ 1 wrm 2 þ : þ 2 s2 s2

^ is deﬁned implicitly by The optimal investment threshold, which maximizes FðPÞ, ^ !b2 ^ y C=I b y 1b1 C=I b1 C=I P=I P=I þ1 þ 2 ¼ þ : b1 b2 wm b1 b2 r wm C=I y1 r y1 ^ is a function only of C=I, r, m, s, w, and f. The ﬁrm will invest only when the market value of the completed Note that P=I ^ ¼ MI, where project is greater than or equal to AðPÞ ^ ^ AðPÞ b 1 P=I b2 C=I b ¼ 1þ 2 2 M¼ I b2 y wm b2 y r b2 y is the real option multiplier M implied by this model. It is a function only of C=I, r, m, s, w, and f. A.3. Key results from Novy-Marx (2007) Firms’ log-capacities are initially uniformly distributed between log q0 and log kq0 . Consider a ﬁrm that owns assets with capacity q. From the proof of Proposition 3, the market value at date t of the ﬁrm’s assets-in-place equals 0 !y1 1 ðb1ÞyqPt @ Pt A, 1 n bðy1ÞðrmÞ Pq where, from p. 1468, 1 wrm b¼ þ þ 2 s2

y ¼ 1þ

P nq ¼

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2r 1 wrm 2 þ , þ 2 s2 s2

1 b, eD ðg1Þ

qg1 bðrmÞkg , ðb1Þðk1Þ

and k 4 1 satisﬁes 0 ¼ ðgygyÞkgygy þ 1 ðgygÞkgygy þ y: Note that the condition deﬁning k can be rearranged to give 0 ¼ gðy1Þðk1Þykð1kðg1Þðy1Þ Þ: At the time this capacity was built, the output price was P t ¼ P nq=k ¼

Pnq ðq=kÞg1 bðrmÞkg ¼ g1 ðb1Þðk1Þ k

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