Dr Emmanuel Hache
email: emmanuel.hache@ifp.fr

Abdullah Suzy

Agafonov Anton

Alkhasli Emil

Al-Menhali Ali

Al- Rashdi Abdulaziz

Ikenna Amadi

Azubuike

Babak Banaei

Salman

Kabchi Bennani

Bock Nassim

Burkov Nathan

Calmels Anton

DUMESNIL Joseph Martin

Haritah Adil

Hernandez Menendez

Nicolas Octavio

Jeanmichel Maxime

KONAN Kaabi Ahmed

Setie Aubin

MAMONKIN Kuczewski-Poray May-Aurore

Morel Fatio

PETROVA Ortega Fabio

Pineros Lucas

Safiullin Timur

Salatt Ali

Sharaf Omar

Tabak Marinus

Usova Tatiana

Vafaiefard Erfan

Volkov Mikhail

Zahran Nahl

Commodities Markets : New paradigm or new fashion?

La collection "Les cahiers de l'économie" a pour objectif de présenter des travaux réalisés à l' IFP et à l'École du pétrole et des moteurs, travaux de recherche ou notes de synthèse en économie, finance et gestion. La forme peut être encore provisoire, afin de susciter des échanges de points de vue sur les sujets abordés. Les opinions émises dans les textes publiés dans cette collection doivent être considérées comme propres à leurs auteurs et ne reflètent pas nécessairement le point de vue de l'IFP ou de l'École du pétrole et des moteurs. Pour toute information sur le contenu, prière de contacter directement l' auteur ; pour toute information complémentaire prière de contacter le

The 1990s, symbolized by the rise of the New Information and Communications Technologies (NICT) sector and dot-coms, marked the end of economies' dependence on raw materials, or so it was believed at the time. Finally liberated, the world economy would, via the Internet and the New Economy, flourish by freeing itself from the materials content of its supply, particularly as commodities1 prices had, by the end of the century, reached the lowest levels ever recorded. The Asian crisis thus caused oil prices to fall to less than $10 per barrel, while at the same time some NICT companies saw their values increase a hundredfold in a just few weeks based solely on the prospect of future gains. And yet, what has the topic of discussion been since the start of the 2000s: business-to-business (BtoB) and start-ups? No, we are talking about oil, reels of copper stolen from construction sites, potential increases in bread prices as a result of soaring wheat prices and so on. In this early part of the 21 st century, the traditional production function -labour, capital and land -is therefore being revisited, bringing to the foreground the basic components of economic activity: raw materials.

The dramatic surge in prices between 2001 and 2007 (industrial raw-material prices up 150% according to the IMF Commodity index) coincides with a new market context, an equation comprising three unknowns: scarcity, price levels and the environment. By again giving rawmaterial markets a central place in economic analysis and forecasting, this trend begs discussion on our part. The why -What imbalance? Which players? -gives way to the how -Is this trend so different from cycles that have occurred since the early 20 th century? Did a raw-material bubble follow the Internet bubble? -and the where are we going -How long will it last?

I-A brief history of supply and demand

Wheat prices at about $7 per bushel, oil at more than $130 per barrel and copper at nearly $8,000 per ton in late 2007-early 2008 represent increases of 100%, 200% and 500%, respectively, over their average 2002 levels. Since the start of the 21 st century, the global landscape of raw-material markets has undergone a transformation, with upward pressure on prices an apparent break from the trend seen since the early 1980s, when prices fell by an average of 1% per year. Although the situations in the various raw materials segments still vary, the upward movement of prices that began in late 2001 has gradually made its way into all commodities markets since 2005. Measured by the IMF Commodity index, commodities prices (excluding energy) have risen, on average, by approximately 140% between 2001 and 2008. Industrial raw materials, the prices of which have increased by 250% on average, have largely driven this movement, with a sharp increase -a tripling of prices -in the non-ferrous metals markets (including copper, aluminium, zinc and nickel). Currently, a ton of copper is traded on the London Metal Exchange (LME) at a price five times higher than that of January 2002, with zinc and aluminium prices higher by some four and two times, respectively. Driven mainly by the situation in the rubber market (traded at $0.60/kg in early 2002 and currently at more than $2.00/kg), agro-industrial raw materials (including cotton, rubber and paper pulp) have not been outdone, with average price increases of roughly 80% over the period. Finally, in the food commodity markets -driven in the first few years (2002)(2003)(2004)(2005) by stronger fundamentals in tropical commodity markets (including cocoa, coffee and sugar) and, since 2006, by grain markets -prices have risen by an average of 75%.

Except in agricultural markets (food and agro-industrial), raw-material prices have reached record highs in current dollars, and in segments like non-ferrous metals in constant dollars.

The key player in this price surge is of course China, and any attempt to minimize its impact is futile. While barely mentioning the upturn in world growth since 2001, it must be noted that, although this increase was of course partly due to stronger growth in the US, it was mainly due to Chinese growth, which exceeded 10% and had strong ripple effects throughout the various economic zones. Supported by expansionist monetary policies in the United States and Europe, world growth has attained levels near or above 5% since 2001, levels not seen since the 1970s for so long a period. China's industrialization model is a defining aspect of the 2000s and the landscape of raw-material markets. Spurred by some 17% growth per year in industrial production, China's growth has required huge amounts of non-ferrous metals (such as aluminium, copper and zinc) to support the development of regions far from coastal zones and to reduce social inequalities between provinces. The construction of roads, highways, airports and other infrastructure is eating up mineral raw materials, as is the boom in the housing sector. In addition, construction related to the 2008 Olympic Games and the Shanghai World Expo in 2010 is fuelling a certain market frenzy. 1 9 5 7 1 9 5 9 1 9 6 1 1 9 6 3 1 9 6 5 1 9 6 7 1 9 6 9 1 9 7 1 1 9 7 3 1 9 7 5 1 9 7 7 1 9 7 9 1 9 8 1 1 9 8 3 1 9 8 5 1 9 8 7 1 9 8 9 1 9 9 1 1 9 9 3 1 9 9 5 1 9 9 7 1 9 9 9 2 0 0 1 2 0 0 3 2 0 0 5 2 0 0 7 IMF Index (Industrial raw materials prices)

IMF Index (All commodities)

Source : IMF A true alchemist, in the course of five years China has become the world's leading consumer in the main non-ferrous metals and agro-industrial raw-material markets. Between 2001 and 2006, it accounted for all additional demand in the lead market, 80% in the cotton and zinc markets, and some 50% in the copper and aluminium markets. It has also begun to change the face of the markets -particularly the rare metals and steel segments -as a leading world producer. This Chinese market boom of the early 2000s had a particular impact as it followed on from the 1990s, a decade marked by a lack of investor interest and a strong downsizing trend in key industrial activities like the non-ferrous metals sector. Between 1997 and 2002, mining companies' exploration budgets were cut in half before being increased in 2004, two years after the upturn in worldwide demand. This cyclical shift is not uncommon in industrial activities. It was exacerbated, however, particularly in the oil and mining markets, by a degree of closure to international investment for these activities, with the focus on more nationalistic management of income derived from increases in raw-material prices. Moreover, in some sectors (such as oil-related industries and mining engineering), the rise in raw-material prices led to a de facto increase in project costs, as well as delaying investment decisions and triggering huge order backlogs, mainly due to a lack of qualified personnel! In this context -demand driven by a synchronized upturn in world growth, propelled by a player (China) capable of causing a strong ripple effect in the industrial sector, and limited supply, which offered a weak response in terms of investment -inventories in key markets were greatly reduced on the LME from eight weeks to less than four weeks of consumption, on average, between 2001 and 2006 -and prices skyrocketed! 1 9 5 7 1 9 5 9 1 9 6 1 1 9 6 3 1 9 6 5 1 9 6 7 1 9 6 9 1 9 7 1 1 9 7 3 1 9 7 5 1 9 7 7 1 9 7 9 1 9 8 1 1 9 8 3 1 9 8 5 1 9 8 7 1 9 8 9 1 9 9 1 1 9 9 3 1 9 9 5 1 9 9 7 1 9 9 9 2 0 0 1 2 0 0 3 2 0 0 5 2 0 0 7 IMF Index (Industrial raw materials prices)

IMF Index (All commodities)

Source : IMF

II-A different cycle in raw-material markets?

A historical study of the cycles observed in raw-material markets demonstrates the similarities and differences between the current trend and those that have occurred since the beginning of the 19 th century. Some studies 2 have shown the existence of "stylized facts" in raw-material markets: (1) phases of falling prices seem to be longer than those of rising prices, (2) in the long run, downward pressure on prices was identified, (3) the hypothesis of joint changes in market prices has never been statistically verified, and (4) markets remain subject to the effects of climatic factors. Although these analyses are overly superficial, as some claim, 3 particularly regarding their effort to make economic policy a factor, they can provide an opportunity to observe and characterize the movement of the 2000s. For example, The Economist's price index study covering 1862 to 1999 revealed short three-to 11-year cycles 4 and long four-to 47-year cycles.

In the history of raw-material markets, three periods of sharp market price increases are generally identified. The first followed the Great Depression of the 1930s and, all in all, simply reflected an upward adjustment in the main markets. The second, which was limited to a few quarters in 1949-1952, stemmed from consequences of the Korean War, particularly in terms of emergency reserve build-ups. During this period, prices of raw materials 5 increased on average by 45%, with a surge in prices of agro-industrial products (+84%) and, to a lesser degree, of metals and minerals (+34%), given that Asia was a key player in these types of commodity at the time. The third, in early 1970, is clearly marked by external factors (oil crises), but its structural origin lies in price rises for agricultural products of +60% between 1971 and 1974 as a result of severe weather conditions in certain production areas. Overall, during this period when oil prices quadrupled, raw-material prices tripled on average.

As a result of external factors (war and OPEC [Organization of Petroleum Exporting

Countries] decisions), the catalyst for price increases during these three periods was accelerated world growth (or sustained growth) of 2.6% to 7.2% between 1949 and 1951 and an average of 5% between 1971 and 1973, driven mainly by an upturn in industrial production. 6 In this regard, the trend observed since the beginning of the 2000s has followed a similar logic, with stepped-up world growth and industrial production.

However, with the exception of this factor, the current trend is quite unusual in several respects. The current increase has been particularly fast and has gradually affected all markets. Moreover, in terms of intensity, it has reached unprecedented levels: +130% for prices of industrial raw materials between 2001 and 2007 versus an increase of 55% between 1971 and 1974. Finally, while price increase cycles have lasted about 27 months on average since the early 1960s, the current cycle began at the end of 2001 and is still not over.

2 [START_REF] Cashin | The Long-Run Behaviour of Commodities Prices : small Trends and Big Variability[END_REF], Cashin P, Scott A, (1999).

3 See [START_REF] Calabre | La dynamique des prix et des marches de matières premières : analyses univariées versus faits stylises analytiques[END_REF]. 4 For simplicity's sake, a cycle will be defined in this article as a period of rising prices (boom) followed by a period of falling prices (slump). 5 Measured by the International Monetary Fund (IMF) index. 6 According to Radetzki, industrial production (North America and Europe in the 1950s, the OECD in the 1970s) rose from -0.3% to 9.3% between 1949 and 1951 and from 6.5% to 8.1% between 1972 and 1973. In terms of composition, the market segments most affected by this price surge are particularly unusual. Indeed, during the 20 th century, up cycles were driven more by strong movements in agricultural products as a result of severe weather conditions (drought) or geopolitical uncertainties (like war), or by structural changes and changes in industrial organization (discontinuation of producer prices in some markets, de-integration of the oil production chain and the creation of OPEC, etc.). The current cycle is largely driven by higher prices for non-ferrous metals, which are outpacing other market segments. Prices of non-ferrous metals have reached record levels in both current and constant terms. In the 1960s to 1980s, the rise in non-ferrous metal prices was sparked by strong growth in demand, which then led to an upturn in investment and a correction of medium-term prices. For its part, the last investment boom in the early 1980s coincided with a drop in the level of consumption of non-ferrous metals, which, along with slumping demand from the former USSR, resulted in latent overcapacities in the sector in the 1990s. The restructuring plans implemented during this decade therefore prevented any attempt to rebalance the markets starting in 2001. The current cycle is therefore also unusual in terms of supply response time (as well as for the reasons mentioned above). As a result, some are quick to wonder whether raw-material markets are currently in the midst of a "supercycle." This term is generally associated with a prolonged upward pressure on prices for several decades as a result of the industrialization or urbanization of a major economy (e.g. the United States in the 19 th century). Clearly, China and its industrialization model are currently behind the supercycle notion. One might also be tempted to compare the consequences of Chinese development with those observed during the industrialization phases of certain European countries (Germany) in the 1960s, Japan in the 1970s and South Korea in the 1980s. However, because of its demographic weight (the Chinese population is twice that of the entire G7), the size of its territory (and therefore the communication infrastructures to be developed) and its development model (based on a dynamic industrial sector), no comparison is possible. The existence of a supercycle is therefore a plausible hypothesis, particularly as studies of per-capita consumption of steel, aluminium and even automobiles in China clearly show that margins by product are high before this country achieves annual growth in industrial-commodity consumption comparable to that of old Europe or the United States. Despite this, however, a certain rebalancing of supply should occur, and prices should settle down; the problem lies in knowing when! Although it is still too early to know whether we are really in the midst of a market supercycle, there is hardly any doubt that the current cycle is unusual and, moreover, has surely been exacerbated by major liquidity in world markets. In fact, some are quick to characterize the current trend as a speculative bubble, much like the one that occurred in the late 1990s with regard to new-technology markets. Although it is clear that raw-material markets are now asset classes in their own right and that certain markets have been euphoric (sugar in 2005, copper in 2006 and nickel in 2007), the single hypothesis of a speculative bubble is not tenable in the medium term. Indeed, in the first few years of price increases, raw-material markets were an ideal refuge for investors that weren't making gains in the stock markets. Sharp price declines observed in the spring of 2004 and 2005, resulting from Chinese authorities' announcements of tightened credit supply in some sectors of the economy, were triggered in part by investment fund flows, without market fundamentals being affected. Since 2004 and the recovery of the world's major stock exchanges, yield differentials have declined, and a certain rebalancing, it can be said, has occurred. It should also be noted that, in the case of a speculative bubble, the sub-prime crisis that affected the financial markets in August 2007 should have brought about more dramatic changes in rawmaterial prices.

III-A new market paradigm?

When some economists announced, starting in 2005, that oil prices would rise to some $300 per barrel by 2015, they were basing their forecasts on a body of analyses developed at the beginning of the 20 th century by Gray 7 (1914), who conducted studies on deferments of production and profit maximization in the mining sector, and, above all, on Hotelling's work 8 (1931) on optimal management of a finite resource in its intertemporal dimension, particularly with the notion of rent due to exhaustibility. The fact that the issue of raw materials has reemerged today in the economic field is thus symptomatic of the movements that have been observed in economic theory for nearly two centuries. From Mirabeau (and the physiocrats) -"all politics emanates from a grain of wheat" -to Malthus' theses on demographic pressures on natural resources, raw materials would be a central theme of economic history until their gradual disappearance under the influence of neoclassical theory. It took until the 1970s and the rise of political ecology to see a resurgence of an ideological body related to raw materials.

What are we now seeing if not an update of the theses, developed by the Club of Rome with the 1972 Meadows report, Limits to Growth, 9 followed in 1974 by Beyond the Age of Waste? At that time, the teams at the Massachusetts Institute of Technology (MIT) issued projections, at the Club of Rome's request, on the risk of a medium-term shortage of raw materials. More generally, these studies focused on the link between the growth of the post-war boom years and the massive use of resources (and their massive waste). In the mid-1970s, certain pressures on agricultural products followed by the first oil crisis were seen as signs of a perpetual increase in market prices. This period had major repercussions, particularly in terms of industrial organization: fears of market cartels and emphasis on the "Small is Beautiful" model. It was also at this time that waste control policies and substitution policies in the energy sector (including expansion of the nuclear industry in France) were introduced. In some developing countries, this period paved the way for the creation of large projects, which were reassessed a few years later in light of the widespread drop in raw-material prices. 7 Gray L.C., "Rent under Assumption of Exhaustibility", Quarterly Journal of Economics, 28, pp. 466-489. 8 Hotelling H., The Economics of Exhaustible Resources, The Journal of Political Economy, Vol. 39, no. 2, pp. 137-175. 9 The Limits to Growth, for the French edition, "Halte à la Croissance? Rapports sur les limites de la croissance." Éd. Fayard, 1972. In the 1980s, these issues would become less relevant because of an overall decline in prices.

The debt crisis and reassessment of development models would therefore come to define the period, as would the financialization of the economy. It was at this time that the financial markets, particularly raw-material markets, began to take off thanks to key players' growing need for protection against price risk. Indeed, despite downward pressure on prices in the medium term, markets were subject to strong volatility. The 1990s would not bring about a major change in this pattern. The gradual dematerialization of the economy, with the emergence of the New Economy, and the globalization of production systems would even cause raw-material constraints to be forgotten in work organization. From "capitalism without factories" to the dot-com era and even the permanent closing of some industrial activitiesnon-ferrous metals such as lead in Europe, coal mining and so on -"business as usual" remained the norm in a world where incentives to reduce the intensity of raw-material usage were few and far between.

However, it was during this period that certain environmental considerations began to fully emerge. Conferences in Rio (1992) and then Kyoto (1997) brought to the forefront the importance of the environment and, more generally, sustainable development in business management. At the time, these conferences did not have the impact that they would begin to have in the early 2000s. Indeed, although the environmental constraint was identified, there was no visible follow-up in the main markets, since the two previous decades had seen downward pressure on prices for key products. Theories about oil and copper production peaks have little impact when the price of oil is less than $10 per barrel and that of copper under $1,000 per ton. However, since the early 2000s, given the rise in some industrial and energy raw-material prices and the fact that this trend has become widespread, scarcity and management of commodity shortage have again become central themes of economic analysis.

We have come full circle, so to speak, since the economy is basically defined as the identification and management of scarce resources!

In the 2000s, raw materials have therefore again become part of the great economic game, and we can now speak of the hypothesis of a new market paradigm or a new market order. Aside from the new raw-material landscape imposed by China's industrial development, the rawmaterial economy has reinvested entire fields in economics, geopolitics, sociology and more. Raw-material Monopoly is therefore being played from three angles now: scarcity of the resource, and thus its price; access to this resource, and its internal geopolitics; and the environmental consequences of its use (from the producer's or consumer's standpoint). Scarcity (depletion of resources) is currently a central theme of the various analyses related to raw-material markets.

In this context, raw-material markets have, as it were, rediscovered the uncertainty of tomorrow's production. This climate tends to exacerbate the slightest news concerning natural resources. It also fosters nationalism related to raw-material resources and closure to international investment, factors which themselves favour upward pressure on prices. This pattern is fuelled by various national, transborder and international tensions related to the control and management of natural resources. Aside from these factors, which are symbols of the new market order, one should add the current climate of uncertainty related to future use of certain products for energy and food purposes. While all types of commodity are feeling the effects of scarcity, there is also a constraint related to sustainable development and efforts to reduce global warming. In some sectors, such as non-ferrous metals, this new context has reinvigorated a sector created in the 1970s: recycling. In others, such as the energy and food sectors, the issue is increasingly that of future choices to be made in the markets. Thus, after the financial unification of the markets, an interdependence related to environmental constraints seems to be emerging among various raw materials. The often-simplistic explanation given for the sharp increase in sugar prices (+40%) between 2005 and 2006, namely higher ethanol production, hardly explains the 30% decrease in prices in an increasingly tense energy context. Moreover, to attribute the current increase in wheat and corn prices exclusively to energy concerns is to simply deny the fact that agricultural production depends first and foremost on climate conditions! Though it is clear that, in light of requirements, choices regarding the use of raw materials for energy and food purposes will need to be made in the long run, these choices are not yet driving market prices.

Something has clearly happened (and is still happening) in raw-material markets since the end of 2001. Given the segments affected (non-ferrous metals markets, and then gradually other markets), its intensity (and length), and the absence of the sort of external crisis (oil, war etc.) we saw in the 1950s and 1970s, the current cycle is quite different from those seen before. In the course of six years, China, the main protagonist in this change, has become a key player in all raw-material markets, to such an extent that some are quick to refer to the current cycle as a supercycle or to speculate that prices will continue to rise, as they did in the 1970s. It is also leaving its mark with regard to raised fears related to scarcity of natural resources and exacerbating certain forms of nationalism in a context where the environment is a factor in each stage of production. This new situation reminds us that the production system is, even now, founded on the production, processing and consumption of raw materials. It must cause us to reflect on the fact that the New Economy, which is based on dematerialization, has not made us more independent of raw materials. The most frequently visited websites in France include those of the SNCF (French national railroad company; www.sncf.com) and Amazon (www.amazon.com), two sites that electronically sell rail, air and automobile transport and various items to be delivered, services that consume energy raw materials for transport and agro-industrial raw materials for packaging.

Finally, it is worth noting that 2007 will be remembered as the year when the worldwide urban population exceeded the worldwide rural population. This gradual urbanization -in a context where, according to UN demographic forecasts, the world's population will be nearly seven billion by 2010 with only 10% additional agricultural area in production (as a best-case scenario!) -will represent a basic variable in understanding the markets, much like the structural changes in the Chinese and Indian development models.

Figure 1 :

 1 Figure 1 : IMF Commodity Index (current terms, in $)

Figure 2 :

 2 Figure 2 : Exploration budgets and Prices of non ferrous metal in the London Metal Exchange

Figure 3 :

 3 Figure 3 : IMF Commodity Index (deflated by Us CPI, in $)

Table 1 :

 1 Commodity prices (current prices)

		1990-2000	2001	2004	2005	2006	2007
	Oil ($/bbl)	19,24	24,51	38,2	54,5	65,2	72,67
	Aluminium ($Us/t)	1 454	1 447	1 716	1 899	2 568	2 638
	Copper ($Us/t)	2 187	1 578	2 866	3 680	6 712	7 122
	Lead($Us/t)	580	476	886	977	1 288	2 583
	Nickel ($Us/t)	7 055	5 958	13 835	14734	24 177	37 228
	Cotton ( US¢/lb)	71	43	62	55	58	63
	Rubber (cts/kg)	96	69	125	141	196	216
	Wood pulp ($Us/t)	607	541	615	610	675	794
	Coffee (US¢/lb)	92	46	62	89	95	107
	Sugar (US¢/lb)	10,5	9	8,6	11,4	15	11,61
	Wheat (US¢/bushel)	3,71	3,28	4,1	4,1	5	6,75
	HWWI index (2000=100)
	Except energy	114	90	127	140	178	209
	Industrial raw materials	107	89	128	147	195	225
	Non ferrous metal	101	88	130	151	240	273
	Food	127	93	126	126	139	174

Sources : Crude oil : Brent, spot price, London US$/barrel ; Non ferrous metal (Aluminium, copper, nickel & lead) : Cash prices, London Metal Exchange, London US$/t; Cotton Middling upland, 1 1/16 inches, contract No 2, nearest month New York US¢/lb ; Rubber Natural rubber, RSS 1, nearest month Kuala Lumpur Malays. cts/kg ; Wood pulp NBSK pulp benchmark index Helsinki US$/t ; Wheat US hard red winter, nearest month Kansas City US¢/bushel ; Coffee ICO composite average indicator price NY,F,D US¢/lb ; Sugar Raw, CSCE, contract No 11, nearest month New York US¢/lb.

Table 2 : Industrial Raw Materials Cycle (based on HWWI Commodity Index)

 2

	Trough	Peak	Peak to Trough	Trough to Peak
	January 1971	April 1974		+ 159%
	November 1975	March 1977	-26%	+ 21%
	November 1977	February 1980	-14%	+ 76%
	November 1982	September 1983	-38%	+ 19%
	March 1985	June 1988	-17%	+ 90%
	December 1993	June 1995	-39%	+ 50%
	October 2001	…..	-40%
	Sources : DB global Markets Research, FMI

Table 3 :

 3 Doubling times

	Growth rate (% per	Approximate doubling
	rate)	times (years)
	0.1	720
	0.5	144
	1.0	72
	2.0	36
	3.0	24
	4.0	18
	5.0	14
	6.0	12
	7.0	10
	10.0	7
	Source : Limits to growth, The 30-Year Update

In this article, we will consider commodities and raw materials as the same thing.

APPENDIX


We suggest a new approach to calculating a project' s net present value, termed the "displaced equity method". Based on a straightforward formula, it analyzes a project partially financed with debt from the perspective that every year the amount of outstanding debt displaces an equivalent amount of equity that otherwise would be tied up in the project. Although they represent distinct shareholders' perspectives, the displaced equity method and the equity residual method lead to the same investment decision. Every year, the project' s value calculated with the displaced equity method is equal to the sum of the project' s debt and equity values. In practice, when the schedule of expected outstanding debt amounts is known, using the displaced equity method is an easy way to estimate the project' s net present value.

INTRODUCTION

To value a project, the key issue is to know if the firm undertaking this project targets a constant debt-to-value ratio. If such is the case, the project' s value is obtained by discounting the project' s operating cash flows at the firm' s constant WACC discount rate (computed with the firm' s target debt ratio). To determine the project' s equity value by using the equity residual method, one must then consider a (theoretical) debt repayment schedule such that every year the project' s debt ratio is equal to the firm' s target debt ratio. In practice, the equity residual method is used when a predetermined stream of debt cash flows can be associated1 with the project under study. The project' s equity value can then be calculated by considering this stream of debt cash flows. In this short note, we stick to the operational viewpoint usually adopted by industrial practitioners who define the project' s cost of equity as being the minimum return-on-equity value required by the firm to undertake the project (under the assumptions made about financing). In other words, we consider that the project' s cost of equity remains unchanged 2 over the project' s lifetime. In this context, we suggest a new equity perspective that leads to a novel and straightforward valuation formula that does not require equity residual cash flows to be computed. Every year, it values an investment project partially financed with debt as the sum of its debt and equity values. This perspective is therefore that of shareholders considering the full project' s economic value, whereas, in the usual equity residual method, shareholders only consider their invested equity. To determine the project' s economic value has a practical interest: it represents the minimum cash amount at which the firm is willing to sell its stake in the project at any given point in time. The paper is structured as follows. In the next section, notations are adopted and the usual equity perspective -i.e., the equity residual method -is briefly recalled. Next, the new equity perspective -i.e., the displaced equity method -is introduced. We prove that the usual and new equity perspectives are consistent. The practical interest of the displaced equity method is emphasized in the last section.

THE USUAL EQUITY PERSPECTIVE: THE EQUITY RESIDUAL METHOD

Let us consider a project generating every year t the after-tax operating cash flow3 t f . The project' s lifetime extends from year 0 to T (i.e., 0

t f = when t T > ).
The project is partially financed with debt. t D denotes the amount of outstanding debt associated with the project at the end of year t (with 0 t D = when t T ≥ , and 1 0 D -= ). This debt is contracted at the annual interest rate d k . The interest payments are assumed to generate tax shields at the corporate tax rate τ . The cost of equity is denoted as e k .

As recalled by Chambers et al. (1982), the equity residual method -also called flow-to-equity method -values the project (or the firm) under study from the invested-equity perspective.

The cash flows considered are those available for distribution to the shareholders (i.e., each year, the after-tax operating cash flow minus the principal repayment and the after-tax interest payment). Each year t, the market value of equity t E is the sum of the present values of subsequent equity residual cash flows:

( ) ( ) ( ) 1 1 1 1 1 T s s d s t s t s t e f D k D E k τ - - = + + -+ - = + (1)
(1) is equivalent to:

( ) ( ) 1 1 1 1 1 t t d t t t e E D k D f E k τ - - + -+ - + = + (2)
with 0

T E = .

THE NEW EQUITY PERSPECTIVE: THE DISPLACED EQUITY METHOD

We now adopt the point of view of shareholders who consider the full project' s value, and not their invested equity only. In this perspective, the debt amount associated with the project every year is considered as just replacing an equivalent amount of equity which, otherwise, would be tied-up in the project. The reinvestment of this displaced equity by the shareholders yields a return equal to their cost of equity4 e k . In other words, in any year 1 t -, the oustanding amount of debt 1 t Dis merely viewed as a means of "freeing" an equivalent amount of equity that is immediately invested by the shareholders for one year at the return e k . The cash flow yielded by this investment in the following year is ( )

1 1 e t k D - +
. The actual gain for the shareholders in year t is given by the difference between this cash flow and the sum of the after-tax interest payment and debt repayment:

( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 e t d t e d t k D k D k k D τ τ - - - + -+ - = --
Therefore, in the displaced equity method, the shareholders' cash flow in year t is:

( ) ( ) 1 1 t e d t f k k D τ - + --
The project' s value t V in year t is consequenty:

( ) ( ) ( ) 1 1 1 1 T s e d s t s t s t e f k k D V k τ - - = + + -- = +
Or, equivalently:

( ) ( ) 1 1 1 1 t t e d t t e f V k k D V k τ - - + + -- = + (3)
In particular, in year 0, by using the notation 1 0 D -= , the project' s net present value computed with the displaced equity method is:

( ) ( ) ( ) 1 0 1 1 T s e d s s s e f k k D k τ - = + -- +

CONSISTENCY OF THE TWO EQUITY PERSPECTIVES

The following propositions are verified:

(i) each year t, the project' s value obtained with the displaced equity method is equal to the sum of the project' s debt and equity values:

( )

t t t
V D E t = + ∀ (4) (ii) the equity residual method and the displaced equity method yield the same net present value and internal rate of return 5 .

Both equity perspectives are therefore consistent, since they lead to the same investment decision, as stated by (ii). However, as shown by (i), the displaced equity method leads to compute the "total" project' s value (and not only the equity value).

Proof of (i) and (ii) (i) is proved by recurrence:

In year T, (4) obviously holds since we have: 0

T T T V E D = + =
Let us now assume that (4) holds in year t: t t t

V E D = + . By subtracting (2) from (3) we obtain:

5 Note that the internal rate of return r yielded by the displaced equity method is given by the following formula:

( ) ( ) ( ) 1 0 1 0 1 T s d s s s f r k D r τ - = + -- = + ( ) 1 1 1 1 1 t t t e t t t e V E D k D V E k - - - --+ + - = + (5)
Since we assume that 0

t t t V E D --= , (5) gives: 1 1 1 t t t V E D - - - - =
The recurrence is over: (4) holds in any year t.

In year 0, (4) gives:

0 0 0 0 0 V f E D f + = + + (6)
The left-hand side of ( 6) is the net present value computed with the displaced equity perpsective and the right-hand side of ( 6) is the net present value obtained with the equity residual method. Both net present values are therefore equal. Since both equity perspectives use the same cost-of-equity discount rate and yield the same net present value, they necessarily yield the same internal rate of return. (ii) is therefore proved.

PRACTICAL INTEREST OF THE DISPLACED EQUITY METHOD

To value a given project, the displaced equity method just requires to know the schedule of the amounts of outstanding debt associated with this project. Every year, the displaced-equity cash flow is the project' s after-tax operating cash flow adjusted in a very simple way. Clearly, when the expected outstanding debt amounts are known, using the displaced equity method is the easiest way to calculate the project' s net present value, since it does not require the equity residual cash flows to be explicitly computed. For this reason, this new method may be of practical interest for those who are in charge of project valuation in industry or financial institutions. As mentioned in the introduction, under certain financing assumptions the financial literature recommends using a cost-of-equity value that varies through time. In theory, the project' s value must then be computed according to a rather delicate and complex process. In this case, the equity residual method, and therefore the displaced equity method, yield projects' values that are equal to those obtained with other methods (see for instance [START_REF] Inselbag | Two DCF approaches for valuing companies under alternative financing strategies (and how to choose between them)[END_REF]), provided that the cost-of-equity rates successively used are correctly estimated. We could have introduced the displaced equity method by considering a cost-of-equity rate varying over the project' s lifetime, but doing this would have rendered this method -as well as the equity residual method -non-operational whereas calculating a project' s equity value and return on equity is widespread practice. For practitioners, assuming a constant cost of equity and using the displaced equity method should most probably be an easy and pragmatic way of estimating a project' s value. Miles, J. and Ezzell, J.R. (1985) Reformulating tax shield valuation: a note. Journal of finance, XL, 1485-1492. Ruback, R.S. (2002) Capital cash flows: a simple approach to valuing risky cash flows. Financial Management, 31, 85-103. Taggart, R.A. (1991) Consistent valuation and cost of capital expressions with corporate and personal taxes. Financial Management,20,[8][9][10][11][12][13][14][15][16][17][18][19][20]

This may happen in various instances: the project is the sole project of the firm (at least in a given class of risk), or it is financed by project finance, or the firm' s stake in the project is consolidated by using the equity method (as defined in IAS

28), i.e. no debt associated with the project appears in the firm' s consolidated balance sheet.2 The financial literature suggests formulas that yield a project' s cost of equity susceptible to change over time when dollar amounts of outstanding debt are targeted; for instance,[START_REF] Inselbag | Two DCF approaches for valuing companies under alternative financing strategies (and how to choose between them)[END_REF] derive a formula consistent with the adjusted-present-value approach, Ruback (2002) produces a formula consistent with the CAPM approach. When a fixed debt ratio is targeted, the cost of equity remains constant (see for instanceMiles and Ezzell (1985),[START_REF] Harris | Risk-adjusted discount rates -extensions from the average-risk case[END_REF] andTaggart (1991)). We further discuss this point in the last section.

Here, this term refers to the cash flow of the project before any financial claims are paid; for tax purposes, the taxable income used is defined as the earnings before interest and taxes, which means that the after-tax operating cash flow includes no interest tax shields; some authors also use the term "free cash flow".

The cost of equity is here viewed as an opportunity cost, as the displaced amount of equity is implicitly assumed to be invested in other assets with a similar risk for the shareholders; this equity displacement remains a mere perspective, in the sense that the cost-of-equity value considered here is that corresponding to the financial structure of the project under study.


Axel Pierru
email: axel.pierru@ifp.frdenis.babusiaux@numericable.fr

Denis Babusiaux

Valuation of investment projects

La collection "Les cahiers de l'économie" a pour objectif de présenter des travaux réalisés à l' IFP et à l'IFP School, travaux de recherche ou notes de synthèse en économie, finance et gestion. La forme peut être encore provisoire, afin de susciter des échanges de points de vue sur les sujets abordés.

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Introduction

Valuing an investment opportunity in oil and gas exploration-production generally requires the use of approaches -decision trees, sensitivity analyses -designed to take into account the uncertainty and flexibility associated with the project in question. Regardless of the approach taken at a given time, a valuation of the project using deterministic hypotheses is nevertheless inevitable. It is therefore necessary to use a method that allows the project's net present value to be accurately determined by factoring in its financing. Investment projects are generally valued on the assumption that the firm must comply with a debt ratio 1 at company level, at least for all projects belonging to a given risk class. This is also the case for a multinational firm investing in various countries. This ratio, called here the "target debt ratio", helps to determine the firm' s cost of equity. A multinational firm may, however, wish to capitalize on the fact that some of its subsidiaries have local access to loans for which the after-tax cost is lower than can be obtained elsewhere. This is particularly true of oil companies. In the exploration and production sector, tax schemes are specific and vary greatly from one country to another, and at times even from one oil field to another. For example, in 2007 the tax rate on oil revenue in Norway -ordinary income tax plus special petroleum tax -is 78%, and interest paid to lenders is fully deductible from taxable income (provided that the amounts borrowed do not exceed 80% of the investment). In Angola, on the other hand, interest associated with loans cannot be recovered under production-sharing agreements. A firm operating in these two countries and seeking to minimize the total amount of taxes paid will therefore borrow more in Norway than in Angola. In reality, an oil company often allocates to a project loans representing more (or less) than the portion corresponding to the target debt ratio that must be complied with at company level (i.e. for all projects).

In this paper 2 , we present and illustrate the generalized ATWACC method, an adaptation of the standard WACC method that can be used to value projects with any debt ratio. This method was proposed by Babusiaux and Pierru (2001a). Here, we prove its validity using a model that maximizes, under constraints, the firm's equity value.

Limitations of project-valuation methods used in the industry

In the industry, a number of methods are used to value investment projects. Here, we consider projects for which debt financing is included in the calculation of the debt ratio targeted by the firm. The methods traditionally used are therefore the standard WACC method and the BTWACC method. 3 The standard WACC method entails discounting after-tax operating cash flows 4 at a rate defined as the firm's after-tax weighted average cost of capital. This calculation reflects the point of view of a department in charge of investment projects for which all financing-related data are included in calculating the discount rate. In the oil exploration and production sector, the BTWACC method is routinely used 5 . This method reflects the perspective of all the firm's funds providers, since it discounts the total cash flow received -or paid out -by shareholders and debtholders (i.e., the sum of the after-tax operating 1 According to [START_REF] Graham | The theory and practice of corporate finance: evidence from the field[END_REF], 80% of firms target a given debt ratio.

This is an abbreviated version of an article published in the OPEC Energy Review (2008, 32(3), p. 197-214). 3 Also called the Arditti-Levy method (1977), the shadow interest method, and, by [START_REF] Ruback | Capital cash flows: a simple approach to valuing risky cash flows[END_REF], the capital cash flow method. 4 Here, the term "after-tax operating cash flow" refers to the cash flow of the project before any financial claims are paid. For tax purposes, the taxable income used is defined as the earnings before interest and taxes, which means that the after-tax operating cash flow includes no interest tax shields. Some authors also use the term "free cash flow". 5 For example, the BTWACC method is the official method of Total. cash flow and the interest tax shields). The discount rate used is the firm's before-tax weighted average cost of capital. In considering a discount rate independent from taxation, this method is particularly appropriate for an international firm in the upstream oil or gas sector subject to tax schemes that vary greatly from one country to another, and even from one permit to another. These often-complex tax schemes are frequently defined on a contractual basis within the scope of production-sharing agreements or service contracts, and can vary over time. The use of both these methods is based on the implicit assumption that each year the project's debt ratio is equal to that of the firm. Both methods therefore lead to the same net present value, as illustrated by [START_REF] Chambers | Treatment of financing mix in analyzing investment opportunities[END_REF]. Neither of these methods can be used to value a project whose debt ratio differs, for tax reasons, from that targeted by the firm. The generalized ATWACC method proposed by Babusiaux and Pierru (2001a) meets this objective.

Presentation of the generalized ATWACC method

Hypotheses and notations

Let us consider a multinational firm subject to different tax schemes from one country to another. For the sake of simplicity, we assume that all the firm's investment projects belong to the same risk class -for example, that they are all oil-field development projects -and are financed using loans contracted at the same 6 annual interest rate, denoted as r. The firm' s cost of equity is denoted as e k . All these data, like those that follow, are expressed in nominal dollars.

We assume that with each project u is associated a given stream of after-tax operating cash flows that goes from year 0 to year T. The after-tax operating cash flow at year n of project u is denoted as , u n F . At each year n, the taxable income of project u is subject to a special tax rate, denoted as ,

u n θ .
We assume a predefined target debt ratio, denoted as w, with which the firm must comply. However, the firm may wish to allocate to a project loans representing more (or less) than the portion corresponding to its target debt ratio w (which must be satisfied at company level). In fact, debt financing giving rise to tax shields in a given country depends on the investments made in that country. In particular, the loans associated with oil-field development projects are usually known. As we pointed out above, tax rules, such as those related to the deductibility of interest payments, and applicable tax rates can vary greatly from one project to another. The firm must therefore seek an optimal allocation of its debt.

Optimal allocation of debt among various projects

In theory, each year the firm must allocate the loans in increasing order of their after-tax cost 7 ( )

, 1 u n r θ - .
When the total amount of loans available at company level (in compliance with the target debt ratio) is fully used up, the after-tax cost of the last loan allocated therefore corresponds to the firm's marginal cost of debt. If the firm's debt capacity were increased by a dollar, the debt would be contracted at this marginal cost. We denote t as the tax rate that applies to the taxable income of the project to which this (last) loan is allocated. At company level, the after-tax marginal cost of debt is therefore ( )

1 t r - ( subsequently
assumed to be constant over time).

This latter assumption can be easily disregarded, as illustrated by Babusiaux and Pierru (2001a). We must take , 0

u n θ = if we consider a concession for which the interest payments are not deductible from the taxable income or a production-sharing agreement for which these payments are not included in cost oil.

The firm's discount rate

In accordance with standard microeconomic reasoning, the discount rate used by the firm is the firm' s marginal cost of capital. As the firm maintains its debt ratio equal to a given target debt ratio, the composition (debt and equity) of the last dollar of capital raised by the firm satisfies this target debt ratio. The firm' s marginal cost of capital is therefore a weighted average of the firm' s marginal after-tax cost of debt and its cost of equity. In the standard case where the firm faces only a single tax rate, to determine the firm' s marginal after-tax cost of debt is not an issue, as the after-tax cost of debt is the same for all projects. Here, the after-tax cost of debt actually paid depends on the project considered. Because of the resulting optimal debt allocation between projects, the firm' s marginal after-tax cost of debt to use is ( )

1 t r - .
The discount rate used here is a WACC computed on a marginal basis. However, unlike the standard case, this WACC discount rate, computed as the weighted average of the firm' s marginal financing costs, is here not equal to the weighted average of the firm' s average financing costs (that would be obtained by considering the average after-tax cost of debt). As a result, the cash flows of all the projects must be discounted at the same 8 rate, i, defined as follows: ( ) ( )

1 1 e i w t r w k = - + - (1)

Net present value of a project

The net present value of project u is given by the following formula:

( ) ) ( , , , 1 ,0 1 1 T u n u n u n u n n F t rB F i θ - = + - + + (2)
where , 1

u n Bis the outstanding amount at the end of year 1 nof the debt allocated to project u.

The cash flow attributed to the project under consideration is therefore equal each year to the sum of the project's after-tax operating cash flow and an interest-tax-shield differential. The economic interpretation of this differential is as follows: at each year 1 n -, compliance with the firm's target debt ratio w implies that, if loan amount , 1 u n Bcould no longer be allocated to project u, it would then be allocated to the project used to define the firm's (after-tax) marginal cost of debt. The corresponding after-tax interest payment would then be ( ) , 1 1

u n t rB - - , instead of ( ) , , 1 1 u n u n rB θ - -
. Therefore, project u generates the following reduction of after-tax interest payment:

( ) ( ) ( ) , 1 , , 1 , , 1 1 1 u n u n u n u n u n t rB rB t rB θ θ - - - - -- = -
.

Other profitability criteria

Other criteria that are used in practice -internal rate of return, profitability index, discounted payback period -can also be computed with the generalized ATWACC method. For instance, a project' s internal rate of return is the value of i that cancels out the project' s net present value given by (2). This return must be compared to the firm' s discount rate given by formula (1).

8 The firm's use of a single discount rate implies that the cost of equity is the same for all projects, regardless of their debt ratio, which may appear to contradict the traditional results of finance theory (for example, the formula proposed by [START_REF] Modigliani | Corporate Income Taxes and the Cost of Capital: a correction[END_REF]). In fact, this assumption is justified here by the existence of the constraint of compliance with the target debt ratio at company level. This constraint "connects" the projects and results in no differentiation in their cost of equity. [START_REF] Pierru | Financing and valuation of a marginal project by a firm facing various tax rates[END_REF] proves this result by considering the adjusted present values of the various projects.

Contractual terms specific to the upstream oil sector

Adapting the generalized ATWACC method to take into account contractual terms specific to the upstream oil sector does not raise any particular problem. Let us consider, for example, a project u carried out under a production-sharing agreement, where the interest payment can be recovered in the form of cost oil. One therefore simply regards this interest payment as taking the place of an equivalent amount of profit oil that otherwise would have been shared between the state and the company. At year n, the actual gain that results from incorporating this interest payment into the cost oil is therefore , , 1

u n u n rB τ -, where , u n τ
is the percentage of the profit oil -or, where relevant, the excess oil -accruing to the state at year n. The cash flow to be considered with the generalized ATWACC method is therefore:

, , , 1
( )

u n u n u n F t rB τ - + -
.

Numerical example

Let us consider a firm using the generalized ATWACC method that computes its discount rate with the following data (expressed in nominal dollars): 15%, 8%, 35%, 40%

e k r t w = = = =
. The firm' s marginal after-tax cost of debt -determined by the firm' s central services as previously describedis therefore:

( )

1 0.35 8% 5.2% - =
According to equation ( 1), the discount rate used by the firm is thus equal to 11.08% . The company is studying an investment project which consists of bringing an oil field into production. The earnings produced by this project are subject to a tax rate u θ of 70%, constant throughout time. We assume that the fiscal system is a concession which allows deductibility of the interest payments from the taxable income.

The project under study has a lifetime of 7 years, the investment outlay ($89 million) being spent in year 0. The after-tax operating cash flow (i.e., revenue minus operating expenses, royalty and tax on earnings before interest) is assumed to remain constant and equal to $18 million each year (from year 1 to year 7). The project is partly financed by a loan of $70 million contracted in year 0 at an annual interest rate of 8%. The loan reimbursement is assumed to be 'as fast as possible' as is often the case in practice. The first step is to compute the schedule of debt reimbursement. For this, it suffices to say that each year the outstanding debt amount will decrease by an amount equal to the difference between the after-tax operating cash flow and the after-tax interest payment (the tax rate considered being 70%, the effective project' s tax rate). The results are given in Table 1.

Table 1 Debt repayment schedule ($ millions)

Each year, the differential of interest tax shields is obtained by multiplying the outstanding debt amount at the end of the previous year by: ( ) The project' s cash flow is computed by adding this interest-tax-shield differential to the after-tax operating cash flow. The results are given by Table 2. The generalized ATWACC method yields a negative net present value of -$0.26 million.

To compare this value with that obtained with a non-rigorous computation, let us now assume that the firm uses the BTWACC method. The discount rate is then computed on a before-tax basis:

( ) ( ) 0.4 8% 0.6 15% 12.2% × + × =
With the BTWACC method, every year, the project' s cash flow is obtained by adding the full interest tax shields to the after-tax operating cash flow (Table 3). The resulting net present value is then $0.75 million. This (positive) value is incorrect as here the project' s debt ratio is not equal to the debt ratio targeted by the firm, which precludes the use of the BTWACC method. [START_REF] Babusiaux | Corporate investment decisions and economic analysis[END_REF] provide other numerical illustrations of the generalized ATWACC method.

A new proof of the generalized ATWACC method

Here we propose a new proof of the generalized ATWACC method. This proof is based on the interpretation of the Lagrange multipliers of a mathematical program that maximizes a firm's equity value in the context of our analysis.

Constrained optimization of a firm's equity value

We consider an international firm involved in the development and production of oil (or gas) fields.

The model, which covers a period of T years, maximizes the equity value of the firm at year 0. The firm has z investment opportunities (i.e., projects) that can be undertaken over this period. The firm has the option of choosing the (non-negative) size x u of each project P u , with: 0

u x ≥ 1,...., u z = (3) 0 u x ≥ 1,...., u z = (1)
This assumption is based on two observations:

-A large oil field is usually developed by an association of several firms that share capital costs (in order to reduce their individual exposure to political risks, for instance); consequently, a given firm can consider a project divisible.

-The decision to develop a large oil field may also include developing smaller surrounding oil fields and the sizing of a collecting pipe (with non-constant returns).

Possible interdependencies between the cash flows of the various projects are not considered in the model. However, even if such interdependencies exist in real-life projects -for instance, the simultaneous development of several oil fields in the same area may give rise to an increase in the corresponding investment costs -their impact on the issue raised here, i.e. the debt allocation process, is likely to be negligible. To sum up, the projects constitute continuous ranges of investment in the model, independent of each other.

The expected after-tax operating cash flow to be produced in year n (

} { 0,1,..., n T ∈ ) by project P u ( } { 1, 2,..., u z ∈
) is denoted as ( ) , u n u F x and is assumed to be differentiable with respect to x u . Given these assumptions, to define bounded project sizes simply consider the functions ( ) , u n u F x so that, in the neighbourhood of the bounds, the investment cost grows very rapidly (or revenues fall sharply). Certain projects can be undertaken beginning in year 0, and others in later years. For a project P q starting in year t, one should then take ( ) , 0 q k q F x = for k t < . The expected terminal value of project P u in year T is given as a function of x u and is denoted as

( ) , u T u V x .
All of the firm' s debt is assumed to be contracted at interest rate r. The loan amount expected to be allocated in year n to project P u is denoted as ,

u n
B . As we assume that the firm cannot lend capital, we must have: , 0 0,...., 1, 1,....,

u n B n T u z ≥ = - = (4)
For each project P u , the debt-related cash flow in year 1 n + amounts to

( ) ( ) , 1 , 1 , 1 1 u n u n u n B r B θ + + -+ - . The interest payment , u n
rB is thus assumed to generate tax shields at tax rate , 1 u n θ + , which depends on the fiscal rules specific to project P u . If the interest payment is neither recoverable nor deductible, as in the Angolan production-sharing contract mentioned in the introduction, then one should simply set out: , 1 0

u n θ + = .
In real-life projects, the amount of debt for which interest payments are deductible from the taxable income cannot exceed a certain limit imposed by local fiscal authorities. This maximum amount can generally be considered a function of the size of the project. For instance, it is equal to 80% of the The values of the Lagrange multipliers associated with the constraints corresponding to the other projects can be deduced from the above equations. Let us first consider the projects to which a loan is allocated at year 0, i.e. the projects for which the constraint of type ( 5) is binding. Equations ( 9) and (10) become: Projects to which no loan is allocated at year 0 have a Lagrange multiplier, associated with the corresponding constraint of type ( 5), with a zero value:

,0 0 u λ = . We write the corresponding equations ( 9) and ( 10):

( ) ( ) ( ) 0 ,0 ,0 ,1 0 1 1 0 1 1 1 1 0 u e u u w k w r µ π π θ µ - + + = - - + - -≤
The first of these two equations (identical to those above) again indicates that:

0 ,0 ,0 u m π π = As for the second, it implies: 0 ,1 ,1 u m θ θ ≤
, which is entirely consistent with the debt allocation process described above: at year 0, no loan will be allocated to a project generating an amount of interest tax shields lower than the amount that would be generated by project 0 m P . This gives a first result that we will generalize to the following years: the Lagrange multipliers associated with the constraints of type ( 7) are equal each year to the discount coefficient used in the generalized ATWACC method.

In fact, at year n, for any project P u , we rewrite equations ( 11) and ( 12):

( ) π π = . Since this hypothesis was verified for year 0, there is in fact equality of these Lagrange multipliers each year. To determine the value, we will again refer to the project used to define the marginal loan for year n in question, denoted as n m P . Once the debt is allocated, the marginal project benefits from a loan without its constraint (5) being active: And finally, for any project:

d F x d B x d V x r d F x d x d x d x d x w k w r w k w r θ θ θ θ + + + - = + + = = + - + + = + - + - + - + - ∏ ∏
The interpretation of this equation is straightforward: the discounted income, determined according to the generalized ATWACC method, of the last euro invested in project u is zero.

There is indeed optimum project dimensioning when the generalized ATWACC method is used. In addition, according to equation ( 13), a project for which 0 λ also indicates the cash flow adjustment to be taken into account, in the form of an after-tax interest expense differential. In fact, if the corresponding constraint (5) were relaxed by one euro, one euro borrowed would be deducted from the "marginal" project and allocated to the project in question. The firm's cash flow would increase at year n+1 by the amount In this paper we focus on the computation of a project' s net present value. Of course, other criteria that are used in practice -internal rate of return, profitability index, discounted payback period -can also be computed with the generalized ATWACC method. Finally, please note that the BTWACC method may be viewed as a specific case of application of the generalized ATWACC method, one in which the firm's after-tax marginal cost of debt is equal to the before-tax interest rate (i.e. 0 t = ). For a firm using the BTWACC method, adopting the generalized ATWACC method therefore simply amounts to changing the value of a parameter and not really changing the method. This remark can prove valuable, considering how resistant organizations are to change.

  At year 0 as for the following years, debt is allocated in increasing order of the loan's after-tax cost (i.e. by decreasing interest tax shields). The corresponding constraints of type (5) are binding. The project to which the last loan is allocated is denoted as 0 m P (project used to define the firm's marginal cost of debt at year 0). The corresponding constraint (5) is not binding, which gives us (Please note that this is the only project that meets both these conditions.) We therefore obtain the following two equations:

ux=

  , i.e. an unrealized project, has a negative or zero marginal net present value. The Lagrange multipliers , the discount coefficients of year n+1 in relation to year 0. The discount rate n i for year n+1 to year n is given by: If the cash flow of a project at year n+1 were increased by one euro, which would amount to relaxing the corresponding constraint (7), the firm's present value would increase by the amount:

  corresponds to an increase in the objective function in the amount:

Table 2

 2 Cash flow with the generalized ATWACC method ($ millions)

	Year	0	1	2	3	4	5	6	7
	Investment	89
	After-tax operating cash flow		18	18	18	18	18	18 18
	Interest-tax-shields differential		1.96	1.5 1.04 0.56 0.07
	Cash flow	-89 19.96 19.5 19.04 18.56 18.07	18 18

Table 3

 3 Cash flow with the BTWACC method ($ millions)

	Year	0	1	2	3	4	5	6	7
	Investment	89
	After-tax operating cash flow		18	18	18	18	18	18 18
	Interest tax shield		3.92 3.01 2.07 1.11 0.13
	Cash flow	-89 21.92 21.01 20.07 19.11 18.13	18 18

investment expenditure in a Norwegian oil concession, as discussed in the introduction (certain intricacies of the Norwegian petroleum tax system are here voluntarily ignored). In the model, this maximum loan amount, denoted as ( ) , l u n u B x , is assumed to be a continuous function, differentiable with respect to x u . The following constraints are therefore included in the model: ( ) , , 0 0,...., 1, 1,....,

Each constraint of type ( 5) means that a debt in excess of ( )

, l u n u B x generates no interest tax shields and, consequently, has no economic interest for the firm. Therefore, such a debt is not contracted through the optimal debt-allocation process (even if this option were explicitly taken into account in the model).

Let , u n V denote the equity value of project P u at the end of year n (present value at year n of the equity cash flows of project P u after year n). At any given year, the equity value of the firm is considered as being equal to the sum of the equity values of the entire set of projects undertaken. The firm' s total value is equal to the sum of the firm' s equity value and debt. As the firm undertakes to satisfy a target debt ratio w on the corporate scale each year, we must have:

The firm' s cost of equity is denoted as e k . ,

V is calculated according to the equity residual method on a stand-alone basis (i.e. by imputing to project P u all cash flows related to the loans that are associated with it):

V is not subject to a non-negativity constraint, since the value of a project can be negative during certain years. Such is the case, for instance, if the last cash flows of the project correspond to the cost for decommissioning and site rehabilitation of a depleted oil field.

The problem of maximization under constraints with which the firm must contend is therefore written as follows:

( 1,...., ) ( 0,...., 1) 0 (5)

( 1,...., ) ( 0,...., 2) 1 1 1 0 (7)

( 1,...., ) 1 1 1 . .

Resolution and economic interpretation

We use Kuhn-Tucker conditions to determine the values of the Lagrange multipliers associated with the various constraints of the problem. Let ,

π -be the Lagrange multipliers associated with constraints of type ( 5), ( 6), ( 7) and ( 8) respectively. To write the dual relations, we do not explicitly consider the Lagrange multipliers associated with the non-negativity constraints of type ( 3) and ( 4). By applying the Kuhn-Tucker first-order optimality conditions, we obtain the following dual relations:

( 1,...., ) 0 ( 13)

Moreover, projects other than project n m P benefiting from a loan at year n have their constraint (5) active. Equation ( 12) then corresponds to an equality, which allows us to write:

∏

The Lagrange multiplier ,

λ is zero for all projects not benefiting from a loan at year n.

We can now replace the Lagrange multipliers by their respective values in equation ( 13): for a realized project benefiting each year from a loan, this equation is written:

If we assume that the rate , n m n θ remains constant and equal to t over the entire period in question, equation (13) becomes:

The left-hand-side term of this equation corresponds to formula (2).

Conclusions: theoretical and practical advantages of the method proposed

A first (theoretical) advantage of the generalized ATWACC method is that it allows the financing decision to be decentralized. In fact, maximizing the value of the project or that of the firm leads to the same decision: a loan will be allocated at year 1 nto project u only if , u n t θ ≥ , i.e. if the after-tax cost of this loan is favourable (which results in a positive cash flow). Otherwise, allocating a loan to the project reduces its net present value. As the point of view of the project and that of the firm are consistent, the financing decision can therefore, in theory, be delegated to the project managers.

From a practical standpoint, the generalized ATWACC method 9 is easy to work with, since calculating the projects' cash flows does not depend on the debt ratio, which, at company level, must be complied with for all projects of the same type. Moreover, even if allocation of the debt takes into account qualitative or strategic elements, the generalized ATWACC method remains applicable. Allocation of the debt does not need to be defined exactly; it must merely be possible to determine the marginal loan (over the amount of which the firm has some flexibility).

It should be noted that the tax schemes to which projects outside the Exploration-Production sector are subject are far less variable. In the plausible case where these tax schemes are similar to the ones used to define the firm's marginal cost of debt, the method proposed comes down to the standard WACC method. Its use therefore allows a single criterion for all the business sectors of an oil group, a standard criterion whose use is easier and much more common than that of the BTWACC method.

During the initial studies of a project's profitability, particularly during any discussions between partners, calculations are generally made without considering a loan; in other words, they are based on tables of operating cash flows. Moreover, the ex-post analysis of financial earnings is often conducted by calculating an overall accounting rate of return that concerns all the invested capital (ROCE: Return On Capital Employed). The accounting revenues used for this purpose include neither corresponding interest expense nor tax shields. This ROCE is therefore comparable to an after-tax cost of capital. In addition, an EVA (Economic Value-Added) method is used by defining value creation for a year as the difference between annual accounting revenue (excluding financial items) and capital costs. The latter also include an after-tax average cost of capital. In these cases, the explicit or implicit reference is the standard WACC method and not the BTWACC method. One advantage of the generalized ATWACC method is therefore that it is based on comparable elements.

An adaptation of the BTWACC method, consistent with the generalized ATWACC method, has also been proposed by Babusiaux and Pierru (2001a); however, this method is difficult to use in practice, since calculating a project's cash flow requires knowing the project's debt ratio each year