Short Term Hedging, Long Term Vertical Integration

Short-term Hedging, long-term Vertical Integration? AnAnalysis of Corporate Risk Management.

ADOUA, Imane1, DE BAZELAIRE, Stanislas2, STOFER, Michel3, YANY-ANICH, Andrs4.

Objectives:

Conventional wisdom suggests that short-term corporate risk management is achieved throughmarket instruments such as forwards and futures while long-term corporate risk management relieson corporate arrangements such as vertical integration or conglomerate acquisitions. The objectiveof this work is to build a model of corporate risk management with a finite horizon and to investigatethe interaction of hedging and vertical integration as two strong tools of corporate risk management.

Issues:

The model will need to combine a simple way of specifying firms concern for risk management,a simple modeling of derivative pricing and an analysis of corporate arrangements such as verticalintegration.

AbstractThe purpose of this article is to develop a model to study the interaction and compare two

tools of corporate risk management: vertical integration and short-term hedging. We begin bystudying some important articles in the literature of Corporate Risk Management. In particular,we focus on two hedging models that we extend for the application of vertical integration. Webuild a two-period model with two markets, an upstream and a downstream. In every market,firms can use hedging with forward contracts to protect their initial wealth against some riskyfactor. We develop two cases, with and without vertical integration. We compare the optimalhedge ratio using each approach. We observe that wealth and correlation effects have a strongimpact on hedging. Moreover, we argue that vertical integration can reduce the need of forwardcontracts under certain conditions given by (a) initial wealth, (b) correlations and volatilitywith respect to a risky variable and (c) a volume effect. We then add more elements to theanalysis, such as a (i) long-term perspective and (ii) the presence of more actors. We try toillustrate our results by particular examples taken from some commodity markets. Then, wedevelop a specific framework to study vertical integration in the electricity market and see underwhich conditions the optimal level of forwards can be reduced.

Keywords: Optimal hedging, vertical integration, changing investment opportunities, forward

contracts, hedging in multiple periods.Option : Corporate FinanceGroup Tutor : Gilles CHEMLA, [email protected] tutor : Romuald ELIE, [email protected] : November 15th to May 29th

1Master of Science at ENSAE Paris-Tech in Corporate Finance2Master in Management at ESCP Europe and Master of Science at ENSAE Paris-Tech in Corporate Finance3Third year student at ENSAE Paris-Tech with major in Corporate Finance4Engineer at Ecole Polytechnique and Third year student at ENSAE Paris-Tech in Corporate Finance

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CONTENTS

Contents1 Introduction 3

2 Corporate risk management in the literature 4

3 A first paradigm of the benefits to hedge 63.1 The model setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Stochastic investment function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.3 Solving the model by a recursive method . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Interaction between vertical integration and hedging 114.1 The non-integrated case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2 The integrated case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.3 Comparison of optimal levels of hedging in the two cases . . . . . . . . . . . . . . . . 13

5 Extensions 155.1 An endogenous weighted average . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.2 A model with several actors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.3 Intertemporal issues: the influence of long-term . . . . . . . . . . . . . . . . . . . . . 17

6 Illustration of the model 206.1 Testing the sign of and the hedging ratio in an oil company . . . . . . . . . . . . . 206.2 Delta Airlines: an example of vertical integration . . . . . . . . . . . . . . . . . . . . 22

7 Comments, critics 237.1 The cost function of raising external funds . . . . . . . . . . . . . . . . . . . . . . . . 237.2 The time horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.3 Perfect price anticipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

8 A model for the electricity market 248.1 Vertical integration in the electricity market . . . . . . . . . . . . . . . . . . . . . . . 248.2 An adapted model for the electricity market . . . . . . . . . . . . . . . . . . . . . . . 25

8.2.1 The setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258.2.2 Problem and optimal solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.2.3 Extension : Vertical integration . . . . . . . . . . . . . . . . . . . . . . . . . . 27

9 Conclusion 29

10 Appendix 3110.1 Equations : First model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3110.2 Equations of the model solution in the case of Cobb-Douglas functions . . . . . . . 3210.3 Effects on the optimal hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

10.3.1 The effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3310.3.2 The endogenous weighted average . . . . . . . . . . . . . . . . . . . . . . . . 34

10.4 The electricity market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.5 The Envelope theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3610.6 Rubinsteins rule (1976) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3610.7 Data of regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

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1 INTRODUCTION

The best Wall Street minds and their best risk-management tools failed to see the crash coming.

New York Times, January 2009.

1 Introduction

The scale of losses in the financial crisis that began in 2007 is unprecedented. The InternationalMonetary Fund estimates these losses to $4.1 trillion and $1.1 trillion to help fix them. These hugenumbers show the depth of the worldwide crisis and the new challenges that are coming. Since then,corporations have become more risk averse and have given strong relevance to risk management. Infact, nothing in the past resembles what we are currently seeing. We are in the presence of eventsthat we have not seen since World War II. This is a period of absolutely exceptional uncertaintythat calls for responses that match the events from both the public and private sector (Jean-ClaudeTrichet, Head of the European Central Bank).

There are several ways to manage corporate risk. Vertical integration has long been consideredas a major tool for that. Causes for vertical integration are diverse: strong volatility of intermediategoods price and demand, the lack of flexibility in some markets, a strong risk aversion. It alsorestores the symmetry between upstream and downstream firms exposure to price and demandvolatility. However, as Carlton (1979) points out, the study of vertical integration has never beena completely-understood phenomenon. Researchers and managers have often concluded that theprominent incentive for vertical integration is a stable supply of inputs for retailers and/or stablecash-flows for producers. For example, DeltaAirlines has recently acquired an oil refinery, justifyingthis strategic move by a significant reduction in fuel costs, the bane of the airline industry.

In a world of increasing uncertainty, corporations have also relied on derivative instrumentsto hedge their risk in a short-term perspective. From now on, by hedging we refer to the use offorward contracts to protect a wealth from a risk. The use of these financial hedging instruments hasstrongly risen in the past few decades (Hull (2003)). A firm can hedge its wealth in order to reducethe need of external financing to fund investments. External financing can be costly for severalreasons well documented in the literature: direct or indirect costs of financial distress, informationalasymmetries, transaction costs and agency costs between managers and outside investors. A goodhedging policy can improve a firms performance by better coordinating financing and investingactivities. It does not mean necessarily that the firm will fully hedge to avoid risk. For instance, if afirms investment opportunities are positively correlated with a risk factor then the firm can engagein partial hedging. It can also engage in under-hedging to increase the exposure to the risk factorif investment opportunities are negatively correlated with the risk.

To which extent can vertical integration reduce the need for forward contracts? The interactionbetween these two risk management tools has not been much studied in the past literature. Theconventional wisdom is that short-term risk management is achieved through hedging with financialinstruments while long-term risk management is achieved through vertical integration. The objectiveof this paper is to investigate the interaction between vertical integration and hedging. This issue isof paramount importance for industries where the price of the input is highly volatile, like commodityand electricity markets. For that purpose, we develop simple models based on reference models thatcombine hedging and vertical integration.

The paper is organized as follows. In section 2 we review the existing literature on hedgingand vertical integration. In section 3, we present the basics of Froot, Scharfstein and Steins model(1993), a prominent model of the coordination between hedging and investment. In section 4 weextend this model to a vertical integration framework and study the interaction between these twotools. In section 5, we test a hypothesis and a prediction of our model on two oil companies. Insection 7 we comment the limits of our work. Section 8 ends by studying an equilibrium-model forthe electricity market. Section 9 concludes.

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2 CORPORATE RISK MANAGEMENT IN THE LITERATURE

2 Corporate risk management in the literatureThis section describes different rationales for corporate risk management in the past literature. Weanalyze different approaches developed in the last years.

Hedging in the economic literatureAlthough there are stories to explain why firms might wish to hedge, it seems there is not yet

a single, accepted framework which can be used to guide hedging decisions (Froot, Scharfstein andStein, 1993).

Managerial motives have been pointed out by several authors. Stulz (1984) argues that hedgingreflects the risk aversion of managers who hold a large portion of their wealth in the firms stocks.Breeden and Viswanathan (1990) and DeMarzo and Duffie (1992) offer a different perspective basedon information asymmetry. They argue that managers may want to influence how they are perceivedby the labor market by taking hedging decisions that reduce the variance of the firms value.

Smith and Stulz (1985) have put forth taxes, costs of financial distress and debt capacity asreasons to vertically integrate. If taxes are a convex function of earnings, a more volatile earningsstream leads to higher expected taxes. If there are costs of financial distress and there is an advantageto having debt, then hedging can be used as a means to increase debt capacity because it reduces thefirms probability of default. In the same spirit, Myers (1977) and Stulz (1990) argue that hedgingcan create value by reducing investment distortions which debt finance sometimes induces.

Close to these debt overhang explanations is the paper by Froot, Scharsfstein and Stein (1993)which we use for our purpose. Their model features a single firm faced with an internal wealthhedging decision and an investment to undertake. The rationale for hedging internal wealth is thatit reduces the variability of external funds to be raised. Reducing the variability of external fundsmakes it possible to increase the firms expected profit provided the cost of external funding isconvex and the production function is concave.

In line with empirical evidence which shows that firms hedge in order to coordinate their in-vestment opportunities with their investment capacities, Geczy, Minton and Schrand (1997), showthat firms with greater investment opportunities and tighter financial constraints are more likely tohedge. Empirical results of Brown (2001) and Nain (2004) support the fact that the hedging policyof a firm depends on its competitors hedging policies. Indeed, a competitors hedging policy affectsits investment opportunities, thus its behavior in the product market, therefore the firms investmentopportunities and ultimately its hedging policy. Loss (2002) studies how the interaction betweenfirms, i.e. the fact that investments are either strategic substitutes or strategic complements, affecttheir hedging policies in a setting of imperfect competition and financial constraint. In equilibrium,when investments are strategic substitutes5 (complements), the level of hedging increases (decreases)with the correlation coefficient between shocks that affect firms internal funds.

Bessembinder and Lemmons model (2002) offers a radically different perspective on hedging.The model features a finite set of producers and retailers in the electricity industry. Hedging is donethrough forward contracts between producers and retailers who also have access to a spot market.The model derives the equilibrium forward price (as electricity cannot be stored, the traditional noarbitrage cost of carry approach is not valid) and the resulting optimal forward positions.

Vertical integration in the economic literatureUntil the mid 1970s, economic thinking on vertical integration was informal. Malmgren (1961)

applied Coases theory of the firm (1937) to argue that vertical integration could reduce a firmstransaction costs. Activities which tended to fluctuate, causing fluctuations in prices and outputs inthe market, could be integrated and balanced against one another . When discussing the reasons forthe formation of the largest companies in the US, Chandler (1969) argues that the initial motivesfor expansion or combination and vertical integration had not been specifically to lower unit costs

5When investments are strategic substitutes (complements), a marginal increase in a firms strategic variabledecreases (increases) a competitors marginal profit.

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2 CORPORATE RISK MANAGEMENT IN THE LITERATURE

or to assure a large output per worker by efficient administration of the enlarged resources of theenterprise. The strategy of expansion had come...from the desire...to have a more certain supplyof stocks, raw materials and other supplies.... In the same spirit, Porter (1980) argues that itis to assure their supply of inputs that firms decide to integrate upstream - this has become thetraditional business explanation.

Green (1974) and Arrow (1975) are considered to be the first economists to have analyticallyinvestigated the effect of uncertainty on the incentives for vertical integration. Arrow analyzed amodel where vertically integrated firms obtain information on the input supplies conditions earlierthan non-integrated firms. This information advantage creates an incentive to fully vertically in-tegrate. Green and Carlton (1979) consider that the inputs price is fixed while demand for it isstochastic. Therefore, rationing can occur and to avoid it, firms integrate to some extent.

Bolton and Whinstons model (1993) features a single firm producing an input used by twodownstream firms. Upstream firms have a random capacity, possibly insufficient to meet down-stream firms demand, resulting in ex post bargaining to transfer the input (incomplete contract-ing). Downstream firms do not appropriate the entire surplus as the upstream firm gets part of itresulting in ex ante inefficiently low investments. Inefficiently low investments create an incentiveto vertically integrate.

Emons model (1996) features a finite set of upstream and downstream firms whose input require-ment is stochastic. Vertical integration is modeled through a downstream firms input productioncapacity, which leaves space for partial vertical integration. The input is transferred downstreamthrough a spot market organized by an auctioneer who chooses between two prices depending onwhether downstream firms demand is larger than upstream firms capacity or not. The main resultis that full integration is an equilibrium market structure: downstream firms will always want tointegrate to some extent in order to cut down on aggregate demand and depress prices.

DiscussionIn the following sections we develop two models which are extensions of two existing risk manage-

ment models: the Froot, Scharstein and Steins model (1993)6 and the Bessembinder and Lemmonsmodel (2002). In these models firms use financial instruments while they do not in the verticalintegration literature. What literature on vertical integration shows however is that a model dealingwith hedging and vertical integration should ideally include an industrial organization dimension aswell.

In FSSs hedging model future wealth is a weighted average of a random component (which

depends on a risk factor) and a non-random one whose proportions are fixed in advance. In theirsetup, a firm hedges to maximize the return on its investment. This return depends on a productionfunction which takes into account the correlation between investment opportunities and the riskfactor. In the authors setup, there is only one firm. We decided to extend the model by addinganother firm in order to have an upstream firm and a downstream one. However, literature onvertical integration suggests that modeling vertical integration should involve a set of upstreamand downstream firms. The reason is that vertical integration affects supply and demand of theintermediary good, therefore its price.This price effect of vertical integration explains why in Emonsmodel downstream firms always choose to integrate vertically to some extent.

Bessembinder and Lemmons model (2002) has this industrial organization dimension we think

is necessary. In their framwork, firms hedge not only to maximize their profit from production,but also because hedging can be a source of revenue in itself. Contrary to FSSs setup where thefirm has no utility function (the increasing cost of marginal external funds is a way to include riskaversion directly in the profit function), in Bessembinder and Lemmons setup market players have a

6From now on, we call FSS to Froot, Scharstein and Stein

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3 A FIRST PARADIGM OF THE BENEFITS TO HEDGE

mean-variance utility function. The model derives the equilibrium forward price of the intermediarygood. Therefore, when market players are not risk averse, they can speculate on the differencebetween the forward price and the expected spot price by entering into forward contracts.

In the following section we start with FSSs basic model and extend it to deal with vertical

integration. We then present and extend Bessembinder and Lemmons model.

3 A first paradigm of the benefits to hedgeIn this section we present several important results of FSS (1993) that we will use later on for ourapplication on vertical integration. We consider a firm facing an investment and financing decisionin a three-period model (we called them period zero, one and two), where we suppose the existenceof a forward market and a costly credit market.

3.1 The model setup

The firm enters period one with internal wealth w. In period one, the firm chooses its investmentI and raises the difference e externally, therefore:

I = w + e (1)

For the sake of simplicity we assume that the interest rate is zero, that is, that outside investorsrequire an expected repayment of e in the second period. However, we assume there are deadweightcosts C(e) associated with external financing. These costs could arise from direct or indirect costs offinancial distress, informational asymmetries, transaction costs or agency issues between managersand outside investors. For further details see Myers and Majluf (1984), Jensen and Meckling (1976),and Myers (1977). We assume that not only these costs are an increasing function of the amountraised externally, but also that these costs increase at the margin, that is, Ce > 0 and Cee > 0.By f(I) we denote the level of output resulting from investing I and assume that the productiontechnology is such that fI > 0 and fII < 0, that is, that there are technological decreasing returnsto scale. By F (I) we denote the net value of the investment7

F (I) = f(I) I (2)In period one, The firm chooses its investment level in order to maximize its profit (w):

(w) = maxIF (I) C(e) (3)The optimal investment level Isatisfies:

fI(I) = 1 + Ce(e) (4)

In period two there are no optimization problems. It is only a period where output from investmentis realized and external investors are repaid.

External financing induces therefore an optimal investment level below the first-best level reachedin the absence of external financing (fI(I) = 1). In this sense, external financing results in under-investment. Using the envelope theorem on (3) and the implicit function theorem on (4), the secondderivative of profits is given by:

ww(w) = (dI

dw)2fII(I

) (dI

dw 1)2Cee(e) (5)

7For the sake of simplicity, we denote F the net value investment function and f the production technology

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As fII < 0 and Cee > 0, ww < 0. Besides, Pw > 0 because w(w) = Ce(e). Thus, the profitfunction is increasing and concave.

The issue of hedging arises when wealth w is random (uncertain) at period one (until now, it was

considered to be given). To understand why full hedging of wealth raises the net expected profitof the firm, lets imagine that w is a random variable with possible states of the world {w1; ...;wn}and associated probabilities {p1; ...; pn} such that

pi = 1. Since is concave:

(

piwi) >

pi(wi) (6)

This last equation clearly shows that before period one occurs, the firm will prefer to hedge fullyto ensure wealth w =

piwi at period one, rather than leave this wealth random.

Figure 1: Raise in value through hedging when the profit function is concave

Proposition n1: full hedging increases the net expected profit of a firm which net value invest-ment function is concave and external financing cost function convex.

3.2 Stochastic investment function

In this part, introduce a linear hedging decision in period zero (forward contracts in practice).We add a period, the period zero, because the firm makes now a hedging decision before raisingexternal funds and investing in period one. Let h denote the hedge ratio chosen by the firm inperiod zero, h [0; 1]. The wealth of the firm in period 1 is therefore:

w = w0(h+ (1 h)) (7)

where is a normally distributed random variable representing the source of uncertainty withmean 1 and variance . For an oil company, could be a shock on the price of oil for instance.

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Figure 2: Timeline in the three-period model

To introduce changing investment opportunities, which is far more realistic, we rewrite the netvalue investment function as follows:

F (I) = f(I) I (8)where = ()+1 and is a measure of the correlation between investment opportunities and

the risk to be hedged. For instance, if stands for a shock on the oil price, a positive means thatthe higher the oil price the more investment opportunities the company has. Here the returns oninvestment are related to the same random variable affecting the value of the assets. This hypothesisis a strong one and deserves some explanations. The variable is a random variable that depends onthe result of the risky variable. It can symbolize a shock to a given production function or in otherwords, a measure of the randomness of investment opportunities. In their article, Froot, Scharfsteinand Stein (1993) define as a variable measuring (or related to) the investment opportunities ofthe firm. They also consider as the variable intermediary price of a product minus a marginalcost.8

3.3 Solving the model by a recursive methodWe solve the model with a recursive method. In period 0 the firm chooses h to maximize expectedprofits E[(w)] where (w) = maxIF (I) C(e), which is the profit earned at period 1. This canbe rewritten as:

maxhE[maxIF (I) C(e)] (9)That is why we say we solve the model by a recursive method. We solve first for t=1 and then

for t=0. Thus for t=1 the optimal level of investment is given by :

ifI 1 = Ce (10)At period 0, the first order condition of this problem is:

E[wdw

dh] = 0 (11)

Using the expression of w, this equation simplifies:

E[w(1 )] = 0 (12)This can be rewritten as:

cov(w, ) = 0 (13)

But on the other hand, using the Rubinsteins rule (1976), we get :8Yet, it is quite unrealistic to think that a the price of a product sold in period 2 can be known one period before.

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cov [, ] = E []E(

)cov [, ] (14)

Then we get :

E [] = 0 (15)

Using the envelope theorem and the implicit function theorem it can be shown that :

= 0(1 h) + fI I

(16)

then taking the expected value we get :

E

[0(1 h) + fI I

]= 0 (17)

Solving for h we get :

h = 1 +

0

E(fII )

E()(18)

and using the envelope theorem we can show that I

= /fII , then

h = 1 + E[fI/fII ]

0E[](19)

where the expressions inside the expected value are evaluated at I = I*().InterpretationsFirst, this expression corroborates the optimal hedge ratio of one (i.e. full hedging) found

previously in the case of a non stochastic investment function (h = 1 when = 0).Second, if investment opportunities and the risk variable are positively correlated ( > 0), the

firm will not hedge fully. Recall that fI > 0, fII < 0 and ww < 0. To understand why, imaginethe firm is an oil company with representing the source of uncertainty arising from volatile oilprices. If is low, the oil firm might be worse-off in terms of internal funds. Nevertheless, becauseinvestment opportunities have deteriorated due to lower oil prices (for instance, investing in someoil fields where extraction costs are relatively high would be not profitable because of low oil prices),the firm can accept this situation at least to some extent, i.e. it does not hedge fully (h [0; 1[).

In fact, if investment opportunities are extremely sensitive to the risk variable , it could be thath < 0, i.e. the firm increases its exposure to the risk variable procyclically. In the oil firm example,this would be the case if high oil prices are associated with so large investment opportunities thatthe sole increase in oil prices would not be sufficient to generate enough internal funds to financethese investments.

Third, if investment opportunities and the risk variable are negatively correlated ( < 0), it couldbe that h > 1, ie: the firm increases its exposure to the risk variable countercyclically. This wouldbe the case if low values of the risk variable are associated with so large investment opportunities(think about low prices pushing some competitors into bankruptcy) that the sole hedging of the riskvariable would not be sufficient to secure enough internal funds to finance these investments.

Proposition n2: a firm with a stochastic and concave investment function will hedge partially

(h [0; 1[) if investment opportunities are positively correlated with the risk variable.Proposition n3: a firm with a stochastic and concave investment function will underhedge

(h < 0), if investment opportunities are strongly positively correlated with the risk variable.

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Proposition n4: a firm with a stochastic and concave investment function will overhedge(h > 1) if investment opportunities are negatively correlated with the risk variable.

To illustrate these propositions, we follow the work by Spano (2001) and evaluate the model using

Cobb-Douglas production and cost functions. The analytical solution is calculated as a second-orderlocal approximation around the expected level of investment and external financing9. With this, wecan plot the optimal hedging as a function of . This is given by :

Figure 3: Optimal hedge ratio as a function of using Cobb-Douglas functions for the cost andproduction.

The intuition concerning proposition 2, 3 and 4 is that the optimal hedging is decreasing withthe correlation. Using the expression of the optimal hedge ratio in (19) it is not possible to concludeabout the sign of the derivative. We must define the form for the cost and the production functionand then solve the model. We can do this using Cobb-Douglas functions for the cost and theproduction. To summarize the previous results in the case of Cobb-Douglas functions, we can say:

Proposition n5: The optimal hedge ratio of a firm with a stochastic and concave investment

function is decreasing in for a small volatility.Proof : The intuition comes from proposition 2, 3 and 4. The idea is that h

E[fI/fII ]0E[] 0 and decreasing if < 0.9See appendix for a complete list of equations

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4 INTERACTION BETWEEN VERTICAL INTEGRATION AND HEDGING

4 Interaction between vertical integration and hedging

This section describes an application of the afore-presented model. We extend the use of theFSSs model (1993) to vertical integration by considering two firms facing a three-period investmentdecision. We consider an upsteam firm (producer) and a downstream firm (distributor) in an certainindustry. We use the same notations as before and again the firms have to manage their risk usingshort-term hedging. We compare the different optimal hedgings in two cases: with and withoutvertical integration. For the case where firms are separated (we call that the non-integrated case),the optimal hedge ratio for the two firms is measured as the weighted average of the optimal hedgeratios of each firms:

hU+D = hU + (1 )hD (20)where [0; 1] is the coefficient on the weighted average10. This coefficient can represent a sort

of relative volume effect in the forward market. In the integrated case, we call hV the optimalhedge ratio when the two firms merges. Our purpose is to see under which conditions verticalintegration reduces the need of short-term hedging (i.e. when hU+D > hV ).

Here, we consider only one source of uncertainty for the two firms, given by . As we said, this

risk can be interpreted in several ways. For example, as a shock on a intermediary price of the twomarkets (for instance, the price of a commodity)11. Also, we could think the intermediary priceas a random variable which is determined by a random negotiation between the two firms. Thisintermediary price fluctuates between a constant marginal cost of an upstream firm and a constantfinal price of the downstream market.

Either the interpretation we consider, our model stays the same. We would like to summarizethe idea of the model built in the following figure

Figure 4: Vertical Integration in a upstream/downstream market

4.1 The non-integrated case

In this subsection we present the non-integrated case. As before, each firm (upstream anddownstream) faces a three-period investment decision (period zero, one and two). For the sake ofsimplicity, we assume that the production function f is the same for both firms. The initial wealthsare noted i0.

10We have chosen an arbitrary coefficient in the weighted average. Nevertheless, a simple case is to analyze = 12

, i.e, the case of a mean.11Nevertheless, the reader must understand that this is only one possible interpretation of the model. The idea of

investment opportunities can have several interpretations.

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We follow a recursive route, which corresponds to the resolution of the model. The little Ustands for Upstream and the little D stands for Downstream. We have i = D,P . Again, in period0 the firm chooses its optimal hedging strategy given by hi in order to maximize the expected profits:

MaxhiE[i(i)]

i(i)

= maxIiFi(Ii) C(ei)

Ii = i + ei

wi = wi0(hi + (1 hi))

(21)

And the investment function are given by :{FU (IU ) = Uf(IU ) IU

U=( ) + 1 (22){FD(ID) = Df(ID) ID

D = ( ) + 1 (23)where is the risk variable which we assume to be normally distributed with mean and a

variance of 2. The same risky shock affects both firms. Here and are a measure of thecorrelation between the investment opportunities and the risky variable.

The random shock

We assume = and > 0. In fact, following the interpretations given in section 2, the invest-ment opportunities of the upstream firm and the downstream firm are correlated in the oppositeway with respect to the random variable. We can see U as the difference between the intermediaryprice and a constant marginal cost and D as the constant final price minus the intermediary price.Then, if we consider the intermediary price to be the only source of uncertainty (like the price ofoil for example), then we would have = and > 0 (an upstream firm positively correlatedwith the intermediary price).

4.2 The integrated case

In the integrated case, the firm has to solve MaxhV E[(V)]

(V)

= maxIV F (IV ) C(eV )

wV = (wU0 + wD0 )(h

V + (1 hV ))(24)

and the investment function is given by

FV (IV ) = V f(IV ) IV (25)Since we have = then V = 12D + 12U = 112the integrated firm faces a non-stochastic

investment function. We consider its initial wealth to be the sum of initial wealth of the downstreamand upstream firms.

12We take a weighted average between D and U . For the sake of simplicity, we give the same weight to everyshock since final results are not influenced

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4.3 Comparison of optimal levels of hedging in the two casesIn this subsection we use the results given in 3.1 and 3.3 to deduce the following propositionsconcerning vertical integration and hedging. As we already said, we focus on comparing the weightedaverage of hedging in a non-vertical integration situation hU+D = hU + (1 )hD to the levelof hedging of an integrated company hV .

Proposition n7: Since an integrated firm faces a non stochastic investment function, it will

hedge fully, i.e. hV = 1.Proof : Using proposition n1, the result is trivial.Proposition n8: In a non integrated market, the weighted average of optimal hedging will

depend on the interaction between anticipated optimal investments, on initial wealths and the weightcoefficient. In particular, we will have :

hU+D = 1 + [E(fIP/

UfII)PU0

(1 )E(fIP/DfII)

PD0

](26)

Proof : Using the results of section 3.3, we add the optimal hedging ratios for every firm.Proposition n9 In a non integrated market, since investment opportunities of the downstream

firm are negatively correlated with the risk variable (i.e. = is negative), the downstream firmwill overhedge (h > 1).

Proof : We set = and > 0 then is negative and using proposition n4, the result is

trivial.Proposition n10: In a non integrated market, if investment opportunities of the upstream

firm are not strongly positively correlated with the risk variable, then the upstream firm will hedgepartially (h[0; 1[).

Proof : We set > 0 then using proposition n2, the result is trivial.Thus we want to compare a weighted average of optimal hedge ratios (one larger than 1, one

smaller than 1) in the non-integrated case hU+D to hV = 1. As we said in proposition 8, severalelements affect the weighted average. For a theoretical proof of the influence of these elements,we need to rely on a specific form for the production and cost functions. For this, as before, wefollow the work by Spano(2001) and use Cobb-Douglas functions. The proofs are given on appendix.However, we give an idea of every effect.

Proposition n11: If investment opportunities of the upstream firm are not strongly positively

correlated with the risk variable and is sufficiently small, the weighted average of the hedge ratiosof non integrated firms will be higher than that of the integrated one and thus vertical integrationwill reduce the need for hedging :

hU+D > hV (27)

Proof: Using proposition 2, we get that hU [0; 1[ and we know that hD > 1 because of

proposition 9. Then, if the downstream firm has a bigger weight in the forward market than theupstream firm, we get the result.

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Proposition n12: Even if investment opportunities are strongly positively correlated with the

risk variable, for a given , the weighted average of the hedge ratios of non integrated firms will behigher than that of the integrated one if the upstream firm has a sufficiently larger initial wealth thanthe downstream firm (U0 D0 ).

Proof: Using proposition 3 and 6 we get the result.InterpretationsWe can see that there are three key elements in the analysis of corporate risk management

through vertical integration and hedging: initial wealths, correlations between investment opportu-nities and the risk variable, the volatility of the risk variable and the volume effect (weight in theforward market). In the following paragraph, we describes this effects.

Initial wealth effectThe initial wealth (which is a constant in the model) has an prominent role in the comparison

between the weighted average of hedgings and the vertical integrated hedge ratio. One condition tohave a beneficial vertical integration (i.e., a smaller vertical integrated hedging) is that the upstreamfirm has to have a sufficiently larger initial wealth than the downstream firm. In fact, a largerinitial wealth will provoke a decrease in the influence of investment opportunities on hedging (i.e.E[fI/fII ]0E[]

decreases with an increase in 0). Then, downstream firms with small intial wealthwill tend to prefer vertical integration as a way to manage risk. Using Cobb-Douglas functions, wesolve the model and plot the hedge ratios as a function of 13. We can see the effects of initialwealth on the optimal hedge ratio (figure 5). If we increase the initial wealth of the upstream firmwith respect to the downstream firm, the weighted average goes up.

Figure 5: Initial wealth effect : Optimal hedge ratios (upstream in red, downstream in blue andweighted average in green) as a function of . Left: Same initial wealth. Right: The initial wealthof the upstream firm is 4 times larger than the downstream firm one (U0 D0 ). The weightedaverage of separated firms is larger than 1 (i.e.hU+D > hV ).

Correlation and volatility effectThe correlation between investment opportunities and the risk to be hedged is fundamental for

the comparison of hedging and vertical integration. A good measure of this correlation is given133 curves: each firm separated and the weighted average. The optimal hedge ratio in the integrated case is constant

and equals to 1 (proposition 7)

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5 EXTENSIONS

by (recall that U=( ) + 1). First, as the integrated firm has no correlation with the riskvariable, this will mean that V does not depend on the risk variable. In other words, the volatilityof the risk to be hedged has no impact on investment opportunities of the integrated firm. Thus,as wV = wV0 (hV + (1 hV )), the integrated firm will hedge totally to avoid completely the riskvariable.

Secondly, the larger is , the smaller will be hU. In other words, the optimal hedge ratio hUis a decreasing function of . The sensitivity of the correlation effect will depend mainly on thevolatility 2 of the risky variable. The higher is the variance of the risky variable, the higher is thesensitivity of the hedging ratio to the parameter 14. This can be demonstrated in the particularcase of Cobb-Douglas functions for the cost and the production. We illustrate this in figure

Figure 6: Volatility and the correlation effect: Optimal hedge ratios (upstream in red, downstreamin blue and weighted average in green) as a function of . At left : 2 = 3. At right : 2 = 6.With a larger volatility the optimal hedge ratio h becomes more sensitive to . (Other parameters:richer downstream firm, upstream firm with bigger weight = 0.6)

The effectRecall that we took an arbitrary and exogenous coefficient for the weighted average of hedging.

This coefficient measures the relative importance of every firm in the forward market. It couldillustrate several elements : a volume effect, a financial wealth effect. It has a strong impact onthe comparison of the hedging levels as we can see in figure 14 in appendix. A lower will giverelatively importance to the downstream firm and hD > 1. Thus, in that case, it will be better toengage in vertical integration.

The interaction between the correlation, the wealth effect and the effect will determine if ver-

tical integration is beneficial from a risk management perspective, compared to short-term hedging.

5 Extensions

5.1 An endogenous weighted average

An interesting case to analyze is when firms are symmetric and we set = U0

U0 +D0

i.e. we calculatethe weighted average taking as coefficient the proportion of initial wealth. This measures definitelya wealth effect. Again, we plot the results in appendix in figure 15. We see that the optimal hedgingratio of an integrated firm is, for every , larger than the endogenous weighted average of non

14See appendix for a proof.

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5 EXTENSIONS

integrated hedgings (i.e. hU+D < hV ). This result has a prominent interpretation: hedging is inthis case a better risk management approach in a short-term perspective. This follows the line ofconventional wisdom, as we saw in the introduction.

5.2 A model with several actors

There is not much research around the extension of FSS to a framework with an outside market.Loss (2012) studies how the interactions between firms affect their hedging strategies. In his model,all the firms hedge their risk and so optimal hedgings are influenced by the interaction with thecompetitors level of hedging. Here we take a different route and suppose the existence of otherfirms in order to justify that our integrated firm continues to sell and buy from other firms in thedownstream and upstream market.

In the previous section we built a model with one upstream firm and one downstream firm thatvertically integrate. The optimal hedge ratio of a integrated firm is full hedging since a situationwith full hedging is equivalent to a situation where the firm knows exactly his future wealth. Thisis true in an integrated firm with no competitors since there is no risk : the intermediary pricehas been completely covered. Nevertheless, this is not the case with more actors.

In the following, we consider the same model as before and we add one upstream firm and onedownstream firm, which are not vertically integrated. Hence, the model is as following :

Figure 7: Vertical Integration in a upstream/downstream market with more competitors

We consider x (resp. 1x) to be the proportion of demand bought by the integrated downstreamfirm to the integrated upstream firm (resp. to the other downstream firm). In the same way, weconsider y (resp. 1 y) to be the proportion of supply sold by the integrated upstream firm to theintegrated downstream firm (resp. to the other upstream firm).

Here, the random shock V will be given by two consecutive linear combinations between invest-

ment opportunities (depending on the sign of and on the proportions sold):

V =1

2

[xVNonstochastic + (1 x)Vstochastic,

]+

1

2

[yVNonstochastic + (1 y)Vstochastic,

](28)

We get:

V =(x y)

2( ) + 1 (29)

Thus, the optimal hedge ratio of the integrated firm will be given by :

hV = 1 + (x y)E[fI/fII ]20E[]

(30)

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5 EXTENSIONS

Proposition n13 : An integrated firm will reduce its level of hedging with respect to the weightedaverage of non-integrated hedgings by selling to the competitor downstream firm and avoiding buyingfrom the competitor upstream firm (vertical foreclosure).

ProofRecall that E[fI/fII ]0E[] < 0. The optimal hedge ratio of the integrated firm depends on the

interaction between the proportions sold and bought by the downstream firm and upstream firm.Recall that our objective is to prove a decrease in hV with respect to the weighted average of nonintegrated hedgings under certain conditions. Thus, an integrated firm will reduce its optimal hedgeratio by selling to the other downstream firm (i.e. increasing 1y thus increasing y) and avoidingbuying from the other upstream firm (i.e. decreasing 1 x, thus increasing x).

InterpretationsAn integrated firm will reduce its level of optimal hedging by engaging a vertical foreclosure.

This strategy consists in selling the majority of the production in the vertical integrated system andthus avoiding the entrance of competitive upstream firms. The firm engages in an abusive use ofits power market, considered an anti-competitive strategy. Also, if the firm buys and sells the sameproportion outside the vertical system (i.e. x = y), then it will engage in an operational hedgingthat will cause a full hedging strategy.

5.3 Intertemporal issues: the influence of long-term

In this subsection, we assume = , > 0 and the same hypothesis as before. We do not addan outside market. The issue we analyze now is whether vertical integration reduces the need forhedging in a long-term perspective. For that purpose, we extend the three-period model by addingmore periods: Each firm will perform two hedging decisions in two different periods, to protect itswealth from two independent risk variables. We will give special importance to the initial wealtheffect, which is the link between the two hedging periods. As a result, we get a five-period model(period zero to four), which is summarized in the following figure:

Figure 8: Five-period model timeline: the problem of each firm

To summarize, this model is equivalent to two of the previous models, one after another. Thelink between the two periods of hedgings is initial wealth (since initial wealth at a period t is theprofit made by the firm at t 1). We wish to compare the optimal hedge ratios and we give specialimportance to the initial wealth effect. We illustrate the new comparison in the following figure:

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5 EXTENSIONS

Figure 9: Comparison of optimal hedge ratios in the two cases: Integrated and non-integrated

At a given point in time, we know that the optimal hedge ration depends on the firms wealthand the optimal investment level. Therefore, h2 depends on I3 and 1 :

h2 = 1 + E[f(I3 )Pww(I

3 )/fII(I

3 )])

1E[Pww(I3 )](31)

with I3 such that:

fI(I3 ) = 1 + Ce(e

3) (32)

and 1 such that:

1 = F (I1 ) C(e1) (33)

Adjusting those results to the upward and downstream firms yields:

hU2 = 1 + E[fI(I

U3 )Pww(I

U3 )/

UfII(IU3 )]

[F (IU1 ) C(eU1 )]E[Pww(IU3 )](34)

hD2 = 1 E[fI(I

D3 )Pww(I

D3 )/

DfII(ID3 )]

[F (ID1 ) C(eD1 )]E[Pww(ID3 )](35)

where > 0 is a measure of the (positive) correlation between investment opportunities ofthe upstream firm and the factor risk . We see that the second optimal hedging depends onthe profits made previously period (the expression F (Ii1 ) C(ei1 ) in the denominator). Recallthat E[fI(I

i3 )Pww(I

i3 )/

ifII(Ii3 )]

E[Pww(Ii3 )]< 0 therefore the sign of hi2 1 will only depend on the sign of

i1 = F (Ii1 ) C(ei1 ), i = U,D.

We may think of the sign of i1 as the ability of a firm to make profitable investments (investmentswhich output covers the cost of the investment and the cost associated with external funding).Note that in this setup a not profitable firm will have debt. Again, the hedge ratio hU+D2 of thenon-integrated firms is the average of their hedge ratios weighted by an arbitrary coefficient ( i.e.hU2 + (1 )hD2 ). Besides, the investment function of the integrated firm being non stochasticwe get hV2 = 1.

Then, we analyze under which conditions hU+D2 < 1 and h

U+D2 > 1. It is important to

understand how the optimal hedging ratio varies as a function of the sign of the initial wealth andthe correlation to the risky variable. We summarize this is the following table:

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5 EXTENSIONS

initial wealth before hedging>0 initial wealth before hedging0 (upstream firm) h < 1 h > 1

corr 1 h < 1

Table 1: Optimal hedging ratio variation and the sign of initial wealth and correlation

Thus we get 4 cases depending on the sign of the initial wealth before hedging i.e. i1 =

F (Ii1 )C(ei1 ), i = U,D. To find the conditions under which hU+D2 > 1, we rely on the propositionsof the previous sections.

Downstream firm\Upstream firm U1 > 0 U1 < 0D1 > 0 h

U+D2 > 1 if

U1 D1 hU+D2 > 1

D1 < 0 hU+D2 < 1. h

U+D2 > 1 if

D1 U1 Table 2: Optimal hedging ratio depending on the sign of previous profits

Case n1: U1 > 0 and D1 > 0. Then hU2 < 1, hD2 > 1, and so h

U+D2 > 1 if

U1 D1

Case n2: U1 > 0 and D1 < 0.Then hU2 < 1 and hD2 < 1, and so hU+D2 < 1.

Case n3: U1 < 0 and D1 > 0. Then hU2 > 1 and hD2 > 1, and so hU+D2 > 1.

Case n4: U1 < 0 and D1 < 0. Then hU2 > 1 and hD2 < 1, and so hU+D2 > 1 if

D1 U1 Hence we summarize the case 1, 3 and 4 in the following proposition.Proposition n14: vertical integration reduces the need for hedging in the long-term in the

following three cases:

the upstream firm is not profitable in the first period while the downstream firm is;

both firms are not profitable in the first period and the downtream firms debt is high enough;

both firms are profitable and the upstream firm is wealthier enough than the downstream firm.

Symetrically:Proposition n15: vertical integration increases the need for hedging in the long-term in the

following three cases:

the upstream firm is profitable in the first period, the downstream firm is not;

both firms are not profitable and the upstream firms debt is high enough;

both firms are profitable and the downstream firm is wealthier enough than the upstream firm.

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6 ILLUSTRATION OF THE MODEL

Intuitions15

The upstream firm is not profitable while the downstream firm is. The not profitableupstream firm (whose investment opportunities are positively correlated with the risk factor) over-hedges its debt (negative wealth) because it wants to make its level sensitive to investment oppor-tunities in such a way that debt decreases (resp. increase) when investment opportunities increase(resp. decrease). The profitable downstream firm (whose investment opportunities are negativelycorrelated with the risk factor) overhedges its wealth in order to increase it when investment oppor-tunities increase. Recall that the downstream firm overhedges because its investment opportunitiesare negatively correlated with the risk factor and that overhedging makes the downstream firmslevel of wealth countercyclical with the risk factor. Therefore, two such firms will have a weightedaverage optimal hedge ratio greater than one, that is, greater than the optimal hedge ratio ofvertically integrated firms.

Both firms are not profitable and the downstream firms debt is high enough. Now,if the downstream firm is not profitable as well (i.e: it has debt), it will either hedge partially orunderhedge depending on how strongly (negatively) correlated are its investment opportunities withthe risk factor. If the downstream firms debt is high enough, the overhedging of the upstream firmwill outweigh downstream firms partial hedging or underhedging, making therefore the optimalhedge ratio of the vertically integrated firm lower than the weighted average of optimal hedge ratiosof separate firms.

both firms are profitable and the upstream firm is wealthier enough than the down-stream firm. Finally, if both firms are profitable (i.e.: they have some wealth), the upstream firmwill hedge partially or underhedge because its investment opportunities are positively correlatedwith the risk factor, while the downstream firm will overhedge because it investment opportunitiesare negatively correlated with the risk factor. If upstream firms wealth is high enough (high enoughgiven the difference in the optimal hedge ratios and the difference in wealth), the optimal hedgeratio of the vertically integrated firm is lower than the weighted average of optimal hedge ratios ofseparate firms.

6 Illustration of the model

6.1 Testing the sign of and the hedging ratio in an oil companyIn this section we try to test some of our results with data taken from an oil company. We examinethe sign of of a downstream oil firm and the hedge ratio of an integrated oil firm. Recall thatwe have assumed that was negative for a downstream firm (and positive for an upstream firm);we have shown that the optimal hedge ratio for a vertically integrated firm was 100%. We test thismodel assumption and this proposition.

The sign of of a downstream firm: the PKN Orlen caseThe parameter is a measure of the correlation between the risk factor and investment oppor-

tunities. The risk factor we consider here is oil price because this is the risk factor which accountsfor vertical integration.

We test the sign of in the case of a specific downstream firm, PKN Orlen, a major16 Europeanoil refiner and petrol retailer. This company perfectly fits such an analysis as it is not verticallyintegrated and data is easily available.

15We interpret only proposition n14 since the n15 is the symetric case.16PKN Orlen is Central Europes largest publicly traded firm

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As a proxy for a companys investment opportunities, we use Tobins Q, i.e., market value17of assets over their book value. Indeed, the higher Tobins Q, the higher the market values assetsin place, that is, the more it expects assets in place to generate cash-flows, that is, the more itexpects good investment opportunities. The risk factor we consider is oil price. We use Brent oil asa benchmark for oil price.

In order to check how PKN Orlens Tobins Q varies when the oil price varies, we run a standardlinear regression analysis where ln(Q) is the explained variable and ln(Oil) is the explanatoryvariable using data ranging from Q4 2004 to Q4 2011 (i.e: 29 periods):

ln(Qi) = + ln(Oili) + i (36)

We obtain the following estimates of (, ) and the following t-statistics (t(), t()):

= 0, 42 = 1, 29t() = 3, 05t() = 2, 18

(37)

We perform a test to check the sign of . The one sided pvalue of t() is 0, 25%. Thus, we canreject the hypothesis that is positive with a probability of 99, 75%. The slope of this regression istherefore very statistically significant.

This regression suggests that when the price of oil increase by 1%, investment opportunities areexpected to decrease by 0, 4%. Our intuitive assumption about the sign of is therefore justified.The data we took for our regression is given in the appendix. We show the regression in the followingfigure :

Figure 10: Tobins Q as a function of oil price for PKN Orlen

17market value of assets = market capitalization + market value of debt 'market capitalization + book value of

net debt

21/41


The hedge ratio of an integrated oil firm: the OMV caseContrary to PKN Orlen, OMV18 is an integrated oil (and gas) firm. Our model predicts there-

fore full hedging of oil production. To test this prediction, we simply checked what proportion ofproduced oil had been hedged by OMV in the past few years (results are presented in the tablebelow).

The good news is that OMV has been fully hedging its oil production for the past three years.The firms rationale for hedging, as the annual reports states it, is to secure cash-flows to financeinvestments and to maintain an investment grade credit rating. Obviously, as in our model, theinvestment aspect of hedging is taken into consideration by the company, as well as the cost ofexternal funds aspect (a credit rating usually determines the cost of external funds). A worse newsis that OMV did not hedge oil production at all in 2007, arguing that cash-flows were strongenough. This is in contradiction with our model which stipulates that full hedging should alwaysoccur (for an integrated firm), whatever the wealth of the company.

Oil Production (Q) Oil Hedginginstruments secured price hedge ratio2011 50000m bbl swaps $97.07/bbl 100%2010 63000m bbl puts $54.20/bbl 100%2009 65000m bbl puts spreads $80/bbl or $95/bbl 100%2008 60900m bbl puts NC 31%2007 59800m bbl no hedging concluded no hedging concluded 0%

Table 3: Oil hedging at OMV. Source : OMV Annual Reports.

As assumed in the model, the sign of was found to be negative in the case of an oil distributor,PKN Orlen. As predicted by our model, the hedging ratio was found to be equal to one in the caseof an integrated oil company, OMV.

6.2 Delta Airlines: an example of vertical integration

Delta Airlines has recently bought for $180M, an oil refinery from Phillips 66 located in Penn-sylvania, USA. Delta bought the refinery in order to reduce the airlines fuel costs. Indeed, 80%of Deltas US fuel needs should be provided by such purchase. If we do the analogy with ourfirst model, the oil refinery represents the upstream firm whereas Delta Airlines is the downstreamfirm. Delta Airlines is a large airline company (Market capitalization of 8.5 billions dollars, earningsgreater than 500 million dollars. . . ) and the refinery a small one (the refinerys purchasing cost isnearly equal to a jumbo jets purchasing cost). Thus, we could say this example corresponds to thecase where the downstream firms volume is much greater than the upstream firms volume. If wetake into consideration the volume effect (given by ), our model predicts that vertical integrationis better in this case.

The aim of the vertical integration is to be protected against oils price volatility. By doing this,the vertical integration allows to guarantee a fixed level of income. Furthermore, Delta Airlines wantsto spend another $ 100M to convert the refinerys operations to maximize jet fuel production. Thisnew spending can be seen as an investment related to good investment opportunities. Therefore,if we follow our model, the vertical integration is totally logical and obvious (here, both wealthand correlation effects lead to concentration) as we bear in mind that the rationale for hedgingin FSSs paper is to protect the firm against external financing assumed to be more costly than

18OMV is Austrias largest listed industrial company in terms of turnover and one of the largest integrated oil andgas groups in Central Europe.

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7 COMMENTS, CRITICS

internal generated funds. However, some analysts have explained they were really skeptical aboutthis purchase. Why? That is one of the weaknesses of the model. Indeed, the model only focuseson the hedging of the stochastic input and not on the global industrial risk.

Therefore, on the one hand, Delta decreases his sensibility to the oil variation which is certainlya way to be protected. But, on the other hand, such strategy increases the break-even point becausethe integration of the refinery will create new fixed costs for the downstream firm (all the fixed costssupported previously by the upstream firm). The strategys impact on the break-even point is alsoa fundamental question in Corporate Finance. The higher the break-even point is, the more riskywill be the firm. What would happen if the oil price suddenly fell? The integrated firm would staywith his high fixed costs due to the upstream firm. A way to improve the model would be to addthis fixed cost.

7 Comments, critics

7.1 The cost function of raising external fundsLiterature offers two different perspectives on the consequences of hedging on external financing.

On the one hand, when capital market imperfections make external finance more costly thaninternal funds, then hedging reduces financing costs because it reduces the variability of externalfunding. On the other hand, DeMarzo and Duffie (1995) show that hedging can reduce informationasymmetry problems by cleaning a firms profits from the noise related to exogenous factors.Hedging therefore improves the quality of the signal about the managers ability.

Recent research 19 has provided empirical evidence showing that hedging firms raise more exter-nal funds than non-hedging firms. This result contradicts somewhat FSSs model.

The cost of external funds function of FSS (1993) is a pillar of their and therefore of our results,and yet there is no consensus about its form and its existence in the literature. Our criticism of thiscost function is first and foremost its exogenity but also, to a lesser extent, its convexity.

Exogenity. In FSSs model (1993), the cost of external funds function is exogenous. This doesnot seem very accurate as the cost function should depend on the hedging strategy and thus beendogenous. In fact, some articles 20consider that the marginal cost of external funds depends onthe effectiveness of hedging.

Convexity. FSS (1993) use this arbitrary characteristic because it is a condition under whichhedging increases value. However, it is difficult to imagine why it should always be the case that thecost function is convex, although there is a large literature which supports theoretically this viewof external financing.

7.2 The time horizon

The rationale for our extension of FSSs paper is to protect an integrated firm against a stochasticinvestment opportunity and thus against external financing. Yet, the purpose of vertical integrationin a hedging perspective is to reduce the risk arising from the inputs price volatility. This rationalecan be used for commodity prices but it is difficult to think this model as a useful modelisation forother particular markets, such as the electricity one, where the intermediary spot price has a strongvolatility in the very short-term.

19S.Magee, (2009), The Effect of Foreign Currency Hedging on External Financing, Macquarie University, AppliedFinance Center

20A. Mello and J. Parsons, 2000, Hedging and Liquidity, The Review of Financial Studies, Vol 13, No 1, pp127-153.

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8 A MODEL FOR THE ELECTRICITY MARKET

The time horizon in this model is therefore that of an investment. This is an important limit ofthis model. With such a time horizon, hedging is not considered as a way to reduce volatilities atthe very short-term, as was seen in the electricity market. Moreover, the simplicity of this model isits strength but also its limitation, since markets need to rely on more specific models. In section8 we extend a framework made exclusively for the electricity market. We will see that this modelmade-to-measure is a more sophisticated way of treating our problem.

7.3 Perfect price anticipationAnother limitation is the interpretation of the random shock . Recall that the model has threeperiods: the firm knows the value of the risk variable in period 1 and then can deduce the valueof . In period 2 it earns the profit given by investment, with the investment opportunity known att = 1. If the variable is said to be interpreted as a price (like in the case of vertical integration),then we are perfectly anticipating prices with one period in advance. This is very unrealistic, inparticular in markets with high volatility.

8 A model for the electricity marketIn this section we do not try to find other interpretations concerning the interaction between hedgingand vertical integration. Instead, we show an extension made for a specific market : the electricitymarket. In particular, we can see that results are easier to interpret. Thus, this framework proposesanother rationale to study the corporate risk management. We begin with a presentation of theelectricity market.

8.1 Vertical integration in the electricity market

In the electricity industry, the question about corporate risk management seems essential forseveral reasons : uncertainty in the demand, spot price volatility 21, lack of market flexibility andrisk aversion. For a long time, the electricity industry has been characterized by an important levelof vertical integration. However, in the last two decades, many market regulatory reforms (Joskow2005, Gilbert and Newberry 2006) have been made to facilitate competition in the upstream and/ordownstream market. According to Aid, Chemla, Porchet and Touzi (2011), throughout these reformsa large number of firms went bankrupt (Enron, Pacific Gas & Electricity in 2001, British Energy in2002). In some countries, like New Zealand where the market has not been subject to regulation,most of retailers that were not integrated had to exit the market, since integrated producers had animportant strategic advantage in terms of risk management.

Table 4 in appendix shows a study made by the IEA22 about the organization and the capitalstructure of the electricitys industry in the 15 countries with largest consumption. We can see thereis a strong level of vertical integration in these markets. In China for instance, several regional firmsare responsible for the electricitys production, transport, local distribution, and sales in restrictedarea. According to Aid, Chemla, Porchet and Touzi (2011), non-integrated retailers can find difficultto stay in the market, unless there is large forward market in which they can rely to manage theirrisk.

In the following, we would like to know to which extent vertical integration has helped managethe risk and how this can affect the forward market. In particular, we focus on the relation betweenforward contracts and vertical integration in the electricity market.

21According to Lemmon and Bessembinder (2002) power prices for day time delivery are typically more than twiceas high as for nighttime delivery

22International Energy Agency, www.iea.org

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8.2 An adapted model for the electricity marketWe rely on the model developed by Bessembinder and Lemmon (2002) 23 to analyze how verticalintegration affects optimal forward positions. We summarize the most important results in the firstpart. Then, we extend the model to a vertical integration study, where a retailer merges with aproducer and we study how the vertical integration influences the forward market. The purpose ofstudying this model is that it is a much more accurate model, more comprehensive and has a betterdescription of reality. Besides, it relies on a general equilibrium between retailers and producers.We will see later on that the simulation of the prices has the shape of the spot price of electricityin reality.

8.2.1 The setup

Bessembinder and Lemmon analyze power production during a single future time period. The modelfeatures NP identical producers who sell power to NR identical retailers in the wholesale market.Retailers sell power to final consumers at a fixed unit price PR. The realized demand for retaileri is an exogeneous random variable denoted QRi . It is assumed that power companies are able toforecast demand in the immediate future with precision (QRi is known at t = 0) and hence knowthe spot price in the immediate future.

Each producer i has a cost function which depends on the total produced quantity:

Ci(QPi) = F +a

c(QPi)

c (38)

where F are fixed costs and QPi is the output of producer i, and c 2 (marginal productioncosts increase with output). In the wholesale spot market, producers sell to retailers who in turndistribute power to customers. Let PW denote the wholesale spot market, QPiW the quantity soldby producer i in the wholesale spot market, QP iF and Q

F

Rjthe quantity that producer i and retailer

j has agreed to deliver (or purchase if negative) in the forward market at the fixed forward pricePF . We also note QD =

QRj and we notice that

QFPi = 0. We summarize the model in the

following figure :

Figure 11: The electricity market by Bessembinder and Lemmon (2002)

23Bessembinder, Lemmon, 2002, Equilibrium Pricing and Optimal Hedging in Electricity Forward Markets, TheJournal of Finance.

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8.2.2 Problem and optimal solution

The model is solved in two steps. First, optimal behavior in the spot market is derived assum-ing forward positions are given. Second, optimal forward positions are derived assuming optimalbehavior in the spot market is given

Optimal behavior in the spot marketThe ex post profit of producer i is given by earning minus costs. Retailers buy on the wholesale

spot market the difference between realized retail demand and their forward positions. The ex postprofit are then given by : {

Pi = PWQWPi

+ PFQFPi Ci(QWPi +QFPi)

Rj = PRQRj + PFQFRj PW (QRj +QFRj )

(39)

Thus, for a given level of forward demand, the first-order conditions and equilibrium conditionslead, to24 : {

PW = a(QD

NP)c1

QWPi =QD

NPQFPi

(40)

To illustrate the quality of the results, we perform a simulation using Microsoft Excel 2007over 10 periods, with 10 producers and 10 retailers. The spot demand is supposed be uniformlydistributed over [0; 1]. In figure 12, we see that the shape of our simulations is similar to the graphicshowing Leipzig Power Exchange electricity spot-price over the period 16 June 200015 October2001:

Figure 12: Leipzig Power exchange electricity spot-price & model simulation with 10 retailers and10 producers

Optimal behavior in the forward market (mean-variance optimization problem)Lets assume that the objective function of producers ans retailers are, with A 0:{

maxQFPiE(Pi) A2 V ar(Pi,)

maxQFRiE(Ri) A2 V ar(Ri,)

(41)

The first order condition yields25:24See appendix for further details25See appendix for further details

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{QFPi =

PFE(Pw)AV ar(PW )

+ Cov(Pi,PW )V ar(PW )QFRi =

PFE(Pw)AV ar(PW )

+ Cov(Ri,PW )V ar(PW )(42)

where Pi and Rj denote respectively the profits of producer i and retailer j with no forwards.We can show that the equilibrium forward price is:

PF = E(PW ) ANP(NP +NR)cax

[cPRCov(P

xW , PW ) Cov(P x+1W , PW )

](43)

8.2.3 Extension : Vertical integration

We assume know that retailer Ri merges with producer Pj and that retailer Ri no longer goes on

the wholesale market to buy the input when producer Pj and retailer Ri merge. We also assume thatproducer Pj faces no capacity constraint. Retailers demand on the wholesale market is QD QRi .As a result, producer Pj produces the quantity of input retailer Ri needs, that is, QRi , and likebefore the merger, an even share of the supply on the wholesale market, that is, (QD QRi)/NP .Let denote P

W the new wholesale equilibrium price. As before, forward contracts are in zero net

supply. Retailers show up on the wholesale market with a demand equal to QD QRi (instead ofQD). The following figure summarizes the idea :

Figure 13: Vertical Integration : Only the retailer leaves the downstream market. The integratedproducer continues to sell electricity to other retailers.

Also, with the previous assumptions, the optimal wholesale spot price is given by :

PW = a(

QD QRiNP

)c1 (44)

Besides, following the merger, the new forward price is:

PF = E(P

W )

NPN cax

[cPRCov(P

xW , P

W ) Cov(P

x+1W , P

W )]

(45)

where N

= (NR +NP 1)/A. The ex post profit with no forwards of the merged entity is:

Ri+Pj = PRQRi + PWQ

WPj F

a

c(QRi +

QD QRiNP

)c (46)

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Using equation (42), the optimal demand for forward contracts of the merged entity is:

QFRi+Pj =PF E(P

w)

AV ar(PW )

+Cov(Ri+Pj , P

W )

V ar(PW )

(47)

In an atomistic market...In an atomistic market, we consider that QD QRi , i.e. the total demand for electricity is much

larger than the demand for electricity of the retailer i. Using (44), this implies that the wholesalespot price is unchanged after the merger, i.e.:

PW ' P W (48)Thus, the wholesale spot price depends only on agregate demand QD. We also assume a large

number of economic agents so as to make the following simplification:

N ' N (49)This approximation and this assumption imply that:

PF ' P F (50)As we did in the previous model, we want to see how the optimal number of forward contracts

varies with or without vertical integration. For this, we can adopt two point of views.

A macroeconomic point of view where we compare the sum of optimal forward contracts tothe integrated case, i.e. QFRi+Pj (QFRi +QFPj ).

A corporate point of view where the retailer compares its demand on forward contract in theintegrated case and in the non integrated case, i.e. QFRi+Pj QFRi .

Looking from a macroeconomic perspective at how vertical integration affects (optimal) forward

positions of two firms which integrate vertically means comparing the optimal forward position ofthe integrated firm with the sum of the optimal forward positions of the separate upstream anddownstream firms and adjusting for intra-companies pre-merger forward positions. As upstreamfirms are symetric and downstream firms as well, an upstream firm enters into forward contractswith all the downstream firms equally and a downstream firm enters into forward contracts with allthe upstream firms equally. Therefore, (optimal) forward contracts of a specific upstream firm Pjwhere the counterparty is a specific downstream firm Ri amounts to 1NRQ

FPj

and forward contractsof a specific downstream firm Riwhere the counterparty is a specific upstream firm Pj amounts to

1NP

QFRi . The variation in forward positions from a macroeconomic perspective is QFRi+Pj

(QFRi +QFPj +

1NR

QFRi +1NP

QFPj ).Using the approximations PW ' P W , PF ' P

F and a large number of economic agents, we can

further approximate the above formula:

QFRi+Pj (QFRi +QFPj +1

NRQFRi +

1

NPQFPj ) '

cov(Ri+Pj Ri Pj , PW )V (PW )

PF E(PW )AV (PW )

(51)

Besides, using (47), from a corporate point of view we will have :

QFRi+Pj QFRi =Cov(Ri+Pj Ri , PW )

V ar(PW )

(52)

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9 CONCLUSION

Ri+Pj Ri ' PWQRi(11

NP) a

c(QRi +

QD QRiNP

)c (53)

QFRi+Pj QFRi will be negative if Ri+Pj Ri and PW vary on average in opposite directions.

Recall that PW = a(QD

NP)c1. Assuming the number of producers fixed, PW increases (decreases) if

aggregate demand increases (decreases). Therefore, QFRi+Pj QFRi will be negative if the increasein aggregate demand induces on average an increase in (variable) production costs which outweighsthe benefit from higher wholesale revenues.

Proposition n 16: In an atomistic market, upstream vertical integration will reduce a retailers

optimal number of forward positions if an increase in aggregate demand induces on average anincrease in production costs which outweighs the benefit from higher wholesale revenues.

The intuition is that if an increase (decrease) in aggregate demand induces on average an increase

(decrease) in production costs which outweighs the benefit (loss) from higher (lower) wholesalerevenues, then the vertically integrated firms revenues will be less sensitive to wholesale spot pricesthan the distributors revenues.

9 ConclusionThe purpose of this paper is to model the interaction bewteen two tools of corporate risk manage-ment: hedging with financial instruments and vertical integration.

We have made several extensions to Froot, Scharfstein and Steins framework. First, we haveadded a firm in order to model a producer-distributor relationship in a short-term model. Theimportant assummption we make is that investment opportunities of the producer are positivelycorrelated with the risk factor, which stands for a shock on the price of the intermediary good, whileinvestment opportunities of the distributor are negatively correlated with this single risk factor.Then, we have considered these two firms in a long-term setup to analyze the impact of both firmswealth after the first investment.Finally, we have introduced an outside market in the short-termsetup to take into account horizontal interactions between firms.

We have shown that in the short-term setup, when both firms have positive initial wealth, theoptimal hedging strategy for an integrated firm is to hedge fully, for a non integrated upstreamfirm, to hedge partially (i.e: limit its exposure to the risk varible) or to underhedge (i.e: increaseits exposure to the risk variable), for a non integrated downstream firm, to overhedge (i.e: have acounter-cyclical exposure to to the risk variable). In the long-term setup, we have shown that theoptimal hedge ratio depends not only on the correlation between the risk variable and investmentopportunities (correlation effect), but also on the profitability of the firms investments (wealtheffect) and the volatility of the risk factor. A profitable (resp. not profitable) upstream firm willhave to hedge partially or under-hedge (resp. overhedge) while a profitable (resp. not profitable)downstream firm will have to overhedge (resp. hedge partially or under-hedge). Therefore, froma macroeconomic point of view (or that of a holding company owning both firms), how verticalintegration will affect the need for hedging will depend both on the sign of their respective wealthsand their relative size. Finally, we have shown that when the market consists of an integrated firmfacing an outside market, its optimal level of hedging will be reduced by selling to other downstreamfirms and avoiding buying from other upstream firms (vertical foreclosure).

We came up with some meaningful results with this model. Some of its hypothesis and predictionshave been tested in the case of two oil companies. Nevertheless, several of its limits must be pointedout. Some of them are a legacy of FSSs model. First, the cost function is assumed to be increasingat the margin. Although this assumption seems intuitive in some external financing cases, likebank financing, we do not really see why this should always be the case. Second, this cost function

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9 CONCLUSION

is exogenous. It should instead rather be determined endogenously. An important limit of thismodel is also the fact that at the time the firm invests, it perfectly knows the return it will geton its investment, as uncertainty about wealth and investment opportunities exists only up to thismoment. Another limit is that the time horizon is that of an investment. With such a time horizon,hedging is not considered as a tool to cope with extreme short-term volatilities such as in theelectricity market. Finally, the simplicity of the model does not make it suitable for the use ina specific industry. To adress this issue, we have extended a model designed with the electricitymarket in mind.

We have applied a prominent model by Lemmon and Bessembinder (2002) to the situationwhere a retailer merges with a producer. Contrary to the previous model, this one has definitely anindustrial organization flavour. An interesting and useful result is that the optimal forward positionof a market player depends on (a) the sensitivity of its profits to the wholesale spot price of theintermediary good (b) the difference between the forward price and the expetected spot price (thefirm can speculate on this difference if it is not risk-averse).

We have shown that in an atomistic market, upstream vertical integration will reduce a retailersoptimal number of forward positions if an increase in aggregate demand induces on average anincrease in production costs which outweighs the benefit from higher wholesale revenues. Theintuition is that if this condition is satisfied, the vertically integrated firms revenues will be lesssensitive to wholesale spot prices than the retailers ones.

Our work could be carried on. In every step we made we found new challenges, new ideas.Some of them were not developed in this article. We have treated every extension (short-termperspective, long-term perspective, outside market) separately. It would be interesting to analyzehow the extensions we made interact. We have also supposed that the producer and the retailerhad their investment opportunities affected with the same magnitude (but in the opposite way)by the risk factor. Research could be extended to the case where firms have their investmentopportunities affected by the risk factor with different magnitudes. This would make the modelmore realistic. Empirical research could also be developped as we have only tested one hypothesisand one prediction in the case of two oil firms. In the long-term perspective, we could link bothshocks by some correlation, creating a new link between the periods.

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10 APPENDIX

10 Appendix

10.1 Equations : First model

I = w + e (54)

F (I) = f(I) I (55)

(w) = maxIF (I) C(e) (56)

fI(I) = 1 + Ce(e) (57)

ww(w) = (dI

dw)2fII(I

) (dI

dw 1)2Cee(e) (58)

w = w0(h+ (1 h)) (59)

F (I) = f(I) I (60)

= ( ) + 1 (61)

maxhE[maxIF (I) C(e)] (62)

ifI 1 = Ce (63)

cov(w, ) = 0 (64)

h = 1 + E[fI/fII ]

0E[](65)

MaxhiE

[i(i)]

i(i)

= maxIiFi(Ii) C(ei)

Ii = i + ei

wi = wi0(hi + (1 hi))

(66)

{FU (IU ) = Uf(IU ) IU

I=( ) + 1 (67){FD(ID) = Df(ID) ID

D = ( ) + 1 (68) MaxhV E[(V)]

(V)

= maxIV F (IV ) C(eV )

wV = (wU0 + wD0 )(h

V + (1 hV ))(69)

FV (IV ) = V f(IV ) IV (70)

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10 APPENDIX

V =1

2D +

1

2U (71)

hU+D = hU + (1 )hD = 1 + [E(fIP/

UfII)PU0

(1 )E(fIP/DfII)

PD0

](72)

FV (IV ) = V f(IV ) IV (73)

V =(x y)

2( ) + 1 (74)

hV = 1 + (x y)E[fI/fII ]20E[]

(75)

fI(I3 ) = 1 + Ce(e

3) (76)

1 = F (I1 ) C(e1) (77)

hU2 = 1 + E[fI(I

U3 )Pww(I

U3 )/

UfII(IU3 )]

[F (IU1 ) C(eU1 )]E[Pww(IU3 )](78)

hD2 = 1 E[fI(I

D3 )Pww(I

D3 )/

DfII(ID3 )]

[F (ID1 ) C(eD1 )]E[Pww(ID3 )](79)

10.2 Equations of the model solution in the case of Cobb-Douglas func-tions

{f(I) = I

11

< 1(80)

{C(e) = e

1+

1+

0 < < 1(81)

Second order approximations around expected values of investment and external financing:{f(I) = aI2 + bI + ka < 0, b > 0,

(82)

{C(e) = c2e

2 + re+ zc > 0, r > 0,

(83)

where the values of a, b and c are calibrated using and .Optimal hedge ratio :

h = 1 +(1 + r c0 bca )((a c)2 + 3a222)

0(a c)((a c)2 + 3ac22) (84)

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10 APPENDIX

Effects on the optimal hedging ratio

Correlation effect The optimal hedging ratio, h, is a decreasing function of the parameter for any 2 lower than a critical value v2 = (ac)

2

32ac

Proof : Given the conditions on the constants, the expression (1 + r c0 bca ) is positive.Besides, ((ac)2 +3a222) is also positive and (ac) < 0, . Then the sign of the fraction dependson ((a c)2 + 3ac22) which is positive for any 2 lower than a critical value v2 = (ac)232ac . Thus,for 2 < 2, the optimal hedging ratio, h, is a decreasing function of the parameter .

Effect of initial wealth Since c > 0 the higher is the initial wealth of the firm, the lower is thesensitivity of the hedging ratio h to the parameter . Therefore, the optimal hedging ratio is anincreasing function of the initial wealth if > 0 and a decreasing function if < 0.

Proof : The factors containing 0 can be insulated:(1+rc0 bca )

0=

(1+r bca )0

c. Given theconditions on the constants, this expression is positive (see Spano (2001) for further details). Henceit is lower as 0 is higher. Hence, the value of his closer to 1 (full hedging) as 0 is higher, for anyvalue of . In other words, h is less sensitive to the relation parameter .

Effect of the volatility h is more sensitive to the correlation parameter when the volatilityincreases.

Proof : When the variance increases, the factor containing the variance in the numerator in-creases by the amount 3a222, while the factor in the denominator decreases by 3a222. Hence,for a given value of , the value of h is farther from 1 as the variance is higher (i.e. h is moresensitive to the correlation parameter ).

10.3 Effects on the optimal hedging10.3.1 The effect

Figure 14: Volume effect : Optimal hedging ratios (upstream in red, downstream in blue andweighted average in green) as a function of . Left: Same weight i.e. = 0.5. Right: the weight ofthe upstream firm is 0.7 ( = 0.7). We see that the weighted average of hedgings moves down (i.e.hU+D < hV ).

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10 APPENDIX

10.3.2 The endogenous weighted average

Figure 15: Endogenous weighted average by initial wealth : Optimal hedge ratios (upstream inred, downstream in blue and weighted average in green) as a function of . The coefficient of theweighted average of the upstream firm is

U0

U0 +D0. We see that the weighted average of hedgings is

below 1 (i.e.hU+D < hV ) for all values of . Other parameters: same initial wealth for both firms.

10.4 The electricity market

Optimal behavior in the spot marketIn the wholesale spot market, producers sell to retailers who in turn distribute power to cus-

tomers. Let PW denote the wholesale spot market, QPiW the quantity sold by producer i in thewholesale spot market, QP iF and Q

F

Rjthe quantity that producer i and retailer j has agreed to

deliver (or purchase if negative) in the forward market at the fixed forward price PF . The ex postprofit of producer i is:

Pi = PWQWPi + PFQ

FPi F

a

c(QPi)

c (85)

whereQPi=Q

WPi +Q

FPi

Retailers buy on the wholesale spot market the difference between realised retail demand andtheir forward positions. The ex post profit of retailer j is:

Rj = PRQRj + PFQFRj PW (QRj +QFRj ) (86)

The first-order condition of (16) gives:

QWPi = (PWa

)x QFPi (87)where

x = 1/(1 c)At equilibrium, total retail demand QD =

QRj equals total production

QWPi and forward

contracts are in zero net supply, ie.:QFPi = 0. This implies:

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10 APPENDIX

PW = a(QD

NP)c1 (88)

and

QWPi =QD

NPQFPi (89)

Optimal behavior in the forward market (mean-variance optimization)Lets assume that the objective function of market players is, with A 0:

maxQF{P,R}iE(pi{P,R}i)

A

2V ar(pi{P,R}i,) (90)

where

pi{P,R}i = {P,R}i + PFQF PWQF (91)

where Pi and Rj

Short Term Hedging, Long Term Vertical Integration

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