PROFIT-AND-LOSS EFFECTS WHEN THE OIL PRICE FALLS AND THE MARKET IS IN BACKWARDATION

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1 Appendix 4.2 Stack-and-Roll Hedge: Profit-and-Loss Effects To better understand the profit-and-loss effects of a stack-and-roll hedge and the risks associated with it, let s assume MGRM sells a string of forward oil contracts and hedges them with a stack of short-dated futures contracts. To simplify our task, let s assume that the string of forward contracts only extends two years into the future. PROFIT-AND-LOSS EFFECTS WHEN THE OIL PRICE FALLS AND THE MARKET IS IN BACKWARDATION Suppose in Year 0 (i.e., when the contracts were initiated), MGRM offers customers fixed-rate forward contracts to purchase one million barrels of oil in Year 1 and Year 2 at prices that are below the current spot rate (i.e., the market is in backwardation). Because the futures market has sufficient liquidity only one year ahead, MGRM hedges the two million barrels of forward contracts by purchasing (i.e., stacking) two million barrels of futures contracts that expire in Year 1, with the expectation of rolling over one million barrels of the contracts at the end of the year so that its Year 2 forward exposure is hedged throughout the two years. The price information for this example is shown in Exhibit A What Happens in Year 0? In Year 0, MGRM would initiate the short forward contracts, which would obligate the company to sell one million barrels of oil in Year 1 and Year 2 for $27 per barrel and $24 per barrel, respectively. To hedge the transactions, MGRM would purchase two million barrels worth of futures contracts that obligate it to buy oil in Year 1 at the futures price of $27 per barrel. What Happens in Year 1? Deliver oil (one million barrels) on the forward contracts maturing in Year 1. MGRM would be obligated under its forward contracts to deliver oil in Year 1 at $27 per barrel, yet, to purchase the physical oil in the spot market, A4.2.1

2 A4.2.2 Appendix 4.2: Stack-and-Roll Hedge: Profit-and-Loss Effects EXHIBIT Price per Barrel of Oil Column 1 Column 2 Column 3 Column 4 Prices/Rates Year 0 Year 1 Year 2 Spot Prices $30 $25 $20 Forward price in Year 0 for $27 NA NA contracts expiring in Year 1 Forward price in Year 0 for $24 NA NA contracts expiring in Year 2 Futures price in Year 0 for $27 NA NA contracts expiring in Year 1 Futures price in Year 1 for NA $22 NA contracts expiring in Year 2 Risk-free interest rate 10% 10% 10% * NA means the information in this cell is not applicable to this example. *The figures under Year 1 and Year 2 are the only unknown variables at the time of the original hedge. it would have to pay $25 per barrel. As a result, MGRM would close out its forward contracts with $2 per barrel gain, which is a total gain of $2 million for the million gallons. Close out the futures contracts (two million barrels) that were taken out in Year 0. In Year 1, MGRM would also close out two million barrels of futures contracts. Because MGRM would be selling each barrel of oil for $25 and buying it for $27, the company would lose on the two million barrels, so the total loss would equal $4 million. Net result in Year 1 MGRM would gain $2 million as a result of its forward transactions and lose $4 million on its futures transactions; therefore, it would lose a net of $2 million (see Exhibit A4.2.2). Invest the profits. The $2 million residual loss in Year 1 would have to be financed for one year at 10%, which would cost MGRM $200,000 (i.e., 10% $2 million).

3 Profit-and-Loss Effects A4.2.3 EXHIBIT A4.2.2 Summary of Net Profit-and-Loss Effects in Year 1 Example: Spot Price Decreases When the Oil Market Is in Backwardation Year 1 Forward contracts expire (1 million barrels) Stack of futures contracts expires (2 million barrels) Net 1 million barrels ($27 $25) $2 million gain 2 million barrels ($25 $27) $4 million loss $2 million loss Roll over the hedge. MGRM would still have the Year 2 forward contracts to hedge, so at the end of Year 1, it would buy futures contracts that matured in Year 2. In Year 1, the futures price per barrel of oil for the Year 2 contract is $22 (see Exhibit A4.2.1). What Happens in Year 2? Deliver oil (one million barrels) on the forward contracts Under the forward contracts that were transacted in Year 0, MGRM would be obligated to deliver oil in Year 2 at $24 per barrel; yet, to purchase the physical oil in the spot market, it would have to pay only $20 per barrel. As a result, MGRM would close out its forward contracts with a gain amounting to $4 per barrel (i.e., it would buy oil for $20 per barrel on the spot market and sell it to the forward counterparties for $24 per barrel). Therefore, the total gain on the million barrels of oil would amount to $4 million. Close out the Year 2 futures contracts that were taken out in Year 1. In Year 2, MGRM would close out the futures contracts that were purchased in Year 1. Because the price of oil had decreased to $20 per barrel, a loss of $2 per barrel would be suffered on the transactions (i.e., the oil could be purchased at the futures price of $22 per barrel and sold in the spot market for $20 per barrel). Therefore, the total loss on the million barrels of oil would amount to $2 million. Net Result in Year 2 MGRM would gain a net of $2 million on its transactions maturing in Year 2, because it would gain $4 million on its Year 2 forward contracts and lose $2 million on its Year 2 futures contracts (see Exhibit A4.2.3).

4 A4.2.4 Appendix 4.2: Stack-and-Roll Hedge: Profit-and-Loss Effects EXHIBIT A4.2.3 Summary of Net Profit-and-Loss Effects in Year 2 Example: Spot Price Decreases When the Oil Market Is in Backwardation Year 2 Forward contracts expire Stack of futures contracts expires Net 1 million barrels ($24 $20) $4 million gain 1 million barrels ($20 $22) $2 million loss $2 million gain (This gain would be partly offset by the $0.2 million cost to finance at 10% the $2 million loss from Year 1.) Overall Result Including Both Year 1 and Year 2 In Year 1, MGRM had a $2 million loss from its forward and futures transactions, which it financed at 10%, and in Year 2, it gained $2 million; thus, overall, the losses from Year 1 (approximately) cancel the gains in Year 2. All that is left over is the $0.2 million financing cost for the loss in Year 1 (see Exhibit A4.2.4). EXHIBIT A4.2.4 Year Year 1 Year 2 Summary of Overall Net Profit-and-Loss Effects in Year 1 and Year 2 Example: Spot Price Decreases When the Oil Market Is in Backwardation Gains/Losses $2 million loss $2 million gain Net $0 (There would be a small $0.2 million cost to finance the loss from Year 1) In this example, everything worked out almost perfectly, because the gains on the futures contracts matched the losses on the forward contracts (i.e., there was only a small amount of residual interest cost). Is this always the case? If not, what special conditions led to this result? Under what conditions could the result turn out worse, and if it could turn out worse, how much worse could it be? In short, what are the chances of large losses occurring with the stack-and-roll hedge? Answering these questions gets to the heart of the risk management problems associated with stack-and-roll hedges.

5 Generalizing the Profit-and-Loss Effect of Stack-and-Roll Hedges A4.2.5 EXHIBIT A4.2.5 Price per Barrel of Oil Column 1 Column 2 Column 3 Column 4 Column 5 Prices Abbreviations Year 0 Year 1 Year 2 Spot Prices for S 0, S 1, S 2 $30 $25 $20 Year 0, Year 1, and Year 2 Forward price in 0 F 1 $27 NA NA Year 0 for contracts expiring in Year 1 Forward price in 0 F 2 $24 NA NA Year 0 for contracts expiring in Year 2 Futures price in 0 F* 1 $27 NA NA Year 0 for contracts expiring in Year 1 Futures price in 1 F* 2 NA $22 NA Year 1 for contracts expiring in Year 2 Risk-free interest I 0, I 1, I 2 10% 10% 10% rate * NA means the information in this cell is not applicable to this example. *The figures under Year 1 and Year 2 are the only unknown variables at the time of the original hedge. GENERALIZING THE PROFIT-AND-LOSS EFFECT OF STACK-AND-ROLL HEDGES To answer these questions, let s go back over the transactions in the earlier section and use symbolic notation to generalize our example. To make the exercise a bit easier, we will ignore the interest cost to finance the Year 1 losses (or the interest earnings from Year 1 gains). Exhibit A4.2.5 is exactly the same as Exhibit A4.2.1, except that Column 2 shows the abbreviations for each of the variables used in our example. Basis is the amount by which the forward price exceeds the spot price, 1 Forward Price Spot Price Basis Equation 1 1 Basis was defined in Appendix 4.1, but is repeated here for the convenience of readers.

6 A4.2.6 Appendix 4.2: Stack-and-Roll Hedge: Profit-and-Loss Effects Therefore, the following abbreviations are used: Basis on a forward contract starting in Year 0 and maturing in Year 1 is 0 B 1, Basis on a forward contract starting in Year 0 and maturing in Year 2 is 0 B 2, Basis on a futures contract starting in Year 0 and maturing in Year 1 is 0 B* 1, and Basis on a futures contract starting in Year 1 and maturing in Year 2 is 1 B* 2. The relationships between the forward price, spot price, and basis, as well as the relationships between the futures price, spot price, and basis are as follows: Forward price in Year 0 for a contract maturing in Year 1 0 F 1 S 0 0 B 1 Forward price in Year 0 for a contract maturing in Year 2 0 F 2 S 0 0 B 2 Futures price in Year 0 for a contract maturing in Year 1 0 F* 1 S 0 0 B* 1 Futures price in Year 1 for a contract maturing in Year 2 1 F* 2 S 1 1 B * 2 In Year 1, the two million barrels of long futures contracts that were taken out in Year 0 mature, and the company earns (or loses) 2 MM 2 (S 1 0 F * 1 ), which is equal to 2 MM [S 1 (S 0 0 B* 1 )]. Offsetting part of these gains (or losses) are the forward contracts for one million barrels that lose (or gain) 1 MM ( 0 F 1 S 1 ), which is equivalent to 1 MM [(S 0 0 B 1 ) S 1 ]. In Year 2, the futures contracts that were taken out in Year 1 mature, and the company gains (or loses) 1 MM (S 2 1 F * 2 ), which is equivalent to 1 MM [S 2 (S 1 1 B * 2 )]. As well, the forward contracts mature, thereby earning (losing) 1 MM ( 0 F 2 S 2 ) which is equivalent to 1 MM [(S 0 0 B 2 ) S 2 ]. Summing the results from Year 1 and Year 2, the hedge nets: 1 MM [( 0 B 1 0 B* 1 ) ( 0 B 2 0 B* 1 1 B* 2 )] Equation 2 All transactions for the two years are summarized in Exhibit A In Equation 2, the only unknown is 1 B * 2, which is the basis of the futures contract that starts in Year 1 and matures in Year 2. Every other element of the equation would be known at the time the original hedge was trans- 2 MM million

7 Generalizing the Profit-and-Loss Effect of Stack-and-Roll Hedges A4.2.7 EXHIBIT A4.2.6 Generalizing the Profit-and-Loss Effects of Stack-and-Roll Hedges Return Return What happens at the end of Year 1? 1. Close out the futures 2 MM (S 1 0 F * 1 ) 2 MM [S 1 (S 0 0 B* 1 )] contracts (two million barrels) that were taken out in Year 0 2. Make delivery on the 1 MM ( 0 F 1 S 1 ) 1 MM [(S 0 0 B 1 ) S 1 ] forward contracts (one million barrels) that were taken out in Year 0 3. Rollover the hedge No effect No effect What happens at the end of Year 2? 1. Close out the futures 1 MM (S 2 1 F * 2 ) 1 MM [S 2 (S 1 1 B* 2 )] contracts (one million barrels) that were taken out in Year 1 2. Make delivery on the 1 MM ( 0 F 2 S 2 ) 1 MM [(S 0 0 B 2 ) S 2 ] forward contracts (one million barrels) that were taken out in Year 0 Net Result: Equation 2 1 MM [( 0 B 1 0 B * 1 ) ( 0 B 2 0 B* 1 1 B* 2 )] acted. 3 Unlike the cash-flow effects of the stack-and-roll hedge that were explained in Appendix 4.1 (see Generalizing the Cash-Flow Effects of Stackand-Roll Hedges), the direction and magnitude of change in the spot price no longer matters. In short, it makes no difference how the oil price changes; the risk is due purely to changes in the basis on the futures contract, when the hedge is rolled over. Let s go back to our original example in this appendix and see why we got a perfect hedge. In Equation 2, ( 0 B 1 0 B * 1 ) is the difference in basis between a Year 1 forward contract and a Year 1 futures contract. These bases should be the same (or nearly the same), because if they were not, arbitragers could make riskless profits by buying in the cheap market and 3 In other words, every element with a 0 subscript would be known in Year 0, when the original hedge was transacted.

8 A4.2.8 Appendix 4.2: Stack-and-Roll Hedge: Profit-and-Loss Effects selling in the dear one. Assuming the two bases were the same, MGRM would earn zero net gains or losses on this part of the hedge. By contrast, if the basis on the short forward contract (i.e., the sell contract) were greater than the basis on the long futures contract (i.e., the buy contract), the company would earn net gains on the hedge, but these gains would be known up front, when the hedge was transacted. The second expression in Equation 2 is ( 0 B 2 0 B * 1 1 B* 2 ). The first term, 0 B 2, is the basis on a two-year forward contract that starts in Year 0 and matures in Year 2. The second two terms, 0 B * 1 and 1 B* 2, are, respectively, the basis on futures contracts that start in Year 0 and mature in Year 1 and the basis on futures contracts that run from Year 1 to Year 2. If the sum of these two yearly futures bases does not equal the two-year basis of the forward contract, then the company will incur a profit or a loss. We saw earlier in this appendix that a positive basis means the forward (or futures) rate is greater than the spot rate, which means the market is in contango. If the reverse were true, then the market would be in backwardation. In terms of Equation 2, a market that moves from backwardation to contango increases the losses on the stack-and-roll hedge. We know this because the only unknown in this hedge is 1 B * 2, which is the basis on a futures contract taken out in Year 1 and maturing in Year 2. If the market turns from backwardation in Year 0 to contango in Year 1 (i.e., 1 B * 2 is greater than zero) then the expression 1 B * 2 becomes a negative and acts to reduce MGRM s profits.