# 9 Hedging the Risk of an Energy Futures Portfolio UNCORRECTED PROOFS. Carol Alexander 9.1 MAPPING PORTFOLIOS TO CONSTANT MATURITY FUTURES 12 T 1)

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2 Helyette Geman c0.tex V - 0//0 :00 P.M. Page Risk Management in Commodity Markets For instance, suppose we wish to construct a -month futures series when the maturity dates are March, June, September and December. On September we use the September and the December contracts, with prices P and P respectively. The number of days between the September contract expiry date and our -month expiry date (which is December on September) is and the time interval between the December contract expiry date and our expiry date is days. Hence, the concatenated price is P + P. We can continue to use the September and December contract prices in the construction of the -month futures price. For instance, on September, our -month expiry date is December, so the concatenated price would be P + P, where P and P are now the prices of the September and December futures contracts on September. Since prices can behave oddly a few days before expiry, we should drop the September contract from our calculations after September, and instead take the -month maturity contract price to be equal to the December contract price for a few days. However, on September we can start using linear interpolation between the December and March futures prices as above, but now with the December contract being the shorter one. As time moves on we decrease the weight of the December price in our calculation of the concatenated futures price and increase the weight on the March futures price. The main advantage of using constant maturity futures as risk factors for commodity portfolios is that the use of spot prices plus discount rates as risk factors assumes futures are always at their fair price, and thus ignores the basis risk due to fluctuations in carry costs and convenience yields, as well as the variation of the market price of the futures within its no arbitrage range. All these sources of uncertainty in the basis are considerable: carry costs due to storage, insurance and transportation are difficult to measure precisely; convenience yields are intangible and even more difficult to assess; and since the spot cannot be shorted the no arbitrage range has no lower bound, whence substantial decoupling of spot and futures prices is often evident when demand surges or supply drops for reasons beyond the control of the market participants. Constant maturity futures provide a long time series of futures prices that can be used to assess the market characteristics. These characteristics vary considerably from market to market. Prices are determined by unpredictable demand and supply factors such as the weather and the economic climate. For instance, the weather affects the supply of corn and the demand for gas and the outbreak of war affects the price of oil. But prices may also be affected by speculative trading, and the herding behaviour of speculative investors can lead to prolonged price trends in futures prices that have nothing to do with demand and supply of the actual commodity. The process of mapping a portfolio to constant maturity futures is very similar to the process of mapping a cash flow to fixed maturity interest rates. This can be done in a present value and volatility invariant manner as the examples below will demonstrate. The methodology is a simple adaptation of the cash flow mapping methods that are described for interest rate sensitive portfolios in Alexander (0) on pages.

3 Helyette Geman c0.tex V - 0//0 :00 P.M. Page Hedging the Risk of an Energy Futures Portfolio Example Mapping commodity forward positions to constant maturity futures Suppose we have just two forward positions on crude oil: a long position with present value \$ million in month and days and a short position with present value \$ million in month and days. How should we map these to equivalent positions on constant maturity crude oil futures at the -month and -month maturity? Solution Suppose there are days in a month. We could simply use linear interpolation and map / / = \$m to the -month vertex and / / = 0 to the -month vertex. But this will change the volatility of the portfolio. Instead we can use a present value and volatility invariant map, depicted in Fig.. For this we need to know the volatilities of the -month and -month futures and their correlation suppose the volatilities are % and % as shown and the correlation is 0.. Using linear interpolation on the variances we infer the volatilities of.0 % for the month day position and.0 % for the month day position. Position PV (\$m) Figure. \$ m days days days (\$ m) mth mths % % Volatility A volatility invariant commodity futures or forwards mapping For the mapped position to have the same volatility as the original position, the proportion x of the long position of \$ million at month days that is mapped to the -month future must satisfy.0 = x + ( x) + 0. x ( x). This quadratic equation has one solution between 0 and, i.e. x = 0.. Similarly, the proportion y of the short position of \$ million at month days that is mapped to the -month future must satisfy.0 = y + ( y) + 0. y ( y), and solving this gives y = 0.. Thus in total we must map 0. \$ \$ = \$ 0

4 Helyette Geman c0.tex V - 0//0 :00 P.M. Page Risk Management in Commodity Markets to the -month future, and 0.0 \$ \$ = \$ to the -month future. This way the present value and the volatility of the mapped position is the same as the volatility of the unmapped position. Example Determining the number of contracts Suppose the mapping in Example was performed on a day when the prices of the crude oil forwards and futures were as shown in Table.. Each forward contract is for 0 barrels and each futures contract is for 00 barrels of crude oil. How many contracts are in the mapped position? Table. Prices of crude oil forwards and futures (\$ per barrel) days days 0 days 0 days Solution The solution is summarized in Table.. The second row of the table shows the unmapped and mapped value of the positions, the number of barrels is this position value divided by the price per barrel and the number contracts is the number of barrels divided by the number of barrels in each contract (00 for the futures and 0 for the forwards). Table. Positions on crude oil forwards and futures Maturity days days 0 days 0 days Price Position Value , Number of Barrels Number of Contracts Of course, there is no need to round the resulting number of futures contracts to the nearest integer, since we are only mapping the portfolio to these non-traded risk factors.. THE PORTFOLIO AND ITS KEY RISK FACTORS Suppose that on August 0 a trader in energy futures holds long and short positions that have been mapped to constant maturity futures as shown in Table.. Each futures contract is for 00 barrels and the minus sign indicates a short position. Note that these positions could result from both straight futures trades and from positions on the two crack spread futures, i.e. unleaded gasoline crude oil and heating oil crude oil. Figures.. show how the daily prices of the constant maturity futures on all three products have evolved over a very long period. All prices spiked before the outbreak of

5 Helyette Geman c0.tex V - 0//0 :00 P.M. Page Hedging the Risk of an Energy Futures Portfolio Table. Number of futures contracts in an energy futures trading book Maturity (months) Crude oil Heating oil Unleaded gasoline m m m m m m Jan- Jan- Jan- Jan- Jan- Jan-0 Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan-00 Jan-0 Jan-0 Jan-0 Jan-0 Jan-0 Jan-0 Figure. NYMEX WTI crude oil constant maturity futures prices the Gulf war in, and since the war in Iraq in 0 prices have risen tremendously. For instance, the crude oil futures prices rose from around \$ per barrel to nearly \$0 per barrel in August 0 and the prices of the futures on the refined products rose even more. The daily price fluctuations of futures of different maturities on each product are always very highly correlated, as are those on futures of different products. We do not need to use years of data for the portfolio risk analysis; in fact looking back into the 0s and beyond may give misleading results since energy markets have become much more volatile during the last few years. But we do need fairly high frequency data, because the trader needs to understand his short term risks. So we shall use daily data between January and August 0. In the spreadsheet for this case study we first calculate the average correlation of daily returns on each of the futures over the sample period. The correlation matrix is too large to be reported in the text, but Table. shows some of these correlations.

6 Helyette Geman c0.tex V - 0//0 :00 P.M. Page Risk Management in Commodity Markets 0 m m 0 m 0 m m 0 m Jan- Jan- Jan- Jan- Jan- Jan-0 Jan- Jan- Jan- Jan- Jan- Figure. Jan- Jan- Jan- m m m m m m Jan- Jan- Jan- Jan- Jan- Jan- Jan-00 Jan-0 Jan-0 Jan-0 Jan-0 NYMEX heating oil constant maturity futures prices Jan-0 Jan- Jan- Jan- Jan- Jan- Figure. NYMEX unleaded gasoline constant maturity futures prices Jan- Jan- Jan- Jan- Jan-00 Jan-0 Jan-0 Jan-0 Jan-0 Jan-0 Jan-0 Jan-0 Jan-0

7 Helyette Geman c0.tex V - 0//0 :00 P.M. Page Hedging the Risk of an Energy Futures Portfolio Table. Daily correlations of futures prices at selected maturities Correlatons m m m m m m m m m m Crude oil (CO) Heating oil (HO) Unleaded gasoline (UL) Cross-correlations m m m m m HO CO UL CO HO UL The crude oil futures behave like a typical term structure, with the correlation being higher at the long end and decreasing as the maturity gap between the futures increases. All correlations are very high indeed. The same comments apply to heating oil and unleaded gasoline futures, and although their term structures are a little less highly correlated than crude oil there is still a very high degree of correlation. The lower part of Table. shows that the cross-correlations between futures on different products are lower than the correlations of futures on one of the products, but they are still very high. Note that the -month futures tend to have slightly lower correlations than futures of months maturity and longer.. IDENTIFYING THE KEY RISK FACTORS We have a total of risk factors. But since they are so highly correlated we should perform a principal component analysis (PCA) to reduce the dimension of the risk factor space. In the spreadsheet we apply PCA to the entire system of risk factors. This way the principal component risk factors capture correlated movements across futures on different commodities, as well as within futures on the same commodity. The PCA may be applied to either the correlation or the covariance matrix of returns, with the latter accounting for any difference between the risk factor volatilities. In each commodity the -month futures have noticeably higher volatilities than the other futures, so we shall perform the PCA on the covariance matrix. The results for the first four components from the PCA on the covariance matrix are displayed in Table.. The entries in Table. are the eigenvectors corresponding to the first, second, third and fourth largest eigenvalues. The percentage of the total variation that is explained by each eigenvalue is shown in the first row of the table. Examining the eigenvalues and eigenvectors we deduce that, between January and August 0: % of the historical variations were a similar and simultaneous shift and tilt in all three term structures; % of the historical variations were when the crude oil and unleaded gasoline futures shift and tilt in opposite directions and the heating oil futures term structure tilts changes convexity; See Alexander (0) Section I.. and Chapter II. for further details of principal component analysis.

8 Helyette Geman c0.tex V - 0//0 :00 P.M. Page Risk Management in Commodity Markets Table. Results of PCA on the futures returns covariance matrix % % % % WTI m WTI m WTI m WTI m WTI m WTI m HO m HO m HO m HO m HO m HO m UL m UL m UL m UL m UL m UL m % of the historical variations were when the crude oil and heating oil futures term structures shift and tilt in opposite directions and the unleaded gasoline futures remain static except at the very short end; % of the historical variations were when the crude oil futures remain almost static, and the heating oil futures and unleaded gasoline futures tilt in opposite directions. The first four principal components are time series that represent the four key risk factors for any portfolio on these oil futures. The common trend principal component risk factor is much the most important, since % of the historical variation in these futures was due to movements of this type. Taking the first four components together captures % of the historical variations in these energy futures since January.. HEDGING THE PORTFOLIO RISK Figure. shows a reconstructed price series for the portfolio, holding the positions constant at the values shown in Table. and revaluing the portfolio using the historical prices of the constant maturity futures. Since the portfolio has short positions its value could become negative, hence we base our analysis on the portfolio P&L and not on portfolio returns. The current value of the portfolio is \$.0 million and its historical P&L volatility based on the reconstructed price series is extremely high, at over \$. million per annum. The % -day historical Values at Risk (VaR) is \$ 0 (measured as minus the % percentile of the daily P&L distribution). Assuming the returns are independent and identically distributed, we apply the square root of time rule to estimate the % -day historical VaR as \$ 0 = \$ 0 0.

9 Helyette Geman c0.tex V - 0//0 :00 P.M. Page Hedging the Risk of an Energy Futures Portfolio 0 Dec- Jun- Dec- Figure. Jun-00 Dec-00 Jun-0 Dec-0 Jun-0 Dec-0 Jun-0 Dec-0 Jun-0 Dec-0 Jun-0 Reconstructed price series for the portfolio (\$ million) Concerned about the high probability of large losses on the portfolio, the trader decides to reduce the volatility of the portfolio returns by hedging. Our question is, which futures should he trade, and how many contracts should he buy or sell? The principal component representation allows a close approximation of the portfolio in terms of its sensitivities to the four major risk factors. We first express the return on each constant maturity futures using its principal component representation. For instance, R WTI m 0.P + 0.P + 0.P + 0.0P, where Rm WTI is the daily return on the -month crude oil futures and P, P, P and P are the first four principal components. Then we map the portfolio return to the principal component risk factors using the portfolio representation shown in Table.. We obtain: P&L =.P.P.0P.P where the coefficients are measured in millions of US dollars. This representation tells that if the first principal component shifts up by % leaving the other components fixed then the portfolio will gain about % of \$. million, i.e. about \$. The first component is where all futures of the same maturities move by approximately the same amount. The largest sensitivity of the portfolio is to the third principal component. Thus a rise in price on the short term heating oil contract is the largest risk exposure of this portfolio. It is not easy to see this from the portfolio composition. Nevertheless, we shall now confirm this by showing that selling the -month heating oil From Table. we know that a % upward shift in the first component implies that the -month crude oil futures price increases by 0. %, the -month heating oil futures price increases by 0. %, the -month gasoline futures price increases by 0. %; and so on for the -month futures. Dec-0 Jun-0

10 Helyette Geman c0.tex V - 0//0 :00 P.M. Page Risk Management in Commodity Markets contract is indeed the best hedge for the portfolio amongst all single contract futures hedges, according to the reduction in P&L volatility that it achieves. We now consider a partial hedge using a single futures contract and targeting a reduction on P&L volatility from over \$. million to less than \$. million per annum. For each of the futures contracts in turn, we calculate how many futures contracts of this type we need to sell to minimize the variance of the hedge portfolio P&L. The results are summarized in Table.. For each commodity we first report the risk factor P&L volatility, and the correlation between this risk factor P&L and the portfolio P&L. Below this we show the number of contracts on the futures that should be sold to minimize the variance of the hedged portfolio s P&L. The futures having the highest correlation with the reconstructed portfolio P&L are the -month heating oil futures (P&L correlation = 0.) and the corresponding minimum variance hedge ratio implies that. contracts on the -month heating oil futures should be sold, and this trade effects the largest possible reduction in volatility compared with any other trade on a single futures contract. Table. Minimum variance hedges to reduce to volatility of the futures portfolio Risk factor WTI m WTI m WTI m WTI m WTI m WTI m Risk factor P&L Volatility (\$) Correlation No. contracts Hedged portfolio P & \$ 0 \$ 0 \$ 0 \$ 0 \$ 0 \$ L volatility (\$) % day historical \$ 0 \$ \$ 0 \$ \$ \$ VaR (\$) Risk factor HO m HO m HO m HO m HO m HO m Risk factor P&L volatility (\$) Correlation No. contracts Hedged portfolio P & \$ 0 \$ 0 \$ 0 \$ \$ \$ L volatility (\$) % day historical \$ \$ 0 \$ \$ 0 \$ \$ VaR (\$) Risk factor UL m UL m UL m UL m UL m UL m Risk factor P&L volatility (\$) Correlation No. contracts Hedged portfolio \$ 0 0 \$ \$ 0 \$ \$ 0 \$ P&L volatility (\$) % day historical VaR (\$) \$ 0 \$ \$ \$ \$ \$ 0 The hedge is effected using exchange traded heating oil futures that are equivalent, under the mapping described in Section., to. contracts on the -month heating oil futures. The result will be a portfolio with a historical P&L volatility of \$. million, compared with over \$. million without this hedge. Similarly, the hedge reduces the % -day historical VaR from \$.0 million to \$. million. Other single contract hedges can also reduce the risk considerably. For instance, taking a position equivalent to selling. of the -month futures contract on unleaded gasoline would reduce the P&L volatility to \$. million and the % -day VaR to \$.

11 Helyette Geman c0.tex V - 0//0 :00 P.M. Page Hedging the Risk of an Energy Futures Portfolio million. Clearly, hedging with more than one futures contract would effect an even greater reduction in portfolio risk. For instance, using both the -month heating oil and -month gasoline futures in a single hedge, we determine the minimum variance hedge ratios using multiple regression. The ordinary least squares estimated model, with t-statistics below the coefficients in parentheses is: P&L =.0 ( 0.0) +. (.) P&L HO m +. (.0) P&L UL m Taking positions equivalent to. contracts on the -month heating oil futures and. contracts on the -month unleaded gasoline futures reduces the portfolio P&L volatility to. million and the % -day VaR to \$. million.. CONCLUSIONS This chapter has used a practical and realistic example to demonstrate how to identify the key risk factors of energy futures portfolios. These are the principal components of a set of constant maturity futures on related energy products in this case, crude oil, heating oil and unleaded gasoline. Due to the very high correlation of these futures, PCA is a statistical tool that serves to reduce the dimensions of the risk factor space considerably. In our case we reduced dimensions from constant maturity futures to just four principal components. Moreover, using PCA we can identify the key risk factors of the portfolio. For instance, we showed that the most common types of movement in energy futures term structures is a similar and simultaneous shift and tilt in crude oil, heating oil and unleaded gasoline futures. Then we considered a particular portfolio, and used PCA to demonstrate that the greatest risk exposure was to short term heating oil futures, a fact that is not evident from a direct examination of the portfolio composition. We demonstrated empirically that hedging with a position that is equivalent to. contracts on the -month heating oil futures reduces the portfolio P&L volatility from over \$. million to \$. million, and reduces the % -day historical VaR from \$.0 million to \$. million. Composite hedging with more than one futures contract reduces the P&L volatility and VaR even further.. REFERENCES Alexander, C. (0) Market Risk Analysis, Volume III: Pricing, Hedging and Trading Financial Instruments. John Wiley and Sons Ltd, Chichster. See Alexander (0), pages.

12 Helyette Geman c0.tex V - 0//0 :00 P.M. Page

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