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Hedging Retail Electricity


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Hedging Retail Electricity

  1. 1. Hedging Retail Electricity Eric Meerdink Director, Structuring & Analytics Hess CorporationDisclaimer: The views and methods described in this presentation are the viewsand methods of the presenter and not endorsed by Hess Corporation.
  2. 2. Risks• Three main areas of risk in retail electricity: – Market Price Risk – Volumetric Risk – Shaping Risk• I will be concentrating on the three risk buckets listed above. How to quantify and measure these risks, and how to hedge them.• Other risks (not part of this talk): – Credit Risk – Operational Risk – Regulatory Risk – Market Structure Risk 2
  3. 3. What is the Appropriate Risk Measure?• Why do firm’s hedge? To reduce risk and the expected costs from financial distress.• What is risk and how do we measure it?• What are the costs of financial distress?• We need a quantitative measure to use as a standard to evaluate risk and evaluate any hedge structure.• This quantitative measure industry specific and needs to fit the specifics of the business.• I propose that Cash Flow at Risk (CF@R) and stress testing are the appropriate risk measures for a retail electricity portfolio. 3
  4. 4. Cash Flow at Risk (CF@R)• Value at risk (V@R) is a probabilistic measure of the reduction in an asset’s value from adverse changes in market prices over a holding period. The holding period is typically anywhere from one day to one week.• V@R assumes that markets are liquid and continuous. All trades can be liquidated at market prices in the measured time horizon thereby limiting losses. Also, notional contractual quantities are fixed and known.• Cash Flow at Risk (CF@R) is a probabilistic measure of the reduction in operating cash flows through delivery from an adverse movement in market prices and usage.• CF@R assumes that all trades are held through delivery. This allows the CF@R model to allow the contractual quantities to vary from the expected, and to be correlated with changes in market prices. 4
  5. 5. CF@R and Retail Contracts• Retail load contracts are bilateral contracts between a supplier and an end user to deliver electricity and other associated products to the end user.• Since these are contracts for the delivery of electricity they are not subject to termination and the supplier has the obligation to meet delivery.• The contracted quantity is a forecast and not known until delivery.• The quantity consumed by the end user is as a rule positively correlated with spot market prices.• The market for retail electricity contracts is not liquid and continuous. All trades must be carried through delivery, and any liquidations would take longer than the typical V@R holding period.• CF@R is a more appropriate measure of risk to use when deciding on a hedging plan and measuring risk. 5
  6. 6. CF@R Calculation• The only practical method to calculate CF@R for a retail business is through simulation analysis (Monte Carlo).• Because we are estimating cash flows through delivery we need to simulate the correlated behavior of spot prices, forward prices and customer load.• Ideally this needs to be done at the hourly level to simulate extreme spot price behavior.• Need to model spot prices and load with the following behavior: – Mean reversion – Jumps and Spikes 6
  7. 7. Stress Testing• CF@R is only as good as our ability to model price-load behavior.• Historic data on the price-load relationship at the hourly level captures the major driver of load and price volatility – weather.• Weather is the major driver of volatility, but not the only driver.• Other drivers exist that can have large impacts on cash flows, but may have either a low probability of occurring, or difficult to model.• One of these is the economy. Economic booms and busts can have large impacts on the demand for electricity and by consequence a large impact on price.• Stress testing is the only practical method of calculating the impact on cash flows of economic impacts.• Any hedging plan needs to incorporate stress testing as a basic tool to judge the effectiveness of the hedge structure. 7
  8. 8. Retail Structures• There are numerous retail pricing structures: – Fixed Price Full Requirements – Fixed Price Block (customer purchases a fixed block at a fixed price) – Floating Index (customer takes the energy at the hourly LMP) – Block & Index (fixed block plus the remainder of the load is index) – Hybrid (incorporate all the above) – Triggers (customer can trigger a block purchase at a known price target)• All of these structures have varying degrees of risk. The structure with the greatest risk for a retailer is the fixed price full requirements contract. I will concentrate my talk on this retail structure. 8
  9. 9. What is a Full Requirements Load Following Contract• Full Requirements Load Following: A fixed price agreement to serve all the electricity load of a customer, and provide all products required to supply the electric load, for a pre-determined interval of time, without restrictions on volume. Typically served at a fixed rate per MWH.• Responsible for the hourly cost to serve the load.• Typical key products to be supplied: – Load Following Energy – Capacity – Transmission – Ancillaries – RECs 9
  10. 10. Load Following Energy• In a load following contract the supplier is obligated to supply the energy demanded (consumed) by the end-user on an hourly basis. Supply equals demand instantaneously.• In most deregulated markets (ISOs) the cost of this energy is determined by the topology of the transmission grid and current generation costs and transmission constraints. Hence, market prices are determined hourly.• The load following cost is the mathematical sum of price times hourly demand for all hours served.• The next slide depicts the load shape of a “typical” end-user. 10
  11. 11. Load Following Energy Diagram 16 15 14 MWH Served 13 Load Following MWH 12 11 10 9 8MW 7 6 5 4 Off-Peak Block On-Peak Block 3 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Hours 11
  12. 12. Expected Cost to Serve Load• Model definitions: – Si = Spot price in hour i (random variable) – Li = Load in hour i (random variable) – Covi = Covariance between S and L in hour i Cov(Si,Li). – i = hours in the month i = 1,…,N – Averages will be denoted with a bar over the variable – Expectations will be taken at time t given information available up to and including time t. Referenced by a subscript t. Sit = E t [ Si ] and Lit = E t [L i ] N 1 St = N ∑ E [ P ] = Forward Value of Power i =1 t i N 1 Lt = N ∑ E [ L ] = Forward Value of Expected Load i =1 t i 12
  13. 13. Expected Cost to Serve Load N• Cost to serve load: Cost = ∑S L i =1 i i  N • Expected cost to serve load : Et [ Cost ] = Et  Si Li    i =1   ∑• Taking expectations and solving we get:  N  N ∑ Et [ Cost ] = Et   Si Li  =  i =1   ∑ {S i =1 i t ⋅ Lit + Cov ( Si , Li )} ∑(L ) N N Et [Cost ] = N ⋅ St ⋅ Lt + i =1 i t − Lt St + i ∑Cov ( S ,L ) i =1 i i 13
  14. 14. Expected Cost to Serve Load• This equation shows how the expected cost to serve the load can be broken down into a block cost, shaping cost and covariance cost. ∑( ) N N (1) Et [ Cost ] = N ⋅ Pt ⋅ Lt + i =1 Lit − Lt Sti + ∑ Cov ( S ,L ) i =1 i i Expected Load Expected Covariance between Expected Block Shaping Cost. Price and Load. Randomness in Cost of Power. Prices and Load increases the Expected Cost to serve Load.• The covariance between price and load enters the expected cost to serve because of the randomness of prices and load.• Load shaping cost is the weighted average price of power where the weight is the deviation of the hourly load around the average load for the period. 14
  15. 15. Expected Cost to Serve Load• Divide equation (1) by the expected MWHs to obtain the cost per MWH. Et [ Cost ] = St + ∑ ( E [ L ] − L ) E [ S ] + ∑ Cov( S , L ) t i t t i i i N ⋅ Lt N ⋅ Lt Forward Price of Forward Cost of Load Shaping Block Power. and Load Following• Divide by the forward price of power: Et [ Cost ] =1 + ∑ ( E [ L ] − L ) E [ S ] + ∑ Cov( S , L ) t i t t i i i NSt Lt NSt Lt Load Shaping Factor (SFt) 15
  16. 16. Expected Cost to Serve Load Equation (1) can be written as in equation (2) below. ( 2) Et [ Cost ] = (1 + SFt ) St N LtThe expected shaping factor, SFt , is the ratio of the sum of theexpected hourly load shaping cost plus the expected hourlycovariance cost to the expected block cost.The shaping factor is the cost (as a percentage) above ourblock cost to cover the expected cost arising from the covariancebetween hourly load and price. 16
  17. 17. Expected Profit Function• The expected profit function for a fixed price load following contract is just the expected revenue less the expected cost.• The expected revenue is the fixed (known) revenue rate times the expected energy. (3) πt = R ⋅ N ⋅ Lt - (1 + SFt ) ⋅ St ⋅ N ⋅ Lt R = Revenue Rate (Contract Price) $/MWh or ¢/KWh N ⋅ Lt = Expected Energy Demand MWh or KWh 17
  18. 18. Market Price Risk: Delta Hedging• The objective of a hedging strategy is to create a portfolio that is riskless over a small interval of time due to changes in the random components.• For a linear contract a delta hedge will provide a hedge against market price movements.• I will show that a load following contract is a non-linear contract where a delta hedge will provide only a first-order approximation to changes in market prices.• Since the cost to serve is a function not only of random prices but random loads, changes in expected load can increase or decrease expected cost.• This section breaks down and explains the hedging components of a load following contract. 18
  19. 19. Delta Hedging the Expected Cost• Start with the expected cost function, equation (1) and take a Taylor series expansion with respect to prices and loads. N (4) ∆Et [ Cost ] = ∑ ( Lit ) ∆Pti + ∆Lit ( Pti ) + ∂Cov i ∆Pti + ∂Cov i ∆Lit +  ∂P i ∂Li i =1 t t• Where  refers to higher order terms. Neglecting these terms we can write the change in expected cost as:  ∂Pti ∂Cov i ∂Pti   N  i ∂Cov i  ∂Lit  ∑( ) N =  N ⋅ Lt +   i =1 Lit − Lt ∂Pt +  ∆P +   P + ∂Pti ∂Pt  t  i =1  t   ∑  ∂Lit  ∂Lt  ∆Lt   (5) Price Hedge Load Hedge The delta on a load following After delta hedging the price risk contract does not equal 1.0 we are left with the first order load risk or Gamma risk. 19
  20. 20. Delta Hedging the Expected Cost• The last equation on the previous slide has two terms that need to be defined.• The first tem represents the change in the expected price at hour i from a change in the forward price of power: ∂P i t ∂Pt• The second term represents the change in the expected load in hour i from a change in the expected average level of load: ∂Li t ∂Lt• Both of these quantities reflect the change in the shape of the hourly price and load shape with respect to a change in the underlying average price and load. 20
  21. 21. Delta of a Load Following Contract• Start with equation (2): (2) E t [ Cost ] = (1 + SFt ) ⋅ St ⋅ N ⋅ Lt• Take the first derivative of equation (2) with respect to price. ∂Cost t ∂SF = (1 + SFt ) ⋅ N ⋅ Lt + St ⋅ N ⋅ Lt ∂ St ∂ St• Disregarding the shaping factor impact the delta of a load following contract equals: (6) Delta = ΔS = (1 + SFt ) ⋅ N ⋅ Lt ≠ N ⋅ Lt In general, the delta of load following contract is greater than the notional quantity when the shaping factor is positive. SF can be anywhere from 0% to 15%, depending on the customer type. 21
  22. 22. What is the Expected Load L ?• What quantity do we use to hedge?• Since load is a random variable we want to hedge the expected quantity.• Why is load random: Weather and economic/business conditions.• Weather has a “known” distribution that can be used to calculate the expected weather (normal weather).• Need to model load as a function of weather.• Economic/business conditions can be either specific to the industry or customer or general economic conditions.• Business conditions specific to the customer can be estimated from past or known customer usage patterns (plant closures, retooling).• General economic conditions include recessions and economic boom periods. 22
  23. 23. Weather• In the short run weather has the most volatility and has the greatest impact on electricity usage.• Weather has a long history of data from which to imply a distribution.• We can estimate the expected load using either of two methods: – 1. Define a normal or expected weather pattern from the data and use the estimated load/weather function to calculate the expected load. – 2. Run the historic weather through the estimated load/weather function to obtain estimates of the load under each year’s historic weather, and then take an average of the estimated loads.• The result is a weather normalized load estimate.• Question: What is the proper historic period to use?• Answer: No agreed upon standard.• The NWS uses a 30-year normal updated every decade. Weather derivative markets use a 10-year average. 23
  24. 24. Economic and Business Conditions• Usage conditions specific to the customer or industry can be applied after the load has been weather normalized.• Specific conditions include planned shutdowns for maintenance and retooling. This type of information can be obtained from the customer or from past usage.• Other specific conditions can arise from an expansion in a customer’s load from favorable economic or business conditions, such as an expansion of a service line or warehouse space.• General economic growth conditions can effect all customers, but is not specific to a particular customer. Need to account for expected load growth (reductions) from growth (decline) in the national or regional economy. 24
  25. 25. Volumetric Risk• After delta hedging the market price risk we are left with the first order load risk or Gamma risk:  N  i ∂Cov i  ∂Lit  ∑ ∆E t [ Cost ] =   Pt +   i =1    ∆L ∂Lit  ∂Lt  t • Gamma risk is a volumetric risk.• Volumetric risk (or Swing risk) is defined as a cash flow risk caused by deviations in delivered volumes compared to expected volumes. The primary cause of these volumetric deviations is weather and economic conditions.• The delivered volumes cause a loss when the deviations in volume are positively correlated with market prices. 25
  26. 26. MW 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 0 6/1/05 9/1/05 12/1/0526 3/1/06 6/1/06 Summer 9/1/06 12/1/06 Winter 3/1/07 6/1/07 9/1/07 12/1/07 3/1/08 Date 6/1/08 9/1/08 12/1/08 3/1/09 June 2005 to December 2010 6/1/09 Cool Summer 9/1/09 Long-Run Demand in PSE&G 12/1/09 Recession 3/1/10 6/1/10 Hot Summer 9/1/10 12/1/10
  27. 27. Long-Run Correlation Between Price and Load 12-Month Rolling Average of Load and Price in PSE&G Zone 5,500 $90.00 $80.00 5,400 $70.00 5,300 $60.00 5,200 $50.00 $/MWHMW $40.00 5,100 MW $30.00 $/MWH 5,000 $20.00 4,900 $10.00 4,800 $0.00 May-06 Sep-06 Jan-07 May-07 Sep-07 Jan-08 May-08 Sep-08 Jan-09 May-09 Sep-09 Jan-10 May-10 Sep-10 Month/Yr 27
  28. 28. Short-Run Correlation Between Price and Load Hourly Load and Price in PSE&G Zone 7/12/10 to 7/17/10 $200.00 12,000 $180.00 Load (MW) Price ($/MWH) 10,000 $160.00 $140.00 8,000 $120.00$/MWH MW $100.00 6,000 $80.00 4,000 $60.00 $40.00 2,000 $20.00 $0.00 0 07/12/10 07/13/10 07/14/10 07/15/10 07/16/10 07/17/10 28
  29. 29. Sources of Swing Risk in Load Following Dispatch Economic Impact (A to B) CurvePower Price $/MWH  Weather – Principal source of swing risk. General Economic Conditions a b Weather Impact between a and b. B A Demand (MW) 29
  30. 30. What is Swing Risk?• Spot prices and quantities are positively correlated.• When quantity and prices both increase the cost to serve load increases. Profit not only becomes negative, but decreases at an increasing rate with an increase in prices.• When quantity and prices both decrease the cost to serve load decreases. Profit becomes positive, but increases at a decreasing rate as prices decrease.• Unlike a typical short sale, the short retail sale is non-linear.• When hedged with a linear instrument, the resulting position is negative and non-linear. 30
  31. 31. Retail Sale and Long Hedge $ Long Hedge Curvature is the product of the Swap (Linear) positive correlation between price and load. +P&L $/MWH Net: Swing Risk “Gamma” - Short Sale Short Retail Sale Non-linear 31
  32. 32. Change in Cash Flow Load greater Load less than Load equals than expected expected load expected load loadPrice less thanexpected price - 0 + Price equalsexpected price 0 0 0 Price greaterthan expected + 0 - price Long Short Hedged Position Position Swing Risk - - - - - - 32
  33. 33. Shaping Factor Risk NCost Function: Cost = ∑SL i =1 i i for hours i to NCan be written as follows after some manipulation: ∑ (L ) NCost = N ⋅ P ⋅ L + i − L Si i =1Adding zero to the above equation we get the following equation: ∑ (L )( ) NCost = N ⋅ P ⋅ L + i − L Si - S i =1 Covariance Function 33
  34. 34. Shaping Factor Risk Cost = N ⋅ S ⋅ L + N ⋅ σ S ⋅ σ L ⋅ ρ SL Shaping Cost = N ⋅ σ s ⋅ σ L ⋅ ρ SL Hourly Correlation between Price and LoadStandard Deviation of Hourly Price Standard Deviation of Hourly LoadHourly shaping cost is a function of the covariance between price and load.How do we hedge this correlation risk? 34
  35. 35. Increase in Shaping Factor Cost $ An increase in the shaping factor shifts the short sale curve downward, and the resulting Long Hedge net position shifts downward. The net position is negative at all price outcomes. +P&L $/MWH - Expected Net Position Realized Net Position Realized Short Sale Expected Short Sale 35
  36. 36. Decrease in Shaping Factor Cost An decrease in the shaping factor shifts the $ short sale curve upward, and the resulting net position shifts upward. The net position Long Hedge is positive at current price levels, but still has negative risk exposure. +P&L $/MWH Realized Net Position - Expected Net Position Realized Short Sale Expected Short Sale 36
  37. 37. Fair Value of a Load Following Contract• A fair price or fair value contract has an expected value of zero.• Fair value contracts require the inclusion of the expected value of the covariance between price and load, not just the expected hourly shaping cost. Excluding this cost component biases the distribution to the left.• But inclusion of the expected covariance in the contract price does not guarantee that the swing risk has been minimized or removed. It only guarantees that the contract is priced fairly.• We are still left with the negative tail risk from large positively correlated price and load movements. 37
  38. 38. Cash Flow Distribution: Swing Risk vs. Swing Cost Excluding the expected covariance produces a distribution with a negative expected value. 0.04 0.03 MeanDensity 0.02 Negative Skew: 0.01 Swing Risk 0.00 -60 -50 -40 -30 -20 -10 0 10 20 30 Cash Flow Swing Cost 38
  39. 39. Volumetric Hedging• Minimum Variance Hedge• Options – Synthetic Gamma Hedge• Slice of System – Gamma Hedge and Shaping Factor Hedge• Rate Design (T&C’s) 39
  40. 40. Minimum Variance Hedge40
  41. 41. Minimum Variance Hedge• Find the hedge quantity that minimizes the variance of the profit (CF@R) function.• Other alternatives, not considered here, are VAR and mean-variance methods.• Expected profit function: πt = R ⋅ N ⋅ Lt − Cost t , Equation (3).• Take the total differential of the profit function and add a hedge consisting of h power swaps.  ∂Cost t  ∂Cost t (7) dπ t =  R ⋅ N −  d L t +  d St + hd St  ∂ Lt  ∂ St• Now set the derivative of the variance with respect to h to zero and solve for h. [ ∂E ( dπ ) 2 = 0 ] ∂h ∂Cost t  ∂Cost t  E(d St d Lt ) (8) h= +  − RN  ∂St  ∂L  E(d S 2 )  t  t 41
  42. 42. Minimum Variance Hedge• We can simplify this equation to: ∂Costt  ∂Costt σ (9) h= +  − R ⋅N L ρ ∂St  ∂L σ  t  P• Where ρ equals the correlation between expected load and expected price.• The standard deviations are for expected load and expected price.• The first term in the equation is the price hedge defined in equation (6).• The second term is the correction to this hedge to take into account the correlation between price and load.• If the correlation is zero then the optimal hedge is the traditional price hedge. 42
  43. 43. Minimum Variance Hedge• Using the optimal hedge the residual unhedged variance is equal to: 2 [ E ( dπ ) 2 ]  ∂Cost t =   ∂L   (1 − ρ 2 ) σ L dt  2  t • If the correlation between expected load and expected price equals 1 or -1 then the residual unhedged variance equals zero. We have eliminated any risk over period dt.• If the correlation equals zero, then the residual unhedged variance equals the full amount of the load effect.• The greater the correlation the greater the effectiveness of the hedge. 43
  44. 44. Minimum Variance Hedge: Application• We need to determine the relationship between expected price and load.• If there is a linear relationship then we can estimated the following equation using OLS. Lt = α + β St + ε t σL• The coefficient on expected price, beta, is equal to β =ρ σS σ• The change in load can then be calculated as d L t = ρ L d St σS• We can now rewrite the hedge quantity for load as the following quantity of MWs. σL [ (10) h = (1 + SFt ) N Lt + (1 + SFt ) St − R Nρ σS ] β 44
  45. 45. Minimum Variance Hedge: Application < 0 if ρ > 0 σL  [ ] (1 + SFt ) St − R N ρ σ = > 0 if ρ < 0  S   0 if ρ = 0 • The first part of this equation is the negative of the expected margin. Sine margin is usually positive this quantity will be negative.• The standard deviations are positive.• If the correlation between expected load and prices is positive, then the optimal hedge adjustment is a negative quantity.• Need to take on a short position to hedge the change in expected load. Since price and quantity are positively correlated, a change in expected quantity will create a long position in the load following contract. The optimal hedge is then a short position.• If the correlation is negative then the optimal hedge adjustment is a positive quantity. Need to go long. Since price and quantity are negatively correlated, a change in expected quantity will create a short position in the load following contract. 45
  46. 46. Example Regression Output On-Peak PSE&G FP Contract Regression Statistics Multiple R 87.27% ˆ σ β July = ρ L = 7.18 + 11.90 = 19.08 R Square Adjusted R Square 76.16% 75.61% σS Standard Error 417.5841127 Observations 445A $1 change is pricesEquals a 7 MW change ANOVAin average daily peak load. df SS MS F Significance F Regression 10 241771008.8 24177100.88 138.6488552 2.4513E-128 Residual 434 75679397.18 174376.4912 Total 444 317450406 Coefficients Standard Error t Stat P-value Intercept 3,698.5486 64.27 57.55 0.00% LMP 7.1820 0.83 8.64 0.00% Mar_P -4.9284 1.02 -4.81 0.00% Apr_P -6.7488 1.01 -6.68 0.00% May_P -5.8316 0.96 -6.08 0.00% Jun_P 6.4907 0.73 8.94 0.00% Interactive dummy variables Jul_P 11.8957 0.73 16.30 0.00% Aug_P 10.2282 0.77 13.28 0.00% Sep_P 4.6888 0.84 5.61 0.00% Oct_P -2.2002 0.92 -2.38 1.77% Nov_P -3.5755 0.99 -3.62 0.03% 46
  47. 47. Hedge Calculation• Using the results of the regression output and equation (10) we can estimate the minimum variance hedge position.• Use July 2010 as an example for PSE&G FP Contract – SF = 5% – N = 336 – L = 3,700 MW h = (1.05) ⋅ 336 ⋅ 3,700 + margin ⋅ 19.08 ⋅ 336 h = 3,885 ⋅ 336 + margin ⋅ 19.08 ⋅ 336• Hedge 3,885 MWs less 19 MWs for each dollar of margin. 47
  48. 48. Static Option Hedge48
  49. 49. Short Gamma Hedge − Γ( P ) How do we create this hedge? + HedgeChange in P&L Monthly Average Price $/mwh gamma - Γ( P ) 49
  50. 50. Creating a Gamma Position from Options Use vanilla calls and puts to construct the gamma position. − Γ( P ) + − Γ( P ) ˆChange in P&L Monthly Average Price $/mwh - 50
  51. 51. Theoretical Model• It has been shown that a static hedge of plain vanilla options and forwards can be used to replicate any European derivative (Carr and Chou 2002, Carr and Madan 2001).• Any twice continuously differentiable payoff function, f (S ), of the terminal price S can be written as: F0 ∞ (11) f ( S ) = f ( F0 ) + f ′( F0 )( S − F0 ) + ∫ f ′′( K )( K − S ) + dK + ∫ f ′′( K )( S − K ) + dK 0 F0 Initial Delta Gamma Hedge: “Swing Risk” P&L Position• Our payoff function is the terminal profit. It can be decomposed into a static position in the day 1 P&L, initially costless forward contracts, and a continuum of out-of-the-money options. F0 is the initial forward price. 51
  52. 52. Theoretical Model• The initial value of the payoff must be the cost of the replicating portfolio. F0 ∞ V0 ( F0 ) = f ( F0 ) e −rT + ∫ f ′′( K ) P ( K , T ) dK + ∫ f ′′( K ) C ( K , T ) dK 0 F0• Where P(K,T) and C(K,T) are the initial values of out-of-the-money puts and calls respectively.• Interpretation of term within the integral: Second derivative of the payoff function representing the quantity of options bought or sold. – R = Fixed revenue rate – SF = Shaping Factor – L(S) = MWH, function of S (spot price of power) f ( S ) = ( R − (1 + SF ) S ) L( S ) f ′′( K ) = 2 (1 + SF ) ∂L ∂S 52
  53. 53. Solving for the Estimated Gamma Function• Select a series of strikes, Ki , and quantities, θi , to create a portfolio of puts and calls.• To estimate the gamma function we need to choose the amount of options for each strike, θi , so as to minimize the distance between the estimated gamma function and the true gamma function.• Estimated gamma function equals: N M (12) − Γ( P ) = ˆ ∑Max ( P − K , 0 ) ×θ + ∑Max ( K i =1 i i i =1 i − P,0 ) ×θi• Choose the optimal quantities by minimizing the sum of the squared errors between the true and estimated gamma function over a set of Q prices. 2 ∑ [Γ( P ) − Γ(P )] Q (13) min ˆ j j θ j =1 53
  54. 54. Estimating the Gamma Function• Need to estimate the relationship between load and price.• Use historic data to estimate the following regression equation. 11 11 Load t = α + β lmpt + ∑λ D + ∑φ D lmp i=1 i i i=1 i i t• The data for this equation is average load (peak, off-peak) and average price (peak, off-peak). LMP is the price, D is a monthly dummy variable, and DXLMP is an interactive dummy variable with price.• Next set up a portfolio of a short load sale and a long hedge using monthly forwards. The fixed rate on the load sale equals the RTC cost of serving the load ($/MWH).• Use the relationship estimated in the regression equation to vary the average monthly load with respect to a change in average monthly price. Use this to estimate the gamma function. 54
  55. 55. Example Regression Output On-Peak PSE&G FP Contract A $1 change is prices equals a 7 MW change in average daily peak load. For July the change equals 19.08 = 7.18 + 11.90. Regression Statistics ∂L Multiple R 87.27% = 7.18 + 11.90 = 19.08 R Square 76.16% ∂S Adjusted R Square 75.61% Standard Error 417.5841127 Observations 445 ANOVA df SS MS F Significance F Regression 10 241771008.8 24177100.88 138.6488552 2.4513E-128 Residual 434 75679397.18 174376.4912 Total 444 317450406 Coefficients Standard Error t Stat P-value Intercept 3,698.5486 64.27 57.55 0.00% LMP 7.1820 0.83 8.64 0.00% Mar_P -4.9284 1.02 -4.81 0.00% Apr_P -6.7488 1.01 -6.68 0.00% May_P -5.8316 0.96 -6.08 0.00% Jun_P 6.4907 0.73 8.94 0.00%Interactive dummy variables Jul_P 11.8957 0.73 16.30 0.00% Aug_P 10.2282 0.77 13.28 0.00% Sep_P 4.6888 0.84 5.61 0.00% Oct_P -2.2002 0.92 -2.38 1.77% Nov_P -3.5755 0.99 -3.62 0.03% 55
  56. 56. Hedge Calculation: Theoretical Calculation• Using the results of the regression output and equation (11) we can estimate the “optimal” static hedging position.• “Closed form” type solution.• Use July 2010 as an example for PSE&G FP Contract – SF = 5% – N = 336 – L = 3,700 MW f ′(F0 ) = ( 1 + SF) N L = (1.05) ⋅ 336 ⋅ 3,700 = 1,305,360 MWH (3,885 MW) ∂L f ′′(F0 ) = 2( 1 + SF ) N = 2 ⋅ (1.05) ⋅ 336 ⋅ 19.08 = 13,463 MWH (40 MW) ∂S F0 ∞ f (S) = f (F0 ) + 3,885 ⋅ 336 ⋅ ( S - F0 ) + ∫ 40 ⋅ 336 ⋅ ( K - S ) dK + 40 ⋅ 336 ⋅ ( S − K ) + dK + ∫ 0 F0 56
  57. 57. Interpretation• Delta hedge is 1.05 times the expected load. Buy 3,885 MWs of costless forward contracts.• Volumetric hedge is 40 MWs for each $1 movement away from the current forward price F0.• Purchase calls and puts (straddle and strangles) in 40 MW increments for each strike. Strikes are $1 increments away from the current forward price.• Sum of the option costs is the Swing or Volumetric hedge cost.• Doing this for the July 2010 PSEG FP Contract as of February 9, 2009 results in an estimated volumetric cost of $1.89/MWH.• The next slide depicts the estimated gamma function. 57
  58. 58. Example of a Gamma Function Estimate Estimated gamma function for July 2010 PSE&G FP load. The option cost equals $1.89/MWH per MWH served. $20,000 $18,000 $16,000 $14,000Change in P&L ($000) $12,000 -Gamma $10,000 Estimate $8,000 $6,000 $4,000 $2,000 $0 $0.00 $20.00 $40.00 $60.00 $80.00 $100.00 $120.00 $140.00 -$2,000 Cost as of February 9, 2009. Market Price 58
  59. 59. Options in Practice• While this is theoretically correct, it is not a practical method to price and hedge the volumetric risk.• First: Market is not liquid enough to carry out the theoretical hedge.• Second: Just calculating the dollar amount and charging the customer does not reduce the risk. An adder will just shift the distribution to the right, increasing the expected profit but not reducing the risk.• Need to purchase the options to reduce the volumetric risk.• Need to find a market executable option positions. This requires a discrete position in straddles and strangles.• Need to look for executable quantities (25MW blocks or larger).• Cannot hedge deal by deal, only practical for a portfolio. 59
  60. 60. Minimizing Cash Flow at Risk• In practice we cannot purchase options in such a way as to create the smooth curves depicted earlier. Instead we need to find discrete strikes so as to minimize the “swing risk”.• Swing risk is here defined as Cash Flow at Risk (CF@R). I am defining CF@R as the difference between the mean of the distribution and the 5 th percentile.• Since we cannot perfectly hedge the swing risk by purchasing a continuum of options we need another objective risk minimization strategy.• Use as a strategy the minimization of the CF@R or an objective level for the CF@R. An example would be to reduce the CF@R by 50%. 60
  61. 61. Reduce Cash Flow at Risk Reduce Cash Flow at Risk 0.06 Accountnig or Actuarial w ith Options 0.05 Accounting Model Hedged with Options 0.04 Delta HedgedDensit y 0.03 0.02 Swing Risk Reduced 0.01 0.00 -50 -40 -30 -20 -10 0 10 20 30 CashXFlow 61
  62. 62. Methodology• Use Monte Carlo simulation to model the load following contract and all hedges.• Run the model to estimate the expected cost to serve the load and establish the fair price of the contract.• Layer in delta hedges to estimate the cash flow distribution and estimate the CF@R.• Determine the amount of risk to be minimized. This is a management decision. Cut the CF@R by 50%.• Determine the portfolio of available options in the market.• Use an available optimization routine to determine the optimal option portfolio that meets the required risk criteria.• Or, try multiple strategies to determine the option strategy that meets your firm’s risk criteria. 62
  63. 63. Simulated Gamma Position Example uses NJ BGS CIEP Load for July. Approximately 80 MWs average load on-peak. $400,000 $200,000 $0 ($200,000) ($400,000)Total P&L ($600,000) ($800,000) ($1,000,000) ($1,200,000) ($1,400,000) ($1,600,000) $0 $50 $100 $150 $200 $250 $300 $350 Average On-Peak LMP 63
  64. 64. Cash Flow Distribution NJ BGS CIEP Load for July Swing Risk64
  65. 65. Cash Flow Distribution with Swing Hedge NJ BGS CIEP Load for July. Objective was to reduce CF@R by 50%. Swing Risk Removed 65
  66. 66. Efficient Frontier Analysis The efficient frontier tells what the minimum option cost would be to achieve a particular level of the 5th percentile. $0 +/- 10% Strangle ($200,000) ($400,000)5th Percentile +/- 30% Strangle ($600,000) ($800,000) ($1,000,000) ($1,200,000) $0 $200,000 $400,000 $600,000 $800,000 $1,000,000 $1,200,000 Option Cost 66
  67. 67. Slice of System67
  68. 68. Slice of System Hedge• Bilateral contract (swap) for the purchase of a fixed percentage of a known and published load shape settled at a fixed price at a known and published node (bus or hub).• Characteristics: – The volume is not fixed, but varies with the known load shape. – Pricing is fixed. – Mirror image of what is sold to the retail customer.• Can be sold as a swap on the load shape, or the buyer of the shape can sell back the block load and have left the residual gamma payoff. 68
  69. 69. How do we Hedge Correlation Risk?• Hourly shaping risk is a function of three factors: – Hourly correlation between load and price – Hourly standard deviation of price – Hourly standard deviation of load• How do we hedge these three factors?• Are there commercially available products?• The typical linear products and also non linear products (options) do not hedge this risk. The correlation exposure is do to underlying hourly price and load behavior which is not represented by options, even hourly options.• Standard deviations are also not traded products ant not reducible to the typical linear and non linear products. 69
  70. 70. Slice of System Hedge• The slice of system hedge is a mirror image of the short retail sale (to a degree).• Since the hedge will not be written on the exact same load, the hedge will always be imperfect. The degree of imperfection is a function of the correlation between the retail load and the hedge load definition.• The closer the correlation between the hedge load and the retail load the better the hedge performance (same for the settlement pricing index).• The next slide demonstrates the net hedge position.• As the retail load changes shape, the hedge load ought to change also if the drivers of the shape change are the same for both loads.• Hence, swing and shaping risk are more perfectly hedged with a slice of system product. 70
  71. 71. Slice of System Hedge $ Slice of Load Hedge+ Net $/MWH- Short Retail Sale 71
  72. 72. Slice of System Hedge: with Block Sale $ Slice of Load Hedge Net+ $/MWH- Block Sale 72
  73. 73. Slice of System Pricing Factors• Need to examine how the two load shapes and pricing nodes are correlated.• How are the three factors correlated – Price – Load – Shaping Factor• No closed form solution, need to use simulation analysis to understand the complexity of this structure.• Required is a load shape and pricing node that are “highly” correlated with the portfolio. 73
  74. 74. Methodology• Run portfolio without the slice of load to determine the CF@R.• Run the slice of system with the portfolio to determine the new CF@R.• Determine by how much the load shape decreases the CF@R in the portfolio. – Does it reduce the CF@R by the required amount? – Adjust volume, load zone or pricing zone. – The amount by which the hedge reduces the risk is a benchmark against which the load shape can be priced.• No objective market. 74
  75. 75. Benefits of a Slice of System• This trade hedges all aspects of the retail trade: – Delta Hedge (Market Price Risk) – Volumetric Hedge (Swing Risk) – Shaping Factor Hedge (Shaping Factor Risk)• This product also has some cons: – It is not a liquid product in the market. – Is a structured product. Customized. – Fewer sellers in the market. The natural sellers are generation owners. – Can be “expensive”. 75
  76. 76. Rate Design76
  77. 77. Rate Design as a Risk Minimization Tool• Rate design can also be a tool used to minimize risk in a retail contract.• Careful design of the terms and conditions in a contract can serve to reduce risk.• An example would be the use of usage band in a contract to protect the supplier from volumetric risk.• Any usage by the customer outside the bands would result in the customer paying for any excess costs not collected in the contracted energy rate.• The supplier needs to determine the amount of risk to be borne by the firm and how much can be transferred to the customer. As much a risk management decision as well as a competitive issue. 77
  78. 78. References• Carr P and Madan R, Optimal Positioning in Derivative Securities, Quantitative Finance Volume 1, 2002• Carr P and Chou A, Hedging Complex Barrier Options, working paper, 2002• De Martini P, A Survey of Volumetric Risk, The Risk Desk, Volume II No. 4• Humphreys B, and Gill R, Using a square peg, Energy Risk, April 2004• Kilinger A, Delta hedging the load serving deal, Energy Risk, September 2006• Kolos S, and Mardanov K, Pricing volumetric risk, Energy Risk, October 2008• Oum Y, Oren S, Deng S, Hedging Quantity Risks with Standard Power Options in a Competitive Wholesale Electricity Market, Navel research Logistics, Vol. 53, July 2006• Oum y, Oren S, Optimal Static Hedging of Volumetric Risk in a Competitive Wholesale Electricity Market, May 2007• Renne G, and Truesdell K, Volume Risk, Energy Politics, Issue IV, Spring 2008• Spencer L, The Risk that Wasn’t Hedged: So What’s Your Gamma Position?, October 2001 78
  79. 79. Contact Info Eric Meerdink Director, Structuring & Analytics Hess Corporation One Hess Plaza Woodbridge, NJ 07095 732-750-6591 emeerdink@hess.com79