Benchmarking the Price Reasonableness of an Electricity Tolling Agreement

Benchmarking the Price Reasonableness of an Electricity Tolling Agreement C.K. Woo is Senior Partner of Energy and Environmental Economics, Inc. (E3) in San Francisco. With more than 20 years in the electric utility business, Dr. Woo has published extensively in electricity economics, applied microeconomics, and applied finance. He received his Ph.D. in Economics from the University of California at Davis. Arne Olson is Senior Consultant at E3. He has more than 10 years of experience in the energy field, including seven years closely following Western electricity and natural gas markets. He holds a joint M.S. in International Energy Management and Policy from the University of Pennsylvania and the French Petroleum Institute. Ren Orans is Managing Partner of E3. Dr. Orans has worked exclusively in the energy business for more than 20 years. His work in utility planning centers on the design and use of area- and timespecific costs for numerous electric utilities. He received his Ph.D. in Civil Engineering from Stanford University.

June 2004

Given a lack of published transaction data for actively traded long-term contracts, the authors propose a regression-based approach to benchmarking the price reasonableness of an electricity tolling agreement. Chi-Keung Woo, Arne Olson, and Ren Orans

I. Introduction Recent research has demonstrated that the lack of long-term procurement policies by local distribution companies (LDCs) aggravated California’s electricity crisis of May 2000 to June 2001 by exposing the LDCs to wholesale spot electricity markets with extremely volatile prices.1 This exposure proved financially ruinous to Pacific Gas & Electric Company (PG&E), one of the largest LDCs in the United States, which declared bankruptcy on Apr. 6, 2001. Looking back from 2004, it is obvious that long-term supply contracts signed before May 2000 would have lessened

the severity of the crisis for California’s ratepayers and economy, and it has since become generally recognized that forward purchases must be an integral part of utility procurement plans. f course, supply contracts must be signed without the benefit of hindsight. However, regulatory policy that emphasizes after-the-fact prudence review may hinder an LDC’s ability to procure forward contracts by exposing it to the risk of cost disallowance, effectively forcing the LDC to choose between market risk and regulatory risk.2 The California Assembly addressed this issue in Assembly Bill (AB) 57, coming down

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squarely on the side of forward contracting. AB57, which became law on Sept. 25, 2002, directs ‘‘the Public Utilities Commission . . . to review each electrical corporation’s procurement plan in a manner that assures creation of diversified procurement portfolio, assures just and reasonable electricity rates, provides certainty to the electrical corporation in order to enhance its financial stability and credit worthiness, and eliminates the need, with certain exceptions, for after-thefact reasonableness reviews of an electrical corporation’s prospective electricity procurement performed consistent with an approved procurement plan’’ (Section 1(c)). n the absence of after-the-fact prudence review, a forwardlooking benchmark of the price reasonableness of electricity supply contracts signed by an LDC is a critical element of a prospective electricity procurement plan. Prudent management of spot and forward purchases suggests that an optimal allocation between spot and forward purchases to meet the LDC’s load obligation generally depends on whether the forward price is ‘‘reasonable’’ when compared to the spot price expectation and volatility.3 It is not always prudent to use forward contracts to completely eliminate procurement cost volatility because of the resulting high cost. At the same time, complete reliance on the highly volatile spot market exposes ratepayers to the possibility of extreme price swings.

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Woo et al. (2001) proposed a regression-based approach4 for benchmarking the price reasonableness of supply offers submitted to LDCs.5 Under this approach, regression analysis is used to develop a forecast of electricity spot prices and an associated variance. Supply offers that are below the expected spot prices over the term of the offer are considered price-reasonable, while supply offers that are sig-

It is not always prudent to use forward contracts to completely eliminate procurement cost volatility because of the resulting high cost.

nificantly above expected spot prices are considered priceunreasonable. Supply offers that are somewhat above expected spot prices may also be considered reasonable because they reduce the buyer’s exposure to spot price volatility. his article extends that approach to the case of a tolling agreement. Under a tolling agreement, an LDC pays the seller a fixed per kilowatt (kW-month or kW-year) payment for making available capacity from a particular generating unit under an agreed-upon set of terms (such as a heat rate, in Btu/kWh). In exchange, the LDC obtains the

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right, but not the obligation, to dispatch the unit to generate electricity using LDC-procured fuel. In effect, the seller leases the unit’s capacity to the LDC, whose cost-minimization objective leads to least-cost dispatch of the leased capacity whenever its variable cost, mainly fuel, is less than the spot electricity price. The per-kW payment is therefore the calloption premium of 1 kW of capacity with a specific heat rate, made available by the seller to the LDC for the term of the agreement. A tolling agreement is attractive to a generation owner because it produces a known positive cash flow, while the buyer assumes the price risks for both the fuel input and electricity output. In exchange for accepting the commodity risks, the buyer gains the ability to dispatch, providing a tool to manage those risks by enabling it to choose whether to obtain electricity by dispatching the tolling unit or by purchasing in the wholesale market. In this article, we develop a method for determining a reasonable capacity payment by an LDC for a tolling agreement with a specific term and heat rate.

II. Least-Cost Dispatch of a Tolling Agreement A tolling agreement gives the LDC the right, but not the obligation, to dispatch a specified generation unit during the term of the agreement. The LDC procures the fuel, typically natural gas, and The Electricity Journal

absorbs the fuel price risk. To minimize the agreement’s ongoing variable cost, the LDC dispatches the unit whenever the spot electricity price exceeds the unit’s per-MWh variable cost, including fuel. The agreement’s daily, per-MWh benefit to the LDC is the positive difference, or the ‘‘spark spread,’’ between the daily spot electricity price and the unit’s daily per-MWh fuel cost, less the non-fuel variable cost. This daily per-MWh benefit is the payoff of a call option with daily strike price equal to the generation unit’s daily per-MWh fuel cost. The daily-varying strike prices over a long contract period, however, make the standard option pricing formulae difficult to apply in determining the agreement’s call-option value.6 This motivates our development of a practical alternative that requires only published data readily available to an LDC. In exchange for the dispatch right, the LDC makes a capacity payment (e.g., in $/kW-month) to the seller, which is the call-option premium paid by the LDC for having 1 kW of capacity available for a month at an agreed-upon heat rate, regardless of the unit’s actual output. Hence, the capacity payment is equivalent to a fixed MWh payment for each MWh obtained by the LDC to meet its load obligation, irrespective of whether that MWh comes from the unit or the spot electricity market. Specifically, the perMWh capacity charge is the per kW-month capacity payment times 1,000 and divided by the June 2004

number of hours (e.g., 744 for January) in the month. For a longterm agreement (say, over 10 years), the capacity payment should approach the LDC’s own cost to build and own the plant.7 east-cost dispatch of a tolling agreement by the LDC always results in a per-MWh variable cost that is lower and less volatile than the spot electricity price (except in the trivial case where the unit’s per-MWh fuel

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Least-cost dispatch of a tolling agreement by the LDC always results in a per-MWh variable cost that is lower and less volatile than the spot electricity price.

cost always exceeds the spot price and the unit is never dispatched). When compared to the spot purchase alternative, the degree of procurement cost and risk reduction is a function of the unit’s heat rate and the relationship between electricity and gas prices. The more frequently the unit is dispatched, the greater the cost and risk reduction.

III. A Statistical Approach to Benchmarking The benefit of a tolling agreement can be illustrated using

electricity forward and gas futures price data. Consider a tolling arrangement under the following simplifying assumptions:
The electricity forward price is $60/MWh for the term of the agreement;
The gas futures price is $4/ MMBtu for the term of the agreement;
The heat rate of the generating unit specified in the tolling agreement is 8,000 Btu/kWh (8 MMBtu/MWh); and
Non-fuel variable operations and maintenance (O&M) costs are negligible. Under these assumptions, the benefit of the tolling agreement is $28/MWh (¼$60/MWh  $4/ MMBtu  8 MMBtu/MWh).8 If the capacity payment is less than or equal to $28/MWh (say, $22/ MWh), the agreement is pricereasonable because the LDC gains $6/MWh (¼$28/MWh  $22/ MWh) by entering into the agreement. Of course, real life presents a number of complicating factors not captured in this simple example. Among these are the following:
Electricity forward prices are available only for a three-year period. No forward price data are available for the later years of a tolling agreement that lasts more than three years.
Electricity forwards typically apply to an on-peak period (e.g., 06:00–22:00, Monday–Saturday), and therefore cannot inform the LDC about the tolling agreement’s per-MWh benefit in the hours outside the on-peak period.

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The calculation of the perMWh benefit in the example assumes no daily price variations over the tolling agreement’s contract period. However, the LDC’s daily dispatch decision is driven by whether the daily spot electricity price exceeds the daily per-MWh fuel cost. Even if the LDC has locked in a gas price via forward/futures contracting, the relevant gas price for the daily dispatch decision is the daily spot gas price, which measures the opportunity cost of gas.9 Hence, per-MWh benefit calculation requires electricity and gas price forecasts and their variances. he uncertainty in daily spot electricity and gas prices in future years implies that a tolling agreement’s price reasonableness must necessarily be statistical in nature. An LDC cannot state with 100 percent certainty the exact benefit of the agreement, even under forward contracting. What the LDC can do, however, is estimate the expected per-MWh benefit and the associated variance, thus permitting a statistical assessment of price reasonableness. Specifically, if a tolling agreement’s per-MWh capacity payment is less than the anticipated per-MWh benefit with a probability of 95 percent or more, the agreement is said to be ‘‘reasonable with almost certainty.’’ This criterion of price reasonableness is an application of the concept of ‘‘value at risk’’ in financial risk management.10 o formalize our approach, we define the following:

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(m1, s21 ) are the forecast of (1) the expectation of the weighted average of daily spot electricity prices by TOD (peak vs. off-peak) period over the tolling agreement’s contract period, and (2) the associated variance. Suppose the on-peak period is 06:00–22:00, Monday–Saturday, as used by the Mid-C spot electricity market. The daily onpeak weight is 2/3 (¼16 hours/ 24 hours) and off-peak weight 1/3

One complicating factor: Electricity forward prices are available only for a three-year period.

(¼8 hours/24 hours) for Monday–Saturday. The on-peak weight is 0.0 and off-peak weight 1.0 for Sunday.
(m2, s22 ) are the forecast of (1) expected average per-MWh variable cost over the tolling agreement’s contract period, and (2) the associated variance. The daily per-MWh variable cost by TOD period is the lesser of (a) the daily spot electricity price by TOD period, and (b) the agreement’s per-MWh fuel cost. The daily perMWh variable cost is then the weighted average of the perMWh variable costs by TOD period. Because of daily least-cost dispatching, the per-MWh vari-

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able cost must be smaller and less volatile than the daily spot electricity price; hence, m2 < m1 and s22 < s21 .
s12 ¼ rs1s2 is the covariance between spot electricity price and per-MWh variable cost of the tolling agreement, with r being the correlation coefficient. Since gas price influences electricity price and vice versa, we expect r > 0.11 Now, the expected average benefit per MWh is m ¼ (m1  m2) > 0, with variance s2 ¼ s21  2rs1s2 þ s22 . Because our focus is the average per-MWh benefit over the entire term of the agreement, we can safely assume, based on the Central Limit Theorem,12 that the per-MWh benefit is normally distributed with mean m and variance s2. Since the derivation of (m, s2) is somewhat lengthy, it is relegated to an appendix. Given (m, s2), an LDC can readily calculate the probability of the ex post but unknown per-MWh benefit B exceeding a per-MWh capacity payment A. This probability is Prob(B > A ¼ m þ zas) ¼ Prob((B  m)/s > za) ¼ (1  a), where za ¼ standard normal variate at (1  a) percent (e.g., za ¼ 1.65 if a ¼ 5% and za ¼ 1.65 if a ¼ 95%). The tolling agreement is price-reasonable if the probability of a net gain (Prob(B > A)) is at least 50 percent, that is, the agreement is expected to reduce LDC procurement costs relative to spot market purchases. However, because a tolling agreement also reduces the LDC’s procurement cost variability, Prob(B > A) < 50% does not always mean the tolling agreement is price-unreasonable. The Electricity Journal

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sing the expectation and variance forecasts, one can compute the maximum cost exposure by procurement alternative under normal circumstances at a 95 percent probability level.13 Under the spot market purchase alternative, the exposure is L1 ¼ m1 þ 1.65s1. Under the tolling agreement, the exposure is L2 ¼ A þ m2 þ 1.65s2. The reasonableness of the tolling agreement is related to the change in exposure of a tolling agreement relative to the spot purchase alternative, or DL ¼ (L2  L1) ¼ [A  (m1  m2)] þ 1.65(s2  s1). The change in the total procurement cost expectation is DC ¼ A  (m1  m2). The DL and DC values help determine price-reasonableness in the following three cases:14
Case 1: DL < 0 and DC < 0. The tolling agreement reduces the procurement cost exposure and the procurement cost expectation. The agreement is per se pricereasonable.
Case 2: DL > 0 and DC > 0. The tolling agreement raises both the procurement cost exposure and the procurement cost expectation. The agreement is per se priceunreasonable.
Case 3: DL < 0 and DC > 0. The tolling agreement reduces the procurement cost exposure but raises the procurement cost expectation. In this case, the agreement’s price-reasonableness hinges on the buyer’s risk tolerance, i.e., the buyer’s perception of whether the risk reduction DL is worth the expected cost increase DC. Our proposed approach is consistent with those used by June 2004

state and federal regulators in the U.S.15 At the state level, it mirrors the avoided-cost pricing principle in the 1978 Public Utility Regulatory Policy Act (PURPA). As the spot market purchase is the LDC’s default supply source, the avoided cost is the tolling agreement’s expected per-MWh benefit (i.e., m ¼ m1  m2). At the federal level, the Federal Energy Regulatory Commission’s (FERC) price benchmarking is a market-price test. In Ocean State II, FERC clearly distinguished the market-based approach from the cost-based one: ‘‘[i]n the case of a market-based formula rate . . . neither the rate (which is the formula itself) nor any of its individual components has to be justified on a cost basis. Rather, if the Commission is satisfied that the rate results from competitive market forces or that the seller does not have market power over the buyer we do not examine the underlying cost structure of the seller.’’16 Our statistical benchmark is developed using spot and futures price data generated by competitive markets, and is therefore a market-based approach consistent with Ocean State II.

IV. Benchmark for a Hypothetical Five-Year Tolling Agreement We conclude by illustrating the practical usefulness of our approach using a hypothetical five-year tolling agreement at a heat rate of 8,000 Btu/kWh, with Mid-Columbia as the delivery point. Based on the statistical results in the appendix, Table 1 presents the expectation and variance forecasts of spot electricity prices and of the agreement’s per-MWh variable cost. Based on Table 1, the tolling agreement’s expected benefit (m ¼ m1  m2) is $3.56/MWh, with a variance of $0.278/MWh and a standard deviation of $0.527/MWh. Table 2 presents the estimated probability of a per-MWh capacity charge exceeding the ex post per-MWh benefit. If the capacity payment is below $2.69/MWh, the tolling agreement is per se price-reasonable because Prob(B > A) > 95%. Indeed, the tolling agreement should be considered price-reasonable whenever the capacity payment is less than the expected benefit of $3.56/MWh. Conversely, if the capacity payment is above $4.43/MWh, the

Table 1: Five-Year (11/03–10/08) Forecast of Expectation and Variance of Spot Electricity Price and a Tolling Agreement’s Per-MWh Variable Cost with Mid-C Delivery Statistic Expectation Variance Standard deviation Correlation coefficient Covariance

Spot Electricity

Per-MWh Variable

Price ($/MWh)

Cost ($/MWh)

37.52

33.96

0.724 0.851

0.303 0.550 0.80 0.38

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Table 2: Price Reasonableness of a Five-Year (11/03–10/08) Tolling Agreement with an 8,000 Btu/kWh Heat Rate Change in Hypothetical Capacity

Probability of Ex Post Benefit

Change in Procurement

Change in Procurement

Procurement Cost Exposure per $1

Payment:

B Exceeding

Cost Expectation:

Cost Exposure:

Change in Cost

Price Reasonableness

A ($/MWh)

K: Prob(B > A)

DC ($/MWh)

DL ($/MWh)

Expectation: DL/DC

Determination

Not needed

$2.75

0.95

$0.81

$1.36

$3.34 $3.56

0.84 0.50

$0.53 $0.00

$1.02 $0.49

$3.74

0.37

$0.18

$0.32

With lower cost exposure and expectation, the tolling agreement is reasonable

1.83

Since a $1/MWh increase in cost expectation yields a $1.83/MWh reduction in cost exposure, the tolling agreement may be reasonable depending on the

$3.93

0.24

$0.13

$0.37

0.34

buyer’s risk tolerance Since a $1/MWh increase in cost expectation yields only a $0.34/MWh reduction in cost exposure, the tolling agreement is unreasonable

$4.08 $4.43

0.16 0.05

$0.53 $0.87

$0.03 $0.37

Not needed

With higher cost exposure and expectation, the tolling agreement is unreasonable

tolling agreement is per se priceunreasonable because Prob(B > A) < 5%. Likewise, a capacity payment of $3.93/MWh should also be considered price-unreasonable, because a $1/MWh increase in cost expectation yields only a $0.34/MWh reduction in cost exposure. apacity payments between $3.56 and $3.81/MWh should be subject to regulatory review that would determine if (DL/DC) < 0 is an acceptable reduction in cost exposure for each dollar increase in cost expectation. Table 2 shows that at a capacity payment of $3.74/ MWh, a $1/MWh increase in cost expectation yields a $1.83/MWh reduction in cost exposure. Whether this level is price-rea-

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sonable depends the buyer’s risk tolerance. The preceding illustration shows that our proposed approach is a practical and useful alternative to other benchmarking alternatives. Whereas FERC’s market-based test requires data for contracts that are comparable and contemporaneous to a long-term tolling agreement,17 our approach uses only readily available spot and futures price data generated by competitive markets. Hence, our approach is not constrained by data limitations due to contract confidentiality and thin trading. hen compared to other computation methods (e.g., complicated option-value

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formulae and Monte Carlo simulation),18 our approach is relatively easy to apply and understand. Even if an LDC can apply other computational methods, it cannot easily make a probabilistic assessment of the price-reasonableness of a tolling agreement. This is notwithstanding that price reasonableness of a tolling agreement before actual delivery is necessarily ex ante and statistical in nature.&

Appendix A. Forecasting Expectation and Variance A.1. Model For the sake of clarity, we describe the model for forecasting The Electricity Journal

expectation and variance with reference to the spot electricity price. Later we shall modify the discussion to capture the tolling agreement’s per-MWh cost. o forecast the expectation and variance of the spot electricity price, consider the following regression that depicts the equilibrium price condition:

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yt  ¼ a þ bxt þ et

(1)

where yt ¼ equilibrium daily average spot electricity price (say, at Mid-C) on day t; xt ¼ gas price (say, at Henry Hub) on day t; and et ¼ error on day t. Equation (1) is the daily price regression on which cross-hedging can be based for the following reasons:
Gas price drives electricity price because a gas price change affects the short-run per-MWh fuel costs of a marginal unit (which is typically gas-fired) dispatched to serve the market demand.
A local spot gas market, like Sumas in the Pacific Northwest, is integrated with Henry Hub, the most active spot and futures market in North America.19
There are no actively traded electricity forward/futures contracts with delivery beyond three years in North America. This precludes us from exploiting electricity forward/futures price data to estimate the spot electricity price expectation and variance for the last two years of a five-year tolling agreement.
Natural gas futures with Henry Hub delivery traded in NYMEX today cover the next 72 months, sufficiently long June 2004

for cross-hedging the spot electricity price risk in a five-year period. Suppose y is the Mid-C spot electricity price on a given day in the future. Cross-hedging against the volatility of y entails buying b MMBtu of gas futures now at price $F/MMBtu for each MWh delivered at Mid-C, taking gas delivery at Henry Hub, and sell-

ing the same gas at Henry Hub spot gas price $x/MMBtu.20 The net price for Mid-C spot electricity is y  bðx  FÞ ¼ a þ bF þ e

(2)

so that the regression’s error term e is the undiversifiable price risk and b is the minimum-variance hedge ratio.21 Since the daily market equilibrium price may not adjust instantaneously to random shocks, we postulate the observed price adjustment is a fraction (l) of the required adjustment:22 

yt  yt1 ¼ lðyt  yt1 Þ

(3)

which can be substituted into equation (2) to yield the estimable

regression: yt ¼ laþlbxt þ ð1  lÞyt1 þlet ¼ yþgxt þ fyt1 þZt (4) where y ¼ la, g ¼ lb, f ¼ (1  l), and Zt ¼ let. Using daily data for Mid-C spot electricity price and Henry Hub spot gas price, we apply the maximum likelihood method to estimate equation (4), so as to correct the potential bias caused by the lagged dependent variable as an explanatory variable in a regression that may have firstorder autoregressive (AR(1)) errors. The estimation will yield: (a) (q, g, f), the estimates for coefficients (y, g, f); (b) the estimate for the AR(1) parameter; (c) the covariance matrix for the coefficient estimates; and (d) the meansquared-errors (MSE) of the daily price regression. We can then infer, for example, the estimates for (a, b) as a ¼ q/(1  f), b ¼ g/ (1  f). Assuming no major structural change in the forecast period that can invalidate our regression estimates, we compute spot electricity price expectation and variance based on equation (2). Suppose Fk is the Henry Hub futures price for month k. Based on the estimated version of equation (2), a daily Mid-C electricity price on day t in that month is: ykt ¼ a þ bFk þ ekt

(5)

where ekt ¼ hkt/(1  f) ¼ forecast error, and hkt ¼ error based on equation (4). We compute the

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expectation of ykt as:23 mkt ¼ a þ bFk :

(6)

The monthly average of daily prices is X mk ¼ ða þ bFk þ ekt Þ=Nk t

where Nk ¼ number of days in the month. Under the Central Limit Theorem, mk is normally distributed. As Fk does not vary within the month, the expectation of mk is (a þ b Fk). Now, the variance of ykt is

tolling agreement’s cost expectation and variance. However, the estimation of equation (4) require data for the per-MWh variable cost of the agreement, which is described in the next section. The expected per-MWh variable cost is m2 and variance s22 , whose computation exactly follows equations (6) through (8b).

varðykt Þ ¼ varðaÞ þ varðbÞFk 2 þ2covða; bÞFk þ varðekt Þ: (7a) We compute var(a), var(b) and cov(a, b) on the right-hand-side of equation (7a) using linear approximation24 and var(ekt) ¼ regression errors’ unconditional variance/(1  f)2. The variance of the month’s average of the daily ykt values is varðmk Þ ¼ varðykt Þ=Nk :

(7b)

Equation (7b) makes sense because within a month, highprice days offset low-price days, implying that a monthly average price expectation should have a smaller variance than a daily expectation. The expectation forecast for a Kmonth period is X m1 ¼ mk =K: (8a) k

The variance forecast for the same period is X varðmk Þ=K2 : (8b) s21 ¼ k

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he preceding computational steps equally apply to a

A.2. Data Our sample contains daily observations reported by Platts on Mid-C on-peak (06:00–22:00, Monday–Saturday) price, Mid-C off-peak (hours outside the onpeak period) price, Sumas gas price, and Henry Hub gas price. The sample period is 07/02/01– 09/02/03, so chosen to avoid the electricity and gas price anomalies in the Western markets during the California energy crisis.25 s described in Section III of the main text, the daily MidC price is the weighted average of the on- and off-peak prices. At an assumed heat rate of 8,000 Btu/ kWh, the tolling agreement’s daily per-MWh variable cost is com-

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puted as follows. Based on historic data for Mid-C electricity price and Sumas spot gas price, we first construct the per-MWh variable cost by time-of-day (TOD) period (on- vs. off-peak) as Min[spot electricity price by TOD period, assumed heat rate  Sumas spot gas price]. The daily per-MWh variable cost is the weighted average of the daily on- and offpeak per-MWh variable costs. Table A.1 reports the summary statistics of the four daily data series and the Augmented Dickey-Fuller (ADF) statistics for testing if a series follows a random walk that may render regression results spurious.26 The summary statistics show that the series are moderately volatile with standard deviations about 40 percent of their respective means. The mean Mid-C spot electricity price is $29.45/MWh, $5.45/ MWh higher than the $24.01/ MWh mean variable cost of the tolling agreement. The electricity price is also more volatile with a standard deviation of $13.6/ MWh, higher than the $10.63/ MWh standard deviation of the per-MWh variable cost. The tolling agreement has lower mean cost and standard deviation than the spot purchase alternative because during 07/02/01–09/02/ 03, the on-peak Mid-C price was lower than the per-MWh fuel cost 68 percent of the time, and the offpeak Mid-C price 51 percent of time. The mean Henry Hub gas price is $3.94/MMBtu, about $0.70/MMBtu higher than the $3.25/MMBtu mean Sumas gas price. Absent economic dispatch, The Electricity Journal

Table A.1: Summary Statistics, ADF Statistics and Pair-wise Correlation Coefficients for Daily Mid-C Spot Electricity Prices, Daily Per-MWh Variable Cost of a Tolling Agreement with an 8,000 Btu/kWh Heat Rate, and Daily Henry Hub Spot Gas Price for the Period of 07/02/01–09/02/03 Tolling Agreement’s Mid-C Spot Electricity

Per-MWh Variable

Henry Hub Spot

Sumas Spot Gas

Price ($/MWh)

cost ($/MWh)

Gas Price ($/MMBtu)

Price ($/MMBtu)

Statistic Mean

29.45

24.01

3.94

3.25

Standard deviation ADF statistic

13.60 4.87

10.63 3.13

1.52 4.15

Mid-C price Tolling agreement per-MWh variable cost

1.0 0.80

0.80 1.0

0.59 0.84

0.67 0.93

Henry Hub gas price

0.59

0.84

1.0

0.94

Sumas gas price

0.67

0.93

0.94

1.0

1.33 2.47

Correlation coefficient

Note: ‘‘*’’ ¼ ‘‘Significant at the 5% level.’’ Data source: Platts.

the tolling agreement’s per-MWh variable cost based on the mean Sumas gas price would have been $26.0/MWh. Except for the Sumas gas price series, the ADF statistics reject the null hypothesis that a daily data series follows a random walk, implying that our Mid-C price and per-MWh variable cost regressions will not be spurious. he Mid-C price is highly correlated with per-MWh variable cost with a correlation coefficient of 0.8. It is moderately correlated with the two gas prices, with correlation coefficients around 0.6. The per-MWh variable cost is highly correlated with Henry Hub and Sumas gas prices, with correlation coefficients of 0.84 and 0.93, respectively.

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A.3. Regressions results Table A.2 presents the daily price and per-MWh variable cost regressions, from which the following observations emerge: June 2004


Both regressions have a good fit. The Mid-C price regression explains 92 percent of the price variance, and the per-MWh variable cost regression 96 percent.
The Henry Hub price is a statistically significant driver of the Mid-C price and the per-MWh variable cost. At Mid-C price equilibrium characterized by equations (1), (3) and (4), each $1/MMBtu increase in Henry Hub price would raise the Mid-C price by the estimated value of the minimum-variance ratio (i.e., b in equation (1)) of $7.8/MWh (¼1.280/(1  0.836)). The same $1/MMBtu increase would raise the equilibrium perMWh variable cost by $6.58/ MWh (¼1.250/(1  0.810)), which is less than the $8/MWh increase sans the LDC’s economic dispatch of the generation unit in the tolling agreement.
The positive difference between the two b estimates

suggests a long position in NYMEX gas futures that is 1.22 MMBtu (¼7.8 MMBtu  6.58 MMBtu) larger under the spot purchase alternative than the tolling agreement alternative, thus confirming our intuition that the former has a higher cost volatility than the latter.
Both the Mid-C price and the per-MWh variable cost data show systematic seasonal decline during April–June because of the spring hydro runoff in the Pacific Northwest. At the market price equilibrium, the Mid-C price decrease for April is 1.373/ (1  0.836) ¼ $8.372/MWh, May 2.194/(1  0.836) ¼ $13.34/ MWh, and June 2.424/(1  0.836) ¼ $14.8/MWh. At the tolling cost equilibrium, the tolling-cost decrease for April is –0.631/(1  0.810) ¼ $3.32/ MWh, May 1.549/(1  0.810) ¼ $8.15/MWh, and June 1.965/(1  0.810) ¼ $10.34/ MWh.

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Table A.2: Maximum Likelihood Estimation of Daily Price and Per-MWh Variable Cost Regressions with Lagged Dependent Variable and AR(1) Errors for the Sample Period of 07/02/01–09/02/03 Dependent Variable Tolling Agreement’s Per-MWh Variable Explanatory Variables and Regression Statistics Intercept Henry Hub price ($/MMBtu)

Mid-C Electricity Price ($/MWh)

0.047 (0.177)

0.021 (0.05) 1.280 (9.68)

Cost at 8,000 Btu/kWh Heat Rate



1.250 (9.68)

¼1, if April; 0, otherwise ¼1, if May; 0, otherwise

1.373 (2.53) 2.194 (3.92)

0.631 (1.74) 1.549 (4.16)

¼1, if June; 0, otherwise Lagged dependent variable

2.424 (4.20) 0.836 (52.2)

1.965 (4.87) 0.810 (42.2)

0.008 (0.04)

0.184 (4.72)

AR(1) parameter Total R-squared Mean squared error ADF statistic for testing H0: regression residuals follow a random walk

0.92 14.50

0.96 4.58

19.0

19.1

Note: ‘‘*’’ ¼ ‘‘Significant at the 5% level.’’ Values in ( ) are t-statistics.


The highly significant lagged dependent variables support using a partial adjustment model to explain the daily variations in the Mid-C spot electricity price and the tolling agreement’s perMWh variable cost. The estimated coefficient for the lagged Mid-C price is 0.836, implying that it takes six days (¼1/(1  0.836)) for the Mid-C price to regain equilibrium after being perturbed by a random shock or a Henry Hub price change. The estimated coefficient for the lagged tolling per-MWh cost is 0.810, implying that it takes five days (¼1/(1  0.810)) for the tolling-cost t to regain equilibrium after being perturbed by a random shock or a Henry Hub price change.
The Mid-C price regression’s errors are not serially correlated

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but the per-MWh variable cost regression’s errors are mildly so.
The Mid-C price regression’s MSE is $14.5/MWh, almost three times the per-MWh variable cost regression’s MSE of $4.58/MWh. The MSE comparison confirms that the spot purchase alternative has much higher undiversifiable risk than the tolling agreement.
The ADF statistics reject the hypothesis that the regression residuals follow a random walk, thus both regressions are not spurious. s a final check, we re-estimate the equation (4) by postulating that the AR(1) errors have time-varying variance that follows a GARCH(1,1) process.27 This more general specification, however, does not yield an unconditional variance suitable for our valuation of a five-year

A

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long-term tolling agreement because the estimated GARCH(1,1) process is non-stationary. Constraining the GARCH(1,1) to be stationary lead to implausibly large unconditional variances. We then restrict the lagged dependent variable coefficient to be zero, resulting in an instantaneous adjustment model. However, this model yields statistically significant and positive coefficient estimates for the monthly dummies. These estimates are implausible because they suggest prices and costs are higher during the spring runoff in April–June. Thus, our expectation and variance computation uses the regression results in Table A.2. A.4. Expectation and variance forecasts Using the NYMEX natural gas futures prices settled on Oct. 15, 2003, we use equation (6) to forecast the monthly electricity price expectation and equation (7b) the associated variance. The tolling agreement’s per-MWh variable cost expectation and variance forecasts are computed in the same manner. e then apply equation (8a) to these monthly expectation forecasts to find a five-year period’s average forecasts whose variance forecasts are determined using equation (8b). The covariance between the price and cost expectation estimates is the product of (a) the standard deviation of the 5-year price expectation, (b) the standard deviation of the tolling agreement’s five-year average

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The Electricity Journal

per-MWh cost expectation, and (c) the coefficient of correlation between the Mid-C spot price and the tolling agreement’s per-MWh variable cost in Table A.1. Endnotes: 1. Chi-Keung Woo, What Went Wrong in California’s Electricity Market?, ENERGY—THE INT’L J., 2001, 26, at 747–758; Ahmad Faruqui, Hung-po Chao, Vic Niemeyer, Jeremy Platt, and Karl Stahlkopf, Analyzing California’s Power Crisis, ENERGY J., 2001, 22 (4), at 29–52; Edward N. Krapels, Determining Prudence in a Hypervolatile and Illiquid Market, ELEC. J., 2001, 14 (4), at 38–40; John L. Jurewitz, California Electricity Debacle: A Guided Tour, ELEC. J., 2002, 15 (4), at 10–29; Frank A. Wolak, Diagnosing the California Electricity Crisis, ELEC. J., 2003, 16 (7), 11–37; James Bushnell, California’s Electricity Crisis: A Market Apart?, ENERGY POLICY, 2004, 32, at 1045–1052. 2. Chi-Keung Woo, Debra Lloyd and William Clayton, Did a Local Distribution Company Procure Prudently during the California Electricity Crisis?, ENERGY POLICY, in press. 3. Chi-Keung Woo, Rouslan Karimov and Ira Horowitz, Managing Electricity Procurement Cost and Risk by a Local Distribution Company, ENERGY POLICY, 2004, 32 (5), at 635–645; Chi-Keung Woo, Ira Horowitz, Brian Horii and Rouslan Karimov, The Efficient Frontier for Spot and Forward Purchases: An Application to Electricity, J. OPERATIONAL RES. SOC., in press. 4. Chi-Keung Woo, Ira Horowitz, and Khoa Hoang, Cross Hedging and Value at Risk: Wholesale Electricity Forward Contracts, ADVANCES IN INVESTMENT ANALYSIS & PORTFOLIO MGMT., 2001, 8, at 283–301. 5. Chi-Keung Woo, Debra Lloyd, Michael Borden, Ron Warrington and Carmen Baskette, A Robust InternetBased Auction to Procure Electricity Forwards, ENERGY—THE INT’L J., 2004, 29 (1), at 1–11. 6. The standard formulae are detailed in JOHN C. HULL, OPTIONS, FUTURES, AND OTHER DERIVATIVES, Third Ed., (Engle-

June 2004

wood Cliffs, NJ: Prentice-Hall, 1997). The formulae for exotic electricity options are in Shijie Deng, Blake Johnson and Aram Sogomonian, Exotic Electricity Options and Valuation of Electricity Generation and Transmission, DECISION SUPPORT SYSTEMS, 2001, 30 (3), at 383–392. Alternatively, a tolling agreement’s option value can be determined through Monte Carlo simulation; see ALEXANDER EYDELAND AND KRZYSZTOF WOLYNIEC, ENERGY AND POWER RISK MANAGEMENT (Wiley, 2003), at 340–341. 7. For computation of ownership cost, see Ren Orans, Chi-Keung Woo and William Clayton, Benchmarking the Price Reasonableness of a Long-Term Electricity Contract, E3 working paper, 2004. This paper is available by emailing C.K. Woo at [email protected]. 8. Gary W. Emery and Qingfeng (Wilson) Liu, An Analysis of the Relationship Between Electricity and Natural Gas Futures Prices, J. FUTURES MARKETS, 2002, 22 (2), at 95–122. 9. To see this point, suppose the spring hydro runoff in Pacific Northwest depresses the Mid-C spot electricity price on a given day to $10/MWh. Unexpected cold weather, however, causes the Sumas (a major delivery point in Pacific Northwest) spot gas price to surge to $5/MMBtu for that day. Further suppose that the LDC must procure for meeting its load obligation and it has two alternatives: (1) Buy spot electricity at $10/MWh from the Mid-C market and sell the contracted gas at $5/MMTU in the Sumas spot gas market; (2) Use the contracted gas to generate electricity at a cost of $40/MWh (¼$5/ MMBtu  8 MMBtu/MWh). The LDC chooses (1) because when compared to (2), it can gain $30/MWh (¼revenue from selling 8 MMBtu gas  cost of buying 1 MWh). 10. PHILIPPE JORION, VALUE AT RISK: THE NEW BENCHMARK FOR CONTROLLING MARKET RISK (Chicago: Irwin, 1997). 11. Edward N. Krapels, Was Gas to Blame? Exploring the Cause of California’s High Prices, PUB. UTIL. FORTNIGHTLY, 2001, 139 (4), at 28–36. 12. ALEXANDER M. MOOD, FRANKLIN A. GRAYBILL AND DUANE C. BOES, INTRODUC-

TION TO THE THEORY OF STATISTICS (New York: McGraw-Hill, 1974), at 195.

13. Chi-Keung Woo, Rouslan Karimov and Ira Horowitz, supra note 3. 14. The fourth case, where DL > 0 and DC < 0 is mathematically impossible since DL ¼ DC þ 1.65 (D2  D1) and D2 < D1. 15. Ren Orans, Chi-Keung Woo and William Clayton, supra note 7. 16. Ocean State Power II, 69 F.E.R.C. ô61,546 (1994). 17. Ren Orans, Chi-Keung Woo and William Clayton, supra note 7; FERC, supra note 15. 18. Shijie Deng, Blake Johnson and Aram Sogomonian; Alexander Eydeland and Krzysztof Wolyniec, supra note 6. 19. Martin King and Milan Cuc, Price Convergence in North American Natural Gas Spot Markets, ENERGY J., 1996, 17 (2), at 17–42 (1996); Won-Woo Lee, U.S. Lessons for Energy Industry Restructuring Based on Natural Gas and California Electricity Incidences, ENERGY POLICY, 2004, 32, at 237–259. 20. Chi-Keung Woo, Rouslan Karimov and Ira Horowitz, supra note 3. 21. Sheng-Syan Chen, Cheng-few Lee and Keshab Shrestha, Futures Hedge Ratios: A Review, Q. REV. ECON. & FIN., 2003, 43, at 433–465. 22. JAN KMENTA, ELEMENTS OF ECONOMETRICS (New York: MacMillan, 1971), at 476. 23. Chi-Keung Woo, Ira Horowitz, and Khoa Hoang supra note 4. 24. Alexander M. Mood, Franklin A. Graybill and Duane C. Boes, supra note 12, at 180–181. 25. Federal Energy Regulatory Commission, Final Report on Price Manipulation in Western Markets, Docket No. PA02-02-000 (2003). 26. ROBERT S. PINDYKE AND DANIEL L. RUBINFELD, ECONOMETRIC MODELS AND ECONOMIC FORECASTS (New York: McGraw-Hill, 1991), at 465–467. 27. CAROL ALEXANDER, MARKET MODELS: A GUIDE TO FINANCIAL DATA ANALYSIS (New York: John Wiley, 2001), at 72–75.

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