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Global Energy uses a stochastic formulation process to value power generation assets. This involves 3 key steps: 1) Establishing an expected long-term price forecast based on fundamental supply and demand analysis; 2) Estimating price uncertainty historically and using this to generate alternative price paths; 3) Evaluating assets against the different price paths. Global Energy captures correlations between electricity and fuel prices to better model asset economics over time. The process involves estimating short-term volatility with mean reversion and long-term random factors to develop stochastic parameters for outage and price drivers.
![Methodology
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Entergy Daily Vol (Monthly) Enteryg Reserve Margin
Higher
Reserve
Margins
Lower Volatility
SOURCE: Global Energy and Power Markets Week.
Volatility in power and fuel markets can be driven by various factors such as weather
patterns, load characteristics, transmission system, generation portfolio, transmission
access, market rules and market players. Some markets are fundamentally more volatile
than the others.
Volatility in power markets has decreased noticeably within the last few years. An influx
of new gas-fired generating units in most of North American markets has caused an
overbuilt market with high reserve margins in most areas. Figure 3-5 illustrates the
relationship between volatility and reserve margin for the Entergy market. The trend
lines clearly show the inverse relationship between reserve margin and the volatility.
Decrease in volatility is a rational outcome of the high reserve margins, because excess
amount of idle generation suppresses any price movement immediately. This will also
increase the mean reversion behavior in the power markets.
In the long term, volatility levels are expected to increase as reserve margins decrease. To
capture this fundamental market change, Global Energy modeled the volatility and mean
reversion rates by incorporating a term structure in stochastic parameters. Initial years’
volatility and mean reversion parameters are estimated by using more recent historical
data. For later years, all available historical data is used. The estimates are done based on
two to three levels of 2-year time intervals. The estimated parameters are summarized in
Appendix A.
Detailed Stochastic Model Description
The discrete time mathematical representation of the two-factor (short-term and long-
term) lognormal model is:
2/][Var)( 1,,,,1,1,,1,,1,, −−−−− −+−+−+= tntn
S
tn
S
tntntntntntntntn SSSLLLSS εσα
(1)](https://image.slidesharecdn.com/brstochasticbestpractice-160310174855/85/Br-stochastic-best-practice-8-320.jpg)
![Methodology
Power Generation BlueBook, 2005 3-9
Equation (8) is a special case of equation (1), when there is a constant equilibrium value,
instead of the general case of a stochastic, time-varying equilibrium value. The Var[ ]/2
term in equation (1) drops out in equation (8) because it is a theoretical “log-bias”
adjustment needed only when the equilibrium value L is stochastic.
Correlations across Commodities
Global Energy’s stochastic model then applies the appropriate correlation among the
short-term shocks for different stochastic entities and among the long-term shocks.
Correlation coefficients, identified in equations (4) and (5) respectively, are input into our
asset valuation model, PLANNING AND RISK™. Cholesky decompositions of the ST and
LT correlation matrices are then used to transform two vectors of independent standard
normal draws for each day into vectors of correlated draws from a multivariate standard
normal distribution.
The Cholesky decomposition is a transformation that may be applied to any positive-
definite matrix. It is sometimes described as a “matrix square root” because like a
traditional square root it can be “multiplied” (in a matrix sense) by itself to arrive back at
the original correlation matrix. If A is a positive-definite matrix, then it has a Cholesky
decomposition matrix C that satisfies
,' ACC = (10)
where C’ is the transposition matrix of C whose columns are the rows of C.
Correlation matrices are symmetric, with ones on the diagonal (since the correlation of a
variable with itself is 1), and with coefficients mnρ between commodities m and n
satisfying .10 << mnρ
The composite correlation is a reflection of the underlying price projections, short run
and long run correlations. Each price iteration will exhibit a different correlation
relationship.](https://image.slidesharecdn.com/brstochasticbestpractice-160310174855/85/Br-stochastic-best-practice-11-320.jpg)
![Methodology
3-10
Table 3-1
Stochastic Parameters
Algonquin New England
Algonquin/New
England Correlation
Alpha Sigma Alpha Sigma
0.079 0.199 0.049 0.108 0.578
Note: Alpha [αααα] Sigma[σσσσ]
SOURCE: Global Energy.
The table above gives the winter stochastic parameter estimation for New England power
and gas. Sigma is the daily volatility, which represents the day-to-day fluctuation of the
prices. As shown, the New England power market has 10.8 percent daily price volatility,
while Algonquin natural gas has 19.9 percent daily price volatility. Typically, natural gas
has higher price volatility in winter, while power has higher price volatility in summer.
Average gas price volatility used in the BlueBook is around 7 percent, and the estimates
vary by season ranging from 3 to 27 percent. The average daily power price volatility is
around 16 percent and the estimates vary by season ranging from 5 to 41 percent. These
figures are daily volatility estimates, and they should not be compared to annual
volatility figures, which are commonly used in commodity markets. In general higher
price volatility gives higher extrinsic (option) value for a power plant.
Alpha, the mean reversion rate, is the second important stochastic parameter, and it is a
factor instead of a percentage. The alpha for power and gas in the above table are 0.049
and 0.079 respectively. 0.079 means the shocked price reverts back to the expected level
in ~13 days (1/0.079), if we assume a linear mean reversion. In the BlueBook we use an
exponential reversion rate. In other words, the mean reversion rate is a factor of the
difference between the current price level and the expected (mean) price level. So a bigger
price shock creates a stronger response in the next time period. As the difference gets
smaller, the response gets weaker. The following graph illustrates the exponential mean
reversion behavior for a difference of 100 percent.
Figure 3-6
Exponential Mean Reversion Behavior
0%
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50%
75%
100%
0 10 20 30 40 50
Days
SOURCE: Global Energy.
In the BlueBook price formation process new stochastic draws around the expected prices
are preformed for each day. So the mean reversion effect is recalculated on a daily basis.](https://image.slidesharecdn.com/brstochasticbestpractice-160310174855/85/Br-stochastic-best-practice-12-320.jpg)
![Methodology
3-12
• Ramp rates and minimum up and down times;
• Energy limited generation;
• Emission functions and costs;
• Limited fuel modeling; and
• Other variable and fixed O&M costs.
Using the two-factor (short run and long run shocks), mean-reversion stochastic models
discussed earlier, Monte Carlo iterations are performed, with random draws used to
simulate stochastic variables as discrete time processes. Antithetic sampling and first
moment (mean) calibration are used to reduce sampling variance from the hypothesized
distributions. Daily draws are taken for average daily prices and loads, and weekly
random draws for forced outages. Within each week, generation units are committed and
dispatched as if they have perfect foresight of future values for that week.
The stochastic parameters are maintained within the PLANNING AND RISK™ data
structures as constant or time varying parameters. The stochastic simulation results are
written to the Monte Carlo output database when a simulation is run.
Combining Deterministic Forecast And Market Uncertainty
What does the extrinsic value mean?
When applying alternative price paths to any given asset there is a distribution of values.
There are three distinct valuation points worth considering.
Value of an inflexible or “base load” plant
First, a completely inflexible plant (e.g., a must run plant) that will generate (sell into the
market) regardless of market price will have a value in each period equal to [spark spread
x MW sold x number of hours].
Figure 3-7
Forward Contract Payoff
Price
Distribution
Profit
Impact
There is a linear relationship between the
price distribution and forward contracts
Break Even
Profit
Loss
$
SOURCE: Global Energy.
In this case, once enough iterations have been run, the average of the stochastic iterations
(the “expected” value) will equal the value of the initial forecast for a plant or contract
with no flexibility. In other words average or expected revenue is linear with expected](https://image.slidesharecdn.com/brstochasticbestpractice-160310174855/85/Br-stochastic-best-practice-14-320.jpg)
Br stochastic best practice
- 1. Best Practice # PaR 2005-1 Global Energy Best Practice: Stochastic Formulation Process Global Energy Modelers Workbench™ Global Energy Modelers’ Workbench is a strategic advisory service providing consulting quality best practice advice and advanced analytics services to enable Global Energy Software clients and Consulting Partners to turn their strategic questions into credible market analytics decision analysis results. Our objective is to leverage the modeling, analytics and market expertise of Global Energy Advisors staff of consultant experts to provide Best Practice advice on performing advanced energy analysis using software from Global Energy. Global Energy uses its PROSYMa fundamentals-based methodology to forecast power prices in each region of North America. Based on its proprietary MARKET ANALYTICS™ system—a proven data management and production simulation model—Global Energy simulates the operation of each region of North America. MARKET ANALYTICS™ is a sophisticated, relational database that operates with a state-of-the-art, multi-area, chronological production simulation model. This Modelers’ Workbench Best Practice summarizes Global Energy Advisors consulting best practice for market price formation using MARKET ANALYTICS™ and discusses in detail how Global Energy develops its long-term price forecast based on the above principles
- 3. Methodology Power Generation BlueBook, 2005 3-1 Overall Approach Global Energy’s valuation and portfolio analysis methodology best practice employs a simulation-based stochastic approach to asset valuation. This means that we create a large number of equally possible future price outcomes for power and fuel, and then value the power generation assets against each of these possible outcomes. The resultant valuations can then be presented in a number of ways including expected value, median value, and percentile values. These can then be compared with a deterministic valuation approach. We shall discuss the merits of different valuation approaches below. Our overall approach is based on three key steps: • Establish the starting long-term price forecast (which is used as the expected price path) taking into account known supply and demand conditions and their expected changes in the future; • Estimate the randomness or uncertainty of these long-term prices using historical data and in turn use this estimate of uncertainty to generate a number of alternative iterations of future prices; and • Evaluate the generation assets under the alternative price paths that have been generated. This process is illustrated in Figure 3-1. This section explains the details of these steps. Figure 3-1 Process Schematic – Valuation Process Hourly Power Prices for 76 Market Areas Individual Asset Valuation (100 Monte Carlo Iterations, detailed plant parameters) Demand (Hourly Load) Supply (Generators) Transmission Hourly Dispatch $/MWh MW Fuel & Emission Prices Supply and Demand Balance Outages Price Forecasting Historical Power & Fuel Prices (Liquid Power and Gas Trading Data) Electricity and Fuel Volatility Estimates (Long & Short Term Daily Volatility, Mean Reversion) Electricity and Fuel Correlation Estimates Stochastic Asset Valuation Intrinsic & Extrinsic Value Expected Average, Distributions, Annual cash flows, NPV Power & Fuel Prices WECC MAPP SPP ERCOT SERC MAIN ECAR MAAC NPCC FRCC CZP26 BC NEW MEXICO NBAJA CSDGE ARIZONA PALO VERDE N NEVADA CSCE UTAH LADWP CO EASTCO WEST CAROLINAS NEBRASKA ALBERTA SOUTH MONTANA WUMS IOWA W-ECAR MINNESOTA ALTW LA OTHER AECI SPPC SPPN SMAIN ENTERGY ALBERTA CENT-N ERCOT NORTH ERCOT SOUTH ERCOT WEST ERCOT HOUSTON SASK POWER MANITOBA SOUTHERN GRID FLORIDA TVA CE_NI WYOMING W DAKOTAS IDAHO NORTHWEST COB CNP15 IID S NEVADA MECS APS AEP FIRST ENERGY VP MARITIMES ONT EC NY WEST ONT MP NY CN NY CITYPJME X NY F NE NORTH PJMW X ONT- NORTH QUEBEC LONG ISLAND SEMA RI NE EASTNE WEST NY GHI NE CTSW ONT-NI ONT WEST KENTUCKY Canada New England New York Southeast TVA FRCC ERCOT MAPP MAIN N California Rockies AZ/NM SPP SoCal ECAR Northwest PJM Entergy ERCOT NORTHEAST WYOMING E Hourly Power Prices for 76 Market Areas Individual Asset Valuation (100 Monte Carlo Iterations, detailed plant parameters) Demand (Hourly Load) Supply (Generators) Transmission Hourly Dispatch $/MWh MW Hourly Dispatch $/MWh MW Fuel & Emission Prices Supply and Demand Balance Outages Price Forecasting Historical Power & Fuel Prices (Liquid Power and Gas Trading Data) Electricity and Fuel Volatility Estimates (Long & Short Term Daily Volatility, Mean Reversion) Electricity and Fuel Correlation Estimates Stochastic Asset Valuation Intrinsic & Extrinsic Value Expected Average, Distributions, Annual cash flows, NPV Power & Fuel Prices WECC MAPP SPP ERCOT SERC MAIN ECAR MAAC NPCC FRCC CZP26 BC NEW MEXICO NBAJA CSDGE ARIZONA PALO VERDE N NEVADA CSCE UTAH LADWP CO EASTCO WEST CAROLINAS NEBRASKA ALBERTA SOUTH MONTANA WUMS IOWA W-ECAR MINNESOTA ALTW LA OTHER AECI SPPC SPPN SMAIN ENTERGY ALBERTA CENT-N ERCOT NORTH ERCOT SOUTH ERCOT WEST ERCOT HOUSTON SASK POWER MANITOBA SOUTHERN GRID FLORIDA TVA CE_NI WYOMING W DAKOTAS IDAHO NORTHWEST COB CNP15 IID S NEVADA MECS APS AEP FIRST ENERGY VP MARITIMES ONT EC NY WEST ONT MP NY CN NY CITYPJME X NY F NE NORTH PJMW X ONT- NORTH QUEBEC LONG ISLAND SEMA RI NE EASTNE WEST NY GHI NE CTSW ONT-NI ONT WEST KENTUCKY Canada New England New York Southeast TVA FRCC ERCOT MAPP MAIN N California Rockies AZ/NM SPP SoCal ECAR Northwest PJM Entergy ERCOT NORTHEAST WYOMING E SOURCE: Global Energy.
- 4. Methodology 3-2 Alternative Approaches The traditional ways of evaluating power generation assets can be seen to have a number of shortcomings. They are: • Many deterministic models do not capture the value associated with the inherent flexibility of assets to respond to future changes in market conditions. As such they may understate asset value, particularly for those that are mid-merit or peaking. • Many “real option” financial models do not capture the complex operational constraints associated with actual plant operations. As such, they may overstate the asset value. • Many models do not directly capture the changing relationship between fuel and power prices over time which is key to asset valuation. As such, they may under or overstate the asset value. Global Energy strongly believes that starting with a consistent price forecast, developing the stochastic parameters, and then running the alternative simulated price paths through a full dispatch model is the most appropriate methodology for generation asset valuation. This stochastic analysis approach relies upon the expected or equilibrium price paths derived from Global Energy’s Price Formation Process to establish the equilibrium price forecasts. The equilibrium price forecast is based on Global Energy’s Power Market Advisory Service, Electricity and Fuel Price Outlook, which is updated every six months. In this outlook Global Energy uses a fundamentals-based methodology to forecast power prices in each region of North America. Based on its proprietary MARKET ANALYTICS™ system—a proven data management and production simulation model—Global Energy simulates the operation of each region of North America. MARKET ANALYTICS™ is a sophisticated, relational database that operates with a state-of-the-art, multi-area, chronological production simulation model. For a complete best practice description of this price formation process see the Global Energy Price Formation Best Practice. Global Energy’s Stochastic Formulation Process Having established the expected or equilibrium price paths from the above results, Global Energy uses its PLANNING AND RISK TM software solution to establish the stochastic parameters for the key drivers of plant outage, electricity price and fuel costs. Volatility and Correlations There has been significant discussion over the last few years on the underlying dynamics of power prices and their impact on potential price paths. Less focus has been placed on the correlation between power and fuel prices that is critical to power plant economics.
- 5. Methodology Power Generation BlueBook, 2005 3-3 The more recent “hybrid” models have stressed the importance of this relationship and Global Energy’s approach directly models this in a three-stage process. First, the underlying correlation between power and gas is linked through the price formulation process. Through this process a mean price stream is created that directly models the relationship between fuel costs and power prices through the forecast period. Second, a long-term random factor is added to these projections. The long-term random factors between power and fuel prices are correlated within each volatility basin. This factor represents the possible drift of the mean price projections over time. Third, a short-term random shock is applied with mean reversion. This means prices will “randomly walk” from the mean, but a reversion factor will be applied that “pulls” them back to the mean projection. These short-term factors are correlated within each volatility basin and shocks are correspondingly synchronized across volatility basins. What Does Stochastic Mean? In Ian Stewart’s Does God Play Dice?, he states the etymology of stochastic in the statement, “The Greek word stochastikos means ‘skillful in aiming’ and thus conveys the idea of using the laws of chance for personal benefit.”1 Generally, stochastic is used to indicate that a particular subject is seen from a point of view of randomness, as part of a probability theory it can predict how likely a particular outcome is. Stochastic is often used as a counterpart of the word “deterministic,” which means that random phenomena are not involved. A single die roll is a probabilistic system—there is a one in six chance that the roll will end with the five facing up. We cannot predict the outcome of the die roll, but we can assign some probability to how often certain events will happen. An important issue is the granularity of the starting price models. In this case we start with the hourly power prices that have been directly linked to the daily gas prices. This allows us to disaggregate volatility and correlations down to the daily level (the minimum gas price period) and ensure these critical profiles are not lost by an averaging process. More importantly we are able to project the changing relationship between gas and power prices through time. There are thus two “random” factors that affect electricity and fuel prices. Long Run Long run (LR) factors such as technology, population changes, and GDP differences will result in a long run random effect on prices. Long-term volatility tends to be small compared to the short-term shocks and these random effects will have a limited effect on individual years particularly in the near term, but will have an increasingly important 1 Stewart, I., 1989, Does God Play Dice? The Mathematics of Chaos, Blackwell Publishers; Second Edition (February 2002).
- 6. Methodology 3-4 affect over the long term. The effect will be to show an increasing variance over time. We assume that LR volatility does not mean revert and follows a standard Brownian motion process. Short Run with Mean Reversion Random factors such as weather, outages, and short run liquidity effects will be captured in the short run volatility parameter. These short run “shocks” are assumed to be temporary deviations from the equilibrium. This process tends to be more significant in driving what is commonly perceived as price volatility and will capture the now infamous price spikes within the electricity price process. Figure 3-3 Stylized Price Diffusion Process Power Price Equilibrium Forecast with uncertainty Gas Price Equilibrium Forecast with uncertainty Prices will vary randomly around the mean Through time Price path cannot randomly walk away from mean Price SOURCE: Global Energy. Global Energy’s analysis and many throughout the industry have concluded that the short run shocks are mean reverting. In other words, after some time they will revert to the equilibrium price. The mean-reverting process can be likened to applying a piece of elastic between the observed price and the equilibrium price. A random factor continues to be applied to the price as it moves through time but as it moves further away from its equilibrium price a proportionately increasing force is applied to it to pull it back. The speed of mean reversion, a key input variable in this process, determines how quickly prices revert to equilibrium. Once we have identified the short run and long run parameters, it is necessary to calculate the related correlations. In this analysis Global Energy is correlating all the fuel and electricity prices within a region for both the short and long run conditions. Model Used Global Energy’s StatTool software was used to describe the stochastic properties of these variables, including their volatility and short-term mean reversion. Eviews is used for multiple variable simultaneous correlation estimations among the historical time series.
- 7. Methodology Power Generation BlueBook, 2005 3-5 Historic price data was input for each price point for power, gas and oil, which then estimated the mean reversion, volatility and correlation parameters used in the simulation. This process is described in detail below. The PLANNING AND RISK™ basic stochastic model is a two-factor model, in which one factor represents short-term or temporary deviations and the other factor represents long-term or cumulative deviations. Some of the important features of the statistical estimation tools and their relation to the stochastic model are summarized below. Figure 3-4 Stochastic Model Process Short-Run (e.g., Daily) Series Fuel prices, electricity prices Long-Run (e.g., Annual) Series Fuel prices, electricity prices StatTool-S StatTool-L Short-Run Parameters Mean reversion Volatility Correlation ,S S mt ntσ σ ,S S mt ntα α S mn ρ Long-Run Process Long-Run Parameters Drift Volatility Correlation ,L L mt ntσ σ ,mt nt µ µ L mn ρ Short-Run Process Long-run (equilibrium) Values Li,t, Lj,t Lag Lag Values Li,t-1, Lj,t-1 Short-run (spot) Values Si,t, Sj,t Lag Values Si,t-1, Sj,t-1 Lag SOURCE: Global Energy. Long- and short-term effects are combined in the two-factor model. First, the equilibrium price (to which the spot price reverts) receives periodic shocks that create a somewhat random or stochastic equilibrium level. Second, short-term factor shocks further cause spot prices to deviate from equilibrium prices. The PLANNING AND RISK™ stochastic model allows multiple entities to be jointly simulated with this two-factor stochastic process, accounting for correlation among the shocks impacting the set of stochastic processes. The entities simulated with this stochastic model in PLANNING AND RISK™ included electricity energy, natural gas, oil, coal, and other fuel prices. Figure 3-5 Volatility and Reserve Margin Relation
- 8. Methodology 3-6 0% 10% 20% 30% 40% 50% 60% 70% 80% Feb-97 Jun-97 Oct-97 Feb-98 Jun-98 Oct-98 Feb-99 Jun-99 Oct-99 Feb-00 Jun-00 Oct-00 Feb-01 Jun-01 Oct-01 Feb-02 Jun-02 Oct-02 Feb-03 Jun-03 Oct-03 Feb-04 Jun-04 0% 20% 40% 60% 80% 100% 120% 140% 160% Entergy Daily Vol (Monthly) Enteryg Reserve Margin Higher Reserve Margins Lower Volatility SOURCE: Global Energy and Power Markets Week. Volatility in power and fuel markets can be driven by various factors such as weather patterns, load characteristics, transmission system, generation portfolio, transmission access, market rules and market players. Some markets are fundamentally more volatile than the others. Volatility in power markets has decreased noticeably within the last few years. An influx of new gas-fired generating units in most of North American markets has caused an overbuilt market with high reserve margins in most areas. Figure 3-5 illustrates the relationship between volatility and reserve margin for the Entergy market. The trend lines clearly show the inverse relationship between reserve margin and the volatility. Decrease in volatility is a rational outcome of the high reserve margins, because excess amount of idle generation suppresses any price movement immediately. This will also increase the mean reversion behavior in the power markets. In the long term, volatility levels are expected to increase as reserve margins decrease. To capture this fundamental market change, Global Energy modeled the volatility and mean reversion rates by incorporating a term structure in stochastic parameters. Initial years’ volatility and mean reversion parameters are estimated by using more recent historical data. For later years, all available historical data is used. The estimates are done based on two to three levels of 2-year time intervals. The estimated parameters are summarized in Appendix A. Detailed Stochastic Model Description The discrete time mathematical representation of the two-factor (short-term and long- term) lognormal model is: 2/][Var)( 1,,,,1,1,,1,,1,, −−−−− −+−+−+= tntn S tn S tntntntntntntntn SSSLLLSS εσα (1)
- 9. Methodology Power Generation BlueBook, 2005 3-7 L tn L tn L tntntntn LL ,, 2 ,,1,, 2/)( εσσµ +−+= − (2) 0, ,,, == LS tn L tn S tn ρεε (3) S tnm S tn S tm ,,,, ρεε = (4) L tnm L tn L tm ,,,, ρεε = (5) where: n = entity (fuel price, or electricity price) t = time period of observation (e.g., day, week, month) nS = logarithm of short run or spot value for commodity n nL = logarithm of long run or equilibrium value for commodity n tn,α = rate of mean-reversion in spot value for commodity n in period t tn,µ = expected rate of growth (drift) of equilibrium value for commodity n in period t , S n tσ = volatility of spot value “returns” for commodity n in period t L nσ = volatility of equilibrium value growth rate for commodity n S ε = normally distributed random vector (mean = 0, s.d.= 1) L ε = normally distributed random vector (mean = 0, s.d.= 1) ,S L ρ = correlation of spot and long run value stochastic changes , S m nρ = correlation of spot price stochastic changes for commodities m and n , L m nρ = correlation of drift rate stochastic changes for commodities m and n Var = variance. The short-term or spot value for entity n, Sn,t, is modeled as following a mean-reverting process in which the “mean” is a time-varying, long run equilibrium level, Ln,t. This process, specified in equation (1), combines the stochastic shocks to the uncertain equilibrium value and short-term deviations around the equilibrium value. The long-term equilibrium value is an unobservable variable towards which the short-term observed spot value Sn,t tends. The long-term value Ln,t is generated by the long-term process specified in equation (2), which describes a random-walk around a time-varying trend rate, tn,µ . In this analysis we have entered the Global Energy Retainer Forecasts as the equilibrium or expected value (mean) forecast, {exp(Ln,1) … exp( Ln,T)}, for periods 1 through the horizon T. Then, a time series of drift rates is calculated by the software for this assumed trajectory of expected values.
- 10. Methodology 3-8 Equation (3) says that short-term and long-term shocks are assumed to be uncorrelated. Equations (4) and (5) allow for a positive or negative correlation between the short-term and long-term shocks, respectively, for any two stochastic entities. The application of the stochastic model summarized by equations (1) – (5) proceeds in two steps: 1. Statistical or judgmental estimation of the parameters, including the short-term mean reversion parameter(s) tn,α , short- and long-term volatilities S tn,σ and L tn,σ , and short- and long-term correlation coefficients S tnm ,,ρ and L tnm ,,ρ ; and 2. The use of these parameters in conjunction with expected value forecasts in Monte Carlo simulations. Mean-Reversion Process The short-term dynamics of prices (and of other stochastic variables) in the PLANNING AND RISK™ stochastic model are a mean-reverting process, in which the variable is assumed to revert through time to an equilibrium or long-term value, while simultaneously being subject to continuing shocks. To focus on understanding just the short-term mean-reverting process, we assume here a simplified form of equation (1), in which time is modeled as continuous and the mean is constant. For a variable x = ln (X) the process can be specified as dWdtxxdx σα +−= )( (6) where x is the mean or equilibrium value towards which the process reverts from a disequilibrium position, and dW is the standard normal increment of a random (Weiner) process over an infinitesimal time increment. The α term in equation (6) is a continuous-time mean-reversion rate. In discrete-time implementations it is expressed in terms of percent per time period. The half-life of a mean-reversion process is a convenient metric to summarize the speed of adjustment of a process. A process with a short half-life is a rapidly mean-reverting process. Given a value of α (including the specification of the time step), the half-life of the process is given as: α )2ln( 2/1 =t (7) where 2/1t is the number of periods required for half of the deviation from a shock to be dissipated. Writing the natural log of the price for a commodity in period t as St, and its mean (or long-term equilibrium value) corresponding to the x term in equation (6) as L, the discrete-time version of equation (6) is tttt SLSS σεα +−=− −− )( 11 (8)
- 11. Methodology Power Generation BlueBook, 2005 3-9 Equation (8) is a special case of equation (1), when there is a constant equilibrium value, instead of the general case of a stochastic, time-varying equilibrium value. The Var[ ]/2 term in equation (1) drops out in equation (8) because it is a theoretical “log-bias” adjustment needed only when the equilibrium value L is stochastic. Correlations across Commodities Global Energy’s stochastic model then applies the appropriate correlation among the short-term shocks for different stochastic entities and among the long-term shocks. Correlation coefficients, identified in equations (4) and (5) respectively, are input into our asset valuation model, PLANNING AND RISK™. Cholesky decompositions of the ST and LT correlation matrices are then used to transform two vectors of independent standard normal draws for each day into vectors of correlated draws from a multivariate standard normal distribution. The Cholesky decomposition is a transformation that may be applied to any positive- definite matrix. It is sometimes described as a “matrix square root” because like a traditional square root it can be “multiplied” (in a matrix sense) by itself to arrive back at the original correlation matrix. If A is a positive-definite matrix, then it has a Cholesky decomposition matrix C that satisfies ,' ACC = (10) where C’ is the transposition matrix of C whose columns are the rows of C. Correlation matrices are symmetric, with ones on the diagonal (since the correlation of a variable with itself is 1), and with coefficients mnρ between commodities m and n satisfying .10 << mnρ The composite correlation is a reflection of the underlying price projections, short run and long run correlations. Each price iteration will exhibit a different correlation relationship.
- 12. Methodology 3-10 Table 3-1 Stochastic Parameters Algonquin New England Algonquin/New England Correlation Alpha Sigma Alpha Sigma 0.079 0.199 0.049 0.108 0.578 Note: Alpha [αααα] Sigma[σσσσ] SOURCE: Global Energy. The table above gives the winter stochastic parameter estimation for New England power and gas. Sigma is the daily volatility, which represents the day-to-day fluctuation of the prices. As shown, the New England power market has 10.8 percent daily price volatility, while Algonquin natural gas has 19.9 percent daily price volatility. Typically, natural gas has higher price volatility in winter, while power has higher price volatility in summer. Average gas price volatility used in the BlueBook is around 7 percent, and the estimates vary by season ranging from 3 to 27 percent. The average daily power price volatility is around 16 percent and the estimates vary by season ranging from 5 to 41 percent. These figures are daily volatility estimates, and they should not be compared to annual volatility figures, which are commonly used in commodity markets. In general higher price volatility gives higher extrinsic (option) value for a power plant. Alpha, the mean reversion rate, is the second important stochastic parameter, and it is a factor instead of a percentage. The alpha for power and gas in the above table are 0.049 and 0.079 respectively. 0.079 means the shocked price reverts back to the expected level in ~13 days (1/0.079), if we assume a linear mean reversion. In the BlueBook we use an exponential reversion rate. In other words, the mean reversion rate is a factor of the difference between the current price level and the expected (mean) price level. So a bigger price shock creates a stronger response in the next time period. As the difference gets smaller, the response gets weaker. The following graph illustrates the exponential mean reversion behavior for a difference of 100 percent. Figure 3-6 Exponential Mean Reversion Behavior 0% 25% 50% 75% 100% 0 10 20 30 40 50 Days SOURCE: Global Energy. In the BlueBook price formation process new stochastic draws around the expected prices are preformed for each day. So the mean reversion effect is recalculated on a daily basis.
- 13. Methodology Power Generation BlueBook, 2005 3-11 The average power mean reversion rate used in the BlueBook is around 0.14 and the estimates vary from 0.013 to 0.65. The average for gas is 0.041, and the range is from 0.002 to 0.31. In general power prices exhibit stronger mean reversion than gas prices. Particularly in fall and spring seasons, the mean reversion for power is faster due to the excess generation available. Higher alpha means faster market response to any kind of price deviation from the expected price levels. This will result in a lower extrinsic value for a plant, since the shocks are absorbed more quickly. The final stochastic factor is the correlation between power and fuel. The winter gas and power correlation in Table 3-1 is 57.8 percent. Correlation estimates vary significantly based on season and the market. For most cases the power and fuel correlations are in the range of 10-20 percent and estimates higher than the 30-40 percent level are generally significant. Higher power and fuel correlation reduces the extrinsic value of assets due to lower spark spread volatility. In other words, if the power and fuel shocks move together, the spark spread for the plant does not change significantly, since every time price moves, the fuel cost moves as well leaving little flexibility to the plant. Asset Valuation Model, PLANNING AND RISK™ The next step in the process, after having established the volatility and correlation parameters that we can evaluate the assets against, is to dispatch asset operation against the alternative price paths that can then be generated. The alternative price paths are developed from a Monte Carlo process that makes random draws from log normal distributions. The dispatch model uses these random paths to optimize asset commitment and dispatch along each random price path. The Global Energy proprietary model, PLANNING AND RISK™, was used to perform Monte Carlo simulation analysis of individual units. PLANNING AND RISK™ is driven by Global Energy’s PROSYM™ chronological market simulation algorithm, which is used to simulate a portfolio’s operation by reflecting pertinent unit operating constraints like start-up costs, ramp rate restrictions, minimum up and down times, and other plant dynamics to provide a credible analysis of asset valuation and risk exposure. Studies have shown asset and portfolio valuation errors of up to 400 percent can occur by ignoring these key details. PLANNING AND RISK™ uses the very detailed, time-varying inputs of the renowned PROSYM™ commitment-dispatch engine to characterize any type of thermal or hydro generating station. A partial list of generation station characteristics includes: • Hourly and seasonal capacity variations; • Dispatching limits; • Spinning reserve capabilities and constraints; • Complete forced and maintenance outage modeling; • Multi-state heat rates with seasonal heat rate variations; • Startup and shutdown costs;
- 14. Methodology 3-12 • Ramp rates and minimum up and down times; • Energy limited generation; • Emission functions and costs; • Limited fuel modeling; and • Other variable and fixed O&M costs. Using the two-factor (short run and long run shocks), mean-reversion stochastic models discussed earlier, Monte Carlo iterations are performed, with random draws used to simulate stochastic variables as discrete time processes. Antithetic sampling and first moment (mean) calibration are used to reduce sampling variance from the hypothesized distributions. Daily draws are taken for average daily prices and loads, and weekly random draws for forced outages. Within each week, generation units are committed and dispatched as if they have perfect foresight of future values for that week. The stochastic parameters are maintained within the PLANNING AND RISK™ data structures as constant or time varying parameters. The stochastic simulation results are written to the Monte Carlo output database when a simulation is run. Combining Deterministic Forecast And Market Uncertainty What does the extrinsic value mean? When applying alternative price paths to any given asset there is a distribution of values. There are three distinct valuation points worth considering. Value of an inflexible or “base load” plant First, a completely inflexible plant (e.g., a must run plant) that will generate (sell into the market) regardless of market price will have a value in each period equal to [spark spread x MW sold x number of hours]. Figure 3-7 Forward Contract Payoff Price Distribution Profit Impact There is a linear relationship between the price distribution and forward contracts Break Even Profit Loss $ SOURCE: Global Energy. In this case, once enough iterations have been run, the average of the stochastic iterations (the “expected” value) will equal the value of the initial forecast for a plant or contract with no flexibility. In other words average or expected revenue is linear with expected
- 15. Methodology Power Generation BlueBook, 2005 3-13 prices. This can be related to a forward contract value as forward contracts have the same economic impact as a must run plant. These contracts have a “delta equivalent” value of 1. Generally, base load plant, such as nuclear or large coal plant, will see almost all their value from this component as they have little flexibility or optionality value. Flexibility Value Second, consider a unit with the ability to respond to market prices. Against the initial forecast (a deterministic run) the unit will now show a higher value than the inflexible unit (as it can now shut down at times of low prices) and avoid running at a loss (against its short run marginal costs). Figure 3-8 Hourly Profile for Flexible Unit Revenue earned equals area above line Plant Runs Plant avoids loss by shutting down during low price periods $ time SOURCE: Global Energy. Figure 3-9 Flexible Unit Payoff Price Distribution Profit Impact With ability to “shut down” average value increases Break Even Profit$ SOURCE: Global Energy. This obvious feature of power stations is the reason for the importance of using sophisticated plant scheduling software. An individual plant with complex dispatch characteristics (for example ramp-up rates, minimum on times, minimum off times) must be “scheduled” accurately against the individual hours throughout the year to ensure a reasonable estimate of its value. Not reflecting the full operating characteristics of the plant can have an impact on plant value of well over 100 percent of net revenue. In this
- 16. Methodology 3-14 case the “shape” of power prices becomes as important as the average price as there may no longer be a simple linear relationship between average prices and plant revenues. It is important to note that this value does not come from price volatility but rather the “granularity” of the forecast being consistent with the asset—a plant that can switch on and off each hour needs to be valued against a price stream that reflects the hourly variability in power prices. As we are still valuing the asset against a single price projection, without taking account of price uncertainty, we refer to this case as the deterministic or intrinsic value of the plant. Optionality or Extrinsic Value Third, we have the stochastic or “real option value.” This reflects the added value of a flexible plant faced with future uncertainty. As we run this plant against the alternative randomly generated projections, the plant will respond differently to each potential price stream. The deterministic value above will give no value to plant that did not run under that particular scenario. However, if you believe there is some uncertainty around that projection, a plant that did not run under the base scenario but could run if prices rose, still has value. The valuation approach to such options has developed significantly since the original work by Black and Scholes; however, the basic principles still hold true. The value of the “option” is proportionate to its probability of being called. Let us take the example of a simple cycle gas plant with an energy cost of $75/MWh. Current power prices are $45/MWh so it does not run and does not receive any revenue. However, analysis of market prices suggests there is a 5 percent chance that prices could move to $100/MWh (we shall ignore intermediate prices for simplicity). In this simple example the plant has an option value of 5 percent times $25/MWh ($100-$75) or $1.25/MWh. Although small, it is a significant increase from zero. Black and Scholes showed that it is possible to set up a riskless portfolio (one with no market risk) consisting of a position in the option (in this case the single cycle plant) and the underlying product (for instance a forward contract). The delta hedge to capture the value is +1 option, -∆ underlying with + being long, and - being short positions. What this means is that with a liquid market you can lock in the $1.25/MWh value by selling the delta equivalent of forward contracts—in this case approximately 5 percent of the total volume. This additional value of plant optionality is referred to as the extrinsic value of the plant while the deterministic value is often referred to as the intrinsic value of the plant. In our analysis we shall show both values for comparative purposes. Another example worth considering is a CC with a variable operating cost just below or at current market prices. In this case a deterministic run may show limited running and little or no value. However, if the price increases by 10 percent over the year (well within the volatility we have recently seen) the plant dramatically increases its load factor and
- 17. Methodology Power Generation BlueBook, 2005 3-15 profitability. Anyone who has looked at an asset sensitivity analysis will have seen this effect. The stochastic approach captures the probability of this positive value and in this case the stochastic value will be significantly greater than the deterministic value. Global Energy provides both the deterministic (intrinsic) and expected value (intrinsic plus extrinsic) of the plant. In addition it provides the full stochastic percentile output that is key to understanding debt or book value coverage ratios. Use of Expected Value and Delta Hedging The extrinsic value identified is thus not purely of academic interest. Even though this value may not be easily identified, and in many cases we do not expect it to materialize in the real world, it may be realized through a hedging program. The ability to delta hedge and to lock in the extrinsic value is key to estimating the relevance of this stochastic approach. This can be through direct hedging of the plant optionality or by placing delta hedges using forward contracts. The delta option is equal to: ∆ Asset Price / ∆ Underlying Market Price Once you have estimated this relationship you can potentially use it to hedge an option with a forward contract. Deltas will vary from –1 to 1. For hedging a power station (a long position in the market) the delta will vary from zero to one (where one is equivalent to running all the time). To hedge this you need to place the opposite hedge (sell to the market) for the delta equivalent. It can be shown that placing delta hedges and continually adjusting them on a regular basis to reflect the changing market prices will lock in the value of that option.2 Figure 3-10 Delta Relationship to Market Prices Strike Price Delta Market Prices SOURCE: Global Energy. Delta hedging is extremely difficult in the power market and as such some risk (and thus potential value erosion) is likely against the expected values even where a sophisticated delta hedging program has been implemented. Lack of liquidity, granularity, non- 2 See John Hull, Options, Futures & Other Derivatives, Prentice Hall.
- 18. Methodology 3-16 standard price distributions and the complexity of the option structures underlying generation assets all contribute to this difficulty. However, there is no doubt that even with the current markets a significant proportion of this value can be captured in the market using a delta equivalent or similar hedging approach and that to ignore this value will significantly undervalue many plants. Many CCs have significantly less value on a deterministic basis. This is because their variable costs (strike price) are often above the expected value price of the market they are selling into. They still, however, have significant delta equivalent value and much of that value (seen in the near-term years) should be able to be captured in the existing forward and contract markets. Probability of Covering Debt While existing CCs have an expected value that reflects their real option value, risk managers and debt holders should be wary of what they provide. Figure 3-11 Cumulative Probability Distribution of Value 0 500 1000 1500 2000 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Probability $/kW 7% Probability of making $1,000/kW 7% 70% Probability of making $177/kW $177/kW Expected (Mean) Value $412/kW Median $310/kW Much of the 'high value' captured is low probability SOURCE: Global Energy. Figure 3-11 shows the cumulative probability of the total value for a typical CC in today’s market (a new entrant cost is around $650/kW). It shows that there is a 7 percent probability of making $1,000/kW or more and a 70 percent probability of making over $200/kW. The expected value in this case is $412/kW, the average of all the simulations. However, the median value, e.g., the point where there is an equal likelihood of prices being higher or lower, is only $310/kW. The $412/kW tells you what value you may be able to capture, it does not provide either a most likely case or the 50/50 case. The deterministic value is only $195/kW. What we are seeing is that significant value comes from a few occasions of very high prices. Unless captured through contracts this value is generally worthless for a debt holder. A debt holder is simply interested in the value being above a minimum threshold that is related to the collateral needed to secure the debt. It is possible that more volatile markets will show both higher valuations and lower
- 19. Methodology Power Generation BlueBook, 2005 3-17 probability of covering their debt. In all cases you should look at the total debt or payment flow and compare that with the probability of the EBITDA value covering that cost. In Figure 3-12 below we show the probability of achieving a range of debt coverage ratios for a new CC costing $650/kW. Figure 3-12 Debt Coverage Ratio for CC 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% GenCC COB GenCC Entergy GenCC ERCOT GenCC FRCC GenCC MAIN PROBABILITY COVERAGE SOURCE: Global Energy. It can be seen in this example that a debt/equity ratio above 40/60 will result in the asset being unlikely to cover its overall debt service. Finally, it is vital to look at the stream of cash flows rather than simply the total expected value.In the example in Figure 3-11, the plant has an expected value of $412/kW, a median value of $310/kW and a deterministic value of $195/kW. With a more detailed analysis of the forward market liquidity you can quickly settle on a value designed for the purpose of any valuation assignment. However, there is one more important issue that cannot be overlooked—the annual cash flow. Figure 3-13 below shows the cash flow associated with the above valuations. If we assume this is a plant with a debt repayment of $50/kW per annum (which would be fairly typical) then we can see the problem—in 2006 and potentially through to 2010, there is a significant chance that plant will not be able to make its debt payments. Regardless of the total value this is perhaps the most critical conclusion.
- 20. Methodology 3-18 Figure 3-13 Annual Cash Flow for a CC Unit 0 20 40 60 80 100 120 2006 2008 2010 2012 2014 2016 2018 2020 2022 2024 AssetValue($/kW-yr) NG-CC Intrinsic NG-CC Extrinsic SOURCE: Global Energy.