Br stochastic best practice

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

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 AssetValuation

(100 Monte Carlo Iterations,

detailed plant parameters)

Demand(Hourly Load)

Supply(Generators)

Transmission

Hourly Dispatch

$/MWh

MW

Fuel &EmissionPrices

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 ValueExpected 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

SPPNSMAIN

ENTERGY

ALBERTA

CENT-N

ERCOTNORTH

ERCOTSOUTH

ERCOT

WEST

ERCOTHOUSTON

SASK

POWERMANITOBA

SOUTHERN

GRIDFLORIDA

TVA

CE_NI

WYOMING

W

DAKOTAS

IDAHO

NORTHWEST

COB

CNP15

IID

SNEVADA

MECS

APS

AEPFIRSTENERGY

VP

MARITIMES

ONT

EC

NY

WEST

ONTMP

NY

CN

NYCITYPJME

X

NY F

NE

NORTH

PJMWX

ONT-

NORTH

QUEBEC

LONG

ISLAND

SEMA

RI

NEEASTNE

WEST

NY

GHI NECTSW

ONT-NI

ONT

WEST

KENTUCKY

Canada

New

England

NewYork

Southeast

TVA

FRCCERCOT

MAPP

MAIN

N

California Rockies

AZ/NMSPP

SoCal

ECAR

Northwest

PJM

Entergy

ERCOTNORTHEAST

WYOMING

E

Hourly Power Prices for 76 Market Areas

Individual AssetValuation

(100 Monte Carlo Iterations,

detailed plant parameters)

Demand(Hourly Load)

Supply(Generators)

Transmission

Hourly Dispatch

$/MWh

MW

Hourly Dispatch

$/MWh

MW

Fuel &EmissionPrices

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 ValueExpected 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

SPPNSMAIN

ENTERGY

ALBERTA

CENT-N

ERCOTNORTH

ERCOTSOUTH

ERCOT

WEST

ERCOTHOUSTON

SASK

POWERMANITOBA

SOUTHERN

GRIDFLORIDA

TVA

CE_NI

WYOMING

W

DAKOTAS

IDAHO

NORTHWEST

COB

CNP15

IID

SNEVADA

MECS

APS

AEPFIRSTENERGY

VP

MARITIMES

ONT

EC

NY

WEST

ONTMP

NY

CN

NYCITYPJME

X

NY F

NE

NORTH

PJMWX

ONT-

NORTH

QUEBEC

LONG

ISLAND

SEMA

RI

NEEASTNE

WEST

NY

GHI NECTSW

ONT-NI

ONT

WEST

KENTUCKY

Canada

New

England

NewYork

Southeast

TVA

FRCCERCOT

MAPP

MAIN

N

California Rockies

AZ/NMSPP

SoCal

ECAR

Northwest

PJM

Entergy

ERCOTNORTHEAST

WYOMING

E

SOURCE: Global Energy.

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.

Methodology


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).

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


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.

Methodology


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

mnρ

Long-Run

Process

Long-Run Parameters

Drift

Volatility

Correlation

,L L

mt ntσ σ

,mt nt

µ µ

L

mnρ

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


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

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)

Methodology


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

Sε = normally distributed random vector (mean = 0, s.d.= 1) L

ε = normally distributed random vector (mean = 0, s.d.= 1) ,S Lρ = 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.

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)

Methodology


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.

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.

Methodology


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;

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

$


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

Methodology


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 $


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

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

Methodology


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

DeltaMarket Prices


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.

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


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

Methodology


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


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.

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

Asset Value ($/kW-yr)

NG-CC Intrinsic NG-CC Extrinsic


Data & Analytics

Br stochastic best practice