Risk Analysis of Interruptible Load Contracts∗
Paula Rocha†
Afzal Siddiqui‡
August 11, 2008
Abstract
Deregulation of electricity industries often poses enormous risk for retailers who
must purchase electricity from spot markets at volatile prices and sell it to end-use
consumers at stable retail rates. Absent any demand-side response, one recourse
available to retailers in order to hedge against this exposure is to offer interruptible
load (IL) contracts. These financial instruments enable the retailer to curtail provision of electricity to the consumer for a fixed duration a given number of times
during a time horizon in return for either a rebate on the retail rate or a per-use
fine. Assuming an exogenous and stochastic electricity price, we set up the retailer’s
decision-making problem as a stochastic dynamic program and solve it using the
least-squares Monte Carlo (LSMC) simulation algorithm. Consequently, we obtain
not only the value of such contracts, but also their optimal interruption policies and
risk measures, such as value-at-risk (VaR) and expected tail loss (ETL).
Keywords: OR in energy, interruptible load contracts, least-squares Monte Carlo
simulation, real options, risk analysis
1
Introduction
Electricity industries have been undergoing liberalisation and deregulation over the recent decades in order to open up their sectors to competition, wherever feasible. As a
∗
Research Report No. 296, Department of Statistical Science, University College London. Date:
August 2008. We are grateful to the feedback received from the attendees of the 2008 Computational
Management Science Conference in London, UK. All remaining errors are our own.
†
Corresponding author, Department of Statistical Science, University College London, London WC1E
6BT, United Kingdom, e-mail: pcmsrocha@googlemail.com
‡
Department of Statistical Science, University College London, London WC1E 6BT, United Kingdom,
e-mail: afzal@stats.ucl.ac.uk
1
Risk Analysis of Interruptible Load Contracts
2
result, state-owned companies are being privatised, and vertically integrated monopolies
are being replaced by firms functionally specialised in the generation, transmission, distribution, or retail sale of power. The deregulation process has been accompanied by
the introduction of competitive spot markets, along with forward and derivative markets
while operation of the transmission network is typical assigned to independent system
operators (ISOs), who are also responsible for ensuring the balance of the system.
The unique characteristics of electricity make its spot price one of the most volatile
among commodity prices. Consequently, market participants, especially retailers, are
more exposed to financial risk than before deregulation took place. It is usually the case
that the retail price charged to the final consumer is time invariant and determined long
before actual consumption occurs, which means that the retailer absorbs the entire risk of
volatile wholesale prices. Since customers are unaware of wholesale price variations, they
have no financial incentive to curtail or reduce electricity demand.
State-owned power companies had the advantage of being able to share financial risk
among all taxpayers. External shocks to fuel prices were moderated by long-term procurement contracts, and by regulations allowing fuel costs to be paid by retail customers via
amortised charges. In the US, regional investor-owned monopolistic utilities were assured
a sufficient rate of return, in exchange for the obligation to supply electricity at prices set
by retail tariffs. Because these tariffs were reviewed periodically, costs shocks and volatile
wholesale prices were averaged and spread out over long periods.
In restructured power markets, risk bearing should be spread out along the supply
chain through forward contracts or financial instruments for hedging, and spot market
transactions should account for only a small fraction of the utilities’ purchases (Wilson,
2002). The retail sector must be able to hedge against the price risk in competitive
wholesale markets by shifting part of it to the customer. Either retailers are able to pass
through the costs of wholesale purchases to customers, or alternatively they can offer their
customers customised contracts that induce price-responsive demand behaviour such as
real-time pricing, peak-load pricing, or interruptible load (IL) contracts. Our analysis
focuses on the latter type of contract, which allows utilities to interrupt a portion of
the customer’s load in exchange for a pecuniary compensation. This contract typically
specifies the maximum number of times that the utility can call for interruptions over the
duration of the contract, the minimum amount of advance notice it must provide, and
the maximum time permitted for each interruption.
IL contracts existed in the regulated electricity industry mainly to prioritise interruption schedules in an emergency. In deregulated competitive markets, they gain more
importance as a risk management tool. Retailers that need to resort to the spot market
Risk Analysis of Interruptible Load Contracts
3
to be able to meet (part of) the electricity demand of its customers benefit from interrupting load sufficiently to avoid paying higher wholesale prices that usually occur during
shortages or system peaks. Furthermore, by reducing the total load to the system, load
interruption may lead to system-wide lower spot prices (Oren and Smith, 1992), which
is crucial since the capacity and reserves requirements of utilities are largely dictated by
peak loads.
Gedra and Varaiya (1993) introduced the concept of a “callable forward” in order to
emulate a simple IL contract. A supplier is able to do so by selling a forward contract
that commits the supplier to deliver one unit of energy at maturity and buying a call
option with the same maturity. If at maturity the spot price exceeds the strike price,
then the supplier exercises the call option, paying the strike price to the customer and
effectively cancelling the forward contract, thereby avoiding peak spot prices. The customer’s discount on the forward price is the option price at the time of contracting. One
shortcoming of this kind of contract is that some consumers with high interruption costs
may not provide a viable strike price for the contract. To solve this problem, Kamat and
Oren (2002) introduced the interruptible service contract with early notification, where
notice of interruption is given prior to curtailment. This contract bundles simple forwards
with exotic call options that have two exercise points with different strike prices. Figueroa
(2006) extended these works by combining a forward contract with an up-swing option,
which allows the holder to exercise the right to curtail electricity supply a fixed number
of times. By exercising the option, the supplier cancels the delivery of electricity (and
compensates the customer with the strike price) on the dates that exercise rights are used.
Since the customer holds the opposite portfolio, the up-swing option value represents the
discount that the customer obtains on the forward contract.
Baldick et al. (2006) developed a framework for valuing IL contracts from the perspective of an electricity retailer by taking the view that optimal interruption reduces
both the demand for electricity and the spot price. Therefore, it proposed a structural
model of the electricity spot market, where prices are determined by matching supply
and demand. The analysis suggests that, in the absence of forward contracts or ownership of generation assets, IL contracts are valuable to the retailer. As more generation is
available at a fixed price, or as the number of competing retailers increases, the value of
the IL contract goes down, and interruption occurs at higher expected loads. However,
risk analysis of the effectiveness of the contracts as hedges against uncertain demand or
prices is not performed. Like Baldick et al. (2006), we focus our attention on two variants
of IL contracts that differ only in the form of the pecuniary compensation offered to the
customer: the pay-in-advance and the pay-as-you-go contracts. While in the former the
Risk Analysis of Interruptible Load Contracts
4
customer receives discounted electricity prices for the entire load, in the latter a lump
sum is paid per interruption to the customer.
Since the number of interruption rights is contractually limited, the problem of optimally exercising these rights must be tackled. In this paper, an algorithm for optimally
exercising interruptible rights for a multiple-exercise IL contract is developed within a
stochastic dynamic programming (SDP) framework. The advantage of this framework is
that it explicitly recognises the recursive nature of this decision problem. The solution
to the SDP yields the value of the multiple-exercise option embedded in the contract and
the thresholds for calling load curtailments. Based on this solution, the value of the entire
IL contract and financial risk measures are determined. Furthermore, insight is given on
pertinent questions faced by the electricity retailer, such as determining the maximum
rebate that can be offered to the customer as an incentive to sign up to an IL contract
and determining whether one type of IL contract is preferred over another in terms of
reducing the risk of extreme losses.
2
Problem Description
An electricity retailer has agreed to provide electricity to satisfy the demand of its customers at a guaranteed fixed rate, pretail , over a time horizon, T (in years). To serve this
load, the retailer purchases electricity at spot price, St , in periods t = 1, 2, . . . , M . For
simplicity, we assume that the retailer has no generation capacity, does not incur any
further costs, and has customers with constant demand.
To hedge against the risk of extreme wholesale prices, rather than offering a normal
load contract, the retailer decides to offer its customers two different types of IL contracts:
pay-in-advance and pay-as-you-go. An IL contract allows the retailer to interrupt part or
the entire supply of electricity a fixed number of times, N , over the life of the contract.
The difference between a pay-in-advance and a pay-as-you-go contract lies only in the
form of the pecuniary compensation offered to the customer as an incentive to sign up. In
the former, the customer receives a discount on the retail price of electricity for its entire
load (thus paying preduced instead of pretail ), while in the latter the customer receives
compensation pf ine per unit of interrupted load. Here, we assume that all loads are either
served or interrupted and that the customer always curtails its load when requested.
IL contracts can be regarded as swing options, which give the holder some flexibility
regarding the delivery amount around the base-load level. These types of contracts are
mainly found in power markets since energy is difficult to store and exhibits price-spiking
behaviour. A swing option allows the holder to exercise repeatedly the right to receive
Risk Analysis of Interruptible Load Contracts
5
greater or smaller amounts of energy, subject to daily and periodic constraints. A right
can be exercised only on discrete dates with at most one right exercised per date. Adapted
to this particular problem, a swing option allows the retailer to exercise the right to curtail
the supply of electricity up to a fixed number of times, N (≤ M , where M is the number
of decision-making periods) in periods t = 1, 2, . . . , M .
At each period t, the retailer must decide whether or not to exercise an interruption
right, given the number of exercise rights remaining, n, at the time. Let this decision be
represented by ant ∈ A, where A is the set of all possible states, namely:
ant =



1 if an interruption right is exercised at time t, given that the number


of exercise rights left before taking the decision is n



 0 otherwise
Given the electricity spot price, St , and the number of exercise rights left, n, the retailer’s
maximum expected discounted (from the swing option obtained) cash flow to go until the
end of the contract, Ctn (St ), for period t = N − n + 1, . . . , M − n, is determined by:
n−an
Ctn (St ) = max
{P (St ; at = ant ) + βEt [Ct+1 t (S̃t+1 )]} , n = 1, . . . , N
n
at ∈A
(1)
Here, β = e−r∆t is the one period discount factor, where r denotes the annual riskT
adjusted rate of return and ∆t = M
is the length of the time period in years (i.e., the
total time horizon divided by the number of decision-making periods). P (St ; at ) represents
the immediate payoff from the decision of exercising at time t an interruption right or
not, and it is specified as follows for t = 1, . . . , M :

 S −K
t
P (St ; at ) = 
0
if at = 1 (if an interruption right is exercised at time t)
if at = 0 (otherwise)
where K represents the strike price. In the case of a pay-in-advance contract, the retailer
considers exercising a curtailment right only if the electricity spot price is above the
reduced retail price preduced . Thus, preduced corresponds to the strike price for this type
of contract. In the case of a pay-as-you-go contract, the retailer obtains a benefit from
exercising a right only if the electricity spot price is above the retail price plus the fine
paid to the customer for curtailing the electricity supply. Therefore, the strike price, K,
is pretail + pf ine .
The intuition behind Equation (1) is that at each period t, given the number of
interruption rights left, n, the retailer bases its decision to exercise an interruption right,
ant , on the comparison of the benefit from exercising an interruption with the benefit
Risk Analysis of Interruptible Load Contracts
6
from not exercising an interruption. If the former is larger than the latter, then the
retailer exercises an interruption right. Using Equation (1), an SDP for the retailer’s
decision-making problem can be specified and solved subject to the following boundary
conditions:
• If all exercise rights have been used, then the cash flows obtained from the swing
option in this case are zero:
Ct0 (St ) = 0
, t = 1, . . . , M
(2)
• When there are n interruption rights remaining, the retailer will decide whether
or not to exercise a right at times t = M − n + 1, . . . , M by determining which
decision maximises the immediate payoff1 , i.e., by maxant ∈A {P (St ; ant )}. Therefore,
the retailer’s maximum expected discounted cash flow to go until the end of the
contract (from the swing option) for these periods is given by:
n−an
Ctn (St ) = max
{P (St ; at = ant )} + βEt [Ct+1 t (S̃t+1 )] , n = 1, . . . , N
n
at ∈A
(3)
In particular, at time M , the expected cash flows from continuation are zero, i.e.,
n
EM [CM
+1 (S̃M +1 )] = 0 ∀n
• At periods t < N − n + 1 (for n = 1, . . . , N ), the exercise of the (N − n + 1)th
interruption right cannot be performed. The reason behind it relates to the number
of exercise rights already used. If, say, only one exercise right is remaining, the
retailer must have already exercised N − 1 rights. Since only one right can be
exercised at each time step, at least the first N − 1 steps must have been used for
this purpose. Therefore, it is not possible to exercise the N th right before time N .
The value of the swing option with N exercise rights corresponds to the maximum expected discounted cash flows to go at time 1, when the number of exercise rights remaining
is N , which is given by βC1N (S1 ).
1
If at time M − 2 there are, say, three exercise rights left, then the retailer bases its decision of
interruption at M − 2, M − 1 and M solely on the payoff from immediate exercise. If the retailer does
not exercise these rights, then they will expire worthless. Thus, the expected value of continuation does
not influence the decision of interruption at these periods.
Risk Analysis of Interruptible Load Contracts
3
7
Solving the Stochastic Dynamic Programme
Options with path-dependent features are priced using numerical schemes since analytical
solutions are not generally available. These numerical schemes fall into one of the following
categories: finite differences, lattice methods, and Monte Carlo simulation. The latter is
becoming more popular due to its ability to cope with uncertainty in a simple way and
also due to the dramatic increase in computational speed. When the underlying process
has a jump component or has multiple stochastic factors, it is easier to value the option
through simulation than through other numerical methods.
The treatment of the early-exercise feature of American-type options had been the
main drawback of Monte Carlo simulation. This problem has been tackled by recent
research, however. Some authors use stratification or parameterisation techniques to
approximate the transition probabilities or the optimal exercise frontier (see Barraquand
and Martineau, 1995). Others propose an approach that consists of estimating by leastsquares a conditional expectation function for the continuation value. One of the most
popular examples of this is the least-squares Monte Carlo (LSMC) simulation algorithm
of Longstaff and Schwartz (2001).
In LSMC simulation, at any exercise time, the holder of an American option compares
the payoff from immediate exercise with the expected payoff from continuation. Thus,
the optimal exercise strategy is determined by the conditional expectation of the payoff
from maintaining the option alive. The key insight of the Longstaff-Schwartz algorithm is
that this conditional expectation can be estimated from the cross-sectional information in
the simulation by using least-squares, more specifically, by regressing ex post discounted
realised payoffs from continuation on basis functions of the values of the state variables.
The fitted value provides a direct estimate of the conditional expectation function. Working backwards through time, for each exercise period, early exercise is performed if the
payoff from immediate exercise exceeds the expected value of continuation. Using this
method, one obtains a complete specification of the optimal exercise strategy along each
path. The American option value is then the average of the time-zero discounted cash
flows over all paths.
For this article, we use a modified version of the Longstaff-Schwartz algorithm to
allow for multiple exercise opportunities.2 The main difference is the addition of an extra
dimension to the algorithm to reflect the number of exercise rights left.3 Consequently,
instead of cash flow matrices, we work with cash flow tensors of three dimensions: number
2
Dorr (2003) provides a detailed description of how to extend the LSMC algorithm to swing options.
This is analogous to the trinomial forest approach for pricing swing options proposed by Jaillet et al.
(2004).
3
Risk Analysis of Interruptible Load Contracts
8
of paths, exercise opportunity times, and exercise rights left.
3.1
LSMC Simulation Algorithm
Before solving the SDP described in Section 2 using LSMC simulation, some terms need
to be defined:
(i)
• Ctn (St ): maximum expected discounted cash flow to go from the swing option at
time t along sample path i, given that the number of interruptions rights left is n.
(i)
(i)
n
• Φnt (St ) ≡ βEt [Ct+1
(St+1 )]: expected continuation value at time t for sample path
i given n remaining interruption rights.
(i)
(i)
n
• Φ̂nt (St ) = f (St )b̂t : estimated expected continuation value at time t for samn
ple path i given that the number of interruptions rights left is n, where b̂t =
[(f(S t ))T (f(S t ))]−1 (f(S t ))T βC nt+1 (S t+1 ) is the parameter vector estimated from the
cross-sectional, least-squares regression of the discounted future cash flows on a
function of the spot prices at time t given the number of interruption rights left, n.
(i)
(i)
(i)2
(i)3
St
St ]
Here, the function of spot prices4 is assumed to be f (St ) = [1 St
Figure 1 describes the algorithm in pseudocode form. It begins by sampling I paths of
(i)
the underlying spot prices at times t = 1, 2, . . . , M , with St representing the electricity
spot price of the ith path at time t. Next, some boundary conditions are set. In line 2,
the cash flows for the case when there are no interruption rights left are set to zero for all
periods and all paths. In line 3, since no further cash flows can occur after the contract
has expired, the cash flows for period M + 1 are also set to zero for every path and every
possible number of interruption rights remaining. In line 4, the estimated continuation
function is initialised to zero.5 The algorithm then works backwards in time (starting at
the end of the contract), proceeding as follows:
(i)
(i)
• line 6: ιt assumes the value 1, if St − K > 0, and 0 otherwise. Its purpose is
to determine at each period t which paths are in-the-money, i.e., in which paths
the spot price exceeds the strike price. Only in those cases will the retailer consider
exercising an interruption right. When the spot price lies below the strike price, i.e.,
4
Longstaff and Schwartz (2001) carried out numerical tests that indicate that the results from the
LSMC simulation algorithm are very robust to the choice of basis functions, so that even simple powers
of the state variables give accurate results. Additionally, they claim that only a few basis functions are
needed to approximate closely the conditional expectation function.
5
In particular, the continuation function for zero exercise rights left will remain zero (since no future
cash flows to go are expected when all interruption rights have been exercised).
Risk Analysis of Interruptible Load Contracts
9
when the option is out-of-the-money, the retailer obtains a profit and, therefore, has
no interest in exercising an interruption right.
• line 9: at periods t > M − n, the decision to exercise an interruption right depends
only on the option’s being in-the-money or not. If n curtailment rights are still
remaining, then the retailer will exercise them at times t = M − n + 1, . . . , M ,
whenever the spot price exceeds the strike price at these periods.
• line 11: more generally, for t ≤ M − n, the response parameter vector for period
t and number of exercise rights left, n, is estimated by least-squares regression of
period t + 1 discounted cash flows to go (given that the number of interruption
n
rights left is n) on the function of the spot prices at time t. To estimate b̂t , only
in-the-money paths at time t are used in the vectors of spot prices.6
• line 12: the estimated continuation function at time t for the number of exercises
left n is obtained by multiplying the response parameter vector for time period t
and the number of exercise rights left n with the function of the time t spot prices,
for all in-the-money paths.
• line 14: the optimal decision at time t for a given number of interruption rights
remaining n is that which maximises the expected discounted cash flow to go using
the estimated continuation function, for all in-the-money paths. Concretely, the
optimal decision at time t for a given number of interruption rights, n, maximises
the immediate payoff from making an interruption decision plus the estimated continuation value of proceeding optimally thereafter from the future state in period
t + 1.7
• line 15: for out-of-the money paths, the optimal decision at time t (irrespective of
the number of exercise rights remaining) is not to exercise an interruption right.
• line 19: the cash flow to go function is updated by using the immediate payoff
(i)
associated to the optimal decision ant ∗ and the actual discounted cash flow to go
of the following period.
• lines 7, 13 and 18: these restrictions guarantee that at periods t < N − n + 1 the
(i)
(i)
decision variable atn ∗ and the cash flow to go function Ctn (St ) are not defined,
since they make no sense. For example, at time 1, there are always N interruption
6
According to Longstaff and Schwartz (2001), using only in-the-money paths allows us to estimate
more accurately the conditional expectation function in the region where exercise is relevant.
7
This term is Φ̂nt (St ) if no interruption right is exercised and Φ̂n−1
(St ) if an exercise right is exercised.
t
Risk Analysis of Interruptible Load Contracts
10
rights remaining, so the cash flows to go and the optimal interruption decision with
any other number of interruption rights left are not defined at time 1.
• line 23: the option value for each path is given by the time-zero discounted cash
flows to go at time 1 when the number of exercise rights remaining is N .
• line 24: averaging the option value over all paths yields the expected value of the
option with N interruption rights.
1
2
3
4
5
6
(i)
GENERATE St
, i = 1, . . . , I, t = 1, . . . , M
(i)
0
SET Ct (St ) = 0 , i = 1, . . . , I, t = 1, . . . , M
(i)
n
SET CM
+1 (SM +1 ) = 0 , i = 1, . . . , I, n = 0, 1, . . . , N
(i)
SET Φ̂nt (St ) = 0 , i = 1, . . . , I, t = 1, . . . , M, n = 0, 1, . . . , N
FOR t = M, . . . , 1
(i)
(i) (i)
SET ιt = arg maxa(i) ∈A {P (St ; at )} , i = 1, . . . , I
t
7 FOR n = max (1, N − t), . . . , N
8
IF t > M − n THEN
(i)
(i)
9
SET ant ∗ = ιt , i = 1, . . . , I
10
ELSE
n
11
SET b̂t = [(f(S t ))T (f(S t ))]−1 (f(S t ))T βC nt+1 (S t+1 )
(i)
(i) n
(i)
12
SET Φ̂nt (St ) = f (St )b̂t
, i WHERE {ιt = 1}
13
IF n > N − t THEN
14
15
16
17
18
SET ant
(i) ∗
= arg max
(i)
(i)
SET ant ∗ = ιt
END IF
END IF
IF n > N − t THEN
(i)
(i)
an
∈A
t
(i)
24 SET C̄ = β
i=1
(i)
(i)
(St )}, i WHERE {ιt = 1}
(i)
, i WHERE {ιt = 0}
(i)
(i)
(i) ∗
n−an
19
SET Ctn (St ) = P (St ; atn ∗ ) + βCt+1 t
20
END IF
21 END FOR
22 END FOR
(i)
23 SET C (i) = βC1N (S1 ) i = 1, ..., I
PI
(i)
n−an
t
(i)
{P (St ; ant ) + Φ̂t
(i)
(St+1 ) , i = 1, . . . , I
C (i)
I
Figure 1: LSMC simulation algorithm
3.2
Optimal Interruption Threshold
The valuation of the curtailment option has been based on the assumption that the
retailer applies the optimal exercise strategy. To be able do so, the retailer must know the
Risk Analysis of Interruptible Load Contracts
11
spot price at each time step above which another curtailment right should be exercised.
From Section 2, it is clear that this decision is dependent on two factors: the number
of curtailment rights remaining and the time period. Let Stn∗ denote the interruption
threshold at period t given the number of exercise rights left, n, where n = 1, . . . , N and
t = 1, . . . , M . For example, if N = 3 and M = 6, then the strategy matrix would look
like:


S∗ = 



N A N A S31∗ S41∗ S51∗ K 

N A S22∗ S32∗ S42∗ K K 

S13∗ S23∗ S33∗ K K K
For t < N − n + 1, the definition of a threshold does not make sense. As explained in
Section 2, the (N − n + 1)th right cannot be exercised before time N − n + 1.
Furthermore, for t > M − n, the optimal threshold is the strike price K since the
respective curtailment rights expire worthless if the retailer does not exercise them.
If the number of paths is sufficiently large, then the threshold values Stn∗ can be
determined by finding the smallest spot price (among the I paths) that implies exercising
∗
an interruption right (i.e., where ant = 1) for each combination of period t and number
of exercise rights remaining, n.
4
Numerical Example and Managerial Insights
To validate the algorithm, we applied it to one-month (T =
30
)
365
pay-as-you-go and pay-
in-advance IL contracts agreed between a hypothetical retailer and a customer with a
constant load subject to the constraints in Section 2. The retailer is allowed to interrupt
the electricity supply up to N = 5 times over the duration of the load contract. Since
1
interruption rights can be exercised on a daily basis (∆t = 365
), there are M = 30 exercise
opportunities. Furthermore, we assume that continuously compounded discounting is
carried out at a rate of 5%. Since demand is assumed to be constant, we will investigate
only the option value per MWh of electricity supplied in this example.8
8
The algorithm was implemented in R. For 100,000 paths it takes approximately 10 seconds to run on
a 2.8 GHz Pentium 4 with 1 GB of RAM. The empirical error (based on 10 simulations of 100,000 paths
each) is approximately 0.45% for the mean value of the swing option.
Risk Analysis of Interruptible Load Contracts
4.1
12
Price Process
For simplicity, suppose that the process followed by the electricity spot price St is a
geometric Brownian motion (GBM)9
dSt = µSt dt + σSt dzt
(4)
where µ is the annual expected growth rate of prices, σ is the annual price volatility, and
{zt , t ≥ 0} is a standard Brownian motion process. In particular, if the volatility of the
spot price were always zero, then this model would imply that:
dSt = µSt dt
(5)
Integrating between t and t + ∆t, we get:
St+∆tυ = St eµ∆t
(6)
where ∆t is the time step expressed in years and υ is a factor that converts the length of
the time period ∆t into days from years. This equation shows that when the volatility is
zero, the price grows at a continuously compounded rate of µ per year.
When the price follows the process in Equation (4), it can be shown (Hull, 2006) using
2
Itô’s Lemma that ln(St ) follows a generalised Brownian motion with drift rate (µ − σ2 )
and variance rate σ 2 :
σ2
d ln(St ) = (µ − )dt + σdzt
(7)
2
In practice, it is usually more accurate to simulate ln(St ) rather than St . Therefore, we
have used the discrete-time version of the process in Equation (7) for constructing paths
for the electricity spot price, which can be written as:
St+∆tυ = St exp [(µ −
√
σ2
)∆t + σ² ∆t]
2
(8)
where ² is a standard normal random variable. Assuming that µ = 0.05, σ = 0.2, and the
initial daily spot price is $21.7 per MWh, we simulated n = 100, 000 (50,000 plus 50,000
antithetic) paths for the spot price according to Equation (8).
9
Note that our approach easily extends to other price processes.
Risk Analysis of Interruptible Load Contracts
4.2
13
Profit
The present value of the total profit per MWh achieved through the IL contract, ΠT , can
be determined by adding the value of the swing option with N interruption rights to the
present value of the profit, ΠN , that would be obtained through that specific contract if
the retailer did not exercise any interruption right:
ΠT
(i)
= C (i) + ΠN
(i)
, i = 1, . . . , I
(9)
The latter profit is given by the price per MWh charged to the customer minus the
electricity spot price. It is defined as:
ΠN
t
(i)
(i)
= pretail − St
, i = 1, . . . , I, t = 1, . . . , M
(10)
in the case of a pay-as-you-go contract or as
ΠN
t
(i)
(i)
= preduced − St
, i = 1, . . . , I, t = 1, . . . , M
(11)
in the case of a pay-in-advance contract. Its present value is simply the sum of these
profits received at each period t discounted back to time zero:
N (i)
Π
=
M
X
β t ΠtN
(i)
, i = 1, . . . , I
(12)
t=1
We then obtain an empirical distribution for ΠN and ΠT since each path does not necessarily lead to the same profit. By taking the average over all paths, we can determine the
expected values Π̄N and Π̄T of ΠN and ΠT , respectively.
4.3
Risk Measures
Based on the empirical distributions described in Section 4.2, we can calculate useful risk
measures such as the probability of loss (given by the proportion of paths with a negative
profit), the value-at-risk (V aR) and the expected tail loss (ET L). For α ∈ [0, 1] the M day, α × 100% value-at-risk (V aRα ) represents the profit that can occur with probability
(1−α)×100% over the next M days. If F (π) is the cumulative distribution function of the
profit, then V aRα is defined from F (V aRα ) = 1 − α. Typical values for α are 0.95 or 0.99,
corresponding usually to a negative profit. V aRα is easily obtained by determining the
(1 − α) × 100% quantile of the empirical distribution of the profit. The M -day α × 100%
expected tail loss (ET Lα ) is the expected loss over a target horizon conditional on the
Risk Analysis of Interruptible Load Contracts
14
losses exceeding V aRα . Formally it is defined as:
R V aRα
πf (π)dπ
E[Π|Π < V aRα ] = R−∞
V aRα
f (π)dπ
−∞
(13)
where f (π) is the probability density function of the profit. Empirically, this measure is
calculated by taking the average of the profits below the V aRα estimate. By convention,
both V aR and ET L are reported as positive values.
V aR tells us the most we can expect to lose if a tail event does not occur, whereas
ET L tell us what we can expect to lose if a tail event does occur. Although V aR is the
most popular measure of risk among regulators and risk managers, ET L is demonstrably
superior.10
4.4
Retail Price
The first problem the retailer is faced with is to determine how much pretail should be.
Assuming that the retailer is a state-owned company, its objective could be to obtain a
zero profit. Using an algorithm to iteratively search for pretail that leads to an expected
value of the profit from a simple load contract (i.e., a contract with no interruption rights)
approximately equal to zero11 , we found that pretail = $21.75. Using the Central Limit
Theorem, a 95% confidence interval for the mean of the profit12 can be constructed and
is given by:
s(ΠT )
(14)
Π̄T ± z0.975 √
I
T
√ ) is its sample standard deviation. For
where Π̄T is the sample average of ΠT and s(Π
I
pretail = $21.75, the 95% confidence interval for the mean of ΠT is [-$0.02, $0.26]. The
probability of loss is 49%, the 30-day 99% Value-at-Risk is $53.05, and the 30-day 99%
expected tail loss is $61.6. In other words, we are 99% certain that the retailer will not
lose more than $53.05 with this simple load contract and that the average amount that
the retailer may lose over the time of the contract, assuming that the 1% worst-case event
occurs, is $61.6.
10
Among other caveats (Dowd, 2002), V aR is uninformative of tail losses (if a tail event occurs, we
can expect to lose more than V aR but we have no indication of how much the loss might be), and it is
not sub-additive (in the sense that aggregating individual risks may increase overall risk).
11
In this sample the closest one could get to a zero profit using prices rounded to two digits was $0.12.
12
In this kind of contract the total profit ΠT is equal to ΠN due to the absence of interruption rights.
Risk Analysis of Interruptible Load Contracts
4.5
15
Rebate and Compensation
Next, the retailer must decide which rebate or compensation to give to the customer as
an incentive to sign up an IL contract. Let us consider first the pay-in-advance contract.
The maximum discount per MWh that the retailer can offer occurs when the entire swing
option value is transferred to the customer. Offering a higher rebate would lead to a
lower mean profit than with a contract without any interruption rights. To determine
the maximum discount, we search for preduced that leads to a mean total profit, Π̄T ,
approximately equal to the mean profit obtained when charging pretail = $21.75 to the
consumer in a normal load contract, which in this sample is $21.66. Thus, the maximum
rebate is pretail − preduced = $0.09. On the one hand, this price increases the probability
of loss to 51%, but on the other hand, it lowers the 30-day 99% V aR to $42.47 and the
30-day 99% ET L to $48.95.
In the case of a pay-as-you-go contract, the maximum compensation per curtailed
MWh corresponds to the case when on average the option is practically worthless, i.e.,
when the compensation is so high that it is more costly for the retailer to exercise an
interruption right than to supply electricity to the consumer. This is the case for penalties
above $5. While the probability of loss, the 30-day 99% V aR, and the 30-day 99% ET L
remain the same, even when offering a large compensation, this type of contract is still
worthwhile to the retailer to hedge against the risk of high unforeseeable spot prices.
Figure 2 contains histograms of the empirical distributions of the total profit under a
simple load contract with pretail = $21.75, under a pay-in-advance contract with preduced =
$21.66 and under a pay-as-you-go contract with pf ine = $3.24 and pretail = $21.75. All
contracts lead to approximately the same mean total profit. Table 1 contains risk measures
for these three contracts.
Contract type
Simple
Pay-in-advance
Pay-as-you-go
30-day 99% VaR (in $)
53.05
42.47
52.75
30-day 99% ETL (in $)
61.6
48.95
60.45
Table 1: Risk measures
Figure 2 and Table 1 show that these IL contracts reduce (or at least do not increase)
the risk of extreme losses, even when transferring the entire profit obtained from the swing
option to the customer.13 In this pay-as-you-go contract, the compensation per interrup13
Baldick et al. (2006) also indicates that IL contracts are very valuable to electricity retailers with
limited amounts of generation available.
Risk Analysis of Interruptible Load Contracts
16
tion paid to the customer is so high that the curtailment rights are seldom exercised,
which explains why the V aR and the ET L improve only marginally. This pay-in-advance
contract leads to a reduction of both extreme losses and extreme profits. By offering a
lower price per MWh for the entire load of the pay-in-advance contract the retailer sees its
“do nothing” profits go down, which explains the reduction of extreme profits. Since this
contract offers the possibility of interrupting the supply of electricity a fixed number of
times, these rights will be exercised when spot prices are particularly high, thus reducing
the risk of extreme losses as compared to a simple load contract.14
Simple load contract
0.010
0.000
Density
mean
30−day 99% VaR
−100
−50
0
50
100
Profit per MWh (in $)
0.010
mean
30−day 99% VaR
0.000
Density
0.020
Pay−in−advance contract
−100
−50
0
50
100
Profit per MWh (in $)
Pay−as−you−go contract
0.010
0.000
Density
mean
30−day 99% VaR
−100
−50
0
50
100
Profit per MWh (in $)
Figure 2: Histograms of total profit
4.6
Comparing Contracts
Suppose that the retailer is interested in comparing a pay-in-advance with a pay-as-yougo contract that leads to the same mean total profit in order to determine which one is
preferable. According to Baldick et al. (2006), the electricity retailer would prefer to
sign contracts that provide compensation upon interruption since interruptible contracts
that provide an up-front discount to the entire customer load can be very costly to the
retailer due to the sunk nature of the compensation. One reason for offering both types of
contracts is to determine which type of contract makes the consumer more likely to sign
up. Fahrioglu and Alvarado (2000) discusses methods based on utility and cost functions
14
In this example, since the initial spot price, S0 , is higher than preduced and spot prices are increasing
on average, it is very likely that the interruption rights are exercised. Therefore, it is not surprising
that the reduction of extreme losses due to the introduction of five interruption opportunities more than
compensates the increase in extreme losses due to the rebate offered to the customer.
Risk Analysis of Interruptible Load Contracts
17
to estimate the demand among customers for IL contracts and describe an incentive
structure that encourages customers to reveal their true value of electricity (and thus, the
value of power interruptibility).
Another criterion for this comparison could be the risk of extreme losses. Given a
certain mean total profit, which type of contract leads to the smallest V aR, ET L, and/or
probability of loss for the retailer? Table 2 contains the outcome of a matching algorithm
that yields the same mean total profit for both types of contracts over the range of rebates
discussed in Section 4.5.15 Figure 3 shows the comparison of risk measures between the
two types of contracts. The smaller the rebate, the higher Π̄T , the 30-day 99% V aR,
and the 30-day 99% ET L. In a pay-as-you-go contract, as pf ine goes up, the retailer will
exercise each interruption right only at higher prices, so it is more likely that the retailer
ends up having more extreme losses. In particular, for large compensations, the retailer
will exercise the interruption rights only on very rare occasions. In a pay-in-advance
contract, as the rebate increases, losses increase because the price paid by the customer
for electricity is lower, but the option value rises because the retailer exercises at lower
prices (i.e., the strike price is lower). The former effect has slightly more weight here,
explaining why the V aR and the ET L increase.
Π̄T
0.13
0.67
1.21
1.76
2.31
preduced
21.66
21.68
21.7
21.72
21.74
pf ine
3.24
1.22
0.72
0.38
0.11
Table 2: Matching profits
In this example, the pay-in-advance contract leads systematically to lower V aR and
ET L. This occurs because the retailer can exercise the interruption rights at lower spot
15
As expected, a larger rebate leads to a lower mean total profit, ceteris paribus. Increasing pf ine in
a pay-as-you-go contract reduces the value of the interruption option C̄ because each interruption right
will be exercised only at higher prices (i.e., the strike price increases). This induces a decrease in Π̄T ,
since Π̄N remains unchanged. On the one hand, charging a lower preduced in a pay-in-advance contract
reduces Π̄N because the retailer receives less for each MWh, yet purchases electricity at the same spot
prices. On the other hand, this increases C̄ because each right can be exercised at lower prices (i.e., the
strike price is lower). This latter effect is weaker than the former, so that Π̄T decreases. To understand
why, recall that over the duration of the contract the electricity retailer will curtail electricity supply at
the most five days, but in the remaining 25 days, the retailer will receive less for each unit of electricity
supplied. Figueroa (2006) also observed that the value of the swing option varies inversely with the strike
price.
Risk Analysis of Interruptible Load Contracts
0.51
pay−in−advance
pay−as−you−go
40
40
0.47
45
0.48
0.49
P(ΠT < 0)
0.50
55
60
pay−in−advance
pay−as−you−go
50
30−day 99% ETL (in $)
55
50
45
30−day 99% VaR (in $)
60
pay−in−advance
pay−as−you−go
18
0.5
1.0
ΠT (in $)
1.5
2.0
0.5
1.0
1.5
2.0
0.5
ΠT (in $)
1.0
1.5
2.0
ΠT (in $)
Figure 3: Comparison of risk measures
prices in comparison to the pay-as-you-go contract. The retailer has already “paid” upfront (by offering a reduced retail price) for the right to interrupt the supply, whereas
in the pay-as-you-go contract the retailer will need to disburse each time to exercise the
interruption rights. Especially when this compensation is quite high, the retailer may end
up not exercising the interruption rights or only exercising them when prices are quite
high.
While in the pay-in-advance contract the V aR and ET L increase nearly linearly as
T
Π̄ decreases, in the pay-as-you-go contract they increase exponentially, leading to an
increasing discrepancy between the risk measures of the two contracts. To achieve the
same mean total profit in the pay-as-you-go contract, we must increase pf ine more substantially than we increase the rebate in the pay-in-advance contract, since the latter is
paid per MWh of electricity supplied and the former is per interruption. As a result, the
discrepancy between the strike prices of the two contracts increases nearly exponentially,
which is reflected in the V aR and in the ET L.
As regards the probability of obtaining a loss, it is nearly the same in both types of
contracts when the rebate per MWh supplied is lower or equal to $0.05 in a pay-in-advance
contract. In this range, this probability decreases as pf ine decreases (since it allows the
retailer to exercise interruption rights at lower prices) and as preduced increases (since a
higher reduced retail price increases ΠN and consequently ΠT ). Offering a discount higher
than $0.05 increases the probability of obtaining losses with a pay-in-advance contract as
compared to a simple load contract because the benefit from the interruption option is
overridden by the decrease in profit due to the reduction of the reduced retail price. In
contrast, in a pay-as-you-go contract this probability will never exceed the one obtained
with a simple load contract because the retailer exercises an interruption right only if its
losses are reduced by doing so.
Risk Analysis of Interruptible Load Contracts
19
In conclusion16 , a pay-in-advance contract with a rebate per MWh supplied lower or
equal to $0.05 is slightly superior to the corresponding pay-as-you-go contract that leads
to the same mean total profit, in the sense that the risk of extreme losses is smaller.
Otherwise, the pay-as-you-go contract may be preferable since the chance of incurring in
a loss does not increase as compared to a simple load contract.
4.7
Sensitivity Analysis
From Section 4.6, one can conclude that in order for the retailer to be prudent and give
as much incentive to the consumer as possible in the form of a rebate, the reduced price,
preduced , proposed in the pay-in-advance contract should be $21.7. Decomposing the mean
total profit of this contract (Π̄T = $1.21) into its two components, we get Π̄N = −$1.38
and C̄ = $2.59. The penalty that leads to the same Π̄T of $1.21 in a pay-as-you-go
contract is $0.72 when pretail is $21.75. In this case, Π̄N = $0.12 and C̄ = $1.09. For the
sensitivity analysis, we will use these parameters and the ones stated at the beginning
of the numerical example, unless otherwise indicated. Furthermore, for consistency, the
same sequence of random standard normal variables is used to simulate spot prices.
4.7.1
Number of Exercise Rights
Figure 4 shows that as the number of exercise rights, N , stipulated in the load contract
increases, both the value of the swing option17 and the mean total profit increase, while
both the 30-day 99% V aR and ET L decrease. The more exercise rights the contract
contains, the more valuable the swing contract is, and thus, the higher the total profit
will be, ceteris paribus.18 Moreover, the retailer is better protected against extreme losses,
so the V aR and the ET L are smaller.
Here C̄ and, consequently, Π̄T of the pay-in-advance contract rise more steeply with
the number of exercise rights when compared to the pay-as-you-go contract, so that for a
number of interruption rights above five the former results in a higher Π̄T than the latter.
Because each interruption right can be exercised at a lower strike price, having an extra
interruption right in the pay-in-advance contract is more valuable than having one more
interruption right in the pay-as-you-go contract.
16
Note that conclusions drawn from this type of analysis depend substantially on the price process
chosen, the initial spot price, and characteristics of the contract (number of exercise rights and number
of exercise opportunities).
17
Figueroa (2006) also observed that the value of the (up-)swing option increases with the number of
exercise rights N .
18 N
Π , the profit obtained if the retailer does not exercise any of the interruption rights, does not change
with the number of interruption rights.
Risk Analysis of Interruptible Load Contracts
2
C (in $)
3
4
pay−in−advance
pay−as−you−go
−1
0
1
2
1
−1
0
ΠT (in $)
3
4
pay−in−advance
pay−as−you−go
20
1
2
3
4
5
6
7
8
1
2
4
5
6
7
8
7
8
45
50
55
pay−in−advance
pay−as−you−go
40
35
40
45
30−day 99% ETL (in $)
50
pay−in−advance
pay−as−you−go
30−day 99% VaR (in $)
3
Number of interruption rights
60
Number of interruption rights
1
2
3
4
5
6
Number of interruption rights
7
8
1
2
3
4
5
6
Number of interruption rights
Figure 4: Number of interruption rights analysis
Figure 4 reveals one of the biggest differences between the profit of the two types of IL
contracts. Pay-as-you-go contracts will never lead to a Π̄T lower than the one obtained
through a simple load contract19 since interruption rights are exercised only if curtailment
is to the benefit of the retailer. By contrast, it is possible that the pay-in-advance contract
leads to a lower Π̄T than a simple load contract. In this example, this would happen if the
pay-in-advance contract contained so few exercise rights that the option value C̄ would
not be high enough to compensate for the negative Π̄N . Recall that since in the pay-inadvance contract the cost of interruption is paid up-front and is sunk, it may happen that
the reduction in income due to the discount on the retail price is higher than the value
added by the IL contract.20 Regarding the V aR and ET L, the pay-in-advance contract
performs better, thereby leading to lower extreme losses.
4.7.2
Volatility
Figure 5 shows that as the volatility σ increases, so does the option value C̄ and consequently the mean total profit Π̄T , since throughout this range of volatilities Π̄N increases
19
20
Recall from Section 4.4 that Π̄T of the simple load contract is $0.12 in this example.
Baldick et al. (2006) comes to the same conclusion.
Risk Analysis of Interruptible Load Contracts
21
only slightly. The interruption rights protect against peak spot prices, which are more
likely to occur when volatility is high, thus explaining the increase in the option value C̄
as volatility increases. Figure 6 shows that the 30-day 99% ET L and V aR also increase
0.1
0.2
0.3
0.4
5
4
3
2
C (in $)
0
−1
0
−1
0.0
pay−in−advance
pay−as−you−go
1
2
ΠN (in $)
3
4
5
pay−in−advance
pay−as−you−go
1
2
−1
0
1
ΠT (in $)
3
4
5
pay−in−advance
pay−as−you−go
0.0
0.1
σ
0.2
0.3
0.4
0.0
0.1
0.2
σ
0.3
0.4
σ
Figure 5: Mean total profit and its components versus price volatility
because it is more likely that the retailer faces high spot prices when volatility is high.21
Comparing both contracts, C̄, Π̄T , V aR, and ET L increase in a very similar fashion thus
leading to quite similar results, except in the case of very low volatilities.22
60
80
100
pay−in−advance
pay−as−you−go
0
20
40
30−day 99% ETL (in $)
60
40
0
20
30−day 99% VaR (in $)
80
100
pay−in−advance
pay−as−you−go
0.1
0.2
σ
0.3
0.4
0.1
0.2
0.3
0.4
σ
Figure 6: Risk measures versus price volatility
21
There may not be enough interruption rights in the contract to safeguard against these high prices.
This is explained by the fact that for very low volatilities it is quite unlikely in the pay-as-you-go
contract that the spot price exceeds the strike price (which is pretail + pf ine = $22.47) over the 30-day
period when the initial spot price is only $21.7. This leads to an option value C̄ very close to zero for very
low volatilities in this type of contract. This phenomenon does not occur in the pay-in-advance contract
since the initial spot price S0 is equal to the reduced retail price preduced .
22
Risk Analysis of Interruptible Load Contracts
4.8
22
Optimal Interruption Threshold
As explained in Section 3.2, when the number of paths is sufficiently large, the optimal
interruption boundary can easily be determined by searching for the smallest spot price
that leads to exercise for each period and number of exercise rights left. Since the spot
price evolves stochastically, we are not able to determine the time at which to exercise the
rights, but instead our exercise rule will take the form of a critical value, Stn∗ , above which
it is optimal to exercise a right. Figure 7 shows the optional interruption threshold23 at
time periods 19 and 22. At a given date, the spot price above which the exercise of one
25.0
24.0
24.5
n
S22
* (in $)
25.0
24.0
24.5
Sn19* (in $)
25.5
pay−in−advance
pay−as−you−go
25.5
pay−in−advance
pay−as−you−go
1
2
3
4
5
1
2
Number of interruption rights left n
3
4
5
Number of interruption rights left n
Figure 7: Optimal interruption boundary for a given time period
interruption right should be performed decreases with the number of remaining exercise
rights, as depicted in Figure 7. The more exercise rights the retailer still has available,
the less prudent the retailer must be when exercising them, so that each right can be used
at lower exceedances over the strike price.
Figure 8 shows the optimal threshold when there are one and five interruption rights
remaining. To understand the shape of the optimal interruption threshold, it is useful
to establish an analogy to the exposition in Dixit and Pindyck (1994). First, we analyse
the case in which there is no uncertainty, i.e., σ = 0. Under no uncertainty, we would
be able to determine the optimal time at which to exercise the interruption rights. From
Equation (6), we know that prices would change at a continuously compounded rate of µ
per year. The present value of the immediate payoff from exercising an interruption right
at period t is then:24
t
t
P (St ; ant = 1) = (S0 eµ υ − K)e−r υ
(15)
23
The threshold was built from the average of the thresholds of ten runs of the LSMC algorithm. Each
run was carried out with 100,000 (50,000 plus 50,000 antithetic) paths.
24
Recall from Section 4.1 that υ scales the time period from years into days, so υ1 scales t from days
to years.
24
25
pay−in−advance
pay−as−you−go
22
22
23
24
St5* (in $)
25
pay−in−advance
pay−as−you−go
23
S1t * (in $)
23
26
26
Risk Analysis of Interruptible Load Contracts
0
5
10
15
20
25
30
0
Period t
5
10
15
20
25
30
Period t
Figure 8: Optimal interruption boundary for a given number of interruption rights left
Supposing that µ ≤ 0, prices would remain constant or fall over time. In that case, it
would be optimal to exercise the interruption rights at the beginning of the contract,
as long as the payoff from immediate exercise were positive. If µ > 0, then there is
a value to waiting before exercising the rights because spot prices are growing. The
reason for delaying the exercise of an interruption right is that in present value terms, the
t
implicit “cost” of exercise decreases by a factor of e−r υ whereas the implicit “revenue”
t
t
from exercise changes by a factor of e(µ−r) υ , i.e., increases by a factor of e(µ−r) υ if µ > r,
t
remains the same if µ = r or decreases by a smaller factor of e−(r−µ) υ if 0 < µ < r.
In other words, it is preferable to delay exercise because the implicit “costs” are being
reduced more than the implicit “revenues.” Note that for µ ≥ r it would be optimal
to delay exercise for as long as possible, but for 0 < µ < r there is an optimal time to
exercise a right.
Figures 9 and 10 show the impact of this phenomenon, called the time value of money,
on the optimal threshold. A lower expected growth rate of prices, µ, or a higher discount
rate, r, lower the thresholds since it becomes even more profitable in terms of present
value to exercise sooner rather than later.
The value to waiting assumes greater importance when there is uncertainty surrounding future prices. In that case, holding an interruption right serves as an insurance against
high prices. Thus, the retailer might delay exercising the interruption right in the hope
of a bigger payoff later on and as an insurance. Figure 11 depicts the effect of volatility
of prices on the threshold. The higher the uncertainty, the more peaked the threshold is.
In the presence of more uncertainty, the retailer should be more prudent when exercising
an interruption right, which explains the higher thresholds.
We are now in a better position to explain the shape of the thresholds. The time
value of money leads to increasing thresholds at the beginning because the “cost” of
Risk Analysis of Interruptible Load Contracts
24
Pay−in−advance contract
Pay−as−you−go contract
−0.15
0.01
0.05
24
22
23
S5t * (in $)
24
22
23
S5t * (in $)
25
µ:
−0.15
0.01
0.05
25
µ:
0
5
10
15
20
25
30
0
5
10
Period t
15
20
25
30
Period t
Figure 9: Impact of expected growth rate of prices on optimal interruption boundary
Pay−in−advance contract
Pay−as−you−go contract
r:
r:
24
22
23
S5t * (in $)
24
22
23
S5t * (in $)
25
0.05
0.4
25
0.05
0.4
0
5
10
15
Period t
20
25
30
0
5
10
15
20
25
30
Period t
Figure 10: Impact of discount rate on optimal interruption boundary
exercise is discounted more heavily than the “revenue” from exercise. For µ ≥ r it would
be optimal to delay exercise as much as possible, but for 0 < µ < r there would be a
point beyond which delaying exercise (from the perspective of the time value of money)
would do more harm than good.25 Because there is uncertainty regarding future prices,
the interruption rights will serve as an insurance against unexpected high prices. The
insurance value increases the wedge between S ∗ and K, that is, the threshold goes up.
The retailer demands a larger immediate payoff for exercising an interruption right. As
the contract approaches the end, exercise is facilitated if there are still some interruption
rights remaining, since there are fewer remaining time periods for higher prices to result
and the available rights may end up not being exercised at all. This explains the reduction
25
Note that this time is larger in the pay-as-you-go contract because the “cost” of exercise is higher
than in the pay-as-you-go contract.
Risk Analysis of Interruptible Load Contracts
25
Pay−in−advance contract
Pay−as−you−go contract
σ:
22
24
S5t * (in $)
28
0.1
0.2
0.4
26
28
26
22
24
S5t * (in $)
σ:
0.1
0.2
0.4
0
5
10
15
20
25
30
0
Period t
5
10
15
20
25
30
Period t
Figure 11: Impact of price volatility on optimal interruption boundary
in the threshold near maturity.
From Figures 7 and 8, it is clear that the optimal exercise threshold of the pay-as-yougo contract is higher than the one of the pay-in-advance contract. In the pay-as-you-go
contract, the implicit “cost” of exercising a right is pretail + pf ine = $22.47 whereas in the
pay-in-advance contract it is preduced = $21.7. In the latter contract, part of the “cost”
has been paid up-front in the form of a rebate per unit of MWh supplied over the entire
load. Since this cost has a sunk nature, it does not affect the threshold. The lower limit
for the optimal interruption threshold of the pay-as-you-go contract is then higher than
the one of the pay-in-advance contract.
5
Summary and Conclusions
In this paper, we have presented an SDP for optimally exercising a limited number of rights
in an IL contract. The optimal exercise strategy is found by maximising the expected
discounted future cash flows obtained through the swing option embedded in the IL
contract. To solve this SDP, we have used LSMC simulation to yield the value of the
swing option and the optimal exercise strategy along each path. By looking for the
smallest spot price that implies exercising a right for each combination of time step and
number of exercise rights left, the threshold for calling load curtailments was determined.
Furthermore, using plain Monte Carlo simulation, the profit obtained through the IL
contract in the case that no interruption rights are exercised can be determined. Combining this result with the swing option value from the LSMC simulation, we obtain the total
value of the IL contract. Based on this empirical distribution, it is possible to calculate
financial risk measures such as the value-at-risk and the expected tail loss.
Risk Analysis of Interruptible Load Contracts
26
Our analysis focused on two types of contracts: pay-in-advance and pay-as-you-go. In
the numerical example, our main aim was to provide some techniques to solve typical
problems faced by an electricity retailer, based on the results from the algorithm. Issues
like determining the retail price, the maximum rebate per MWh supplied in a pay-inadvance contract and the maximum compensation per curtailment in a pay-as-you-go
contract were addressed. In addition, for a plausible range of rebates, we compared a
pay-in-advance contract with the corresponding pay-as-you-go contract that leads to the
same mean total profit with respect to their risk measures to determine whether one type
was preferable to the other and within which range. A sensitivity analysis was carried out
for one plausible example of each type of contract to establish how the total profit and
its components, and the risk measures are affected by the number of interruption rights
and the spot price volatility. Finally, we determined the optimal threshold for calling
load curtailments for each combination of time period and number of interruption rights
remaining, and examined how it is affected by changes in the parameters of the price
process.
Our analysis suggests that, in the absence of ownership of generating assets, IL contracts are rather valuable to electricity retailers since those retailers would need to rely on
the spot market to provide electricity at a fixed price to its customers, and interruptible
contracts mitigate the need to resort to the spot market. A priori, one expects that given
a choice between different types of IL contracts, pay-as-you-go contracts are preferable
to the retailer (Baldick et al., 2006). Pay-as-you-go contracts will never lead to a total
profit lower than a simple load contract since the retailer exercises interruption rights
only when it is to its own benefit. In contrast, pay-in-advance contracts may lead to a
lower mean total profit in cases where the retailer offers a discount on the retail price
too high to the customer so that the reduction in income is higher than the value added
by the option of curtailment. For very large rebates, the probability of obtaining losses,
the V aR, and the ET L may even increase in comparison to a simple load contract. It
is, therefore, important to choose the rebate in pay-in-advance contracts wisely to avoid
these cases, as discussed in Sections 4.5 and 4.6. In our example, we showed that pay-inadvance contracts can produce lower V aR and a ET L than a simple load contract, when
the rebate offered to the customer is not too large. When comparing both contracts, we
also concluded that, for small rebates, a pay-in-advance contract may be preferable to a
pay-as-you-go contract that leads to the same mean total profit in the sense that the risk
of extreme losses is smaller and the probability of loss is not higher.
Another important issue for the retailer addressed in this article is to know the spot
price above which another interruption right should be exercised, at each period and
Risk Analysis of Interruptible Load Contracts
27
given the number of exercise rights remaining. In our example, we concluded that as
the number of interruption rights available decreases, interruption occurs at higher spot
prices since the retailer needs to be more prudent in exercising the remaining interruption
rights. For a given number of interruption rights remaining, the optimal exercise threshold
goes up at the beginning and decreases as the contract approaches its end. Though some
of the conclusions drawn are specific to our examples, this type of analysis can easily be
extended to other price processes and different contract characteristics, and may be useful
for retailers to help them solve pertinent questions related to IL contracts.
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