Forecasting the Mean and the Variance of Electricity Prices in

Transcription

Forecasting the Mean and the Variance of
Electricity Prices in Deregulated Markets
16
Abstract
A fundamental bid-based stochastic model is presented to predict electricity hourly prices and average price in a given period. The model captures both the economic and physical aspects of the pricing
process, considering two sources of uncertainty: availability of the units and demand. This work is
based on three oligopoly models —Bertrand, Cournot and Supply Function Equilibrium (SFE) due to
Rudkevich, Duckworth, and Rosen— and obtains closed form expressions for expected value and
variance of electricity hourly prices and average price. Sensitivity analysis is performed on the number
of firms, anticipated peak demand and price elasticity of demand. The results show that as the number of firms in the market decreases, the expected values of prices increase by a significant amount.
Variances for the Cournot model also increase. But the variances for the SFE model decrease, taking
even smaller values than Bertrand’s. Thus if the Rudkevich model is an accurate representation of the
electricity market, the results show that an introduction of competition may decrease the expected
value of prices but the variances may actually increase. Finally, using a refinement of the model it
has been demonstrated that an accurate temperature forecast can reduce significantly the prediction
error of the electricity prices.
Resumen
Se presenta un modelo estocástico basado en los procesos técnicos de despacho de energía eléctrica
y en las estrategias de oferta de precios en subastas, para predecir los precios de electricidad: precios
horarios y precio promedio en un periodo determinado. El modelo captura los aspectos económicos
y físicos del proceso de fijación de precios, considerando dos fuentes de incertidumbre: la disponibilidad de las unidades generadoras y la demanda. Este trabajo está basado en tres modelos de
oligopolios: modelo de Bertrand, modelo de Cournot y modelo del Equilibrio de Funciones de AbasReprinted with authorization from IEEE Transactions on Power Systems.
Claudio M. Ruibal is with the Universidad de Montevideo, Montevideo, Uruguay (email:[email protected]).
Mainak Mazumdar is with the University of Pittsburgh, Pittsburgh, PA, USA (e-mail: [email protected]).
Revista de Ciencias Empresariales y Economía
Mainak Mazumdar and Claudio M. Ruibal
17
FORECASTING THE MEAN AND THE VARIANCE OF ELECTRICITY PRICES IN DEREGULATED MARKETS
El modelo de Bertrand se toma como referencia. En el modelo de Cournot también aumentan las varianzas; pero en el modelo de Rudkevich las varianzas disminuyen, llegando a tomar valores menores
a los del modelo de Bertrand. Por lo tanto, si el modelo de Rudkevich fuera una representación fiel
del mercado de electricidad, los resultados muestran que una mayor competencia puede disminuir el
valor esperado de los precios pero aumentar su varianza. Finalmente, utilizando un ajuste del modelo, se demuestra que un pron´ostico preciso de la temperatura ambiente puede reducir significativamente el error de predicción de los precios de energía eléctrica.
Index Terms—Electricity Prices, Deregulated Electricity Markets, Electricity Price Variance, Cournot
Model, Bertrand Model, Supply Function Equilibrium, Rudkevich and Duckworth and Rosen’s Formula, Stochastic Load, Hourly Prices, Average Prices, Edgeworth Expansion, Method of Cumulants.
1. Notation
Capacity of unit i
Sum of capacities of the first i units
Total system demand at time t as a function of p
Variable cost of generating unit i
Derivative of D(t, p) with respect to price at time t
Marginal cost at time t
Marginal unit at time t
Cumulant of order # of Xj (t)
Bivariate cumulant of order (i, j) of [Xm (r), Xl (t)]
Actual electricity load at time t
(assumed to be normally distributed)
L J(t) (t) Equivalent load at time t
M
Marginal unit at daily peak
n
Number of companies in the market
N
Number of generating units in the market
p(t)
Price of electricity at time t
p(I,H) Average price between hours I and H
Proportion of time that generating unit i is up
pi
pml (r, t) Joint probability of [J(r) > m, J(t) > l]
qi
Proportion of time that generating unit i is down
(known as Forced Outage Rate, FOR)
wt
Averaging weight of the hourly load at time t
Excess of load not met by the available generated
Xj(t)
power up to generating unit j at time t
Yi(t)
Generating unit i state at hour t
(=1 if working, =0 in case of outage)
−1
Mean time to failure of generating unit i
i
−1
Mean time to repair of generating unit i
i
t
Mean of load at time t
Correlation coefficient
ml ( ) Correlation coefficient between X (r) and X (t)
m
l
t2
Variance of load at time t
r,t
Covariance between the loads at time r and t
ci
Ci
D(t, p)
di
Dp(t)
dJ(t)
J(t)
K#j
Kij
L(t)
18
2. Introduction
In this paper we outline a procedure for forecasting the mean and variance of the average price
of electricity over a specified time interval in a deregulated market. Such information would be
found useful in financial forecasts, risk management, derivative pricing, investment and operational
decisions. The computations are based on a system model in which the physical and engineering
processes and the bidding strategies are simultaneously considered. The price of electricity depends
on physical factors such as production cost, load, generation reliability, unit commitment, and transmission constraints. It also depends on economic factors such as strategic bidding and load elasticity.
We consider here a model that captures the dependence of the price on costs, load, reliability, and
bidding strategies. Many of these factors are stochastic in character which we have characterized
by their probability distributions. A potential advantage of our approach is that it can be used to
consider changes in system’s structure over time (e.g., entry of additional generators or a change in
load.) We do not consider factors related to transmission congestion, transmission outages, and unit
commitment. While these factors should be accounted for in a full and complete description of the
movement of electricity prices, we believe that the system based approach in this paper is an important first step in the construction of a comprehensive model. The inclusion of unit commitment and
transmission constraints will make the system model very complicated.
The degree of complexity that will ensue can be comprehended by referring to Hobbs, Metzler and
Pang [6] and to the papers contained in Hobbs, Rothkopf, O’Neil and Chao [7]. The emphasis in the
current paper is on the use of analytical methods to forecast the statistical distributions of prices.
When unit commitment and transmission constraints are included in the system model it appears
that there will remain no alternative other than using Monte Carlo methods for such forecasts. The
approach that we have taken for modeling the prices is as follows. First we have provided the formulation for computing the mean and variance of price for any specified hour given its generation
and load characteristics using a traditional production costing type model. Then we have given the
framework for computing the mean and variance of the time-average of the price over a specified
time interval. This latter step is much more difficult because in addition to the results obtained from
the first step, estimates on covariances of hourly loads and production quantities become necessary.
In this effort we were aided by the method of cumulants based formulation (in Valenzuela [16]) used
for finding means and variances of production costs over a given interval. The estimates of variances
in addition to those of the expected values would allow computation of prediction intervals for the
price as well as individual firm’s profits. By comparing these prediction intervals yielded by different
models with the actually realized prices a judgment can be made about the accuracy of the individual
models. This is the essence of what is known as backcasting (Paehlke, R. [11]). Also an estimate of
variance will be useful for the purposes of risk management, for example, in the computation of the
value-at-risk and conditional value-at-risk indices (Pilipovi´c [12], Rockafellar and Uryasev [13]). Its
use in the derivative market is also apparent.
We consider three bidding models for the market price of electricity. In each model we assume that
the electricity traded within the region of interest is unconstrained by transmission. The market under
consideration is one where marginal bid pricing is assumed to prevail. We consider three bidding
models: Bertrand model [1] in which firms offer their marginal costs, the Cournot model [2] in which
firms offer quantities that maximize expected profits, and a Supply Function Equilibrium (SFE) model
(Klemperer and Meyer [10]) in which firms offer a supply curve (quantity vs. price) based on Rudkevich, Duckworth and Rosen’s equilibrium formula [14].
1
These three models are based in Nash equilibrium solutions for different bidding strategies. Other
papers, like Kang et al. [9], explicitly include the bidding strategies in the model. We assume that the
deregulated market will eventually arrive at an equilibrium. So, using the equilibrium solutions becomes a more realistic analysis. The first two models have been considered extensively in the context of
deregulated electricity markets. The elegant analytical expression developed in the Rudkevich model
1 Nash equilibrium is a profile of strategies such that each player’s strategy is an optimal response to the other players’ strategies.
tecimiento (SFE por su sigla en inglés) de Rudkevich, Duckworth y Rosen. Se obtienen expresiones
analíticas tanto del valor esperado como de la varianza, de los precios horarios y del precio promedio
de electricidad. Se realiza un análisis de sensibilidad sobre el número de firmas competidoras, el pico
de demanda anticipada para un periodo y la elasticidad de la demanda respecto al precio. Los resultados muestran que a menos firmas compitiendo, los valores esperados de los precios aumentan
considerablemente.
MAINAK MAZUMDAR / CLAUDIO M. RUIBAL
19
Only the large firms participating in the market are considered identical. Even if the firms were actually identical, generator outages would break the theoretical and assumed symmetry. We have
performed sensitivity analysis on the statistics related to electricity prices varying the number of firms,
anticipated peak demand and price elasticity of demand. The results show that as the number of
firms in the market decreases, the expected values of prices increase by a significant amount. Variances for the Cournot model also increase. But the variances for the SFE model decrease, taking even
smaller values than Bertrand’s. Thus if the Rudkevich model is deemed to be an accurate representation of the electricity market, the results suggest that an introduction of “perfect” competition may
decrease the expected value of prices but the variances may actually increase. We also address the
following question: can the electricity prices be more accurately predicted in the sense of obtaining
smaller prediction intervals if accurate predictions of ambient temperature are available for the period
for which the average price is being computed?
The paper is organized into the following sections. Section III briefly describes the three economic
models. Section IV gives the formulas for the mean and variance of hourly price. The expressions for
the mean and variance of the time averaged price over a given interval are given in Section V. Section
VI describes a stochastic model for factoring in the forecast information about ambient temperatures.
Section VII gives the numerical results for a hypothetical market. Section VIII states the conclusions.
3. Basic Models on Electricity Pricing
In the current literature three major models are in use for (imperfect) electricity markets: the Bertrand
model, Cournot model and Supply Function Equilibrium (SFE) model. Cournot and Bertrand models
constitute the two often used paradigms of imperfect competition. In the Bertrand (1883) model
firms compete in price. They simultaneously choose prices and then must produce enough output
to meet demand after the price choices become known. Under the assumption that each firm has
enough capacity to meet demand, the Nash equilibrium price in this model is the marginal cost which
is the same as the case of perfect competition. The other basic non-cooperative equilibrium is the
Cournot (1838) model. In this model competition is in quantities. Firms simultaneously choose the
quantities they will produce, which they then sell at the market-clearing price (the price for which
demand is met by supply). Firms choose the quantities that optimize their profit. Cournot model is
a more accurate representation of the market. The assumption underlying the Bertrand model that
competition is over prices and the firms have enough capacity to meet demand is not sustainable.
Cournot models prevail over Bertrand models in the current literature on electricity markets.
20
Yet another model has been used in the recent literature. This approach is based upon the work of
Klemperer and Meyer [10] and was applied to a pool model by Green and Newbery [5]. In this model
competition is neither over price (as in Bertrand models) nor quantity (as in Cournot models) but in
supply functions. A supply function relates quantity to price. It shows the prices at which a firm is
willing to sell different quantities of output. The SFE model applies very well to the market structure
of many restructured electricity markets, such as New Zealand, Australia, Pennsylvania-New JerseyMaryland Interconnection (PJM) and California Power Exchange. In these markets the bid format is
precisely a supply function. SFE models can better explain the markups of electricity prices which
empirical studies have shown to be above the Bertrand equilibrium but below the Cournot model.
The problem with the use of SFE models is that in general there is not a unique equilibrium. There
is often an infinite number of solutions lying between the Cournot and Bernard equilibria, which
represent their upper and lower limits in price respectively. The existence of many equilibria makes
it difficult to predict the likely outcome of strategic interaction among players. There are some factors that reduce the range of feasible equilibria: uncertainty of demand and capacity constraints are
among them. Rudkevich, Duckworth and Rosen [14] calculated the electricity prices that would result
from a pure pool market with identical profit-maximizing generating firms, bidding stepwise supply
functions. Under the assumption that the price at peak demand is the marginal cost of the peak
marginal unit, they obtain the unique Nash equilibrium market-clearing price of electricity in a pool,
formula expression.
given by a closed
4. Mean and Variance of the Hourly Price
Ignoring unit commitment constraints, it is assumed that the system consists of N+1 generating
units, which are dispatched in an ascending merit order, based on the offered price of each one. Utilities will offer energy (quantity and price), unit by unit, to the Independent System Operator (ISO). The
latter will order the units by offered price, and dispatch the units from the cheapest to the more expensive ones, until the demand is met. This
is the case with PJM and many other electricity markets.
We will condition on the marginal unit2 J(t) to get the expected value and the variance of the price.
The expected value can be written as follows
N
+1
E
ptJt jP r
Jt j
>?@
E
pt j=1
where j is the merit order index.
j =1,2,3, ..., N + 1.
The probability mass function
of
J(t)
is
needed.
Following Valenzuela and Mazumdar [18] we will
express
j−1
j !"
j being the cumulative distribution function of the auxiliary variable
j
j
j
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function
of Xj(t):
K32j
K3
K4j
1
1
3
5
3
61 K2j (t)j3/2 (1−z2 )+ 24
j 2 (3z−z ) 72 K2 (t)3 (−15z+10z −z ) K2
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j (
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is applicable when a) the competing firms are identical in every respect, b) the load has zero elasticity
with respect to price, and c) the price at the maximum load is equal to the marginal cost of supplying
it. Because of our desire to use the simplified closed form expressions given by the Rudkevich formula,
we have retained the assumptions used in this work. No doubt the first assumption is far removed
from reality. It can perhaps be partially justified by noting that only a few firms realistically influence
the price: namely, those that usually own the marginal unit. It is assumed that those are not the small
firms which are price-takers, but the large ones.
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(
฀
฀
฀
%(
t for the Rudkevich
variance
฀
%(
'(
'(
%(
฀
'(
(
%(
(
฀
(
฀
( %( ฀
(
'(
(
฀
฀
฀
฀
(
'( ฀
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(
฀
฀ ( '(
฀
%(
฀
(
%(
(
฀
%(
(
฀
%(
'(
฀ ( %( ฀
฀ ฀
'(
฀
฀
฀
฀
(
'(
value and variance of hourly load
'( '(
ecomomic
under
'( the
three
Now we compute the expected
models.
฀
'(
฀
฀
'(
'(
(4)
(
'(
฀ ฀
%(
'(
฀
(11)
฀
'(
฀ ฀ ฀
%( '( ฀ (
฀
฀ ฀
'(
cost.
'(
Under the Bertrand model, the Nash equilibrium market-clearing price is the marginal
฀
(
is
'(
'( ฀ %( That
฀ (
%(
฀
฀ %(
฀
฀
฀
฀
฀
฀
฀
%(
฀
%(
฀
฀
the฀the
(11)
฀ ฀
฀value
'(
'( Nowwe
compute
expected
and variance
of
dj฀
฀ hourly
฀ . Considering
al cost. That is %(
฀
฀
Considering
the
’s to
known
and%(
deterministic
constants,
expected
฀
฀
฀ ฀ ฀
be
%(the
%(
฀ ฀
฀ ฀ ฀
฀ ฀
%(
฀
฀
฀
฀
฀
'%(
฀ '%(
'%(
฀
%(
constants,
฀
฀
฀
'(
'( '%( ฀ '%(
฀
'%(
฀ ฀
be known
and%(
deterministic
the expected %(
฀
฀
฀
฀
%(
฀
'%(
'( '%( ฀
'%(
฀
price
at hour t for the Bertrand
model
are:
and variance
฀
฀
'%(
model,
the
Nash
equilibrium
market-clearing
price
is
the
(
(
(
(
฀
'%(
(
฀
((
฀
฀
(
฀
(
฀
. Considering
the '(
(
฀
%(
฀
%(
฀ %(
฀
%(
'(
%(
฀
'(
'(
฀ %(
฀
%(
(
฀
'(
'(
฀
฀ '(
฀
(12)
฀
฀
'%(
'%(
%(
%(
'(
฀
%(
’s
to
be
known
and
deterministic
constants,
the
expected
฀
฀
฀
'(
฀
฀
%(
%(
฀
฀
฀
'(
฀
%(
%(
฀
%(
%(
฀
฀
'(
฀
'( %(
฀
'(
฀ ฀ '(
%( %(
'(
฀ ฀
%(%( ฀ '%(
฀
'(
%(
'(
(
฀
(5)
'(
(5)
%(
'( %(
฀ (
'( %( ฀
฀
%( ฀ '(
฀
฀
'%(
'%(
%( ฀ '( (
'(
'( ฀
฀
%(
%(
(12)
((
'(
'( ฀
฀
%(
%(
'(
%(
฀
฀
'(
%(
'(
%(
'
'%(
'(
฀
%(
'
'(
฀
%(
'%(
฀
5.
Mean
and
Variance
of
the
Time-Average
Price
฀
'
฀ %(
'(
฀
'(
฀
฀
%(
( '฀ '%(
'
฀
%( %(
'%(
'( ฀ %(
฀ '%(
'( %(
฀
'
฀
'(
'(
฀
%(
'%(
'
'(
'%(
฀ '
'฀
'(
%(
'(
฀ ฀
'฀ ฀ '%(
฀ '( ฀ %(
%(
'(
%(
฀
'( '%( '฀ (6)
฀ %(
This section derives formulas for the expected value and variance of average prices. The objective is
'
'(
฀
%(
'%(
฀
to
predict
accurately the average for some specific hours of a given day.
'( ฀ %(
'%( '฀ (6)
'(
฀
%(
'
'%(
a sym A load-weighted average is considered, as a more general approach. The weighted average price bet the
฀
Green and ฀
Newbery
[5]
derive
the
Cournot
model
Nash
equilibrium
price
for
basic
case
of
'
'(
฀
%(
'%(
฀
฀
฀
฀
'
'(
฀
%(
'%(
฀ price for ฀
the
basic
duopoly:
฀ ฀ ฀ &(
case
฀ %( ฀ ฀ &(
฀ &( %(
฀
ofa฀
symmetric
฀ equilibrium
%(
฀
&(
฀
฀ ฀
฀
H is
is . The expected
expected value and variance
฀ween the&(
&(
฀
฀฀
%(
initial
hour I and
hour
฀฀
฀
฀ ฀ ฀
and
฀
฀
%(
metric
duopoly:
derivative
of the
to- %(
where
&(
is the(
฀
%
฀
%(
&(
final
฀
%(
&(
&(
฀ ฀ (
%
฀
฀ ฀ (
฀
฀ %
(
%
value
and
variance
can
be
expressed
as:
can
be
expressed
as:
&(
฀ %( the
(
%
(
%
฀ derivative
%Following
the total
system
demand (
% with
( %
(
%p) withofrespect
tal system demand(
D(t,%
to(
price.
Green
[4], itrespect
is assumed in this฀work that
฀
(
%
฀
฀
&(
฀ ฀
%(
&( ฀
฀ ฀
%(
฀
the total demand D(t, p) is a linear function of price.
฀
฀ % &(
%(
(
฀
฀
฀ ฀
&(
฀
%(
(
%
(13) as a random independent variable. This basic case i
฀
&( %( ฀ ฀ (
%
The outage of units has the same effect on price as a shift upwards of the load, in the same
amount
( %
%
(
model 1 in this section and in the following section. A q
arises at this moment: Can we predict electricity pric
of the power that cannot be delivered. In order to consider the uncertainty of the load and the avai
฀ ฀ ฀
฀
฀ (
accurately
if we can better explain the variability of d
(14)
&
฀ ฀ ฀
฀
lability of the units simultaneously,
we use the equivalent load LJ (t)(t) defined in equation (3) instead
฀
Some previous work (see Valenzuela, Mazumdar and
of the actual load LJ(t). It can be seen that the Nash equilibrium market-clearing price for the Cournot
(14) [19])
showed that part of the load variance can be ex
where the expressions for and
have been
model can be generalized to
where the expressions for E[p(t)] and
฀ (
Var[p(t)]
have been derived in the preceding section for each
Note that
p(r) and p(t) are correlated
(7)
%( ฀ ฀
bidding model.
for any pair r, t. To get an expression for
(7)
#
Cov[p(r), p(t)] we need to
E[p(r)p(t)]. Conditioning
on
compute
J(r) and J(t), the following expres holds
sion
Once again, conditioning on(
J(t) and using the approximation given by (4) the expected price and
(
(
฀ ฀ ฀ ฀ ฀ ฀ ฀
฀
฀ ฀ ฀ ฀ ฀ ฀ ฀ 23 22
'( %( ฀
#
฀
฀ ฀฀ ฀
฀
฀ ฀
฀ ฀
฀ ฀ ฀
฀
฀฀ ฀ ฀
฀
฀
฀
฀
฀ ฀
฀฀ ฀
฀ ฀
฀฀ ฀
฀
฀฀
฀฀ ฀฀฀
฀฀฀
฀
฀
฀
฀
฀
฀
฀
฀
฀ ฀ ฀ ฀ ฀
฀ ฀ ฀฀
฀ ฀
฀
฀
฀
฀
฀
฀
฀
฀ ฀
฀
฀
฀
฀
฀ ฀ ฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀ ฀ ฀
฀฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀ ฀ ฀ ฀
฀
฀
฀
฀ ARIANCE
OF
฀ ฀ ฀ ฀MARKETS
PRICES
฀
฀ ฀
฀
฀ ฀ EREGULATED
MAINAK MAZUMDAR / CLAUDIO
THE
D
IN
FORECASTING
฀฀
฀฀ ฀
฀
MEAN
AND THE V
M.
ELECTRICITY
฀
RUIBAL ฀
฀
฀฀
฀฀ ฀ ฀
฀
฀
฀฀฀฀฀฀฀
฀ ฀
฀
฀
฀
฀฀
฀฀ ฀฀
฀฀฀฀ ฀
฀
฀
฀
฀
฀฀
฀ ฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀ ฀ ฀
฀
฀ ฀
฀฀฀฀
฀
฀
฀
฀฀
฀ ฀ ฀
฀฀ ฀
if and
(15)
remaining
where
the
remaining
term.
where and , the
term.
฀
฀
฀ if฀ ฀
฀
฀ ฀ ฀ ฀
฀
฀
฀
฀
฀
฀
฀
฀
฀ ฀
฀ ฀
฀ ฀ ฀
฀ ฀
฀
฀
฀
฀
฀ ฀฀
฀
฀ ฀
฀
฀
฀
฀ ฀ (15)
฀
฀ ฀฀ ฀ ฀฀ ฀฀
฀
฀
฀
฀
฀
฀ ฀
฀
Valenzuela
= m, approximate
expression
for
฀ has
฀ ฀Pr[J(r)
฀
=
l]฀
฀ J(t)
฀ an
obtained
฀
[16]
฀
฀
฀
We
study
two
models
based
on
this
expression:
model
2
considers
x(t)
as
normally
distributed
for
฀
฀
฀
฀
using
฀
฀
฀
The
฀ ฀฀
฀
฀
below.
Edgeworth
which
฀
฀the
isexpansion
฀
which
is summarized
pansion
formula,
summarized
below.
events
฀
฀
formula,
฀ each t; model
฀ ฀
฀
฀
events
The
3 considers
x(t) with a time series
in which x(t) is correlated
to x(t - 1) and to
฀
฀
฀
฀
using
the
variables
฀
So,
approach,
are
are
equi ฀
and
฀ equivalent.
and
฀
฀
฀
฀
to com
฀
฀
other
terms
of
the
series.
In
model
2,
the
effect
of
temperature
is
subtracted
from
the
load,
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
are variables
฀ defined
defined
and
before,
฀
defined
before,
defined
So,
using
the
thevariables
valent.
using
and
฀
variance
฀ before,
defined
before,
฀
฀ t. Temperature is considered
฀
฀฀฀฀and
฀฀ ฀
equivalent.
before,
฀ ฀
฀
฀฀
So,
฀ ฀ ฀ ฀ ฀
remaining
x(t) for each
to
฀
฀
฀
by
฀ ฀ ฀ ฀ pute the expected value
฀ and
of the
฀ .฀ ฀Denoting
฀ ฀probability
the
฀joint
฀ ฀ ฀ the
฀ of
฀
bytwo
฀฀฀฀฀the
joint
joint
฀ events,
฀
฀฀฀
.
Denoting
฀฀฀
฀
.
Denoting
by
฀
by
฀ ฀
Denoting
the
the
fo฀
the
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
so all
฀
comes
฀ a฀
฀ variable,
฀
new
expected
฀the
฀ ฀
deterministic
randomness
from
x(t).
The
value
฀ ฀ ฀
฀ ฀ ฀
฀ ฀ ฀
฀
฀ ฀฀฀
฀ ฀฀
฀be
฀
฀
฀
฀฀
฀
฀ ฀฀ ฀term
฀฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀ ฀ ฀ ฀฀ ฀ ฀ ฀ ฀ ฀ of
฀
฀
฀
probability
two
equality
฀ ฀฀ probability
ofthe
the
two
events,
thefollowing
following
holds
฀ events,
the
holds
฀equality
llowing
holds
by฀ ฀
฀
covariances
฀฀
load
฀
฀ and
฀ ฀
฀฀฀฀
฀
฀
฀ ฀
฀
฀฀
฀ ฀
equality
฀฀฀ ฀฀฀
฀ ฀฀ ฀
฀ are
฀
฀
฀฀ ฀
฀
฀
฀
are given
given
of
hourly
฀฀
฀variance
฀฀
฀
฀
฀ ฀ ฀ ฀ ฀
฀฀ ฀
฀ ฀฀ ฀
฀
฀฀฀฀ ฀฀฀
฀฀
฀ ฀
฀
฀
฀
฀and
฀ by
฀ covariances
฀
฀
฀฀
฀฀฀฀ ฀฀฀
฀ ฀ ฀ ฀ ฀
฀฀ ฀ ฀ ฀฀
฀
฀
฀
฀
฀฀ ฀
฀ ฀฀
฀฀
฀฀
฀฀ ฀
฀฀฀฀
฀
฀
฀
฀
฀
฀ (16)
฀฀ ฀ ฀
฀฀
฀฀
฀
฀ ฀฀ ฀
฀
฀
฀฀
฀฀
฀
฀฀
฀฀
฀฀฀ ฀฀
฀
฀ ฀฀฀฀฀฀฀฀
฀
฀฀ ฀
฀฀ ฀฀
฀ ฀
฀฀
฀
฀
฀ ฀ ฀฀฀฀฀
฀฀ ฀฀฀
฀
฀฀
฀฀ ฀
฀ ฀฀฀ ฀ ฀ ฀ ฀฀ ฀
฀ ฀ ฀ ฀ ฀
฀
฀
฀ ฀
฀ ฀
฀
฀
฀฀
฀
฀฀ ฀฀ ฀ ฀ ฀ ฀฀ (16)
฀฀ ฀฀฀
(16)
฀ ฀฀ ฀฀ ฀ ฀ ฀ ฀ ฀ ฀฀ ฀ ฀฀
฀ ฀
฀฀ ฀
฀ ฀฀
฀
฀ ฀ ฀
฀ ฀
฀฀
฀฀ ฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
(17)
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀ ฀ ฀ ฀ ฀
฀ ฀ ฀ ฀ ฀฀฀
฀฀฀ ฀ ฀ ฀
฀ ฀฀
฀฀
฀
฀ ฀
฀ ฀
฀
฀
฀
฀
can
be
฀
฀฀ ฀ ฀฀ ฀฀ ฀ ฀
฀ ฀ ฀฀ ฀
฀ ฀ ฀฀
฀ ฀฀฀฀ ฀฀
฀ ฀ ฀
฀ ฀ ฀฀ ฀ ฀฀฀ ฀ ฀
฀
฀฀฀ ฀ ฀expansion.
฀
฀
฀฀฀฀฀฀Edgeworth
฀
฀฀฀฀฀฀฀ ฀
฀
฀
฀฀using
฀
Iyengar
฀ ฀ ฀ ฀฀฀
฀ ฀
฀
฀
฀฀ ฀
approximated
by
the
and
฀฀
฀
฀
(18)
฀ ฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
can
be
The joint
joint
probability
distribution
can
be
approximated
by
using
the
Edgeworth
The
probability
distribution
of
฀give
Mazumdar
฀฀ ฀Edgeworth
฀ ฀ ฀฀ ฀฀
฀
฀expansion
฀of
฀฀ ฀ ฀
฀
the
pro approximate
฀
฀ give
฀[8]
Edgeworth
and approximate
approximated
by using
the
Iyengar
Mazumdar [8]
give
the Iyengar
Edgeworth
approximate
expansion.
[8]expansion.
฀
expansion
the
the
of
Edgeworth
and
฀ expansion
฀
(19)
of
joint
Mazumdar
฀ ฀ ฀ ฀ ฀ the joint probability
฀ distribution
฀฀
฀ ฀ ฀
฀
of ฀ :
฀
฀ ฀฀ ฀฀bability distribution
฀
฀
฀
฀
฀
฀ ฀ ฀
฀ ฀ ฀ ฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
฀
The
฀
฀
฀
once
thefrom
effect
ofhistorical
temperature
is set,
removed.
On
the
hand,
arecomputed
computed
reonce
other
฀ temperature
3isconsiders
thethe
historical
data
effect
from
are
฀ ฀ the
฀ of
฀ model
฀
data
set,
฀ The
฀ ฀ ฀ ฀ the
฀ ฀On
other
฀
the
ofother
temperature
hand,
฀
฀
฀฀ ฀฀moved.
฀฀ ฀
฀ 33 considers
ashand,
time
series
—denoted by
bythe
thesymbol
model
฀
฀฀ ฀
฀
model
aa time
series
—denoted
symbol — following an A
฀
once
effect
isremoved.
Onthe
฀
฀
฀
฀ ฀ ฀ ฀
฀
฀
฀
฀
฀฀ ฀฀
฀
following
฀฀฀
฀ ฀ ฀ ฀
฀
฀ form
฀฀ ฀฀
฀
฀
฀฀฀฀ ฀฀฀฀ ฀
฀ ฀
(1,120,0)
theform
(1,120,0)
process
is Gaussian
an
of
where
฀ ฀process
฀฀ ฀ ฀ of
where
form
฀ ฀
฀ where isis฀Gaussian
฀ ฀ ฀ following
฀
฀ ฀—
฀
the
an ARIMA
ARIMA
฀ where
฀฀฀ ฀฀ ฀ ฀ ฀
฀ ฀฀ ฀ ฀ ฀
฀
and
is
the autocorrelation
noise
zero and variance
and
ismean
the
coefficient
for
Gaussian
white
noise
with
mean
zero
and
variance
, and
,, and
with
mean
zero
and
variance
฀
฀ with
฀
฀
฀
฀
฀ ฀ white
฀ ฀
฀
฀ ฀ 1-hour
฀ ฀
฀
฀ ฀ has
been
model
has
by Valenzuela
฀ ฀This
model
autocorrelation
coefficient
for
1-hour
lag. This
This and
has [17]. Given a temperature
฀
validated
Mazumdar
autocorrelation
for
model
฀
฀ ฀
฀ ฀ ฀ ฀a฀
฀
฀
aa1-hour
฀฀฀ ฀ ฀฀฀ ฀
฀
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stochastic
to September
during March
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The total nominal capacity of the system is 18000 MW. The model assumes that infinite amount of
is the
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hourlytemperature
temperature F) is
equations in which the hourly load L(t) is the
andand
thethehourly
energy can be bought outside the system; four ownership scenarios of the system: 3, 4, 6, and 12
independent
variable.
identical firms, with 4, 3, 2, and 1 8-unit groups each respectively; and that all the firms forecast the
฀ ฀ ฀ ฀ ฀ ฀ ฀ load with the same accuracy. It is assumed that the generators are dispatched in a pre-arranged merit
฀ ฀ ฀ ฀ ฀ ฀ ฀ order, based on the offered prices. There exists a positive correlation between bids and production
costs.
24
฀
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Load data from PJM for weekdays of Spring 2002 (March 21 to June 20, 2002) is used in these illustrations. Table II shows the mean and standard deviation of the hourly load. It is assumed that hourly
loads follow a normal probability distribution. Standard deviations are small enough with respect
to the mean so that the probability of negative loads can be neglected. The data points used in the
model were scaled by a factor of 0.75 to fit into the supply model. For the Cournot model, a non-zero
price elasticity of the demand is proposed.
A linear demand function is used having the form D(t, p) = a(t) + pDp with Dp < 0 being deterministic
and constant across all hours t, and a(t) being a random variable different for every hour. For the Rudkevich model, zero price elasticity of demand is required. A code written in Matlab is used to run
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the model. Sensitivity analysis is then performed on number of firms (3 – 12); demand slope for Cournot models (-300 – -100 (MWh)2/$); and peak-demand-to-full-capacity ratio (PDFCR) for Rudkevich
model (0.6 – 1.0). The PDFCR expresses the belief about the anticipated peak demand with respect to
full capacity. Bertrand model results play the role of benchmarks. The number of firms in the market
affects the expected value of the price and the variance. In all the cases, when the number of firms
increases, the results tend to the Bertrand solution.
Figure 1 shows that the expected values and variances of hourly prices for the Cournot model follow
a similar profile to the Bertrand model, but always stay above it. Only two cases of demand elasticity
are shown (the highest and the lowest), to keep the graphics clear. The other three cases fall between
them. Both expected values and variances can reach high values when the elasticity of demand is low.
For different values of demand elasticity, expectations and variances with twelve firms remain around
one half those when there are only three firms in the market. For the Rudkevich model, for low
26
Fig. 1. Expected values and variances of hourly prices (Cournot model)
anticipated peak demand, market concentration does not affect the results in both expectations and
variances to a marked degree. The expected values and variances of hourly prices for the Rudkevich
model are shown in figure 2. Only two cases of anticipated peak load are given for the sake of clarity.
The results for PDFCR=0.6 were equal or very close to the Bertrand solution; therefore they are not
included. In figure 2.a, for PDFCR=0.8 the curve of expected values is flatter than in the other cases,
and between hour 9 and hour 22, the differences between ownership scenarios are very small. As
was expected, all the prices are above Bertrand hourly prices.
When the PDFCR is high (close to 1), then the differences are striking. Rudkevich expected prices for
low demand hours are more affected by the fluctuation in demand than the expected prices for peak
hours. This produces the effect of leveling of prices. Rudkevich model variances of hourly prices (figure 2.b) are less disparate for the different ownership scenarios for PDFCR=0.8. The lower the PDFCR
is, the closer the solutions are to Bertrand’s curve. Note that, for peak hours, except in the case of 12
firms, Rudkevich’s variances are smaller than Bertrand’s.
Fig. 2. Expected values and variances of hourly prices (Rudkevich model)
!&
27
The reason for this is that Rudkevich’s supply functions have smaller slopes than the marginal cost
at peak hours. Furthermore, the fewer the number of firms in Rudkevich’s model, the higher are the
expected values, but the smaller are the variances, something which is not intuitive.
Figure 3 shows the expected values and variances of average price between hours 13 and 16 for the
three bidding models, with sensitivity analysis. Results for the Bertrand model are insensitive to all the
parameters, and are taken as references.
Fig. 4. Expected values and variances of hourly prices
As could be intuited, in all the cases the expected values increase when the number of firms decreases. When the number of firms is large the behavior tends to the perfect competition case. As also
could be expected, Cournot average prices (figure 3.b) increase when demand is more inelastic (i.e.,
Dp decreases in absolute value). The increase may be very high with respect to Bertrand prices. Rudkevich expected values (figure 3.a) increase with the PDFCR, implying that a big peak load for a given
day will drag up all the hourly prices of that day. But markups are not that large for anticipated peak
loads less than or equal to 90% of total capacity.
For the Cournot model, variances of average prices (figure 3.d) are always above the Bertrand model
case. They also increase when the number of firms decreases, and when demand is more inelastic.
On the contrary, Rudkevich model variances of average prices (figure 3.c) have quite a different behavior.
28
The first thing to point out is that the variances are below that for the Bertrand model for most of
the chosen values of peak-demand-to-full-capacity ratio. Second, the variances increase with the
number of firms. The explanation for this is again that when the number of firms increase, the market
tends to the perfect competition scenario, so the variances get close to that of the Bertrand model.
In addition, with fewer companies in the market, Rudkevich prices go up and flatten more quickly
and, therefore, the slopes of the supply curves are smaller for relatively higher values of the peakdemandto-full-capacity ratio. Third, for values of PDFCR increasing from 0.6 to 0.9, the variances of
average prices decrease in this range of on-peak load.
Figure 5 depicts the expected values and variances of average prices between hour 13 and hour 18
for the three load models and the three bidding models. In this case, all Rudkevich’s and Cournot’s
scenarios are shown. There are no big changes in expected values of average prices across load models. Rudkevich model’s expected value of average price increases a lot for a forecasted peak demand
close to full capacity. Also, in this case, variances of average price are much larger for model 1. As
was anticipated, temperature plays an important role in the expected value and variance of hourly
prices and average prices. Forecasting temperature accurately can reduce the variance of prices considerably.
Fig. 3. Expected values and variances of average prices between hours 13 and 16
Considering the load stochasticity, and the effect of temperature on the load, after running the three
load models described in section VI, the outputs are compared to extract some conclusions. Figure 4
shows the expected values and variances of hourly prices under three load models and three bidding
models. Rudkevich model was selected with a PDFCR of 0.8 and Cournot model was selected with
a demand-to-price slope of Dp = -200 (MWh)2 /$. Rudkevich model’s expected values are close to
Cournot model’s for low demand hours, and closer to Bertrand model’s for peak hours. There do
not appear to be great differences between load models. On the contrary, the variances of hourly
prices show a huge difference between load models. Variances at peak hours are extreme for model
1, being twice to five times larger than in the other two models. Across bidding models, Rudkevich
model’s variances for hours 6 to 23 are half of Cournot model’s.
29
Fig. 5. Expected values and variances of average prices
8. Conclusions
The numerical results of the previous section are detailed enough to derive some conclusions in the
following respects: price behavior with regard to market concentration, price reaction to demand
elasticity and installed capacity. Computations made using the models of section VI allow us to arrive
at some conclusions on the effect of temperature on expected prices and variances. Market concentration is an important factor in the determination of the expected value and the variance of hourly
and average prices, especially in the Cournot model for all values of demand elasticity. In the Rudkevich model, the greater the number of firms, the lower the prices are and the greater the variances.
With a small number of firms in the market, prices tend to level off across hours.
30
The Cournot model helps us to understand and to measure the effect of price elasticity of demand.
As is to be expected, a large elasticity brings the prices down and the variances as well. A significant
part of the demand is totally inelastic because it is needed irrespective of price. The remaining part of
the demand shows more elasticity. A key factor is to design the market structure in such a way that
it provides this elasticity. In order to do this, it is necessary to allow the end consumers to react to
different prices in the wholesale market even though they buy energy in the retail market. This change should be carefully considered by the market designers as an important part of the deregulation
process. The Rudkevich model has the advantage of showing the effect of the entire supply system
on the prices. Prices are affected by the costcapacity structure of the market, even by those units that
are not running in a given hour. It is clear that if the market has much more capacity than needed it
can assure a better service because it has a lot of energy reserve, and the buyers will appreciate that
up to a certain point. Eventually, the firms will charge a bit more to compensate for the investment
on the excess capacity. Even for a market which does not have a large excess capacity, the perception
of the firms about the daily peak demand affects the price, under the Rudkevich model.
The difference between an anticipated peak demand of 90% and of 100% turns out to be important.
Then, the question to think about is how to influence the firms’ beliefs.
Temperature can explain in great part the variance of the load. In the example shown in the previous
section, for on-peak hours temperature explains up to 75% of the variance of the prices. Temperature
plays a more important role in determining the hourly load than the load in the preceding hour. What
is also true is that the hourly temperatures are very correlated among themselves.
[1]
J. Bertrand. Theorie mathematique de la richesse sociale. Journal des Savants, 45:499–508,
1883.
[2]
Augustin Cournot. Recherches sur les Principes Mathematiques de la Theorie des Richesses.
Hachette, Paris, 1838. English translation by N. T. Bacon published in Economic Classics [Ma
cmillan, 1897] and reprinted in 1960 by Augustus M. Kelly.
[3]
H. Cramer. Mathematical Methods of Statistics. Princeton University Press, Princeton, NJ,
1946.
[4]
R.J. Green. Increasing competition in the british electricity spot market.
Journal of Industrial Economics, 44(2):205–216, 1996.
[5]
R.J. Green and D.M. Newbery. Competition in the british electricity spot market. The Journal
of Political Economy, 100(5):929–953, October 1992.
[6]
B.F. Hobbs, C.B. Metzler, and J.S. Pang. Strategic gaming analysis for electric power system:
an MPEC approach. IEEE Transactions on Power Systems, 15(2):638–645, 2000.
[7]
B.F. Hobbs, M.P. Rothkopf, R.P. O’Neil, and H-P. Chao, editors. The Next Generation of Electric Power Unit Commitment Models. Kluwer Academic Publishers, Boston, 2000.
[8]
S. Iyengar and M. Mazumdar. A saddle point approximation for certain multivariate tail pro
babilities. SIAM Journal on Scientific Computing, 19:1234–1244, 1998.
[9]
C. Kang, L. Bai, Q. Xia, J. Jiang, and J. Zhao. Incorporating reliability evaluation into the
uncer tainty analysis of electricity market price. Electric Power Systems Research, 73(2):205–
215, February 2005.
[10]
P.D. Klemperer and M.A. Meyer. Supply function equilibria in oligopoly under uncertainty.
Econometrica, 57:1243–77, November 1989.
[11]
R. Paehlke. Conservation and Environmentalism: An encyclopedia. Garland Publishing Inc.,
New York, 1995.
[12]
D. Pilipovic. Energy Risk. Valuing and Managing Energy Derivatives. McGraw-Hill, 1997.
[13]
R.T. Rockafellar and S. Uryasev. Optimization of conditional value-atrisk. The Journal of Risk,
2(3):21–41, 2000.
[14]
A. Rudkevich, M. Duckworth, and R. Rosen. Modeling electricity pricing in a deregula
ted generation industry: The potential for oligopoly pricing in a poolco. The Energy Journal,
19(3):19–48, 1998.
[15]
C. Ruibal. On the Variance of Electricity Prices in Deregulated Markets. PhD thesis, De
partment of Industrial Engineering, University of Pittsburgh, 2006.
[16]
J. Valenzuela. Stochastic Optimization of Electric Power Generation in a Deregulated Market.
PhD thesis, Department of Industrial Engineering, University of Pittsburgh, 2000.
[17]
J. Valenzuela and M. Mazumdar. Statistical analysis of electric power production costs. IIE
Transactions, 32:1139–1148, 2000.
9. References
31
[18]
J. Valenzuela and M. Mazumdar. A probability model for the electricity price duration curve
under an oligopoly market. IEEE Transactions on Power Systems, 20(3):1250–1256, August
2005.
[19]
J. Valenzuela, M. Mazumdar, and A. Kapoor. Influence of temperature and load forecast un
certainty on estimates of power generation production costs. IEEE Transactions on Power
Systems, 15(2):668–674, May 2000.
Original Sin and Redemption:
Rebalancing the Currency Structure of
Uruguayan Public Debt
Umberto Della Mea
[email protected]
Antonio Juambeltz
[email protected]
1
Debt Management Unit
Ministry of Economy and Finance
Colonia 1089, 3rd Floor, 11100 Montevideo, URUGUAY
Abstract
This document discusses the optimal structure of the Uruguayan public debt given a set of current
parameters of cost and risk. In a traditional portfolio setting, the model decomposes and develops
alternative ways to reestimate the covariance matrix in order to internalize a changing environment.
Policy and institutional innovations, as well as the countercyclical properties of nominal debt, are thus
reflected in the parameters of the model on a forward-looking basis.
32
Resumen
Este documento discute la estructura óptima del endeudamiento público uruguayo en función de un
conjunto de parámetros de costo y riesgo. En el marco de un modelo tradicional de portafolios, esta
aproximación innova formas alternativas de reestimar la matriz de covarianzas, de modo de internalizar cambios e innovaciones institucionales y de política. Así, las propiedades contracíclicas de la
deuda nominal se reflejan en los parámetros del modelo sobre una base forward-looking.
Clasificación JEL: H63
Palabras clave: Gestión de Deuda Pública, Pecado Original, Deuda Indexada
1. The opinions here expressed are those of the authors and do not necessarily represent those of the Debt Management Unit.
JEL Classification: H63
Keywords: Public Debt Management, Original Sin, Indexed Debt
33
ORIGINAL SIN AND REDEMPTION: REBALANCING THE CURRENCY STRUCTURE OF URUGUAYAN PUBLIC DEBT
UMBERTO DELLA MEA / ANTONIO JUANBELTZ
1. Introduction
There seems to be a wide consensus in the literature about the main drivers of sovereign debt creditworthiness. These drivers, which often tend to feedback, are in general related to the overall quality
of the country, characterized by its degree of economic, social and institutional development. However, there is also a particularly important role played by the public debt management. In this regard,
aspects like the maturity profile, the currency composition or the intrinsic quality of the asset-liability
management are key factors to determine the current levels of sovereign risk.
Fig. 2: Changes in
Non Financial Public Sector Debt
70%
In Uruguay, the currency composition of the public debt has been the main explanation to the strong
variations observed in the overall debt service and debt stock, relative to the domestic level of activity. A highly dollarized economy, with a high share of foreign-currency denominated public debt,
is sensitive to real exchange rate fluctuations and -in particular- to the level of the exchange rate as
compared to the level of domestic prices. Any appreciation in the currency of denomination of the
public debt, relative to domestic inflation, will increase the debt burden relative to GDP, deteriorating
the solvency indicators and producing an increase in the risk premium that might in turn trigger a
debt crisis.
Since the early 70s, when Uruguay started to open the capital account, the currency choice in favor
of US Dollars covered a wide range of domestic financial instruments in the banking sector and the
capital markets, reaching even the means of payment of the economy. Della Mea (2007) pointed out
some of the reasons that explained this trend over the last decades: the status of legal tender granted
to the foreign currency, setting an equal treatment relative to the domestic currency; the small size
and degree of openness of the economy, where most durable goods were imported and their prices
were quoted in foreign currency to provide a natural hedge in absence of a developed financial market; a long history of medium to high inflation and lack of institutional commitment to price stability;
and finally, the absence of alternatives providing inflation protection, like CPI-indexed instruments.
The public debt was no exception to this process. Still at the beginning of this decade, almost 100%
of the Non-Financial Public Sector debt was foreign currency denominated. This factor was determinant to fuel the financial crisis of 2002, whose aftermath -after a strong real depreciation of the
domestic currency and a deep recession- was a soaring Debt/GDP ratio which more than doubled in
one year, surpassing 100%. A similar consequence is observed over the debt service 2. Fig. 1 illustrates
the pattern of the debt stock and the debt service, relative to GDP, in the current decade:
Fig. 1: Debt Stock and Debt Service
110%
9%
100%
In this respect, the objective of this paper is to set up a model on the optimal currency composition
of the Uruguayan public debt. This model brings into consideration not only the expected costs and
risks associated to a set of possible units of denomination, but also the way these factors are correlated as well as their macroeconomic properties.
8%
90%
34
80%
5%
70%
4%
Debt/GDP
Interest/GDP
7%
6%
60%
3%
50%
2%
40%
1%
0%
30%
2000
2001
Interest/GDP
2002
2003
2004
2005
The crucial question is how debt managers should behave in order to avoid this kind of exposure,
in particular when the country is likely to be subject to periodical real exchange rate shocks. Foreign
currency debt might be more risky, but at the same time it might also be cheaper on average, especially when domestic capital markets are not developed or when there is uncertainty about future
development of inflation and other key domestic variables. The past choice in favor of foreign currency may have had sensible foundations. The question is how to assess this choice at present and
balance foreign currency liabilities with other available domestic alternatives, in particular nominal
and CPI-indexed debt.
2006
Debt/GDP (RHS)
2. Foreign-currency denominated debt and debt service are valued at end-of-period exchange rates and compared to nominal GDP, valued at
market prices.
The rest of the document is organized as follows: Section 2 surveys some recent economic literature
about optimal composition of public debt. A model of portfolio selection based on the traditional
mean-variance approach is presented in Section 3. This model, in spite of working on a somewhat
standard setting, introduces significant changes in the way of estimation of the covariance matrix
of expected returns, in order to produce more forward-looking parameters. Section 4 discusses the
results obtained in different scenarios and finally, Section 5 concludes.
(Gross Non Financial Public Sector Debt)
10%
In turn, Fig. 2 discloses the factors of variation of the Non-Financial Public Sector debt into a series of
basic components. The major source of variation is the depreciation of the nominal effective exchange rate, which more than compensated a very low pass-through to the GDP deflator in a context of
strong capital outflows. Moreover, real GDP behavior tends to be negatively correlated with the real
exchange rate, feeding back the process and worsening the overall scenario. Strange as it may seem,
fiscal factors did not seem to contribute in a significant way to explain changes in this ratio over the
period under consideration.
35

Forecasting the Mean and the Variance of Electricity Prices in

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