QuantReg UKValue at Risk Modelling for Energy Commodities using Volatility Adjusted Quantile Regression
Energy Finance Conference
Sjur Westgaard
Email: sjur.westgaard@iot.ntnu.no
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www.ntnu.no/iot
Phone: +47 73598183 or +47 91897096
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Price-driving fundamentals
From Burger et al. (2008)*:
Liberalisation of energy markets (oil, gas, coal, el, carbon) has fundamentally changed the way power companies do business
Competition has created both strong incentives for improving operational efficiency and as well as the need for effective financial risk management
Why Value at Risk Modelling for Energy Commodities using Volatility Adjusted Quantile Regression?
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More volatile energy markets and complex trading/hedging portfolios (long and short positions) has increased the need for measuring risk at
Individual contract level
Enterprise level
Understanding the dynamics and determinants of volatility and risk (e.g. Value at Risk and Expected Shortfall) for energy commodities are therefore crucial.
We therefore need to correctly model and forecast the return distribution for energy futures markets
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Conditional return distributions various across energy commodities
Conditional return distributions for energy commodities varies over time
This makes risk modelling of energy futures markets very challenging!
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The problems with existing “standard” risk model (that many energy companies have adopted from the bank industry) are:
RiskmetricsTM VaR capture time varying volatility but not the conditional return distribution
Historical Simulation VaR capture the return distribution but not the time varying volatility
Alternatives:
CaViaR type models
Although these models works fine according to several studies, there is a problem of calibrating these non-linear models and therefore they are only used to a very limited extent in practice
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In this paper we propose a robust and “easy to implement” approach for Value at Risk estimation based on;
First running an exponential weighted moving average volatility model (similar to the adjustment done in RiskmetricsTM) and then
Run a linear quantile regression model based on this conditional volatility as input/explanatory variable
The model is easy to implement (can be done in a spreadsheet)
This model also shows an excellent fit when backtesting VaR
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trader having a short position in German
front Quarter Futures.
Oil prices ($/Ton)
One important task is to model and forecast the upper and lower tail of the price distribution using standard risk measures such as Value at Risk and Conditional Value at Risk for different quantiles (e.g. 0.1%, 1%, 5%, 10%, 90%, 95%, 99%, 99.9%).
Measuring Value at Risk / Quantiles
Loss for a producer or
trader having a long position in German
front Quarter Futures.
Previous work/literature of risk modelling of (energy) commodities
European Energy Futures Markets (crude oil, gas oil, natural gas, coal, carbon, and electricity) and Data/Descriptive Statistics
ICE
ICE-ENDEX
EEX
RiskMetricsTM
Andriosopoulos and Nomikos (2011)
Quantile regression in general and applications in financial risk management:
Alexander (2008)
Taylor (2000, 2008a, 2008b,2011)
We want to fill the gap in the literature by performing Value at Risk analysis for European Energy Futures using volatility adjusted quantile regression.
Discussion weaknesses RiskMetrics and Historical Simulation during Financial Crises
Sheedy (2008a, 2008b)
Bunn D., 2004, Modelling Prices in Competitive Electricity Markets , Wiley
Burger, M., Graeber B., Schindlmayr, G. 2008, Managing energy risk – An integrated view on power and other energy markets, Wiley.
Eydeland A. and Wolyniec K., 2003, Energy and power risk management, Wiley
Geman H., 2005, Commoditites and commodity derivatives – Modelling and pricing for agriculturas, metals and energy, Wiley
Geman H., 2008, Risk Management in commodity markets – From shipping to agriculturas and energy, Wiley
Kaminski V., 2005, Managing Energy Price Risk – The New Challenges and Solutions, Risk Books
Kaminski V., 2005, Energy Modelling, Risk Books
Kaminski V., 2013, Energy Morkets, Risk Books
Pilipovic D., 2007, Energy risk – Valuation and managing energy derivatives, McGrawHill
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ICE: www.theice.com
ICE-ENDEX: www.iceendex.com
EEX: www.eex.com
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Data and descriptive statistics
Returns are calculated as relative price changes ln(Pt/Pt-1) for crude oil, gas oil, natural gas, coal, carbon, month and quarter base contracts for UK, Nederland, Germany, and the Nordic market
Returns when contract rolls over are deleted
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Data and descriptive statistics
Mean, Stdev, Skew, Kurt, Min, Max, Empirical 5% and 95% quantiles are estimated for all series in the following periods:
13Oct2008 to 30Dec2008
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Data and descriptive statistics
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High return volatility for energy commodities. Annualized values much higher than for stocks and currency markets, specially for short term natural gas and electricity contracts
Fat tails for all energy commodities. Some energy commodities have mainly negative skewness (e.g. crude oil), others positive (e.g natural gas)
Return distribution of energy commodities varies a lot over time, hence the evolution of empirical VaR over time
Data and descriptive statistics
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Quantile regression was introduced by a paper in Econometrica with Koenker and Bassett (1978) and is fully described in books by Koenker (2005) and Hao and Naiman (2007)
Applications in financial risk management (stocks / currency markets) can be found by Engle and Manganelli (2004), Alexander (2008), Taylor (2008)
Quantile regression
0.1, 0.5, and 0.9 quantile regression lines.
The lines are found by the following minimizing the weighted absolute distance to the qth regression line:
Yqt = αq+βqXt+εqt
1) Calculate Exponentially Weighted Moving Average of Volatility
2) Run a Quantile Regression Regression with the Exponentially Weighted Moving Average of Volatility as the explanatory variable
3) Predict the VaRqt+1 from the model
Volatility Adjusted Quantile Regression
Example El_Ger_Q
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Todays EWMA vol is 2% on a daily basis.
What is 5%, 95% 1 day VaR given our model?
Var5% = -0.006589 % - 1.290804*2% = -2.59%
Var5% = -0.002108 % + 1.817529 *2% = 3.63%
Similar equations are found for all the quantiles
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VaR Models
Kupiec Test
Christoffersen Test
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It is easy to estimate VaR once we have the return distribution
The only difference between the VaR models are due to the manner in which this distribution is constructed
Comparing Value at Risk models for energy commodities
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VaRα=-1(α)*σt
Volatility changes dynamically over time and needs to be updated. For this reason many institutions use an exponentially weighted moving average (EWMA) methodology for VaR estimation, e.g. using EWMA to estimate volatility in the normal linear VaR formula
These estimates take account of volatility clustering so that EWMA VaR estimates are more risk sensitive
RiskMetricsTM
Historical simulation
Makes no assumptions about the parametric form of the return distribution, but do not make the distribution conditional upon market conditions / volatility
Comparing Value at Risk models for energy commodities
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Historical Simulation
Choose a sample size to reflect current market conditions (Banks usually use 3-5 years)
Draw returns from the empirical distribution and calculate VaR for each simulation
Use the average VaR for all simulation (you alsp get the distribution of VaR as an outcome)
Also called bootstrapping
Volatility Adjusted Quantile Regression
Allows for all kinds of distributions (semi non-parametric method) and allow the distribution to be conditional upon the volatility/changing market condition
Comparing Value at Risk models for energy commodities
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1) Calculate Exponentially Weighted Moving Average of Volatility
2) Run a Quantile Regression Regression with the Exponentially Weighted Moving Average of Volatility as the explanatory variable
3) Predict the VaRqt+1 from the model
Volatility Adjusted Quantile Regression
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Backtesting refers to testing the accuracy of a VaR model over a historical period when the true outcome (return) is known
The general approach to backtesting VaR for an asset is to record the number of occasions over the period when the actual loss exceeds the model predicted VaR and compare this number to the pre-specified VaR level
Backtesting VaR Models
In sample analysis
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Backtesting Value at Risk Models
A proper VaR model has:
The number of exceedances as close as possible to the number implied by the VaR quantile we are trying to model
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To validate the predictive performance of the models, we consider two types of test:
The unconditional test of Kupiec (1995)
The conditional coverage test of Christoffersen (1998)
Kupiec (1995) test whether the number of exceedances or hits are equal to the predefined VaR level. Christoffersen (1998) also test whether the exceedances/hits are randomly distributed over the sample
Backtesting VaR Models
Backtesting Value at Risk Models – Kupiec Test
The Kupiec (1995) test is a likelihood ratio test designed to reveal whether the model provides the correct unconditional coverage.
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Backtesting Value at Risk Models – Kupiec Test
Under the null hypothesis of correct unconditional coverage the test statistic is
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Backtesting Value at Risk Models – Christoffersen test
In the Kupiec (1995) test only the total number of ones in the indicator sequence counts, and the test does not take into account whether several quantile exceedances occur in rapid succession, or whether they tend to be isolated.
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Backtesting Value at Risk Models – Christoffersen test
The test statistic is defined as follows:
where nij represents the number of times an observations with value i is followed by an observation with value j (1 is a hit, 0 is no hit). Π01=n01/(n00+n01) and Π11=n11/(n11+n10). Note that the
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Example Front Quarter Futures Nordic
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Model Comparison
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Model Comparison
Preliminary results show:
RiskMetricsTM capture the conditional coverage for crude oil, gas oil, natural gas, coal, carbon but not for most of electricity contracts.
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Stylized facts shows that European Energy Futures returns have very different dynamics regarding the return distribution.
For a given energy commodity, the conditional distribution changes over time.
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Non-Linear Copula Quantile Regression for investigating the returns-volatility relationship following ideas of Alexander (2008)
Multivariate risk analysis for portfolios (e.g. many electricity futures contracts) using Borger et al (2007) as a starting point. One idea is to apply the following setup:
First run Principal Component Analysis for the factors
Then run volatility filtering (e.g. EWMA) on the components
Then run Quantile Regression on the filtered components
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IVIIIIIIIVIIIIIIIVIIIIIIIVIIIIIIIVIIIIII
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CRUDEOIL
60
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IVIIIIIIIVIIIIIIIVIIIIIIIVIIIIIIIVIIIIII
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COAL
0
5
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IVIIIIIIIVIIIIIIIVIIIIIIIVIIIIIIIVIIIIII
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CARBON
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IVIIIIIIIVIIIIIIIVIIIIIIIVIIIIIIIVIIIIII
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GASOIL
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NATURALGAS
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IV
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IV
IIIIII
IV
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IV
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IV
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el-UK-M
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IV
IIIIII
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el-UK-Q
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el-Ned-M
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IIIIII
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el-Ned-Q
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el-Ger-M
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el-Ger-Q
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el-Nor-M
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el-Nor-Q
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el-Ger-Q
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30354045505560657075808590
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CRUDEOIL
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GASOIL
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NATURALGAS
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IVIIIIIIIVIIIIIIIVIIIIIIIVIIIIIIIVIIIIII
20092010201120122013
COAL
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IVIIIIIIIVIIIIIIIVIIIIIIIVIIIIIIIVIIIIII
20092010201120122013
CARBON
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IVIIIIIIIVIIIIIIIVIIIIIIIVIIIIIIIVIIIIII
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el-UK-Mel-UK-Q
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el-Ned-Mel-Ned-Q
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el-Ger-Mel-Ger-Q
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el-Nor-Mel-Nor-Q
EL_GER_Q
2009-0.20 %1.68 %0.014.81-8.50 %-2.56 %2.30 %7.23 %
20100.08 %1.20 %1.414.30-2.87 %-1.36 %2.53 %6.64 %
2011-0.10 %1.06 %-0.265.72-5.97 %-1.52 %1.51 %4.82 %
2012-0.05 %1.06 %2.4016.31-3.08 %-1.44 %1.30 %8.11 %
02Jan13 - 30Sep13-0.04 %0.93 %1.105.88-2.66 %-1.38 %1.38 %5.39 %
13Oct08 - 30Sep13-0.09 %1.28 %0.386.63-8.50 %-1.87 %1.94 %8.11 %
EL_NOR_M
20090.01 %2.72 %-0.343.20-14.21 %-4.31 %4.27 %7.97 %
20100.31 %3.09 %-0.192.05-10.66 %-4.48 %5.62 %10.57 %
2011-0.49 %3.17 %0.022.41-12.76 %-5.53 %4.97 %10.61 %
2012-0.16 %3.62 %-0.192.20-16.71 %-6.29 %4.81 %12.22 %
02Jan13 - 30Sep130.09 %2.18 %-0.172.20-7.80 %-3.29 %3.45 %7.75 %
13Oct08 - 30Sep13-0.07 %3.06 %-0.162.44-16.71 %-4.88 %5.09 %12.22 %
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Display all coefficients
Quantile CoefficientStd. Errort-StatisticProb.