Predicting Time Periods of Excessive Price Volatility:
The Case of Rice
Dr. Ramon Clarete Alfonso Labao
University of the ...
Flow of Presentation
Overview
Methodology
Empirical Results
Conclusion and Constraints
Overview:
Overview:
Importance of being forewarned of extreme food price-volatility:
Overview:
Importance of being forewarned of extreme food price-volatility:
provides time to undertake cooperation
prevent ...
Overview:
Importance of being forewarned of extreme food price-volatility:
provides time to undertake cooperation
prevent ...
Our Methodology:
Our Methodology:
- Use daily rice future prices
(5407 days from Sep 1991 to Mar 2013)
Our Methodology:
- Use daily rice future prices
(5407 days from Sep 1991 to Mar 2013)
- Identify time-periods of excessive...
Our Methodology:
- Use daily rice future prices
(5407 days from Sep 1991 to Mar 2013)
- Identify time-periods of excessive...
First Two Parts:
- Use daily rice future prices
(5407 days from Sep 1991 to Mar 2013)
- Identify time-periods of excessive...
Actual Prices:
Snapshot of actual prices:
Converted into Price Returns:
Converted into Price Returns:
Converted into Price Returns:
where price returns refer to day-to-day price movements, or:
Converted into Price Returns:
where price returns refer to day-to-day price movements, or:
price return = ln(
price of tod...
Create Thresholds via MTY’s two-step methodology:
Create Thresholds via MTY’s two-step methodology:
Create Thresholds via MTY’s two-step methodology:
Methodology:
MTY’s Two-step method:
Methodology:
MTY’s Two-step method:
Construct a nonparametric trend via spline-backfitted-kernel:
rt = m0 +
d
a=1
ma(Xta) +...
Methodology:
MTY’s Two-step method:
Construct a nonparametric trend via spline-backfitted-kernel:
rt = m0 +
d
a=1
ma(Xta) +...
How are Time Periods of Excessive Price Volatility (EPV) defined?
How are Time Periods of Excessive Price Volatility (EPV) defined?
Martins-Filho, Maximo Torero and Feng Yao’s definition of ...
How are Time Periods of Excessive Price Volatility (EPV) defined?
Martins-Filho, Maximo Torero and Feng Yao’s definition of ...
1st (Jul 1993 - Feb 1994) and 2nd (Jul 1994 - Oct 1994) EPV cluster:
1st (Jul 1993 - Feb 1994) and 2nd (Jul 1994 - Oct 1994) EPV cluster:
3rd (Mar 1999 - Sep 1999) and 4th (Jun 2001 - Aug 200...
5th (Jul 2002 - Aug 2002) and 6th (May 2003 - Oct 2003) EPV cluster:
5th (Jul 2002 - Aug 2002) and 6th (May 2003 - Oct 2003) EPV cluster:
7th (Mar 2008 - Nov 2008) and 8th (Apr 2009 - May 200...
9th (Jul 2011) EPV cluster:
In Summary:
In Summary:
501 days of excessive price-volatility (EPV)
Nine (9) clusters of EPV
In Summary:
501 days of excessive price-volatility (EPV)
Nine (9) clusters of EPV
In Summary:
501 days of excessive price-volatility (EPV)
Nine (9) clusters of EPV
... Of the above clusters of EPV, only f...
Third Part:
- Use daily rice future prices
(5407 days from Sep 1991 to Mar 2013)
- Identify time-periods of excessive pric...
Next Task: Looking for a Good Early Warning Signal
Next Task: Looking for a Good Early Warning Signal
Next Task: Looking for a Good Early Warning Signal
Next Task: Looking for a Good Early Warning Signal
Next Task: Looking for a Good Early Warning Signal
Next Task: Looking for a Good Early Warning Signal
Next Task: Looking for a Good Early Warning Signal
Next Task: Looking for a Good Early Warning Signal
We set the mirror-image’s lag at 60 days prior to an EPV cluster...
Next Task: Looking for a Good Early Warning Signal
Next Task: Looking for a Good Early Warning Signal
Next Task: Looking for a Good Early Warning Signal
Parameters of the Early Warning Signals:
Next Task: Looking for a Good Early Warning Signal
Parameters of the Early Warning Signals:
Window of Observation
Next Task: Looking for a Good Early Warning Signal
Parameters of the Early Warning Signals:
Window of Observation
Lower-Or...
Next Task: Looking for a Good Early Warning Signal
Parameters of the Early Warning Signals:
Window of Observation
Lower-Or...
Next Task: Looking for a Good Early Warning Signal
Parameters of the Early Warning Signals:
Window of Observation
Lower-Or...
Next Task: Looking for a Good Early Warning Signal
Parameters of the Early Warning Signals:
Window of Observation
Lower-Or...
Next Task: Looking for a Good Early Warning Signal
... the scope’s time-coverage
What makes a good early warning signal:
What makes a good early warning signal:
accurate: scope’s time-coverage covers an EPV cluster, minimal false alarms
What makes a good early warning signal:
accurate: scope’s time-coverage covers an EPV cluster, minimal false alarms
good l...
What makes a good early warning signal:
accurate: scope’s time-coverage covers an EPV cluster, minimal false alarms
good l...
What makes a good early warning signal:
accurate: scope’s time-coverage covers an EPV cluster, minimal false alarms
good l...
We expressed the criteria (accurate, comprehensive, good lead time), into a single
additive objective function that can be...
We expressed the criteria (accurate, comprehensive, good lead time), into a single
additive objective function that can be...
We expressed the criteria (accurate, comprehensive, good lead time), into a single
additive objective function that can be...
We expressed the criteria (accurate, comprehensive, good lead time), into a single
additive objective function that can be...
We expressed the criteria (accurate, comprehensive, good lead time), into a single
additive objective function that can be...
We use R’s PSO package to minimize the objective function...
We use R’s PSO package to minimize the objective function...
PSO: Particle Swarm Optimization
PSO is a meta-heuristic desi...
We use R’s PSO package to minimize the objective function...
PSO: Particle Swarm Optimization
PSO is a meta-heuristic desi...
We use R’s PSO package to minimize the objective function...
PSO: Particle Swarm Optimization
PSO is a meta-heuristic desi...
After 1000 iterations, we obtain the following optimal solution:
After 1000 iterations, we obtain the following optimal solution:
Optimal Solution:
Window of Observation: 11.61 future day...
After 1000 iterations, we obtain the following optimal solution:
Optimal Solution:
Window of Observation: 11.61 future day...
After 1000 iterations, we obtain the following optimal solution:
Optimal Solution:
Window of Observation: 11.61 future day...
Just a review...
Just a review...
What makes a good early warning signal:
accurate: scope’s time-coverage covers an EPV cluster, minimal fa...
We obtain these Performance Statistics:
We obtain these Performance Statistics:
accuracy of ews: 94.44 percent predictive power
comprehensiveness: 99.40 percent o...
Below is a summary of the different EPV clusters and the lead-times of the ews:
Below is a summary of the different EPV clusters and the lead-times of the ews:
Total of 125 early warning signals activate...
Below is a summary of the different EPV clusters and the lead-times of the ews:
Total of 125 early warning signals activate...
Below is a summary of the different EPV clusters and the lead-times of the ews:
Average of 43 days between earliest ews and...
Below is a summary of the different EPV clusters and the lead-times of the ews:
Average of 43 days between earliest ews and...
Below is a summary of the different EPV clusters and the lead-times of the ews:
Average of 43 days between earliest ews and...
Below is a summary of the different EPV clusters and the lead-times of the ews:
Average of 43 days between earliest ews and...
for illustration...
1st (Jul 1993 - Feb 1994) and 2nd (Jul 1994 - Sep 1994) EPV Clusters:
1st (Jul 1993 - Feb 1994) and 2nd (Jul 1994 - Sep 1994) EPV Clusters:
3rd EPV Cluster (Mar 1999 - Sep 1999)and 4th (Jun 20...
5th (Jul 2002 - Aug 2002) and 6th (May 2003 - Oct 2003) EPV Clusters:
5th (Jul 2002 - Aug 2002) and 6th (May 2003 - Oct 2003) EPV Clusters:
7th (Mar 2008 - Nov 2008) and 8th (Apr 2009 - May 20...
5th (Jul 2002 - Aug 2002) and 6th (May 2003 - Oct 2003) EPV Clusters:
7th (Mar 2008 - Nov 2008) and 8th (Apr 2009 - May 20...
9th (Jul 2011) EPV Cluster:
Constraints:
Constraints:
usage of daily future prices, not daily spot prices
sensitivity analysis
application to other commodities
Thank You.
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Predicting Time Periiods of Excessive Price Volatility: The Case of Rice- Ramon Clarete and Alfonso Labao

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Predicting Time Periiods of Excessive Price Volatility: The Case of Rice- Ramon Clarete and Alfonso Labao

  1. 1. Predicting Time Periods of Excessive Price Volatility: The Case of Rice Dr. Ramon Clarete Alfonso Labao University of the Philippines, School of Economics September 25, 2013
  2. 2. Flow of Presentation Overview Methodology Empirical Results Conclusion and Constraints
  3. 3. Overview:
  4. 4. Overview: Importance of being forewarned of extreme food price-volatility:
  5. 5. Overview: Importance of being forewarned of extreme food price-volatility: provides time to undertake cooperation prevent herding and self-fulfilling crises avoid repetition of 2008 rice crisis prevent welfare costs for the poor
  6. 6. Overview: Importance of being forewarned of extreme food price-volatility: provides time to undertake cooperation prevent herding and self-fulfilling crises avoid repetition of 2008 rice crisis prevent welfare costs for the poor G20 report stresses importance of accurate and timely market information
  7. 7. Our Methodology:
  8. 8. Our Methodology: - Use daily rice future prices (5407 days from Sep 1991 to Mar 2013)
  9. 9. Our Methodology: - Use daily rice future prices (5407 days from Sep 1991 to Mar 2013) - Identify time-periods of excessive price volatility (via Martins-Filho, Maximo Torero, and Feng Yao’s (IFPRI, 2010) methodology: Two-step spline-backfitted-kernel estimation / GPD distribution)
  10. 10. Our Methodology: - Use daily rice future prices (5407 days from Sep 1991 to Mar 2013) - Identify time-periods of excessive price volatility (via Martins-Filho, Maximo Torero, and Feng Yao’s (IFPRI, 2010) methodology: Two-step spline-backfitted-kernel estimation / GPD distribution) - Develop Early Warning Signals (via Particle Swarm Optimization (PSO))
  11. 11. First Two Parts: - Use daily rice future prices (5407 days from Sep 1991 to Mar 2013) - Identify time-periods of excessive price volatility (via Martins-Filho, Maximo Torero, and Feng Yao’s (IFPRI, 2010) methodology: Two-step spline-backfitted-kernel estimation / GPD distribution) - Develop Early Warning Signals (via Particle Swarm Optimization (PSO))
  12. 12. Actual Prices:
  13. 13. Snapshot of actual prices:
  14. 14. Converted into Price Returns:
  15. 15. Converted into Price Returns:
  16. 16. Converted into Price Returns: where price returns refer to day-to-day price movements, or:
  17. 17. Converted into Price Returns: where price returns refer to day-to-day price movements, or: price return = ln( price of today price of yesterday )
  18. 18. Create Thresholds via MTY’s two-step methodology:
  19. 19. Create Thresholds via MTY’s two-step methodology:
  20. 20. Create Thresholds via MTY’s two-step methodology:
  21. 21. Methodology: MTY’s Two-step method:
  22. 22. Methodology: MTY’s Two-step method: Construct a nonparametric trend via spline-backfitted-kernel: rt = m0 + d a=1 ma(Xta) + (h0 + d a=1 ha(Xta)1/2 ) t Estimate the high-order 95 percent quantile via Generalized Pareto Distribution (GPD): ˆqt(α) = ˆk+1 + ˆβt ˆγt (( (1 − α) (k/N) )−ˆγt − 1)
  23. 23. Methodology: MTY’s Two-step method: Construct a nonparametric trend via spline-backfitted-kernel: rt = m0 + d a=1 ma(Xta) + (h0 + d a=1 ha(Xta)1/2 ) t Estimate the high-order 95 percent quantile via Generalized Pareto Distribution (GPD): ˆqt(α) = ˆk+1 + ˆβt ˆγt (( (1 − α) (k/N) )−ˆγt − 1) Estimated 95 percent Conditional Quantiles: ˆqt(α/rt−1, rt−2) = ˜m(rt−1, rt−2) + [(˜h(rt−1, rt−2))1/2 ˆqt(α)]
  24. 24. How are Time Periods of Excessive Price Volatility (EPV) defined?
  25. 25. How are Time Periods of Excessive Price Volatility (EPV) defined? Martins-Filho, Maximo Torero and Feng Yao’s definition of time periods of EPV: EPV Definition: Time-Periods whereby the preceding 60 days experienced a significantly high amount of extreme positive price returns over the 95 percent conditional quantile.
  26. 26. How are Time Periods of Excessive Price Volatility (EPV) defined? Martins-Filho, Maximo Torero and Feng Yao’s definition of time periods of EPV: EPV Definition: Time-Periods whereby the preceding 60 days experienced a significantly high amount of extreme positive price returns over the 95 percent conditional quantile. ... at least 7 instances of extreme positive price returns within a span of 60 days
  27. 27. 1st (Jul 1993 - Feb 1994) and 2nd (Jul 1994 - Oct 1994) EPV cluster:
  28. 28. 1st (Jul 1993 - Feb 1994) and 2nd (Jul 1994 - Oct 1994) EPV cluster: 3rd (Mar 1999 - Sep 1999) and 4th (Jun 2001 - Aug 2001) EPV cluster:
  29. 29. 5th (Jul 2002 - Aug 2002) and 6th (May 2003 - Oct 2003) EPV cluster:
  30. 30. 5th (Jul 2002 - Aug 2002) and 6th (May 2003 - Oct 2003) EPV cluster: 7th (Mar 2008 - Nov 2008) and 8th (Apr 2009 - May 2009) EPV cluster:
  31. 31. 9th (Jul 2011) EPV cluster:
  32. 32. In Summary:
  33. 33. In Summary: 501 days of excessive price-volatility (EPV) Nine (9) clusters of EPV
  34. 34. In Summary: 501 days of excessive price-volatility (EPV) Nine (9) clusters of EPV
  35. 35. In Summary: 501 days of excessive price-volatility (EPV) Nine (9) clusters of EPV ... Of the above clusters of EPV, only four (4) are severe
  36. 36. Third Part: - Use daily rice future prices (5407 days from Sep 1991 to Mar 2013) - Identify time-periods of excessive price volatility (via Martins-Filho, Maximo Torero, and Feng Yao’s (IFPRI, 2010) methodology: Two-step spline-backfitted-kernel estimation / GPD distribution) - Develop Early Warning Signals (via Particle Swarm Optimization (PSO))
  37. 37. Next Task: Looking for a Good Early Warning Signal
  38. 38. Next Task: Looking for a Good Early Warning Signal
  39. 39. Next Task: Looking for a Good Early Warning Signal
  40. 40. Next Task: Looking for a Good Early Warning Signal
  41. 41. Next Task: Looking for a Good Early Warning Signal
  42. 42. Next Task: Looking for a Good Early Warning Signal
  43. 43. Next Task: Looking for a Good Early Warning Signal
  44. 44. Next Task: Looking for a Good Early Warning Signal We set the mirror-image’s lag at 60 days prior to an EPV cluster...
  45. 45. Next Task: Looking for a Good Early Warning Signal
  46. 46. Next Task: Looking for a Good Early Warning Signal
  47. 47. Next Task: Looking for a Good Early Warning Signal Parameters of the Early Warning Signals:
  48. 48. Next Task: Looking for a Good Early Warning Signal Parameters of the Early Warning Signals: Window of Observation
  49. 49. Next Task: Looking for a Good Early Warning Signal Parameters of the Early Warning Signals: Window of Observation Lower-Order Quantile Level
  50. 50. Next Task: Looking for a Good Early Warning Signal Parameters of the Early Warning Signals: Window of Observation Lower-Order Quantile Level Frequency of Breach
  51. 51. Next Task: Looking for a Good Early Warning Signal Parameters of the Early Warning Signals: Window of Observation Lower-Order Quantile Level Frequency of Breach Scope
  52. 52. Next Task: Looking for a Good Early Warning Signal Parameters of the Early Warning Signals: Window of Observation Lower-Order Quantile Level Frequency of Breach Scope Together, these parameters generate a time-based variable, namely...
  53. 53. Next Task: Looking for a Good Early Warning Signal ... the scope’s time-coverage
  54. 54. What makes a good early warning signal:
  55. 55. What makes a good early warning signal: accurate: scope’s time-coverage covers an EPV cluster, minimal false alarms
  56. 56. What makes a good early warning signal: accurate: scope’s time-coverage covers an EPV cluster, minimal false alarms good lead time: there’s reasonable lead time before EPV cluster
  57. 57. What makes a good early warning signal: accurate: scope’s time-coverage covers an EPV cluster, minimal false alarms good lead time: there’s reasonable lead time before EPV cluster comprehensive: signals pre-empt almost all EPV clusters
  58. 58. What makes a good early warning signal: accurate: scope’s time-coverage covers an EPV cluster, minimal false alarms good lead time: there’s reasonable lead time before EPV cluster comprehensive: signals pre-empt almost all EPV clusters Basically... We create a new set of trends (using spbk) and solve for parameters that meet above requirements
  59. 59. We expressed the criteria (accurate, comprehensive, good lead time), into a single additive objective function that can be minimized: Objective Function for Minimization: f (X) = [( count1 laghvp ) ∗ 1.35] + [ count2 hvp ] + [(accuracy) ∗ 1.35]+ [( totalews totaldays ) ∗ 1.25] + [ totalscope totaldays ]
  60. 60. We expressed the criteria (accurate, comprehensive, good lead time), into a single additive objective function that can be minimized: Objective Function for Minimization: f (X) = [( count1 laghvp ) ∗ 1.35] + [ count2 hvp ] + [(accuracy) ∗ 1.35]+ [( totalews totaldays ) ∗ 1.25] + [ totalscope totaldays ] Where: count1: total no. of lagged mirror-images of time periods of EPV not covered by the scopes of the signals laghvp: total no. of lagged mirror-images of time periods of EPV count2: total no. of actual time periods of EPV not covered by the scopes’ time-coverage of the early warning signals hvp: total no. of actual time periods of EPV accuracy: predictive accuracy of the early warning signal with following ratio: [1-((no.of high.vol.periods captured by scope’s time-coverage) / (hvp)]. total ews: total no. of early warning signals total scope: total no. of days covered by the scopes’ time-coverage of the early warning signals total days: total no. of days within the sample timeframe (1991 to 2013): 5407 future days
  61. 61. We expressed the criteria (accurate, comprehensive, good lead time), into a single additive objective function that can be minimized: Objective Function for Minimization: f (X) = [( count1 laghvp ) ∗ 1.35] + [ count2 hvp ] + [(accuracy) ∗ 1.35]+ [( totalews totaldays ) ∗ 1.25] + [ totalscope totaldays ]
  62. 62. We expressed the criteria (accurate, comprehensive, good lead time), into a single additive objective function that can be minimized: Objective Function for Minimization: f (X) = [( count1 laghvp ) ∗ 1.35] + [ count2 hvp ] + [(accuracy) ∗ 1.35]+ [( totalews totaldays ) ∗ 1.25] + [ totalscope totaldays ] First two (2) terms - optimize good lead-time and comprehensiveness
  63. 63. We expressed the criteria (accurate, comprehensive, good lead time), into a single additive objective function that can be minimized: Objective Function for Minimization: f (X) = [( count1 laghvp ) ∗ 1.35] + [ count2 hvp ] + [(accuracy) ∗ 1.35]+ [( totalews totaldays ) ∗ 1.25] + [ totalscope totaldays ] First two (2) terms - optimize good lead-time and comprehensiveness Last three (3) terms - minimize false alarms and improve accuracy.
  64. 64. We use R’s PSO package to minimize the objective function...
  65. 65. We use R’s PSO package to minimize the objective function... PSO: Particle Swarm Optimization PSO is a meta-heuristic designed for functions that are hard to optimize using traditional optimization procedures (due to several peaks, non-linearities, long-forms, etc..)
  66. 66. We use R’s PSO package to minimize the objective function... PSO: Particle Swarm Optimization PSO is a meta-heuristic designed for functions that are hard to optimize using traditional optimization procedures (due to several peaks, non-linearities, long-forms, etc..) PSO Parameter Search Range:
  67. 67. We use R’s PSO package to minimize the objective function... PSO: Particle Swarm Optimization PSO is a meta-heuristic designed for functions that are hard to optimize using traditional optimization procedures (due to several peaks, non-linearities, long-forms, etc..) PSO Parameter Search Range: Window of Observation: from 1 to 40 future days Quantile Level: from 50 percent to 99 percent quantile Frequency of Breach: from 1 to 5 Scope of Early Warning Signal: from 60 to 250 future days
  68. 68. After 1000 iterations, we obtain the following optimal solution:
  69. 69. After 1000 iterations, we obtain the following optimal solution: Optimal Solution: Window of Observation: 11.61 future days Quantile Level: 91.31 percent Frequency of Breach: 4.18 Scope of Early Warning Signal: 107.06 future days
  70. 70. After 1000 iterations, we obtain the following optimal solution: Optimal Solution: Window of Observation: 11.61 future days Quantile Level: 91.31 percent Frequency of Breach: 4.18 Scope of Early Warning Signal: 107.06 future days Given these parameters, an early warning signal will be activated...
  71. 71. After 1000 iterations, we obtain the following optimal solution: Optimal Solution: Window of Observation: 11.61 future days Quantile Level: 91.31 percent Frequency of Breach: 4.18 Scope of Early Warning Signal: 107.06 future days Given these parameters, an early warning signal will be activated... ... once the 91.31 percent conditional quantile is breached five (5) times within 11.61 future days..
  72. 72. Just a review...
  73. 73. Just a review... What makes a good early warning signal: accurate: scope’s time-coverage covers an EPV cluster, minimal false alarms good lead time: there’s reasonable lead time before EPV cluster comprehensive: signals pre-empt almost all EPV clusters
  74. 74. We obtain these Performance Statistics:
  75. 75. We obtain these Performance Statistics: accuracy of ews: 94.44 percent predictive power comprehensiveness: 99.40 percent of time periods of EPV are covered lead-time before EPV cluster: average of 43 days from the earliest signal to start of high-volatility time-period
  76. 76. Below is a summary of the different EPV clusters and the lead-times of the ews:
  77. 77. Below is a summary of the different EPV clusters and the lead-times of the ews: Total of 125 early warning signals activated from Sep 1991 to Mar 2013.
  78. 78. Below is a summary of the different EPV clusters and the lead-times of the ews: Total of 125 early warning signals activated from Sep 1991 to Mar 2013. This is only 2.3 percent of the time.
  79. 79. Below is a summary of the different EPV clusters and the lead-times of the ews: Average of 43 days between earliest ews and start of an EPV cluster
  80. 80. Below is a summary of the different EPV clusters and the lead-times of the ews: Average of 43 days between earliest ews and start of an EPV cluster But among the four (4) severe EPV clusters, there is an average lead-time of 61 days
  81. 81. Below is a summary of the different EPV clusters and the lead-times of the ews: Average of 43 days between earliest ews and start of an EPV cluster But among the four (4) severe EPV clusters, there is an average lead-time of 61 days The unflagged 2011 EPV cluster is not severe..
  82. 82. Below is a summary of the different EPV clusters and the lead-times of the ews: Average of 43 days between earliest ews and start of an EPV cluster But among the four (4) severe EPV clusters, there is an average lead-time of 61 days The unflagged 2011 EPV cluster is not severe.. All EPV are sufficiently covered from beginning till end by the scopes of the ews..
  83. 83. for illustration...
  84. 84. 1st (Jul 1993 - Feb 1994) and 2nd (Jul 1994 - Sep 1994) EPV Clusters:
  85. 85. 1st (Jul 1993 - Feb 1994) and 2nd (Jul 1994 - Sep 1994) EPV Clusters: 3rd EPV Cluster (Mar 1999 - Sep 1999)and 4th (Jun 2001 - Aug 2001) EPV Clusters:
  86. 86. 5th (Jul 2002 - Aug 2002) and 6th (May 2003 - Oct 2003) EPV Clusters:
  87. 87. 5th (Jul 2002 - Aug 2002) and 6th (May 2003 - Oct 2003) EPV Clusters: 7th (Mar 2008 - Nov 2008) and 8th (Apr 2009 - May 2009) EPV Clusters:
  88. 88. 5th (Jul 2002 - Aug 2002) and 6th (May 2003 - Oct 2003) EPV Clusters: 7th (Mar 2008 - Nov 2008) and 8th (Apr 2009 - May 2009) EPV Clusters:
  89. 89. 9th (Jul 2011) EPV Cluster:
  90. 90. Constraints:
  91. 91. Constraints: usage of daily future prices, not daily spot prices sensitivity analysis application to other commodities
  92. 92. Thank You.

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