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RISK FRAMEWORKS AND APPLICATIONS

Ever since I started running risk management training workshops and teaching Executive MBA students, I had been looking for a textbook that walked the middle ground between plain English and as few mathematical equations as possible.  It had to be concise and to the point yet cover the basics of quantifying risk, implementing risk policies, derivative pricing, product types and real life market applications.  Easy to read but deep enough to be a handy desk reference, perhaps a cross between a HBS case and a customized text booke.

Towards the end of 2008 we started work on a project that resulted in the product you now hold in your hands.  Let me be honest and upfront, we had to let in a few formulae in (Paul Petty will never forgive me) since the simplest pricing model also required elementary mathematics.  But we compensated this failing by including as many rich cases as we could find that would help students understand and appreciate the world risk managers and derivative traders live in.

This is a book inspired by questions, not answers. I hope that it helps you ask the right questions and interests you enough for you to make the effort to find the right answers.

Many clients, colleagues, peers and family members contributed and sacrificed to make this project real. While their contribution added value, any mistakes and oversights are completely our responsibilities.

Jawwad Farid

 

Dedications

In the long list of individuals who helped me decipher the code behind risk management and derivative pricing three names stand out.

The first is Paul Petty.  Paul was the Executive Director on Goldman’s London Prime Brokerage desk who introduced me to Value at Risk and showed me how to write simply without numbers, formulae or complex terminology.  He helped me build bridges with technology and firm wide risk and brought Bernie Minski into our many risk debates.  Without Paul, Goldman and the fall of 1998, I may have never walked the route that led to Alchemy.

The second is Mark Broadie at Columbia Business School.  Mark taught me how much I didn’t know about security pricing and Excel despite 20 odd cover to cover readings of John C. Hull and 14 years of building spread sheets.  Once we acknowledged our ignorance Mark opened the doorway to building simple models for complex term sheets.  The elegance of his pricing and modelling approach added a new dimension of brilliance to Mark’s work.

The third and final is late Sir Aziz-ullah at the BVS Parsi High School. He ignited the fire for numbers in his high school students like no other teacher I have ever met and inspired just as many to become teachers themselves.

Without these three light bearers this book would have never been written.


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Table of Contents from Risk Frameworks and Applications

PART 1: Introduction to Risk Management Frameworks
  • Chapter 1 – Framework
  • Chapter Two – Applications
  • Chapter Three – Calculating Value at Risk

    PART 2: Derivatives
  • Chapter One – Terminology
  • Chapter Two – Products & Pricing
  • Chapter Three – Variations

    PART 3: DERIVATIVE PRICING

    PART 4: Understanding Commodities Risk

    PART 5: Risk Metrics

    Bibliography

    Index

    Table of Figures

    To download the full Table of contents, click here.


    Bibliography from Risk Frameworks and Applications

    1. Numerical Option Pricing, Mark Broadie and Paul Glasserman, Columbia University
    2. TN 96-02. Risk Neutral Pricing in Discrete Time, D. M. Chance
    3. The boy’s guide to pricing and hedging, Emmanuel Derman, Columbia University
    4. The principles and practices of verifying derivative prices, Emmanuel Derman,
      Goldman Sachs & Co.
    5. Refresher in Option Pricing, Howard Corb, J.P.Morgan
    6. Options, Futures and Other Derivatives, John C. Hull, 7th Edition, Prentice Hall, 2009
    7. Case Problems in Finance, Kester, Fruhan, Piper, Ruback, 11th Edition, Irwin/ McGraw-Hill, 1997
    8. The Handbook of Fixed Income Securities, Frank J. Fabozzi, 7th Edition, McGraw-Hill, 2005
    9. Risk Management and Financial Institutions, John C. Hull, Low Price Edition,
      Pearson Education, Inc., 2007
    10. Exotic Equity Derivatives Manual, Salomon Smith Barney, August 1998
    11. Understanding Market, Credit and Operational Risk- The Value at Risk Approach, Linda Allen, Jacob Boudoukh and Anthony Saunders,Blackwell Publishing, 2004
    12. Mind on Statistics, Utts and Heckard, Duxbury 2002
    13. Beyond Value at Risk, The New Science of Risk Management, Kevin Dowd, John Wiley & Sons, 1998
    14. Higher-Order Simulations: Strategic Investment Under Model-Induced Price Patterns, Gilbert Peffer and Bàrbara Llacay, Journal of Artificial Societies and Social Simulation vol. 10,
      no. 2, 6, 2007
    15. Oil Prices and Inflation: Learning the Arithmetic Anew, Dr Benjarong Suwankiri and Dr Sutapa Amornvivat, Bangkok Post Business, February 2010
    16. Commodities: 2010 Outlook, Goldman Sachs, 18 December 2009
    17. Weekly Comment: Things To Ponder Over The Christmas Break, Economics, Goldman Sachs, 18 December 2009
    18. Global Oil & Gas Daily, J. P Morgan, 24 December 2009
    19. Global Metals & Mining Weekly Wrap-Up, J. P Morgan, 18 December 2009
    20. Gold & Precious Metals, J. P Morgan, 18 December 2009
    21. Precious metals as alternative FX reserve diversifiers, Global Metals Weekly, Bank of America Merrill Lynch, 18 December 2009
    22. 8 Reasons To Own Gold, Tony Daltorio
    23. A Forex Trader's View Of The Aussie/Gold Relationship, Richard Lee
    24. Commodity Prices And Currency Movements, Kathy Lien
    25. Oil Price Patterns Portend Shrinking Of Excess Stockpiles, Brian Baskin of Dow Jones Newswires, Wall Street Journal, 30 September 2009
    26. Energy outlook: Crude correlations and what comes next, Christine Birkner, Futures Mag.Com, October 2009
    27. Monthly Review on Price Indices, Statistics Division, Federal Bureau of Statistics, Government of Pakistan, December 2009 & January 2010
    28. VaR Applications: Setting VaR-based Limits, Carlos Blanco and Sally Blomstrom, Financial Engineering Associates, Inc., May 1999
    29. Commonly Used Market Risk Limits, Guidelines on Risk Management of Derivatives and other traded instruments, Annex D
    30. Establishing exposure limits for a credit Portfolio, James V. Lentino, The RMA Journal,
      July-August 2005
    31. Quantitative Finance, Second Edition, Paul Wilmott, John Wiley&Sons, Ltd., 2006
    32. Investments, 4th Edition, Zvi Bodie, Alex Kane, Alan J. Marcus, Irwin McGraw-Hill, 1999
    33. Risk Management and Capital Allocation, Reto Gallati, McGraw Hill, 2003
    34. Integrating Corporate Risk Management, Prakash A. Shimpi, Texere, 1999,2001
    35. Modern Portfolio Theory and Investment Analysis, 5th Edition, Edwin J. Elton, Martin J. Gruber, John Wiley & Sons, Inc.
    36. The Black Swan, the impact of the highly improbable, Nassim Nicholas Taleb, Random House, 2007
    37. Dynamic Hedging, Managing Vanilla and Exotic Options, Nassim Taleb, John Wiley & Sons, Inc.,1997
    38. Currency Derivatives, Edited by David F DeRosa, John Wiley & Sons, Inc.,1998
    39. Understanding Commodities Risk, First Edition, Jawwad A. Farid, Alchemy Publications,
      March 2010
    40. Pakistan Risk Review, Volume 01, January 2010, Jawwad A. Farid, Alchemy Publications
    41. Pakistan Risk Review, Volume 04, September 2009, Jawwad A. Farid, Alchemy Publications
    42. Pakistan Risk Review, Volume 03, July 2009, Jawwad A. Farid, Alchemy Publications
    43. Pakistan Risk Review, Volume 02, May 2009, Jawwad A. Farid, Alchemy Publications
    44. Pakistan Risk Review, Volume 01, April 2009, Jawwad A. Farid, Alchemy Publications

    Table of Figures from Risk Frameworks and Applications

    Figure 1: Risk Framework
    Figure 2: Risk profile - Crude input price changes
    Figure 3: Histogram of returns
    Figure 4: Various Distributions
    Figure 5: Time line - From changes in input prices to P&L impact
    Figure 6: Crude Oil price change distribution
    Figure 7: Output product price change distribution
    Figure 8: Margin change distribution
    Figure 9: WTI prices & volatilities
    Figure 10: Expected Input Price & Margin Shortfall ranges at different odds of occurrence
    Figure 11: Expected Input Price & Inventory Losses ranges at different odds of occurrence
    Figure 12: Limit monitoring and review process
    Figure 13: Risk reporting process
    Figure 14: Risk Framework revisited
    Figure 15: Capital Loss calculation example
    Figure 16: Allocation of actual stop loss limits by lines of investment
    Figure 17: Example of slippage
    Figure 18: Odds, Confidence Levels & Limits
    Figure 19: Example - Setting VaR limits for sub portfolios
    Figure 20: Determination of multiple for setting an upper bound on VaR limit
    Figure 21: Setting Duration & Convexity limits
    Figure 22: Setting PVBP limits
    Figure 23: Limits applicable to Money Market Products
    Figure 24: Limits applicable to Capital Market Products
    Figure 25: Value at Risk
    Figure 26: Distribution of returns
    Figure 27: Variance Co-variance Approach
    Figure 28: Historical Simulation Method
    Figure 29: Monte Carlo Simulation Method
    Figure 30: Comparison of VaR Methods
    Figure 31: Daily time series data
    Figure 32: Interpolated Interest Rates.
    Figure 33: Derived Bond Price Series
    Figure 34: Daily return series
    Figure 35: Calculation of portfolio weights
    Figure 36: Portfolio daily return series
    Figure 37: Squared differences of the returns series
    Figure 38: Calculation of weights under EWMA
    Figure 39: Scaling of weights
    Figure 40: Calculation of EWMA variance
    Figure 41: SMA and EWMA daily VaRs
    Figure 42: Ordered portfolio return series
    Figure 43: Holding Period VaR
    Figure 44: Payoff profiles for forwards & futures contracts
    Figure 45: Payoff profiles for option contracts
    Figure 46: Quadrant IV - Payoff profiles for a long forward contract
    Figure 47: Payoff profile for a long forward contract
    Figure 48: Payoff profile for underlying security
    Figure 49: Comparative downsides of forward & option  payoffs
    Figure 50: Payoff profile for the holder of a call option
    Figure 51: Payoff profile for the writer of a call option
    Figure 52: Payoff profile for the holder of a put option
    Figure 53: Payoff profile for the writer of a put option
    Figure 54: Building blocks for synthetic configurations
    Figure 55: Synthetic forward contract creations
    Figure 56: Comparative payoff profiles for calls & forwards
    Figure 57: Comparative payoff profiles for calls, puts & forwards
    Figure 58: Payoff profile for a synthetic long forward
    Figure 59: Template for evaluating derivatives
    Figure 60: Comparative look at derivatives
    Figure 61: Payoff to the buyer & seller of a call option
    Figure 62: Payoff to the buyer & seller of a put option
    Figure 63: Binomial tree - prices of the underlying asset
    Figure 64: Using a Binomial tree to calculate the price of an American Call Option
    Figure 65: Payoffs for the buyer & seller of a forward contract
    Figure 66: Plain Vanilla Interest Rate Swap
    Figure 67: Interest Rate Cap
    Figure 68: Interest Rate Floor
    Figure 69: Payoffs for the buyer of a call & seller of a put i.e. of a synthetic forward contract
    Figure 70: Binomial tree of the underlying assets prices
    Figure 71 : Binomial tree for the option
    Figure 72: Binomial tree of underlying asset’s prices with 2 time steps
    Figure 73: Binomial tree of a European call option with 2 time steps
    Figure 74: Prices of the underlying asset at expiry
    Figure 75: Terminal payoffs of the  European call option
    Figure 76: Formulas for spreadsheet implementation of binomial tree for a European call option
    Figure 77: Spreadsheet for European call option example
    Figure 78: Spreadsheet for European put option example
    Figure 79: Formulas for spreadsheet implementation of binomial tree for an American call option
    Figure 80: Spreadsheet for American call option example
    Figure 81: Formulas for spreadsheet implementation of binomial tree for an American put option
    Figure 82: Spreadsheet for American put option example
    Figure 83: Formulas for spreadsheet implementation of binomial tree for capped calls with automatic exercise
    Figure 84: Spreadsheet for capped calls with automatic exercise example
    Figure 85: Formulas for spreadsheet implementation of binomial tree for down & out call options
    Figure 86: Spreadsheet for down & out call option example
    Figure 87: Good values of ‘n’ for Barrier Options
    Figure 88: Formulas for spreadsheet implementation of binomial tree for down and in call options assuming option not previously knocked in
    Figure 89: Spreadsheet for down and in call options assuming option previously knocked in
    Figure 90: Spreadsheet for down and in call options assuming option not previously knocked in
    Figure 91: Excess capacity and WTI spot prices
    Figure 92: Excess supply and WTI spot prices
    Figure 93: Production & spare capacity of OPEC nations
    Figure 94: USD - EUR exchange rate & WTI spot prices
    Figure 95: S&P 500 index & crude oil prices for the period Jan 08 - Aug 09
    Figure 96: S&P 500 index & Crude Oil prices for the period Jan 01 to Sep 09
    Figure 97: Contract spreads & WTI prices
    Figure 98: AUD - USD exchange rate & gold prices
    Figure 99: 90 day trailing correlation between gold prices & AUD - USD exchange rates
    Figure 100: Scatter plot of AUD - USD exchange rate against gold prices
    Figure 101: AUD - USD exchange rate & WTI spot prices
    Figure 102: Scatter plot of AUD - USD exchange rate against WTI spot prices
    Figure 103: USD – EUR exchange rate & gold prices
    Figure 104: Exchange rates
    Figure 105: Gold &crude oil prices.
    Figure 106: Ratio of gold to crude oil prices
    Figure 107: AUD - USD exchange rates & gold to oil ratio
    Figure 108: Brent Crude Oil  & West Texas Crude Oil prices
    Figure 109: Scatter plot - Brent Crude Oil & West Texas Crude Oil prices
    Figure 110: Correlations between Brent Crude & West Texas Crude Oil prices
    Figure 111: Brent Crude Oil & Fuel Oil prices
    Figure 112: Scatter plot - Brent Crude Oil & Fuel Oil prices
    Figure 113: Correlations between Brent Crude Oil & Fuel Oil prices
    Figure 114: Brent Crude Oil  & Diesel Fuel prices
    Figure 115: Scatter plot - Brent Crude Oil & Diesel Fuel prices
    Figure 116: Correlations between Brent Crude Oil & Diesel Fuel prices
    Figure 117: Brent Crude Oil & Natural Gas prices
    Figure 118: Scatter plot - Brent Crude Oil & Natural Gas prices
    Figure 119: Correlations between  Brent Crude Oil & Natural Gas prices
    Figure 120: Brent Crude Oil & Gold prices
    Figure 121: Scatter plot - Brent Crude Oil & Gold prices
    Figure 122: Correlations between  Brent Crude  Oil & Gold prices
    Figure 123: Brent Crude Oil & Silver prices
    Figure 124: Scatter plot - Brent Crude Oil & Silver prices
    Figure 125:Correlations between  Brent Crude  Oil  & Silver prices
    Figure 126: Brent Crude Oil & Platinum prices
    Figure 127: Scatter plot - Brent Crude Oil & Platinum prices
    Figure 128: Correlations between  Brent Crude  Oil & Platinum prices
    Figure 129: Brent Crude Oil & Aluminium prices
    Figure 130: Scatter plot - Brent Crude Oil & Aluminium prices
    Figure 131: Correlations between  Brent Crude  Oil & Aluminium prices
    Figure 132: Brent Crude Oil & Steel prices
    Figure 133: Scatter plot - Brent Crude Oil & Steel prices
    Figure 134: Correlations between Brent Crude  Oil & Steel prices
    Figure 135: Brent Crude Oil & Copper prices
    Figure 136: Scatter plot - Brent Crude Oil & Copper prices
    Figure 137: Correlations between Brent Crude  Oil & Copper prices.
    Figure 138: Brent Crude Oil & Cotton prices
    Figure 139: Scatter plot - Brent Crude Oil & Cotton prices
    Figure 140: Brent Crude Oil & Cotton prices
    Figure 141: Brent Crude Oil & Corn prices
    Figure 142: Scatter plot - Brent Crude Oil & Corn prices
    Figure 143: Correlations between Brent Crude  Oil & Corn prices
    Figure 144: Brent Crude Oil & Wheat No.2 prices
    Figure 145: Scatter plot - Brent Crude Oil & Wheat No.2 prices
    Figure 146: Correlations between Brent Crude  Oil & Wheat No.2 prices
    Figure 147: Brent Crude Oil & Wheat No.1 prices
    Figure 148: Scatter plot - Brent Crude Oil & Wheat No.1 prices
    Figure 149: Correlations between Brent Crude  Oil & Wheat No.1 prices
    Figure 150: Brent Crude Oil & Coffee prices
    Figure 151: Scatter plot - Brent Crude Oil & Coffee prices
    Figure 152: Correlations between Brent Crude  Oil & Coffee prices.
    Figure 153: Brent Crude Oil & Sugar prices
    Figure 154: Scatter plot - Brent Crude Oil & Sugar prices
    Figure 155: Correlations between Brent Crude  Oil & Sugar prices
    Figure 156: Brent Crude Oil & Corn Oil prices
    Figure 157: Scatter plot - Brent Crude Oil & Corn Oil prices
    Figure 158: Correlations between Brent Crude  Oil  & Corn Oil prices
    Figure 159: Brent Crude Oil & Soybean Oil prices
    Figure 160: Scatter plot - Brent Crude Oil & Soybean Oil prices
    Figure 161: Correlations between Brent Crude  Oil & Soybean Oil prices
    Figure 162: Brent Crude Oil & Crude Palm Oil prices
    Figure 163: Scatter plot - Brent Crude Oil & Crude Palm Oil prices
    Figure 164: Correlations between Brent Crude  Oil & Crude Palm Oil prices
    Figure 165: Data used in analysis
    Figure 166: Crude Palm Oil futures contracts prices
    Figure 167: SMA Volatility & 3 – Sigma band for FCPO prices
    Figure 168: EWMA Volatility & 3 – Sigma band for FCPO prices
    Figure 169: 10 Day Holding VaRs
    Figure 170: FCPO volatilities over time
    Figure 171: Correlation between FCPO & other commodity prices
    Figure 172: FCPO prices & USD – EUR exchange rates
    Figure 173: Scatter plot - FCPO prices &  USD – EUR exchange rates.
    Figure 174: FCPO prices & MYR – USD exchange rates
    Figure 175: Scatter plot - FCPO prices &  MYR – USD  exchange rates
    Figure 176: FCPO prices & Gold prices
    Figure 177: Scatter plot - FCPO prices & Gold prices
    Figure 178: FCPO & WTI prices
    Figure 179: Scatter plot – FCPO & WTI prices
    Figure 180: FCPO & Wheat No.2 prices
    Figure 181: Scatter plot - FCPO & Wheat No.2 prices
    Figure 182: FCPO & Corn prices
    Figure 183: Scatter plot - FCPO & Corn prices
    Figure 184: Composition of the Consumer Price Index (CPI)
    Figure 185: Impact of CPI components on inflation
    Figure 186: Composition of the Wholesale Price Index (WPI)
    Figure 187: Impact of WPI components on inflation
    Figure 188: Crude Oil prices & the Consumer Price Index
    Figure 189: Food & Beverages CPI & Crude Oil prices
    Figure 190: Apparel, Textiles & Footwear CPI & Crude Oil prices
    Figure 191: House Rent CPI & Crude Oil prices
    Figure 192: Fuel & Lighting CPI & Crude Oil prices
    Figure 193: Household, Furniture & Equipment CPI & Crude Oil prices
    Figure 194: Transport & Communication CPI & Crude Oil prices
    Figure 195: Recreation & Entertainment CPI & Crude Oil prices
    Figure 196: Education CPI & Crude Oil prices
    Figure 197: Cleaning, Laundry & Personal Appearance CPI & Crude Oil prices
    Figure 198: Medicare CPI & Crude Oil prices
    Figure 199: Crude Oil prices & the Wholesale Price Index (WPI)
    Figure 200: Food WPI & Crude Oil prices
    Figure 201: Raw Materials WPI & Crude Oil prices
    Figure 202: Fuel, Lighting & Lubricants WPI & Crude Oil prices
    Figure 203: Manufactures WPI & Crude Oil prices
    Figure 204: Building Materials WPI & Crude Oil prices
    Figure 205: Inflation & Bond yield data for Pakistan and India
    Figure 206: CPI & WPI Inflation in India
    Figure 207: CPI & WPI Inflation in Pakistan
    Figure 208: Inflation based on WPI in India and Pakistan
    Figure 209: Inflation based on CPI in India and Pakistan
    Figure 210: CPI Inflation in Pakistan, 3 – year Bond Yields & Spreads
    Figure 211: 3 – year Bond Yield Spreads with CPI & WPI inflation in Pakistan
    Figure 212: CPI Inflation in Pakistan, 5 – year Bond Yields & Spreads
    Figure 213: 5 – year Bond Yield Spreads with CPI & WPI inflation in Pakistan
    Figure 214: CPI Inflation in Pakistan, 7 – year Bond Yields & Spreads
    Figure 215: 7 – year Bond Yield Spreads with CPI & WPI inflation in Pakistan
    Figure 216: CPI Inflation in Pakistan, 10 – year Bond Yields & Spreads
    Figure 217: 10 – year Bond Yield Spreads with CPI & WPI inflation in Pakistan
    Figure 218: CPI Inflation in India, 3 – year Bond Yields & Spreads
    Figure 219: 3 – year Bond Yield Spreads with CPI & WPI inflation in India
    Figure 220: CPI Inflation in India, 5 – year Bond Yields & Spreads
    Figure 221: 5 – year Bond Yield Spreads with CPI & WPI inflation in India
    Figure 222: CPI Inflation in India, 7 – year Bond Yields & Spreads
    Figure 223: 7 – year Bond Yield Spreads with CPI & WPI inflation in India
    Figure 224: CPI Inflation in India, 10 – year Bond Yields & Spreads
    Figure 225: 10 – year Bond Yield Spreads with CPI & WPI inflation in India
    Figure 226: 3 – Year nominal bond yields & spreads between the Indian & Pakistani markets
    Figure 227: 5 – Year nominal bond yields & spreads between the Indian & Pakistani markets
    Figure 228: 7 – Year nominal bond yields & spreads between the Indian & Pakistani markets
    Figure 229: 10 – Year nominal bond yields & spreads between the Indian & Pakistani markets
    Figure 230: Spreads between 5 – Year & 7 – Year bond yields
    Figure 231: Correlation between India and Pakistan bond yield spreads
    Figure 232: Spreads between 3 – year & 10 – year bond yields
    Figure 233: Correlation between India and Pakistan of 3 – year & 10 – year bond yield spreads
    Figure 234: 5 – year WPI Inflation adjusted bond yields in India and Pakistan
    Figure 235: 5 – year CPI Inflation adjusted bond yields in India and Pakistan
    Figure 236: Correlations between Indian & Pakistani 3 – year bond adjusted yields
    Figure 237: Annualized volatilities in 2008 & 2009
    Figure 238: Crude Oil volatility levels
    Figure 239: Refined Products volatility levels
    Figure 240: Precious Metals volatility levels
    Figure 241: Other Metals volatility levels
    Figure 242: Fibres & Textiles volatility levels
    Figure 243: Grains & Feeds volatility levels
    Figure 244: Food volatility levels
    Figure 245:Fats & Oils volatility levels
    Figure 246: 3 – Sigma band for commodity prices
    Figure 247: Commodity details
    Figure 248: Correlation matrix (1-1-2008 to 15-4-2009)
    Figure 249: Correlation matrix (1-1-2008 to 31-12-2009)
    Figure 250: Natural Gas consumption by end use in USA
    Figure 251: Natural Gas prices
    Figure 252: Working Gas in underground storage in USA
    Figure 253: Dry Natural Gas production & consumption in USA
    Figure 254: Dry Natural Gas consumption in USA,  China, India & Pakistan
    Figure 255: WTI and Brent Crude Oil prices
    Figure 256: OPEC Crude Oil supply
    Figure 257: Crude Oil & Gasoline prices in USA
    Figure 258: Crude Oil production & consumption in USA
    Figure 259: Petroleum consumption in USA, China, India & Pakistan
    Figure 260: Spot Price Ratio: WTI Crude Oil to Henry Hub Natural Gas
    Figure 261: Total energy consumption in China
    Figure 262: Total Oil production and consumption  in China
    Figure 263: Dry Natural Gas production &  consumption in China
    Figure 264: Coal production & consumption in China
    Figure 265: Net Coal Exports - China
    Figure 266: Production & consumption of Crude Oil in India
    Figure 267: Natural Gas production & consumption in India
    Figure 268: Total Oil production & consumption in Pakistan
    Figure 269: Dry Natural Gas production &   consumption in Pakistan.
    Figure 270: Crude Oil & Gasoline prices in Pakistan
    Figure 271: Rules of Thumb for interpreting correlations
    Figure 272: Table of Critical values for a two tailed test of significance

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