2. 2005 Canadian Annual Derivatives Conference August 17 - 20, 2005 Québec, Canada. Ranjan Bhaduri, BSc (Hons), MBA, MMath, PhD.

3. Overview The Mathematics of Risk Portfolio Risk Elements of Risk Aftermath

4. The Mathematics of Risk How do we define risk? Entanglement between randomness, probability, and risk Mathematical tools to measure risk & performance, and improve security (cryptography)

5. Portfolio Risk Tail Analysis (extreme risk) Can NOT just sweep non- normality under the rug Must look at higher moments & journey to the tail Omega function very useful as risk tool

6. What is the Omega function? Invented by mathematicians (Shadwick & Keating) in 2002 Can be thought of as the quality of an investment on a return above a certain level (threshold) A rankings function that encodes return, variance, skew, kurtosis, and all of the higher moments - without penalizing for upside volatility

7. Mathematical Definition of Omega Where F is the cumulative distribution of returns, and r is the threshold chosen by the investor.

8. Omega - the Finance Intuition R is the threshold value (and the strike) C(R) and P(R) are prices of one period European call and put prices; the underlying is the security’s RETURN, not the security’s price. numerator = E [ max (x – R, 0)] denominator = E [ max (R – x, 0)] Can be thought of as the quality of an investment on a return above a given level (threshold); “quality” is upside versus downside R is the threshold value (and the strike) C(R) and P(R) are prices of one period European call and put prices; the underlying is the security’s RETURN, not the security’s price. numerator = E [ max (x – R, 0)] denominator = E [ max (R – x, 0)] Can be thought of as the quality of an investment on a return above a given level (threshold); “quality” is upside versus downside

9. Omega Graphs Omega analysis

10. How can Omega be used in Risk Management? Portfolio construction Risk monitoring Leverage setting tool Performance review Comparative Studies Robustness of portfolio Fine-tuning the tail Portfolio construction Risk monitoring Leverage setting tool Performance review Comparative Studies Robustness of portfolio Fine-tuning the tail

11. Elements of Risk Market Risk Credit Risk ”Credit Risk arises from the simple fact that there are an infinite number of people who wish to borrow money, but only a finite number of people capable of paying it back.” - Nobel Laureate Joseph Stiglitz Operational Risk Market Risk Credit Risk ”Credit Risk arises from the simple fact that there are an infinite number of people who wish to borrow money, but only a finite number of people capable of paying it back.” - Nobel Laureate Joseph Stiglitz Operational Risk

12. Elements of Risk Liquidity Risk Geo-Political Risk Model Risk Leverage - upping the stakes Liquidity Risk Geo-Political Risk Model Risk Leverage - upping the stakes

13. Aftermath Quantitative tools to be used qualitatively (not auto-pilot) Derivatives to hedge specific exposures Be on top of the capital markets Quantitative tools to be used qualitatively (not auto-pilot) Derivatives to hedge specific exposures Be on top of the capital markets

14. Aftermath Don’t fall for the pretty pictures! Lots of phony stuff out there. Don’t follow the flock! Be tough! (how has it helped in actual investment actions? has the tool been vetted?) Integrity Act in the light of intelligence, guided by experience. Don’t fall for the pretty pictures! Lots of phony stuff out there. Don’t follow the flock! Be tough! (how has it helped in actual investment actions? has the tool been vetted?) Integrity Act in the light of intelligence, guided by experience.

15. Good Risk Management is Alpha A good offence is better with a strong defence... Every good trading strategy is better with proper risk management! Guy Lafleur!! A good offence is better with a strong defence... Every good trading strategy is better with proper risk management! Guy Lafleur!!

16. Acknowledgments Denis Taillefer, Mx Christiane Lavallée, Mx James Vandenberg, apostrophe.ca Gunter Meissner, Derivatives Software / HPU Oliver King, Harvard University Nipa Banerjee, CIDA

17. References “ The Jungles of Randomness - A Mathematical Safari” - Ivars Peterson, (Wiley, 1998) “MFA’s 2005 Sound Practices for Hedge Fund Managers” - Managed Funds Association, August 2nd 2005 (www.mfainfo.org) Managing Financial Risk - Guide to Derivative Products, Financial Engineering, and Value Maximization - Charles Smithson (McGraw-Hill, 1998) “Credit Derivatives” - Gunter Meissner (Blackwell, 2005) “Inconsistency and Interest Rate Model Risk” - Anthony Di Silvestro (McMaster, 2004)