2005 Canadian Annual Derivatives Conference
August 17 - 20, 2005 Québec, Canada.
Ranjan Bhaduri , BSc (Hons), MBA, MMath, PhD.
Overview
•The Mathematics of Risk
•Portfolio Risk
•Elements of Risk
•Aftermath
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)
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
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
Mathematical Definition of Omega
Mathematical Definition of Omega
•Where F is the cumulative distribution of returns, and r is the threshold chosen by the investor.
•Where F is the cumulative distribution of returns, and r is the threshold chosen by the investor.
Omega - the Finance IntuitionOmega - 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
Omega GraphsOmega Graphs
Omega analysisOmega analysis
How can Omega be used in Risk Management?
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
Elements of RiskElements 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
Elements of RiskElements of Risk
Liquidity Risk
Geo-Political Risk
Model Risk
Leverage - upping the stakes
Liquidity Risk
Geo-Political Risk
Model Risk
Leverage - upping the stakes
AftermathAftermath
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
AftermathAftermath Don’t fall for the pretty pictures! Lots of phonystuff out there. Don’t follow the flock!
Be tough! (how has it helped in actual investmentactions? has the tool been vetted?)
Integrity
Act in the light of intelligence, guided by experience.
Don’t fall for the pretty pictures! Lots of phonystuff out there. Don’t follow the flock!
Be tough! (how has it helped in actual investmentactions? has the tool been vetted?)
Integrity
Act in the light of intelligence, guided by experience.
Good Risk Managementis Alpha
Good Risk Managementis 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!!
AcknowledgmentsAcknowledgmentsDenis Taillefer, Mx
Christiane Lavallée, Mx
James Vandenberg, apostrophe.ca
Gunter Meissner, Derivatives Software / HPU
Oliver King, Harvard University
Nipa Banerjee, CIDA
ReferencesReferences“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)