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)