Musings of a Mathematician About the Hedge Fund Space

M U S I N G S O F A M AT H E M AT I C I A N A B O U T T H E H E D G E F U N D S PA C E

O C T O B E R 2 2 N D , 2 0 1 4 .

California State University Dominguez Hills Mathematics Colloquium

By: Ranjan Bhaduri PhD CFA CAIA M.Math MBA BSc. (Honours) Chief Research Officer at Sigma Analysis and Management

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Abstract

The hedge fund space has grown into a multi-trillion dollar business, and there are several quantitative and systematic hedge funds in existence. In addition, certain mathematical techniques are invoked in the hedge fund industry. This talk gives some insights about the mathematics utilized in the hedge fund world. In addition, it gives some nuggets of wisdom to students (both undergraduate and graduate) looking to have success in the business and finance world.

Sigma Analysis & Management Ltd. www.sigmanalysis.com

• 1. Introduction– Jim Simons, Renaissance Technologies– Quantitative Trading Strategies / Mathematical Techniques– Some misconceptions about hedge funds

• 2. The Mathematics of Liquidity

• 3. The Omega Function

• 4. Assorted Remarks about Mathematical Techniques in Hedge Funds

• 5. Some Nuggets of Wisdom – Success in industry / business world

• Appendix– Who is Sigma Analysis & Management?– Bio of Ranjan Bhaduri– Contact details

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CONTENTS

I would have written a shorter letter, but I did not have the time. - Blaise Pascal


http://www.sigmanalysis.com/

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1. INTRODUCTION

INTRODUCTION


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Jim Simons – Renaissance Technologies

• Jim Simons, PhD – Mathematician– World-class Mathematician (PhD at 23 from Berkeley, BSc from MIT)– Chairman & Professor of Mathematics at SUNY– A cryptanalyst at the Institute of Defense Analyses in Princeton– He received the American Mathematical Society Veblen Prize in

Geometry in 1975– Chern-Simons Invariants applications in Theoretical Physics– http://www.nytimes.com/2014/07/08/science/a-billionaire-mathem

aticians-life-of-ferocious-curiosity.html?_r=0

– Founder, CEO of Renaissance Technologies (investment firm), Medallian hedge fund (1982).


http://www.nytimes.com/2014/07/08/science/a-billionaire-mathematicians-life-of-ferocious-curiosity.html?_r=0



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MATHEMATICAL TECHNIQUES / TRADING STRATEGIES

- Systematic Trading Strategies- Statistical Arbitrage- Short-term CTAs- Trend-Following CTAs- Quant Equity- Market-Neutral Equity- Most good hedge funds invoke some ideas from mathematical

reasoning, either in portfolio construction, risk management or trading methodology.


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Some Mathematical techniques used in the hedge fund world …

- Some Mathematical techniques- Differential Equations- Machine Learning- Kelly Criterion- Gambler’s Ruin- Probability Theory- Game Theory- Matrix Theory / Linear Algebra- Cryptography / Cyber-security- Statistical Pattern Recognition- Bioinformatics- Bayesian Statistics- Fourier Analysis- Numerical Methods- Monte Carlo Analysis- Regression- Statistical Distributions- Chaos Theory- Randomness- Fractals- Signal Processing- Statistical Analysis (Big Data)


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Some Basic Facts about Hedge Funds

• Many Sovereign Wealth Funds, Pensions, Endowments, and Foundations invest in Hedge Funds

– Corollary: Hedge Funds can have a positive impact for many people, and is does not just benefit the super-wealthy.

• Hedge Fund Managers are entrepreneurs, and in some sense, the “small business owners” in the financial landscape (compared to banks, governments, etc.)

• Hedge Fund Managers – meritocracy • Hedge Funds did not take any money during the bail-out of 2008-09.• Hedge Funds usually are not harmful to the environment• Hedge Funds help to create jobs and efficiency.Sigma Analysis & Management Ltd. www.sigmanalysis.com

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2. The Mathematics of Liquidity

THE MATHEMATICS OF LIQUIDITY


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BALLS IN THE HAT GAME

Balls in the Hat Game "Business is a game.“ - IBM founder Thomas J. Watson Consider the following game (Balls in the Hat Game)

– There’s a hat with 6 black balls and 4 white balls.

– At each turn you choose whether to draw out a single ball at random, without replacement.

– You gain $1 for each white ball drawn, and lose $1 for each black ball drawn.

– The game ends when you choose not to remove any further balls or when the hat is empty.

Would you want to play this game? Why?If someone were to play this game,

would you expect him/her to lose money?Sigma Analysis & Management Ltd. www.sigmanalysis.com

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The Value of Liquidity

Joint work with Dr. Niall Whelan, Scotia Capital .

Answer to the Balls in the Hat Game ….– Consider the hat of size six black balls and four white balls.

– Intuitively, one might think that it is not worth playing since there are more black balls than white balls.

– Surprisingly the expected value of this game is positive; equal to 1/15.

– Thus, it makes sense to play the game!

– WHY IS IT POSITIVE??

– The reason is that the ability of being able to stop at any time overcomes this imbalance of black balls to white. There is a value of the player's right to stop.

– Liquidity risk arises from not being able to pull one’s money out of an investment instantaneously at “fair” price

– Being able to stop playing the game at any time is analogous to liquidity Sigma Analysis & Management Ltd. www.sigmanalysis.com

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The Power of Liquidity


Hat with six black balls and four white balls has a positive expected value of +1/15

Power of liquidity (i.e. being able to stop playing any time) overcomes the negative imbalance of black balls to white balls

Whelan and Bhaduri (2008)

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Solution Template to Balls in the Hat Game


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Hotel California

Hotel California – “You can check out anytime, but you can never leave…”


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Statistical Analysis of Investment Choices

Liquidity – the Forgotten Dimension in statistical analysis?

Consider the following Scenario:

– Hedge Fund A has a 2-year Lock-up, annual redemptions, and trades in illiquid instruments (distressed debt, structured credit, OTC derivatives)

– Hedge Fund B has no Lock-up, monthly redemptions, and trades in illiquid instruments

– Hedge Fund C, which is quantitative, has no Lock-up, monthly redemptions, and trades in liquid instruments

– All three hedge funds have a 5-year track record.


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Statistical Analysis of Investment Choices

Liquidity – the Forgotten Dimension in statistical analysis?

Consider the following Scenario:

– Is it fair just to compare the statistics (return, volatility, skew, kurtosis, omega, Sharpe, etc.) and the risk factors of these three hedge funds?

• If so, then liquidity is getting a value of zero. (i.e. the value of liquidity once again being underestimated

– But Hedge Fund C has the best liquidity and liquidity has a value! (Fund A has bad liquidity, Fund B has a liquidity mismatch which is a risk)

– Mistake: Liquidity ignored in statistical analysis of investment decisions

– Liquidity Risk is a composition of both how onerous the lock-up & redemption terms are as well the volume & complexity of the underlying instruments that it trades


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Model Risk – No Model is Perfect

Know Model Risk


“Nothing at MIT had ever reminded me of my lab at home. I suddenly realized why Princeton was getting results. They were working with the instrument. They built the instrument; they knew where everything was, they knew how everything worked.”

- Late Nobel Laureate Richard Feynman on why Princeton’s cyclotron was getting better results than MIT’s.

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Connection Between Liquidity Risk and Model Risk

–Liquidity Risk and Model Risk are entangled

–In general, the less liquid the instruments that are traded, the MORE the hidden risk, and the more dangerous model risk becomes.

–Example: credit crisis and financial crisis of 2008

–Liquid Hedge Funds tend to have less hidden risks. (Exchange-Traded => no valuation/accounting issues, no smoothing)

“The market can stay irrational longer than you can stay solvent.” - John Maynard Keynes


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Liquidity Solutions

Liquidity and Portfolio Management

– Risk and Return are two sides of the same coin

– In portfolio construction, one MUST take liquidity of the underlying investments into consideration

– Liquidity buckets

– Liquid instruments tend to have less “hidden” risks

– Liquidity mismatches

– Liquidity vultures

– Liquidity derivatives?


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Liquidity Buckets and Liquidity Index

Liquidity Buckets furnishes a simple, yet useful way for portfolio managers to assess their portfolio with a liquidity lens.

1. List the investments in ascending order via liquidity

2. Partition the list into liquidity buckets

3. Calculate the average statistics over a common time interval for each of the liquidity buckets


Liquidity buckets applied to hedge funds in June 2004 – June 2007 (i.e. before the financial crisis) showed that hedge funds that were less liquid were not statistically better than those that were more liquid. (AIQ – Second Quarter 2008, Bhaduri & Art)

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LIQUIDITY DURATION


•

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Pattern of Derivatives

“A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.” ― G.H. Hardy, A Mathematician’s Apology

Derivatives – take a risk, isolate it, and redistribute it

– Equity Derivatives (equity risk)

– Foreign Exchange Derivatives (currency risk)

– Interest Rate Derivatives (interest rate risk)

– Credit Derivatives (credit risk)

– Weather Derivatives (weather risk)

– Real Estate Derivatives (real estate risk)

Next… – Liquidity Derivatives! (liquidity risk) … this relates somewhat to futurisation


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Why Quant Strategies are the Liquid Alpha strategies

• Statistical Arbitrage, Market-Neutral Equity, Quantitative Equity, Systematic CTAs (aka Managed Futures), and other Quantitative and Statistical strategies whose trading domain is contained in equities, futures, options, and FX – are all liquid.

• By trading liquid, exchange traded instruments (or deep FX), these managers are able to utilize data that is clean and true data (not marked in a way that is artificial).

• BIG Data, advanced computing power, all lends itself well to a lot of testing, developing, on both the alpha generation, as well as stress testing, scenario testing, and risk management.

• Hence, Quantitative strategies tend to have: Less model risk (since dealing with liquid instruments) More ability to invoke an impressive array of mathematical techniques More ability to do testing and development

• Do not mistake illiquidity for alpha! (illiquid strategies are fine – just make sure that you are being paid properly for them)


24Sigma Analysis & Management Ltd. www.sigmanalysis.com

Portfolio Construction Technique: Overlay/Underlay Alternatives Blend“Tradition is a guide, not a jailer.” – W. Somerset Maugham

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Cash Allocation to HF

Sweep for the CTA/HF blend. CTA Funding factor: 2

Cash Allocation to CTA

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Blending Quant Equity Hedge Funds and CTAs is efficient

Sources: Portfolio Construction Technique: Overlay/Underlay Alternatives Blend, Bhaduri & Lobachevskiy, Alternative Investment Quotient, Sep 2011 Kat, Harry. "Managed Futures and Hedge Funds: A Match Made in Heaven“, 2004

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LIQUIDITY IS THE FIRST LINE OF DEFENSE

“Liquidity is the first line of Defense.”- Daniel MacDonald, CFA, Portfolio Manager, Alternative Investments

Ontario Teachers Pension Plan


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LIQUIDITY MATTERS

– Dangerous to trust one’s “intuition” (i.e. laziness) with regards to the value of liquidity

– Behavioral Finance – easy to underestimate the value of liquidity

– Mathematical subtleties in liquidity

– In illiquid investments, model risk gets magnified. Liquid hedge funds have less hidden risks.

– In liquid hedge funds, there is more data (by definition), and the data is more meaningful

– Quant strategies dovetail well with liquid instruments (more data, better quality data)

– Blending quantitative equity hedge funds with CTAs in an overlay/underlay strategy is an effective portfolio management technique (as always, rigorous due diligence needed)

– Proper and prudent portfolio management and risk management gives REAL attention to liquidity


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3. The Omega Function

THE OMEGA FUNCTION


The Need for a better Performance Measure

• Do not ignore the Non-Normality!

• Alternative investments (hedge funds, private equity, commodities, and real estate) typically

• have non-normal distributions. • Sharpe Ratio = ( µ - rf) / σ

• Problems: • Look at its formula – the Sharpe Ratio only uses mean & variance. This approach totally• discards skew, kurtosis, and all of the other higher statistical moments! Thus, it does not • capture all of the risk-reward features unless the distribution is normal.

• WHY RELY ON A TECHNIQUE THAT RESTS ON AN ASSUMPTION THAT WE KNOW TO• BE FALSE?

• NO distinction between upside volatility and downside volatility!

Information Ratio - cousin of the Sharpe Ratio Not much info!

• If the information ratio is higher does that mean it is better?

• Fund A beats benchmark by 0.5% each month.

• Fund B beats benchmark by 0.8% in half of the months and 1% in

• half of the months.

• IF all else equal, which Fund do you prefer?

• But... Fund A has the higher information ratio.

Information Ratio = (AnnRtn(r1, ...,rn) - AnnRtn(s1, ...,sn)) / AnnStdDev(e1, ...,en)

“Modern” Portfolio TheoryUsing cutting-edge techniques from the 1950s

Why not utilize the computing power available today?

– “Modern” Portfolio Theory using Markowitz mean-variance optimization

(1952) was based on computing power available in the 1950s.

– It does not distinguish between upside and downside volatility

– It does not take into account skew, kurtosis, or any of the higher moments (hence

the name “mean-variance optimization)

•

Oversimplifies the statistical distribution, does not take into account fat tails or skew, and does not distinguish between upside volatility or downside volatility.

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• Markowitz - Essentially a re-run of the problems with the Sharpe Ratio

• The Omega Function does NOT suffer from any of these problems!

• Applying better statistical methods can give a competitive edge. (Moneyball)

“Tradition is a guide and not a jailer.” W. Somerset Maugham


What is the Omega Function

– Invented by mathematicians Keating & Shadwick in 2002, it can be thought of as the quality of an investment on a return above a given level (threshold)

– Essentially a ranking function that captures return, variance, skew, kurtosis, and of the higher statistical moments – without penalizing for upside volatility. (i.e. all the higher moments are encoded in the Omega function)

– Has been referred to as “a sharper Sharpe” – Omega does not reward smoothing (unlike Sharpe)

• An Omega value of less than 1 implies that the quality of the investment is low with respect to the selected threshold, and that the selected threshold is higher than the mean of the investment’s return series.

• An Omega value of greater than 1 implies that the selected threshold is less than the mean of the investment’s return series.

• An Omega value equal to 1 implies that the selected threshold is equal to the mean of the investment return series.

Mathematical Definition of Omega

• Where F is the cumulative • distribution of returns, and r is the • threshold chosen by the investor.

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

– Kazemi, Schneeweis, and Gupta proved that the mathematical definition of Omega is equivalent to the finance definition above.

– Math-Finance Duality

Mathematical Proof that Omega at the Mean is One

Omega Graphs – Geometry of Risk?

Using Sharpe Ratio leaves Investor vulnerable to smoothing by Illiquid Hedge Funds

• Let σReported = λ*σTrue , where 0 < λ ≤ 1 (if λ=1 then no smoothing is being done)

• • Then the SharpeReported = (E(r) – rf)/ σReported = (E(r) – rf)/ λ*σTrue = (1/ λ) * SharpeTrue

• • Thus, the reported Sharpe increases by a factor of (1/ λ), which can be fairly substantial in• some cases.

• The effect of smoothing on an Omega function is mixed. Smoothing will increase• the Omega value for lower thresholds, but decrease the Omega value for higher

thresholds.• This is intuitive from the mathematical definition of Omega. Essentially the Omega graph

will• have a higher y-intercept but will have a steeper slope, and thus a lower robustness

coefficient• [e(dΩ /dr)].

How Can the Omega Function and Omega Graphs be used?

– Performance Review

– Peer Analysis

– Comparative Analysis

– Quantitative Due Diligence

– Risk Monitoring

– Quantitative Leverage-setting Tool

– Robustness of Portfolio = e(dΩ /dr)

– Fine-tuning the tail

– Portfolio optimization

– Three-dimensional Omega – the geometry of risk?

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4. Assorted Remarks about Mathematical Techniques in Hedge Funds

Assorted Remarks about Mathematical Techniques in Hedge Funds


Know Model Risk

“Nothing at MIT had ever reminded me of my lab at home. I suddenly realized why Princeton was getting results. They were working with the instrument. They built the instrument; they knew where everything was, they knew how everything worked.”

- Late Nobel Laureate Richard Feynman on why Princeton’s cyclotron was getting better results than MIT’s.

44102309CA

Nuggets of Wisdom … No Model is Perfect … understand the limitations, input and output – be able to explain it

Model Risk – Anscombe’s Quartet

Don’t regress too much!Each data set has the same Mean, Variance, t-stat, etc.,and leads to the same regression line!

Some Nuggets of Insights – The Viciousness of Percentages

Percent Loss

Percent Gain Needed to Get Back to Even

5% 5.26%10% 11.11%20% 25%25% 33.33%30% 42.86%35% 53.85%40% 66.67%45% 81.82%50% 100%55% 122.22%60% 150%65% 185.71%70% 233.33%75% 300%80% 400%85% 566.67%90% 900%95% 1900%96% 2400%97% 3233.33%98% 4900%99% 9900%

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• This table helps to demonstrate the importance of cash preservation, and the risk involved in trying to make an investment that “swings for the fences”.

• The formula used to generate the above is that a loss of k, requires a gain of k/(1-k).

• Living by the sword can lead to dying by the sword. It becomes increasingly difficult of recovering from a large loss.

• The above table does not take the time value of money into consideration (i.e. that a dollar two years from now, is worth less than a dollar today), and consequently is conservative in its assessment of the increase required in order to come back.

The Gretzky Rule: Don’t Chase Returns!

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Black-Scholes-Merton Equation

• S is the price of the stock• V (S, t)is the price of a derivative as a function of time and stock price.• σ the standard deviation of the stock's returns; this is the square root of

the quadratic variation of the stock's log price process.• r the annualized risk free interest rate, continuously compounded.


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Cryptography

• Focused Research• Centre for Applied Cryptographic Research (CACR): A joint project between the

University of Waterloo, the Federal Government of Canada, and a number of corporations. (www.cacr.math.uwaterloo.ca)

• University Cryptography Departments• University of Waterloo: CrySP (crysp.uwaterloo.ca)• McGill University: CQIL (crypto.cs.mcgill.ca)• University of Calgary: CISaC (cisac.ucalgary.ca)• Columbia University: CryptoLab (www.cs.columbia.edu/crypto)

• Mathematicians (Number Theory, Computer Science, Cryptography)• Alex Stanoyevitch, California State University Dominguez Hills• Claude Levesque, Université Laval• Neal Koblitz, University of Washington• Alfred Menezes, University of Waterloo


http://www.cacr.math.uwaterloo.ca/

http://crysp.uwaterloo.ca/

http://crypto.cs.mcgill.ca/

http://cisac.ucalgary.ca/

http://www.cs.columbia.edu/crypto

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5. Some Nuggets of Wisdom – Success in finance industry / business world

Some Nuggets of Wisdom – Success in finance industry / business world


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Some Nuggets of Wisdom for the Business / Finance World

1. (M) In the math world, one has to be precise, and not much BS, since one has to prove any statement (a proof is forever). (B/F) In the Business world / finance world – people (more frequently) make incorrect statements, and sometimes do so with a lot of confidence

2. (M) In the math world, substance trumps everything. (B /F) In the business / finance world, style matters (substance is still the most important, but style is important).

3. (M) In the math world, intellectual curiosity is a very key ingredient (B/F) In the business/finance world, while this is still a very useful trait to have, one needs to remember that most people only care about two things:

– How much money does it make me?– How does it make my life easier?


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• MOST IMPORTANT– Good integrity & ethics– Right thing to do, plus good for business (where is Enron today?)– Your reputation is important

• Work hard (competitive world)– Care about the quality of your work– Be reliable and responsible– Be dependable

• Communicate well (TA / Teach / Tutor)• Someone above you should be a supporter• Do not do “bad” politics• Do not be afraid to rap a little• Network / keep in touch• Keep learningSigma Analysis & Management Ltd. www.sigmanalysis.com

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• Surviving Success (it’s one thing to have success, it’s another to survive it)– “You’re only as good as your last shift” – old hockey axiom– Go into the corners (or in the slot)– Respect the markets (no crystal ball – Amaranth example)

• Research is absolutely critical (firms that do not invest sincerely in research, don’t or won’t last)

• Diversification – the only free lunch in finance (if you don’t invest in hedge

funds, then you should be forced to justify why in writing)

• The Human Element

• Act in the light of intelligence, guided by experience with reason.Sigma Analysis & Management Ltd. www.sigmanalysis.com

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Acknowledgements

THANKS!

• Dr. Alex Stanoyevitch, California State University Dominguez Hills, Department of Mathematics

• Wai Yan Pong, California State University Dominguez Hills, Department of Mathematics

• All of you!


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Appendix – Who is Sigma Analysis?

• Sigma Analysis & Management Ltd. is a Toronto-based firm with operational excellence, world-class hedge fund research with an elite quant group, that delivers high-touch, value-add customized service in the hedge fund space.

• Sigma Analysis was founded in 1999 by Dave Rudd and Dr. Luis Seco to help service institutional investors in its hedge fund portfolio construction and risk monitoring. Dr. Luis Seco has PhD in Mathematics from Princeton University and is a Professor of Mathematics at the University of Toronto.

• Professionals on Sigma Analysis have global experience from well known firms including: Goldman Sachs, Morgan Stanley, State Street, Citco Fund Services, Ernst & Young, and the Montreal Exchange.

• Sigma Analysis services include:

– Customized Managed Account & Transparency services operations, technology, risk monitoring, smart risk analytics, reporting, & research

– Customized Research & Advisory services due diligence support, hedge fund origination, asset allocation analysis, structural alpha analysis, portfolio construction,

quantitative analytics, and mathematical and statistical analysis, & commissioned research studies on specialized topics in hedge funds, portfolio management, and risk management

– Customized Education & Training services Educational workshops and/or Customized Seminars on a variety of topics (including due diligence, hedge fund strategies,

structural alpha, portfolio construction, risk management, managed accounts, operations, quantitative finance, risk management, governance, ethics & integrity); Training of Analysts (due diligence, hedge fund operations, risk management, & hedge fund research).

• Sigma Analysis is a pioneer in hedge fund research, smart risk analytics, transparency services, and managed accounts.



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Appendix - Sigma Analysis - Transparency

• Transparency requires Technology – Sigma Analysis has built proprietary software which helps to

empower the client– Sigma Analysis continues to invest in research & technology– Protect against fraud, strategy drift, style drift, concentration risk– Gain structural alpha

• Sigma Analysis will furnish truly customized solution

• Sigma Analysis is client-focused with excellent service

Sigma Analysis & Management Ltd. www.sigmanalysis.comF O R E D U C A T I O N A L A N D D I S C U S S I O N P U R P O S E S O N L Y


Bio: Dr. Ranjan Bhaduri

Dr. Ranjan Bhaduri is the Chief Research Officer. Prior to joining Sigma in 2014, he was the Chief Research Officer and the Head of Product Development at AlphaMetrix Alternative Investment Advisors. At AlphaMetrix, Dr. Bhaduri designed and led an institutional due diligence and research program. Dr. Bhaduri worked closely with institutional clients on portfolio matters.

Prior to joining AlphaMetrix, he was a Vice President and on an Investment Committee at Morgan Stanley where he conducted due diligence and helped design customized portfolios of Alternatives. Earlier, he was at a Canadian Fund of Funds, and at a multi-billion dollar capital management firm where he was involved in all aspects of its fund of hedge funds and structured finance business. He has also worked with two major Canadian investment banks in the Financial Strategy Consulting Group and in Global Risk Management & Control, respectively.

Dr. Bhaduri has held advisory roles at the East-West Center, a leading think tank on the Asia-Pacific region, and at ClassMouse, an early stage software company. He has taught finance and mathematics at several universities and lectured on Derivatives for the Montreal Exchange. Dr. Bhaduri has published papers on, and been invited to speak worldwide regarding hedge fund issues, and advanced portfolio and risk management techniques.

Dr. Bhaduri has a M.Math in Combinatorics & Optimization from the University of Waterloo (1992), a PhD in Mathematics from the University of Hawaii (1999) and an MBA from McMaster University (2002). Dr. Bhaduri holds both the CFA and CAIA charters. He is a member of the American Mathematical Society, the Mathematical Association of America, the Toronto CFA Society, and the Global Association of Risk Professionals (GARP). Dr. Bhaduri has previously served as a member of the All About Alpha Editorial Board, and as a CAIA Chicago Chapter Executive.

Contact Info.

Ranjan Bhaduri PhD, CFA, CAIA, MMath, MBA, BSc (Honours) Chief Research Officer

Sigma Analysis & Management, Ltd. MaRS Discovery District 101 College Street Suite 345 Toronto ON M5G1L7 Canada.

Email: [email protected] Phone: +1-416-716-0341

Sigma Analysis & Management Ltd. www.sigmanalysis.com 58

mailto:[email protected]


Notice of Confidentiality

This document has been prepared by Sigma Analysis & Management Ltd, ("SIGMA") and is being made available to you on a confidential basis for informational purposes only. By receiving this document you acknowledge and agree that (i) you will not disclose or distribute this information, in whole or in part, to any third-party without the prior written approval of SIGMA; and (ii) such information is being provided to you for informational purposes only as a current or potential investor/client quali fied to invest in hedge funds or other alternative investments. This document is a description of potential services and is not an advertisement nor is it intended for public use or further distribution. It is neither an offer to sell nor a solicitation of an offer to buy interests in any investment. This document is intended solely for the use of the person to whom it is provided. It may not be modified, reproduced, or further distributed without the prior written consent of SIGMA.

Sigma Analysis & Management Ltd. www.sigmanalysis.com 59


Musings of a Mathematician About the Hedge Fund Space

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