Musings of a Mathematician About the Hedge Fund Space

1. Musings of a Mathematician About the Hedge Fund Space California State University Dominguez Hills Mathematics Colloquium By: RanjanBhaduriPhD CFA CAIA M.Math MBA BSc. (Honours) Chief Research Officer at Sigma Analysis and Management October 22nd, 2014.

2. 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

3. CONTENTS I would have written a shorter letter, but I did not have the time. - Blaise Pascal • 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 RanjanBhaduri • Contact details Sigma Analysis & Management Ltd. www.sigmanalysis.com

4. 1. INTRODUCTION INTRODUCTION Sigma Analysis & Management Ltd. www.sigmanalysis.com

5. 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 • Acryptanalyst 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-mathematicians-life-of-ferocious-curiosity.html?_r=0 • Founder, CEO of Renaissance Technologies (investment firm), Medallian hedge fund (1982). Sigma Analysis & Management Ltd. www.sigmanalysis.com

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

7. 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) Sigma Analysis & Management Ltd. www.sigmanalysis.com

8. 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

9. 2. The Mathematics of Liquidity THE MATHEMATICS OF LIQUIDITY Sigma Analysis & Management Ltd. www.sigmanalysis.com

10. 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

11. 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

12. 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) Sigma Analysis & Management Ltd. www.sigmanalysis.com

13. Solution Template to Balls in the Hat Game Sigma Analysis & Management Ltd. www.sigmanalysis.com

14. Hotel California Hotel California– “You can check out anytime, but you can never leave…” Sigma Analysis & Management Ltd. www.sigmanalysis.com

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

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

17. Model Risk – No Model is Perfect “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. Know Model Risk Sigma Analysis & Management Ltd. www.sigmanalysis.com

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

19. 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? Sigma Analysis & Management Ltd. www.sigmanalysis.com

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

21. LIQUIDITY DURATION Sigma Analysis & Management Ltd. www.sigmanalysis.com

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

23. 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) Sigma Analysis & Management Ltd. www.sigmanalysis.com

24. Blending Quant Equity Hedge Funds and CTAs is efficient Portfolio Construction Technique: Overlay/Underlay Alternatives Blend“Tradition is a guide, not a jailer.” – W. Somerset Maugham 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 Sigma Analysis & Management Ltd. www.sigmanalysis.com

25. LIQUIDITY IS THE FIRST LINE OF DEFENSE “Liquidity is the first line of Defense.” - Daniel MacDonald, CFA, Portfolio Manager, Alternative InvestmentsOntario Teachers Pension Plan Sigma Analysis & Management Ltd. www.sigmanalysis.com

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

27. 3. The Omega Function THE OMEGA FUNCTION Sigma Analysis & Management Ltd. www.sigmanalysis.com

28. 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!

29. 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)

30. “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.

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

32. 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.

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

34. 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

35. Mathematical Proof that Omega at the Mean is One

36. Mathematical Proof that Omega at the Mean is One

37. Mathematical Proof that Omega at the Mean is One

38. Mathematical Proof that Omega at the Mean is One

39. Mathematical Proof that Omega at the Mean is One

40. Omega Graphs – Geometry of Risk?

41. 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)].

42. 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?

43. 4. Assorted Remarks about Mathematical Techniques in Hedge Funds Assorted Remarks about Mathematical Techniques in Hedge Funds Sigma Analysis & Management Ltd. www.sigmanalysis.com

44. Nuggets of Wisdom … No Model is Perfect … understand the limitations, input and output – be able to explain it 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. 43 102309CA

45. 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!

46. Some Nuggets of Insights – The Viciousness of Percentages • 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.

47. The Gretzky Rule: Don’t Chase Returns!

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

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

50. 5. Some Nuggets of Wisdom – Success in finance industry / business world Some Nuggets of Wisdom – Success in finance industry / business world Sigma Analysis & Management Ltd. www.sigmanalysis.com