The seminar takes place in the Spring of 2024, Tuesdays and Thursdays, 7:40 pm — 8:55 pm.

Location: 312 Mathematics Building.
For directions, please see Directions to Campus and Morningside Campus Map.

Organizer: Lars Tyge Nielsen

Schedule of Presentations

Scroll down for the Schedule of Past Presentations.

Tuesday, April 9, 2024 – Thursday, April 25, 2024: Lecture Series by Ioannis Karatzas

Ioannis Karatzas studied at the Technical University of Athens and at Columbia University, where he is Professor of Mathematics and Statistics. He works and publishes in Probability Theory and Its Applications, including Stochastic Control, Mathematical Physics, Mathematical Economics, and Finance. He has co-authored with Steven Shreve the books “Brownian Motion and Stochastic Calculus” (1988), “Methods of Mathematical Finance” (1998); and with Constantinos Kardaras the monograph “Portfolio Theory and Arbitrage” (2021).

Abstract:

The goal of this series of six lectures is to introduce the Stochastic Theory of Portfolios, a rich and flexible framework for analyzing portfolio behavior and equity market structure; and to explore some of its applications to portfolio management and performance measurement.

The first lecture is intended for a general audience; very little background is assumed. The second and third lectures will introduce the basic notions and results of the Markowitz-Fernholz theory of portfolios. The fourth and fifth lecture will be of particular interest to portfolio managers who wish to learn new methods for measuring the impact of size on their equity portfolios. The final lecture will explore implications for the mathematical foundations of Finance.

Tuesday, April 16, 2024

Lecture 3: Functional Generation of Portfolios

Abstract:

Simple optimization problems. Portfolios generated by smooth concave functions. Examples: the equal-weighted and entropy-weighted portfolios.

Thursday, April 18, 2024

Lecture 4: Applications – Smart Beta

Abstract:

Notions of Arbitrage and Outperformance; “Smart Beta” weighting schemes. Examples.

Tuesday, April 23, 2024

Lecture 5: Applications – Size and the Distribution of Capital

Abstract:

Stability of markets; the size effect; rank-based portfolio selection. Market diversity, diversity-weighted portfolios. The distributional component of returns.

Thursday, April 25, 2024

Lecture 6: Implications for the Mathematical Theory of Finance

Abstract:

We survey briefly a theory of Finance based on a simple and intuitive absence-of-arbitrage principle. This posits that it should not be possible to fund a non-trivial liability, starting with initial capital arbitrarily close to zero. We discuss the various equivalent incarnations of this principle, and show that it leads to a theory powerful enough to deal with all fundamental questions of this field (hedging, valuation, portfolio optimization).

Past Presentations

Tuesday, January 16, 2024

Title: Some of the latest thoughts on credit portfolio models.

Speaker
: David Xianglin Li, Columbia University

David is currently a professor at Shanghai Advanced Institute of Finance, Shanghai Jiaotong University

Abstract:

We will present an overview of credit portfolio modeling, the origin of the problem, two different approaches: copula function and reduced form, and their comparison. We will investigate the theoretical approach to credit portfolio pricing models via replication and equilibrium approaches. Using the pricing of the first -to-default contract we argue that the credit market is usually not complete as we do not have enough trading instruments to replicate the first-to-default contract for its default risk. Finally, we present an equilibrium approach based on Buhlmann’s model and show a risk-neutralized Gaussian copula credit portfolio model.

Thursday, January 18, 2024

Title: Deep Learning VS Reinforcement Learning for Portfolio Management

Speaker:
Alexander Fleiss, Rebellion Research

Alexander Fleiss is CEO of Think Tank & Robo Advisor Rebellion Research. Rebellion Research received over 1 million readers for November, 2023. Mr. Fleiss has 20 published research papers and is an Editor of the Journal of Financial Data Science, an Advisory Board Member of Cornell Financial Engineering, a Board Member of The New York Food Truck Association & FLYJETS. Mr. Fleiss holds a Bachelor’s Degree from Amherst College.

Abstract:

Tuesday, January 23, 2024

Title: Thesis Driven Digital Asset Investing

Speaker:
Kelly Ye, Head of Research and Portfolio Manager, Decentral Park Capital

Abstract:

Liquid venture investing in crypto and investment themes for 2024

Thursday, January 25, 2024

Title: Navigating the Currents: Key Aspects of Oil and Gas Trading

Speaker:
Samvel Gevorkyan

Samvel Gevorkyan is a quantitative researcher at Freepoint Commodities. He is managing a systematic book, trading crude oil and natural gas and has been building algorithmic strategies and analytics for the past 6 years

Abstract:

We are going to give a brief introduction to commodities markets, walk through the main components of Supply and Demand, and consider some aspects of systematic strategy design in order to extract risk premia based on physical as well as financial drivers. We will attempt to look at energy futures markets through the eyes of discretionary macro portfolio managers, CTA funds, and relative value strategists to gain insights into the movement of capital.

Tuesday, January 30, 2024

Title: Using AI to Understand Wallet Profiles in Cryptoverse

Speaker: Irene Aldridge, Able Markets

Irene Aldridge is an academic, entrepreneur, and speaker. She teaches at Cornell Financial Engineering Program, and is the founder and CEO of two companies: AbleMarkets AI, a financial technology company, and AbleBlox, a blockchain company. Aldridge lives in New York. She is an advocate for innovation and technology, and has been featured in many publications, including the Wall Street Journal, Forbes Magazine, and Bloomberg News. She has given talks at numerous conferences and universities around the world. She is also the author of many research articles and several books, including High-Frequency Trading: A Practical Guide Algorithmic Strategies and Trading Systems, Real-Time Risk (with Steve Krawciw, 2017) and Big Data Science in Finance (with Marco Avellaneda, 2021). Aldridge is passionate about promoting the development of innovative technologies that can benefit society.

Abstract:

Security in Crypto is one of the key topics for the market participants as well as regulators. Using advances in Artificial Intelligence, we show how transacting wallets in Cryptoverse can be identified with key attributes in near-real-time. The results offer an advance toward seamless and transparent real-time global markets with agent screening.

Thursday, February 1, 2024

Title: Dynamic Leverage: a Risk Measure Incorporating Time Horizon and Illiquidity

Speaker: Mikhail Smirnov, Columbia University

Abstract:

We introduce a Dynamic Leverage which is a VAR extending risk measure taking into account the investment time horizon and illiquidity. We introduce this measure, and give a formula for it. Dynamic leverage among other things can be used in a single hedge fund setting and in a fund of funds setting. Dynamic leverage depends on the level of fund volatility, time horizon and distance in terms of NAV to a pre-defined critical liquidation level for a fund. Thus dynamic leverage incorporates the minimal holding time of investment and the risk associated with it. This reflects the fact that investments in hedge funds have lockups and limited redemption frequency. We present a variety of models for dynamic leverage to illustrate some of the differing structural features of hedge funds. For an investment fund with dynamically controlled risk exposure and certain risk inertia, we demonstrate the existence of a critical NAV level below which the efficacy of de-leveraging is compromised. We give examples of historical critical liquidation levels for hedge funds following different strategies. Using the Dynamic Leverage risk measure we propose a modification of Black-Jones-Perold portfolio insurance.

Tuesday, February 6, 2024

Title: Strategic Asset Allocation with Private Assets

Speaker: Xinyang Li, BlackRock

Xinyang Li is a quantitative researcher at BlackRock, focusing on equity and cross-asset allocation. She graduated from Columbia University’s MAFN program in 2017 before pursuing PhD in Mathematical Finance at Boston University.

Abstract:

Thursday, February 8, 2024

Title: The current state of Bitcoin investments

Speaker: CK Zheng, ZX Squared Capital

Dr. CK Zheng is a Co-Founder and CIO of ZX Squared Capital, a crypto option hedge fund since July 2021. Dr. Zheng has over 30 years of experiences on Wall Street and holds a Ph.D. in Finance from University of Chicago, with a specialization in derivatives pricing and risk management. Before founding ZX Squared, Dr. Zheng was previously Managing Director and Global Head of Risk and Valuation at Credit Suisse for over 17 years after a 5-year career at Morgan Stanley as Executive Director. Before that, he actively traded options at Bank of America and Susquhanna Investment Group. Dr. Zheng’s market insights were widely featured on main media such as CNBC, Forbes, CoinDesk, The Block, Blockworks and others.

Abstract:

We will discuss the current state of Bitcoin investments, why Bitcoin is here to stay, the impacts of the recent SEC approval of Bitcoin spot ETFs, the bitcoin options market, the crypto custodian services.

Tuesday, February 13, 2024

Title: Volatility Surface Construction

Speaker: Bryan J. Liang, Bloomberg L.P. and Columbia University

Bryan Liang is a senior quant researcher at Bloomberg L.P. and adjunct professor at Columbia University. He joined the Bloomberg quant research team in 2011 and has been working extensively on various aspects of derivatives modelling, including pricing, hedging, risk management, structuring, market making, trading strategies, parallel computing and applications of machine learning techniques in derivatives. Prior to joining Bloomberg, He worked for derivatives analysis group at Goldman Sachs, as an interest rate derivatives quant. Bryan received his Ph.D. in mathematics from University of Michigan and was a faculty member at Northwestern University and UC Davis before he moved to finance.

Abstract:

The implied volatility surface is a basic building block for nearly all tasks in derivatives business. An accurate and robust calibration of implied vol surface is of fundamental importance to market makers, traders, quants and risk managers. In this talk we present a survey of common algorithms and issues that arise from the construction of implied and local volatility surfaces, including some of the most recent developments and the author’s own research. We cover the fitting of one maturity, multiple maturities while preserving convex order, arbitrage-free maturity interpolations, and global calibration by neural networks via Monte Carlo simulation and PDE.

Thursday, February 15, 2024

Title: Introduction into MBS market, MBS instruments, Structured MBS, and MBS Derivatives

Speaker: Ilya Zhokhov

Ilya Zhokhov is currently Managing Director in ALM portfolio management group at PNC Bank focusing on mortgage servicing rights portfolio management. Prior to this role, Ilya was Executive Director at JP Morgan where he was revenue CFO in mortgage bank most recently and head of MSR risk management in Chief Investment Office prior to that. Before JP Morgan Ilya spent a number of years at Blackrock and was responsible for managing relationships with banks and financial institutions. His team provided risk management and strategic advisory services to many of the nation’s largest banks and financial institutions.

Abstract:

Introduction and overview of MBS market, key MBS investors, the structure of MBS market and key investment risk associated with investing in MBS as well as overview of MBS modeling process and key assumptions.

Tuesday, February 20, 2024

Title: (Almost) Everything They Teach You About Quantitative Finance is Wrong! Here’s Why…

Speaker: Graham Giller, Giller Investments

Graham Giller is one of Wall Street’s original data scientists. Starting his career at Morgan Stanley in the UK, he was an early member of Peter Muller’s famous PDT group and went on to run his own investment firm, in addition to taking data science leadership positions at Bloomberg, JP Morgan and Deutsche Bank. He is the author of “Adventures in Financial Data Science,” which chronicles the research he has done over his career, and the “Essays on Trading Strategy,” which is focused on optimal trading theory. He currently leads a team that supports Bloomberg Intelligence in quantitative analytics and data science and, for the last two years, he has written a regular column in Wilmott Magazine.

Abstract:

Practitioners often don’t use the key results of quantitative finance in their work, which is because it fails to perform as promised. Quants often fixate on one of several solutions to explain the defects of canonical theory, and ignore the fact that most of the problems can be understood as caused by the inappropriate application of Normal distribution theory in a venue where it simply, empirically, does not apply. Instead of replacing conventional, but incorrect, theory with ad hoc procedures and rules, in this talk I will show how moving away from the Normal distribution, in a manner which is controlled and understandable, delivers practical results that explain the observed defects of canonical theory and replace it with procedures that justify some of the heuristics actually used by actual practitioners in the markets, thereby placing them on a sounder theoretical foundation than usually thought.

Thursday, February 22, 2024

Title:

Speakers: Peter Cai and Ritesh Bansal

Peter Cai is a member of the Risk Management Executive Committee, and the global head of Risk Data, Analytics, Reporting & Technology (DART). As the head of Risk Data, Analytics, Reporting & Technology, Peter oversees risk data aggregation, controls, reporting and analysis across all risk teams.

Prior to joining Citi, Peter was at Barclays where he oversaw risk in the global asset-liability profile and investments, responsible for the ALM/IRRBB framework, quantitative analytics and regulatory requirements. Prior to Barclays Peter was the CRO at Global Atlantic (formerly Goldman Sachs Reinsurance) and was a MD at Morgan Stanley in Enterprise Risk Management and Lehman Brothers in Global Credit Trading/Fixed Income.

Peter has a Ph.D. degree in Materials Science from Pennsylvania State University and a B.S. in Mathematics and Applied Mechanics from Fudan University in China.

Ritesh Bansal leads Solution Engineering, and is responsible for quant products, infrastructure and innovation for Data, Analytics, Reporting and Technology in Risk at Citi. Before Citi, Ritesh founded VerusAI which pioneered a ML platform for real estate valuation and analytics. During this time, he also served on the standards committee for the Appraisal Institute on automated valuation models. Prior to VerusAI, he was a Stat and Index Arb trader at Ronin Capital and Millennium Partners and began his career at LTCM. Ritesh received his B.S in Math and Computer Science from Carnegie Mellon University.

Abstract:

Peter will provide analysis and historical perspective on the recent turmoil in the Banking sector, and what we can learn from it. He will delve into how shortcomings of any of the elements that go into Risk Management – models, processes, risk culture and regulations – can lead to failures. Peter will draw lessons from the recent public failures in the banking sectors and how Risk Management needs to evolve to address them.

Tuesday, February 27, 2024

Title: The Cross-Section of Hedge Fund Returns

Speaker: Christos Koutsoyannis

Christos Koutsoyannis, PhD, is the Chief Investment Officer of Atlas Ridge Capital, an emerging quantitative hedge fund focusing on innovation in the areas of alternative data and scientific modeling. Christos is also an Adjunct Professor at NYU Courant. Christos is a senior portfolio manager with significant experience in the design and day–to–day management of very diverse quantitative long–short investment strategies, in equities and across asset classes. Prior to Atlas Ridge, Christos was a Portfolio Manager at Alliance Bernstein, a Senior Portfolio Manager at State Street Global Advisors, and the Deputy Head of Quant Research at Old Mutual Asset Managers (now Jupiter) in London. Christos sits on the Board of Directors of the Society of Quantitative Analysts (SQA) since 2013, was on the committees and Board of Directors of Inquire (UK) from 2005 – 2016, and is the Chair of Trinity in New York (Cambridge University alumni). Christos holds a B.A. Hons Economics from the University of Cambridge, UK, and a PhD in Financial Economics from Birckbeck College, University of London, UK.

Abstract:

The presentation will consist of two parts:

First, Christos will discuss current employment and recruitment trends in Hedge Funds.

Second, Christos will discuss the cross-section of Hedge Fund returns. Using a novel dataset, Christos will discuss hidden commonalities and the underlying drivers of hedge fund performance, across asset classes. He will discuss structural differences between HFs with larger vs smaller Assets Under Management (AUM). Finally, he will discuss a novel measure of crowding in HF investment strategies.

Thursday, February 29, 2024

Title: Quants, Quantum Mechanics and Quantos

Speaker: Luca Capriotti

Abstract:

Using the path-integral formalism we develop an accurate and easy-to-compute semi-analytical approximation to transition probabilities and Arrow-Debreu densities for arbitrary diffusions. We illustrate the accuracy of the method by presenting results for the Black-Karasinski model for which the proposed approximation provides remarkably accurate results, even in regimes of high volatility and for multi-year time horizons. The accuracy and the computational efficiency of the proposed approximation makes it a viable alternative to fully numerical schemes for a variety applications, ranging from maximum-likelihood estimation in econometrics to derivatives pricing.

Tuesday, March 5, 2024

Title: Skin in the Game: Risk Analysis of Central Counterparties

Speaker: Samim Ghamami, U.S. Securities and Exchange Commission

Samim Ghamami is currently an economist at the U.S. Securities and Exchange Commission, the Division of Economic and Risk Analysis (DERA), a senior researcher and an adjunct professor of finance at New York University, a senior researcher at UC Berkeley Center for Risk Management Research and the Department of Economics, and a senior advisor at SOFR Academy.

Ghamami has been a senior economist, a senior strategist, and a senior vice president at Goldman Sachs. He has also been an adjunct associate professor of economics at Columbia University. In 2019, Ghamami moved to the Financial Services Forum through Goldman Sachs, where he was the senior economist and managing director. Ghamami has also been an associate director and a senior economist at the U.S. Department of the Treasury, Office of Financial Research, and an economist at the Board of Governors of the Federal Reserve System.

Ghamami’s work has broadly focused on financial economics, quantitative finance, and on the interplay of finance and macroeconomics. His work on banking, asset management, risk management, economic policy, financial stability, financial regulation, and central clearing has been presented and discussed at central banks. He has been an advisor to the Bank for International Settlements and has worked as an expert with the Financial Stability Board on post-financial crisis reforms in 2016 and 2017. Ghamami also served on the National Science Foundation panel on Financial Mathematics in 2017 and 2018.

Abstract:

This paper introduces an incentive compatibility framework to analyze agency problems linked to central counterparty (CCP) risk management. Our framework, which is based on a modern approach to extreme value theory, is used to design CCP skin-in-the-game (SITG). We show that under inadequate SITG levels, members are more exposed to default losses than CCPs. The resulting risk management incentive distortions could be mitigated by using the proposed SITG formulations. Our analysis addresses investor-owned and member-owned CCPs, we also analyze multilayered and monolayer default waterfalls. Viewing the total size of SITG as the lower bound on CCP regulatory capital, the framework can be used to improve capital regulation of investor-owned and member-owned CCPs. We also demonstrate that bank capital rules for CCP exposures may underestimate risk. The broader central clearing mandate of U.S. Treasuries may take place under monolayer CCPs. These clearinghouses may need to allocate more of their own capital to the default waterfall.

Thursday, March 7, 2024

Title: Local Volatility

Speaker: David-Antoine Fournie

David-Antoine Fournie is Managing Director, Head of Equity Structured Products Trading at Bank of America. He previously worked at Morgan Stanley, Deutsche Bank and Deauville Capital Management. He is a graduate of Ecole Polytechnique and Ph.D. in Mathematics from Columbia University. His work included an extension of Ito’s formula to functional spaces.

Abstract:

We will explain the construction of the Local Volatility model which is the most important model in Equity Derivatives. We will examine relaxing assumptions like deterministic rates and deterministic dividends.

Tuesday, March 12, 2024 – SPRING BREAK
Thursday, March 14, 2024 – SPRING BREAK
Tuesday, March 19, 2024

Title: Portfolio Construction in Quant Trading: Theory, Practice and New Development

Speaker: Rui Yang, Verition Fund Management

Rui Yang is currently a quantitative researcher at Verition Fund Management, where he develops and manages systematic equity strategies. Before joining Verition, Rui was a founding member of Omnis Quantitative, a Millennium platform company. Prior to that, he worked as a fixed income quant at Nomura and Credit Suisse. He holds a Ph.D. in Industrial Engineering from the University of Illinois at Urbana-Champaign, specializing in portfolio optimization and a B.S. in Applied Mathematics from Peking University.

Abstract:

Thursday, March 21, 2024

Title: A Conversation with Nigol Koulajian

Speaker: Samvel Gevorkyan and Nigol Koulajian

Nigol Koulajian is the Founder and Chief Investment Officer of Quest. Mr. Koulajian founded Quest in March 2001 to pursue his passion for quantitative investment research and strategy development, which he has focused on from the beginning of his career in the early 1990’s. After lengthy research, Mr. Koulajian identified specific strategies using proprietary techniques that have been continuously enhanced over the past twenty-four years and became the basis for the growth of Quest. The firm, based in New York, manages $2.4 billion in assets and employs forty people as of March 2024.

Abstract:

In this discussion we delve into the intricacies of mid-frequency systematic trading strategies. The conversation will focus on the firm’s approach to investing with positive convexity, exploring how they capitalize on market trends across various asset classes. We will also examine the role of macroeconomic factors in shaping their investment decisions and how they navigate the ever-changing liquidity landscape. Nigol Koulajian will share insights into delivering alpha, risk management techniques, and the unique challenges and opportunities presented by the current market environment. Additionally, we will touch upon the evolution of the CTA industry, the impact of deep learning advances on systematic strategies, and the potential future directions for momentum style premium.

Tuesday, March 26, 2024

Title: An ML perspective on Munis

Speaker: Roberto Strepparava, CME Group

Roberto Strepparava is currently quant researcher at CME Group where he develops and manages risk systems and valuation for fixed income and rate (linear, nonlinear) products. Prior to that he was a fixed income quant researcher at JPMorgan, at Guggenheim Partners under Marcos Lopez De Prado and at Bloomberg LP in BVAL. Roberto holds a M.Sc. in Theoretical Physics from Milan University and a Ph.D. in Mathematics from Padua University, both on dynamical systems theory.

Abstract:

Munis are an asset class particularly amenable to data driven approaches, given the sheer size of the universe and their trading in baskets. We give results on Benchmark curves and Muni Comparables.

Thursday, March 28, 2024

Title: Topics in modeling options

Speaker: Thomas Feng, Graham Capital Management, L.P.

Tom Feng is Chief Investment Officer of Quantitative Strategies at Graham Capital Management, where he has worked for 15 years and is responsible for developing and managing systematic trading strategies in global macro asset classes. Prior to joining Graham, Tom was a quantitative portfolio manager at Fortress Investment Group and RBS Greenwich Capital, respectively. Tom started his career in quantitative finance at RBS Greenwich Capital, in research roles culminating in Head of Interest Rate and Mortgage Derivatives Research, before transitioning into quantitative investing where he has accumulated close to 25 years of experience.

Tom holds BS and MS degrees in Mathematics from Yale and a PhD in Mathematics from Princeton.

Abstract:

Option modeling comes up in different contexts in quantitative finance. Various standard methodologies include arbitrage-free models and equilibrium models that evolve future paths of asset prices, to time series models that extract information from historical price movements. We seek to combine features from various approaches to build a model that provides fair values across the entire term structure of volatility. We will use quantitative techniques including time series analysis, Monte Carlo simulation, variance reduction and neural networks.

Tuesday, April 2, 2024

Title: Introduction to Factor Investing

Speaker: Natalia Zvereva, JP Morgan Asset Management

Natalia Zvereva, Executive Director, is a portfolio manager in the Quantitative Solutions group at JP Morgan Asset Management, based in New York. Her primary responsibilities include portfolio management of strategic beta products and overseeing the implementation of quantitative investment strategies. An employee since 2011, prior to joining Quantitative Solutions team, Natalia was an Investment Director within Multi-Asset Solutions, overseeing the group’s investment portfolios, as well as spending over 5 years as a risk manager covering Strategic Beta Strategies and the Alternatives businesses. Previously, she worked in Credit Portfolio Solutions Derivatives and Derivatives risk management within the Investment Bank, where she performed quantitative analysis of cross-asset derivatives portfolios. Natalia is on the leadership team on Asset Management Solutions Women organization and mentorship program. Natalia holds a Master’s Degree in Financial Mathematics from Columbia University, BBA in Finance & Investments from Baruch College, and is a CAIA charterholder.

Abstract:

In this lecture, we will cover practical application of factor-based investing, drawing on real-world case studies, risk management, and industry trends. We will go over various types of factors used in factor models, gain insights into construction and evaluation of factor strategies, as well as discuss how factor-based investing can be used to enhance returns, manage risk, and achieve diversification objectives across different market conditions.

Thursday, April 4, 2024

Title: Exotic Equity Derivatives

Speaker: Faousi Batouche, BNP Paribas

Faousi Batouche is currently a Vice President since 2019 in equity derivatives structuring at BNP Paribas, mostly responsible for designing, pricing and pitching complex flow products to institutional clients. Prior to this role, Faousi was a market risk manager at Goldman Sachs covering different trading desks.

Faousi holds a master of engineering from Centrale Supelec and a Master of Arts from Columbia university (MAFN 2015).

Abstract:

We will be discussing about common equity derivatives products. Major banks provide a wide range of equity derivatives payoffs for institutional and retail clients. These products can be sophisticated and their risk management heavily relies on the use of Greeks of 1st and 2nd order as well as the understanding of their dynamic. For example, we will be discussing barrier options and the Greeks deformation / hedging for this payoff.

Lecture Series by Ioannis Karatzas
Tuesday, April 9, 2024

Lecture 1: Overview of Models for Stock Prices

Abstract:

Introduction of some basic notions: probabilities, random variables, means and variances. Random walks and Brownian motion, models for stock prices. Arithmetic and logarithmic rates of return.

Thursday, April 11, 2024

Lecture 2: Portfolios

Abstract:

Notion of proportional investment. Portfolios, their rates of return and growth. The market and related portfolios. Notion of Excess Growth Rate. Relative growth rates and the numeraire invariance of the excess growth rate. The excess growth rate of the market portfolio.