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Papers by robert fernholz
arXiv (Cornell University), Aug 22, 2016
arXiv (Cornell University), Feb 27, 2016
arXiv (Cornell University), Jul 13, 2017
The Annals of Applied Probability, 2011
arXiv (Cornell University), Feb 15, 2013
Annals of Applied Probability, Feb 1, 2018
In Chapter 2 we examined certain characteristics of the distribution of capital, but since the an... more In Chapter 2 we examined certain characteristics of the distribution of capital, but since the analytical tools at our disposal were limited, the results were incomplete. In this chapter we shall apply the methods introduced in Chapters 3 and 4 to achieve a deeper understanding of the structure of the capital distribution. In particular, local times related to the market weight processes permit us to develop a detailed model for a stable capital distribution.
The distribution of capital is of fundamental importance in stochastic portfolio theory, as are f... more The distribution of capital is of fundamental importance in stochastic portfolio theory, as are functionally generated portfolios. In this chapter we shall combine these two concepts.
Probability Theory and Related Fields, May 9, 2012
arXiv: Mathematical Finance, 2020
A market portfolio is a portfolio in which each asset is held at a weight proportional to its mar... more A market portfolio is a portfolio in which each asset is held at a weight proportional to its market value. A swap portfolio is a portfolio in which each one of a pair of assets is held at a weight proportional to the market value of the other. A reverse-weighted index portfolio is a portfolio in which the weights of the market portfolio are swapped pairwise by rank. Swap portfolios are functionally generated, and in a coherent market they have higher asymptotic growth rates than the market portfolio. Although reverse-weighted portfolios with two or more pairs of assets are not functionally generated, in a market represented by a first-order model with symmetric variances, they will grow faster than the market portfolio. This result is applied to a market of commodity futures, where we show that the reverse price-weighted portfolio substantially outperforms the price-weighted portfolio from 1977-2018.
Research Papers in Economics, Feb 1, 2016
Research Papers in Economics, Jun 1, 2016
arXiv: Mathematical Finance, 2020
A market portfolio is a portfolio in which each asset is held at a weight proportional to its mar... more A market portfolio is a portfolio in which each asset is held at a weight proportional to its market value. A swap portfolio is a portfolio in which each one of a pair of assets is held at a weight proportional to the market value of the other. A reverse-weighted index portfolio is a portfolio in which the weights of the market portfolio are swapped pairwise by rank. Swap portfolios are functionally generated, and in a coherent market they have higher asymptotic growth rates than the market portfolio. Although reverse-weighted portfolios with two or more pairs of assets are not functionally generated, in a market represented by a first-order model with symmetric variances, they will grow faster than the market portfolio. This result is applied to a market of commodity futures.
arXiv: Mathematical Finance, 2015
Long-term relative arbitrage exists in markets where the excess growth rate of the market portfol... more Long-term relative arbitrage exists in markets where the excess growth rate of the market portfolio is bounded away from zero. Here it is shown that under a time-homogeneity hypothesis this condition will also imply the existence of relative arbitrage over arbitrarily short intervals.
SSRN Electronic Journal, 2018
A set of data with positive values follows a Pareto distribution if the log-log plot of value ver... more A set of data with positive values follows a Pareto distribution if the log-log plot of value versus rank is approximately a straight line. A Pareto distribution satisfies Zipf's law if the log-log plot has a slope of -1. Since many types of ranked data follow Zipf's law, it is considered a form of universality. We propose a mathematical explanation for this phenomenon based on Atlas models and first-order models, systems of positive continuous semimartingales with parameters that depend only on rank. We show that the stable distribution of an Atlas model will follow Zipf's law if and only if two natural conditions, conservation and completeness, are satisfied. Since Atlas models and first-order models can be constructed to approximate systems of time-dependent rank-based data, our results can explain the universality of Zipf's law for such systems. However, ranked data generated by other means may follow non-Zipfian Pareto distributions. Hence, our results explain w...
We study Atlas-type models of equity markets with local characteristics that depend on both name ... more We study Atlas-type models of equity markets with local characteristics that depend on both name and rank, and in ways that induce a stable capital distribution. Ergodic properties and rankings of processes are examined with reference to the theory of reflected Brownian motions in polyhedral domains. In the context of such models we discuss properties of various investment strategies, including the so-called growth-optimal and universal portfolios.
For a functionally generated portfolio, there is a natural decomposition of the relative log-retu... more For a functionally generated portfolio, there is a natural decomposition of the relative log-return into the log-change in the generating function and a drift process. In this note, this decomposition is extended to arbitrary stock portfolios by an application of Fisk-Stratonovich integration. With the extended methodology, the generating function is represented by a structural process, and the drift process is subsumed into a trading process that measures the profit and loss to the portfolio from trading.
arXiv (Cornell University), Aug 22, 2016
arXiv (Cornell University), Feb 27, 2016
arXiv (Cornell University), Jul 13, 2017
The Annals of Applied Probability, 2011
arXiv (Cornell University), Feb 15, 2013
Annals of Applied Probability, Feb 1, 2018
In Chapter 2 we examined certain characteristics of the distribution of capital, but since the an... more In Chapter 2 we examined certain characteristics of the distribution of capital, but since the analytical tools at our disposal were limited, the results were incomplete. In this chapter we shall apply the methods introduced in Chapters 3 and 4 to achieve a deeper understanding of the structure of the capital distribution. In particular, local times related to the market weight processes permit us to develop a detailed model for a stable capital distribution.
The distribution of capital is of fundamental importance in stochastic portfolio theory, as are f... more The distribution of capital is of fundamental importance in stochastic portfolio theory, as are functionally generated portfolios. In this chapter we shall combine these two concepts.
Probability Theory and Related Fields, May 9, 2012
arXiv: Mathematical Finance, 2020
A market portfolio is a portfolio in which each asset is held at a weight proportional to its mar... more A market portfolio is a portfolio in which each asset is held at a weight proportional to its market value. A swap portfolio is a portfolio in which each one of a pair of assets is held at a weight proportional to the market value of the other. A reverse-weighted index portfolio is a portfolio in which the weights of the market portfolio are swapped pairwise by rank. Swap portfolios are functionally generated, and in a coherent market they have higher asymptotic growth rates than the market portfolio. Although reverse-weighted portfolios with two or more pairs of assets are not functionally generated, in a market represented by a first-order model with symmetric variances, they will grow faster than the market portfolio. This result is applied to a market of commodity futures, where we show that the reverse price-weighted portfolio substantially outperforms the price-weighted portfolio from 1977-2018.
Research Papers in Economics, Feb 1, 2016
Research Papers in Economics, Jun 1, 2016
arXiv: Mathematical Finance, 2020
A market portfolio is a portfolio in which each asset is held at a weight proportional to its mar... more A market portfolio is a portfolio in which each asset is held at a weight proportional to its market value. A swap portfolio is a portfolio in which each one of a pair of assets is held at a weight proportional to the market value of the other. A reverse-weighted index portfolio is a portfolio in which the weights of the market portfolio are swapped pairwise by rank. Swap portfolios are functionally generated, and in a coherent market they have higher asymptotic growth rates than the market portfolio. Although reverse-weighted portfolios with two or more pairs of assets are not functionally generated, in a market represented by a first-order model with symmetric variances, they will grow faster than the market portfolio. This result is applied to a market of commodity futures.
arXiv: Mathematical Finance, 2015
Long-term relative arbitrage exists in markets where the excess growth rate of the market portfol... more Long-term relative arbitrage exists in markets where the excess growth rate of the market portfolio is bounded away from zero. Here it is shown that under a time-homogeneity hypothesis this condition will also imply the existence of relative arbitrage over arbitrarily short intervals.
SSRN Electronic Journal, 2018
A set of data with positive values follows a Pareto distribution if the log-log plot of value ver... more A set of data with positive values follows a Pareto distribution if the log-log plot of value versus rank is approximately a straight line. A Pareto distribution satisfies Zipf's law if the log-log plot has a slope of -1. Since many types of ranked data follow Zipf's law, it is considered a form of universality. We propose a mathematical explanation for this phenomenon based on Atlas models and first-order models, systems of positive continuous semimartingales with parameters that depend only on rank. We show that the stable distribution of an Atlas model will follow Zipf's law if and only if two natural conditions, conservation and completeness, are satisfied. Since Atlas models and first-order models can be constructed to approximate systems of time-dependent rank-based data, our results can explain the universality of Zipf's law for such systems. However, ranked data generated by other means may follow non-Zipfian Pareto distributions. Hence, our results explain w...
We study Atlas-type models of equity markets with local characteristics that depend on both name ... more We study Atlas-type models of equity markets with local characteristics that depend on both name and rank, and in ways that induce a stable capital distribution. Ergodic properties and rankings of processes are examined with reference to the theory of reflected Brownian motions in polyhedral domains. In the context of such models we discuss properties of various investment strategies, including the so-called growth-optimal and universal portfolios.
For a functionally generated portfolio, there is a natural decomposition of the relative log-retu... more For a functionally generated portfolio, there is a natural decomposition of the relative log-return into the log-change in the generating function and a drift process. In this note, this decomposition is extended to arbitrary stock portfolios by an application of Fisk-Stratonovich integration. With the extended methodology, the generating function is represented by a structural process, and the drift process is subsumed into a trading process that measures the profit and loss to the portfolio from trading.