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Papers by robert fernholz

arXiv (Cornell University), Jan 19, 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. Functionally generated portfolios are portfolios for which the logarithmic return relative to the market portfolio can be decomposed into a function of the market weights and a process of locally finite variation, and this decomposition is convenient for characterizing the long-term behavior of the portfolio. A permutation-weighted portfolio is a portfolio in which the assets are held at weights proportional to a permutation of their market values, and such a portfolio is functionally generated only for markets with two assets (except for the identity permutation). A reverse-weighted portfolio is a portfolio in which the asset with the greatest market weight is assigned the smallest market weight, the asset with the secondlargest weight is assigned the second-smallest, and so forth. Although the reverse-weighted portfolio in a market with four or more assets is not functionally generated, it is still possible to characterize its long-term behavior using rank-based methods. 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.

arXiv (Cornell University), Aug 22, 2016

The capitalization-weighted total relative variation d i=1 • 0 µ i (t)d log µ i (t) in an equity ... more The capitalization-weighted total relative variation d i=1 • 0 µ i (t)d log µ i (t) in an equity market consisting of a fixed number d of assets with capitalization weights µ i (•) is an observable and nondecreasing function of time. If this observable of the market is not just nondecreasing, but actually grows at a rate which is bounded away from zero, then strong arbitrage can be constructed relative to the market over sufficiently long time horizons. It has been an open issue for more than ten years, whether such strong outperformance of the market is possible also over arbitrary time horizons under the stated condition. We show that this is not possible in general, thus settling this long-open question. We also show that, under appropriate additional conditions, outperformance over any time horizon indeed becomes possible, and exhibit investment strategies that effect it.

arXiv (Cornell University), Feb 27, 2016

An Atlas model is a rank-based system of continuous semimartingales for which the steady-state va... more An Atlas model is a rank-based system of continuous semimartingales for which the steady-state values of the processes follow a power law, or Pareto distribution. For a power law, the log-log plot of these steady-state values versus rank is a straight line. Zipf's law is a power law for which the slope of this line is −1. In this note, rank-based conditions are found under which an Atlas model will follow Zipf's law. An advantage of this rank-based approach is that it provides information about the dynamics of systems that result in Zipf's law.

arXiv (Cornell University), Jul 13, 2017

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 strictly positive continuous semimartingales with parameters that depend only on rank. We show that the stationary 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 why Zipf's law holds for word frequency, firm size, household wealth, and city size, while it does not hold for earthquake magnitude, cumulative book sales, the intensity of solar flares, and the intensity of wars, all of which follow non-Zipfian Pareto distributions.

The Annals of Applied Probability, 2011

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.

arXiv (Cornell University), Feb 15, 2013

A first-order model for a stock market assigns to each stock a return parameter and a variance pa... more A first-order model for a stock market assigns to each stock a return parameter and a variance parameter that depend only on the rank of the stock. A second-order model assigns these parameters based on both the rank and the name of the stock. First-and second-order models exhibit stability properties that make them appropriate as a backdrop for the analysis of the idiosyncratic behavior of individual stocks. Methods for the estimation of the parameters of second-order models are developed in this paper.

Annals of Applied Probability, Feb 1, 2018

The capitalization-weighted cumulative variation d i=1 • 0 µ i (t)d log µ i (t) in an equity mark... more The capitalization-weighted cumulative variation d i=1 • 0 µ i (t)d log µ i (t) in an equity market consisting of a fixed number d of assets with capitalization weights µ i (•), is an observable and a nondecreasing function of time. If this observable of the market is not just nondecreasing but actually grows at a rate bounded away from zero, then strong arbitrage can be constructed relative to the market over sufficiently long time horizons. It has been an open issue for more than ten years, whether such strong outperformance of the market is possible also over arbitrary time horizons under the stated condition. We show that this is not possible in general, thus settling this long-open question. We also show that, under appropriate additional conditions, outperformance over any time horizon indeed becomes possible, and exhibit investment strategies that effect it.

Stable Models for the Distribution of Capital

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.

Portfolios of Stocks Selected by Rank

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.

Springer eBooks, 2002

Library of Congress Cataloging-in-Publication Data Fernholz, Erhard Robert. Stochastic portfolio ... more Library of Congress Cataloging-in-Publication Data Fernholz, Erhard Robert. Stochastic portfolio theory I E. Robert Fernholz. p. cm.-(Applications of mathematics ; 48) Includes bibliographical references and index.

Probability Theory and Related Fields, May 9, 2012

For given nonnegative constants g, h, ρ, σ with ρ 2 + σ 2 = 1 and g + h > 0, we construct a diffu... more For given nonnegative constants g, h, ρ, σ with ρ 2 + σ 2 = 1 and g + h > 0, we construct a diffusion process (X 1 (•), X 2 (•)

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

An Atlas model is a rank-based system of continuous semimartingales for which the steady-state va... more An Atlas model is a rank-based system of continuous semimartingales for which the steady-state values of the processes follow a power law, or Pareto distribution. For a power law, the log-log plot of these steady-state values versus rank is a straight line. Zipf's law is a power law for which the slope of this line is −1. In this note, rank-based conditions are found under which an Atlas model will follow Zipf's law. An advantage of this rank-based approach is that it provides information about the dynamics of systems that result in Zipf's law.

Research Papers in Economics, Jun 1, 2016

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: 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

It has been widely observed that capitalization-weighted indexes can be beaten by surprisingly si... more It has been widely observed that capitalization-weighted indexes can be beaten by surprisingly simple, systematic investment strategies. Indeed, in the U.S. stock market, equal-weighted portfolios, random-weighted portfolios, and other naïve, nonoptimized portfolios tend to outperform a capitalization-weighted index over the long term. This outperformance is generally attributed to beneficial factor exposures. Here, we provide a deeper, more general explanation of this phenomenon by decomposing portfolio log-returns into an average growth and an excess growth component. Using a rank-based empirical study we argue that the excess growth component plays the major role in explaining the outperformance of naïve portfolios. In particular, individual stock growth rates are not as critical as is traditionally assumed.

Annals of Finance

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. Functionally generated portfolios are portfolios for which the logarithmic return relative to the market portfolio can be decomposed into a function of the market weights and a process of locally finite variation, and this decomposition is convenient for characterizing the long-term behavior of the portfolio. A permutation-weighted portfolio is a portfolio in which the assets are held at weights proportional to a permutation of their market values, and such a portfolio is functionally generated only for markets with two assets (except for the identity permutation). A reverse-weighted portfolio is a portfolio in which the asset with the greatest market weight is assigned the smallest market weight, the asset with the secondlargest weight is assigned the second-smallest, and so forth. Although the reverse-weighted portfolio in a market with four or more assets is not functionally generated, it is still possible to characterize its long-term behavior using rank-based methods. 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.

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.

arXiv (Cornell University), Jan 19, 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. Functionally generated portfolios are portfolios for which the logarithmic return relative to the market portfolio can be decomposed into a function of the market weights and a process of locally finite variation, and this decomposition is convenient for characterizing the long-term behavior of the portfolio. A permutation-weighted portfolio is a portfolio in which the assets are held at weights proportional to a permutation of their market values, and such a portfolio is functionally generated only for markets with two assets (except for the identity permutation). A reverse-weighted portfolio is a portfolio in which the asset with the greatest market weight is assigned the smallest market weight, the asset with the secondlargest weight is assigned the second-smallest, and so forth. Although the reverse-weighted portfolio in a market with four or more assets is not functionally generated, it is still possible to characterize its long-term behavior using rank-based methods. 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.

arXiv (Cornell University), Aug 22, 2016

The capitalization-weighted total relative variation d i=1 • 0 µ i (t)d log µ i (t) in an equity ... more The capitalization-weighted total relative variation d i=1 • 0 µ i (t)d log µ i (t) in an equity market consisting of a fixed number d of assets with capitalization weights µ i (•) is an observable and nondecreasing function of time. If this observable of the market is not just nondecreasing, but actually grows at a rate which is bounded away from zero, then strong arbitrage can be constructed relative to the market over sufficiently long time horizons. It has been an open issue for more than ten years, whether such strong outperformance of the market is possible also over arbitrary time horizons under the stated condition. We show that this is not possible in general, thus settling this long-open question. We also show that, under appropriate additional conditions, outperformance over any time horizon indeed becomes possible, and exhibit investment strategies that effect it.

arXiv (Cornell University), Feb 27, 2016

An Atlas model is a rank-based system of continuous semimartingales for which the steady-state va... more An Atlas model is a rank-based system of continuous semimartingales for which the steady-state values of the processes follow a power law, or Pareto distribution. For a power law, the log-log plot of these steady-state values versus rank is a straight line. Zipf's law is a power law for which the slope of this line is −1. In this note, rank-based conditions are found under which an Atlas model will follow Zipf's law. An advantage of this rank-based approach is that it provides information about the dynamics of systems that result in Zipf's law.

arXiv (Cornell University), Jul 13, 2017

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 strictly positive continuous semimartingales with parameters that depend only on rank. We show that the stationary 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 why Zipf's law holds for word frequency, firm size, household wealth, and city size, while it does not hold for earthquake magnitude, cumulative book sales, the intensity of solar flares, and the intensity of wars, all of which follow non-Zipfian Pareto distributions.

The Annals of Applied Probability, 2011

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.

arXiv (Cornell University), Feb 15, 2013

A first-order model for a stock market assigns to each stock a return parameter and a variance pa... more A first-order model for a stock market assigns to each stock a return parameter and a variance parameter that depend only on the rank of the stock. A second-order model assigns these parameters based on both the rank and the name of the stock. First-and second-order models exhibit stability properties that make them appropriate as a backdrop for the analysis of the idiosyncratic behavior of individual stocks. Methods for the estimation of the parameters of second-order models are developed in this paper.

Annals of Applied Probability, Feb 1, 2018

The capitalization-weighted cumulative variation d i=1 • 0 µ i (t)d log µ i (t) in an equity mark... more The capitalization-weighted cumulative variation d i=1 • 0 µ i (t)d log µ i (t) in an equity market consisting of a fixed number d of assets with capitalization weights µ i (•), is an observable and a nondecreasing function of time. If this observable of the market is not just nondecreasing but actually grows at a rate bounded away from zero, then strong arbitrage can be constructed relative to the market over sufficiently long time horizons. It has been an open issue for more than ten years, whether such strong outperformance of the market is possible also over arbitrary time horizons under the stated condition. We show that this is not possible in general, thus settling this long-open question. We also show that, under appropriate additional conditions, outperformance over any time horizon indeed becomes possible, and exhibit investment strategies that effect it.

Stable Models for the Distribution of Capital

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.

Portfolios of Stocks Selected by Rank

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.

Springer eBooks, 2002

Library of Congress Cataloging-in-Publication Data Fernholz, Erhard Robert. Stochastic portfolio ... more Library of Congress Cataloging-in-Publication Data Fernholz, Erhard Robert. Stochastic portfolio theory I E. Robert Fernholz. p. cm.-(Applications of mathematics ; 48) Includes bibliographical references and index.

Probability Theory and Related Fields, May 9, 2012

For given nonnegative constants g, h, ρ, σ with ρ 2 + σ 2 = 1 and g + h > 0, we construct a diffu... more For given nonnegative constants g, h, ρ, σ with ρ 2 + σ 2 = 1 and g + h > 0, we construct a diffusion process (X 1 (•), X 2 (•)

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

An Atlas model is a rank-based system of continuous semimartingales for which the steady-state va... more An Atlas model is a rank-based system of continuous semimartingales for which the steady-state values of the processes follow a power law, or Pareto distribution. For a power law, the log-log plot of these steady-state values versus rank is a straight line. Zipf's law is a power law for which the slope of this line is −1. In this note, rank-based conditions are found under which an Atlas model will follow Zipf's law. An advantage of this rank-based approach is that it provides information about the dynamics of systems that result in Zipf's law.

Research Papers in Economics, Jun 1, 2016

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: 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

It has been widely observed that capitalization-weighted indexes can be beaten by surprisingly si... more It has been widely observed that capitalization-weighted indexes can be beaten by surprisingly simple, systematic investment strategies. Indeed, in the U.S. stock market, equal-weighted portfolios, random-weighted portfolios, and other naïve, nonoptimized portfolios tend to outperform a capitalization-weighted index over the long term. This outperformance is generally attributed to beneficial factor exposures. Here, we provide a deeper, more general explanation of this phenomenon by decomposing portfolio log-returns into an average growth and an excess growth component. Using a rank-based empirical study we argue that the excess growth component plays the major role in explaining the outperformance of naïve portfolios. In particular, individual stock growth rates are not as critical as is traditionally assumed.

Annals of Finance

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. Functionally generated portfolios are portfolios for which the logarithmic return relative to the market portfolio can be decomposed into a function of the market weights and a process of locally finite variation, and this decomposition is convenient for characterizing the long-term behavior of the portfolio. A permutation-weighted portfolio is a portfolio in which the assets are held at weights proportional to a permutation of their market values, and such a portfolio is functionally generated only for markets with two assets (except for the identity permutation). A reverse-weighted portfolio is a portfolio in which the asset with the greatest market weight is assigned the smallest market weight, the asset with the secondlargest weight is assigned the second-smallest, and so forth. Although the reverse-weighted portfolio in a market with four or more assets is not functionally generated, it is still possible to characterize its long-term behavior using rank-based methods. 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.

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.