# JOHN NUTTALL

This site has information and opinions on a number of topics that interest me. I welcome comments on any of these topics.

Click on the descriptive heading to go to a summary of the topic. Click on the Section number to go directly to the full report.

### SECTION 2 - MATHEMATICS

• SECTION 2.1 - RIEMANN HYPOTHESIS

## INTRODUCTION TO THE VARIOUS TOPICS

Below are found brief introductions to the topics discussed on this web site. Click on the Section number to go to the full report.

### SECTION 1 - TOPICS IN INVESTING

SECTION 1.1 - QUANTITATIVE INVESTMENT MANAGEMENT

This section considers approaches to quantitative equity investing, where information is analyzed and predictions are made with the help of a computer.

SECTION 1.1.1 - General Discussion

A structure is described that encompasses most methods of investment management. Special attention is paid to the application of Machine Learning in predicting returns.

SECTION 1.1.2 - Machine Learning and Short-Term Equity Return Prediction

In an article written with R. Yan and C. Ling, we explain how an intelligent use of Machine Learning can improve predictive ability in a particular case.

Cooper showed how a filter method could be used to predict equity returns for the next week by using information about returns and volume for the two previous weeks. Cooper's method may be regarded as a crude method of Machine Learning. Over the last 20 years Machine Learning has been successfully applied to the modeling of large data sets, often containing a lot of noise, in many different fields. When applying the technique it is important to fit it to the specific problem under consideration. We have designed and applied to Cooper's problem a practical new method of Machine Learning, appropriate to the problem, that is based on a modification of the well-known kernel regression method. We call it the Prototype Kernel Regression method (PKR).

In both the period 1978-1993 studied by Cooper, and the period 1994-2004, the PKR method leads to clear profit improvement compared to Cooper's approach. In all of 48 different cases studied, the period pre-cost average return is larger for the PKR method than Cooper's method, on average 37% higher, and that margin would increase as costs were taken into account. Our method aims to minimize the danger of data snooping, and it could plausibly have been applied in 1994 or earlier.

There may be a lesson here for proponents of the Efficient Market Hypothesis in the form that states that profitable prediction of equity returns is impossible except by chance. It is not enough for them to show that the profits from an anomaly-based trading scheme disappear after costs. The proponents should also consider what would have been plausible applications of more sophisticated Machine Learning techniques before dismissing evidence against the EMH.

SECTION 1.1.3 - New Approach to Market Dynamics

This is an attempt to approach the question of asset pricing from a new direction. More study is needed to determine whether anything useful will arise in practical situations.

Equity portfolio managers typically convey instructions to their traders in the form of target portfolio weights for the various shares in their portfolio. We present a set of differential equations that allows the calculation of the share prices, number of shares, and value of each manager's portfolio over time, in terms of share weights. It is also necessary to know the amount of cash flowing into each portfolio and the number of each type of shares outstanding.

We suggest some potentially useful information that might be derived from this formalism, such as a quantitative estimate of the main driver of share price changes, the influence of index investing on the market, and the origin of the equity premium.

We believe that this realistic method could be the basis for a better understanding of how financial markets operate, as compared with the conventional academic approach. In our view standard asset pricing theory makes implausible assumptions about the existence of stochastic processes, the ability of participants to foretell the future, and their capacity to make sound deductions from the information they have. Even an imperfect alternative should be better than that house of cards.

SECTION 1.2 - FALSE DOCTRINES

In my view the subject of financial economics, particularly as it is applied to investing in portfolios of equities, is noteworthy for the number of unrealistic assumptions made to construct the dominant theories. Here we present some of the many existing examples of what we consider to be dubious conclusions, including two of several written by Fama.

SECTION 1.2.1 - Identification of Individual Skilled Fund Managers

Can we identify individual mutual funds with skill?

Cuthbertson et al have recently described a method that is claimed to be able to identify individual fund managers who exhibited skill over a long period in the past. The only input to the process is monthly fund returns. We suggest that a critical step in the Cuthbertson method is flawed. This step involves the study of the order statistics of period average fund returns. We construct a simple model to which the Cuthbertson method should apply. Simulations with the model conclusively demonstrate that the method fails to detect many funds with skill, and also erroneously identifies many funds as having skill they do not possess.

SECTION 1.2.2 - The Importance of Asset Allocation

There is a widespread belief in the truth of statements such as "studies show that asset mix determines 93.6% of the return of a portfolio". This belief apparently arises from an article by Gary Brinson and colleagues published in 1986. If you search for "asset allocation" you will find a lot of variants of the statement. Many of them specifically give the source as Brinson and colleagues. Almost all these statements misquote the Brinson results. Not only that, there are serious problems with the results themselves.

Read the whole story, which seems to suggest that thousands of people in the investment industry in the US and Canada have been misleading the public for years.

Note that our report was written in 2000. Recently Brinson has published his reflections on 35 years in investment management [Financial Analysts Journal, July/August 2005, p. 24], where he makes some interesting remarks that are worthwhile reading. It is perhaps significant that there is not a word on the importance of asset allocation, no doubt the subject for which he is best known.

SECTION 1.2.3 - The Efficient Market Hypothesis According to Fama, Interpreted

We present five quotes or pairs of quotes from Fama's original papers on the Efficient Market Hypothesis (EMH). After each quote we give our paraphrase of Fama's statement in the quote. There follows in plain language our deductions from the statements, sometimes with a reason. The logical result is the conclusion that Fama has effectively conceded that there is little or nothing of practical value in the EMH concept.

SECTION 1.2.4 - A Questionable Market Equilibrium Model Proposed by Fama

This report shows how to determine in analytic form the security prices implied by the market equilibrium model described by Fama in his book "Foundations of Finance", Chapter 8, Section III. The model assumes that all investors agree on the expected values and covariances of the random final prices at the end of an investment period. The investors interact so that the initial prices lead to an efficient portfolio with security weights proportional to the total initial value of the corresponding firm.

If we assume that the expected portfolio return or the risk-free rate is specified, we find that there is a one-dimensional continuum of sets of initial prices satisfying the conditions of the model. It is unclear how to resolve this ambiguity about which model is correct.

Note that the book was published in 1976, so that it might be reasonable to conclude that Fama may have since changed his mind on this matter. However, in notes for a recent course, he effectively reaffirms what he wrote then on the question.

SECTION 1.3 - OTHER

SECTION 1.3.1 - Equity Premium Puzzle

A mathematician might say that the fundamental theorem of investing, as put forth by almost all the investment industry, is "In the long run stocks will always return more than bonds, perhaps 6%/year more." Here is an introduction to the study of this issue.

SECTION 2.1 - RIEMANN HYPOTHESIS

A famous problem that is over 150 years old. Below are several recent contributions to the determinantal approach to the problem that was initiated by Csordas, Norfolk and Varga in 1986.

SECTION 2.1.1 - ON THE RIEMANN HYPOTHESIS - PART 1

This report applies the Laplace approximation method for estimating the Riemann determinants D(n,r) for large values of n.

SECTION 2.1.2 - ON THE RIEMANN HYPOTHESIS - PART 2

This report describes some interesting relations for determinants, etc.. It turns out that most of them are very old, going back to Hadamard, but one may be new. Among other things they are related to the problem of finding zeros of a power series or a meromorphic function. See Gutknecht and Parlett, IMA Journal of Numerical Analysis (2011) Vo. 31, 741.

SECTION 2.1.3 - ON THE RIEMANN HYPOTHESIS - PART 3

This report describes an improvement to Section 2.1.1

SECTION 2.1.4 - DETERMINANTAL INEQUALITIES AND THE RIEMANN HYPOTHESIS

This report extends the method of Csordas, Norfolk and Varga from 1986, 1988, to prove the determinantal inequalities of order 3 that must hold if the Riemann Hypothesis is correct.

SECTION 2.1.5 - RIEMANN DETERMINANTAL INEQUALITIES OF ORDER 3

This report descibes a more systematic method for proving the result of Section 2.1.3.

SECTION 2.1.6 - DETERMINANTAL APPROACH TO A PROOF OF THE RIEMANN HYPOTHESIS

This document makes suggestions, including three significant conjectures, and proves results on five topics related to the Determinantal Approach to the Riemann Hypothesis. The conjectures are supported by numerical calculations, but even if they were all correct, there would still be substantial problems to be solved before the Riemann Hypothesis was verified. The first two pages present a summary of the report.

SECTION 2.1.7 - THE SINGLE MOMENT METHOD IN THE DETERMINANTAL APPROACH TO THE RIEMANN HYPOTHESIS

This report describes an improved method of showing that some Riemann determinants are positive.

SECTION 2.1.8 - LECTURE NOTES ON THE RIEMANN HYPOTHESIS

Notes for a lecture given to the New York Number Theory Seminar, Oct. 20, 2011.

SECTION 2.1.9 - WRONSKIANS AND THE RIEMANN HYPOTHESIS

This report describes some speculations about the sign-regularity properties of an approximation to the main function used in Sec. 2.1.6, with numerical support.

SECTION 2.1.10 - A MOMENT DETERMINANTAL FRAMEWORK TO STUDY THE RIEMANN HYPOTHESIS AND RELATED QUESTIONS

An extension of Sec. 2.1.7.

SECTION 2.1.11 - BASIC WRONSKIANS IN THE RIEMANN HYPOTHESIS

A simplified version of Sec. 2.1.6.

SECTION 2.1.12 - STRUCTURE OF WRONSKIANS RELATED TO THE RIEMANN HYPOTHESIS

An extension of the method used in Sec. 2.1.5.

SECTION 2.1.13 - NUMERICAL PATTERNS RELATED TO THE RIEMANN AND CONREY/GHOSH PHI-FUNCTIONS

An extension of the method used in Sec. 2.1.10.

SECTION 2.1.14 - WRONSKIANS, CUMULANTS AND THE RIEMANN HYPOTHESIS

An extension of the method used in Sec. 2.1.5.

SECTION 2.1.15 - PROGRESS REPORT ON THE RIEMANN AND THE EIGHT CONREY/GHOSH PHI-FUNCTIONS

SECTION 2.1.16 - CUMULANTS, THE RIEMANN HYPOTHESIS, AND SIMILAR PROBLEMS

For one Conrey/Ghosh case, a proof of the Cumulant conjecture of Sec. 2.2.6.

### SECTION 3 - PERSONAL

Some items of personal interest.

SECTION 3.1 - Brief Biography

A few facts about the author of this web site.

SECTION 3.2 - Smith's Prize

Many years ago I was lucky enough to win this prize, which has been awarded to Cambridge students since 1769. Among previous winners are scientists Kelvin, Maxwell, Rayleigh, J. J. Thomson and Turing.

SECTION 3.3 - St. Lawrence String Quartet

A well-travelled group of string players based at Stanford University. Perhaps you can guess why I have this link.

SECTION 3.4 - My Fitness Story

The story of my resumption of regular exercise after a very long layoff.