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School of Education, Culture and Communication

Division of Applied Mathematics

MASTER THESIS IN MATHEMATICS / APPLIED MATHEMATICS

Socio-Economically Responsible Investing and Income Inequality in the

USA

by

David Brown

Masterarbete i matematik / tillämpad matematik

DIVISION OF APPLIED MATHEMATICS

MÄLARDALEN UNIVERSITY SE-721 23 VÄSTERÅS, SWEDEN

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School of Education, Culture and Communication

Division of Applied Mathematics

Master thesis in mathematics / applied mathematics

Date:

2016–06–10

Project name: Socio-Economically Responsible Investing and Income Inequality in the USA Author:

David Brown

Version:

12th June 2017

Supervisor(s):

Lars Pettersson and Anatoliy Malyarenko

Reviewer: Daniel Andrén Examiner: Linus Carlsson Comprising: 30 ECTS credits

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Abstract

To add to the tools currently available to combat income inequality in the United States an in-vestment fund type is proposed, justified, described, and created using historical asset returns from 1960 to 2015. By focusing on two socio-economic indicators of poverty, inflation and unemployment rates, this fund, when marketed to investors who live near, at, or below the poverty line, seeks to increase returns during times of increased strain on the economies of the poor. Multiple hurdles are proposed and affirmatively answered to this end and a fund type and corresponding four factor model that realized hypothetical excess returns fitting the require-ments of a successful investment strategy was developed and evaluated. With the increasing importance of socially responsible investment practices an investment bank who maintains a fund of this type could potentially see financial and reputational benefits.

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Contents

Introduction 6

1 Formulation of the problem 9

1.1 Characteristics of the portfolio . . . 9

1.1.1 Simplifications and Presumptions . . . 10

1.2 Literature review . . . 11

1.2.1 The Markowitz Model . . . 11

1.2.2 The Single Index Model . . . 12

1.2.3 The Multi Index Model . . . 13

1.2.4 Asset selection: P/E, PEG, and PEGY ratios . . . 14

1.2.5 The Beta Puzzle . . . 15

1.2.6 Portfolio evaluation techniques . . . 16

1.3 The mathematical tools . . . 16

1.3.1 Principal Component Analysis . . . 17

1.3.2 Non-linear Iterative Partial Least Squares Regression . . . 18

1.4 Methodology . . . 20

1.4.1 Asset selection . . . 20

1.4.2 First asset filter . . . 21

1.4.3 Second asset filter . . . 21

1.4.4 Third asset filter . . . 22

1.4.5 Portfolio creation and evaluation . . . 22

1.4.6 Model creation and evaluation . . . 22

2 Solution 23 2.1 Asset selection process . . . 23

2.1.1 Asset selection . . . 23

2.1.2 First asset filter . . . 24

2.1.3 Second asset filter . . . 25

2.1.4 Third asset filter . . . 25

2.2 Portfolio creation and evaluation . . . 26

2.2.1 Create portfolios . . . 27

2.2.2 Evaluate each portfolio . . . 27

2.2.3 Select an optimum portfolio . . . 29

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3 Conclusion 36 3.0.1 Improvements and ideas for future research . . . 38 3.0.2 Applications . . . 38

Bibliography 39

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List of Figures

1 Congressional Budget Office - Income growth rates from 1946 to 2015 in the US as a percentage of 1973 levels. Source: Center on Budget and Policy Priorities . . . 7 2 Congressional Budget Office - After-tax income gains since 1979 in the US.

Source: Center on Budget and Policy Priorities . . . 7

2.1 Asset diversification scree plot with the variance explanatory power of prin-ciple components shown as a percentage of total variance of the data. . . 26 2.2 Graphical analysis of percent excess return to variability measure with the 4

highest ranked portfolios designated with a red trend line. . . 28 2.3 Principle component variance explanatory analysis of Max Sharpe with Short

Sales portfolio. . . 29 2.4 Principle component variance explanatory analysis of Max Sharpe with Short

Sales with all Assets portfolio. . . 31 2.5 Principle component variance explanatory analysis of Max Sharpe with Short

Sales with all Assets and 15% investiture portfolio. . . 31 2.6 Principle component variance explanatory analysis of Minimum Variance with

Short Sales with all Assets and 15% investiture portfolio. . . 32 2.7 MPIU 4 factor multi-index model distribution with realized return Z-scores. . 35

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List of Tables

2.1 Differential return with risk measured by standard deviation analysis where MinV means minimum variance, SS means short selling allowed, NSS means no short selling allowed, MaxR means maximum return, MaxS means max-imum Sharpe Ratio, and BD means best diversitfication selection. . . 28 2.2 Asset weights for the 4 efficient portfolios with the best predicted performance. 30 2.3 Index model volatilities for January 2015 . . . 33 2.4 Index model average returns for 2015 . . . 33 2.5 Aggregate Cumulative Distribution Function values evaluated at model Z-Scores 34 2.6 Cumulative real returns for Jan to Dec 2015. . . 35

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Introduction

According to the Center on Budget and Policy Priorities, over the last 30 years the gap between the annual income of the top 5% of US households and the bottom 60% of US households in the United States has grown at a rate higher than at any time in history. Over that time period annual household income has increased by approximately 75% for the top 5% and decreased by approximately 4% for the bottom 60% of US households with the lowest 20% realizing a decrease in annual household income of 12% [33].

In 2013 the percentage of people ages 18-65 whose income fell below 50% of the median income for the United States was 15.3% [11]. When compared to other western countries the US is 29th in terms of poverty rate and according to the US Census Bureau that equates to 46.2 million people who live near, at, or below the poverty line [7]. While the poverty rate has gone down in the recent past the growing number of people, in total population, who comprise the poverty class has risen dramatically and disproportionately to the increase in US population over the last 30 years [27]. According to the Congressional Budget Office (CBO), from the late 1940’s until the early 1970’s income growth for all income levels roughly doubled in inflation adjusted terms whereas from the late 1970s until the present that growth has significantly declined for the bottom 80% of income earners while greatly increasing for the top 20% of income earners. According to the CBO and illustrated in figure 1 income gains, as a percentage of 1973 levels, of the top 5% of familes have seen an increase in real family income nearly 70% higher than the lowest 20% of families with the top 1%, as seen in figure 2, seeing after tax income gains of over 150% above the lowest 20% of families [33].

Combating the increasing number of poverty level families in the US is currently a hot topic political issue. Time magazine’s Rana Foroohar wrote an article for their business sec-tion and Nobel laureate Joseph Stiglitz wrote a report for the Roosevelt Institute that illustrate the point [12, 32]. There are many private organizations in the United States that specialize in anti-poverty assistance. Organizations such as Oxfam America, a relief organization that works to create lasting solutions to poverty, and Children’s Defense Fund, a non-profit child advocacy agency that champions policies that lift children out of poverty, and many other private organizations are active in addressing the problem of poverty and income inequality [1]. The government, both state and federal, has created programs, for example social se-curity; unemployment insurance; and refundable tax credits, that target the problem in many ways such as increased job training, attempts to raise the minimum wage, improvements in education, and federal subsidies, to name a few [15].

Socially responsible investment (SRI) practices are investment practices that utilize cur-rent political and social sustainability issues to create investment standards and guidelines that

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Figure 1: Congressional Budget Office - Income growth rates from 1946 to 2015 in the US as a percentage of 1973 levels. Source: Center on Budget and Policy Priorities

Figure 2: Congressional Budget Office - After-tax income gains since 1979 in the US. Source: Center on Budget and Policy Priorities

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meet the needs of the general population in terms of ethics, environment, and economic well-being. This paper addresses the issue of combating increased poverty levels by proposing an investment strategy that aims to benefit poverty level investors by creating an SRI fund that capitalizes on the unique characteristics of poverty level families that can be marketed as an alternative to investing in a simple interest bearing savings account.

The inspiration for this approach to investing stems from the understanding that to affect an increase in wealth tomorrow one must possess things of value that appreciate over time rather than depreciate. This concept is not new however its practice is often beyond the means of a typical poverty level family. As mobile smart phones are becoming a common possession of even the poorest families the world of investing in mutual funds has finally come within the means of poverty level families who live paycheck to paycheck with no effective disposable income [30]. Investment banks and online investment companies have finally managed to open their markets to poverty level families through the help of mobile phone apps and other online features that allow for very small amounts of money to be invested in mutual funds.

This paper will detail the steps to creating such a fund from both financial and mathem-atical viewpoints from initial asset selection through to portfolio creation. This paper will also create and evaluate the performance of such a portfolio using data collected from Ya-hoo! Finance. Chapter 1l contains the information needed to understand the portfolio creation and evaluation process. Section 1.1l describes and justifies the characteristics of the portfolio: dividends, poverty rate, inflation, low market covariance, and unemployment. Section 1.2 con-tains a literature review of all applicable financial principes and section 1.3 all applicable math-ematical principles used for the creation and evaluation of the portfolio: principle component analysis and the NIPALS algorithm, single and multi-index models, portfolio evaluation tech-niques, investment ratios, and the Beta puzzle. Section 1.4l contains a brief description of the methodology of the creation and evaluation of the portfolio. Chapter 2l illustrates the step by step process of fund creation using raw data from Yahoo! Finance and the statistical analysis software R as well as a detailed evaluation of the created fund.

There are 3 critical questions that must be answered in this investigation:

1. Are there enough stocks available on the market, that fit the socio-economic character-istic requirements, to create a sufficiently diversified portfolio?

2. Can an optimum portfolio be created from those stocks that has predicted performance that is not below a designated performance level standard?

3. Can a sufficiently predictive model be created to facilitate and simplify the implement-ation and upkeep of the fund?

Sections 2.1, 2.2, and 2.3, respectively, addresses these questions and their answers. Chapter 3 presents the results of Chapter 2 in a conclusion as well as provides insight into further work that can be done to continue with this topic and applications in the context of the current financial world.

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Chapter 1

Formulation of the problem

To begin to address the primary purpose of this study we must first identify and justify the poverty related social and economic metrics used to determine the characteristics of the final portfolio: dividend payments, inflation rate, unemployment rate, and poverty rate. Further-more, I will highlight the simplifications and presumptions made along with their potential affect on the conclusion. I will present the financial tools needed to understand the methodo-logy and solution: The Markowitz, single index, and multi-index models; the asset selection ratios P/E, PEG, and PEGY ratios; the Beta Puzzle; and two portfolio evaluation techniques, excess return to variability measure and risk measured by standard deviation. I will present the mathematical tools needed to understand the methodology and solution: principal component analysis and non-linear iterative partial least squares regression. Finally, I will enumerate and describe in brief each of the 20 steps in the methodology that will be used in Chapter 2 to create an investment portfolio and evaluate the results.

1.1

Characteristics of the portfolio

According to a 2016 survey by Bankrate.com and a following article published in Forbes magazine 63% of poverty level and middle class families find themselves without enough sav-ings to cover a financial emergency of $500 [24]. That being the case it is unreasonable to expect those families to have the financial freedom to invest money that they cannot access when fiscal emergencies arise. Therefore this portfolio will concentrate exclusively on fin-ancial instruments that pay dividends as a means to supplement the wages of investors. This also provides the fund with two added benefits: in the event of a loss to the fund dividends in excess of the interest rate earned on a simple saving account can be reinvested into the fund to help counter the loss and all dividends paid are tax free as dividends paid to families that do not cause their overall income to pass into taxable levels are taxed as standard income which is zero.

Many publications available through the UC Berkeley Labor Center highlight that families who exist at or below the poverty line are largely dependent on wage income which helps propagate poverty [8]. Wage income is affected by inflation more so than other forms of in-come resulting in an effective decrease in family inin-come each year when wages are stagnant.

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This phenomena is illustrated in a study conducted by the Federal Reserve Bank of St. Louis which found that over the past 55 years wage growth has fallen behind inflation growth [28]. Another aspect of those living at or near poverty is that when they do spend their limited in-come they purchase items that immediately depreciate in value such as consumables, common retail goods, and general livelihood upkeep expenses which further compound the negative ef-fects of inflation on the poor [5]. With the effect on wage income with the increase of the inflation rate this portfolio will be designed to increase returns when the inflation rate rises.

When unemployment rises it affects families who do not have enough money in savings to weather the financial crisis. As unemployment increases middle class families who are on the cusp of the poverty line tend to fall below the poverty line and those families who already exist in poverty require further assistance from their investments [23]. With an increase in the number of poverty level families as well as the increase in financial demand from those fam-ilies already in poverty this fund will be designed to increase returns when the unemployment rate rises.

Beta-neutral funds are funds in which the returns of the fund are minimally covariant with some market index. Historical performance records of beta-neutral funds show that they outperform high beta funds in regard to long term performance [9]. Aside from the long term positive performance results of beta-neutral portfolios being beta-neutral provides additional benefits to the fund in this study. This fund is specifically designed to hedge against times of financial hardship which are generally expected to correlate more closely with drops in the overall market rather than increases. By striving for beta-neutrality this fund hopes to add increased fund protection from drawdown periods at the cost of reduced performance during market increases. When financial times are good as measured by an increase in an appropriate market index families near, at, or below the poverty line can be expected to benefit similarly to everybody else so increased fund performance is not required. For this reason this fund will aim to be minimally covariant with its market index.

Lastly this fund will be designed to have increased returns when the poverty level rises. The purpose for this is twofold. The main reason for this is that when the poverty rate increases the fund needs increased returns to meet the demands of more families in poverty. The second reason focuses on one of the base assumptions of investing which is that individual investors do not affect the market. We have seen from experience that this is not true as currently existing mutual funds that are extremely large do indeed effect the market as discussed in a March 2016 The New Yorker article [19]. That being the case when poverty level families invest their money it is in their best interests to invest their money in companies that perform well during times of the investor’s greatest need, when the poverty rate increases.

1.1.1

Simplifications and Presumptions

To facilitate the initial investigation and research into developing an investment fund type and model as described in the previous subsection there are several simplifications and presump-tions that have been made. Firstly, to simplify the universe of available instruments from which to choose for investment only corporate equities available from the online resource Ya-hoo! finance are used. This simplification decreases the potential efficiency and performance of the resulting fund by excluding fixed income instruments, financial derivatives, and other

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exotic instruments while also excluding many internationally traded and all OTC traded in-struments. Secondly, to increase the simplification of the study and decrease the amount of analysis required the instruments available for asset selection were not corrected for survivor-ship bias and, by the nature of Yahoo! finance stock screener, are 3 year growth biased. Each of these two biases potentially impacts the future performance of the resulting fund in the pos-itive direction. Moreover, the historical data used was smoothed from daily data to monthly data using simple averages instead of moving avergaes. This had the result of potentially los-ing some information regardlos-ing monthly average returns but was necessary due to limitations in the computational power of the programs and hardware being used for data analysis. And lastly, the investment strategy used to evaluate model performance is a very naive strategy where no seurity weight or model index adjustments were made over the time period of the evaluation. The effects of this simplification is that the resulting evaluation analysis loses ac-curacy from the first month under evaluation to the last but the time required to include that level of complexity in the evaluation was not available. In addition to these simplifications historical returns are presumed to be random and normally distributed and the S&P500 index is presumed to be appropriate to represent a broad view on the general economy of the United States.

1.2

Literature review

In this section, we describe the financial and related mathematical principles that will be used in this paper. I explain the historical financial principles in chronological order as the later advances in portfolio theory build on the principles of previous research.

1.2.1

The Markowitz Model

In March of 1952 Harry Markowitz revolutionised asset portfolio investment when he presen-ted to the world his research on portfolio optimization of risk vs. return through asset selection. He proved that the maximization of discounted returns as a principle investment strategy was inefficient. By careful consideration of the cooperative movement between paired assets an in-vestor could realize higher returns for comparable, or even lower, risk for comparable returns. A portfolio with this characteristic is considered efficient [22].

The expected return of a portfolio is the weighted sum of the expected returns of each individual asset, ¯ Rp= N

i=1 XiR¯i, (1.1)

where ¯Rp is the portfolio expected return, Xi is the weight of the ith asset, ¯Ri is the expected

return of the ith asset, and N is the total number of assets. The volatility of a portfolio, σp2, is the expectation of the squared demeaned weighted returns of the portfolio’s individual assets,

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Recalling that (X +Y + Z)2= X2+Y2+ Z2+ 2XY + 2X Z + 2Y Z we can represent the volatility of a portfolio thusly, σp2= N

i=1 Xii2+ N

i=1 N

j=1 XiXjσi j, (1.3)

with j = 1, 2, ..., N, j 6= i, σi2= volatility of the ith asset, and σi j is the covariance between the

ith and jth assets.

This seminal work by Markowitz utilizes the covariance structure of returns to maximize diversification of returns in an efficient portfolio. The means to calculate the efficient portfolio is a quadratic programming problem. The Critical Line Method developed by Markowitz is useful in solving this problem [22]. This paper uses the Excel solver add-in to compute asset weights for individual efficient portfolios directly from the variance-covariance matrix and historical expected returns. The Excel solver uses a Generalized Reduced Gradient Algorithm (GRGA) for optimising nonlinear problems. GRGAs are algorithms that attempt to minimize a general form nonlinear function using specified constraints [18].

1.2.2

The Single Index Model

William Sharpe, in a paper presented in 1963, proposed a means to estimate future expect-ations of a portfolio that simplified the variance covariance matrix calculation process used in the Markowitz model. He posited that by regressing returns against a shared index as an approximation for all co-movement between returns a reasonable approximation for future ex-pectations could be reached. This model simplified the covariance matrix by eliminating the non-diagonal values of the covariance matrix, assets do not co-vary independent of the index, from the needed calculations thus it is named the Diagonal Model [29].

Sharpe’s decomposition of an ith asset’s return, Ri, consists of asset specific parameters, Ai

and Bi, and random component, Ci, with an expected value of zero and variance Qi,

Ri= Ai+ BiI¯+Ci. (1.4)

where ¯Iis the expected value of some time series index I. The index I can be decomposed into a single constant parameter, An+1, and a random component, Cn+1, with an expected value of

zero and a variance of Qn+1,

I= An+1+Cn+1,

¯

I= E[An+1] + E[Cn+1] = An+1.

(1.5)

Therefore the expected return of the ith asset, ¯Ri, is,

¯ Ri= Ai+ BiI,¯ (1.6) and, σI2= Qn+1, σC2i= Qi, σi2= B2iσI2+ σC2i, σi j= BiBjσI2. (1.7)

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Making the substitutions from the Diagonal Model equation (1.1) becomes, ¯ Rp= N

i=1 XiAi+ N

i=1 XiBiI,¯ (1.8)

and equation (1.3) becomes,

σp2= N

i=1 N

j=1 XiXjBiBjσI2+ N

i=1 XiC2i. (1.9)

For the purposes of this paper the significant aspects of the Single Index Model, apart from the fundamental principles of portfolio analysis through indexes, is estimation of the Ai and

Biparameters. Ai, Bi, and σC2i cannot be directly observed from the market using historic data

but can be estimated using regression analysis. The estimations are, by definition,

Bi= E[(Ri− ¯Ri)(I − ¯I)]/E(I − ¯I)2= σiI/σI2, (1.10)

Ai= ¯Ri− BiI,¯ (1.11)

σC2i= σi2− B2iσI2. (1.12) When this estimation is used on the market as the index it is traditionally named Beta and represented by the Greek letter of the same name however Sharpe clarified that any appropriate index could be used. For our purposes the Bicomponents for all assets are found using proxy

indexes constructed from time series data regression [29].

1.2.3

The Multi Index Model

Professionals and academics alike have noted that contrary to the basic principle of the Single Index Model of a single shared return index multiple indexes better reflect the reality of as-set return changes. The multi index belief is evidenced by the numerous averages available on the market such as industrials, rails, and utilities [10]. In 1966 Benjamin King conduc-ted an in-depth study on the Multi Index Model with the aim to discover and “explain” the various market indexes with the goal of isolation and consolidation. King concluded that his analysis supported the hypothesis that multiple market and industry indexes can be useful in predicting asset return movement [16]. The formulation of the Multi Index model is a fairly straightforward extension of the Single Index model. As is the case with the previous model the assumption that the covariance of individual asset returns not related to model indexes is zero [10].

In the Multi Index model framework equation (1.4) becomes,

Ri= Ai+ Bi1I1+ Bi2I2+ ...BiLIL+Ci, (1.13)

for indexes 1, 2, ..., L and volatility of the Lthindex, σIL2, equations (1.6) and (1.7) become, ¯

Ri= Ai+ Bi1I¯1+ Bi2I¯2+ ...BiLI¯L,

σi2= B2i1σI12 + B2i2σI22 + ... + B2iLσIL2 + σC2i, σi j= Bi1Bj1σI12 + Bi2Bj2σI22 + ... + BiLBjLσIL2,

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with parameters Ai, BiL, and σC2i, BiL= σiIL/σ 2 IL Ai= ¯Ri− Bi1I¯1− Bi2I¯2− ... − BiLI¯L σC2i= σi2− B2i1σI12 − B2i2σI22 − ... − B2iLσIL2. (1.15)

For the multi index model portfolio entire for instruments 1, 2, ..., N equations (1.8) and (1.9) become, ¯ Rp= N

i=1 XiAi+ N

i=1 L

k=1 XiBikI¯k, σp2= N

i=1 N

j=1 L

k=1 XiXjBiBjσI2k+ N

i=1 XiC2i, σC2p = N

i=1 σi2− N

i=1 L

k=1 B2ikσk2. (1.16)

The multi index model proposed in this paper utilizes four orthogonal indexes constructed as proxies for real world data. The four orthogonal indexes in the 4 Factor model represent the return rate of the S&P500, the US inflation rate, the US unemployment rate, and the US poverty rate, all from 2011–2014.

¯ Rp= N

i=1 XiAi+ N

i=1 XiBiMIM¯ + N

i=1 XiBiPI¯P+ N

i=1 XiBiII¯I+ N

i=1 XiBiUI¯U, σp2= N

i=1 N

j=1 (XiXjBiMBjMσM2 + XiXjBiPBjPσP2+ XiXjBiIBjIσI2+ XiXjBiUBjUσU2) + N

i=1 XiC2i, σC2p= N

i=1 σi2− N

i=1 (B2iMσM2 + B2iPσP2+ BiIσI2+ B2iUσU2), (1.17)

where M, P, I, and U represent the Market, Poverty, Inflation, and Unemployment rate proxies respectively [16].

1.2.4

Asset selection: P/E, PEG, and PEGY ratios

How to strategically narrow the available field of assets in which to include in one’s investment portfolio is a very robust area of research with many different approaches. On the one side you have such scientific methods as picking the asset whose ticker symbol is the initials of the investor’s favourite niece while on the other side you have a complete and detailed analysis of a company’s earnings, dividends, risk, the cost of money, and future growth rate. Common tools for evaluating a company’s true value are Discounted Cash Flow Models. These models strive to compute a company’s true value by analyzing discounted future characteristics, such

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as dividends or earnings, which Miller and Modigliani discussed in their work in 1961 [25]. Many investor’s, however, do not have the time or resources to conduct thorough cash flow valuations particularly when the number of available assets for inclusion in one’s portfolio is vast.

One method to facilitate access to the asset valuation process is the use of ratios which distill asset specific information into easily understood and comparable numbers. Arguably chief among these ratios is the Price to Earnings ratio (P/E ratio). The P/E ratio measures the price investors in the market are willing to pay for one dollar of a company’s earnings, the P/E Ratio = Price per Share/Earnings per Share [35].

Another common ratio used by investor’s is a modified version of the P/E ratio known as the PEG ratio. This ratio includes information on a company’s growth rate, either historical or future, in an attempt to provide a broader perspective on a company’s investment value. A general rule of thumb is that a PEG ratio below one is desirable where the PEG Ratio = P/E Ratio ÷ Annual Earnings Per Share Growth [21].

The ratio utilised in this paper is a modification of the PEG ratio known as the PEGY ratio. The PEGY ratio adds a company’s dividend yield to the earnings per share growth of the PEG ratio [21]. This ratio attempts to capture whether a dividend paying company is traded favourably for the investor on the market with respect to price appreciation. The standard PEGY ratio, P/E Ratio ÷ (Annual Earnings Per Share Growth + Dividend Yield), information is not available on Yahoo! Stock Screener however by modifying the PEGY ratio, (PEG−1+Yield/(P/E))−1, it can be used as a search parameter provided the PEG and P/E ratios are not zero.

1.2.5

The Beta Puzzle

Beta as measured in the Single Index Model, equation(1.8), plays a key role in many invest-ment strategies. By increasing the Beta of a portfolio through inclusion of high Beta assets an investor hopes to capture greater exposure to up markets while leveraging against down markets. A white paper co-authored by David Cowen and Sam Wilderman in 2011 for GMO, a private investment firm, details how it is low Beta investment portfolios that have historically outperformed both the market and high beta portfolios. By eschewing the added premium im-posed on up side exposure assets in high demand, high Beta assets, an investor can potentially outperform the market with substantially lower realised volatility and smaller draw-down peri-ods. The counterintuitive nature of buying higher returns with discounted volatility is what is popularly referenced as the Beta Puzzle. Through an analogous examination of the options market the authors provide a reasonable explanation for Beta Puzzle behaviour [9]. A port-folio’s Beta can be optimized directly after initial asset selection but by filtering out high Beta assets as part of the asset selection process the portfolio has increased protection against inclusion of high Beta assets that might get otherwise included in a post portfolio creation optimization.

For the purposes of this paper a deeper understanding of the Beta Puzzle is enlightening but unnecessary. Based solely on the historical performance of low Beta portfolios outperforming the market the stock selection process prefers a low covariance, a Beta < 1 is consider less

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volatile than the market, with the market proxy index.

1.2.6

Portfolio evaluation techniques

Apart from the process of creating an investment portfolio using principles grounded in port-folio theory an investor must evaluate the performance of their investment strategy. There are many ways to evaluate portfolio and investment manager performance. An historically com-mon approach, though crude, was to simply measure return performance, “Did an investment strategy increase returns over the period of the investment?” With the increasing importance of Modern Portfolio Theory in the management of investment holdings evaluation techniques have moved away from simple investment return evaluation and now include in-depth analysis of risk exposure versus return potential as well as other forms of statistical and empirical ana-lysis. This paper utilises two rather straightforward approaches to evaluate the performance of the created portfolios, the Excess Return to Variability Measure and the Differential Return with Risk Measured by Standard Deviation.

The Excess Return to Variability Measure evaluation process involves investigating all potential portfolios of interest and ranking them according to the unit increase in excess return, over the risk free lending and borrowing rate, versus the portfolio’s variability. The portfolio with the highest unit return per unit increase in variability, measured in standard deviation, is the preferred portfolio with the desired level of risk obtained through lending and borrowing of the risk free asset. The name of this ratio is the Sharpe ratio [10].

Given a portfolio with return ¯Ri and variance σi2 the preferred portfolio is the portfolio with the maximum Sharpe ratio, ¯Ri/σi.

The Differential Return with Risk Measured by Standard Deviation evaluation process involves comparing the performance of an investment portfolio against the naive strategy of buying and holding an appropriately representative index fund. By comparing the difference in return from the portfolio being evaluated and a risk adjusted index fund all portfolios of interest can be ranked comparably to the naive strategy [10].

¯

Rp− ¯RRAI = Differential Return, (1.18) where ¯Rp is the expected return on a portfolio and ¯RRAI is the risk adjusted expected return, measured by that index’s historical return performance, on some ideally performing bench-mark index fund. The initial model investment portfolios in this exercise are measured against the S&P500 and the final optimum investment portfolio measures realized portfolio returns against the US Federal Reserve yearly interest rate available at https://www.federalreserve.gov/.

1.3

The mathematical tools

In this Section, we explain the mathematical tools that will be used in this paper. I have presented the math in chronological order as the later work is built upon important aspects of the previous work.

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1.3.1

Principal Component Analysis

Modern principle component analysis (PCA) is a multivariate statistical technique first for-mulated by Karl Pearson [26] in 1901 and later formalized by Harold Hotelling in 1933 [13]. PCA is used across almost all scientific fields as a statistical processing tool for data tables populated with inter-correlated observations described by several dependent variables. By analyzing inter-correlated observation data tables with PCA the original data is approximated as the product of two different matrixes that provide simplification of the information and potential data reduction for large amounts of data [36]. This simplification and reduction is done by an orthogonal linear transformation to a new subspace whose axes are the principle components of the data which are by construction uncorrelated [14].

Let ξ1, ξ2, . . . , ξn be n random variables. Assume that E[ξ2j] < ∞ for all j = 1, 2, . . . , n.

We observe each variable m times and write down the observations into a matrix, ˜X, with m rows and n columns. The matrix ˜X is called the data table. The goals of PCA are [4].

1. extracting the most important information from the data table;

2. compressing the size of the data set by keeping only this important information;

3. simplifying the description of the data set as a set of new random variables η1, η2, ..., ηp

which are uncorrelated; and

4. analyzing the structure of the observations and the variables.

Before analysis the variables and observations of matrix ˜X, ξ s and m’s respectively, re-quire preprocessing that serves to calibrate the information in such a way as to minimize the effects of statistically outlaying observatons like localized extreme values and differenes in measurement units and to make the data more symmetrically distributed [36]. For the pur-poses of this paper one preprocessing step is taken: the utilization of logarithmic residuals of the observations to make them more symmetrically distributed. After preprocessing the data table can be understood as ˜X= ˜1 ¯x+ ˜Y+ ˜E where ˜1 is the identity matrix, ¯xis a row vector of

˜

X and ˜Y is a new matrix containing the variance information data set from the original data table observations and ˜E is an error matrix.

In order to extract the principal components from the variance matrix ˜Y the singular value decomposition (SVD) method is used. The SVD theorem states that a rectangular matrix

˜

Y = ˜U ˜S ˜VT where ˜U is an m × m matrix, ˜V is an n × n matrix, ˜U and ˜VT are orthogonal, and ˜S is a diagonal matrix with the dimensions of ˜Y whose non-zero elements are unique [31].

Given a matrix ˜ Y =    a11 a12 ... ... a1n a21 ... ... ... .. . ... ... ... am1 ... ... ... amn   

we solve for the singular values by computing the matrices ˜Y ˜YT and ˜YTY˜ whose eigenvectors make up the columns of matrices ˜U and ˜VT respectively. The matrices ˜Y ˜YT and ˜YTY˜ have eigenvectors ¯xuand ˜xvand eigenvalues λuand λvsuch that ( ˜Y ˜YT− λu˜1) ¯xu= ( ˜YTY˜− λv˜1) ¯xv=

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0. For these equalities to be true the det( ˜Y ˜YT − λu˜1) = det( ˜YTY˜ − λv˜1) = 0. Solving these

equations for λuand λv and then ¯xuand ¯xvresults in

˜ U =     xu11 xu12 ... xu 1m xu 21 ... ... .. . ... ... xum1 ... ... xumm     , ˜ V =     xv11 xv12 ... xv1n xv21 ... ... .. . ... ... xvn1 ... ... xvnn     , and ˜ S=     √ λ11 012 ... 01n 021 √ λ22 ... .. . ... ... 0m1 ... ... √ λmn     where λ11> λ22> · · · > λmn.

The SVD constructs a linear combination of principal components which model the vari-ance matrix ˜Y such that elements of ˜U ˜S are orthogonal factor scores, ˜F, that define the axes of a subspace upon which the variables, or columns, of ˜Y are projected onto with loadings represented by ˜V. The magnitude of each element in ˜V measures the relative contribution of the associated variable to the factor score. This relationship can be more easily viewed as a series of equations [4]

˜

F = ˜U ˜S= ˜U ˜S ˜VTV˜ = ˜Y ˜V. (1.19) To simplify the model the least significant factor scores in the linear combination can be removed and replaced with an error term matrix, ˜E, [36]

˜

Y = ˜U ˜S ˜VST+ ˜E. (1.20)

where ˜VST is ˜VT with the least significant factor scores removed.

A standard guildeline for determining how may factor scores to include in the PCA model is by using a scree plot. A scree plot is a line graph that plots the variance explanatory power of each factor score to see if there is a factor score where the slope of the line goes from steep to flat like in figure 2.1. This bend in the scree plot is called an elbow and it delineates how many factor scores provide significant explanatory power [36].

1.3.2

Non-linear Iterative Partial Least Squares Regression

Non-linear estimation by iterative least squares is the pioneering work in partial least squares regression done by Herman Wold in the late 1960’s. It was originally created for use in the field of social sciences but was improved upon and renamed non-linear iterative partial least squares (NIPALS) by Svante Wold and Esbensen in 1987 [36]. NIPALS is an algorithm used on high dimensional datasets for computing principal components [2, 17].There are several

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variants of the NIPALS algorithm however the variant that transforms variables X and Y to have mean zero is considered to be one of the most efficient [3].

The NIPALS algorithm constructs a series of variables called latent variables (weights, scores, loadings etc) as linear combinations of the dataset. This means that NIPALS, similar to Principle Component Analysis, decomposes both ˜X and ˜Y as a product of a common set of orthogonal factors and a set of specific loadings so that there is no correlation between factors used in the predictive regression model. So we have the model as,

˜

X = ¯t ¯p0+ ˜E, ˜

Y = ¯uc¯0+ ˜G, (1.21)

where ˜X is the predictor matrix and ˜Y is the response matrix. ¯t, ¯u, ¯p, and ¯care the vectors of X-scores, Y -scores, X -loadings, and Y -loadings respectively. ˜E and ˜Gare error term matrices assumed to be independent and identically distributed random variables.

The pre-step in the algorithm is to find the first or preliminary Y -scores vector, ¯u. The vector ¯ucan either be initialised as random values or chosen from any column of ˜Y. Later this parameter is then updated. The latent variable matrices; ˜T, ˜U, ˜P, and ˜Cwill be built one vector at a time through the following projections with constraints that ¯w0w¯ = 1 and ¯t0¯t = ˜I where ˜I is the identity matrix [2, 17].

Step1 w¯= ˜X0u/( ¯¯ u0u)¯ estimate X− weights Step2 ¯t = ˜Xw¯ estimate X− scores Step3 c¯= ˜Y0¯t/(¯t0¯t) estimate Y− weights Step4 u¯= ˜Yc/( ¯¯ c0c)¯ estimate Y− scores Step5 p¯= ˜X0¯t/(¯t0¯t) estimate X− loadings Step6 E˜1= ˜X− ¯t ¯p0:= ˜Xupdated partial out component

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The procedure is re-iterated an arbitrary number of times, A, until we obtain the desired number of components approximated from ˜X (represented by vectors ¯t ¯p0). While the number of re-iterations is arbitrary the maxium number of components is limited to the least value of the variables in the predictor and reponse matrices, ˜X and ˜Y. These components or scores are orthogonal to each other and contain as much unique variance of ˜X as possible.

For clarification, let us examine how step 6 works. After the first component, represented by ¯t1p¯01, has been extracted from ˜X the outer product, ¯t10p¯01, can then be subtracted from ˜X

leaving the residual matrix ˜E1. This residual matrix will then be used as the updated matrix ˜

X to calculate subsequent components as steps 1 to 6 are repeated but this time we use the Y -scores derived in step 4 as ¯u. The whole procedure is repeated until we obtain A components. This procedure dramatically reduces computational time since calculation of the covariance matrix is avoided.

According to the above NIPALS algorithm that gives A vectors for the X -scores, ta, and

X-loadings, pa, the model X in element form is,

Xmn=

a

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where enkare X -residuals, ˜E. The X -scores are also estimates of ˜Y when multiplied with the

Y-weights, cain,

Yn=

a

catna+ fn, (1.24)

where fn are the Y -residuals. The factors ta and ua contain information about the predictors

and how they relate to each other. The weights wa and ca give us information on how the

variables combine to form the quantitative relation between the X and Y . The element form of X -factors, ta, is also given as,

tna=

k

wkaxnk. (1.25)

As in most multivariate regression methods, the purpose of NIPALS is to build a linear model in the form,

y=

β x + e (1.26)

So far we have derived all necessary elements and the PLS representation of Y is described,

ynm =

a cma

m wmaxnm+ fnm, (1.27) or in matrix form, Y = XWC0+ F (1.28)

Let βabe the beta coefficients of the linear model with the same number of columns as the

matrix ˜Y giving us the vector ¯β in our case. The estimated ˜β is derived from the combination of weights and is the basis for predicting new Y -values, ˜Y from the new X -data, ˜Xnew,

˜

Y = ˜Xnewβ .˜ (1.29)

In this study the NIPALS algorithm used in R is superior to PCA using the SVD, described in subsection 1.3.1, as it allows for calculation of loadings and factor scores for data tables that have missing data. The time series for the assets analized in this study have different starting dates and therefore when combined in a data table there is significant missing data.

1.4

Methodology

This section describes the 20 step process developed to select appropriate assets, create and evaluate a simple equity investment portfolio, and the creation of single- and multi-index factor models along with their evaluation.

1.4.1

Asset selection

Step 1: Find all stocks available which fit the following criteria: must pay a dividend and have a positive PEG ratio.

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Step 2: Calculate the PEGY ratio for all stocks currently selected and select those which have a PEGY ratio that is below the average PEGY ratio for the entire selection.

Step 3: Obtain the complete history of daily return data for each of the stocks selected.

Step 4: Smooth the data for all stocks from daily to monthly to smooth out the random price fluctuations of the asset returns as well as to standardize the time periods of each asset for time series analysis..

1.4.2

First asset filter

Step 5: Collect monthly data on the socio-economic factors used for indexes. Collect daily data on the market index used to represent the general economy and convert to monthly data using the same smoothing technique as on the individual asset return series. Cal-culate the monthly return rate for the market index.

Step 6: Use PCA, with the matrix ˜X composed of the market and socio-economic indexes and ¯ua vector of random values, and R statistical software to produce socio-economic and market rate proxies that are uncorrelated yet retain the original data’s variability characteristics. Identify the proxy indexes by comparing their correlations and covari-ances with the original indexes and analyze them for appropriateness using correlation and covariance ratios, number of elements, and mean relative difference.

Step 7: Calculate the covariance of each stock selected with the uncorrelated market proxy index and select the stocks which have a covariance less than 1 to fulfill our low market index covariance fund requirement.

Step 8: Calculate the covariance of each stock from step 7 with one of the socio-economic rate proxies and select the remaining stocks which have a positive covariance to fulfill our requirement that the fund experience increased performance during times of increas-ing movement in that socio-economic indicator.

Step 9: Repeat step 8 for each successive socio-economic rate proxy.

1.4.3

Second asset filter

Step 10: Collect yearly data on the poverty index.

Step 11: Convert the stock data from step 9 to a time series that coincides with the data for the poverty index.

Step 12: Calculate the covariance of each stock from step 8 with the poverty index and select the stocks that have a positive covariance to fulfill our requirement that the fund experience increased performance during times of increasing poverty.

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1.4.4

Third asset filter

Step 13: Use the NIPALS algorithm on the adjusted returns of the portfolio stock picks and evaluate diversification by comparing factor loadings.

Step 14: Remove stocks which overly contribute to any secondary principle components identified in step 13 and repeat until exposure to more than one principle component is within acceptable levels according to scree plot analysis.

1.4.5

Portfolio creation and evaluation

Step 15: Calculate the geometric mean return for all asset returns using only the previous 3 years of available data and calculate the effective risk free rate using a monthly risk free rate calculated from government treasury bonds.

Step 16: Create a variance covariance matrix of the surviving assets for use in the optimiza-tion problem in step 17.

Step 17: Calculate individual stock weights, portfolio excess dividend adjusted returns, and risk for: a minimum variance portfolio, a maximum return portfolio, a maximum Sharpe ratio portfolio with both short selling and no short selling with maximum 30% short selling allowed for both unconstrained investiture in any 1 stock and constrained, 0.001 minimum investment. Using generalized reduced gradient non-linear optimization op-timize the weights for each portfolio according to minimum variance as measured by standard deviation, maximum return, and maximum Sharpe ratio individually.

Step 18: Evaluate the portfolios from step 17 using both the Excess Return to Variability Measure Evaluation and the Differential Return with Risk Measured by Standard Devi-ation. Select the portfolio that performs best according to the intended purpose of the fund which is to minimize risk while

1.4.6

Model creation and evaluation

Step 19: Using the proxy indexes for the market, poverty rate, and socio-economic factor rates create single and multi-index models to predict portfolio performance. Measure the predictive quality of each model by aggregate absolute value z-scores evaluated using the cumulative distibution function.

Step 20: Measure the models from step 19 against realized asset return performance data for the portfolio selected from step 18. Use the same evaluation techniques from step 18.

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Chapter 2

Solution

This chapter focuses on the application of the theoretical framework outlined in Chapter 1 to create a hypothetical investment portfolio. Each step in the methodology will be addressed seperately with references to the appropriate sections and equations from Chapter 1. All data collected was available for free online at the time of this paper’s creation from the sites refer-enced in each section. Similarly all programs used in the data collection and analysis of this investigation were also free and available online with the single exception of Microsoft Excel.

2.1

Asset selection process

The following section will detail the asset selection process for inclusion in a portfolio using all asset return data available from Yahoo! Finance Stock Screener.

2.1.1

Asset selection

The only requirements for the first selection of assets are that each asset pay a dividend and have positive growth as detailed in section 1.3.4 to ensure calculation of the modified PEGY ratio is possible. This introductory search returned 1387 assets for consideration with a simple average PEGY ratio of 1.7915. There were 981 assets with a PEGY ratio below the average and those assets were selected. The entire daily asset price history for each of the 981 selected assets was downloaded using an Excel workbook script, Multiple Stock Quote Downloader for Excel, created by Samir Khan. Using historical data it is necessary to use the adjusted closing price of each asset rather than the actual closing price of each asset. This avoids incorrectly measuring stock splits and dividend payments as devaluation of the associated company’s stock. Simple averages of each asset’s daily returns over each month were calculated in an attempt to smooth out the random price fluctuations of the asset returns as well as a way to standardize the time periods of each asset for time series analysis.

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2.1.2

First asset filter

The monthly unemployment and inflation rates of the United States were collected from the United States Bureau of Labour Statistics from 1960-2016. Historic data for the S&P500, to be used as a market index, was collected from Yahoo! Finance. Simple monthly averages were calculated from daily adjusted close values of the index to maintain consistency with the calculations of the stock data and the S&P500 monthly rates were calculated.

Using the statistical software analysis program R, with code available online [6], the cor-relations of these three rates were calculated and the three series of rates were stored in an R matrix, with the results,

cor(C) #The indexes are correlated with each other

[,1] [,2] [,3]

[1,] 1.00000000 -0.0223683 0.02334898 [2,] -0.02236830 1.0000000 -0.12195340 [3,] 0.02334898 -0.1219534 1.00000000

where C is the series’ matrix in R, series 1 is the market index rate, series 2 is the inflation rate, and series 3 is the unemployment rate.

The PCA component of the nipals function in R was used on the matrix of index rates, C, to take advantage of the uncorrelating effect on the resulting factor scores from equations (1.22). To verify that the resulting indexes are adequate to serve as proxy indexes for our original rates we analyse several metrics: number of elements, element difference, correlation, and covariance.

The number of elements in the original indexes and the proxy indexes must be the same to facilitate time series analysis,

#Test if the number of elements (rows) are equal > all.equal(nrow(C),nrow(my_nipals3$scores)) [1] TRUE

The elements in the proxy indexes must be adequate representations of the original ele-ments,

#Test if the individual elements are comparably equivalent > all.equal(mark.new,UncorrMARK)

[1] "Mean relative difference: 0.2591693" > all.equal(inf.new,UncorrINF)

[1] "Mean relative difference: 0.1295227" > all.equal(test.uep,UncorrUEP)

[1] "Mean relative difference: 0.003486381"

where the market, inflation, and unemployment proxy index elements are an average 25.92%, 12.95%, and 0.35% different respectively.

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cor(my_nipals3$scores) #The scores are uncorrelated with each other

t1 t2 t3

t1 1.000000e+00 -5.545987e-05 1.955042e-10 t2 -5.545987e-05 1.000000e+00 -3.478283e-06 t3 1.955042e-10 -3.478283e-06 1.000000e+00

Compute the correlations and covariance ratios of the original indexes with their proxies,

#Correlations and covariances between original data and uncorrelated data

> diag(cor(C,UncorrC)) #Values approaching 1

indicate an increasingly strong linear relationship [1] 0.9997062 0.9923318 0.9999997

> abs(diag(cov(C))-diag(cov(UncorrC))/diag(cov(C))) [1] 0.9983121 0.9847190 0.9671737

where the amount of variance in the original series explained by the proxy indexes is 99.83%, 98.47%, and 99.99% for market, inflation, and unemployment respectively. Using the proxy indexes as acceptable representatives for the original series of the market, inflation, and unemployment rates the covariance ratios of each of the 981 stocks selected through the first selection round were calculated using the process described in equation (1.10) with all information up to December 2014. Any assets with a market proxy covariance ratio less than one, as explained in section 1.3.5 The Beta Puzzle, and inflation and unemployment proxy covariance ratios of greater than zero, as explained in section 1.1 Characteristics of the Portfolio were selected. This process returned 236 stocks with the required characteristics.

2.1.3

Second asset filter

The monthly returns of the 236 viable stock selects from the second round selection must be converted to yearly data with the year starting in September and ending in August to coincide with how the annual poverty rate is calculated by the US Census Bureau. Unlike the other series in the process a proxy series is not calculated to represent the poverty rate. As we use monthly data for the first three indexes and yearly data for the poverty rate an uncorrelated poverty rate proxy was not created. Calculating covariance ratios per equation (1.10) between the assets and the poverty rate and selecting all assets whose poverty rate covariance ratio is greater than one, as explained in section 1.1 Characteristics of the Portfolio, returned 66 assets viable for inclusion in our financial portfolio. Covariance ratios are included in Appendix A with reference (A, 1-10).

2.1.4

Third asset filter

Taking the previous three years of returns from the 66 assets from the third round selection process and converting them to log returns, available in Appendix A with reference (B, 1-10),

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Figure 2.1: Asset diversification scree plot with the variance explanatory power of principle components shown as a percentage of total variance of the data.

allows for index factor analysis using the NIPALS algorithm. According to equation (1.29) the loadings, Xnew, are the weights, or factor loadings, of each asset to the principle components

of the total return on all assets.

head(Log_nipals$loadings) #Loadings for the first six assets

p1 p2 ECOL 0.0820753 0.1869120 SQM 0.1198656 -0.3136985 QSII 0.1300255 0.1215666 AIN 0.1970745 0.1215075 SLW.TO 0.1256239 0.1432741 UGP 0.1656027 -0.2220545

By analysing the loadings of each asset and removing the assets which exhibit comparably large sensitivity to the second principle index component, the scores, overall sensitivity to the second and subsequent indices can be reduced. The end state of this process is a combination of assets which have been diversified across all principle components that are orthogonal to the primary component. The filtering process was conducted until the primary index contained at least four times more explanatory power than the second largest index (see figure 2.1).

This process returned a portfolio selection universe of 45 viable assets.

2.2

Portfolio creation and evaluation

This section details the creation and evaluation of multiple portfolios, each with different return and risk characteristics, using the mean variance technique of the Markowitz model according to equations (1.1) and (1.3). Minimum variance, maximum return, and maximum Sharpe ratio portfolios were created under the following constraints: each portfolio must be

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fully invested, both with and without shortselling (shortselling allowed up to a maximum of 30% of the initial investment), with and without a requirement that at least 0.1% must be invested in each available asset. A single portfolio restricting investment in any single asset to 15% of the initial investment with short selling allowed was also created. These portfolios were then evaluated using the methods detailed in section 1.3.6.

2.2.1

Create portfolios

Geometric mean returns and sample standard deviations were calculated for each of the 45 assets from the third asset filter process from data going back three years available in Appendix A with reference (G, 11). Asset correlations were computed and a variance covariance matrix was created and is available in Appendix A with reference (C, 11).

Monthly dividend yields for each asset were estimated as the discounted yield rate for each asset paid quarterly or annually, dependent upon the historical dividend payment structure, and added to the geometric mean returns of all assets that were not sold short in each portfolio calculation. The monthly risk free rate was taken as the discounted 3 month US treasury bond rate for December 2014. Only the annual excess return rate for each portfolio includes the risk free rate information which is needed for portfolio evaluation. The monthly and annualised standard deviations, mean returns, and Sharpe ratios of each portfolio, excluding the equally weighted portfolio, were solved using generalized reduced gradient non-linear optimization [20] available in the Excel Solver add-on and are presented in Appendix A with reference (D, 11). The asset weights associated with each portfolio are available in Appendix A with reference (E, 11).

2.2.2

Evaluate each portfolio

Figure 2.2 is a graphical analysis of the 15 portfolios from table 2.1 according to the standards of Excess Return to Variability Measure outlined in section 1.3.6 which displays trend lines for portfolio returns in excess of the risk free rate according to the Sharpe Ratio. The four most efficient portfolios are the upper left most red trend lines.

Table 2.1 displays a computational analysis of the 15 portfolios, using equation (1.18) and the S&P500 from January 2011 to December 2014, according to the standards of the Differen-tial Return with Risk Measured by Standard Deviation outlined in section 1.3.6 returns. Note the data is in the order of the evaluation method results and not in the order of each portfolios optization method.

Comparing the top four optimum portfolios from the latter evaluation technique returns the same portfolios as the former technique namely: Maximum Sharpe Ratio with Short Selling, Maximum Sharpe Ratio with Short Selling and 0.1% Minimum Investment, Maximum Sharpe Ratio with Short Selling, 0.1% Minimum Investment, and 15% Maximum Investment, and Minimum Variance with Short Selling, 0.1% Minimum Investment, and 15% Maximum In-vestment.

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Figure 2.2: Graphical analysis of percent excess return to variability measure with the 4 highest ranked portfolios designated with a red trend line.

Portfolio Differential Returns Monthly Returns Risk Adjusted Market Returns MaxS SS 1,5533% 0,0264% 0,0109% BD MaxS SS 0,9842% 0,0221% 0,0123% BD MaxS SS 15% 0,8442% 0,0229% 0,0145% BD MinV SS 15% 0,5527% 0,0192% 0,0137% MinV SS 0,3032% 0,0109% 0,0079% BD MinV SS 0,2759% 0,0120% 0,0092% MaxS NSS 0,1845% 0,0185% 0,0166% BD MaxS NSS 0,1178% 0,0184% 0,0173% MinV NSS -0,1898% 0,0104% 0,0123% BD MinV NSS -0,2080% 0,0106% 0,0127% BD MaxR SS 15% -0,3632% 0,0326% 0,0362% MaxR SS -2,4580% 0,0573% 0,0819% BD MaxR NSS -2,9228% 0,0399% 0,0691% MaxR NSS -3,0411% 0,0412% 0,0716% BD MaxR SS -4,0466% 0,0552% 0,0956%

Table 2.1: Differential return with risk measured by standard deviation analysis where MinV means minimum variance, SS means short selling allowed, NSS means no short selling al-lowed, MaxR means maximum return, MaxS means maximum Sharpe Ratio, and BD means best diversitfication selection.

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Figure 2.3: Principle component variance explanatory analysis of Max Sharpe with Short Sales portfolio.

2.2.3

Select an optimum portfolio

The four top performing portfolios according to both evaluation metrics in the previous section were selected for further evaluation. A follow-up study could benefit from evaluating more than the top four portfolios but the time constraints of this study made such a broad study prohibitive. This evaluation involves a detailed comparative exploration of the diversification of each portfolio using weighted returns and variances through principle component analysis of log returns using equations (1.23). The weights for each asset in each portfolio are shown in Table 2.2.

Including the weights for each portfolio into the principle component analysis the percent variance explained by the principle components of the log returns are graphically represented (see figures 2.3-2.6)

To explain at least 70% of the variance, marked with a solid line, in each portfolio requires 8 principle components in the Max Sharpe with Short Sales portfolio (figure 2.3), 10 with the Max Sharpe with Short Sales with all Assets (figure 2.4), 10 with the Max Sharpe with Short Sales with all Assets and 15% investiture (figure 2.5), and 10 principle components with the Min Variance with Short Sales with all Assets and 15% investiture (figure 2.6).

As described in section 1.1 a portfolio that caters to the needs of investors whose income is near, at, or below the poverty line must be resilient to unexpected outcomes. Of the best peforming portfolios in regard to return the portfolio with the least variance best serves the needs of the investors with regard to risk. In addition, a portfolio that retains exposure to all assets used to provide adequate diversification adds increased risk protection against

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unex-MaxS SS BD MaxS SS BD MinV SS 15% BD MaxS SS 15% ECOL 0,0911 0,1190 0,1053 0,1106 SQM 0,0000 0,0010 0,0010 0,0010 QSII -0,1120 0,0010 0,0010 0,0010 AIN 0,0000 0,0010 0,0010 0,0010 SLW.TO 0,0000 -0,0194 -0,0052 -0,0058 UGP 0,0000 0,0010 0,0010 0,0010 RPM 0,0000 0,0010 0,0010 0,0010 PNNT 0,0000 0,0255 0,0010 0,0010 SWK 0,0000 -0,0010 -0,0010 -0,0010 CCU -0,0127 -0,1152 -0,0076 -0,0111 FSC 0,0000 0,0010 0,0722 0,0144 EBMT 0,3655 0,3765 0,1500 0,1500 GHM 0,0000 0,0010 0,0010 0,0010 UNH 0,0647 0,0044 0,0939 0,0541 RBCAA 0,0000 0,0120 0,0010 0,0010 WTS 0,0000 0,0010 0,0010 0,0010 MEOH 0,0000 -0,0010 -0,0436 -0,0010 CPSI 0,0586 -0,0010 0,0401 0,0256 TRIB 0,0000 0,0010 0,0010 0,0010 CCO.TO 0,0000 0,0010 0,0010 0,0010 CCJ -0,0373 -0,0691 -0,0638 -0,0985 GSM -0,0267 0,0010 0,0010 0,0010 NDSN 0,0000 -0,0010 -0,1087 -0,0760 PCAR 0,0503 0,0191 0,0959 0,1100 CMI 0,0000 0,0010 0,0010 0,0010 VAC 0,1581 0,1072 0,1161 0,1500 FLIR 0,0000 0,0010 -0,0010 -0,0010 DFT 0,1100 0,1474 0,1202 0,1500 ASH 0,0000 0,0010 0,0010 0,0010 OZM 0,0165 -0,0132 -0,0010 -0,0010 ENI -0,0047 -0,0761 -0,0650 -0,1016 FINL -0,0102 -0,0010 -0,0010 -0,0010 SWM 0,0000 0,0010 0,0010 0,0010 HEP 0,0000 -0,0010 -0,0010 -0,0010 NTES 0,0413 0,0111 0,0422 0,0420 CFNL 0,0983 0,0423 0,0308 0,0423 ARCC 0,0425 0,1419 0,1500 0,1500 RNF -0,0940 0,0010 0,0010 0,0010 MERC 0,0000 0,0010 0,0010 0,0010 RCKY -0,0025 0,0619 0,0780 0,0651 FGP 0,0675 0,0910 0,1268 0,1500 MRE.TO 0,0327 -0,0010 -0,0010 -0,0010 CYD 0,0723 0,0105 0,0010 0,0010 AHT 0,0196 0,0485 0,0010 0,0010 AUO 0,0109 0,0649 0,0593 0,0669

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Figure 2.4: Principle component variance explanatory analysis of Max Sharpe with Short Sales with all Assets portfolio.

Figure 2.5: Principle component variance explanatory analysis of Max Sharpe with Short Sales with all Assets and 15% investiture portfolio.

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Figure 2.6: Principle component variance explanatory analysis of Minimum Variance with Short Sales with all Assets and 15% investiture portfolio.

pected outcomes which is an added level of risk protection not achieved as a consequence of building a minimum variance portfolio. Lastly, to hedge against added exposure due to over investiture in a single asset a portfolio that is limited to a maximum 15% initial investment into any one asset is also recommended. The Minimum Variance portfolio that allows Short Selling with a maximum of 15% investiture into any individual asset is the optimum portfolio with 10 principle components, a predicted annual Sharpe Ratio of 1.72, and predicted monthly differential returns of 0.5527% or similarly annual differential returns of 6.8377%.

2.3

Create factor models and evaluate

The factor index models representing the Minimum Variance portfolio that allows Short Selling with a maximum of 15% investiture in any individual asset from equations (1.17), using all 15 combinations of the indexes (M = market, P = poverty, I = inflation, and U = unemployment), have values for ¯Rpand σp2to describe a one month portfolio outlook, specifically January 2015 (see table 2.3).

Updating the model’s average monthly return using actual return data results in 12 itera-tions of each model January through December with monthly average returns as seen in table 2.4 with model specific volatilities available in Appendix A with reference (F, 12).

Actual monthly returns over the 12 month period occurring with the normal cumulative distribution function evaluated at each absolute valued z-score and then summed with the lowest value being the model that has the most predictive value are listed in table 2.5.

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Portfolio Mean 0.019197586 Index Volatility M 0.000426289 MP 0.000426596 MI 0.000529371 MU 0.000439076 MPI 0.000529678 MPU 0.000439384 MIU 0.000542158 MPIU 0.000542465 P 0.000425145 PI 0.000528227 PU 0.000437933 PIU 0.000541014 I 0.00052792 IU 0.000540707 U 0.000437625

Table 2.3: Index model volatilities for January 2015

Month Monthly Model Avg Returns

January 0.019197586 February 0.018233725 March 0.018380559 April 0.018401081 May 0.017897566 June 0.017364617 July 0.017552135 August 0.016971648 September 0.015324841 October 0.014427061 November 0.012818688 December 0.011912556

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Indexes Aggregate CDF(Z-Score) M 10.1225 MP 10.1261 MI 9.8912 MU 10.0924 MPI 9.8842 MPU 10.0951 MIU 9.8613 MPIU 9.8602 P 10.1269 PI 9.8918 PU 10.0972 PIU 9.8613 I 9.8918 IU 9.8613 U 10.0979

Table 2.5: Aggregate Cumulative Distribution Function values evaluated at model Z-Scores

The model with the most predictive power at an aggregate CDF value of 9.8602 is the 4 index, MPIU, model. Standardized realized monthly returns inlcuding dividends, Z-scores, are distributed across the MPIU model’s distribution as shown in figure 2.7 with 58.33% of realized monthly returns falling within 1 standard deviation, 91.67% falling within 2 standard deviations, and 100% within 3 standard deviations from the mean.

The cumulative actual returns of the fund are as follows assuming no reinvestment of dividends are shown in table 2.6.

Individual investors would see a dividend return of 0.2% on average per month or a cumu-lative 2.43% over the entire 12 month period with a realized annual Sharpe Ratio of 0.12 and an annual differential return over the maximum US Federal Reserve interest rate, 0.34%, for the period from January 2015 to January 2016 of 2.09%.

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Figure 2.7: MPIU 4 factor multi-index model distribution with realized return Z-scores.

Month Cumulative Real Returns

January -0.023481699 February -0.002247421 March 0.010987029 April 0.011598867 May 0.012387613 June 0.014706748 July 0.01580313 August 0.01360746 September 0.014008271 October 0.013320109 November 0.013754328 December 0.016489762

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Chapter 3

Conclusion

The purpose of this investment portfolio creation and analysis was to help address the problem of income inequality for families who live near, at, or below the poverty line. By creating an investment fund strategy that caters to the specific financial needs of the poor and capitalizing on fluctuations of the social and economic indicators of poverty it is hoped that another tool can be brought to bear on the issue. Beginning with the portfolio requirements, assumptions, and simplifications discussed in sections 1.1 and 1.1.1 there were several critical problems that needed to be addressed to reach a reasonable conclusion regarding the overall purpose of this investigation. Those critical problems were specifically:

1. Are there enough stocks available on the market, that fit the socio-economic character-istic requirements, to create a sufficiently diversified portfolio?

2. Can an optimum portfolio be created from those stocks that has predicted performance that is not below a designated performance level standard?

3. Can a sufficiently predictive model be created to facilitate and simplify the implement-ation and upkeep of the fund?

Sections 2.1, 2.2, and 2.3 analyse the answers to each of these questions respectively.

Sections 2.1.1 and 2.1.2 discuss the requirements for initial stock selection: it must pay a dividend and it must be below the average PEGY ratio of the selection set. Initially 1387 assets were identified with an average PEGY ratio of 1.7915 of which 981 stocks were se-lected. Further, of those 981 stocks 236 were found to fit the covariance requirements of the proxy index rates resulting from NIPALS analysis of the unemployment, inflation, and market return rates. The proxy indexes being representative of the real rates with a mean relative difference of 25.92% for the market, 12.95% for the inflation, and 0.35% for the unemploy-ment rates. The proxy indexes were also highly covariant with their respective real rates at 99.83%, 98.47%, and 99.99% for market, inflation, and unemployment respectively. Section 2.1.3 illustrates that of the 236 viable remaining stock picks 66 assets were appropriately co-variant with the annual poverty rate while the diversification analysis in section 2.1.3 details, using the NIPALS algorithm, that of the 66 remaining assets a total of 45 specific assets are needed to create a sufficiently diversified fund when evaluated using eigenvalues represented

Figure

Figure 1: Congressional Budget Office - Income growth rates from 1946 to 2015 in the US as a percentage of 1973 levels
Figure 2.1: Asset diversification scree plot with the variance explanatory power of principle components shown as a percentage of total variance of the data.
Figure 2.2: Graphical analysis of percent excess return to variability measure with the 4 highest ranked portfolios designated with a red trend line.
Figure 2.3: Principle component variance explanatory analysis of Max Sharpe with Short Sales portfolio.
+6

References

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