**Company Profile**

Portfolio Selection Systems is a provider of modeling and decision support for investment funds, financial professionals and financial organizations. The PSS software package offers substantial capability for active and passive investment managers and pension sponsors to analyze and optimize their portfolios of assets.

The PSS software package enables users to find optimal portfolios of investments using both portfolio theory and fundamental analysis with the added dimension of timing. The digital signal processing approach is not available elsewhere. PSS Release 2.0 does not use technical analysis but instead takes advantage of know calendar anomalies such as the quarterly effect, summer effect, January effect, and the presidential effect to accomplish portfolio selection. Portfolios can be selected based on the users required holding period and purchase dates.

**A Brief History**

The portfolio network and digital signal processing model developed in the book and software package is a direct descendent of the Modern Portfolio Theory first developed by Markowitz. The concepts of Digital Portfolio Theory where presented for the first time in a 1983 seminar and appear in the proceeding of the American Institute of Decision Sciences. The portfolio network model in conjunction with signal processing was the product of Professor Jones’ Ph.D. dissertation published in 1986. In 1992 Ken Jones published his first theoretical textbook Portfolio Management, (McGraw-Hill) outlining the author’s contribution to that point. In 1997 Dr. Jones began publication of a portfolio selection software package, PSS, based on the Digital Portfolio Theory concepts presented in the book. Since the book was published there have been considerable theoretical and empirical advances. In December, 2001 the author’s article “Digital Portfolio Theory” appeared in the Journal of Computational Economics.

**The Evolution of Portfolio Theory**

The Markowitz essential insight was to realize that investment analysis in the 1950’s specified that the investor should only be concerned with expected value or return on securities. John Burr Williams and Graham & Dodd had not presented any analytic framework for diversification. The value of a portfolio was nothing more than the present value of the future dividends and portfolio diversification was not considered. Diversification has always been a common practice but there was no formal model to reflect the importance of the correlation between assets to effectively diversify. Markowitz’s Modern Portfolio Theory reflected two important characteristics. It incorporated in analytic form the definition of portfolio risk and secondly it showed how this formulation of risk could be used in an optimization model to find efficient portfolios.

**Investors have Memory**

In 1982, 30 years after Markowitz’s first article, Ken Jones reflected that while Markowitz had added a dimension of risk to portfolio theory there was still a large disparity from what the average investor was doing with regard to their portfolios and the theory. No one can argue about the importance of time to the investor. How many investors have lost or gained fortunes because of timing decisions. If this is the case, what are the components of these timing decisions? Jones distinguished two significant factors missing from portfolio theory. First investors have memory, in the Modern Portfolio Theory model no memory of any past performance or event was utilized other than the estimate of the securities variance and covariance. For example, today’s investor might observe based on his or her memory “every January since I can remember the stock market has done better than in December or in February.” Additionally, the small companies have contributed most to this superior performance. Our memory provides a rational justification for timing decisions. The Jones insight is that today’s investor when making a portfolio acquisition or divestiture decision does not have an analytic measure of this memory nor is there a model to maximize portfolio return that incorporates memory. While the investment professional must make timing decisions every day no quantification of the memory of past events is available to make such decisions. Instead an assortment of short term technical analysis and forecasting methods are applied on an ad hoc basis.

**The Holding Period is Missing in Modern Portfolio Theory**

The second important missing factor of the single period portfolio decision is how the purchase date and holding period of the portfolio interacts with memory. If it is September and I expect to make a decision to buy a portfolio and hold it for six months, how can the memory of recent and distant past risk levels be quantified and applied? For example, an investor may remember many unfortunate downturns in October as well as many upturns in January. In addition to the investor’s October memory, there are in fact many other memory characteristics of a security that may be important even though the investor’s personal memory may not have taken notice. For example, do you remember that the first two years of presidential terms have seen lower returns than the last two years? The investor may not remember this, but if he or she has a portfolio selection model that does, the investor will profit from this knowledge. While past experience is no guarantee of future performance this does not mean that we should pretend that our memory is irrelevant to our current decision. Yet this was the state of portfolio theory when Jones set about examining the problem as a topic for his Ph.D. dissertation in 1982.

**The Golden Age of Digital Signal Processing**

It happens that one of the newest and most efficient technologies of our time, Digital Signal Processing has a solution to this problem of measuring memory. Jones recognized that Digital Signal Processing offers a solution to the quantification of the memory of risk for long and short periods. By using long periods (4- or 8-years) of security returns as digital signals, the memory related to time intervals can be accurately measured and incorporated into the portfolio theory model as time interval dependent risk estimates. This new Digital Portfolio Theory not only remembers effects that happened on an annual basis but also remembers events that occurred on quarterly, six month, 2-year and 4-year ( and even 8-year) basis. In addition, the Digital Portfolio Theory model remembers the effects of events on risks that were not related to regular calendar, or yearly events. This non-calendar memory is unrelated to regular time intervals and therefore can not be used to benefit timing decisions. Markowitz’s Modern Portfolio Theory added a dimension of risk to the return maximization problem that allowed the diversification decision to be quantified. Jones’ Digital Portfolio Theory is a mathematical model that adds the dimension of memory to the dimensions of risk and return. Additionally, the Digital Portfolio Theory model offers a rational framework for basing portfolio decisions that are dependent on the point in time and on the holding period of the investor.

**The Father of the Digital Age of Finance**

If Markowitz is the father of Modern Portfolio Theory, Jones is the father of Digital Portfolio Theory. As Modern Portfolio Theory suggested the CAPM, an equilibrium pricing relationship for all assets based on a linear relation to their betas. Digital Portfolio Theory also implies an equilibrium model based on a linear relationship to multiple betas related to the strength of memory effects at multiple time periods. This model is concisely presented in Dr. Jones’ 1992 textbook and these calendar Betas are used in the Portfolio Selection System software package available at portfolionetworks.com.

**Milton Friedman Rejected the Field of Operations Research**

The economist Milton Friedman’s objected that the Markowitz, Modern Portfolio Theory model was “not math, it’s not economics, and its not even business administration”. Friedman’s criticism also implicated Koopmans and the field of linear programming and operations research. This comment has been repeated to Professor Jones many times and continues to this day. Initially Modern Portfolio Theory generated relatively little interest. In fact the portfolio theory of Markowitz was not taught in graduate courses until the late 60’s and then only at a few select universities. One of the biggest drawbacks of Modern Portfolio Theory was that it required a quadratic programming solution. The Markowitz model contained a large number of nonlinear terms so that specialized software was required to find the recommended portfolio. Today practical commercially available software to find the Markowitz efficient frontier from large universes is still not available to the average investor. Dr. Jones has made the new Digital Portfolio Theory financial optimization software PSS available to the investment industry. Digital Portfolio Theory unlike Modern Portfolio Theory is a completely linear model that maximizes return subject to multiple risk constraints. Digital Portfolio Theory does not require a specialized algorithm to find the optimal portfolio. Because of the facility with which Digital Portfolio Theory allows constraints to be added, the PSS software package includes an efficient new method of fundamental analysis as well. The stochastic portfolio network optimization model because it is linear can accommodate very large asset and liability universes. The flow in these portfolio networks is a monetary flow. It will be possible to construct and solve for optimal monetary flow in portfolio optimization models that have millions, or even billions of assets and liabilities. Thus economy wide portfolio networks are a real possibility. Ironically Friedman’s monetarists may in the end utilize the very operations research methodologies they had rejected earlier.

**The Unexpected Consequences of Digital Portfolio Theory**

The Jones application of Digital Signal Processing (DSP) to the portfolio theory problem had three unexpected consequences. First, the optimization problem formulation was no longer nonlinear! Digital Portfolio Theory allows linear programming to be directly employed to find the solution in one iteration. The investor can easily create his or her own computer model to try many different risk levels. Secondly, Modern Portfolio Theory required a very large number of terms to represent the variance/covariance matrix. The number of terms increases by the square of the size of the security universe. Digital Portfolio Theory represents the variance, covariance and autocorrelation with a much smaller number of terms that increases linearly with the security universe size. It is therefore much easier to manage large portfolio universes. Finally, one problem of Modern Portfolio Theory is that solutions are often unreliable because they are based on historical mean and variance to estimate expected return and risk. The nonlinear form of Modern Portfolio Theory greatly magnified these distortions. To correct this problem Modern Portfolio Theory requires users to adjust estimates before optimization. DPT(Digital Portfolio Theory) uses the latest small sample Digital Signal Processing methods currently being employed in speech recognition. These DSP techniques require 16 year monthly return histories to estimate input parameters. The new DSP methods utilize multiple overlapping realizations of the security return process resulting in more significant estimates of the inputs to the portfolio optimization model. Since the solution variables in the Digital Portfolio Theory model are linear, more confidence can be placed in the solution portfolios and the distortions commonly associated with Modern Portfolio Theory are greatly reduced.

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