Parallax US Equity Valuation Neural Network Model

8/19/2019 Parallax US Equity Valuation Neural Network Model

1/84

THE APPLICATION OF NEURALNETWORKS TO U.S. EQUITY

PRICING AND CLASSIFICATION

by

S. Kris Kaufman, PresidentDouglas Frick, Vice President

Alex Kaufman, Research AssociateParallax Financial Research, Inc.

2004

White Paper for Parallax Financial Research, Inc. products:

Price Wizard TM

Dynamic Price Wizard ™

ValueNet TM

Beta Predictor™

Wizard Explorer TM


2/84

Abstract

Neural network modeling technology has been applied to the problem of determining the famarket price for a company based on all significant stores of value as reported through requireSEC filings and current economic conditions. No attempt was made to use analyst estimates oother future looking assumptions. Significant biases due to trend persistence, price, economconditions, and company size were reduced using proprietary splitting and sampling techniquSector and industry biases were eliminated by producing one network for each of 16 economsectors, each of which included binary flags to signify industry membership. Data used for thwork came from Zacks databases of US corporate data between 1990 and 2001. After a

preparation step which separated the data samples into training and test sets, and applied somoutlier elimination filters, the NGO product by BioComp was used to train the networks.Training for most sectors resulted in R squared numbers above 0.8 as follows:

Sector R 2 SamplesConsumer Staples .88 16,413Consumer Discretionary .79 12,943Retail/Wholesale .75 21,511

Medical .77 14,321 Auto/Tires/Trucks .90 3,891Basic Materials .86 13,893Industrial Products .87 16,935Construction .89 8,619

Multi-Sector Conglomerates .93 4,795

Computer And Technology .63 27,991 Aerospace .88 3,387Oils/Energy .82 11,035Finance .91 32,000Utilities .96 15,567Transportation .88 6,451Business Services .75 4,065

The final neural models return an estimated price. There are numerous applications for thes price estimates such as pricing IPO’s, portfolio selection, macro modeling of sectors and industrstock market index pricing, asset allocation, and construction of enhanced indices. The asse

allocation application required an additional network be built to classify “value” and “growth”stocks. Another stock screening application required that the recent rates of change of valuatioand price be combined with our estimated price in a dynamic neural network model. Beta is aoften used management variable, and we wished to estimate beta to the S&P 500 as well asbeta to an industry average. A beta predictor neural net was constructed for this purpose. Tesresults show that our methodologies significantly improve the stock screening process at all leve


3/84

TABLE OF CONTENTS

List of Figures...................................................................................................................... iiList of Tables....................................................................................................................... iv

Acknowledgements.............................................................................................................vGlossary................................................................................................................................viIntroduction .........................................................................................................................1Chapter I: Building an Equity Pricing Model (Price Wizard TM ) ................................4

The Search for Value...................................................................................................4Data Preparation .........................................................................................................7Neural Network Background ..................................................................................12Neural Net Training Results ....................................................................................17

Value/Growth Classification Network (ValueNet TM

).........................................26Predicting Beta to the S&P500 and Industry Indices (Beta Predictor™) .......27

Adding Rate-of-Change to Price Wizard (F% or Dynamic Price Wizard™).28Chapter II: Model Delivery & Visualization ................................................................30

Model Processing & Delivery..................................................................................30Interactive Price Calculator (Wizard Explorer TM ) ................................................32

TradeStation TM Historical Graphing.......................................................................35 TradeStation TM Aggregate Portfolio Graphing .....................................................37EXCEL TM Database and Summary Spreadsheets................................................44

Chapter III: Historical Testing Methods .....................................................................48Survivorship and In-Sample Considerations ........................................................48

Chapter IV: Applications.................................................................................................60Stock Selection Screening.........................................................................................40CSFB Holt versus Parallax Equity Pricing Model ...............................................40

Aggregate Valuation Statistics..................................................................................40 Aggregate Portfolio Valuation.................................................................................40Sector and Industry Valuation .................................................................................40Index/ETF Valuation ...............................................................................................40Enhanced Index Applications .................................................................................40Long/Short Applications .........................................................................................40

Asset Allocation Applications..................................................................................40Chapter V: Client Performance Examples...................................................................60

The Influence of Price Trend Persistence.............................................................63Equity Screening Example 1 (Marque Millennium)............................................63Equity Screening Example 2 (Corporate Consulting).........................................63

Glossary...............................................................................................................................73Bibliography .......................................................................................................................75Index....................................................................................................................................78

Appendix A: Parallax Financial Research.....................................................................77 Appendix B: Zacks Historical Data...............................................................................78


4/84

ii

LIST OF FIGURES

Number Page1. Zacks Current Database......................................................................................5

2. Three Layer Neural Network ...........................................................................12

3. Neural Network Activation Functions ...........................................................14

4. Network Training for the Finance Sector........................................................16

5. Predicted Price vs. Actual for the Test Data .................................................17

6. Predicted Price vs. Actual for the Training Data ..........................................17

7. Consumer Staples Network Training Results ....................................................188. Utilities Network Training Results....................................................................18

9. Retail/Wholesale Network Training Results.....................................................19

10. Finance Network Training Results....................................................................19

11. Medical Network Training Results....................................................................20

12. Oil/Energy Network Training Results..............................................................20

13. Neural Response Surface...................................................................................21

14. Training Statistics for Best Value/Growth Network...................................26

15. Neural Response Curve for Value/Growth Network.................................27

16. Price Calculator Data Retrieval ........................................................................32

17. Price Calculator Data Editing and Recalculation..........................................33

18. TradeStation Display of Estimated Prices .....................................................34

19. CSCO Estimated Prices shows Bubble Formation......................................35

20. June 1996 Relative Valuation Histogram...................................................................36

21. March 2001 Relative Valuation Histogram....................................................37

22. Percentage of Undervalued Stocks in the Banking Sector..........................3923. Percentage of Undervalued Stocks in the Retail Sector...............................40

24. Percentage of Undervalued Stocks in the Oil/Energy Sector....................41


5/84

iii

25. Percentage of Undervalued Stocks in the Nasdaq 100................................42

26. Percentage of Undervalued Stocks by Sector ................................................43

27. Percentage of Undervalued Stocks by Market Cap Band ...........................4428. Percentage of Undervalued Stocks by Industry ............................................45

29. Percentage of Undervalued Stocks by Asset Class.......................................46


6/84

iv

LIST OF TABLES

Number Page1. List of Fundamental Factors used to Estimate Price.....................................5

2. Top Price Factors for Consumer Discretionary Stocks ..............................22

3. Neural Network Training Results....................................................................23

4. TradeStation Inputs for Indicator PFR_Wizard...........................................35

5. Monthly EXCEL Spreadsheet of Estimated Equity Prices........................43

6. Monthly EXCEL Spreadsheet with the Statistical Valuation Summary...44


7/84

v

ACKNOWLEDGMENTS

Work on this product began in 1990 and has gone through three major revisions

over the intervening 14 years. The author wishes to express sincere appreciation

to Edward Raha, Doug Frick, Phillip Zachary, Gary Gould, John Rickmeier, Bill

Meckel, Ken Garvey, Diana Kaufman, and Alex Kaufman for their support and

assistance in the development of these products and related applications. This

work was partially supported by contracts with Bankers Trust, Daiwa Securities,

Managed Quantitative Advisors, and Marque Millennium Capital Management


8/84

vi

GLOSSARY

Neural Network . A mathematical modeling technique which has the capacity tolearn by example. It is also referred to as a “non-parametric” modelingtechnique.

ETF . Exchange traded funds are portfolios of stocks that have similarcharacteristics and are available for trading as a single entity.

IPO . Initial public offering of a stock.

EXCEL . Microsoft Office spreadsheet software program

TradeStation . Stock charting and back testing software program from TradeStation Securities

Sector . One of 16 divisions of the US economy by corporate characteristics

SEC . The Securities and Exchange Commission.

NGO . Neurogenetic optimizer software produced by Dr. Carl Cook ofBioComp Systems ( www.biocompsystems.com )

Zacks Investment Research is a Chicago based firm with over24 years of experience in providing institutional and individual investors with theanalytical tools and financial information necessary to the success of theirinvestment process ( www.zacks.com )

Russell 3000. The Russell 3000® Index offers investors access to the broad U.S.equity universe representing approximately 98% of the U.S. market. The Russell3000 is constructed to provide a comprehensive, unbiased, and stable barometerof the broad market and is completely reconstituted annually to ensure new andgrowing equities are reflected ( www.russell.com/US/Indexes/US/3000.asp ).

I/B/E/S. Institutional Brokers Estimate System. A system that gathers andcompiles the different estimates made by stock analysts on the future earnings forthe majority of U.S. publicly traded companies ( www.firstcall.com )


9/84

1

INTRODUCTION

“Intrinsic value is the investment concept on which our views ofsecurity analysis are founded. Without some defined standards ofvalue for judging whether securities are over- or under-priced in themarketplace, the analyst is a potential victim of the tides of pessimismand euphoria which sweep the security markets.”

...“Intrinsic value is therefore dynamic in that it is a moving targetwhich can be expected to move forward but in a much less volatilemanner than typical cyclical or other gyrations of market price. Thus,if intrinsic value is accurately estimated, price will fluctuate about it.”Graham and Dodd, Security Analysis (pg. 41 and 43, Fifth Ed.)

The price of a stock is an opinion, or more correctly, the sum total of thousands

of opinions which are expressed through buy and sell decisions every day in the

marketplace. These opinions are based on published corporate and economic

fundamentals, as well as imprecise factors such as future estimates, recent trends,

news, and lawsuits. They are also subject to irrational investment behavior,rumors, individual biases, and to some extent random fluctuations. Since price is

dependent on so many factors, it is natural to try and separate out the effect of

the most intrinsic components first. In fact, our goal was to find a mathematical

way of pricing a stock that may not have been priced by the marketplace. As they

say though, if the problem were that easy, it would have been solved already.

Even the intrinsic components of a stock price depend on many interrelated

factors, and just as identical real estate varies in price by location, the same

corporate earnings numbers will result in different valuations within different

sectors or in different economic conditions. Our model had to be based on

known fundamentals and economic conditions, compensate (normalize) for


10/84

2

sector and industry, and properly weight the many stores of value that give a

company an intrinsic worth.

Mathematical models are normally built by making a-priori assumptions about the

functional form of the solution. These are called parametric models, and are

solved by regression methods to determine a number of coefficients. This is fine

if you know that the solution must be a straight line or some other simple, well-

known function. But in the real world, relationships are not necessarily simple,

and we don’t always know the form of the solution. Inputs and outputs could

even be related in a non-linear fashion. If you don’t have to guess the functional

form of the answer, you have a big advantage. We chose to use neural network

modeling for this reason.

A neural network is a mathematical modeling tool which has the capacity to learn

by example. This is an extraordinarily useful ability, especially in financial

modeling, where the inputs & outputs are usually well documented, and there are

countless examples. Networks are “trained” by being presented with thousands

of facts, each fact consisting of inputs and corresponding outputs. Through a

unique feedback process, the network learns how those inputs are related to the

outputs, and develops a general model to describe the relationship.

In our case, fundamentals were fed in, and the corresponding stock price was

used as the output. Again, if this problem were so easy…. There were significant

sources of bias and data error which had to be dealt with, and some of our

techniques are so innovative that they must remain proprietary. Data preparation

steps were critical to the success of the model, and will be discussed in as much

depth as possible.


11/84

3

Once trained, the model knows how to form a price opinion that should be close

to a reasonable consensus opinion based solely on economic and fundamental

report data. We expect that when the model price is lower than the actual price,then there is probably a degree of optimism that things will be far better in the

future. On the flip side, if actual price is lower than the model, then there may be

significant pessimism. Whatever the cause of the discrepancy, it is the analyst’s

job to try and understand why it exists.

A successful model will have many applications depending on the kind of client

being served. Our prices could be used to evaluate portfolios of all sorts, from

individual to professional funds. Model prices could even be used to assess the

economy based on broad collections of stocks. For clients that need investment

products, our model would be useful at all levels. The prices could be used to

rank stocks from under-valued purchase candidates to over-valued sell or short

sale candidates.


12/84

4

C h a p t e r 1

BUILDING AN EQUITY PRICING MODEL

The Search for Value

The search for corporate value is complex. Do earnings alone determine a fair

price? How about cash flow, book value, sales, or dividends? How important are

interest rates, debt, or the industry in which the company operates? These are allgood questions, and the answer is that they all matter. The complexity of judging

how much to weight each factor, and how the factors change relative to each

other is hopelessly complex. Add to that the challenge of remaining unbiased by

the times in which we live. It was rare, for example, to find an analyst who was

willing to say that tech stocks were overvalued in 1999. Instead, many chose to

adjust their models to reflect an irrational reality. So Parallax has chosen

mathematical modeling for its ability to provide consistent results to an

unmanageably complex problem.

Our goal was to build a computer model with no assumptions about the

relationship between price and fundamental factors. We wanted the model to

learn from example and be able to generalize what it had learned to any other set

of corporate data. More specifically, our equity-pricing model used neural

network technology to learn how a stock’s fundamental balance sheet data had

been translated to a market price in the past, within the context of its industry

group and economic sector. Then, going forward in time, the trained model

simultaneously evaluates all these factors as they change month to month, in

order to produce an estimated stock price.


13/84

5

To accomplish this ambitious goal we needed to include the factors that havebeen cited by the financial community as key to stock valuation. This also

included such economic numbers as PPI, CPI, GDP growth and interest rates.

The data we chose for this project came from Zacks corporation’s DBDQ,

DBCM, and ECON databases, and are listed below along with a screenshot of

the DBCM database reference guide from which all of the values and their

explanation were gleaned:

Figure 1 – Database Items reference page for the Zacks DBCM current database. The tablebelow was compiled from this document.

Inputs Input Description

stale This is an integer value representing the number of months oldthe data is

roi Return on Invested capital. Calc: (Income before extras &discontinued operations)/(Long-term debt, convertible debt +non-current capital leases + mortgages + book value preferredstock + common equity)

sales_q Sales quarterly per share

inc_bnri12 Twelve month Income before non-recurring items per share

book/share Book value of common equity per share


14/84

6

trend_bvsg Trend book value per share growth rate calculated from betaof an exponential regression on the last 20 quarters’ of fiscalbook value per share

cash/share Cash flow per common share. Calc: CASH FLOW/ SHARESOUT

op_margin (Income before extraordinary items and discontinuedoperations for the previous 12 month period/Sales for theprevious 12 month period) X 100

net_margin (Net income for the previous 12 month period/Sales for theprevious 12 month period) X 100

pretax_mrg Trailing four quarter pretax profit margin. Calc: (PRETAXINC/SALES) x 100

curr_ratio Current Ratio: Current Assets/Current liabilities

payout_rat Dividend payout ratio. Calc: INDICATED ANNUALDIVIDEND/ACT EPS 12

inventory Inventory value per share

tot_c_asst Total current assets per share

tot_c_liab Total current liabilities per share

beta Stock return volatility relative to the S&P 500 over the last 60months (includes dividends)

roe_12m Return on Equity; 12-month EPS/Book Value per Share

roa_12m Return on Assets; 12-month EPS/Total Assets per Share

div_yield% Dividend yield, based on IND AN DIV and PRICE

ttl_lt_dbt Total Long-term debt quarterly; Debt due more than one yearfrom balance sheet date; long-term debt, convertible debt,non-current capital leases and mortgages

act_eps_q Diluted quarterly actual EPS before non-recurring items.

act_eps_12 Diluted quarterly actual earnings per share before non-recurring items

sales_12mo Sales during the last 12 months

12msls/-4q Last 12 months of sales divided by the last 12 month sales 4quarters ago

12meps/-4q Last 12 months of earnings divided by the last 12 monthearnings 4 quarters ago

quick_rati Quick Ratio – Most Recent (Current Assets-Inventory)/Current Liabilities

gdp_growth Real Gross Domestic Product divided by GDP one quarterago

Ppi_fin_gds Producer price Index – Finished Goods Seasonally AdjustedIndex


15/84

7

Cpi Consumer Price Index – All Urban Consumers – NotSeasonally Adjusted

TBond Yield Yield on Long-term Treasury Bonds

TBill Discount Discount rate on new issues of 91-day Treasury bills

Prime_Rate Average prime rate charged by banks

Industry Zacks industry classification

Table 1. List of all fundamental factors used to estimate stock price via neural network.

We selected this list for a few reasons besides the obvious connection to

valuation. First, these items had the highest historical availability for US stocks in

the Zacks databases, and the more samples the better. Some of the items are

primary and some are expressed as formulas. There are certainly items that aredependent on each other, and even a few that are redundant. We found that

neural net training is less likely to become biased if the inputs are chosen in this

manner.

At the end of training we can look at what the neural net learned about the

relative importance of factors in general, and a bit about how price varies with

each factor. It is essential that what the network learns also makes sense. Price

should be positively correlated with earnings, cash flow, and book value forinstance. Choosing a good set of inputs is important, but data quality and

preparation is critical. In the next section we will discuss how training and testing

data were prepared.

Data Preparation

The most significant source of bias is due to sector and industry norms.

Different types of companies show their worth in different ways. For example, we can’t hope to evaluate utility and tech stocks with the same model, since their

business structures are just too different. For this reason we split the data up by


16/84

8

Zacks economic sector. Twelve years of monthly data samples for 2,547 stocks,

were physically divided into sixteen EXCEL spreadsheet files by sector (223,000

facts in all). The Zacks sectors are as follows:

Then, within each sector file, columns were inserted and filled with a one or zero

binary flag to signify industry membership. In the end, the full impact of sector

and industry membership was preserved and this source of bias removed. The

following is the complete list of all industry classifications placed within their

respective sectors:


17/84

9


18/84

10


19/84

11

The trickiest bias to remove was price. Stocks priced less than $1 are simply not

valued in quite the same way as those at $100, and what about BerkshireHathaway which is priced in the tens of thousands of dollars. The proprietary

trick we developed to eliminate price bias also had the side benefit of stopping

economic bias. Data errors and extreme outliers had to be eliminated also. We

used a multiple pass Gaussian filter for this step. Lastly, cycles of optimism and

pessimism are also a source of bias. We were careful to balance price trend

persistence over the twelve year period in order to deal with this problem.


20/84

12

Neural Network Background

The NeuroGenetic Optimizer (NGO) software by BioComp Systems, facilitates

the creation of artificial neural networks by employing genetic algorithms in order

to modify neural net structure. The term neural network could refer to the real

life networking of neurons in various biological organisms, but it is used here

(and in most cases) to reference the concept of an artificial neural network, which

is a model created inside the computer that mimics the most fundamental

functions in a biological neural network.

In real world neural networks, it is believed that “learning” takes place when the

strength of an axon connection is increased by virtue of its usage. For instance,

when a small child is learning how to define a tree, he could be shown a pine tree,

and the connections supporting a tree as a tall, green, spiky, thing with brown

bark are reinforced and strengthened. As the concept of “tree” evolves, green

will be given more weight, and spiky will be given less weight, because, while

most trees he is shown are still green, they do not all have spikes (i.e. – a maple

tree). This is a real-world example of a neural network, in order for the boy toaccurately predict what is and is not considered a “tree”; he must have seen

enough examples of different kinds of trees . This way, he can determine the

connection between the various characteristics of the tree (independent variables)

and whether or not it is a tree (classification output).

This real-world example might be solved with a “classification” neural network.

For the young boy’s learning, he had to classify objects as ‘tree’ or ‘not tree’ by

looking at the variables that are important. He knew which variables were

important by experience and training with many different sorts of trees. If he had

only seen pine trees all his life, and classified them as ‘tree’, then he would not be


21/84

13

able to accept that a palm tree is also a ‘tree’. For this project, the computer was

simply asked to learn how corporate fundamentals contribute to stock price,

which is a “function approximation” problem.

The NeuroGenetic Optimizer allows the user to train a number of different

application types: function approximation, diagnosis, clustering, time series

prediction, and classification. Classification neural networks look at examples

with multiple outputs or categories, and correlate that to the input data provided.

Once a classification net has been properly trained, it will be able to look at a set

of data and determine what category it falls into. Neural networks are powerful

tools, but nobody has placed biological neurons inside the computer, and they are

certainly not comparable to human consciousness, so how exactly is this machine

learning achieved?

In order to understand neural networks it is important to understand the ‘hidden

layer’ of nodes in between the input and output layers. Input nodes could be

compared to the neurons in the eyes, and output nodes could be the neurons

leading to the vocal chords where an answer or result is spoken aloud. The

hidden layer simulates the set of neurons in between, where both learning and

problem solving occur. All the input nodes send information to all the hidden

nodes, and all the hidden nodes send data to all the output nodes. Here is a

simplified diagram of just such a basic three-layer neural network:


22/84

14

Figure 2. Picture provided from Louis Francis’ article “The Basics of Neural NetworksDemystified” in Contingencies magazine, < www.contingencies.org/novdec01/workshop.pdf >

In the diagram, the term “feedforward” simply references the direction of data

flow from input layer to hidden layer to output layer. A quote from the magazinearticle from which this diagram was taken goes a long way to summarize the

“learning” that goes on in the hidden layer:

Neural networks “learn” by adjusting the strength of the signal coming fromnodes in the previous layer connecting to it. As the neural network betterlearns how to predict the target value from the input pattern, each of theconnections between the input neurons and the hidden or intermediate neuronsand between the intermediate neurons and the output neurons increases ordecreases in strength… A function called a threshold or activation functionmodifies the signal coming into the hidden layer nodes… Currently,activation functions are typically sigmoid in shape and can take on any valuebetween 0 and 1 or between -1 and 1, depending on the particular functionchosen. The modified signal is then output to the output layer nodes, whichalso apply activation functions. Thus, the information about the pattern beinglearned is encoded in the signals carried to and from the nodes. These signalsmap a relationship between the input nodes (the data) and the output nodes(the dependent variable(s)). (3)

Just as with our analogy of the young boy and the trees, the neural network takes

each independent variable (input) and weights them according to their

importance in discerning the dependent variable(s) (output). The activation

functions that are triggered in each hidden node of the neural network record the


23/84

15

signal strength and thereby encode the patterns correlating input and output data.

In summary, this program “evolves” to a point where it can continually solve

similar problems over and over again. These tools are extraordinarily useful,because they can “learn” about how to solve a problem that may be non-linear,

such as stock market price estimation.

The trained network is the answer to the problem. If one were to distill the

neural network down to a corresponding equation it probably would be pages in

length and be too confusing to comprehend. The weights and activation

functions of the hidden layer can be viewed by the user, since they are written

into the file, but only when each of the weights and interconnections are taken

together in their entirety can the pattern be discerned by the neural network.

Understanding how and why the chosen variables, the activation functions, and

the connection weights interact to produce the correct answer is next to

impossible for a human being to discern by simply looking at the net. Neural

networks are often referred to as “black box” modeling, because the correlation

that’s determined is not easily discernible, but the trained neural network will

allow the user to look at “response curves.” These response curves show agraphed correlation between data variables used for input, and allow the user to

judge whether or not the neural network has learned about the independent input

variables adequately. There will be an elaboration on response curves in the

results section.

The last bit of information that is important to know concerns the activation

functions. The NGO utilizes three different types of activation functions, which

are the functions in the hidden layer that specify the relationship between inputs

and outputs. The different types of functions used by the NGO are as follows:


24/84

16

1. “Lo” – Logistic Sigmoid

2. “T” – Hyperbolic Tangent

3. “Li” – Linear

Figure 3. Three types of neural network activation functions used by NGO.

These functions adjust during training so that the neural net as a whole can

simulate the learning process. Depending on the problem being solved, differentcombinations of these activation functions will be employed, and the sum of the

dynamics of these functions will produce the trained network.


25/84

17

Neural Net Training Results

Recall from the data preparation discussion that we prepared training and test

data containing thousands of facts for all sixteen sectors being considered. Each

fact contained a monthly snapshot of our fundamental inputs and the

corresponding month-end price as output. The facts were run through NGO

directly from EXCEL on a modest 1GHz Pentium IV PC with 512M Ram. It

typically took several hours to complete the training and evaluation of each sector

net.

A successfully trained network will understand how to interpret new information.

In all neural net training runs, some data is held back for testing and additional

validation. If the net has not learned anything, and merely memorized its training

data, then it will score well when presented with the training data, but poorly

when presented with new fresh data. A successfully trained net performs virtually

the same on both sets.

The network was presented with test facts to gauge its progress after everytraining loop. The error criterion chosen for training was r-squared. Low r-

squared numbers close to zero signify a poor training result, while high r-squared

numbers close to one signify good results. Poor results are also confirmed when

the training set has a high r-squared and the test set a low one. This signifies

memorization occurred instead of true learning. We were hoping for r-squared

results that were consistent and in excess of 0.8. An r-squared of 1.0 means that

the predicted stock prices exactly matched the actual prices. We know this is not

possible simply because of random noise and other effects that we did not model.


26/84

18

The picture shown below is an actual screenshot of the training run for the

Finance sector network. Notice the training and test r-squared results were

almost identical at 0.88.

Figure 4. This is a screenshot taken during training of the finance sector network. Notice howsimilar the r-squared figure is between the test and training sets.


27/84

19

Figure 5. Predicted price for the test set fits the actual results very well. This indicates the networkhas learned how to estimate price.Figures 5 and 6 are actual screenshots of “Predicted vs. Actual” prices from the

Finance sector training run. Notice that the quality of fit on the training set is

almost identical to the fit on the test set. This indicates that our network is

learning instead of memorizing.


28/84

20

Figure 6. Predicted price fits the actual training prices, which indicates that the fundamentalfactors relate strongly to market price.

Figure 7. This picture shows the predicted versus actual prices we found from the ConsumerStaples network training (r-squared = 0.8866):

Figure 8. This figure shows the network training results for the Utilities sector (r-squared=0.9594).


29/84

21

Figure 9. Network training results for the Retail/Wholesale sector (r-squared=0.7529)

Figure 10. Network training results for the Finance sector (r-squared=0.9077)


30/84

22

Figure 11. Network training results for the Medical sector (r-squared=0.7657)

Figure 12. Network training results for the Retail/Wholesale sector (r-squared=0.7529)


31/84

23

A response curve, or “neural response surface”, provides information about how

two or more variables relate to one another in the prediction of a particular

output. These response curves are generated by isolating one or two of theinputs and changing them over a particular range of values in order to see what

relationship the net perceives between the variables and the output. All other

inputs are left at average values. At this point a graph is produced so the

relationship can be visualized. This tool can be used to show whether or not the

connections that the neural network has made between input and output

variables make sense.

Figure 13. Does the result make sense? A neural response surface shows the relationship betweensome inputs and an output.

One such relationship that can be easily verified with response curves is that price

should increase as book value and earnings rise. The figure above shows that the

network has learned this lesson from all the examples it was fed.


32/84

24

Another way to analyze the neural net training results is by a sensitivity analysis.

This test determines which of the inputs have the most influence on the output,

and if they are positively or negatively correlated. Here are the top factors for theConsumer Discretionary sector network:

Causal Variable (Input) Direction Pct.Effect Cum.Effect

----------------------------- ---------- ---------- ----------

act_eps_12 Positive 9.07% 9.07%

inc_bnri12 Positive 6.98% 16.05%

book/share Positive 5.35% 21.40%

payout_ratio Positive 4.41% 25.80%

Table 2. Top factors effecting price for Consumer Discretionary sector stocks

If a net didn’t train well, it could mean that a few bad facts made the process

difficult. It could also mean that the problem has no solution. Our results were

excellent in terms of high r-squared numbers, the equivalence of training and test

r-squared, and most importantly, the response curves made economic sense.

Price did in fact correlate positively with earnings, cash flow, book value, etc.

Sector R^2 Samples TrendlineConsumer Staples .88 16,413 A=0.97P+6.70Consumer Discretionary .79 12,943 A=0.98P – 0.68Retail/Wholesale .75 21,511 A=0.92P+4.16

Medical .77 14,321 A=0.97P+0.39 Auto/Tires/Trucks .90 3,891 A=0.99P+0.95

Basic Materials .86 13,893 A=0.98P+2.34Industrial Products .87 16,935 A=1.00P – 1.09Construction .89 8,619 A=0.98P+1.70

Multi-Sector Conglomerates .93 4,795 A=0.99P+1.08


33/84

25

Computer And Technology .63 27,991 A=0.97P+0.11 Aerospace .88 3,387 A=1.00P+0.70Oils/Energy .82 11,035 A=0.96P+9.47Finance .91 32,000 A=0.99P - 0.22Utilities .96 15,567 A=0.99P+0.17Transportation .88 6,451 A=0.96P+2.66Business Services .75 4,065 A=0.92P+4.66

Table 3. The figure shows the sector r-squared error, number of samples, and best regressiontrendline found during neural net modeling

The table above contains the r-squared results for all sample facts used during

training and testing. It also includes the formula for a best fit regression trendline

through the results graph. If the fundamental model could completely explain

price, the trendline formula would simply be A (actual price) = P (predicted

price). Our results are very close to this result. Slopes were slightly less than 1.0

and most intercepts slightly more than 0, which could be interpreted as slight

tendency for predicted to be less than actual price at the lower price end. This

effect is canceled by the method we use for constructing a final estimate, since we

sample the model numerous times across the whole price spectrum

The sector networks were successfully trained and are now ready to accept fresh

corporate data and produce price estimates. No further training is needed,

although we expect to retrain every 5 years in order to benefit from the additional

data. In later chapters we will explore the monthly processing, distribution,

visualization, investment uses, and historical test results for this pricing tool we

call Price Wizard TM


34/84

26

Value/Growth Classification Network

Before examining the test results for our equity pricing model, we found the need

for one more piece of information. The world of equity management is subject

to numerous product divisions based on broad classifications of stocks as “value”

or “growth”. We needed to build a neural network that would be able to look at

a company’s fundamental data and determine whether it should be classified as a

value stock or a growth stock, or maybe somewhere in between. Value stocks are

thought to be safer, growing slowly, but having solid intrinsic value and often

paying dividends to compensate investors for the slower growth. Growth stocksoften do not pay dividends, may have little intrinsic value, but have been growing

very fast. It is useful to have a consistent method of classifying stocks as value or

growth since it enables us to build asset allocation tools and better evaluate client

portfolios.

Training data was taken from the Russell 3000 value and growth indices and the

fundamental data from Zacks. Data preparation was the same as for the price

network except that all sixteen sector training files were combined into one, withIndustry membership replaced by Sector membership. The Russell method

includes I/B/E/S earnings estimates in their formula, which Parallax found to be

misleading in past studies. We expect that by leaving expectations out of our

training, that the classification system produced may even outperform the Russell

technique.


35/84

27

Figure 14 – Training statistics for the best Value/Growth neural network found

The best network used 16 of the possible 42 inputs for value vs. growth

prediction as follows:

1. book value of common equity per share,2. trend book value per share growth rate,3. cash flow per common share,4. operating margin,5. pretax margin,6. inventory value,7. total current assets per share,8. total current liabilities per share,9. beta,10. return on assets,11. dividend yield,12. 12 month sales divided by 12 month sales 4 quarters back,13. presence in Sector 1 (Consumer Staples),14. presence in Sector 3 (Retail/Wholesale),15. presence in Sector 6 (Basic Materials), and

16. presence in Sector 10 (Computers and Technology).


36/84

28

Results show that our model was able to match 92% of the Russell classifications

without the IBES estimates, which is sufficient for our purposes. Response

curves again help us verify that the model makes sense. One such relationshipthat can be easily verified with a response curve is between book value per share,

dividend yield per share, and the value stock designation. Generally when a

company has high dividend yield and a high book value then it will be designated

as a value stock. However, if the opposite is true, and both the dividend yield

and book value are low, then it is more likely going to be listed as a growth stock.

Here is a picture of the response surface that shows that the neural network

learned to make that distinction.

Figure 15 – The response surface showing the relationship between dividend yield, book value per share, and the value stock designation.


37/84

29

A few other things can be inferred from the graph as well. For instance, when

dividend yield is really high, an increase in the book value will have little effect on

the stock being a value stock. If dividend yield is low however, book value has alarge impact on the classification. In this way human beings can check the

reasoning of the neural network and declare whether it is marred by confusion or

has the same common sense perceptions about market variables as an analyst.

The end product of this training is our ability to assign a rank; we call it Value%,

to every stock we process. A Value% of 0 means the stock is a growth stock;

while a rank of 100 means it is a pure value selection. Value% in between 0 and

100 represents a stock with qualities of both classes.


38/84

30

Beta Predictor Network

Beta is a measure of a stock’s price volatility relative to a chosen benchmarkindex. It is often calculated in relation to the S&P 500 Index in order to judge

whether a security is more or less volatile than the market. A stock with a beta of

1.00 will tend to move higher and lower in tandem with the S&P 500. Securities

with a beta greater than 1.00 tend to be more volatile than the S&P 500, and

those with betas below 1.00 tend to be less volatile than the underlying index.

Securities with betas of zero generally move independently of the overall market.

Just as price is dependent on fundamental factors, the volatility of price also is

dependent upon fundamental factors. It isn’t hard to imagine that a company

with high debt and poor earnings might be subject to bigger price swings than a

company that is on a more solid financial footing. Our goal was to train a neural

network to predict beta to the S&P 500 and beta to corresponding sector indices.

Zacks provides us with a 60-month beta to the S&P 500 which includes

dividends. We have added sector betas using Zacks sixteen sectors.

As in the pricing network, we used all the data items in Table 1 except beta, to

train the sixteen sector nets. Response curves help us verify that the model

makes sense. One such relationship that can be easily verified with a response

curve is between current ratio and predicted beta. The current ratio is the ratio of

current assets over current liabilities, which measures the company’s liquidity.

The lower the current ratio, the less liquid and the higher we might expect beta to

become. Here is a picture of the response surface that shows that the neural

network learned to make that distinction.


39/84

31

Figure 15 – The response surface showing the relationship between current ratio andbeta to the s&p 500.

The next two pictures show the predicted versus actual betas for the consumer

staples sector after training.


40/84

32


41/84

33

The end product of this training is our ability to assign a predicted beta to the

S&P 500 as well as a predicted beta to each corresponding sector. The following

table lists the r-squared results from training each sector and the number ofsamples used for training. Note that as expected it is easier to predict betas to the

sector than to the S&P 500:

SectorS&P5005YrBeta

R 2

Sector1YrBeta

R 2 Samples

Consumer Staples 0.68 0.83 8645Consumer Discretionary 0.71 0.63 6207

Retail/Wholesale 0.59 0.69 12159 Medical 0.56 0.86 7839 Auto/Tires/Trucks 0.83 0.98 2037Basic Materials 0.50 0.75 6591Industrial Products 0.68 0.76 7593Construction 0.78 0.80 4535

Multi-Sector Conglomerates 0.79 0.85 2627Computer And Technology 0.54 0.73 15170

Aerospace 0.67 0.81 2001Oils/Energy 0.64 0.81 6249Finance 0.61 0.72 18319Utilities 0.50 0.66 7939Transportation 0.68 0.78 3521Business Services 0.38 0.74 1929


42/84

34

C h a p t e r 2

EQUITY PRICING MODEL DELIVERY & VISUALIZATION

Model Processing and Delivery

On the third Friday following the last trading day of the month, Zacks sends a

custom report file to Parallax that contains the most recent fundamentals for

about 8,600 stocks. We chose Zacks because from our experience they have thelongest historical databases with the highest quality of any corporate data vendor

(see Appendix B). Despite this, only about 3,600 stocks (Aug 2004 count) have

all the data required for pricing. Some of the “dropout” is linked to new

companies, since we need company records to go back at least 5 years. It seems

though, that most of the dropout is due to incomplete SEC filings. We will come

back to this topic later in the results section.

The pricing process for each stock is actually quite involved. Numerous priceestimates are retrieved from the trained networks by repeatedly splitting the per

share data, presenting the artificial data to the net, retrieving the price estimate,

and then un-splitting the estimate. If the model is internally consistent and the

stock is relatively easy to value, then the price estimates will all be about the same

and independent of split factor. In practice, non-linearities and biases that remain

in the model can cause scattered or even unstable estimates. We have even

noticed cases of “dual” stable solutions that are widely separated in price. Our

technique identifies these cases. The benefit of this type of error analysis is that it

samples different regions of the valuation model for a single stock. A big error


43/84

35

also may signal that the stock’s balance sheet is a bit atypical. Negative earnings,

cash flow, ROE, etc. can cause low or unstable prices.

We produce price estimates in this manner for all stocks with at least one stable

solution. Monthly results are delivered to clients via three products:

1. A monthly database of estimated prices for each stock in an EXCEL

spreadsheet format

2. A summary report in EXCEL format containing statistics on various

collections of stocks. For example, we list the percentage of large capstocks that are undervalued, and by what average and median amount.

3. A database file for the TradeStation TM charting program, so that all price

estimates can be displayed. This same database also powers our

interactive price calculator program called Wizard Explorer TM

Clients are notified monthly by email that these products are available for

download. A message such as the following is sent from our operations center inHawaii:

The August Wizard database is now available in:

ftp://ftp.pfr.com/users/dfrick/private/wiz_mwx/wizdb200408.exe

This is a self-extracting Zip archive that will put the

database files into their proper location.

The Wizard spreadsheet and summary report are in:ftp://ftp.pfr.com/users/dfrick/private/wiz_mwx/wizrep200408.zip


44/84

36

Interactive Price Calculator

The interactive price calculator software we call Wizard Explorer TM

, is thesimplest way for a client to explore our neural network pricing model. When the

calculator appears, enter the stock symbol, month, and year that you wish to

examine in the top three boxes and then press the “Get Data” button. All of the

fundamental factors will be read from our database and displayed in the edit

windows.

The windows with a white background may be edited, including the pulldown list

of industry names. If for instance you want to see what Aetna is worth as aninsurance company instead of a health care company, just change the industry

before pressing the “Re-Calculate” button at the bottom. This button invokes

our neural net model and causes the estimated price to be calculated. The price

above this estimate is the month-end price from the prior month, so if you

selected August 2004 at the top, the month-end price date would be July 31,

2004.

Sometimes fields display a “N/A” which means that no data was available fromZacks for this field. Just type in an estimate for the missing data and then re-

calculate. If all the input data is present and the price estimate is listed as N/A,

then you know that a stable solution could not be found. Our database goes

back to 1990, but the displayed historical data is split corrected to September

2001 only. We have placed the appropriate split factor in an edit box on the

lower right side for splits that have occurred since then.


45/84

37

Figure 16. Neural network price calculator for “what-if” exploration. Enter stock information atthe top and press “Get Data” to populate the fields below.

Enter symbol, month,year, and then press“Get Data” button tobring in fundamentaldata.


46/84

38

Figure 17. Edit any of the fields with a white background and recalculate. Compare the actualmonth end price with our estimated price.

Edit any of the fields witha white background andre-calculate the estimatedprice


47/84

39

TradeStation TM Historical Graphing

The best way for clients to view our Price Wizard pricing model results is within

the TradeStation charting program from TradeStation Technologies. The graph

below shows our estimated stock valuations in purple for the stock IBM. The

solid purple line is the published record, while the dashed line refers back to the

company quarter from which the data was taken. Additional text is included

which describes the average valuation of the Industry and Sector. In the graph

below we show only valuations from 1998-2000. The text describes IBM on Dec

27, 2002. At that time we estimated it was worth $143.

Figure 18. TradeStation display of estimated prices from our neural net model

In TradeStation this feature is an indicator called PFR_Wizard and has the

following inputs:

Price Wizardhistorical stockprice estimates

Notice how overvalued IBMbecame


48/84

40

INPUTNAME

DESCRIPTION DEFAULT

WizText Prints descriptive text if true TRUE

CoRepLine A dashed line is drawn if this is truethat leads to the quarter end dateupon which the valuation is made

TRUE

DumpFile Dump valuation data to an ASCIIfile if True (C:\IBM_Wiz.csv”)

FALSE

Split Wizard is published once a month,and price is split corrected to theend of the month prior to thepublished date. If your stock splitsince publication, then you mustchange this factor. A 2:1 splitrequires Split=2.0

1

DBS Reserved for testing otherdatabases

“Wizard”

Table 4. TradeStation inputs for Parallax indicator PFR_Wizard.

The stock price bubble of the late nineties is best seen in the chart of Cisco

Systems.:

Figure 19. Model prices of Cisco clearly showed the bubble forming


49/84

41

TradeStation TM Aggregate Portfolio Graphing

All of the stocks we price each month are tagged with an industry, sector, marketcap, and now a value/growth rank. We can calculate the percentage that they are

away from the actual market price and plot histograms of relative valuation for

many different collections of stocks. Statistics can be used on these distributions

to characterize the collection. For instance, we may choose to look at the relative

valuations of large cap finance stocks, or maybe small cap growth stocks. The

figure below is a histogram of relative valuation from June of 1996. 55% of all

the stocks we priced were under their estimated price at that time.

Figure 20. June 1996 relative valuation histogram shows 55% of all stocks are undervalued.


50/84

42

By March 2001 however, only 25% of stocks were under-valued. What came

next was an enormous stock market drop.

Figure 21. March 2001 relative valuation histogram shows 25% of all stocks are undervalued.


51/84

43

The percentage of undervalued stocks is one useful statistic, another is the

average amount of that relative valuation histogram, the median and standard

deviation may also be important. We will come back to the question of whichstatistics work best in the testing chapter. As far as visualization is concerned, we

have built several tools in TradeStation that plot “percent under-valued” graphs

for stock collections by sector (PFR_WizardSector), industry

(PFR_WizardIndustry), and any arbitrary list of stock symbols (PFR_WizardList).

In the following figures, the size of the “percent undervalued” bar at the bottom

of the chart represents the sampling error. Arrows mark extreme readings, which

can be used to guide buy and sell decisions.


52/84

44

Figure 22. The figure shows the bank index and the percentage of stocks that were undervalued inthe bank industry over time. The size of the bar represents the sampling error. Arrows markextreme readings, which can be used to guide buy and sell decisions.


53/84

45

Figure 23. The figure shows the retail index and the percentage of stocks that were undervalued inthat sector over time. The size of the bar represents the sampling error. Arrows mark extremereadings, which can be used to guide buy and sell decisions.


54/84

46

Figure 24. The figure shows the oil index and the percentage of stocks that were undervalued in

the oil sector over time. The size of the bar represents the sampling error. Arrows mark extremereadings, which can be used to guide buy and sell decisions.


55/84

47

Figure 25. The figure shows the percentage of stocks that make up QQQ (Nasdaq 100) that areunder their fair value.


56/84

48

EXCEL TM Database and Summary Spreadsheets

The results of our stock pricing model are sent to clients each month in several

forms, as was mentioned before. The following picture shows the EXCEL

database format for the August 2004 report:

Table 5. EXCEL spreadsheet containing estimated prices for all stocks with sufficient data.

The fields listed across the top include symbol, company name, sector name,

industry name, market cap, month-end price from the prior month, estimated

price (called “netprice”), and the percent change between the month-end price

and our estimated price.


57/84

49

Another EXCEL report contains statistics on various collections of stocks. The

August 2004 report is shown below. At the top of this report is the statistical

analysis of the stocks by common industry. For instance, in the apparel industry,

there are 37 stocks, of which 16% are undervalued. The average price is 18%

over the actual market price, and the median price is 20% over the market price.

Not a very attractive industry. The report looks at sectors, market cap ranges,

value versus growth, and industry breakdowns. The next two pages containpictures created in EXCEL and derived from this August 2004 summary report.

Table 6. EXCEL spreadsheet containing valuation statistics by sector, industry, market cap, and value/growth rank.


58/84

50

Figure 26.

Figure 27 .


59/84

51

Figure 28 .

Figure 29 .


60/84

52

C h a p t e r 3

HISTORICAL TESTING

Survivorship and In-Sample Considerations

Historical testing errors are often the undoing of great looking investment

management systems. There are four main sources of these errors, all dangerous;

curve fitting, clairvoyance, random data errors, and survivorship bias.

We have already discussed curve fitting, although it is called memorization in

neural network lingo. It means that a neural net has not really learned anything

about the facts presented, but because of the size of the network, it was able to

just remember every fact. We have shown that this did not happen in our work.

Both the training and test sets scored the same when processed by our networks.

An additional memorization test requires we compare how well the networks

performed in the “out-of-sample” period versus the “in-sample” period. Ourout-of-sample period began in October 2001, so we will examine results before

and after that date.

Clairvoyance in our case means that information used to construct historical price

estimates was not available on or before the publishing date for those estimates.

We have confidence that Zacks data is date consistent, since the databases we

used had been carefully vetted over many years.


61/84

53

We do know, however, that occasionally corporate data is entered into these

databases with a decimal place in the wrong spot. These random errors are

worked out of a database over time, but would have been present in real time.

We do need to set some limits for our testing so that data errors are a bit more

limited. Toward that end, we have chosen a set of all stocks, sampled monthly,

which has a month-end price prior and subsequent to the published valuation

report which were above $5.00 a share at the time. We assume that if a stock is

found mid-month that it is bought at month end if it is still above $5.00. Further,

we have restricted market capitalizations to be above 250 million as of the prior

month-end. We are also requiring that the database list an earnings report date.

In order to evaluate subsequent performance, we require at least one additional

price point within the following 12 months. Performance for missing price

periods is interpolated, but never extrapolated. Missing prices for the end of the

evaluation period are simply the last known price carried forward. This set of

filters produces 350,215 records spanning the date range from Oct-1989 through

Aug-2004 (181 months). Out of these, a subset of 193,467 have been pricedusing our model.

The average 12 month performance of all stocks during the period was 9.23%.

We were not able to price every stock because of missing data. The absence of

data turns out to be a bit of an advantage however. The 12 month performance

of stocks that had valuations was 9.67%, while the average was only 8.69% for

the stocks that had none. This means that just having a valuation gives an

advantage of 98 basis points over those that didn’t, or 44bp over the set of allstocks, as shown in the graph below (from w1_2.xls):


62/84

54

Figure 30. This shows the average percentage gain for all stocks, all priced stocks, and all unpricedstocks during our test period from Oct-1989 to August 2004 (250 million min market cap). Notethere is a 44 basis point advantage at 12 months just because the company reported sufficient datato allow pricing.

The next pictures show the average 12 month performance of all priced stocks

during the test period that were undervalued by different percentages at the time

of hypothetical purchase. The first chart shows the straight average, while the

second shows the excess performance above the average. Note that just being

undervalued by any amount resulted in a performance gain of 400 basis points.


63/84

55

Figure 31. This shows the average percentage gain for all priced stocks that were undervalued bysome percentage at the time of hypothetical purchase during our test period from Oct-1989 to

August 2004 (250 million min market cap). Note there is a 400 basis point advantage at 12months just by being undervalued by any amount.


64/84

56

Figure 32. This shows the 12-month excess average percentage gain in basis points for all pricedstocks that were undervalued by different percentages at the time of hypothetical purchase duringour test period from Oct-1989 to August 2004 (250 million min market cap). Note there appearsto be an optimal 12-month performance of 600 basis points achieved by being undervalued by atleast 20% at purchase.


65/84

57

Figure 33. This shows the average percentage gain for all priced stocks that were either over orundervalued by some percentage at the time of hypothetical purchase during our test period fromOct-1989 to August 2004 (250 million min market cap).


66/84

58

Figure 34. This shows the average percentage gain for all priced stocks that were undervalued bysome percentage at the time of hypothetical purchase during our test period from Oct-1989 to

August 2004 (5 billion min market cap). Note there is a 325 basis point advantage at 12 monthsjust by being undervalued by any amount. However, large cap stocks more than 40% undervaluedperformed worse than average at 12 months.


67/84

59

Figure 35. This shows the average excess percentage gain by year for all priced stocks that wereundervalued by some percentage at the time of hypothetical purchase during our test period fromOct-1989 to August 2004. Note that during the bubble years of 1998 and 1999, undervaluedstocks underperformed the index, while after the bubble in 2000, they soared.


68/84

60

Information Coefficient

Information Coefficient or IC is an accepted measure of skill for ranking stocks. IC is calculatedas follows: 1) Rank the stocks from best long ideas (forecasted highest return) to best short ideas(forecasted lowest return). 2) Rank the subsequent period actual returns, e.g. one month or 6month from highest return to lowest return. 3) Calculate the correlation between those 2 series of

numbers. The higher the IC, the better is the skill. A good IC is typically in the .05-.06 range, andabove .07 is generally considered excellent. IC is a measure of the slope of the graph of forecastedreturns compared with actual returns. The weakness of IC is that it’s a linear calculation.

Therefore, you could have a high IC, but lower accuracy on the tails of the distribution – best longand short recommendations – or have a low IC, yet with excellent forecast ability on the tails.


69/84

61

Figure 36. This shows the average percentage gain per month for all priced stocks, andundervalued stocks with varying predicted betas, during our test period from Oct-1989 to August2004 (using a 5 billion minimum market cap). Note that having a predicted beta from 0.8 to 1.2and being undervalued, performed better than stocks with predicted betas outside of this range.


70/84

62

Predictor Statistical Testing Methods

Parallax has been producing neural-network based financial predictors since 1990

and so it is an integral part of our business to validate these predictors using

reliable statistical methods. With each predictor, we need to answer the following

set of questions:

1. What price behavior is being predicted?

2. What is the effective duration of the predictor?

3. How statistically significant is the predictor at each time step forward?

4. Is the predictor effective on all time scales?

5. Is the predictor effective on all financial series?

6. What combinations of predictors are the most effective?

The following section is an overview of a simple and reliable statistical method

and an 8-year analysis of the Price Wizard predictor. In order to carry out this

analysis, it is necessary to measure the price action during the time period

immediately following each prediction event. We call this the “post-predictor” or

“outrun” period.


71/84

63

Post-Predictor Z Scores

We are interested in characterizing the post-predictor time period using a scale-independent method that allows comparisons between financial series. The “Z

Score” is the most appropriate measure for this job. A Z-Score is the measure of

how many standard deviations price has moved away from its price at the

prediction event, assuming that the probability of either an up or down move is

random at 50%.

By measuring local volatility at the prediction event, a normal probability

distribution can be drawn going forward in time that acts as a roadmap forsubsequent price moves. The map is centered at the closing price of the

prediction event. Each day the map widens according to normal diffusion

velocities, which are proportional to 1/ √ Time, representing the region where the

future price is most likely to be found. An example of such a probability map is

shown below for the stock Home Depot on Feb 3, 2005:


72/84

64

Viewed from the side at two time steps, diffusion acts to spread out the region

where we might find the stock price as time elapses:

At each time step, there is a larger standard deviation and the same mean. If we

represent the actual price achieved at each step in terms of that standard

deviation, we produce a series of Z-Scores. For example, if the price at a

prediction event is $5, and then moves to $7 on day ten with a standard deviation

of $1.60, then the Z-Score on day ten would be (7-5)/1.6 = 1.25. This means

that price moved 1.25 standard deviations above the price at the prediction event.

Since the standard deviation continues to increase, in order to maintain the same

Z, price would have to increase by the same relative amount.

Diffusion causesexpected prices tospread out overtime


73/84

65

If all post-predictor prices for all financial series are converted to Z-Scores, and if

the predictors do not work, and if markets are random , then at every time

step, a histogram of all the Z-Scores would be expected to result in a“standardized” normal distribution, with mean of zero and a standard deviation

of one N(0,1) as shown below:

We of course hope that our predictors actually predict non-random pricebehavior, so the degree of deviation from the normal curve is critical. We will use

a Chi-squared test to determine if the distributions are significantly different.

Are Financial Series Random?

We have assumed that price movement is best characterized by a random walk

model, but this might not be the case. There is gathering evidence for other

characteristic distributions such as a Biased Random Walk, Truncated Levy

Flights, or Cauchy distribution. Our solution is to produce a quantitative

background distribution based on randomly selected dates across our entire 3000


74/84

66

day test period, and for all of the 2500 stocks being considered. The figure below

shows this background distribution of Z-scores corresponding to random

prediction dates, and a normal distribution based on a random walk assumptionplotted together. It is clear from this figure that a strict random walk assumption

is inappropriate during our 8 year test period. Instead there appears to be a

positive return bias.

To illustrate this another way, we could ask what percentage of buy predictions

are winners (Z>0) if the timing is random, and plot this percentage each day

following the randomly selected purchase dates. Normally this would be 50%,

but since the background distributions is positively biased, the figure below

shows the percent winners for randomly selected buys climbs steadily over time.

The reverse is of course true for randomly selected sells (mirror image).


75/84

67

On a weekly scale the effect is even more pronounced, as shown in the next

figure. I guess this could be called the dartboard effect, in that even a random

selection of stocks during this period showed a 57% win rate at 30 weeks afterpurchase, and shorting was decidedly unprofitable.


76/84

68

We will use these curves as benchmarks for our predictors over this test period

and stock set. In order to have a non-random buy predictor then, we need to see

a win rate in excess of 57% at 30 weeks for example, and sells would need a win

rate better than 43%.

So far we have examined the background distribution of Z-Scores and the

percentage of winners. What we still need to know is how much gain is possible

from randomly selected buys and sells, so that each predictor’s excess gain profile

can be evaluated. The pictures below show the positively biased background gain

performance for randomly selected buys and sells:


77/84

69


78/84

70

Price Wizard™ as a PredictorPrice is often out of sync with value, being bid up or down based on future

expectations and irrational trend persistence. The Price Wizard model discussed

in this paper incorporates all major stores of corporate value, normalized

simultaneously for sector, industry, and economic factors. It predicts what price

should be, and if it works, then we should be able to detect a significant effect.

The figure below shows the Z-Scores 30 weeks following a stock moving to

undervalued from overvalued. Note the significant positive bias between the

background distribution and the undervalued stock Z distribution. A Chi-

squared test is designed for comparing distributions and a p


79/84

71

These two graphs show the percentage of winners, daily and weekly, for buys andshorts, which exceed the background “dartboard” rate.


80/84

72

The next four graphs show the median and average percentage gain, daily and weekly, for longs and shorts, which exceed the background “dartboard” rate.


81/84

73


82/84

74

Dynamic Price Wizard™ (F% for short) as a Predictor

The knowledge of how much a stock is undervalued or overvalued is of primary

importance, but the rate of change in value and price during the preceding year isalso important. The Dynamic Price Wizard neural net (we call the indicator F%

for short) adds value by capturing these dynamics explicitly. The figure below

shows the Z-Scores 30 weeks following a stock moving to F% greater than a rank

of 90 out of 100. Note the significant positive bias between the background

distribution and the undervalued stock Z distribution.

These two graphs show the percentage of winners, daily and weekly, for buys andshorts, which exceed the background “dartboard” rate.


83/84

75

The next two graphs show the median and average percentage gain weekly, forlongs and shorts, which exceed the background “dartboard” rate.


84/84

Date post:	07-Aug-2018
Category:	Documents
Upload:	alex-bernal
View:	214 times
Download:	0 times

Parallax US Equity Valuation Neural Network Model

Documents