MFC-092 AksoyLeblebicioglu.DOC

Modelling the Ise100 Index by Using Fuzzy Logic and Neural

Fuzzy Systems

Hakan AKSOY (corresponding author)

Ph.D. Candidate (in Finance) Senior Portfolio Manager

Department of Management Koc Asset Management

Bogazici University, Istanbul, Turkey Besiktas, 34349, Istanbul, Turkey

Email: [email protected]

Phone: +90 (532) 706 53 66

Fax: +90 (212) 213 71 56

Prof. Dr. Kemal LEBLEBICIOGLU

Electrical Engineering Department

Middle East Technical University

ODTU, 06531, Ankara, Turkey

Modelling the Ise100 Index by Using Fuzzy Logic and Neural

Fuzzy Systems

Abstract

Fuzzy logic and neural fuzzy models has been constructed to forecast the

monthly returns of the ISE100 Index of Turkey by using the following financial

variables: price over earning ratio, dividend price ratio, equity transaction ratio,

volatility, foreign investment over the ISE100 Index, foreign investments over

market capacity of the ISE100 Index, technical analysis of the ISE100 Index, the

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mailto:[email protected]

Dow Jones Index of the New York Stock Exchange, gross national product,

industrial production index, capacity utilisation index, balance of payments, FX

reserves, external debt stock, FX rate, inflation rate, domestic debt stock, budget

deficit, repo rate, bond price index, domestic risk and foreign risk. Each of the

fuzzy logic and neural fuzzy method has its good and bad aspects and they are

complementary methods in designing advanced models. Fuzzy logic is better than

neural fuzzy systems in mathematical modelling and expert knowledge but worse

than in learning and optimization ability.

Consequently, this study compares the modellings of the ISE100 Index of

Turkey are demonstrated by using the fuzzy logic and neural fuzzy systems in

system view approach.

Keywords:Stock Market, Efficient Market, Anomalies, Classifier

Systems,Learning, Fuzzy Logic, Dynamic Games, Optimization.

JEL Classification: C45, C53, C82, G14.

1. SYSTEM VIEW OF THE TURKISH ECONOMY

Economy can be viewed as a system of financial and real variables. These

variables are related not only to each other, but also to the domestic and global

security and commodity markets. In Turkey, there are three main security markets

to trade in TL terms: the stock market, the repo market and the bond market. The

stock market in Turkey is the most risky one, an observation that is valid for the

markets in the world as well.

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The Istanbul Stock Exchange (ISE) is the only stock market in Turkey

whereby stocks may be bought and sold. Like every investment, investing in

stocks entails some degree of risk. There are two types of risk; systematic risk and

nonsystematic risk. The nonsystematic risk can be overcome by a sound

investment strategy, called diversification. However by using the variables in the

system of the economy, the systematic risk can be minimised with some

nonparametric methods if it is not eliminated (Lo and MacKinlay, 1988).

In the past decade, fuzzy systems have been used with conventional

techniques in many scientific applications and engineering systems, especially in

system theory. Fuzzy sets, introduced by Zadeh (1965) as a mathematical way to

represent vagueness in linguistics, are different than the classical set theory. In a

classical nonfuzzy set, element of the universe either belongs to or does not belong

to the set, in other words the membership of an element is crisp. A fuzzy set is a

generalisation of an ordinary set in that it allows the degree of membership for

each element in a unit interval.

Fuzzy logic and neural networks are complementary methods in designing

advanced models. Each method has its good and bad aspects. Fukuda and Shibata

(1994) presented the comparison of these techniques. Fuzzy logic is better than

neural networks in mathematical modelling, knowledge representation and expert

knowledge but worse than neural networks in learning and optimization ability. In

this study the bad and good aspects of the fuzzy logic and neural fuzzy systems

will be demonstrated by using the modellings of the ISE100 Index.

In order to design various types of stock market models, the title of the

paper by Lo and MacKinlay (1988) will come to the rescue: ‘Stock Prices do not

follow random walks’. In their paper they present that considerable evidence exists

justifying this statement and show that stock returns are to some extent predictable.

Human reasoning can be modelled as if the thought process is described by

the application of fuzzy logic (LeBaron et al, 1999). Traders are capable of

handling a large number of rules for mapping of market states into expectations.

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Tay and Linn (2001) shows this by allowing agents the ability to compress

information into a few fuzzy notions which they can in turn process and analyse

with fuzzy logic.

Kooths (1999) developed a macroeconomic model to realise an alternative

to conventional expectation hypothesis. The experience and rule based

expectations can be used in forecasting behaviour that is characterised by explicit

rule orientation (theory foundation), vague formulation (bounded rationality) and

learning process (acquisition of experience).

Consequently, to characterise the ISE100 Index, there may be easily

implemented fuzzy models to explain to some extent predictable behaviour

patterns. Thus, this kind of mathematical advanced model for the ISE100 Index

has been used for the first time in the literature.

In the next section, information about investment instruments, stock market

and financial variables is presented. Section 3 presents general concepts about

fuzzy logic, membership functions, rule generation and the description of financial

time series modelling for the ISE Index. In section 4, a similar model for the

ISE100 Index is constructed and optimized by using neural fuzzy systems,

followed by the conclusion and studies to be undertaken in the future.

2. THE ISE100 INDEX OF TURKEY AND THE DEPENDENT

VARIABLES

Every investment has some degree of risk; it requires a certain sacrifice at

the present for an uncertain future benefit. When we look at the risk return analysis

of the remaining assets; namely stock, repo and bonds: Stock market has the

highest level of risk. In Turkey the average fluctuations of bond and repo market

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are almost similar and significantly less risky with respect to the stock market,

because the markets are affected by the similar politic and macroeconomic events.

The ISE price indices are computed and published throughout the trading

session while the return indices are calculated and published only at the close of

the session. The ISE100 Index is used as a main indicator of the ISE. The ISE100

is composed of market companies except investment trusts. The contents of the

ISE100 Index are selected on the basis of predetermined criteria directed for the

companies to be included in the indices as well as in consideration of their ability

to represent relevant sectors. The market capitalisation is weighted by the publicly-

held portion of each constituent stock kept in custody (Takasbank) (except those

kept in non-fundable accounts). The basic formula for calculating of the ISE100

index is as follows:

(2.1)

where Pit is the closing price stock 'i' at period 't', Nit is the total number of shares

of the stock 'i' at period 't' (Paid-in capital/1,000), Fl it is the flotation weight

(publicly-held portion) of the stock 'i' kept in custody (except those kept in non-

fungible accounts) at period 't', and Dt is the value of divisor at period 't' (adjusted

base market value). In the calculation of the index only registered prices are taken

into account.

While forecasting the stock market that is the most risky market, the

systematic risk should be reduced by using the variables in the system of economy.

These financial variables are price over earning ratio, dividend price ratio, equity

transaction ratio, volatility, foreign investment over the ISE100 Index, foreign

investments over market capacity of the ISE100 Index, technical analysis of the

ISE100 Index, the Dow Jones Index of the New York Stock Exchange, gross

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national product, industrial production index, capacity utilisation index, balance of

payments, FX reserves, external debt stock, FX rate, inflation rate, domestic debt

stock, budget deficit, repo rate, bond price index, domestic risk and foreign risk. In

this study, modelling depends on three basic variables. These are risk variables,

stock market variables and economic variables.

2.1 Risk Variables

Risk variables consists of two factors:

1. Domestic Risk Factor:

All of the risk factors were triggered or instigated by the Turkish

government, social and financial groups or instruments. Some examples are the

Korkmaz Yiğit case in the final quarter of 1998 and the row between the President

and the Prime Minister in February 2001. In each of the cases the stock market

was drastically hit.

2. International Risk Factor:

Similarly, all of the risks were instigated or caused by external factors such

as international politics or social and financial groups. In the beginning of the

second half of 1998, the devaluation in Russia threatened the international

investors in Turkey and the stock market decreased by 60.6% from 4615 to 1819.

The devaluation in Brazil in 1999 is another significant example of international

risk factors.

2.2 Stock Market Variables

Stock market variables can be grouped into three factors:

1. Technical Analysis of the ISE100 Index:

Technical analysis is based on the widely accepted premise that security

prices are determined by the supply of and the demand for securities. Typically,

technical analysts record historical financial data on charts and study these charts

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in search of meaningful patterns and they aim to use the patterns to predict future

prices. For this variable, I created a survey and gave it to a number of professional

investors. For every month from 1997 to 2000, 30 days moving average, 10 days

moving average, relative strength index, volatility and total volume of the ISE100

Index were calculated and given to the professional investors in the survey.

Investors were asked to forecast the next month's return by considering these

parameters from -2 to 2.

2. Dow Jones Index in New York Stock Exchange:

It is observed that there is a significant correlation between the Dow Jones

and the ISE100 Index in the last ten years. The Turkish Stock Market is an

emerging market and is heavily influenced from foreign investors because the

significant portion of the market consists of foreign investors. Since foreign

investors take their positions by considering worlds largest stock market; the New

York Stock Exchange, there is a significantly strong correlation between the Dow

Jones and the ISE100.

3. Stock Market Ratios:

Not only the prices but also some other financial ratios and anomalies

affect the stock market significantly. The most important factors are:

3.A. Price over Earnings Ratio: Price over earnings ratio fluctuates inversely to

the risk of an asset. For this variable from 1997 to 2000, monthly price over

earning ratios are calculated by the ISE and they are used in the simulation for the

modelling.

3.B. Dividend to Price Ratio:The return on stocks, or the yield to a holder of a

stock, is equal to the dividend (as percent of price) plus the capital gain. There are

more than 300 stocks in the ISE. It is used as a variable for the weighted average

of each stock's dividend performance calculated by the ISE.

3.C. Volume / ISE100 Index Level: Next to the prices, trading volume is the

second most important statistic that an analyst should follow. Because of the high

inflation rates and the stock exchange's growth in Turkey, equity turnover ratio

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gives more precise results. Equity turnover ratio, in other words trading volume

over the ISE100 Index level gives the average transaction rate in the ISE. The

basic reasons for the importance of the equity turnover ratio are as follows:

Active investors want liquid markets, in which a large volume of contracts is

traded. This is desirable because it allows those who are trading large positions

to buy or sell at any time without causing a significant price change.

In any case of danger such as a need for liquidity or insider knowledge of bad

news, investors need to sell the stock if the demand is enough.

The demand for well performing stocks is always higher than the ones that

cause more transactions and higher equity turnover ratio.

After making profit, selling the stock is easier than after stopping the loss,

selling the stock in human psychology. Thus, in bullish market (market in

positive trends) volume and transaction rate increase. However, in bearish

market (market in negative trends) generally investors expect that the market

will turn back. As a result they prefer to hold the stock.

For this variable, the ratio of the monthly trading volume over the monthly average

of the ISE100 Index level is calculated.

3.D. Volatility of the monthly returns of the ISE100 Index: Volatility is defined

as a measure of the risk of the stock market investments. Corporations use stock

markets to raise capital for investments. The most important function of the stock

market is to raise capital for corporations. If stock prices rationally reflect

fundamental values, the stock market can then serve as a forecasting signalling for

firms and investors to guide the process of capital allocation. If stock prices

deviate from real values because of noise trading or volatile movements, investors

may not want to hold equities, which tend to increase capital cost and affect

investment negatively. The standard deviation of the monthly returns of the

ISE100 Index is calculated and used as a variable.

3.E. Foreign Investments / ISE100 Index Level: In emerging markets, foreign

investments are high relative to the domestic investments. From 1997 to 2000, the

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foreign investments in the ISE change between $3.07 billion to $15.36 billion that

is quite important for the Turkish economy. Since the emission, the total money in

the Turkish Economy is around $3-5 billion, the decision of the foreign investors

can affect the Turkish economy deeply. Consequently, the ratio of foreign

investments over the ISE100 Index is very important to analyse how the market

will behave; because, the decision-makers of the foreign investments are very

professional in managing their portfolios. They can follow the news faster than the

regular investor and can affect the market.

3.F. Foreign Investments / Market Capacity of ISE100: Free float market

capacity of the ISE changes between approximately $7 billion to $24billion. As

written above, the foreign investments in the ISE change between $3.07 billion to

$15.36 billion. As seen from the numbers, the market leader of the ISE is foreign-

investors. However, even foreign-investors have faced big losses in the ISE. As

also mentioned earlier, if the investors' movements are synchronised with the

market, the index fluctuates normally. If the big players of the ISE have large

amount of losses, they will not allow the market to soar sharply. Consequently, the

ratio of foreign investments to market capacity of the ISE100 is also a good

indicator to analyse the trend of the index.

2.3 Economic Variables

The economic variables can be grouped into four factors in which the last

one, TL market has high correlation with the ISE. Thus, the importance of the TL

market is more than the other parts in the modelling part. To group them into their

categories, these four factors are:

1. Real Economic Indicators:

Macroeconomic performance in Turkey can be measured by three broad

measures: GNP, industrial production and capacity utilisation level. News of these

three variables makes the headlines because these issues affect our daily lives.

They also dominate the research agenda in macroeconomics.

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1.A. Gross National Product (GNP): When the growth rate is high, the production

of goods and services is rising, making possible an increased standard of living.

With the high growth rate typically go lower unemployment and the availability of

more jobs. High growth is a target and hope of most securities. The growth rate of

real GNP is the most important of all the macroeconomic indicators by which to

judge the economy's long run performance. In Turkey, quarterly real GNP values

are available. By using linear interpolation method, the data are converted to the

monthly real GNP values. Because of the seasonality, the each month's data are

divided by the previous year's same month's data as a variable for the models.

1.B. Industrial Production Level: Growth rates of total factor productivity differ

widely across sectors. It also raises doubts about how effectively output is

measured in some service industries, such as banks or the government. While

analysing the growth of a country, it should be better to consider the measurement

of industrial productivity. The State Institute of Statistics releases the industrial

production data every month. As a variable, because of the seasonal effects, the

change in the ratio of the industrial production level with respect to the previous

year's same month is calculated.

1.C. Capacity Utilisation Level: Production depends on the amount of output

produced in an economy to the inputs of factors of production and to the state of

technical knowledge. For the production, any company in the economy works in

full capacity or less than full capacity, depending on the external and internal

effects. Consequently, capacity level of a company affects the production and the

growth. SIS releases the data of the capacity utilisation level every month. As a

variable, the difference of the data with respect to the previous year's same month

is used.

2. Foreign Exchange (FX) Market Indicators:

Because of the heavy influences of the foreign investors in Turkish markets, it is

important to analyse the foreign exchange (FX) market. These are:

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2.A. Balance of Payments: The balance of payments is the record of the

transactions of the residents of a country with the rest of the world. The simple

rule for balance of payments accounting is that any transaction that gives rise to a

payment by a country's resident is a deficit item in that country's balance of

payments. Thus, imports of cars or oil, use of foreign shipping, gifts to foreigners,

purchase of land in the other country, or making a deposit in a Bank in Switzerland

are all deficit items. Examples of surplus items, by contrast, would be sales of cars

abroad, payment by foreigners, pensions from abroad or foreign purchases of

stock. When there is a net outflow, balance of payments is in deficit. To control

the economy and have the healthy growth in a country, the surplus in balance of

payments is necessary. When the foreign investors leave the country, it will turn

into a deep deficit and affect the economy worse.

As a variable, the total previous 12 months balance of payments' surplus or

deficit are calculated.

2.B. FX Reserves: The Central Bank of Turkey holds reserves that they would sell

in the market when there was an excess demand for dollars. Conversely, when

there was an excess supply of dollars, they would buy the dollars. As a variable,

the total intervention of the Central Bank of Turkey in the previous 12-month

period is used.

2.C. Total External Debt Stock: The easiest way to finance the government's and

the private companies' budget deficit is to find an external debt. The interest rate

and the cost in external debt are relatively lower than the domestic debt. Thus, the

government prefers to borrow external debt. Besides, long term debt issues in

external debt stock are more frequent than the issues in domestic debt stock. In

Turkey, the data for the external debt stock is released in every quarter of the year.

By using linear interpolation method, the quarterly data are converted to monthly

data. As a variable, the change of the debt stock with respect to the previous year's

same month is calculated.

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2.D. FX Rate: FX rate is the value of one unit of foreign money in TL terms. A

rise in the exchange rate means that foreign prices have increased relative to the

prices of goods produced here. Goods abroad have become more expensive

relative to the prices of goods at home, which other things equal, implies people

are likely to switch some of their spending to goods at home. This is often

described as an increase in the competitiveness of our products, as our goods

become cheaper relative to foreign goods. In this case of devaluation, the stock

market will be cheaper and have comparative advantage in prices, while other

things being constant. Monthly devaluation rate in United States Dollar is used as

a variable in the models.

3. Price Indicators:

The budget deficit and the domestic debt show how the government can turn its

debt relaxed. These variables cause high inflation.

3.A. Inflation: The inflation rate is the percentage rate of increase of the level of

prices during a given period. The inflation is one of the main concerns of citizens,

policy makers, and macroeconomists. During periods of inflation, the prices of

goods people buy are rising. Partly for this reason, inflation is unpopular, even if

people's incomes rise along with the prices. Inflation is also unpopular because it is

often associated with other disturbances to the economy such as the oil price

increases that would make people worse off. Mainly, there are two types of

inflation wholesome price index (WPI) and consumer price index (CPI). These

price indexes depend on different categorical variables. To sum up, CPI is more

individual needs' price increase oriented. However, in an economy the price

changes are generally measured by WPI because it gives the large amount of

goods' price index. As a variable, the increase in WPI with respect to the previous

year's same month is used.

3.B. Total Domestic Debt Stock: When the budget is in deficit, the national debt

increases. National debt is the result of past budget deficits. The Treasury sells

securities more or less continuously. Long term debt issues are less frequent.

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Issues of the Treasury debt are not only made for the purpose of financing the

budget deficit but also made to refinance parts of national debt that are maturing.

The Treasury sells the amount of Treasury Bills or Government Bonds it has

offered at the auction to the bidders who offer the highest prices, or lowest interest

rates. If the debt is in Turkish Lira terms, it will be the total domestic debt. When

the foreign investors leave the country in case of crisis, the Treasury has

alternative debtors, the domestic investors. However in domestic borrowing the

interest rate is relatively high. As a variable, the ratio of the total change of the

domestic debt over the average WPI prices in 12 months is calculated.

3.C. Budget Deficit: Like private companies, the government has a budget too.

When the government is spending more than it receives, the budget is in deficit.

The size of the budget deficit is affected by the government's fiscal policy

variables; such as government purchases, transfer payments, and tax rates. To

finance the deficit, the government has to borrow debt from creditors or the budget

should be in surplus by increasing revenues or decreasing expenditures in the next

periods. As a variable, the ratio of the total budget deficit over the average WPI

prices in 12 months is calculated.

4. TL Market Indicators:

In Turkey, there are other markets to invest in TL-terms. These are repo and bond

market. Although their volatility is less than the ISE, they are almost influenced by

similar effects. Thus, the correlation between each other is relatively high.

4.A. REPO Rates: Repurchase agreements (repos) are instruments used to help

finance part of their inventories of marketable securities for one or few days. For

instance, if an investor ends a day of trading with an increase of $1 million in its

inventory of marketable securities, a repo may be sold to finance the $1 million

inventory overnight. The investor is essentially making a short-term loan to the

other investor with his inventory serving as collateral. Repos that last longer than

overnight, called term repos, can span 30 days or even longer. Term repos are

marketable securities, actively traded between the money market trading desks of

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large banks and brokerage houses. In another words, repo is a short-term loan

instrument to find money to enter the stock market. In the ISE, it is calculated the

average monthly repo rates and used as a variable for cost of carrying money.

4.B. Treasury Bills' and Government Bonds' Price Index in the ISE (DIBS): A

bond is a promise by a borrower to pay the lender a certain amount at a specified

date and pay a given amount of interest per year. The interest rates on bonds issued

by different borrowers reflect the differing risks of default. Default occurs when a

borrower is unable to meet the commitment to pay interest or principal.

Government Bonds represent the indebtedness of government. The owners of the

bonds are creditors; government is the debtor. Government Bonds are of such high

quality that their yield is often used as an example of a riskless, default free or

interest rate. In the ISE government bonds are also traded. There are

approximately 20-30 government bonds traded in the ISE and these bonds change

according to the maturity date and interest rates in the meantime. The ISE

calculates the weighted average price of the government bonds, the DIBS Index.

As a variable, the monthly return of the DIBS Index is used.

3. MODELING, OPTIMIZATION AND RESULTS FOR THE

ISE100 INDEX BY USING FUZZY LOGIC

Fuzzy logic has been applied very successfully in many areas where

conventional model based approaches are difficult or expensive to implement for

the design and the learning. However, as the system complexity increases, reliable

fuzzy rules and membership functions used to describe the system behaviour

become difficult to determine. Furthermore, due to the dynamic nature of

economic and financial applications, rules and membership functions must be

adaptive to the changing environment in order to continue to be useful. It has been

considered as one of the most attractive strategies in identifying complex systems,

particularly for nonlinear systems with imprecise and uncertain knowledge of

system information and behaviour. In contrast to a regular knowledge base in

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expert system, the fuzzy rules, being structured knowledge, can be coded into

relevant explicit numerical algorithms or a fuzzy model with fuzzy identification

or fuzzy reasoning based on system non-fuzzy input or data by Lin and Lee

(1996). Fuzzy logic can be applied to system modelling, estimation, optimal and

optimization of control and adaptive control problems. Only system and input-

output data are required.

If the number of variables are too much, the number of rules will increase

much faster. This will cause a rule explosion in the model. In order to prevent the

rule explosion, an expert should make a hierarchical modelling in fuzzy logic. In

the beginning of this study, a hierarchical modelling is used too. In the model,

there is also middle layers to combine related variables in groups. The membership

functions of the input and middle variables are optimized. Finally, in this section a

model for the stock exchange will be developed to estimate the monthly returns of

the ISE100 Index with 22 financial input variables.

3.1 Prepositional Logic and Rule Generation

In crisp logic, such as binary logic, variables are true or false, black or

white, 1 or 0. An extension to binary logic is multi-valued logic, where variables

have many crisp values. Prepositional logic on the other hand, is defined with

uncertain terms. The next months' return of the ISE100 index is estimated as 6%

increase. In some aspects it may be a good increase, in another it may be a very

good increase. Thus, with some degree it is a good increase and a very good

increase.

In the example given above, very bad decrease, bad decrease, neutral, good

increase and very good increase are the linguistic terms which can be converted

from the real data with some degree.

A relationship is defined to express the distribution of truth of a variable.

For example, 'good increase' may be defined as a distribution around a value 'x'.

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Any value within the distribution may be interpreted as 'good increase', although

with different degrees of truth or confidence.

Theoretically, a fuzzy set F of a universe of disclosure X={x} is defined as

a mapping, by which each x is assigned a number in range [a,b]. This indicates the

boundaries of the attribute F of x. If x is the monthly return of the ISE100 index,

'good increase' may be considered as a particular value of the fuzzy variable, and

each x is assigned a number in the range of real numbers. Ugood inc.(x) is element of

[a,b] that indicates the extent to which that x is considered to be good increase.

Ugood inc.(x) is element of [a,b] is called a membership function. Let X be a time-

invariant set of objects x. A fuzzy set F in X may be expressed by a set of ordered

pairs F = {(x,U(x)) | x is an element of X}, where U is the membership function

that maps X to the membership space M, and U(x) is the grade of membership or

degree of truth of x in F. In addition when M contains the values 0 and 1 only,

then F is non-fuzzy and U is the characteristic function of a non-fuzzy set. Thus,

binary and multi-value logic are extreme sub-cases of fuzzy logic. As an example,

let the monthly returns of the ISE Index in 2000 be 2.1%, 9.4%, -4.5%, 16.7%, -

2.3%, -5.3%, 2.8%, 2.5%, -11.3%, 1.5%, 2.2% and 4.1%, respectively. Increases

around 10% are considered Very-Good-Increase, around 5% Good-Increase,

around 1% Neutral, around -1% Bad-Decrease and -5% Very-Bad-Decrease. In the

example there are not more then two membership-functions for each x as seen in

the below (While constructing the architectures of the models there are

similarities).

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0

0.25

0.5

0.75

1

-12% -9% -6% -3% 0% 3% 6% 9% 12% 15% 18%

very bad bad neutral good very good

Figure 3.1 Membership Functions

As a result of the membership functions the data can be converted easily.

Simplicity, piecewise linear membership functions have been assumed.

Membership functions can be continuous curves of many different shapes (Lin

and Lee, 1996).

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0

0.25

0.5

0.75

1

-12% -9% -6% -3% 0% 3% 6% 9% 12% 15% 18%

very bad bad neutral good very good

Table 3.1 Fuzzification of the Variables

Fuzzification of the five crisp variables as in the figure above, causes

spreading of the variables with a distribution profile that suits the problem.

In fuzzy problems, the rules are produced based on experiences.

Concerning problems that deal with fuzzy engines or fuzzy control, all possible

input-output relationships should be in fuzzy terms. The input output relationships

or rules are expressed with 'if then' statements such as:

If (a is in A1) and (b is in B1), then (c is in C5); or

If (a is in A1) and (b is in B2), then (c is in C1); or

If (a is in A2) and (b is in B5), then (c is in C3).

The A's and the B's are fuzzified inputs, and the C's are the actions for each

variable. For 2 variables with 5 membership functions for each, total number of

rules should be 25 rules (5x5) as seen in the below table.

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Date

Monthly Return of

ISE100

Very- Good-

IncreaseGood-

Increase NeutralBad-

Decrease

Very- Bad-

Decrease00/01 2.10% 0% 0.275% 0.725% 0% 0%00/02 9.40% 0.880% 0.120% 0% 0% 0%00/03 -4.50% 0% 0% 0% 0.125% 0.875%00/04 16.70% 1% 0% 0% 0% 0%00/05 -2.30% 0% 0% 0% 0.675% 0.325%00/06 -5.30% 0% 0% 0% 0% 1%00/07 2.80% 0% 0.450% 0.550% 0% 0%00/08 2.50% 0% 0.375% 0.625% 0% 0%00/09 -11.30% 0% 0% 0% 0% 1%00/10 1.50% 0% 0.125% 0.875% 0% 0%00/11 2.20% 0% 0.300% 0.700% 0% 0%00/12 4.10% 0.775% 0.225% 0% 0% 0%

Table 3.2 Rules

The rules become more difficult to tabulate if the fuzzy statements are

more in number, that is if there are 3 variables with 5 membership functions for

each, total number of rules would increase to 125 rules (5x5x5).

3.2 Defuzzification

In fuzzy logic, the values are not crisp, and their fuzziness exhibits a

distribution described by the membership function. If one tries to get two fuzzy

variables, what will the output be? The question has been addressed by various

fuzzy logics such as:

Min operation (Mamdani).

Max-min operation (Zadeh).

In general, defuzzification is the process where the membership functions

are sampled to find the grade of membership. The grade of the membership is used

in the fuzzy logic equation and after an outcome region is defined, the output is

deduced.

Several techniques have been developed to produce an output, such as:

Center of Area (COA).

Mean of Maximum (MOM).

As an example by using min rule (Mamdani) for fuzzy logic operation and

COA for defuzzification:

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VAR. B

B1 B2 B3 B4 B5

A1 C5 C1 C2 C2 C5

A2 C4 C4 C1 C3 C3

A3 C2 C3 C4 C1 C5

A4 C1 C2 C4 C4 C1

A5 C1 C1 C3 C5 C2

VA

R. A

Let, if (a is in A4) and (b is in B1) then (c is in C1);

if (a is in A5) and (b is in B1) then (c is in C1);

if (a is in A4) and (b is in B2) then (c is in C2);

if (a is in A5) and (b is in B2) then (c is in C1).

UA4(a)=0.282, UA5(a)=0.718, UB1(b)=0.191 and UB2(b)=0.809.

Then, Min(UA4(a), UB1(b))=0.191 for (c is in C1);

Min(UA5(a), UB1(b))=0.191 for (c is in C1);

Min(UA4(a), UB2(b))=0.282 for (c is in C2);

Min(UA5(a), UB2(b))=0.718 for (c is in C1).

Since there is one question but four answers, the defuzzification process

will be done via taking the COA of the four rectangles as seen in the graphics

below, and will give the value of the output.

Figure 3.2 Defuzzification

Fuzzy system can express knowledge but can not learn to adapt by itself. In

many applications, partial understanding of the knowledge is available, but a

complete set of rules is not. However input-output data may be available to

determine the rest. What is needed is technology that can work with a partial

knowledge base and can also learn from the additional data in order to perform the

21

0

0.2

0.4

0.6

0.8

1

Membership Functions: C1(0.19), C1(0.19), C1(0.72) and C2(0.28)

task correctly. Fuzzy logic handles the explicit knowledge, whereas optimization

handles the knowledge implicit in the data. A fusion of both these into one

provides a better way of resolving problems.

3.3 Modelling

One of the main advantages of fuzzy logic is that it more closely models

the kind of reasoning that a person engages in when dealing with issues or

elements that are not precisely defined and that involve aspects of degree or

judgement. However it may not be possible to think in terms of sharp boundaries

for modelling. Thus, the disadvantages of fuzzy logic systems are the same as

those found in traditional knowledge based systems: Someone has to write rules,

which means that expert knowledge has to be available and formalised. These

systems can not learn on their own, nor can they adapt to changing market

conditions, except by manually rewriting the rules, adjusting the membership

functions or other specific rule-finding methodologies.

According to professional investors, the stock markets are generally

affected by four main criteria (Virtual Trading, 1995):

Stock Market Factors

Macro Economic Factors

Political Risk Factors

Inter-market Factors.

In the model designed for the ISE100 and the investors, the inter-market

factors (effects of other markets) are included in macro economic factors and stock

market factors part. Thus, the models in the following pages generally consist of

three main parts; finance, economy and political risk.

The Economy section has two main parts: Economic ratios and TL market.

TL market, consists of repo and DIBS and is also part of Inter-market factors.

Economic ratios part consists of three parts: Real economy, FX market and prices.

22

Real economy shows the fundamental changes in Turkey. These are GNP,

industrial production index and capacity utilisation ratio. The FX market consists

of FX rate and FX ratios. FX ratios depend on balance of payments, FX reserve

and total foreign debt in Turkey. FX rate is the monthly percent changes in the

ratio of USD to TL.

The stock market section has three main parts: The stock market ratios,

technical analysis of the stock market and the Dow Jones Index in United States.

The Dow Jones is the most popular stock market in the world and has the highest

volume. Thus, it is also a good indicator for the inter-market factors. Technical

analysis sometimes explains what the fundamental analysis can not explain. The

stock market ratios section consists of three parts: financial ratios, transactions and

foreign investments. Financial ratios depend on the ISE100’s average price over

earnings ratio and dividend rate. Transactions depend on the ratios of average

monthly volume to the ISE100 Index and monthly standard deviation. Foreign

investments depend on the ratio of foreign investments to the ISE100 Index and

the ratio of foreign investments to market capacity of the ISE100 Index.

Finally, in the political risk section, there are only two parts: Domestic

political risk and external political risk.

Consequently, the inputs used in all of the models are summarised in the

table below. Foreign debt and GNP variables are originally in quarterly format. To

convert them into monthly format, linear interpolation method is used. Thus, there

are 22 input variables, all of which are in monthly format.

23

Table 3.3 Variables Used in the Models

For each variable the maximum, minimum, average and the standard

deviations are given in the following table. By using these values with expert

opinion, the groups of the fuzzy membership functions of the variables are given

in the next table. All of the variables have five membership functions. As

explained before, if there are two variables with five membership functions for

each, number of rules will be 25. Number of rules will be 125 for three variables

with five membership functions.

To have five membership functions, there should be seven numbers. As

seen in Figure 3.1, the inner three membership functions are triangles and the outer

two membership functions are trapezoids. In Table 3.4; 1, 2 and 3 are the border

points of the first trapezoid; 2, 3 and 4 are the border points of the first triangle; 3,

4 and 5 are the border points of the second triangle; 4, 5 and 6 are the border points

of the third triangle and 5, 6 and 7 are the border points of the last trapezoid. The

24

V1 P/E Price over Earning Ratio V2 Dividend to Price Ratio Dividend RatioV3 Volume/ISE100 Equity Turnover Ratio V4 Standard Deviation VolatilityV5 Foreign Investment/ISE Foreign Investment/ISEV6 Foreign Investment/MCAP Foreign Investment/Market Capacity of the ISEV7 Technical Analysis Technical Analysis of the ISEV8 Dow Jones % Change in Dow Jones of the New York Stock ExchangeV9 GNP % Change in Gross National Product (y-o-y)V10 Industrial Production % Change in Industrial Production (y-o-y)V11 Capacity Utilization % Change in Capacity Utilization (y-o-y)V12 Balance of Payments Central Bank Balance of Payments (y-o-y)V13 FX Reserves Increase in FX Reserve of Central Bank (y-o-y)V14 Foreign Debt Increase in Foreign Debt Stock of Turkey (y-o-y)V15 FX Rate US Dollar/TL ParityV16 Inflation % Change in Wholesome Price IndexV17 Budget Deficit Budget Deficit/Wholesome Prices (y-o-y)V18 Domestic Debt Increase in Domestic Debt/Wholesome Prices (y-o-y)V19 Repo Rates Monthly Average of Repo RateV20 DIBS Government Bills and Bonds Index (according to the ISE)V21 External Risk Risk from outside of TurkeyV22 Domestic Risk Risk from inside of Turkey

middle number for each triangle is the head and the others are the bottom numbers.

The last number for the first trapezoid and the first number for the last trapezoid

are the bottom numbers.

Table 3.4 Statistical and Membership Range of the Variables

After having the variables with membership functions, the rules should be

produced for the last step of the modelling. As seen in Figure 3.3, there are 22

input variables and the other 14 middle variables are produced from these 22

variables. Half of the 14 middle variables are produced from two different

fuzzified variables (25 rules for each variable). The other 7 variables are produced

from three different fuzzified variables (125 rules for each variable).

Finally, there should be 1050 rules for the overall model. The rules for each

variable are given in the following two tables.

25

MAX MIN AVE STDEV 1 2 3 4 5 6 7Y 0.798 -0.390 0.057 0.196 0.798 0.526 0.254 0.057 -0.139 -0.265 -0.390A1 -7.329 -39.020 -18.258 7.559 -7.329 -9.014 -10.700 -18.258 -25.817 -32.418 -39.020A2 4.070 0.620 2.169 1.009 4.070 3.624 3.178 2.169 1.160 0.890 0.620A3 -197.098 -1072.839 -480.452 172.129 -197.098 -252.710 -308.323 -480.452 -652.581 -862.710 -1072.839A4 -0.073 -0.431 -0.183 0.072 -0.073 -0.091 -0.110 -0.183 -0.255 -0.343 -0.431A5 6.312 3.123 4.410 0.750 6.312 5.736 5.159 4.410 3.660 3.391 3.123A6 0.135 0.089 0.110 0.011 0.135 0.128 0.121 0.110 0.099 0.094 0.089A7 2.000 -2.000 0.167 1.548 2.000 1.857 1.715 0.167 -1.382 -1.691 -2.000A8 0.102 -0.151 0.012 0.051 0.102 0.083 0.063 0.012 -0.039 -0.095 -0.151A9 1.056 0.404 0.734 0.212 1.056 1.001 0.947 0.734 0.522 0.463 0.404A10 0.213 -0.121 0.031 0.081 0.213 0.162 0.112 0.031 -0.050 -0.085 -0.121A11 6.500 -7.200 -0.658 3.437 6.500 4.640 2.779 -0.658 -4.096 -5.648 -7.200A12 4329.000 -10411.000 -2122.208 3782.136 4329.000 2994.464 1659.927 -2122.208 -5904.344 -8157.672 -10411.000A13 10048.000 -4623.500 1552.525 3272.174 10048.000 7436.349 4824.699 1552.525 -1719.649 -3171.574 -4623.500A14 12150.000 3747.500 7118.146 2328.215 12150.000 10798.180 9446.361 7118.146 4789.931 4268.715 3747.500A15 0.071 -0.008 0.040 0.019 0.071 0.065 0.059 0.040 0.021 0.007 -0.008A16 0.071 0.004 0.039 0.016 0.071 0.064 0.056 0.039 0.023 0.014 0.004A17 1093.099 327.127 676.431 280.966 1093.099 1025.248 957.397 676.431 395.464 361.296 327.127A18 214.214 112.876 154.373 27.876 214.214 198.231 182.248 154.373 126.497 119.687 112.876A19 1.927 0.214 0.677 0.255 1.927 1.429 0.932 0.677 0.422 0.318 0.214A20 -102.410 -107.700 -104.546 1.571 -102.410 -102.693 -102.975 -104.546 -106.118 -106.909 -107.700A21 2.000 -2.000 0.313 0.938 2.000 1.625 1.250 0.313 -0.625 -1.313 -2.000A22 2.000 -1.500 -0.177 0.884 2.000 1.354 0.707 -0.177 -1.061 -1.281 -1.500

Table 3.5 Rules in 2 by 2 Tables for the Modelling of the ISE100 Index

26

FINANCIAL RATIOSDivident Ratio

B1 B2 B3 B4 B5A1 C1 C1 C2 C2 C2

Price over Earning A2 C2 C2 C2 C3 C3A3 C2 C3 C3 C3 C4A4 C3 C3 C4 C4 C4A5 C4 C4 C4 C5 C5

TRANSACTIONStandard Deviation

B1 B2 B3 B4 B5A1 C1 C1 C2 C2 C2

Volume / ISE100 A2 C2 C2 C2 C3 C3A3 C2 C3 C3 C3 C4A4 C3 C3 C4 C4 C4A5 C4 C4 C4 C5 C5

FOREIGN INVESTMENTForeign Inv/MCAP

B1 B2 B3 B4 B5A1 C1 C1 C2 C2 C2

Foreign Inv/ISE100 A2 C2 C2 C2 C3 C3A3 C2 C3 C3 C3 C4A4 C3 C3 C4 C4 C4A5 C4 C4 C4 C5 C5

TL MARKETDIBS

B1 B2 B3 B4 B5A1 C1 C2 C2 C3 C4

Repo A2 C1 C2 C3 C3 C4A3 C2 C2 C3 C4 C4A4 C2 C3 C3 C4 C5A5 C2 C3 C4 C4 C5

RISKDomestic Risk

B1 B2 B3 B4 B5A1 C1 C2 C2 C3 C4

External Risk A2 C1 C2 C3 C3 C4A3 C2 C2 C3 C4 C4A4 C2 C3 C3 C4 C5A5 C2 C3 C4 C4 C5

FX MARKETFX Ratios

B1 B2 B3 B4 B5A1 C1 C1 C2 C2 C2

FX Rate A2 C2 C2 C2 C3 C3A3 C2 C3 C3 C3 C4A4 C3 C3 C4 C4 C4A5 C4 C4 C4 C5 C5

ECONOMYEconomy Ratios

B1 B2 B3 B4 B5A1 C1 C1 C2 C2 C2

TL Market A2 C2 C2 C2 C3 C3A3 C2 C3 C3 C3 C4A4 C3 C3 C4 C4 C4A5 C4 C4 C4 C5 C5

Table 3.6 Rules in 3 by 3 Tables for the Modelling of the ISE Index

Consequently, the final model for the ISE is shown in Figure 3.3. The

result of the model by using the rules above with fuzzy logic algorithm is quite

accurate, 55.18%, which is the sum of the squares of the difference of the real

returns from the estimated returns divided by the real returns of the stock market.

27

REAL ECONOMY Capacity Utilisation (C) Industrial Production (B)

B1 B2 B3 B4 B5 B1 B2 B3 B4 B5 B1 B2 B3 B4 B5 B1 B2 B3 B4 B5 B1 B2 B3 B4 B5A1 D1 D1 D2 D2 D2 D2 D2 D2 D3 D3 D2 D2 D3 D3 D3 D3 D3 D3 D4 D4 D3 D3 D4 D4 D4

GNP A2 D1 D2 D2 D2 D3 D2 D2 D2 D3 D3 D2 D3 D3 D3 D4 D3 D3 D3 D4 D4 D3 D4 D4 D4 D5A3 D1 D2 D2 D2 D3 D2 D2 D3 D3 D3 D2 D3 D3 D3 D4 D3 D3 D4 D4 D4 D3 D4 D4 D4 D5A4 D2 D2 D2 D3 D3 D2 D2 D3 D3 D3 D3 D3 D3 D4 D4 D3 D3 D4 D4 D4 D4 D4 D4 D5 D5A5 D2 D2 D2 D3 D3 D2 D3 D3 D3 D4 D3 D3 D3 D4 D4 D3 D4 D4 D4 D5 D4 D4 D4 D5 D5

FX RATIOS Foreign Debt (C)FX Reserve (B)


Bal. of A2 D1 D2 D2 D3 D3 D2 D2 D3 D3 D4 D2 D2 D3 D3 D4 D2 D2 D3 D3 D4 D2 D3 D3 D4 D4Payments A3 D2 D2 D3 D3 D4 D2 D2 D3 D3 D4 D2 D3 D3 D4 D4 D2 D3 D3 D4 D4 D2 D3 D3 D4 D4

A4 D2 D3 D3 D4 D4 D2 D3 D3 D4 D4 D2 D3 D3 D4 D4 D3 D3 D4 D4 D5 D3 D3 D4 D4 D5A5 D2 D3 D3 D4 D4 D3 D3 D4 D4 D5 D3 D3 D4 D4 D5 D3 D3 D4 D4 D5 D3 D4 D4 D5 D5

PRICES Domestic Debt (C)Budget Deficit (B)


Inflation A2 D2 D2 D2 D3 D3 D2 D2 D2 D3 D3 D2 D2 D3 D3 D3 D2 D2 D3 D3 D3 D2 D3 D3 D3 D4A3 D2 D2 D3 D3 D3 D2 D3 D3 D3 D4 D2 D3 D3 D3 D4 D3 D3 D3 D4 D4 D3 D3 D3 D4 D4A4 D3 D3 D3 D4 D4 D3 D3 D3 D4 D4 D3 D3 D4 D4 D4 D3 D3 D4 D4 D4 D3 D4 D4 D4 D5A5 D3 D3 D4 D4 D4 D3 D4 D4 D4 D5 D3 D4 D4 D4 D5 D4 D4 D4 D5 D5 D4 D4 D4 D5 D5

STOCK MARKET RATIOS Foreign Investments (C)Transactions (B)


Financial A2 D2 D2 D2 D2 D2 D2 D2 D2 D2 D3 D2 D2 D3 D3 D3 D3 D3 D3 D3 D3 D3 D3 D3 D3 D4Ratios A3 D2 D2 D2 D3 D3 D2 D3 D3 D3 D3 D3 D3 D3 D3 D3 D3 D3 D3 D4 D4 D3 D4 D4 D4 D4


ECONOMY RATIOS Real Economy (C)FX Market (B)


Prices A2 D2 D2 D2 D3 D3 D2 D2 D2 D3 D3 D2 D2 D3 D3 D3 D2 D2 D3 D3 D3 D2 D3 D3 D3 D4A3 D2 D2 D3 D3 D3 D2 D3 D3 D3 D4 D2 D3 D3 D3 D4 D3 D3 D3 D4 D4 D3 D3 D3 D4 D4A4 D3 D3 D3 D4 D4 D3 D3 D3 D4 D4 D3 D3 D4 D4 D4 D3 D3 D4 D4 D4 D3 D4 D4 D4 D5A5 D3 D3 D4 D4 D4 D3 D4 D4 D4 D5 D3 D4 D4 D4 D5 D4 D4 D4 D5 D5 D4 D4 D4 D5 D5

STOCK MARKET Stock Market Ratios (C)Dow Jones Index (B)


Tech. A2 D2 D2 D2 D2 D2 D2 D2 D2 D2 D3 D2 D2 D3 D3 D3 D3 D3 D3 D3 D3 D3 D3 D3 D3 D4Analysis A3 D2 D2 D2 D3 D3 D2 D3 D3 D3 D3 D3 D3 D3 D3 D3 D3 D3 D3 D4 D4 D3 D4 D4 D4 D4


ESTIMATED OUTPUT Economy (C)Stock Market (B)


Risk A2 D2 D2 D2 D2 D2 D2 D2 D2 D2 D3 D2 D2 D3 D3 D3 D3 D3 D3 D3 D3 D3 D3 D3 D3 D4A3 D2 D2 D2 D3 D3 D2 D3 D3 D3 D3 D3 D3 D3 D3 D3 D3 D3 D3 D4 D4 D3 D4 D4 D4 D4A4 D3 D3 D3 D3 D3 D3 D3 D3 D3 D4 D3 D3 D4 D4 D4 D4 D4 D4 D4 D4 D4 D4 D4 D4 D5A5 D3 D3 D3 D4 D4 D3 D4 D4 D4 D4 D4 D4 D4 D4 D4 D4 D4 D4 D5 D5 D4 D5 D5 D5 D5

C5C1 C2 C3 C4

C5C1 C2 C3 C4

C5

C1 C2 C3 C4 C5

C1 C2 C3 C4

C5

C1 C2 C3 C4 C5

C1 C2 C3 C4

C5C4C3C1 C2

Figure 3.3 Fuzzy Logic Model for the ISE100 Index

3.4 Optimization and Results

After having 55.18% explanation, the question should be: Is there any

improvement in the accuracy? Thus, the next work is to find the optimum stock

market model, which is developed by changing the border points of the

membership functions by using steepest descent algorithm. By minimising the

square of the error, which is the difference of real stock market returns from

estimated stock market returns, the inner five border points of the membership

functions are changed. Firstly, this process is done for the last three variables (risk,

28

stock market and economy) and the output. Secondly, the membership functions of

the 22 input variables are changed according to the same algorithm. Lastly, the

same process is done to the last three variables and the output again. Finally, the

new model is a little different than the first model with membership functions of

the 22 input variables, the last 3 variables and the output.

The learning of the model in Pentium III computer after 72 minutes is with

%69.73 accuracy, which is the sum of the squares of the difference of the real

returns from the estimated returns divided by the real returns of the stock market.

The result of the optimization will end with 14.55% improvement in the modelling

of the ISE100 Index.

Figure 3.4 Real and Estimated Output with 69.73% Accuracy

The estimated and the real output is always observed at the same directions

(When the real output is negative, the estimated output is also negative and when

the real output is positive, estimated output is also positive). In February and

November 1999, the difference of the real and the estimated data are the largest.

This causes the accuracy of the model to decrease to 69.73%. If these two data are

discarded, the accuracy will increase to above 90%.

29

-40%

-20%

0%

20%

40%

60%

80%

Jan-97 Oct-97 Jul-98 Apr-99 Jan-00 Oct-00

Real Y Estimated Y

Besides, it will be better to compare these models with the data that are not

used in the optimization process. The monthly data used in the optimization are

taken between 1997 and 2000. The validity test is done with the monthly data of

the first six months of 2001.

Figure 3.5 Validity with 75.92% Accuracy

As seen in Figure 3.5, the accuracy is 75.92%. Thus, the validity test of the

model is better than the optimization part. Besides, all the data are estimated in the

correct direction. For the data of April 2001, there is a little difference between the

real and estimated output as in the results of the optimization part.

4. MODELING, OPTIMIZATION AND RESULTS FOR THE

ISE 100 INDEX BY USING NEURO FUZZY SYSTEMS

Fuzzy logic and neural networks are complementary technologies in the

design of intelligent systems. They are used in an environment to improve the

30

-40%

-20%

0%

20%

40%

60%

Jan-01 Feb-01 Mar-01 Apr-01 May-01 Jun-01

Real Y Estimated Y

intelligence of systems working uncertain, imprecise and noisy environments.

Each method has good and bad aspects. In 1994, Fukuda and Shibata presented the

comparison of these techniques.

Table 4.1 Comparison of the Fuzzy Systems with respect to the other methods

Fuzzy systems are better in mathematical modelling, knowledge

representation, expert knowledge and nonlinearity. However in optimization and

learning ability, neural networks are better. In this chapter, it is focused on the

rationale of integrating fuzzy logic and neural networks into a working functional

system.

This happy marriage of these two techniques results with the neuro-fuzzy

system method with benefits of both neural networks and fuzzy logic system.

These are, the neural networks provide the distributed representation properties

and the learning abilities to the fuzzy logic systems and the fuzzy logic systems

provide the neural networks with high level fuzzy rule thinking and reasoning.

In this section, neuro-fuzzy modelling is tried because there may be other

important but unknown rules. In fuzzy logic modelling of the ISE100 Index, there

are 1050 rules. If it was used similar membership functions; three triangles and

31

Fuzzy Systems

Neural Networks

Genetic Alg.Control Theory

Symbolic AI

Math. Model Slightly Good Bad Bad Good Slightly BadLearning Ability Bad Good Slightly Good Bad BadKnowledge Rep. Good Bad Slightly Bad Slightly Bad GoodExpert Knowledge Good Bad Bad Slightly Bad GoodNonlinearity Good Good Good Bad Slightly BadOpt. Ability Bad Slightly Good Good Slightly Bad BadFault Tolerance Good Good Good Bad BadUncertainty Tol. Good Good Good Bad BadReal-time Op. Good Slightly Good Slightly Bad Good Bad

two trapezoids in the neuro-fuzzy modelling and if the output of the model was

directly affected by all of the 22 initial variables, there would be

2,384,185,791,015,620 (= ) rules. As seen in expression 4.2, a Gaussian

function is used for the modelling subsection, which makes the number of rules

infinite. Consequently, with infinite number of rules in neuro-fuzzy modelling,

there may be some rules, which are not included in fuzzy logic modelling with

finite number of rules.

4.1 Modelling

Unlike fuzzy logic, neuro-fuzzy modelling of the ISE Index is not so

semantically complicated. The estimation only depends on a function;

(4.1)

where x is the input data, R is the rules, w is the constant number and y is the

output.

In neuro-fuzzy modelling rules are not available explicitly. Before starting

to the simulations number of rules are set to a number, which is large enough. In

this case, number of rules are set to 50 or in other words n is equal to 50.

Rules are exponential functions;

(4.2)

where c is the index of the rule, is the input data, and are specific

parameters for the rule. If the number of variables is equal to n, there are (n+2)

32

unknown numbers for each rule. ( N unknowns because of , 1 unknown

because of and 1 unknown because of ).

If the number of variables is equal to k and number of variables is equal to

n, there are k times (n+2) unknowns.

By using steepest descent algorithm, the optimum stock market model

developed by minimising the square of the error, which is the difference of real

stock market returns from estimated stock market returns. Finally, the model is

used to estimate the real monthly return of the stock market by 22 financial

variables.

4.2 Optimization and Results

The optimization of the model in Pentium III computer with 50 rules ends

after 18 days 9 hours with %98.99 accuracy, which is the sum of the squares of the

difference of the real returns from the estimated returns divided by the real returns

of the stock market. The result seems to be better; however, some of the rules are

negligibly small. Thus, step by step the rules are discarded and the model is

optimized again. Consequently, the final result was calculated with 8 rules in

Pentium III after 3 days 18 hours with almost same accuracy, %96.66.

33

-40%

-20%

0%

20%

40%

60%

80%

Jan-97 Aug-97 Mar-98 Oct-98 May-99 Dec-99 Jul-00

Real Y Estimated Y

Figure 4.1 Real and Estimated Output with 96.66% Accuracy

According to these eight rules, some of the rules can be more effective than

the other rules. In the formula presented in 4.2, there are three parameters; c,

and and these parameters are different for each of the eight rules. Some of the

variables can be more important than the other variables because of the differences

in the data and the parameters. If the square of the ratio of the difference between

and the mean of to is close to zero, the inversely correlated power of

the exponential will be very small and this variable for that rule is not important.

Each of the eight rules’ figures with respect to the time are given in the appendix

part.

All of the rules, accept rule 6 and 8 are controlled by the similar rules. In

rule 1, external risk and domestic risk variables are the most important variables.

In rule 2, GNP is the most important variable. In rule 3, the ratio of foreign

investment to the ISE 100 Index has the dominant affect. In rule 4, technical

analysis is the most important variable. In rule 5, equity per share and price over

earning ratios are the dominant variables. In rule 7, capacity utilisation ratio is the

most important variable. In rule 6 and 8, there is no unique variable to affect the

results.

Consequently, the optimization result of the neuro-fuzzy model is better

than the fuzzy logic model. However, it will be better to compare these models

with the data not used in the optimization process. The monthly data used in the

optimization are taken between 1997 and 2000. Like the previous fuzzy logic

models, the validity test is done with the monthly data of the first six months of

2001.

34-40%

-20%

0%

20%

40%

60%

Jan-01 Feb-01 Mar-01 Apr-01 May-01 Jun-01

Real Y Estimated Y

Figure 4.2 Validity with 56.61% Accuracy

As seen in the accuracy, 56.61%, the validity test of the model is not as

good as the optimization part. Besides, to compare the validity test of the neuro-

fuzzy model with the fuzzy logic model, this model does not work as good as the

first one. As seen in the figure, half of the data are estimated in the wrong direction

which means while the real output is negative, estimated output is positive.

Consequently, fuzzy logic model works better and the dominant rules in

neuro-fuzzy modelling are also dominant in the fuzzy logic modelling (see figure

3.3). Thus, there is not a radical change in the fuzzy logic modelling.

5. CONCLUSION & FUTURE WORK

The classical economic estimation models are too slow and they can easily

be affected by seasonal trends in the data, also they are very sensitive to the

disturbance in the data. On the contrary there are better ways developed to

estimate the data. The most important improvement is the learning and dynamic

ability of the algorithms. In this study, models of the ISE100 Index are tried to be

35

implement in the field of economics via fuzzy logic. Especially while modelling

the players’ behaviour, some of the outputs are not changed and remain flat on the

graphics. As a result, the rules for the models may not be enough to explain the

real outputs. There may be some other rules to explain the behaviours more

explicitly and the models can be developed and can be redesigned better with the

new rules.

Similarly, there may be some new variables to explain the characteristics of

the behaviours better. In modellings of this study there is not included any lagged

variables of the inputs. There may be one or more period lagged variables to

extend the models’ modelling perspectives. Besides, the data set can be enlarged

by including other new variables (such as population growth or GNP per capita)

and by using the weekly or daily data to test the models more accurately.

For the ISE100 Index, another model is also implemented by using neural

fuzzy systems. The second model can learn better than the first model in the

optimization part. However, testing both of the models with the data not used in

the optimization part (validity test) proves that the first model works better than

the second model. The architecture of the second model shows that there are other

related rules for the estimation of the ISE100 Index. Consequently, the models

produced by using fuzzy logic can be developed and redesigned by observing the

new rules obtained from the models by using neural fuzzy systems.

Finally, the relations between the financial variables have not derived yet.

Except the domestic and foreign risk variables, all of the variables can be accepted

as endogenous variables and all of the variables have positive or negative

correlations between each other. After deriving the rough relations between the

endogenous variables used in the models, the stock market model can also be

constructed for the simulations, might be happened in the future.

36

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APPENDIX

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Figure A.1 Rule 1 of the Neuro-Fuzzy Modelling


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Figure A.4 Rule of the Neuro-Fuzzy Modelling


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