# Performance Evaluation of Mutual Funds Via Single Valued Neutrosophic Set (SVNS) Perspective: A Case Study in Turkey.

1 Introduction

A feedback based on the performance evaluation of an investment is quite significant for the success of any investment. It is important to determine whether the portfolios created for the purpose of risk distribution, either by institutional or individual investors, have the capacity to create the desired benefit or if the there exists a healthy risk-return relationship.

Evaluations of the performance of investment funds, managed by professionals, and composed of various securities (just as a portfolio), will have an impact on the success of the investment fund. When fund managers determine the assets that they intend to invest in, it is extremely important to measure the fund performance by considering the return and risk on the assets alongside other indicators. When determining the fund performance, another important point is to determine whether or not the performance realized is the result of any chance factor. The right decisions by the fund managers will lead to the selection of the right securities for the fund as well as the inclusion to the fund of securities that move in tandem with an emerging market. They are also able to make changes to the securities making up the fund in time by including securities that change slower than the market in case of declining markets.

2 Fund Performance Evaluation

When determining fund performance, the basic emphasis should be on risk and return. The total return expected from the fund should be compared to the level of risk that the fund is exposed to. The first thing that should be done therefore is the determination of the risk and return of the fund. The risk and return of the fund will vary according to the risks and returns of the assets in the fund pool.

2.1 Calculating the Morningstar Return

The Morningstar system is fundamentally made of two parts; while the first part works out the 'Morningstar Return', the second part determines the 'Morningstar Risk'. In calculating the Morningstar Return in the first stage, the monthly returns used should be calculated from the monthly closing price per share of selected funds. This simple calculation of the return can be expressed as:

Rp = (Vt - Vt-1) / (Vt-1) (1)

Where Rp is the monthly return of the fund, Vt is the monthly closing price per share, V-t is the monthly closing price per share for the previous month.

After calculating of the monthly returns of the fund, Morningstar obtains the value for 'adjusted return' for each month by subtracting, from the monthly returns, the monthly costs charged on the fund, such as commissions, expenses, and management fees among others. The monthly excess return for each fund is thus calculated based on the adjusted returns. In other words, the additional earnings by the fund above the risk-free rate. Since Morningstar always presents investors with investment alternatives to the risk-free assets, the excess return earned by a fund is now understood better as either above or below the risk-free rate. Consequently, the return of the fund is calculated by subtracting the risk-free interest rate for that month from the adjusted monthly return.

ERp = Adjusted Fund Return- Rf (2)

Where ERp is the monthly excess return of the fund and Rf is risk-free rate.

In the next stage, Morningstar divides the funds into categories. When Morningstar initially put forward this system, it grouped the funds into the four basic asset classes, as mentioned earlier, and classified (assigned) the funds structures into one of these asset classes according to the investment strategies. However, these asset classes were later revised to allow better evaluation and to avoid the comparison of apples and oranges. In 1996, Morningstar introduced categories that grouped funds into narrower classes. However, these categories were not integrated into the Morningstar Star Rating System until mid-2002. With the initial being 48 categories, this number rose to 64, and 81 by August 2009, and finally to 122 in April 2016. After the fund categories have been defined, the monthly 'category average return' for every fund category is calculated. To do this, the adjusted monthly returns of all the funds in the category are added and divided by the number of funds in the category, resulting into the determination of the monthly category average return for that fund category. After this is done, Morningstar compares the category average return to the risk-free rate, resulting in the Morningstar Return.

Morningstar Return = ERp / (category average return - Rf) or Rf (3)

As can be seen, the ERp obtained by subtracting the risk-free rate-based return from the adjusted monthly return of the fund forms the numerator of the equation and is divided by the greater of the (category average return - Rf) or Rf. Also, it will be noticed that while the numerator essentially shows the excess return of the fund, the denominator shows a comparison of the average excess return of the category in which the fund is found and the risk-free rate. Thus, the denominator of this equation may change from month to month, i.e., it may be the category average return--Rf expression in some months and only made of the risk-free return for other months. Morningstar divides the result by one of these two variables to avoid distortions due to low or negative average excess returns in the denominator of equation [7].

2.2 Calculating the Morningstar Risk

Once the Morningstar return has been calculated, the fund's Morningstar risk should be determined. In this regard, it should be noted that Morningstar is not based on the risk values obtained using measures of risk such as the standard deviation or beta coefficient, but it is determined on the basis of the downward risk to investors, which is the financial risk associated with losses, and which is believed to be investors' biggest fear. For this operation, first, the fund's adjusted monthly return and the risk-free interest rate are compared. At this point, months with reported negative monthly excess returns are identified. These negative returns are then summed up and divided by the total number of months in the period. The aim here is to determine the opportunity cost incurred by the investors in terms of monthly average (the monthly average loss) as a result of not investing at the risk-free rate. The same method is then applied to the fund category.

The average return of the category is compared with the risk-free interest rate, the months in which the monthly excess returns of the category in which the fund is found are reported as negative are determined, and these negative returns are summed and divided by the total number of months in the period. The average monthly loss of the category of funds, i.e. category risk, is calculated. The Morningstar risk of the fund is expressed as follows:

Morningstar Risk = AMLp / AMLc (4)

Where AMLp is the Average Monthly Loss of the fund and AMLc is the Average Monthly Loss of the category in which the fund is found.

2.3 Calculation of Raw Return

After obtaining the required data on Morningstar return and the Morningstar risk the next step is the calculation of the Morningstar raw return for each fund. This value is obtained by subtracting the Morningstar risk from the Morningstar return of the fund.

Raw Return = Morningstar return--Morningstar risk (5)

2.4 Treynor Ratio

The Treynor Ratio (Index) introduced by Jack Treynor in 1965 measures portfolio performance using the beta coefficient, which measures the systematic risk, instead of the standard deviation which measures the total risk. The beta coefficient indicates how sensate the securities or portfolio are to the market. Treynor argued that although non-systematic risks could be eliminated through portfolio diversification, systematic risk cannot be removed in any way [8]. Treynor thus divided the investment risk on a diversifiable portfolio into two parts: general market fluctuations, and fluctuations in securities within the portfolio. Treynor argued that the first risk is valid for all stocks and that it cannot be eliminated while the second risk can be eliminated or reduced by appropriately diversifying the portfolio. Just like in the Sharpe Ratio, the risk premium is also calculated here. However, unlike the Sharpe Ratio, the premium calculated here is based on the beta coefficient and not standard deviation. In this way, there is a residual return for every unit of systematic risk assumed, i.e. earning above the risk-free rate of return. The Treynor ratio can be expressed as follow [11]:

[T.sub.n] = ([R.sub.n] - [R.sub.f])/[[beta].sub.n] (6)

Where [T.sub.n] = Treynor Ratio

[R.sub.f] = Risk Free Rate

[R.sub.n] = Return of Fund

[B.sub.n] = Beta coefficient of the fund.

Treynor explained his portfolio performance based on the Security Market Line (SML), not the Capital Market Line (CML) used by Sharpe [24]. If the evaluated portfolio is above this level, then it has performed worse than the market, a performance below this line implies it is better than the market.

2.5 Sharpe Ratio

The Sharpe Ratio which measures portfolio performance based on the total risk was developed by William Forsyth Sharpe in 1966. This is one of the most commonly used and simplest methods for measuring portfolio performance. The idea that it is necessary for investors to invest in the index funds through the market portfolio in order to avoid the non-systematic risks that have been forecast by the Sharpe Ratio rests on the assumption that it would be erroneous to try to obtain more returns from the market, as stocks in active markets always reflect the prices correctly [22].

In his study, Sharpe tried to subject the measures presented by Treynor to the empirical test by evaluating their predictive powers. He also sought to advance Treynor's work, as well as make more explicit the relationship between the recent developments in capital theory and alternative models of investment fund performance and then subjecting these alternative models to more empirical tests. He analysed the annual return rates of 34 open-end investment funds between 1954 and 1963. In the study, the performance of 23 investment funds was found to be lower than the Dow Jones Industrial Average (DJIA), which was considered as the indicator. On the other hand, when relatively poorly diversified portfolios are considered, the Treynor Index, due to its diversification weakness and failure to address some of the variables, may lead the results to vary considerably. This led the conclusion by Sharpe that, whereas it may be a good measure for predicting future performance, the same reasons make it a poor measure for past performance [19].

The model introduced by Sharpe shows the extra return over the risk-free rate that the investor seeks to achieve for the amount of total risk undertaken, i.e. the additional return expected for every unit of total risk. The Sharpe Ratio is based on the Capital Market Line (CML) and assumes that the investment fund affects the portfolio [26]. The Capital Market Line can thus be said to be the indicator. When the Sharpe Ratio calculated for the portfolio that an investor holds is found to be greater than the Sharpe Ratio of the market portfolio, it can be concluded that the portfolio has performed better than the market [4].

The calculation of the Sharpe Ratio which is based on the total portfolio risk can be expressed as follows [22]:

S = ([R.sub.n] + [R.sub.f])/[[sigma].sub.n] (7)

Where S = Sharpe Ratio,

[R.sub.p] = Return of Fund

[R.sub.f] = Risk-Free Rate

[[sigma].sub.p] = Standard Deviation of the Fund (Total Risk of the Fund)

The difference between the return on the portfolio and the risk-free interest rate, the numerator of the equation, is known as the risk premium. This premium shows the reward or the residual return (excess return, return on risk-free interest rate) that the investor gets for undertaking risk. The denominator of the function indicates the total risk made of both systematic and non-systematic risks.

2.6 Jensen Ratio

This ratio was developed by Michael Cole Jensen in 1968 and uses just a single value in measuring portfolio performance. The Jensen Ratio measures the deviation of any portfolio from the Securities Market Line [18].

The Ratio based on the Capital Assets Pricing Model (CAPM) is the difference between the realized return of the portfolio and the expected return (with assumptions) above the Securities Market Line. This ratio also acts as an examination of the skills of the portfolio manager in formulating the portfolio. Since Jensen expressed the difference mentioned in terms of Alpha Coefficient, the ratio is also known as the Jensen Alpha. This ratio can also be defined as the supernormal return above the expected return according to the CAPM.

In his study, Jensen asserted that he had created a method (criterion) that measures the predictive skills of portfolio managers as well as their contribution to the returns of the funds known as the Jensen Alpha. He sought to measure the predictive skills of 115 mutual fund managers between 1945 and 1964. He noted that the performance of the 115 mutual funds, and by extension the skills of the managers in predicting the stock prices did not exceed the average by much [10].

The Jensen Ratio is expressed as follows [1]:

[[alpha].sub.n] = [R.sub.n] - [[R.sub.f] + [[beta].sub.n] ([R.sub.m] - [R.sub.f])] (8)

Where [[alpha].sub.n] = Jensen (Alpha) Ratio of the fund

[R.sub.p] = Return of the Fund

[R.sub.m] = Return of the Indicator Index (Market)

[R.sub.f] = Risk Free Rate

[[beta].sub.n] = Beta coefficient of the fund.

3 Methodology

3.1 Neutrosophic Set

Neutrosophic Sets (NS) is proposed by Smarandache (1998) having with degree of truth, indeterminacy and falsity membership functions in which all of them are totally independent [20,21]. Let U be a universe of discourse and x [member of] U. The neutrosophic set (NS) N can be expressed by a truth membership function [T.sub.N] (x), an indeterminacy membership function [I.sub.N](x) and a falsity membership function [F.sub.N](x), and is represented as N = {< x : [T.sub.N](x), [I.sub.N](x), [F.sub.N](x) >, x [member of] U}.. Also the functions of [T.sub.N](x), [I.sub.N](x) and [F.sub.N](x) are real standard or real nonstandard subsets of ][0.sup.-],[1.sup.+] [, and can be presented as T, I, F : U [right arrow] ][0.sup.-], [1.sup.+] [. There is not any restriction on the sum of the functions of [T.sub.N](x), [I.sub.N](x) and [F.sub.N](x), so:

[0.sup.-] [less than or equal to] sup [T.sub.N](x) + sup [I.sub.N](x) + sup [F.sub.N](x) [less than or equal to] [3.sup.+] (9)

The complement of a NS N is represented by [N.sup.C] and described as below [6] :

[T.sup.C.sub.N] (x) = [1.sup.+] [??] [T.sub.N](x) (10)

[I.sup.C.sub.N] (x) = [1.sup.+] [??] [I.sub.N] (x) (11)

[F.sup.C.sub.N] (x) = [1.sup.+] [??] [F.sub.N](x) for all x [member of] U (12)

There are applications of the neutrosophic set such as MCDM problems of supplier selection [1], strategic planning [2], logistic center location selection [17], teacher recruitment in higher education [13] and school choice [14].

3.2 Single Valued Neutrosophic Sets (SVNS)

Single-Valued Neutrosophic Set (SVNS) which is a case of NS was developed by Smarandache (1998) and Wang, Smarandache, Zhang, and Sunderraman (2010) in order to deal with indeterminate, inconsistent and incomplete information. The interval [0,1] was considered rather than ][0.sup.-],[1.sup.+][ for better representation and application to real-world problems. Let U be a universe of discourse and x [member of] U. A single-valued neutrosophic set B in U is described by a truth membership function [F.sub.B](x), an indeterminacy membership function [I.sub.B](x) and a falsity membership function [F.sub.B](x). When U is continuous, an SVNS B is depicted as [mathematical expression not reproducible]. When U is discrete an SVNS B can be represented as B = [[summation].sup.n.sub.i=1] < [T.sub.B]([x.sub.i]),[I.sub.B]([x.sub.i]),[F.sub.B]([x.sub.i])/[x.sub.i]: [x.sub.i] [member of] U [15]. The functions of [T.sub.B] (x), [I.sub.B] (x) and [F.sub.B] (x) are real standard subsets of [0,1] that is [T.sub.B](x) : U [right arrow] [0,1], [I.sub.B](x) : U [right arrow] [0,1] and [F.sub.B](x) : U [right arrow] [0,1]. Also, the sum of [T.sub.B](x), [I.sub.B](x) and [F.sub.B](x) are in [0,3] that 0 [less than or equal to] [T.sub.B](x) + [I.sub.B](x) + [F.sub.B](x) [less than or equal to] 3 [5].

For simplicity two SVNSs such as [B.sub.1] = ([t.sub.1], [l.sub.1], [f.sub.1]) and [B.sub.2] = ([t.sub.2], [l.sub.2], [f.sub.2]) then summation between [B.sub.1] and [B.sub.2] can be described as below:

[B.sub.1] [direct sum] [B.sub.2] = ([t.sub.1], + [t.sub.2] - [t.sub.1],[t.sub.2],[l.sub.1],[l.sub.2],[f.sub.1],[f.sub.2]) (13)

Two SVNSs such as [B.sub.1] = ([t.sub.1], [l.sub.1], [f.sub.1]) and [B.sub.2] = ([t.sub.2], [l.sub.2], [f.sub.2]) then multiplication between [B.sub.1] and [B.sub.2] can be described as below:

[B.sub.1] [cross product] [B.sub.2] = ([t.sub.1][t.sub.2],[l.sub.1] + [l.sub.2] - [l.sub.1][l.sub.2],[f.sub.1] + [f.sub.2] - [f.sub.1][f.sub.2]) (14)

For an SVNS as B = (t, l, f) and [lambda] [member of] R an arbitrary positive real number then,

[lambda]B = [(1 - (1 -t).sup.[lambda]],[l.sup.[lambda]],[f.sup.[lambda]]),[lambda] > 0 (15)

The complement of an SVNS B is represented by [sub.c](B) and is described as follow [9] :

[T.sub.c] (B)(x) = F (B)(x) (16)

[I.sub.c] (B)(x) = 1 -1(B)(x) (17)

[F.sub.c] (B)(x) = T(B)(x) for all x [member of] U (18)

The union of two SVNS namely [B.sub.1] and [B.sub.2] is an SVNS [B.sub.3] denoted by [B.sub.3] = [B.sub.1] [union] [B.sub.2] and its truth, indeterminacy and falsity membership functions are shown below [23]:

T ([B.sub.3])(x) = max(T ([B.sub.1])(x),T ([B.sub.2])(x)) (19)

I ([B.sub.3])(x) = min (I ([B.sub.1])(x), I ([B.sub.2])(x)) (20)

F([B.sub.3])(x) = min(F([B.sub.1])(x),F([B.sub.2])(x)) for all x [member of] U (21)

The intersection of two SVNS namely [B.sub.1] and [B.sub.2] is an SVNS [B.sub.3] denoted by [B.sub.3] = [B.sub.1] [intersection] [B.sub.2] and its truth, indeterminacy and falsity membership functions are shown below [12]:

T([B.sub.3])(x) = min(T([B.sub.1])(x),T([B.sub.2])(x)) (22)

I ([B.sub.3])(x) = max(I([B.sub.1](x),I([B.sub.2])(x)) (23)

F([B.sub.3])(x) = max(F([B.sub.1])(x),F([B.sub.2])(x)) for all x [member of] U (24)

3.3 Single Valued Neutrosophic Sets (SVNS) Entropy Based Decision Making

A new single-valued neutrosophic sets (SVNS) entropy based multi-attribute decision making (MADM) was proposed by Nirmal and Bhatt (2016) and composed of steps seen as follows [16]:

1-Type of decision problem (ranking, evaluation, sorting etc.) is identified in the first step.

2-Then alternatives with regard to criteria having qualitative or quantitative values are identified.

3-Decision matrix involving criteria and alternatives with respect to decision-making problem is constructed.

4-Qualitative information is transformed into fuzzy numbers by means of matrix normalization techniques shown as Table 1:

5-Elements of input matrix in the classic or fuzzy set are conversed to single-valued neutrosophic sets by means of conversion rule for beneficial and non-beneficial criteria explained as below:

a) For beneficial criteria: Positive ideal solution (PIS) is constructed as < [T.sup.*.sub.max](x), [I.sup.*.sub.min](x), [F.sup.*.sub.min](x) >. Normalized input matrix beneficial criteria are created as the degree of truthness [T.sub.L](x), the degree of indeterminacy and degree of falsehood are considered as [I.sub.L](x) = [F.sub.L](x) = 1 - [T.sub.L](x) respectively.

b) For non-beneficial criteria: Negative ideal solution (NIS) is constructed as < [T.sup.*.sub.min](x), [I.sup.*.sub.max](x), [F.sup.*.sub.max](x) >. Normalized input matrix non-beneficial criteria are created as the degree of indeterminacy and falsehood as [I.sub.L](x) = [F.sub.L](x), the degree of truthness is considered as [T.sub.L](x) = 1 - [I.sub.L](x) = 1 - [F.sub.L](x).

c) Entropy value for the jth attribute is calculated according to Eq.(25) as shown below:

[mathematical expression not reproducible] (25)

6-Entropy weight for the jth attribute is calculated as below [24]:

[mathematical expression not reproducible] (26)

Weight vector W = [([w.sub.1], [w.sub.2[, [w.sub.3], ..., [w.sub.n]).sup.T] of attributes, K = {[K.sub.j], j = 1,2, ..., n} with [W.sub.j] [greater than or equal to] > 0 and

[n.summation over (j=1)] [W.sub.j] = 1.

7-Value of each alternative is calculated as follows:

[mathematical expression not reproducible] (27)

Where for beneficial attribute PIS=< [T.sup.*.sub.max](x), [I.sup.*.sub.min](x), [F.sup.*.sub.min](x) >=<1,0,0>, and for non-beneficial attribute NIS=< [T.sup.*.sub.min](x), [I.sup.*.sub.max](x), [F.sup.*.sub.max](x) >=<0,1,1>.

8- Each alternative is ranked according to the descending order of [L.sub.w].

4 Data Set

This study investigated mutual funds which operated continuously and without merging with any other funds for the five-year period between 2012 and 2016, and found security funds that could be grouped under 58 "equity umbrella funds" according to the established criterion. Data from the relevant pages of the Turkish Capital Markets Board website were utilized in the determination of these funds. The unit share price, and consequently the total portfolio value of one of the funds- Alkhair Portfolio Participating Equity Fund (Equity intensive fund)on 31.07.2013 was found to be zero. Since this had the potential to affect the return and risk of the fund, this fund was removed from the study, leaving only 57 funds for analysis. These crisp data are converted to neutrosophic values because of the superiority of neutrosophy over crisp ones.

5 Analysis

A decision matrix was constructed for crisp data drawn from the equity funds and within the framework of the four criteria (Morningstarrating, Sharpe ration, Treynor ratio, and Jensen ratio) as seen in Table 2. After the decision matrix for crisp data, vector normalization process was implemented to obtain the normalized decision matrix. After that, the normalized decision matrix was transformed into the SVNS decision matrix comprised of the degree of truthness [T.sub.L](x), indeterminacy [I.sub.L](x), and falsehood [F.sub.L](x) using the conversion rule for beneficial and non-beneficial criteria. This step is shown in Table 3.

After constructing SVNS decision matrix, entropy values ([E.sub.j]) and weights ([W.sub.j]) for each criterion werecalculated as shown in Table 4.

The findings presented in Table 4 above show that the entropy weights based on the different evaluation criteria were close to each other. The weights based on the Morningstar rating system were found to be the highest leading to the assumption of Morningstar as the most significant evaluation criteria. Jensen ratio was found to have the least weights prompting assumption of its weakness as evaluation criteria for equity funds. Finally, the value of each equity fund ([L.sub.w]) was computed and ranked as highlighted in Table 5 below.

According to the findings in Table 5, Yapi Kredi Asset Management Foreign Technology Sector Equity Funds had the highest [L.sub.w] value and ranked in the first position followed, respectively, by Finans Asset Management Turkey Large-Cap Banks (Equity Intensive) Exchange Traded Fund and is Asset Management BIST Bank Index Share Fund (Equity Intensive Fund). On the other hand, is Asset Management BIST Technology Capped Index Share Fund (Equity Intensive) Fund had the lowest [L.sub.w] value and ranked in the last position.

Conclusion

In this study, SVNS entropy-based decision making is used to rank equity funds traded in Turkey under incomplete, indeterminate and inconsistent information by using four evaluation criteria. To the best of our knowledge, it is the first study in evaluating equity funds traded in Turkey from the neutrosophic set perspective. According to the entropy weights results, Morningstar rating was found to be the most significant evaluation criteria. This indicates the importance of Morningstar weighted scores for equity funds traded in Turkey. On the contrary, Jensen-based weighting was found to be the least significant evaluation criteria for equity funds. Yapi Kredi Asset Management Foreign Technology Sector Equity Fund had the highest [L.sub.w] value and ranked first, followed by Finans Asset Management Turkey Large-Cap Banks (Equity Intensive) Exchange Traded Fund and is Asset Management BIST Bank Index Share Fund (Equity Intensive Fund). On the other hand, is Asset Management BIST Technology Capped Index Share Fund (Equity Intensive) Fund had the lowest [L.sub.w] value and ranked in the last position.

In future studies we suggest the use of other neutrosophic logic-based techniques and normalization methods (out of vector normalization) to analyze this concept. SVNS based entropy technique may also be applied in evaluating equity funds of different countries by considering other criteria. This methodology may also be applied to different sectors.

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[24] Wang, J., and Zhang, Z. Multi-criteria decision making method with incomplete certain information based on intuitionistic fuzzy number. Control and Decision, 24(2009), 226-230.

[25] Yildiz, A. Yatirim fonlari performansinin veri zarflama analizi yontemiyle degerlendirilmesi. Ankara Universitesi SBF Dergisi, 61(2006), 211-234.

[26] Yolsal, H. A tipi yatirim fonlarinin performansi: Banka ve araci kurum fonlari uzerine bir inceleme. Marmara Universitesi iktisadi ve idari Bilimler Dergisi, 32(2012), 343-364.

Received: November 8, 2018. Accepted: December 3, 2018

Serpil Altinirmak (1), Yavuz Gul (2), Basil Oluoch Okoth (3) Caglar Karamasa (4)

(1) Eskicehir Vocational School, Anadolu University, Yunus Emre Campus, Eskicehir, 26470, Turkey. E-mail: saltinirmak@anadolu.edu.tr

(3) Graduate School of Social Sciences, Anadolu University, Yunus Emre Campus, Eskicehir, 26470, Turkey. E-mail: basils.okoth@yahoo.com

Serpil Altinirmak, Yavuz Gul, Basil Oluch Okoth, Caglar Karamasa. Performance Evaluation of Mutual Funds Via Single Valued Neutrosophic Set (SVNS) Perspective: A Case Study in Turkey
```Table 1. Matrix normalization techniques

Normalized
Normalization    beneficial           Normalized non-
technique        value                beneficial value

Linear scale     N.sub.ij] =          N.sub.ij] =
transformation   [x.sub.ij]/          [x.sub.i min]/
max method       [x.sub.i max]        [x.sub.ij]/

Linear scale     N.sub.ij] =          N.sub.ij] =
transformation   [x.sub.ij] -         max [x.sub.ij] -
max-min method   min [x.sub.ij]/      [x.sub.ij]/
max [x.sub.ij] -     max [x.sub.ij] -
min [x.sub.ij]       min [x.sub.ij]

Linear scale     N.sub.ij] =          N.sub.ij] =
transformation   [x.sub.ij]/          1 - [x.sub.ij]/
sum method       [m.summation         [m.summation
over (i=1)]          over (i=1)]
[x.sub.ij]           [x.sub.ij]

Vector           N.sub.ij] =          N.sub.ij] =
normalization    [x.sub.ij]/          1 - [x.sub.ij]/
method           [square root of      [square root of
([m.summation        ([m.summation
over (i=1)]          over (i=1)]
[x.sup.2.sub.ij])]   [x.sup.2.sub.ij])]

Table 2. Decision matrix for crisp data in terms of equity funds

Morningstar   Sharpe       Treynor
weighted      weighted     weighted
Equity Funds           point         point        point

Ak Asset Management      0,692148      0,228627    -0,0363502
America Foreign
Equity Fund

Ak Asset Management      -0,24477      0,113154     -0,0522
Europe Foreign
Equity Fund

Ak Asset Management      -1,447455    -0,001768    -0,000950
Asia Foreign
Equity Fund

Ak Asset Management      -3,683878     0,008855     0,000668
BIST 30 Index
Equity Fund (Equity
Intensive Fund)

Ak Asset Management      -3,82955      -0,03114     -0,00174
BIST Banks Index
Equity Fund (Equity
Intensive Fund)

Ak Asset Management      -1,730383     0,035271    0,0076763
BRIC Countries
Foreign Equity Fund

Ak Asset Management      -3,50801      -0,01854     -0,00094
Equity Fund (Equity
Intensive Fund)

Ak Asset Management      -0,120508     0,169739    -0,093322
Foreign Equity Fund

Ata Portfolio First      -3,113982     0,089787    0,0056179
Equity Fund (Equity
Intensive Fund)

Azimut PYC First         -2,553792     0,045385    0,0035996
Equity Fund (Equity
Intensive Fund)

Bizim Portfolio          -2,432883    -0,095570    -0,010140
Energy Sector
Participation Equity
Fund (Equity
Intensive Fund)

Bizim Portfolio          -3,161174     0,039539    0,0027570
Construction
Industry
Participation Equity
Fund (Equity
Intensive Fund)

Deniz Portfolio          -3,694172     0,004020     0,000370
BIST 100 Index
Equity Fund (Equity
Intensive Fund)

Deniz Portfolio          -3,517794     0,010906     0,000791
Equity Fund
(Equity
Intensive Fund)

Finans Portfolio         -3,907524     0,021992     0,001424
BIST 30 Index
Equity Intensive
Fund Exchange

Finans Asset             -3,296607     0,052358    0,0034596
Management First
Equity Fund

Finans Asset             -3,799676    0,0257993    0,0016396
Management
Dow Jones Istanbul
20 (Equity
Intensive)
Fund

Finans Asset             -3,479966    0,0508895    0,0031856
Management
Second Equity
Fund

Finans Asset             -5,244110    -0,0010259   0,0000074
Management Turkey
Large-Cap Banks
(Equity Intensive)
Fund

Fokus Asset              -2,486771     0,041656    0,0028337
Management Equity
Fund (Equity
Intensive Fund)

Garanti Asset            -3,877312    -0,000583    0,0000974
Management BIST
30 Index Equity
Fund (Equity
Intensive Fund)

Garanti Asset            -3,378613     0,040647    0,0025761
Management Equity
Fund (Equity
Intensive Fund)

Gedik Asset              -3,148063    -0,010759    -0,000589
Management First
Equity Fund (Equity
Intensive Fund)

Gedik Asset              -0,604129     0,107282     0,120414
Management G-20
Countries Foreign
Securities (Equity
Intensive Fund)

Gedik Asset              -1,946579     0,121667    0,0103163
Management Second
Equity Fund (Equity
Intensive Fund)

Global MD Asset          -3,418733    -0,042653    -0,0025318
Management First
Equity Fund (Equity
Intensive Fund)

Global MD Asset          -3,266048    -0,063557    -0,0037824
Management Second
Equity Fund (Equity
Intensive Fund)

Halk Asset               -3,673393    0,0238852    0,0015849
Management Equity
Fund (Equity
Intensive Fund)

HSBC Asset               -3,819283    -0,0068613   -0,000271
Management BIST 30
Index Equity Fund
(Equity Intensive
Fund)

HSBC Asset               -3,818957     0,036008    0,0022634
Management Equity
Fund (Equity
Intensive Fund)

ING Asset                -3,510678    -0,0138847   -0,000641
Management First
Equity Fund (Equity
Intensive Fund)

Asset Management         -3,380190    -0,012627    -0,000613
Dividend Paying
Corporations Share
Fund (Equity
Intensive Fund)

Istanbul Portfolio       -1,818028    -0,092841    -0,006290
Equity Intensive
Fund (Equity
Intensive Fund)

Is Asset Management      -3,798493    -0,0012180   0,0000549
BIST-30 Index Share
Fund (Equity
Intensive Fund)

Is Asset Management      -3,871759    0,0208728    0,00137201
BIST 30 equity
intensive Exchange
investment fund

Is Asset Management      -4,291139    -0,0092844   -0,000475
BIST Bank Index
Share Fund (Equity
Intensive Fund)

Is Asset Management      -0,875815     0,283976     0,034661
BIST Technology
Capped Index Share
Fund (Equity
Intensive Fund)

Is Asset Management      -3,407564    -0,014655    -0,000716
Share Fund (Equity
Intensive Fund)

Is Asset Management      -2,531887    0,0536240    0,0037697
Bank Subsidiaries
Fund (Equity
Intensive Fund)

Is Asset Management      -1,323817    -0,0959215   -0,0099110
Participation Share
Fund (Equity
Intensive Fund)

Is Portfolio Banking     -3,334418    -0,0018027   0,0000289
Private Equity Fund
(Equity Intensive
Fund)

Kare Asset Management    -2,874745     0,143354    0,0088840
Equity Fund (Equity
Intensive Fund)

Qinvest Asset            -2,518660    0,0033631     0,000253
Management Equity
Fund (Equity
Intensive Fund)

Strateji Asset           -2,653906     0,117171    0,0081913
Management Second
Equity Fund (Equity
Intensive Fund)

Strateji Asset           -0,999745     0,074355    0,0089824
Management Second
Equity Fund (Equity
Intensive Fund)

Seker Asset              -2,965969     0,005848     0,000374
Management Equity
Fund (Equity
Intensive Fund)

Tacirler Asset           -1,524716     0,041657    0,00270103
Management Equity
Fund (Equity
Intensive Fund)

TEB Asset                -3,568576     0,024865     0,00166
Management Equity
Fund (Equity
Intensive Fund)

Vakif Asset              -3,749163    -0,0009460   0,0000723
Management BIST 30
Index Equity Fund
(Equity Intensive
Fund)

Yapi Kredi Asset         -3,688064    0,0097042     0,000707
Management BIST 30
Index Equity Fund
(Equity Intensive
Fund)

Yapi Kredi Asset         -3,602030    0,00260078    0,000302
Management BIST
100 Index Equity
Fund (Equity
Intensive Fund)

Yapi Kredi Asset         -3,445713    0,0078333     0,000610
Management First
Equity Fund
(Equity
Intensive Fund)

Yapi Kredi Asset         -2,185636    0,1336988    0,0085835
Management Koc
Holding Affiliate
and Equity Fund
(Equity Intensive
Fund)

Yapi Kredi Asset         0,152437      0,239112    -1,015735
Management Foreign
Technology Sector
Equity Fund

Ziraat Asset             -3,734672    0,0036299     0,000338
Management BIST 30
Index Equity Fund
(Equity Intensive
Fund)

Ziraat Asset             -3,091296    -0,008640    -0,000496
Management Equity
Fund (Equity
Intensive Fund)

Ziraat Asset             -2,095050    0,0000948    0,0000245
Management Dividend
Paying Corporations
Equity Fund (Equity
Intensive Fund)

Jensen
weighted
Equity Funds           point

Ak Asset Management     0,0096924
America Foreign
Equity Fund

Ak Asset Management      0,004424
Europe Foreign
Equity Fund

Ak Asset Management     0,0000765
Asia Foreign
Equity Fund

Ak Asset Management      0,000955
BIST 30 Index
Equity Fund (Equity
Intensive Fund)

Ak Asset Management      0,00126
BIST Banks Index
Equity Fund (Equity
Intensive Fund)

Ak Asset Management     0,0019280
BRIC Countries
Foreign Equity Fund

Ak Asset Management      -0,00049
Equity Fund (Equity
Intensive Fund)

Ak Asset Management     0,0071872
Foreign Equity Fund

Ata Portfolio First      0,005439
Equity Fund (Equity
Intensive Fund)

Azimut PYC First         0,002600
Equity Fund (Equity
Intensive Fund)

Bizim Portfolio         -0,0036231
Energy Sector
Participation Equity
Fund (Equity
Intensive Fund)

Bizim Portfolio         0,0024940
Construction
Industry
Participation Equity
Fund (Equity
Intensive Fund)

Deniz Portfolio         0,0006699
BIST 100 Index
Equity Fund (Equity
Intensive Fund)

Deniz Portfolio         0,0009917
Equity Fund
(Equity
Intensive Fund)

Finans Portfolio         0,001816
BIST 30 Index
Equity Intensive
Fund Exchange

Finans Asset            0,0032420
Management First
Equity Fund

Finans Asset            0,0019916
Management
Dow Jones Istanbul
20 (Equity
Intensive)
Fund

Finans Asset            0,0032607
Management
Second Equity
Fund

Finans Asset             0,000465
Management Turkey
Large-Cap Banks
(Equity Intensive)
Fund

Fokus Asset             0,0020580
Management Equity
Fund (Equity
Intensive Fund)

Garanti Asset            0,000431
Management BIST
30 Index Equity
Fund (Equity
Intensive Fund)

Garanti Asset           0,0026067
Management Equity
Fund (Equity
Intensive Fund)

Gedik Asset             -0,0001846
Management First
Equity Fund (Equity
Intensive Fund)

Gedik Asset             0,0041053
Management G-20
Countries Foreign
Securities (Equity
Intensive Fund)

Gedik Asset             0,0058834
Management Second
Equity Fund (Equity
Intensive Fund)

Global MD Asset         -0,0016431
Management First
Equity Fund (Equity
Intensive Fund)

Global MD Asset         -0,0024167
Management Second
Equity Fund (Equity
Intensive Fund)

Halk Asset              0,0018096
Management Equity
Fund (Equity
Intensive Fund)

HSBC Asset               0,000007
Management BIST 30
Index Equity Fund
(Equity Intensive
Fund)

HSBC Asset              0,0026858
Management Equity
Fund (Equity
Intensive Fund)

ING Asset                -0,00026
Management First
Equity Fund (Equity
Intensive Fund)

Asset Management        -0,000216
Dividend Paying
Corporations Share
Fund (Equity
Intensive Fund)

Istanbul Portfolio      -0,0022918
Equity Intensive
Fund (Equity
Intensive Fund)

Is Asset Management     0,0003845
BIST-30 Index Share
Fund (Equity
Intensive Fund)

Is Asset Management     0,0017460
BIST 30 equity
intensive Exchange
investment fund

Is Asset Management     -0,0001311
BIST Bank Index
Share Fund (Equity
Intensive Fund)

Is Asset Management     0,0166801
BIST Technology
Capped Index Share
Fund (Equity
Intensive Fund)

Is Asset Management     -0,000307
Share Fund (Equity
Intensive Fund)

Is Asset Management     0,0025575
Bank Subsidiaries
Fund (Equity
Intensive Fund)

Is Asset Management     -0,0022049
Participation Share
Fund (Equity
Intensive Fund)

Is Portfolio Banking     0,000307
Private Equity Fund
(Equity Intensive
Fund)

Kare Asset Management   0,0087556
Equity Fund (Equity
Intensive Fund)

Qinvest Asset            0,000378
Management Equity
Fund (Equity
Intensive Fund)

Strateji Asset           0,006719
Management Second
Equity Fund (Equity
Intensive Fund)

Strateji Asset          0,0021746
Management Second
Equity Fund (Equity
Intensive Fund)

Seker Asset             0,00060018
Management Equity
Fund (Equity
Intensive Fund)

Tacirler Asset          0,0011679
Management Equity
Fund (Equity
Intensive Fund)

TEB Asset               0,0018599
Management Equity
Fund (Equity
Intensive Fund)

Vakif Asset              0,000391
Management BIST 30
Index Equity Fund
(Equity Intensive
Fund)

Yapi Kredi Asset         0,000994
Management BIST 30
Index Equity Fund
(Equity Intensive
Fund)

Yapi Kredi Asset         0,000586
Management BIST
100 Index Equity
Fund (Equity
Intensive Fund)

Yapi Kredi Asset         0,000827
Management First
Equity Fund
(Equity
Intensive Fund)

Yapi Kredi Asset        0,0066485
Management Koc
Holding Affiliate
and Equity Fund
(Equity Intensive
Fund)

Yapi Kredi Asset        0,0091293
Management Foreign
Technology Sector
Equity Fund

Ziraat Asset             0,000647
Management BIST 30
Index Equity Fund
(Equity Intensive
Fund)

Ziraat Asset            0,0000372
Management Equity
Fund (Equity
Intensive Fund)

Ziraat Asset            0,00027349
Management Dividend
Paying Corporations
Equity Fund (Equity
Intensive Fund)

Table 3. SVNS Decision Matrix

Morningstar          Sharpe
weighted             weighted
Equity Funds            point                point

Ak Asset Management       (0.0299,0.97,0.9    (0.37,0.6299,0.6
America Foreign                  7)                 299)
Equity Fund

Ak Asset Management              (-            0.183,0.816,0.8
Europe Foreign           0.01,1.010,1.010)           16)
Equity Fund

Ak Asset Management              (-                  (-
Asia Foreign              0.062,1.062,1.0      0.002,1.002,1.0
Equity Fund                     62)                  02)

Ak Asset Management              (-           (0.014,0.985,0.9
BIST 30 Index             0.159,1.159,1.1            85)
Equity Fund (Equity             59)
Intensive Fund)

Ak Asset Management              (-                  (-
BIST Banks Index          0.165,1.165,1.1      0.05,1.05,1.05)
Equity Fund (Equity             65)
Intensive Fund)

Ak Asset Management              (-           (0.057,0.942,0.9
BRIC Countries            0.07484,1.074,1            42)
Foreign                        .074)
Equity Fund

Ak Asset Management              (-                  (-
Equity Fund (Equity       0.151,1.151,1.1      0.03,1.03,1.03)
Intensive Fund)                 51)

Ak Asset                  (0.00521,1.005,     (0.274,0.725,0.7
Management Foreign             1.005)                25)
Equity Fund

Ata Portfolio First              (-           (0.145,0.854,0.8
Equity Fund (Equity       0.134,1.134,1.1            54)
Intensive Fund)                 34)

Azimut PYS First                 (-           (0.07,0.926,0.92
Equity Fund (Equity       0.11,1.11,1.11)            6)
Intensive Fund)

Bizim Portfolio                  (-                  (-
Energy Sector             0.105,1.105,1.1      0.154,1.154,1.1
Participation Equity            05)                  54)
Fund (Equity
Intensive Fund)

Bizim Portfolio                  (-           (0.063,0.936,0.9
Construction              0.136,1.136,1.1            36)
Industry                        36)
Participation
Equity Fund (Equity
Intensive Fund)

Deniz Portfolio BIST             (-           (0.006,0.993,0.9
100 Index Equity          0.159,1.159,1.1            93)
Fund (Equity                    59)
Intensive Fund)

Deniz Portfolio                  (-           (0.017,0.982,0.9
Equity Fund (Equity       0.152,1.152,1.1            82)
Intensive Fund)                 52)

Finans Portfolio                 (-           (0.035,0.964,0.9
BIST 30 Index             0.169,1.169,1.1            64)
Equity Intensive                69)
Fund Exchange

Finans Asset                     (-           (0.084,0.915,0.9
Management First          0.142,1.142,1.1            15)
Equity Fund                     42)

Finans Asset                     (-           (0.041,0.958,0.9
Management Dow            0.164,1.164,1.1            58)
Jones Istanbul 20               64)
(Equity Intensive)
Fund

Finans Asset                     (-           (0.082,0.917,0.9
Management Second         0.15,1.15,1.15)            17)
Equity Fund

Finans Asset                     (-                  (-
Management Turkey         0.226,1.226,1.2      0.001,1.001,1.0
Large-Cap Banks                 26)                  01)
(Equity Intensive)
Fund

Fokus Asset                      (-           (0.067,0.932,0.9
Management                0.107,1.107,1.1            32)
Equity Fund                     07)
(Equity
Intensive Fund)

Garanti Asset                    (-                  (-
Management BIST           0.167,1.167,1.1      0.00095,1.0009,
30 Index Equity                 67)                1.0009)
Fund (Equity
Intensive Fund)

Garanti Asset                    (-            0.065,0.934,0.9
Management Equity         0.146,1.146,1.1            34)
Fund (Equity                    46)
Intensive Fund)

Gedik Asset                      (-                  (-
Management First          0.136,1.136,1.1      0.017,1.017,1.0
Equity Fund                     36)                  17)
(Equity
Intensive Fund)

Gedik Asset                      (-           (0.173,0.826,0.8
Management G-20           0.026,1.026,1.0            26)
Countries Foreign               26)
Securities (Equity
Intensive Fund)

Gedik Asset                      (-           (0.196,0.803,0.8
Management Second         0.0841,1.084,1.            03)
Equity Fund                     084)
(Equity
Intensive Fund)

Global MD Asset                  (-                  (-
Management First          0.147,1.147,1.1      0.069,1.069,1.0
Equity Fund                     47)                  69)
(Equity
Intensive Fund)

Global MD Asset                  (-                  (-
Management Second         0.141,1.141,1.1      0.102,1.102,1.1
Equity Fund (Equity             41)                  02)
Intensive Fund)

Halk Asset                       (-           (0.038,0.961,0.9
Management Equity         0.158,1.158,1.1            61)
Fund (Equity                    58)
Intensive Fund)

HSBC Asset                       (-                  (-
Management BIST           0.165,1.165,1.1      0.011,1.011,1.0
30 Index Equity                 65)                  11)
Fund (Equity
Intensive Fund)

HSBC Asset                       (-           (0.058,0.941,0.9
Management Equity         0.165,1.165,1.1            41)
Fund (Equity                    65)
Intensive Fund)

ING Asset Management             (-                  (-
First Equity Fund         0.151,1.151,1.1      0.022,1.022,1.0
(Equity                         51)                  22)
Intensive Fund)

Is Asset Management              (-                  (-
Dividend Paying           0.146,1.146,1.1      0.0204,1.0204,1
Corporations Share              46)                .0204)
Fund (Equity
Intensive Fund)

Istanbul Portfolio               (-                  (-
Equity Intensive          0.078,1.078,1.0      0.15,1.15,1.15)
Fund (Equity                    78)
Intensive Fund)

Is Asset Management              (-                  (-
BIST-30 Index Share       0.164,1.164,1.1      0.0019,1.0019,1
Fund (Equity                    64)                .0019)
Intensive Fund)

Is Asset Management              (-           (0.033,0.966,0.9
BIST 30 equity            0.167,1.167,1.1            66)
intensive Exchange              67)
investment fund

Is Asset Management              (-                  (-
BIST Bank Index           0.185,1.185,1.1      0.015,1.015,1.0
Share Fund                      85)                  15)
(Equity
Intensive Fund)

Is Asset Management              (-           (0.459,0.54,0.54
BIST Technology           0.037,1.037,1.0            )
Capped Index Share              37)
Fund (Equity
Intensive Fund)

Is Asset Management              (-                  (-
Share Fund (Equity        0.147,1.147,1.1      0.023,1.023,1.0
Intensive Fund)                 47)                  23)

Is Asset Management              (-           (0.086,0.913,0.9
Is Bank Subsidiaries      0.109,1.109,1.1            13)
Fund (Equity                    09)
Intensive Fund)

Is Asset Management              (-                  (-
Participation             0.057,1.057,1.0      0.155,1.155,1.1
Share Fund                      57)                  55)
(Equity
Intensive Fund)

Is Portfolio                     (-                  (-
Banking Private           0.144,1.144,1.1      0.0029,1.0029,1
Equity Fund (Equity             44)                .0029)
Intensive Fund)

Kare Asset                       (-           (0.232,0.767,0.7
Management Equity         0.124,1.124,1.1            67)
Fund (Equity                    24)
Intensive Fund)

Qinvest Asset                    (-           (0.0054,0.994,0.
Management Equity         0.108,1.108,1.1           994)
Fund (Equity                    08)
Intensive Fund)

Strateji Asset                   (-           (0.189,0.81,0.81
Management Second         0.114,1.114,1.1            )
Equity Fund (Equity             14)
Intensive Fund)

Strateji Asset                   (-           (0.12,0.879,0.87
Management Second         0.043,1.043,1.0            9)
Equity Fund (Equity             43)
Intensive Fund)

Seker Asset                      (-           (0.0094,0.99,0.9
Management Equity         0.128,1.128,1.1            9)
Fund (Equity                    28)
Intensive Fund)

Tacirler Asset                   (-           (0.067,0.932,0.9
Management Equity         0.065,1.065,1.0            32)
Fund (Equity                    65)
Intensive Fund)

TEB Asset                        (-           (0.04,0.959,0.95
Management Equity         0.154,1.154,1.1            9)
Fund (Equity                    54)
Intensive Fund)

Vakif Asset                      (-                  (-
Management BIST           0.162,1.162,1.1      0.00153,1.0015
30 Index Equity                 62)              3,1.00153)
Fund (Equity
Intensive Fund)

Yapi Kredi Asset                 (-           (0.015,0.984,0.9
Management BIST           0.159,1.159,1.1            84)
30 Index Equity                 59)
Fund (Equity
Intensive Fund)

Yapi Kredi Asset                 (-           (0.004,0.995,0.9
Management BIST           0.155,1.155,1.1            95)
100 Index Equity                55)
Fund (Equity
Intensive Fund)

Yapi Kredi Asset                 (-           (0.012,0.987,0.9
Management First          0.149,1.149,1.1            87)
Equity Fund                     49)
(Equity
Intensive Fund)

Yapi Kredi Asset                 (-           (0.216,0.783,0.7
Management Koc            0.094,1.094,1.0            83)
Holding Affiliate               94)
and Equity Fund
(Equity
Intensive Fund)

Yapi Kredi Asset          (0.0065,0.993,0.    (0.387,0.612,0.6
Management Foreign              993)                 12)
Technology Sector
Equity Fund

Ziraat Asset                     (-           (0.005,0.994,0.9
Management BIST           0.161,1.161,1.1            94)
30 Index Equity                 61)
Fund (Equity
Intensive Fund)

Ziraat Asset                     (-                  (-
Management Equity         0.133,1.133,1.1      0.013,1.013,1.0
Fund (Equity                    33)                  13)
Intensive Fund)

Ziraat Asset                     (-            (0.000154,0.999
Management Dividend       0.09,1.09,1.09)         ,0.999)
Paying Corporations
Equity Fund (Equity
Intensive Fund)

Treynor             Jensen
weighted            weighted
Equity Funds            point               point

Ak Asset Management             (-           (0.323,0.676,0.6
America Foreign           0.035,1.035,1.0           76)
Equity Fund                     35)

Ak Asset Management             (-           (0.147,0.852,0.8
Europe Foreign            0.05,1.05,1.05)           52)
Equity Fund

Ak Asset Management             (-                  (-
Asia Foreign              0.00092,1.0009      0.0025,1.0025,1
Equity Fund                 2,1.00092)            .0025)

Ak Asset Management       0.00064,0.999,     (0.0318,0.968,0.
BIST 30 Index                 0.999)               968)
Equity Fund (Equity
Intensive Fund)

Ak Asset Management             (-                  (-
BIST Banks Index          0.00169,1.0016      0.042,1.042,1.0
Equity Fund (Equity         9,1.00169)              42)
Intensive Fund)

Ak Asset Management       (0.00745,0.992,    (0.0642,0.935,0.
BRIC Countries                0.992)               935)
Foreign
Equity Fund

Ak Asset Management             (-                  (-
Equity Fund (Equity       0.00091,1.0009      0.016,1.016,1.0
Intensive Fund)             1,1.00091)              16)

Ak Asset                        (-           (0.239,0.76,0.7
Management Foreign        0.0905,1.0905,1           6)
Equity Fund                   .0905)

Ata Portfolio First      (0.005,0.994,0.9    (0.181,0.818,0.8
Equity Fund (Equity             94)                 18)
Intensive Fund)

Azimut PYS First         (0.003,0.996,0.9    (0.086,0.913,0.9
Equity Fund (Equity             96)                 13)
Intensive Fund)

Bizim Portfolio                 (-                  (-
Energy Sector             0.0098,1.0098,1     0,12,1.12,1.12)
Participation Equity          .0098)
Fund (Equity
Intensive Fund)

Bizim Portfolio          (0.0026,0.997,0.    (0.083,0.916,0.9
Construction                   997)                 16)
Industry
Participation
Equity Fund (Equity
Intensive Fund)

Deniz Portfolio BIST      (0.00035,0.999,    (0.022,0.977,0.9
100 Index Equity              0.999)                77)
Fund (Equity
Intensive Fund)

Deniz Portfolio           0.00076,0.999,     (0.033,0.966,0.9
Equity Fund (Equity           0.999)                66)
Intensive Fund)

Finans Portfolio         (0.0013,0.998,0.    (0.06,0.939,0.93
BIST 30 Index                  998)                 9)
Equity Intensive
Fund Exchange

Finans Asset             (0.0033,0.996,0.    (0.108,0.891,0.8
Management First               996)                 91)
Equity Fund

Finans Asset             (0.0015,0.998,0.    (0.066,0.933,0.9
Management Dow                 998)                 33)
Jones Istanbul 20
(Equity Intensive)
Fund

Finans Asset             (0.003,0.996,0.9    (0.108,0.891,0.8
Management Second               96)                 91)
Equity Fund

Finans Asset              0.00000722,0.9     (0.015,0.984,0.9
Management Turkey            99,0.999)              84)
Large-Cap Banks
(Equity Intensive)
Fund

Fokus Asset              (0.0027,0.997,0.    (0.068,0.931,0.9
Management                     997)                 31)
Equity Fund
(Equity
Intensive Fund)

Garanti Asset             (0.0000946,0.99    (0.0144,0.985,0.
Management BIST              9,0.999)              985)
30 Index Equity
Fund (Equity
Intensive Fund)

Garanti Asset            (0.0025,0.997,0.    (0.086,0.913,0.9
Management Equity              997)                 13)
Fund (Equity
Intensive Fund)

Gedik Asset                     (-                  (-
Management First          0.00057,1.0005      0.006,1.006,1.0
Equity Fund                 7,1.00057)              06)
(Equity
Intensive Fund)

Gedik Asset              (0.116,0.883,0.8    (0.136,0.863,0.8
Management G-20                 83)                 63)
Countries Foreign
Securities (Equity
Intensive Fund)

Gedik Asset              (0.01,0.989,0.98    (0.196,0.803,0.8
Management Second               9)                  03)
Equity Fund
(Equity
Intensive Fund)

Global MD Asset                 (-                  (-
Management First          0.0024,1.0024,1     0.054,1.054,1.0
Equity Fund                   .0024)                54)
(Equity
Intensive Fund)

Global MD Asset                 (-                  (-
Management Second         0.0036,1.0036,1     0.08,1.08,1.08)
Equity Fund (Equity           .0036)
Intensive Fund)

Halk Asset               (0.0015,0.998,0.    (0.0603,0.939,0.
Management Equity              998)                939)
Fund (Equity
Intensive Fund)

HSBC Asset                      (-           (0.002,0.997,0.9

Management BIST           0.00026,1.0002            97)
30 Index Equity             6,1.00026)
Fund (Equity
Intensive Fund)

HSBC Asset                (0.00219,0.997,    (0.089,0.91,0.91
Management Equity             0.997)                )
Fund (Equity
Intensive Fund)

ING Asset Management            (-                  (-
First Equity Fund         0.00062,1.0006      0.008,1.008,1.0
(Equity                     2,1.00062)              08)
Intensive Fund)

Is Asset Management             (-                  (-
Dividend Paying           0.00059,1.0005      0.0072,1.0072,1
Corporations Share          9,1.00059)            .0072)
Fund (Equity
Intensive Fund)

Istanbul Portfolio              (-                  (-
Equity Intensive          0.0061,1.0061,1     0.076,1.076,1.0
Fund (Equity                  .0061)                76)
Intensive Fund)

Is Asset Management       (0.0000533,0.99    (0.0128,0.987,0.
BIST-30 Index Share          9,0.999)              987)
Fund (Equity
Intensive Fund)

Is Asset Management       (0.00133,0.998,    (0.058,0.941,0.9
BIST 30 equity                0.998)                41)
intensive Exchange
investment fund

Is Asset Management             (-                  (-
BIST Bank Index           0.00046,1.0004      0.0043,1.0043,1
Share Fund                  6,1.00046)            .0043)
(Equity
Intensive Fund)

Is Asset Management      (0.033,0.966,0.9    (0.556,0.443,0.4
BIST Technology                 66)                 43)
Capped Index Share
Fund (Equity
Intensive Fund)

Is Asset Management             (-                  (-
Share Fund (Equity        0.00069,1.0006      0.01,1.01,1.01)
Intensive Fund)             9,1.00069)

Is Asset Management      (0.0036,0.996,0.    (0.0852,0.914,0.
Is Bank Subsidiaries           996)                914)
Fund (Equity
Intensive Fund)

Is Asset Management             (-                  (-
Participation             0.0096,1.0096,1     0.0735,1.0735,1
Share Fund                    .0096)              .0735)
(Equity
Intensive Fund)

Is Portfolio              0.0000281,0.99     (0.01,0.989,0.98
Banking Private              9,0.999)               9)
Equity Fund (Equity
Intensive Fund)

Kare Asset               (0.0086,0.991,0.    (0.291,0.708,0.7
Management Equity              991)                 08)
Fund (Equity
Intensive Fund)

Qinvest Asset             (0.00024,0.999,    (0.012,0.987,0.9
Management Equity             0.999)                87)
Fund (Equity
Intensive Fund)

Strateji Asset           (0.0079,0.992,0.    (0.224,0.775,0.7
Management Second              992)                 75)
Equity Fund (Equity
Intensive Fund)

Strateji Asset           (0.0087,0.991,0.    (0.0725,0.927,0.
Management Second              991)                927)
Equity Fund (Equity
Intensive Fund)

Seker Asset               (0.00036,0.999,    (0.02,0.979,0.97
Management Equity             0.999)                9)
Fund (Equity
Intensive Fund)

Tacirler Asset            (0.00262,0.997,    (0.0389,0.961,0.
Management Equity             0.997)               961)
Fund (Equity
Intensive Fund)

TEB Asset                 (0.00161,0.998,    (0.062,0.937,0.9
Management Equity             0.998)                37)
Fund (Equity
Intensive Fund)

Vakif Asset               (0.0000702,0.99    (0.013,0.986,0.9
Management BIST              9,0.999)               86)
30 Index Equity
Fund (Equity
Intensive Fund)

Yapi Kredi Asset          (0.00068,0.999,    (0.033,0.966,0.9
Management BIST               0.999)                66)
30 Index Equity
Fund (Equity
Intensive Fund)

Yapi Kredi Asset          (0.00029,0.999,    (0.019,0.98,0.98
Management BIST               0.999)                )
100 Index Equity
Fund (Equity
Intensive Fund)

Yapi Kredi Asset          (0.00059,0.999,    (0.0275,0.972,0.
Management First              0.999)               972)
Equity Fund
(Equity
Intensive Fund)

Yapi Kredi Asset         (0.0083,0.991,0.    (0.221,0.778,0.7
Management Koc                 991)                 78)
Holding Affiliate
and Equity Fund
(Equity
Intensive Fund)

Yapi Kredi Asset                (-           (0.304,0.695,0.6
Management Foreign        0.986,1.986,1.9           95)
Technology Sector               86)
Equity Fund

Ziraat Asset              (0.00032,0.999,    (0.021,0.978,0.9
Management BIST               0.999)                78)
30 Index Equity
Fund (Equity
Intensive Fund)

Ziraat Asset                    (-           (0.0012,0.998,0.
Management Equity         0.00048,1.0004           998)
Fund (Equity                8,1.00048)
Intensive Fund)

Ziraat Asset              (0.0000238,0.99    (0.009,0.99,0.99
Management Dividend          9,0.999)               )
Paying Corporations
Equity Fund (Equity
Intensive Fund)

Table 4. Entropy values ([E.sub.j]) and weights [W.sub.j])
for each evaluation criteria

Entropy       Entropy
value         weight
Criteria            ([E.sub.j])   ([W.sub.j])

Morningstar          1.243136      0.307845
weighted point
Sharpe               0.891585      0.220789
weighted point
Treynor              1.033546      0.255943
weighted point
Jensen               0.869916      0.215422
weighted point

Table 5. Value of each equity funds ([L.sub.w]) and ranking

Value
Equity Funds             ([L.sub.w])   Ranking

Ak Asset Management       1,402621       56
America Foreign
Equity Fund

Ak Asset Management       1,471126       53
Europe Foreign
Equity Fund

Ak Asset Management       1,516942       45
Asia Foreign
Equity Fund

Ak Asset Management       1,568346       20
BIST 30 Index
Equity Fund (Equity
Intensive Fund)

Ak Asset Management       1,587197        4
BIST Banks Index
Equity Fund (Equity
Intensive Fund)

Ak Asset Management       1,507607       46
BRIC Countries
Foreign Equity Fund

Ak Asset Management       1,574065       11
Equity Fund (Equity
Intensive Fund)

Ak Asset Management       1,465676       54
Foreign Equity Fund

Ata Portfolio First       1,523178       44
Equity Fund (Equity
Intensive Fund)

Azimut PYC First          1,526375       42
Equity Fund (Equity
Intensive Fund)

Bizim Portfolio            1,57504       10
Energy Sector
Participation
Equity Fund (Equity
Intensive Fund)

Bizim Portfolio            1,54372       36
Construction
Industry
Participation
Equity Fund (Equity
Intensive Fund)

Deniz Portfolio           1,570483       15
BIST 100 Index
Equity Fund (Equity
Intensive Fund)

Deniz Portfolio           1,563529       25
Equity Fund
(Equity Intensive
Fund)

Finans Portfolio           1,56876       17
BIST 30 Index
Equity Intensive
Fund Exchange

Finans Asset              1,542249       38
Management First
Equity Fund

Finans Asset              1,564664       23
Management Dow
Jones Istanbul 20
(Equity Intensive)
Fund

Finans Asset              1,547307       35
Management Second
Equity Fund

Finans Asset              1,611662        2
Management Turkey
Large-Cap Banks
(Equity Intensive)
Fund

Fokus Asset               1,527103       41
Management Equity
Fund (Equity
Intensive Fund)

Garanti Asset             1,576842        8
Management BIST
30 Index Equity
Fund (Equity
Intensive Fund)

Garanti Asset              1,54881       34
Management Equity
Fund (Equity
Intensive Fund)

Gedik Asset               1,562302       28
Management First
Equity Fund
(Equity
Intensive Fund)

Gedik Asset                1,41159       55
Management G-20
Countries Foreign
Securities (Equity
Intensive Fund)

Gedik Asset               1,483426       51
Management Second
Equity Fund (Equity
Intensive Fund)

Global MD Asset           1,580536        6
Management First
Equity Fund (Equity
Intensive Fund)

Global MD Asset           1,583647        5
Management Second
Equity Fund (Equity
Intensive Fund)

Halk Asset                1,562333       27
Management Equity
Fund (Equity
Intensive Fund)

HSBC Asset                1,577772        7
Management BIST
30 Index Equity
Fund (Equity
Intensive Fund)

HSBC Asset                 1,56098       29
Management Equity
Fund (Equity
Intensive Fund)

ING Asset                  1,57242       13
Management First
Equity Fund
(Equity
Intensive Fund)

Is Asset Management       1,568706       18
Dividend Paying
Corporations Share
Fund (Equity
Intensive Fund)

Istanbul Portfolio        1,554046       32
Equity Intensive
Fund (Equity
Intensive Fund)

Is Asset Management       1,575,108       9
BIST-30 Index
Share Fund (Equity
Intensive Fund)

Is Asset Management       1,568286       21
BIST 30 equity
intensive Exchange
investment fund

Is Asset Management       1,590875        3
BIST Bank Index
Share Fund (Equity
Intensive Fund)

Is Asset Management       1,384472       57
BIST Technology
Capped Index Share
Fund (Equity
Intensive Fund)

Is Asset Management       1,570112       16
Share Fund (Equity
Intensive Fund)

Is Asset Management       1,524021       43
Bank Subsidiaries
Fund (Equity
Intensive Fund)

Is Asset Management       1,543436       37
Participation Share
Fund (Equity
Intensive Fund)

Is Portfolio Banking      1,563628       24
Private Equity Fund
(Equity Intensive
Fund)

Kare Asset Management     1,495966       49
Equity Fund (Equity
Intensive Fund)

Qinvest Asset              1,54147       39
Management Equity
Fund (Equity
Intensive Fund)

Strateji Asset            1,501294       48
Management Second
Equity Fund (Equity
Intensive Fund)

Strateji Asset            1,479225       52
Management Second
Equity Fund (Equity
Intensive Fund)

Ceker Asset                1,55172       33
Management Equity
Fund (Equity
Intensive Fund)

Tacirler Asset            1,504807       47
Management Equity
Fund (Equity
Intensive Fund)

TEB Asset                 1,559298       31
Management Equity
Fund (Equity
Intensive Fund)

Vakif Asset               1,573768       12
Management BIST 30
Index Equity Fund
(Equity
Intensive Fund)

Yapi Kredi Asset          1,568154       22
Management BIST
30 Index Equity
Fund (Equity
Intensive Fund)

Yapi Kredi Asset          1,568682       19
Management BIST
100 Index Equity
Fund (Equity
Intensive Fund)

Yapi Kredi Asset          1,562843       26
Management First
Equity Fund
(Equity
Intensive Fund)

Yapi Kredi Asset          1,485724       50
Management Koc
Holding Affiliate
and Equity Fund
(Equity
Intensive Fund)

Yapi Kredi Asset          1,816721        1
Management Foreign
Technology Sector
Equity Fund

Ziraat Asset              1,571666       14
Management BIST
30 Index Equity
Fund (Equity
Intensive Fund)

Ziraat Asset              1,559821       30
Management Equity
Fund (Equity
Intensive Fund)

Ziraat Asset              1,531766       40
Management Dividend
Paying Corporations
Equity Fund (Equity
Intensive Fund)
```
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