3 Mart 2012 Cumartesi




2.5.1 Definition

Jensen assumes that CAPM is empirically valid. He builds his theory mainly on this assumption and comes with the following formula in terms of realized rates of return.

Rjt = Rf  + βj ( Rm - Rf ) + ujt                                                                                     (17)

Subtracting Rf  from both side he obtains:

Rjt - Rf  = βj ( Rm - Rf ) + ujt                                                                                      (18)

( Rm - Rf ) : market risk premium
ujt random error term

This formula indicates that risk premium earned on jth portfolio is equal to the market risk premium times  plus a random error term. In this form, one will not expect an intercept for the regression equation, if all securities are in equilibrium. But if certain superior portfolio managers can persistently earn positive risk premiums on their portfolios, the error term  will always have a positive value. This also proves the success of the portfolio manager. In such a case, an intercept value  must be included in the equations as follows.  measures positive differences from the model. Note that these positive differences are a result of the performance of the management.


Jensen usesas his performance measure. A superior portfolio manager (successful) would have a significant positive  value because of the consistent positive residuals. Inferior managers (unsuccessful), on the other hand would have a significant negative . Average portfolio managers who have no forecasting ability-but still cannot be considered inferior- would earn as much as one can expect on the basis of CAPM. His profit accessing risk free investment will be only risk premium times beta value of the portfolio. The residual terms would randomly be positive and negative, and this would give an intercept value, which is insignificantly different from zero.

2.5.2 The Foundations of the Jensen Model

a)      As I mentioned above Jensen derives his measure and index from application of Capital Asset Pricing Model combined with the studies of Treynor and Shape and also Lintner. Lintner’s studies are out of the scope of this study so they are not covered. Jensen adopts the 7 assumptions that are used by the previous studies.
b)      All investors are risk avers.
c)      All investors are single period-expected utility of terminal wealth maximisers.
d)     All investors have identical decision horizons and homogeneous expectations regarding investment opportunities.
e)      All investors are able to choose among portfolios solely on the basis of expected returns and variance of returns.
f)       Transaction costs and taxes are ignored.
g)      All assets are infinitely divisible.

Jensen has an additional assumption
·         The capital market is in equilibrium.
All three models yield the following expression for the expected one period return, , on any security (or portfolio) j:
*        the one-period risk free interest rate
the measure of risk (hereafter called systematic risk) which
                                the asset pricing model implies is crucial in determining the
                                prices of risky assets.
the expected one-period return on the “market portfolio” which consists
                of an investment in each asset in the market in proportion to its fraction
                of the total value of all assets in the market.

Thus equation (20) implies that the expected return on any asset is equal to the risk free rate plus a risk premium given by the product of the systematic risk of the asset and the risk premium on the market portfolio. Note that since  is a constant for all securities the risk of any security is just . But since , we can conclude the variance of the market portfolio is just , and thus we are really measuring the riskiness of any security relative to the market portfolio. Hence the systematic risk of the market portfolio, ,is unity, and thus the dimension of the measure of systematic risk has a convenient intuitive interpretation. The risk premium on the market portfolio is the difference between the expected returns on the market portfolio and the risk free rate.

“Equation (20) then simply tells us the amount that any security (or portfolio) can be expected to earn given its level of systematic risk, . If a portfolio manager or a security analyst is able to predict security prices he will be able to earn higher returns that those implied by equation (20) and the riskiness of his portfolio.” (Jensen, 1968: 391).

Later Jensen shows how (20) can be adapted and extended to provide an estimate of the forecasting ability of any portfolio manager. Equation (20) is stated in terms of the expected returns on any security or portfolio j and the expected returns on the market portfolio. However according to the theory and the assumptions these expectations are strictly unobservable. This rises as a serious doubt of the ability of equation (20) to measure realizations of any individual portfolio j and market portfolio M objectively.

“In equation (20) it was shown that the single period models of Sharpe, Lintner and Treynor can be extended to a multiperiod world in which investors are allowed to have heterogeneous horizon periods and in which the trading of securities takes place continuously through time. These results indicate that we can generalize equation (20) and rewrite it as
where the subscript t denotes an interval of time arbitrary with respect to length and beginning (and ending) dates.” (Jensen 1968: p392)

The market model is;
                                j = 1,2,…,N                                        (21)
is a parameter, which may vary from security to security
is an unobservable “market factor” which to some extent affects the returns on all securities
N is the total number of securities in the market.

The measure of risk in (20a), , is approximately equal to the coefficient  in the “market model” (It is also shown by equation (24) and (30)).

Jensen also displays the “diagonal model” (21a), which is identical to the “market model” (21) and descriptions of (22a)-(22d). The “diagonal model” is usually stated as
*is some index of market returns,
 is a random variable uncorrelated with ,
and  are constants.

“The differences in specification between (21) and (21a) are necessary in order to avoid the over specification which arises if one chooses to interpret the market index I as an average of security returns on the market portfolio, M. That is, if  (equivalent to (22c)) cannot hold since  contains .” (Jensen 1968: 392)

The variables  and the  are assumed to be independent normally distributed random variables with
                   j = 1,2,…,N                                                                          (22b)
           j = 1,2,…,N                                                                          (22c)
j = 1,2,…,N                                                   (22d)

Furthermore to a close approximation the return on the market portfolio can be expressed as

Since evidence given in (20, 30) indicates that the market model, given by equation (21) and (22a) (22d), holds for portfolios as well as individual securities, we can use (21) to recast (20a) in terms of ex post returns. Substituting for  in (23) plugging it in equation (9a) and adding  to both sides of Jensen has

As one will observe from (21) that the left hand side of (24) is just . So (24) reduces to:

Thus assuming that the asset-pricing model is empirically valid, (see equation 30) , equation (25) implies that the realized returns on any security or portfolio can be expressed as a linear function of four indicators. Its systematic risk, the realized returns on the market portfolio, the risk free rate and a random error, , which has an expected value of zero.

The term  can be subtracted from both sides of equation (25), and since its coefficient is unity the result is

The left hand side of (26) is the risk premium earned on the j’th portfolio. (Jensen, 1968: 393)

2.5.3 The Measurement of Fund Performance

Jensen index is basically based on the comparison of  values. The reason and the steps of derivation of  value is explained below.

Equation (26) may be used directly for empirical estimation. If we wish to estimate the systematic risk of any individual security or of an unmanaged portfolio the constrained regression estimate of  in equation (26) will be an efficient estimate (in the statistical sense of term) of this systematic risk. However, the situation is not exactly the same for managed portfolios. If the manager is a superior forecaster (perhaps because of special knowledge not available to others) he will tend to systematically select securities, which have residual value of > 0. Hence manager’s portfolio will earn more than the normal risk premium for its level of risk. We must allow for this possibility in estimating the systematic risk of a managed portfolio.

Allowance for such forecasting ability can be made by simply not constraining the estimating regression to pass through the origin. Then it will pass through the positive side of y-axis. That is, we allow for the possible existence of a non-zero constant. Plugging this constant in equation (26) by using the following equation (27) as the estimating equation.

 : is the constant added for return of a managed portfolio other than the proportion achieved from simple risk premium.
 : is the new error term, which will now have , and should be serially independent.

Thus if the portfolio manager has an ability to forecast security prices, he certainly will benefit out of it and the intercept, , in equation (27) will be positive. Indeed, it represents the average incremental rate of return of the portfolio per unit time, which is due solely to the manager’s ability to forecast future security prices. And it is surely the performance we are trying to measure by using performance indices. It is interesting to note that a naïve random selection buy and hold policy,  will be negative.

“At first glance it might seem difficult than a random selection policy, but such results may very well be due to the generation of too many expenses in unsuccessful forecasting attempts.”(Jensen, 1968: 394)

Fortunately the model outlined above will also measure the success of these market forecasting or timing activities as long as it can be assumed that the portfolio manager attempt on average to maintain a given level of risk in his portfolio. More formally as long as we can express the risk of the j’th portfolio at any time t as

 is the target risk level, which the portfolio manager wishes to maintain on average through time,
 is a normally distributed random variable (at least partially under the manager’s control) with  = 0.
The variable  is the vehicle through which the manager may attempt to capitalize on any expectation he may have. If we say that  is the behaviour of the market factor, we can conclude that manager has an expectation that depends on the behaviour of the market factor  in the next period. For example if the manager correctly perceives that there is a higher probability that p will be positive next period, he will be able to increase the returns on his portfolio by increasing its risk. That is possible by making  positive this period. On the other hand he can reduce the losses and therefore increase the average returns on the portfolio by reducing the risk level of portfolio -making  negative- when the market factor p is expected to be negative. Thus if the manager is able to forecast market movements to some extent, we can find a positive relationship between random variables  and market behaviour. We can state this relationship formally as:
 : is the error term assumed to be normally distributed with . Basically above  represents the manager’s effect
: represents the unknown or uninformed part of the market that the manager tries to foreseen.

Jensen substitutes from (28) into (27) for a more general model, that appears as

“Now as long as the estimated risk parameter  is an unbiased estimate of the average risk level  the estimated performance measure  will also be unbiased.” (Jensen, 1968: 398). Under the assumption that the forecast error  is uncorrelated with , Jensen shows that the expected value of the least squares estimator  is:

However, if the manager does not have information or he is not a well manager, has not an ability to forecast market movements, constant  in equation (31) tends to be positive. The portfolio or asset will have an estimated risk parameter less than it usually will. Estimated risk parameter will be biased downward. This means, of course if we remember the CAPM line that the estimated performance measure  will be biased upward (since the regression line must pass through the point of sample means).

“Hence it seems clear that if the manager can forecast market movements at all we most certainly should see evidence of it since our techniques will tend to overstate the magnitude of the effects of this ability. That is, the performance measure, , will be positive for two reasons:
(1)   The extra returns actually earned on the portfolio due to the manager’s ability.
(2)   The positive bias in the estimate of  resulting from the negative bias in our estimate of .” (Jensen, 1968: 398)

One question that the method of Jensen rises is related to his method and CAPM. As seen in Figure 6, empirical estimates of the CAPM line have a higher intercept and a lower slope than the theoretical CAPM line. So “Utilizing the empirical line rather than the theoretical line would result in portfolios with Betas less than one having smaller differential returns and portfolios with Betas greater than one having larger differential returns” (Elton/Gruber : p 660) This may result negative for a fund although it would result positive if theoretical line is used


Theoretical Line




Figure 6 : Theoretical/Empirical CAPM line\
Source : ELTON E., GRUBER M.; “Modern Portfolio and Investment Analysis”; 660


2.6.1 Definition

Graham& Harvey recently suggested that the performance of a portfolio should be measured by its excess return over the return of a “market index/risk-free-asset combination” with a standard deviation equal to that of the portfolio. Therefore, if the standard deviation of a portfolio is different from the market standard deviation, the latter must be increased or decreased to the level of portfolio standard deviation by forming an appropriate combination of market index and risk-free-asset.

2.6.2 GH1 Index

Average Return

                                                                                                                      x Fund B
                                                                                               x  market
                                                                          x Fund A
                         x  Treasury Bill
Figure 7 :  GH1 Index
Source : KARATEPE Y., KARACABEY A.. “A-Tipi Fonların Performansının Yeni Bir Yöntem Kullanılarak Değerlendirilmesi: Graham-Harvey Performans Testi

GH1 deals with the adjustment of risk index to the risk level of analysed fund’s risk. In this model, if index’s risk is below the mutual fund’s risk it is increased to the level of mutual fund. If it is above the mutual fund’s risk then it is decreased to the level of mutual fund.

As it is observed in Figure 7, fund A’s risk and return levels are below market index which is considered as market index. According to Graham-Harvey method comparison of fund’s portfolio with the market’s portfolio will not give a meaningful result. Because there is difference between risk and return of fund’s portfolio and market’s portfolio. In order to make a comparison, market’s risk level should be decreased to the level of fund’s risk. For the purpose of comparison a new portfolio of market index and treasury bills is established. The weight of treasury bills in the new portfolio is set at a level that is required to equal the risk of the new “market and treasury bills” portfolio to the fund A’s portfolio. Returns of the two portfolios are compared.


 : Fund A’s return
 : Fund B’s return
 : Return of the new risk adjusted portfolio of market index and treasury bills.

If the difference between the fund’s return and new portfolio of market and treasury bills is negative, we conclude fund displays low performance. If the difference between the fund’s return and new portfolio of market and treasury bills is positive, we conclude fund displays high performance.

2.6.3 GH2 Index

GH2 has a significant difference than GH1. Here, fund’s risk is adjusted to the market index’s risk. In order to do that we form a new portfolio including fund A and treasury bills.

As shown in Figure 8, fund B’s risk and return are above the market index’s risk and return. To compare the fund’s performance with the market, risk of the fund will be adjusted to the market index by adding treasury bills. After the risk of new portfolio of fund B and treasury bills and market index are adjusted, we compare their returns. If the return of the new fund and treasury bill portfolio is still above the return of the market portfolio we may conclude that fund performances better than the market. If it the return of the new fund and treasury bill portfolio is below the market return we say fund perform worse than the market.

         Average Return

                                                                                                                      x Fund B
                                                                                                       GH2 (B)
                                                                                                    x  Market
                                                                                         GH2 (A)
                                                                x   Fund A
                      x      Treasury Bill
Figure 8 :  GH2 Index
Source : KARATEPE Y., KARACABEY A.. “A-Tipi Fonların Performansının Yeni Bir Yöntem Kullanılarak Değerlendirilmesi: Graham-Harvey Performans Testi


 : Adjusted Fund A’s return
 : Adjusted Fund B’s return
 : Return of the market index.

If the difference between the fund and treasury bill portfolio’s return and portfolio of market is negative, we conclude fund displays low performance. If the difference between the fund and treasury bill portfolio’s return and portfolio of market is positive, we conclude fund displays high performance.

2.6.4. An Illustration of G&H Method:
Market return () of 15 % with a standard deviation () of 0.2
A portfolio return () of 25% with a standard deviation () of 0.34
And a risk-free rate () of 10 % with a standard deviation () of 0.02
Graham&Harvey would make 100 % levered portfolio of which standard deviation is also 40 % using the equation of ;
0.34 = 0.02  + 0.2
And we solve it for
 = 2. Plug it in the following equation
 = 10*(1-2) + 15 (2)
 = 20 => 20%
Since the return of this combination would be 20 % excess return of the portfolio would be measured as 5 %
(25 % - 20 %) = 5 %.
The higher the excess return, the better the portfolio performance.
Graham-Harvey (1997) method can be categorized as GH1 and GH2 indexes.


The common point of the four of the methods is that combinations of any portfolio and the riskless asset lie among a straight line in either expected return beta space or expected return standard deviation space to evaluate performance. Roll criticizes saying that, this approach requires identifying the correct index. The index has to be efficient. If the index is not efficient, then fund performances become a function of particular selected index. Therefore these four methods are called as single-index method.

The problem mentioned above led to the development of multi-index performance measures. Although they are out of the scope of this study, as a comment on single index methods and for further researches it is valuable to give brief explanation about them. Other than Roll’s critique about the efficiency of reference index, development of Asset Pricing Theory rise some commands about the single indices. Asset Pricing Theory “postulates that expected returns can be expressed as a linear function of sensitivities to more than one index” (Elton/Gruber, 1999, p. 452)

“Finally, returning to viewing the manager relative to a passive fund or funds, it is clear that comparing active funds in total or particular active fund to a passive strategy is inappropriate if these funds hold very different types of securities from the passive portfolios with which they are being compared” (Elton/Gruber, 1999, p. 425)

As a result of the his studies, Ippolito comes up with a new version of Jensen model
where  is the return on an index of small stocks in period t and other terms as before. As a proof to his model Ippolito compares the results he received with the same data set using Jensen model and the rewritten form of it. As a result of analysis done by using Jensen model, funds with higher expenses had at least as good performance as the others. Plus load funds outperformed no load funds. Both these results are against the general theory. However  both of the results were turned the opposite way as they are repeated by using Equation 39 above.

Ippolito’s two index model rose the question about the number of indexes. Recent researches were done related to this question. “For example, Elton/Gruber/Blake(1996) employ a four-index model involving the S&P 500 Index, a size-related index, a bond index, and a growth-value index to explain the return on domestic non-specialized mutual funds. Sharpe uses a 12-index model to explain the return on a broader set of mutual funds including domestic as well as international bond funds and stock funds” (Elton/Gruber, 1999, p 453)

Elton/Gruber/Blake (1996) later show that no more than five indexes are sufficient enough and Sharpe formalize a general alternative measurement for a multi-index world based on his old method. In his new approach he uses the ratio between the average return on the portfolio and the benchmark portfolio to the standard deviation of difference.

These multi-index models that are generated after the second half of ‘90s have also affected the issues of forecasting and timing. Both Sharpe (1994) and Elton/Gruber/Blake(1996) about forecasting and timing, also Hendricks (1993), Grinblatt and Titman (1992) about forecasting show that past performance when risk adjustment is done carefully through the use of an appropriately designed multi-index model allows investors to select funds that have superior future performance and “the use of a single or multiple index model to judge performance and to forecast performance assumes that betas (sensitivities) are reasonably constant and that management does not change them to gain added return through market timing.” (Elton/Gruber, 1999, p 454)


4.1  DATA

Time period of the study is between January 1998-June 2000. The reasons for choosing this time interval are;
1.      Fund market is fairly new in Turkey. It’s around 16 years old. I aimed to keep my set as large as possible by choosing a time interval beginning from the data of the last available data and going back.
  1. Time interval is aimed to be chosen such that it would cover at least one bull and one bear market (one increase and a decrease following it). By doing that manager’s performances are tested under both market situation which is I believe more reliable.

No sampling method is used to create a sample set of funds. All of the funds which are displaying continuity with no more than 2 missing data that follow each other and any rapid change for the term of January 1998-June 2000 are included in the data set. The reason for eliminating the fund without continuous data is due to the observation that resources other than Capital Board do not Markets follow the pattern and therefore considered to be unreliable. Including those data would also effect the variance of the fund returns. Using an average value method could be an option but I avoided using it since an average of the values that are reflecting past and future two week of the missing value could not be reliable for the volatility of the market. As a result 4 of the funds are excluded from the data set.

Finally 133 mutual funds are included in the data set. Friday closing values of each fund for January 1998-June 2000 term were picked from the daily data. The reasons for working on weekly data are;
  1. Daily changes would not be a good estimate for performance measurement because time interval is too short to give any idea about the manager’s effect on the fund performance.
  2. Monthly data would leave only 30 observations for each fund which I believe was too few for a significant statistical testing.
  3. Monthly observations may display too volatile data and rapid changes.

The reason of picking Friday values of the week is because Friday closing prices reflect all available information of the week in the market. For any missing value of the Friday closing prices, Thursday’s closing values were taken in order to achieve the price that reflects the most information that otherwise would be reflected by the Friday closing price. For situations where neither Friday nor Thursday values are available, closing price of the following Monday is taken in order to limit the effect of extra available information. Doing that any possible change is tried to be limited by only a few information of the weekend and Monday’s change that is only one more exercise day.

            Data are organized for calculation in four different time period.
1.      From January 1998 to December 1998
2.      From January 1999 to December 1999
3.      From January 2000 to June 2000
4.      From January 1998 to June 2000 (entire period)

Return of the fund for the i’th week is calculated as
 where p refers to the price of the fund.

“Market rate of return – Rm” which is required for performance index calculations are obtained from ISE-100 index. It is taken from Metastock data base and again only Fridays are chosen. Market return for the week is calculated as
An example of ISE-100 column to give an idea about the calculation of Rm values is displayed below in Table 1 for the period of Jan-Feb 2000

Table 1

Sample of Weekly Cahnge Data of ISE-100

Weekly Change

For the calculations of risk free rate Turkish Treasury Bills data are obtained from Istanbul Stock Exchange Data Base. All exercises for any T-Bill with a maturity date less than 15 days or more than 365 days were deleted. T-Bills, which have a maturity date less than 15 days are observed to display results with a high volatility. Possible reason of this may be the effect of transaction costs. As time to maturity becomes shorter, transaction costs for exercise tend to carry bigger proportion compared to the return of the T-Bill. So the exercise price of the T-Bill increases in order to cover this costs. At some points return of the T-Bill results more than 130% which is way above the market rate and inflation rate for the same term.

Setting a certain risk free market rate value would be meaningless because that would also mean predicting a certain risk free return rate. That would make all the testing biased and insufficient. Therefore a limit of 15 days was chosen which seemed reasonable based on the results. To limit the maturity with 365 days is simple to understand. By definition T-Bills are investment tools with a maturity date less than a year. Rest of the exercises with longer maturity date were considered as Treasury Bond (by definition).

After choosing T-Bill exercises that meet the limitations above, the following calculations are made to obtain the risk free return rate for the week. As the first step assumption of nominal value of a treasury bill is 100.000 TL is made for all of the exercises. Risk free return value for every single exercise is calculated as displayed in the example;

Nominal value of the treasury bill: 100.000 TL. (given data by ISE)
Days to maturity : 20 days
Exercise price in market : 97.200 TL.

Rf value of the treasury bill :


The application above was applied to the rest of the exercise in the treasury bill market for that certain week. The average of the risk free return is determined as the risk free return rate.

Let’s assume that there were 6 treasury bill exchanges during the weeks of Jan 9th / Jan 13th and 5 treasury bill exchanges Jan 16th / Jan 20th . They are listed in Table 2.


If we use the same logic that we used for the calculation of the weekly return of ISE-100 and individual fund for the calculation of the weekly return of T-Bills for the week of January 16th /January 20th, we come up with the result of +0.03% (58.4-58.1).


Funds are divided according to their type and the time periods. So I had two groups of fund –A Type and B Type - and each group was organized in four different time table as mentioned above –yearly and entire period-. For each column –ISE-100, , funds- on each time table the following are calculated.
a)      Averages weekly return for four periods.
b)      Standard deviation of the returns for the four periods.
c)      Beta coefficient of each fund, ISE-100 for the four periods.
Values of return, standard deviation and betas are given in Section 8 under the title “Mean/Standard Deviation/Slope Values Table”

Sharpe, Treynor, Jensen and Graham-Harvey indices were calculated for each A Type and B Type portfolio as well as ISE-100 index based on the formulas and explanations given under the title of each index.

Tables of performance results of all mutual funds are prepared for four of the methods. Funds are ranked with respect to their performance results measured by four of the methods. The fund with the worst performance was given rank number of “1”. Both the results of indices and rank results are given in Section 8.

In order to test whether the four different methods rank the portfolios similarly or different, Spearman rank correlation analysis is applied and rank correlation coefficients were calculated for each pair of ranking criteria. Results of the correlation analysis are used to answer if there is any significant difference between the performance results of each mutual funds when they are ranked according to these four different methods. The reason for choosing the Spearman rank correlation among the available rank correlations of the “Statistica” package is it was used for both Sharpe and Jensen for the comparison of their results with the previous results.

At this point I observed that data and the results suggest to test the validity of capital market theory in Turkey. Therefore the subject of the thesis is enlarged to test validity of the capital market theory in Turkey besides the test of significancy between the rankings of the performance measurement tests.

In order to compare portfolio performance of A Type and B Type funds with those of T-Bills, and ISE-100 to see the significance of the differences and validity of capital market theory in Turkey over the same period :

  1. Weekly average risk premiums on A Type, B Type mutual funds and ISE-100 indices are displayed in Figure 9.
  2. Average of the Sharpe coefficients of A Type mutual funds, B Type mutual funds and ISE-100 indices were compared and the statistical significance of the differences were tested.



Figure 9 depicts the behaviour of average weekly risk premiums  on A Type funds, B Type funds and the ISE-100 index. Under normal capital market conditions these risk premiums would always be expected to be positive. But, this was not the case in Turkey over the analysis period. Negative risk premiums mean that T-Bills were a better investment tool than the other three instruments in almost half of the observation periods. In other words, most of the funds could not beat the market. This is obviously the financial market implication of unfavorable macroeconomic conditions prevailing in Turkey over those years.

            Figure 9 : Risk Premiums
Figure 9 permits us to make following observations as well:
a)      B Type funds were not a good investment at all. Their average risk premium was negative, but their variation was greater than zero.
b)      A Type funds were successful in reducing the portfolio risk below the market risk. However their risk premiums were below the market risk premium. Therefore the rationale of A Type funds can be commented upon only after evaluating them against ISE-100 index depending upon Sharpe coefficients because Sharpe coefficients take standard deviation as an indicator of performance. I think standard deviation can be the best indicator since it also carries great importance for the three other performance measurement techniques. This will be done below.
c)      A Type funds have provided higher risk premiums than B Type funds as expected.


Table 3

Results of “Spearman r” Correlation Analysis

*, **, ***  indicates significance at 10, 5, 1% significance level respectively using two-tailed test.

A Type funds and B Type funds were ranked according to Sharpe, Treynor, Jensen, and Graham&Harvey criteria, and Spearman rank correlation coefficients were calculated for each pair of indices. The results are summarized in Table 3.

In A Type funds, the calculated “Spearman r”s for the entire period as well as the three sub-periods are quite high and significant at 1% α level. This means that index used in evaluating A Type funds does not matter.

In B Type funds Spearman rank correlation coefficients are much lower. But they are still significant at 1% α level in 18 cases, at 5 % α level in 3 cases, and at 10 % α level in one case. Only in two cases in Year 1999, r was found insignificant.


Having seen that it is highly correlated with other indices, and given the fact that it measures the success in diversification as well, the Sharpe index was chosen to compare the performance of alternative investment media included in the research.

Figure 10 : Sharpe Coefficients

Weekly Sharpe indices of T-Bills, A Type Funds, B Type Funds and ISE-100 indices are graphed in Figure 10. Sharpe index for T-Bills is zero by definition, and coincides with X axis. For other instruments, a negative Sharpe index means that return on the instrument is less than T-Bill rate. Figure 10 indicates that there are as many negative Sharpe indices as positive ones, and this is against the expectation.

Averages of weekly Sharpe indices of the four categories are given in Table 4.

Table 4
Average Sharpe Index (%)

Entire Period
ISE-100 Index
Type A Funds
Type B Funds

According to Table 4, for the entire period as well as the years 1998 and 2000/I, T-Bill was the best investment, followed by ISE-100 index, B Type funds and A Type funds respectively. Only in 1999, performances of ISE-100 and Type B funds were superior to T-Bill . The sign and rank of Type A funds, however, remained the same.

Table 5 shows the number of A Type funds with Sharpe coefficients greater than that of ISE-100 index. The figures on the diagonal of the matrix represent the total number of A Type funds that exceeded ISE-100 in Sharpe coefficient. Other figures in the same row tells us how many of them were better than ISE-100 in other periods as well. For example in the entire period 7 A Type funds performed better than ISE-100. Of this 7, 2 in 1998, 4 in 1999, 6 in 2000/I also outperformed ISE-100.

Table 5

Number of A Type funds That Beat The Market

Entire Period
Number of Funds That Beat The Market

 Table 5 figures are not promising with respect to the performance of A Type funds.

Table 6, on the other hand, provides information on B Type funds, which outperformed T-Bills:

Table 6

Number of B Type Funds That Beat The Market

Entire Period
Number of Funds That Beat The Market

Table 6 figures reflect poor performance of B Type funds over the analysis period as well as in the sub-periods.

In order to see whether the average Sharpe indices of the four investment alternatives given in Table 4 were significantly different from each other, the standard Z test was applied to the calculated Wilcoxon’s W statistics. The findings are summarized in Table 7 .

Z Test Results For The Significance of Mean Differences

ISE-100 /
Type A /
Type B /
Type A /
Type B/
Type A /
Type B
Mean Difference
- 0.02

Mean Difference

Mean Difference

Mean Difference

*, **, ***  indicates significance at 10, 5, 1% significance level respectively using two-tailed test.


Figures listed in the Table 7 implies that:

a)      For the entire analysis period, and in years 1998 and 1999 the differences between the Sharpe coefficients are statistically significant. This means that performance ranking in Table 4, which is against the expectations under normal capital market conditions based on CAPM, is reliable. Only two observations, both in 1999, are in line with expectations of capital market theory: In that year ISE-100 performed better than B Type funds, and B Type funds better than T-Bills. However the differences between B Type funds, T-Bills and ISE-100 index and B Type funds were not found to be statistically significant. The difference between ISE-100 and T-Bills, on the other hand, is significant at 10% α level.
b)      Table 4 ranking is valid in 2000 as well, but Z-values are insignificant except the one pertaining to A Type funds- T-Bills difference.

I use the Wilcoxon signed-rank test as my principal method of testing for significant differences between variables. This procedure tests whether the median difference in variable values in two samples is zero. I base my conclusions on the standardized test statistic Z, which for samples of at least 10 follows approximately a standard normal distribution. Wilcoxon signed-rank test computes the differences between the two variables for all cases and classifies the differences as either positive, negative, or tied. If the two variables are similarly distributed, the number of positive and negative differences will not differ significantly. The Wilcoxon signed-rank test considers information about both the sign of the differences and the magnitude of the differences between pairs.


There are some possible suggestions related to the study.

Time interval can certainly be expanded. One can expand the time interval as long as sample data set reflects the whole set.

In some event studies and time series studies related to the money markets and tools I observed that Wednesday or Monday were chosen as the data day of the week interval. Monday is open to the question of “Monday effect”, which is studied. However Wednesdays can be chosen for a retesting.

For the application of performance indices and ranking ISE-100 is used as  indicator. ISE-100 can certainly be used for measuring the performance of A Type funds (Since at least 25% of them are invested in securities), but ISE-100 may not be the best indicator for B Type funds. Some of the B Type funds are invested in tools are than securities and they may be compared to the similar tools, which they hold.

I am aware that calculations of risk free rate rise many questions. 15 day rule is the first one may come up with. Second is the weight of the exercises among all of the exercises are not taken into account. I suggest further studies to use DIBS (Government Domestic Borrowing Index) index, which is generated by ISE. The problem with the DIBS index is that it has been generated in the January of 1996. Therefore it can be an indicator for time intervals after January 1996.

Assuming that investment world is global and managers have access to any tool throughout the world, American T-Bills can be another indicator. Having the fact that worldwide investors had chosen to invest to American T-Bills and gold after the WTA attack on September 11th. These two tools can be considered as the risk free tools throughout the world. Here we must distinguish one difference between American T-bills and gold. Gold is even not open to the effects of financial decisions of USA and therefore has a competitive advantage as an risk free rate indicator. However CAPM theory has to be discussed from this point of view if gold is taken as risk free rate indicator.

Few of the funds displayed both negative risk premium rate and negative beta value. Since Treynor is basically risk premium per beta, they came up with positive Treynor performance measurement results. This problems occurs because of the character of that Turkish market displayed during the time interval of the study. However this points out the fact that “Treynor may not be a well performance indicator for Turkish market”.


This study leads us to the following issue.

First, none of the performance indices measures fund performance significantly better than others. They rank the funds not significantly different from one another. Therefore we cannot say either one of the four methods measures the fund performance better than the others.

Relative performances of T-Bills, B Type funds, A Type funds and ISE-100 index were measured through the Sharpe index. It was found that, over the entire analysis period as well as in the three sub-periods T-Bills were the best investment. It was followed by ISE-100 index, B Type funds and A Type funds respectively. The reliability of this ranking was tested through standard Z test applied to Wilcoxon Signed-Rank Test Statistic calculated for each pair of investment media included in the analysis over the entire period and for each of the sub-periods. Z test offers us the results which supports our thesis.

Therefore, it was concluded that the efforts to form A Type and B Type funds in expectation of reaching superior performance to T-Bills, and ISE-100 index totally failed over the analysis period. This is interpreted as the financial market implication of adverse macroeconomic conditions prevailing in the country during the same period.

Second, the study rises some doubts and questions on the issues of fund managers performance to outperform the market, diversification of the portfolio, the effort and money spent for management activities of mutual funds. Results seem to be not convincing in favour of the managers. Most of the funds could not beat the market between 1998/2000(I). We think that this issue is open to further researches.

One other issue that is pointed out by the research is the performance of government T-Bills compared to other 3 instruments. T-Bills are supposed to be the investment instruments that carry the lowest risk and provide the lowest return with respect to their risk. That is what the theory of finance suggests. However the research shows us just an opposite result for the research period. T-Bills in Turkish investment markets perform better than average funds and stocks. This points out the important impact of Turkish government on domestic investment markets, and rises doubts about the safetiness of the portrait displayed, the riskless structure of T-Bills and default risk possibilities. We suggest default risk of Turkish government T-Bills and government securities in general to be found as another issue to be studied in further researches.

Treynor Index is found to be less reliable for Turkish market because both risk premium and beta value results negative for sme funds which mean a positive indeice result.

One last point to indicate is “How reasonable can it be to make long term investment strategies in Turkish investment market while some well known theories such as CAPM are not applicable and far from being able to reflect the characteristic of the market?”

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