5
JENSEN INDEX
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)
Where
(
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.
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.
(19)
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.
as 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:
, on any security (or portfolio) j:
(20)
where
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.
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).
. 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
(20a)
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)
where
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)).
, 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
(21a)
where
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)
(equivalent to (22c))
cannot hold since contains .” (Jensen 1968: 392)
The variables and the 
are assumed to be
independent normally distributed random variables with
and the are assumed to be
independent normally distributed random variables with
(22a)
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
(23)
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
(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 
(24)
As one will observe from (21) that the left
hand side of (24) is just 
. So (24) reduces to:
. So (24) reduces to:
(25)
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.
, 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
can be subtracted from
both sides of equation (25), and since its coefficient is unity the result is
(26)
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.
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.
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.
(27)
where
: 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.
, 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
(28)
where
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 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:
(29)
where
: 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
(30)
“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:
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:
(31)
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).
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:
, 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)
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
 |
|
Ri |
M
 |
βi
Figure 6 : Theoretical/Empirical
CAPM line\
Source : ELTON
E., GRUBER M.; “Modern Portfolio and Investment Analysis”; 660
.
2.6 GRAHAM HARVEY METHOD
2.6 GRAHAM HARVEY METHOD
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
GH1(B)
x
x
market
x
GH1(A)
x Fund A
x Treasury Bill
Standard
Deviation
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.
(32)
(33) : 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
x
GH2 (B)
x Market
GH2 (A)
x 
x Fund A 
x Treasury Bill
Standard
Deviation
Figure 8 : GH2 Index
Source : KARATEPE Y., KARACABEY A.. “A-Tipi Fonların Performansının Yeni Bir
Yöntem Kullanılarak Değerlendirilmesi: Graham-Harvey Performans Testi
(34)
(35)
: 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:
Assuming;
Market
return (
) of 15 % with a standard deviation (
) of 0.2
) of 15 % with a standard deviation () of 0.2
A
portfolio return (
) of 25% with a standard deviation (
) of 0.34
) of 25% with a standard deviation () of 0.34
And a
risk-free rate (
) of 10 % with a standard deviation (
) of 0.02
) 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 ;
(36)
0.34 =
0.02 
+ 0.2 
+ 0.2
And we solve it for
= 2. Plug it in the
following equation
(37)
= 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 %
would be measured as 5 %
(38)
(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.
3.
COMMENTS ON FOUR SINGLE INDEX METHODS
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
(39)
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.
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. RESEARCH AND METHODOLOGY
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.
- 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;
- 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.
- Monthly data would leave only
30 observations for each fund which I believe was too few for a
significant statistical testing.
- 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
|
Dates
|
ISE-100
|
Weekly
Change
|
|
07.01.00
|
15837
|
0.0413
|
|
14.01.00
|
19110
|
0.2067
|
|
21.01.00
|
17258
|
-0.0969
|
|
28.01.00
|
18172
|
0.0530
|
|
04.02.00
|
16871
|
-0.0716
|
|
11.02.00
|
15598
|
-0.0755
|
|
18.02.00
|
15364
|
-0.0150
|
|
25.02.00
|
15618
|
0.0165
|
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;
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 :
(39)
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).
4.2
METHODOLOGY
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.
, 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 :
- Weekly average risk premiums
on A Type, B Type mutual funds and ISE-100 indices are displayed in Figure
9.
- 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.
5. RESEARCH FINDINGS
5.1 RISK PREMIUMS
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.
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.
5.2 PORTFOLIO RANKINGS
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.
5.3
COMPARISON OF SHARPE INDICES
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
|
1988
|
1999
|
2000/I
|
|
T-Bills
|
0
|
0
|
0
|
0
|
|
ISE-100 Index
|
-0.02
|
-0.29
|
0.27
|
-0.11
|
|
Type A Funds
|
-0.45
|
-0.77
|
-0.23
|
-0.27
|
|
Type B Funds
|
-0.09
|
-0.36
|
0.20
|
-0.15
|
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
|
1998
|
1999
|
2000/I
|
|
Number of Funds That Beat The
Market
|
7
|
2
|
4
|
6
|
|
2
|
6
|
1
|
3
|
|
|
4
|
1
|
9
|
7
|
|
|
6
|
3
|
7
|
24
|
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
|
1998
|
1999
|
2000
|
|
Number of Funds That Beat The
Market
|
3
|
0
|
2
|
2
|
|
0
|
0
|
0
|
0
|
|
|
2
|
0
|
14
|
3
|
|
|
2
|
0
|
3
|
5
|
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 .
Table-7
Z Test
Results For The Significance of Mean Differences
|
|
|
ISE-100 /
T-Bill
|
Type A /
T-Bill
|
Type B /
T-Bill
|
Type A /
ISE-100
|
Type B/
ISE-100
|
Type A /
Type B
|
|
Entire
|
Mean Difference
|
- 0.02
|
-0.45
|
-0.09
|
-0.43
|
-0.07
|
-0.36
|
|
|
Z-Statistics
|
0.57
|
7.85***
|
1.26
|
5.56***
|
1.84*
|
6.1***
|
|
1988
|
Mean Difference
|
-0.29
|
-0.77
|
-0.36
|
-0.48
|
-0.07
|
-0.41
|
|
|
Z-Statistics
|
2.05**
|
6.24***
|
2.62***
|
3.88***
|
0.80
|
4.30***
|
|
1999
|
Mean Difference
|
0.27
|
-0.23
|
0.20
|
-0.50
|
-0.07
|
-0.43
|
|
|
Z-Statistics
|
1.74*
|
3.40***
|
1.35
|
4.07***
|
1.51
|
4.78***
|
|
2000/I
|
Mean Difference
|
-0.11
|
-0.27
|
-0.15
|
-0.16
|
-0.04
|
-0.12
|
|
|
Z-Statistics
|
0.82
|
2.79***
|
1.04
|
0.82
|
0.93
|
0.58
|
*, **, *** 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.
6.
QUESTIONS AND SUGGESTIONS
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.
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”.
7.CONCLUSION
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?”
