Evaluating Mutual Fund Performance

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					                       Evaluating Danish
                    Mutual Fund Performance

                               Michael Christensen*
                              Department of Finance
                           The Aarhus School of Business
                                 Fuglesangs Allé 4
                                   8210 Aarhus V.
                                     Denmark

                                E-mail: mic@asb.dk



          Abstract
          To date analyses of Danish mutual fund performance have been
          few and mainly pursued by mutual funds themselves or by the
          Association of Danish Mutual Funds (www.ifr.dk). The purpose of
          this paper is to provide the first independent performance analy-
          sis of Danish mutual funds, which includes 44 mutual funds that
          have been in operation since October 1994. In this analysis we
          focus on the Jensen measure of performance considering a single
          index model and a multi-index model, respectively. Further-
          more, we analyse the timing ability of the Danish mutual funds
          pursuing both the quadratic regressions of Treynor & Mazuy
          (1966) and the option approach suggested by Henriksson & Mer-
          ton (1981). Finally, we consider the influence of management
          fees and other mutual fund expenses on performance.


Keywords: Mutual funds, performance evaluation, market timing, expense ratios.

JEL classification: G12, G14, G23
1. Introduction
Since Jensen (1968) the research in mutual fund performance has increased signifi-
cantly and similarly the popularity of mutual fund investments among private in-
vestors has grown dramatically during the last 30 years. In Denmark the market
value of mutual funds has increased from approximately USD 11 billion in 1997 to
more than USD 34 billion in 2002, which amounts to a yearly increase of more than
25%. Despite this huge increase in market value there has only been few analyses
of Danish mutual fund performance, and the analyses have mainly been pursued by
the mutual funds themselves or by the Association of Danish Mutual Funds. There-
fore, the purpose of this study is to provide the first independent analysis of Danish
mutual fund performance. The sample includes 44 funds of which 27 funds are eq-
uity funds and 17 funds are fixed income funds. The sample period is October 1994
to January 2002.


A fascinating aspect of the earlier mutual fund research was the simultaneous de-
velopment of performance evaluation methods. Today, however, there is general
agreement that the Jensen measure (Jensen’s α) is superior to the Sharpe and
Treynor measures for a number of reasons. Firstly, the interpretation of the Jensen
measure is the risk-adjusted excess return measured in percentage points, which is
much easier to communicate to private investors than the Sharpe and Treynor
measures that measure the risk-adjusted excess return as a ratio. Secondly, the
Jensen measure can easily be estimated from an asset pricing regression, which
furthermore provides us with a measure of statistical significance. Thirdly, the Jen-
sen measure is seen relatively to a benchmark1. Fourthly, determining the Jensen
measure from a regression, one can take account of a non-constant risk-free rate,
whereas the Sharpe and Treynor measure use a time average of the risk-free rate.
Moreover, Friend & Blume (1970) argue that the Jensen measure (and the Treynor
measure) is superior to the Sharpe measure because the Jensen and Treynor meas-
ure can be applied to both efficient and inefficient securities and portfolios,
whereas the Sharpe measure can only be applied to efficient securities and portfo-
lios. Finally, the Jensen (and the Treynor) measure is the appropriate performance
measure to apply to a mutual fund performance analysis, since the Jensen measure
is based on the security market line, which assumes well diversified portfolios con-

                                          2
trary to the Sharpe measure, which measures the portfolio manager’s ability to di-
versify the unsystematic risk.


An important drawback of the Jensen measure is, however, that any conclusion
reached about fund performance rests on the asset pricing model chosen. Earlier
analyses applied the CAPM model, thus recognising the problems associated with
the choice of benchmark following the Roll critique, and newer research concludes
that the choice of benchmark has important consequences for the performance
evaluation. Lehmann & Modest (1987) show that the Jensen measure differs signifi-
cantly between a CAPM and an APT asset pricing model. Similarly, Elton et al.
(1993) find that a multi-index benchmark seems to be superior to a single-index
benchmark. Also Grinblatt & Titman (1994) find that mutual fund performance is
sensitive to the choice of benchmark, and Gruber (1996) argues that a four-variable
multi-index model is appropriate. Finally, Carhart (1997) argues in favour of a four-
factor multi-index model, which besides the market index includes factors for size,
book-to-market equity and the one-year momentum in stock returns.


The general conclusion reached in the literature e.g. Jensen (1968), Malkiel (1995)
and Detzler (1999) is that mutual funds in the US net of expenses (net returns)
have not been able to generate excess returns. However, using gross returns by
adding back expenses to the net asset value, superior performance can be identi-
fied, but this superior performance is just about equal to their expenses implying a
cost elasticity of approximately –1, see Blake et al. (1993) and Detzler (1999). This
conclusion is very much in line with the Grossman & Stiglitz (1980) theory of infor-
mationally efficient markets, where informed investors are compensated for their
information gathering.


Most research on mutual fund performance has been on US mutual funds, see e.g.
Malkiel (1995), whereas there have only been few analyses of non-US mutual funds.
Exceptions are analyses by Dermine & Röller (1992) for French mutual funds, Wit-
trock & Steiner (1995) for German mutual funds, Ter Horst et al. (1998) for Dutch
mutual funds, Cesari & Panetta (2002) for Italian mutual funds, Blake &
Timmerman (1998) for UK mutual funds, and Dahlquist et al. (2000) for Swedish
mutual funds. Furthermore, the first European cross-country analysis has been per-
                                         3
formed by Otten & Bams (2002), which includes France, Italy, the UK, Spain, Ger-
many and the Netherlands.


The overall picture of the Otten & Bams analysis is that they confirm the US evi-
dence for the period 1991 to 1998, but with two exceptions. Firstly, there seems to
be evidence for France, the UK and the Netherlands that small caps have out-
performed their benchmarks even after expenses have been deducted, and sec-
ondly UK mutual funds exhibit significant positive Jensen α’s for net returns as well
as for gross returns. However, for Germany and Italy Otten & Bams conclude in fa-
vour of the informationally efficient market conclusion, and this evidence is con-
firmed by Wittrock & Steiner (1995) and Cesari & Panetta (2002), respectively.
However, the Otten & Bams analysis differs significantly from the Blake &
Timmermann study, which conclude that UK mutual funds in general have been un-
der-performing their passive benchmarks. This difference may, however, be due to
the fact that the Otten & Bams analysis only includes European funds that are pure
domestic equity funds.


In a recent study by Dahlquist et al. (2000) they consider Swedish mutual fund per-
formance for 210 equity, bond and money market funds. Their sample is, however,
restricted to funds investing only domestically. They show that special equity
funds, bond and money market funds have neutral to significantly negative per-
formance, whereas regular equity funds seem to have obtained over-performance.


To date no serious evidence on Danish mutual fund performance exits. This is
partly because the analyses pursued by the mutual funds themselves and the Asso-
ciation of Danish Mutual Funds are focussed purely on Sharpe ratio comparisons
rather than on the Jensen measure, and partly because these studies only compare
the mutual funds to each other rather than comparing them to relevant bench-
marks.


The study in this paper includes equity as well as fixed income funds. Dependent
on the actual investment objective of each fund, we relate an appropriate bench-
mark to each fund and estimate the Jensen measure. First, we consider the stan-
dard CAPM security market line regression and then a multi-index model is esti-
                                         4
mated for each fund as well as for equally-weighted portfolios. Our general conclu-
sion is that net of expenses none of the 44 Danish mutual funds have been able to
obtain superior performance. Most funds have insignificant Jensen α’s and 5 funds
(11%) have significantly negative Jensen α’s at the 5% level.


Generalising the single index model to take account of market timing pursuing both
the quadratic regression of Treynor & Mazuy (1966) and the option approach of
Henriksson & Merton (1981) does not, however, alter this conclusion that Danish
mutual funds have performed neutrally. Moreover, concerning market timing we
are not able to document significant timing ability on the part of Danish mutual
funds. Out of our sample of 44 mutual funds the quadratic regressions find that
only 7 funds (16%) show significant market timing ability, whereas the option re-
gressions only find that 5 funds (11%) have possessed market timing ability.


The paper is organised as follows. In the next section we describe our data. Section
3 presents the results of the standard CAPM security market line regressions, and in
Section 4 we present the results of the multi-index model. In Section 5 we analyse
the relevance of timing ability and in Section 6 we consider the influence of ex-
pense ratios on fund performance. Section 7 concludes.



2. Data description
The sample in this study consists of 44 Danish mutual funds in operation since Oc-
tober 1994, and the sample period is October 1994 to January 2002. The sample is
split between 27 equity funds and 17 fixed income funds. In Table 1 the 44 funds
are categorised into homogeneous groups according to their investment objectives.


                       Table 1. Mutual Fund Categorisation


In Table 1 funds are categorised into equity and fixed income funds. Equity fund
are further categorised into Danish equity funds and foreign equity funds, where
we distinguish between various areas as Europe, Eastern Europe, World, Pacific,
Japan, North America, and Latin America, and the fixed income funds are further
categorised into Danish fixed income funds and foreign fixed income funds. This

                                          5
classification is not based on any statistical analysis, but follows the classification
used by the Association of Danish Mutual Funds. An almost similar categorisation is
used by Cesari & Panetta (2002) for Italy, although they use cluster analysis.


We use net asset values obtained from the Copenhagen Stock Exchange through
Bonnier A/S and returns are determined as log returns using monthly observations,
which amount to a total of 87 observations. The net asset values have been cor-
rected to include dividends, and we have assumed that dividends are reinvested
the day after the day where dividends have been declared. Excess returns are ob-
tained by subtracting the 1-month CIBOR rate from the net asset returns.


Based on the different categories in Table 1 a number of benchmarks will be ap-
plied. For Danish equities we use the KFX index, which includes the 20 Danish equi-
ties with the highest market value. However, the KFX index supplied by the Copen-
hagen Stock Exchange, is not corrected for dividends, and therefore we use a divi-
dend-corrected KFX index obtained from Belter & Tanggaard (2001).


Most of the benchmarks applied for the foreign equity funds are obtained from
Morgan Stanley. These include MSCI Europe, MSCI EM Eastern Europe, MSCI World,
MSCI AC Pacific Free Ex. Japan, MSCI Japan, and MSCI EMF Latin America. For the
North America fund we use the S&P 500 index. The MSCI indices are obtained form
Morgan Stanley, and the S&P 500 index is obtained from Datastream. All indices are
measured in Danish Kroner, dividends are included and assumed reinvested, and
excess returns are again determined by subtracting the 1-month CIBOR rate.


On the fixed income side the benchmark applied for the funds, which invest in Dan-
ish fixed income is the J. P. Morgan Denmark Government Bond index, whereas the
J. P. Morgan Global Broad ex. Denmark index is used for the foreign fixed income
funds. These indices include coupon payments, are measured in Danish Kroner, and
are obtained from Datastream. In order to check the robustness of the results, we
have also applied the EFFAS indices obtained from Bloomberg. Although we have
distinguished between short term funds and long term funds using the EFFAS 1-3
years maturity index and the EFFAS all government bond > 1 year index, respec-
tively, the results are almost identical to the results obtained using the J. P. Mor-
                                          6
gan indices with the exception that the goodness-of-fit using J. P. Morgan indices
are in almost all cases marginally superior to the goodness-of-fit obtained using the
EFFAS indices. Therefore, only results based on the J. P. Morgan indices will be re-
ported below.


An important issue that has been discussed thoroughly in the literature, see e.g.
Brown et al. (1992), is survivorship bias. As argued by Malkiel (1995) the analysis
will significantly overstate the returns if one systematically ignores the non-
surviving funds. We believe, however, that this study of Danish mutual funds is free
of survivorship bias, because no funds have defaulted during the analysis period
and therefore we include virtually all the funds that have been in operation during
the period October 1994 to January 2002.


In Table 2 we provide summary statistics for the whole sample of 44 mutual funds
and the benchmarks used in this analysis.


                            Table 2. Summary statistics

For each of the categories and benchmarks we present the excess return and the
standard deviation measured annually. Furthermore, the Jarque-Bera statistic for
normality and its probability value have been calculated for each fund and the
benchmarks. Dybvig & Ross (1985) and Grinblatt & Titman (1989) show that the
Jensen measure is biased if the fund and benchmark returns are not jointly normal
or are non-linear. This motivates the test for normality in this sample.


From Table 2 we conclude that for half of the funds and for 60% of the benchmarks
normality cannot be rejected. This means that normality is rejected for approxi-
mately half of the sample, and we therefore need to be careful when interpreting
the inference of the Jensen measure in the empirical analysis below.


Although we use excess returns, we have also tested for stationarity applying the
Augmented Dickey-Fuller test. In our case the critical 5% value is 3.46, and from
Table 2 we infer, as expected, that in all cases we cannot reject stationarity.



                                            7
3. Performance using the single index model
In this section we estimate the Jensen measure of performance based on the stan-
dard CAPM security market line. We estimate the security market line for each of
the 44 mutual funds as well as for equally-weighted portfolios within each invest-
ment category. The regressions can be formalised as:


      rit – rft = αi + βi(rmt – rft) + εit                                        (1)


where rit, rft and rmt are the returns at month t of the i’th fund (the i’th equally-
weighted portfolio), the risk-free return and the benchmark return, respectively, αi
is the Jensen measure, and βi is a measure of the systematic risk of fund (portfolio)
i. Finally, εit is a white noise error term. Equation (1) indicates that the excess re-
turn on fund (portfolio) i is linearly related to the excess return on the benchmark.


Equation (1) is estimated by OLS but in order to account for potential serial corre-
lation and heteroscedasticity, we use Newey-West corrected standard errors. In
Table 3 we present the estimation results obtained from equation (1), but only the
average results within each fund category is presented. However, taking simple av-
erages within each group does not yield meaningful standard errors and t-statistics.
Therefore, the reported t-statistics, the β’s and the adjusted coefficients of de-
termination (R2-adj) are obtained from the equally-weighted portfolios based on
the mutual funds in each category.


The general conclusion in Table 3 is that none of the 44 mutual funds have been
able to over-perform their passive benchmarks. The Jensen measure is in most
cases negative and not statistically significant. From the individual regressions we
find that 5 funds have a significantly negative Jensen measure at the 5% level. No
funds have a significantly positive Jensen measure at the 5% level. We can there-
fore conclude that net of expenses the performance of Danish mutual funds has
been neutral.




                                             8
               Table 3. Summary statistics of the Jensen measure
                              The single-index model

This is further confirmed by Figure 1, which presents the distribution of the α’s,
and as expected we infer that the distribution is close to being symmetric around
zero.


             Figure 1. Frequency distribution of estimated Jensen α’s


Thus, these preliminary results indicate that Danish mutual funds net of expenses
have not been able to beat the market. On the other hand, most funds have per-
formed neutrally i.e., they have been able to obtain gross returns just sufficient to
cover their expenses, leaving the fund members with net returns that are not sig-
nificantly different from the passive benchmark returns. In order to validate the
robustness of this conclusion, the asset pricing model is extended to a multi-index
model following Elton et al. (1993) and Gruber (1996) in the next section.



4. Performance using the multi-index model
The choice of benchmark has been discussed intensively in the literature. In the
single index security market line model only one benchmark is applied, which im-
plicitly assumes that the fund has a well-defined investment objective, which can
be represented by one single benchmark. Simultaneously, we assume that the rele-
vant underlying asset pricing model is the CAPM.


Kon & Jen (1978) compare the CAPM with Black’s zero beta model and argues in fa-
vour of the zero beta model. Similarly, Lehman & Modest (1987) find that the Jen-
sen measure differs significantly between a CAPM and an APT model. Grinblatt &
Titman (1989) and (1994) use alternative benchmarks in their analyses and con-
clude that the Jensen measure differs considerably between these benchmarks.
Opposed to these results Ippolito (1989) argues that different alternative bench-
marks do not alter his basic conclusion that 143 US mutual funds over the period
1965-1984 outperformed passive index funds. The Ippolito conclusion was contro-
versial, when it was published, and Elton et al. (1993) show, that his conclusion

                                         9
rests on his choice of benchmark. In fact, Ippolito analyses mutual fund perform-
ance relative to the S&P 500 index despite including funds that invest in non-S&P
500 stocks. When Elton et al. (1993) include a non-S&P 500 benchmark, Ippolito’s
conclusion is reversed. Besides including the non-S&P 500 benchmark, Elton et al.
(1993) also include a bond index in their analysis. Similarly, Gruber (1996) argues in
favour of including both equity and bond indices as benchmarks as well as he in-
cludes factors for size and growth following Carhart (1997), who finds that factors
for size, book-to-market equity and the one-year momentum in stock returns affect
mutual fund returns. Finally, Kothari & Warner (2001) argue that standard per-
formance measures depend on the benchmarks ability to mimic the fund style, and
therefore benchmarks must be selected carefully.


From Table 3 we inferred that the R2-adj statistics were between 0.70 and 0.96 in-
dicating that the chosen benchmarks are not able to fully explain the mutual fund
returns. In Denmark the legislation regulating mutual funds states that 75% of the
investments performed by the funds must be within the main investment category.
This means that the individual mutual fund is free to invest within any investment
area for up to 25% of its asset value2. If Danish mutual funds in fact use this oppor-
tunity, a single index model is obviously not appropriate. In order to account for
this opportunity to invest up to 25% of the asset values in other investment areas
than the primary area, we will formulate a multi-factor model based on Elton et al.
(1993) and Gruber (1996) including both equity and fixed income indices.


The multi-factor model reads as:


      rit – rft = αi + βim(rmt – rft)


              + βiKFX(rKFXt – rft) + βiW(rWt – rft) + βiD(rDt – rft) + βiG(rGt – rft) + εit   (2)


The main motivation for equation (2) is that a fund whose main investment objec-
tive is to invest in Danish stocks can according to Danish legislation invest up to 25%
of its assets in foreign equities, in Danish bonds or in foreign bonds. In order to
measure these different possibilities we include the following relevant factors,
where:
                                                10
   -   rKFXt: the return on the Danish KFX index at month t
   -   rWt:   the return on the MSCI world index at month t
   -   rDt:   the return on the J.P. Morgan Denmark Government Bond index
              at month t
   -   rGt:   the return on the J.P. Morgan Global Broad ex. Denmark index at
              month t


In the case of a mutual fund whose main purpose is to invest in Danish equities, rmt
= rKFXt, and equation (2) becomes a four-factor model, but in the case of a mutual
fund that primarily invests in Japan, say, rmt equals the excess return on MSCI Ja-
pan, and equation (2) becomes a five-factor model etc.


In Table 4 we present the results of the multi-index model. Again standard errors
are Newey-West corrected and t-statistics and R2-adj. are obtained from the equal-
ly-weighted portfolios.


                Table 4. Summary statistics of the Jensen measure
                              The multi-index model

The general conclusion to be drawn from Table 4 is that the results are almost
similar to the results obtained from the single index CAPM in Table 3. Most of the
Jensen α’s are negative and not significantly different from zero. From the individ-
ual regressions we find that 4 funds have performed significantly negative at the 5%
level compared to 5 funds in Table 3. 3 of these funds are the same.


However, making a close inspection of the individual regressions (not reported), we
find that in special cases the multi-index model provides superior information to
the single index model. E.g. the KFX index plays a positive and significant role for
two of the world equity funds, and the world equity index has a significant and
positive influence on the Japanese and the US funds. Further, for 4 out of the 14
Danish fixed income funds the J.P. Morgan Global Broad ex. Denmark index plays a
positive and significant role, and likewise the J.P. Morgan Government Denmark in-
dex has a positive and significant influence on 2 of the 3 global fixed income funds.
There is, however, no indication that equity funds have been investing in fixed in-

                                         11
come products as well as we find no evidence that fixed income funds have been
investing in equities. These results are very much in line with our expectations,
since otherwise the funds will be registered as balanced funds, which will affect
the way the funds are taxed, see note 2. However, it is worth pointing out that al-
though the return of some funds is related to more than just one index, we infer
from Table 4 that the goodness-of-fit is only marginally higher compared to the
goodness-of-fit in Table 3 indicating that the standard security market line model
does the job almost as well as the multi-index model.


Accordingly, the main conclusion is not altered that is, the multi-index model con-
firms that in general Danish mutual funds have performed neutrally, and mutual
fund investors have obtained net returns that are not significantly different from
the passive benchmark returns.


Although the multi-index model validates the results of the single index model, we
will examine the robustness of the results further by considering whether mutual
funds have been able to time the market that is, we test their timing ability. But as
shown in this section, the results of the multi-index model are indistinguishable
from the results obtained applying the standard CAPM single index model, and
therefore the next section will proceed adopting the CAPM single index model as
our reference asset pricing model.



5. Timing and selectivity
In the previous sections the performance evaluation has focused on the selectivity
of the Danish mutual funds that is, whether Danish mutual funds have been able to
obtain significant and positive Jensen measures. This kind of selectivity is often re-
ferred to as micro forecasting or security analysis as opposed to macro forecasting,
which concerns forecasts of price movements of the general market as a whole,
which is also called market-timing, see Fama (1972).


If fund managers change the fund beta (β) according to their expectations of bull
and bear markets, βi becomes a decision variable, which will not be constant, see
Kon & Jen (1978). In this case mutual funds are expected to be able to time the

                                         12
market, which has important implications for the performance evaluation per-
formed in the previous sections. Although this was recognised as early as in the
study by Jensen (1968), he argued that timing ability would make us overestimate
the true α’s. But as shown by Grant (1977), in fact, market timing implies that the
estimate of Jensen’s α becomes downward biased that is, our estimate of Jensen’s
α will be less than the true α, and we are inclined to underestimate the actual per-
formance of mutual funds.


Empirically, a number of studies have analysed the ability of mutual funds to time
the market, and most of these analyses seem to agree that mutual funds do not
possess timing ability. For example, Treynor & Mazuy (1966) find using a quadratic
equation that for only one out of 57 mutual funds the hypothesis of no timing abil-
ity could be rejected, and Veit & Cheney (1982) conclude that in general mutual
funds do not change their characteristic lines in bull and bear markets and specifi-
cally a majority of those few funds that did change their characteristic lines, their
timing was unsuccessful. In fact, only 3 out of 74 mutual funds obtained a success-
ful timing. These conclusions are confirmed by Henriksson (1984), who applies pa-
rametric as well as non-parametric techniques developed by Merton (1981) and
Henriksson & Merton (1981). His sample consists of 116 US open-end mutual funds,
and he finds no evidence of market timing ability for the period 1968 to 1980. Us-
ing an extended version of the Henriksson & Merton model, Connor & Korajczyk
(1991) and Hendricks et al. (1993) confirm that US mutual funds do not possess tim-
ing ability. Finally, Grinblatt & Titman (1994) analyse performance using different
measures as the Jensen measure, the Treynor & Mazuy measure and their own PPW
measure. However, they conclude that the results are insensitive to the measure
used and the simple Jensen measure performs as well as the Treynor & Mazuy
measure and consequently inferences based on a pure selection basis is as good as
inferences based on the quadratic regression, where an explicit distinction is made
between selection and timing ability.


Exceptions to the general conclusion that mutual funds have no timing ability are
Kon & Jen (1978), who argue using a switching regression model that for most of
the 49 US mutual funds analysed, they cannot reject nonstationarity of the system-
atic risk, and Lee & Rahman (1990), who find that out of 93 US mutual funds, 17%
                                         13
show significant timing ability and 15% show a positive and significant Jensen
measure that is, selectivity. In fact, 10 funds (11%) have both significant timing and
selection ability.


A number of alternative methods have been suggested in the literature to test the
timing ability of mutual fund managers, and in this analysis we will apply two dif-
ferent methods to validate the robustness of the results on Danish mutual fund per-
formance.


The first specific test of the timing skill of mutual funds was developed by Treynor
& Mazuy (1966), who argue that if the mutual fund manager acts as if he can time
the market, he will hold a greater proportion of the market portfolio, when he ex-
pects the return on the market will be high and vice versa. In fact, he will adjust
the portfolio β according to the return on the market portfolio as:


       βit = bi0 + bi1(rmt – rft)                                                  (3)


and substituting equation (3) into equation (1), we find:


       rit – rft = αi + bi0(rmt – rft) + bi1(rmt – rft)2 + εit                     (4)


which gives us the quadratic Treynor & Mazuy equation. Compared to the standard
single index model, equation (4) includes a new term, which is the excess market
return squared. From estimates of the parameters in equation (4), we are able to
distinguish between selection and timing abilities. If αi is positive and significantly
different from zero, we identify selection ability, as in the single index model, and
if bi1 is positive and significant, the mutual fund manager possesses timing ability.


Jensen (1972) develops a model in which he defines a signal given as πt = [(rmt-rft) -
E(rmt-rft)], where E(rmt-rft) is the unconditional expected value of the market excess
return. The signal πt can be interpreted as the market excess return forecast error.
Under various assumptions Jensen shows that the portfolio β is determined by the
sum of a target portfolio β and a proportion of the expected signal, and it can be

                                                  14
shown that this model is consistent with the Treynor & Mazuy quadratic equation,
see e.g. Lee & Rahman (1990). The quadratic equation of Treynor & Mazuy (1966),
Jensen (1972), Lee & Rahman (1990) and others is one of the different specifica-
tions, which will be applied below to test timing and selectivity ability of the Dan-
ish mutual funds.


The second specification applied in this study follows the Merton (1981) and Hen-
riksson & Merton (1981) option approach. In their model the mutual fund manager
is assumed to receive a binary signal, which can take two distinct values depending
on the true outcome of the market return. Based on these two distinct signals, the
mutual fund manager chooses one of two values of the portfolio β, and they show
that this extends the standard CAPM security market line specification to:


       rit – rft = αi + βi(rmt – rft) + γiMax[-(rmt – rft) ; 0] + εit              (5)


where the new term represents an informational advantage represented by a no
cost put option on the market portfolio. Henriksson & Merton (1981) argue that it is
possible to distinguish between selectivity and timing ability in equation (5). For a
positive and significant αi, we identify selection skills, as in the single index model,
and for a positive and significant γi the mutual fund manager possesses timing abil-
ity.


The two approaches promoted by Treynor & Mazuy (1966) and Henriksson & Merton
(1981) focus on testing whether mutual funds possess timing ability, and both tests
are consistent with the standard CAPM security market model with a time-varying
systematic risk factor, β. In the general case, however, even for passively managed
mutual funds the systematic risk is likely to vary over the market cycle, since the
portfolio beta is a market-weighted average of the individual security betas, and
market price changes will affect the weighting and accordingly the portfolio beta
will be time-varying. Further to this observation, Merton (1973) develops his in-
tertemporal CAPM, which gives a theoretical motivation of why beta is time varying
over the investment cycle. Following this line of thought Christopherson et al.
(1999) argue that not only may the systematic risk (β) be time-varying, but also one


                                                 15
would suspect that the Jensen measure is time-varying. They ague that both α and
β are functions of market information variables, and in their empirical test they
use dividend yield and the detrended level of short-term Treasury yields as market
information variables, and they find some support for their hypothesis of time-
varying alphas (α).


At this point it is worth mentioning that the Treynor & Mazuy quadratic regression
in fact is consistent with the hypothesis of a time-varying Jensen measure. If we
assume following the definition of the time-varying βit, see equation (3) that αit =
ai0 + ai1(rmt – rft) that is, the Jensen measure is linearly related to the excess market
return, we find substituting into equation (4) that:


       rit – rft = ai0 + [ai1 + bi0](rmt – rft) + bi1(rmt – rft)2 + εit             (6)


To the econometrician equation (4) cannot be distinguished from equation (6), and
therefore equation (4) is consistent with the more general specification, where
both alpha and beta are functions of the excess market return.


The two different specifications given by equations (4) and (5) are able to provide
us with an estimate of the performance (selection ability) of the Danish mutual
funds analysed. Furthermore, we can separate selection and timing ability. We be-
lieve that these two models are able to explore important potential differences in
the behaviour of Danish mutual funds concerning selection and timing ability.


Firstly, we will test the selection and timing ability of Danish mutual funds pursuing
the Treynor & Mazuy quadratic regression approach. Again, we only report the re-
sults of the 10 different investment groups as categorised in Table 2, where the t-
statistics are obtained from equally weighted portfolios within each of the 10 cate-
gories.


The estimation results are presented in Table 5. Since we focus on selection and
timing ability, we only present the estimates of αi and bi1, see equation (4). Again
t-statistics are based on Newey-West corrected standard errors to correct for po-

                                                  16
tential serial correlation and heteroscedasticity. In this case, it is particularly im-
portant to obtain heteroscedasticity consistent standard errors, because adding a
quadratic term to the regression equation imposes a heteroscedasticity type of
problem into the model.


            Table 5. Summary statistics of selection and timing ability
                      The Treynor & Mazuy quadratic model



Compared to the previous analyses, Table 5 provides a number of new and interest-
ing results. Firstly, we infer that although most of the estimated α’s are still nega-
tive and not significantly different from zero, we see that the Japanese fund has
been able to obtain a significant and positive α and based on the portfolio standard
error the Danish fixed income funds have as a group obtained a significant and
negative α, despite that none of the 14 individual funds have significant α’s. How-
ever, we see no significant differences between the absolute values of the α’s
across tables 3, 4 and 5, and evidently, we cannot confirm the observation by
Grant (1977), who argued that ignoring market timing, we tend to underestimate
the true alphas.


Secondly, Table 5 shows that most of the equity fund managers and global fixed in-
come managers did not possess market timing ability, whereas 2 out of 5 Pacific
equity fund managers and 5 out of 14 Danish fixed income fund managers seem to
have been able to time the market. The market timing parameter is in most cases
insignificant for equity funds and global fixed income funds and even significantly
negative in 5 cases. However, we cannot conclude that Danish fixed income man-
agers in general are able to time the market, since the majority that is, 9 out of 14
funds, show an insignificant positive timing ability parameter. But taking a closer
look at the 5 Danish fixed income funds that show a significant timing ability, 4 out
of these 5 funds obtained significantly negative selection ability in terms of signifi-
cantly negative α’s


The main conclusion to be drawn from the quadratic model is that taking account
of a potential market timing ability on the part of Danish mutual fund managers,

                                          17
our estimates of their selection ability parameter α is insignificant, and still the
overall picture is that most funds have performed neutrally. Exceptions are 5 funds
that have performed significantly negative and 1 fund that has performed signifi-
cantly positive. Furthermore, we find no general tendency that mutual funds have
been able to time the market. Exceptions are 7 (16%) Pacific equity funds and Dan-
ish fixed income funds out of a total sample of 44 mutual funds.


To validate the robustness of these results we will now consider the option ap-
proach of Henriksson & Merton (1981) that is, we will estimate equation (5) for the
44 individual funds as well as for the category portfolios. The results are presented
in Table 6.


              Table 6. Summary statistics of selection and timing ability
                       The Henriksson & Merton option model

The evidence in Table 6 as far as selection ability is concerned is almost identical
to the evidence of the quadratic model in Table 5. The estimates of α are margin-
ally higher in Table 6, but again most α’s are insignificant and again the Japanese
equity fund shows up as an exception with a significant and positive α. A few funds
have obtained a significant and negative α, and the 4 Danish fixed income funds
with significantly negative α’s are the same funds in Table 5 and Table 6.


Concerning timing ability we also find almost identical results in the quadratic and
the option models. In Table 5 the quadratic regression indicated that 7 funds had
been successful market timers, whereas the option model is able to identify 5
funds with timing ability. 5 of the 7 funds in Table 5 are identical to the 5 funds in
Table 6.


Although we analyse the selection and timing ability of the Danish mutual funds
applying two different models, the results are very similar, and furthermore the re-
sults concerning selectivity are identical to the results obtained from the single in-
dex and the multi-index models. On this basis, we believe that our conclusion that
the Danish mutual funds have performed neutrally with no particular selection and
timing ability is robust.
                                          18
6. Performance and expenses
As mentioned in the Introduction Blake et al. (1993) and Detzler (1999) find that a
1% increase in mutual fund expense tends to lower the performance by approxi-
mately 1%. In order to analyse the response from expense ratios to performance for
the Danish mutual funds, we will follow Detzler (1999), and perform the following
regression:


      αi = ci0 + ci1[expensei] + εit                                            (7)


where i = 1, 2 …, 44, αi is the estimated Jensen measure for fund i, and expensei is
the expense ratio for fund i. Equation (7) postulates a simple linear relationship
between the performance measure and the expense ratio, and the US evidence is
ci1 ≈ -1. The expense data have kindly been provided by Jesper Kirstein.


We estimate equation (7) using OLS with Newey-West corrected standard errors for
each of the 4 different models applied in this study that is, the standard CAPM sin-
gle index model, the multi-index model, the Treynor & Mazuy quadratic model, and
finally the Henriksson & Merton option model. The estimation results are presented
in Table 7.


         Table 7. Summary statistics of performance and expense ratios


The conclusion to be drawn from Table 7 is clearly that irrespectively of the model
used, we find absolutely no relationship between mutual fund performance and
mutual fund expense ratios. Although the sensitivity parameter ci1 is negative in all
four cases, it is not significantly different from zero at any conventional signifi-
cance level. Consequently, we are not able confirm the US evidence.



7. Conclusions
Although Danish mutual fund assets have grown by more than 25% annually during
the last 6 years, this study is the first independent analysis of Danish mutual fund
performance. We have analysed a survivorship free sample of 44 funds divided into


                                         19
27 equity funds and 17 fixed income funds that have been in operation since Octo-
ber 1994. Applying various different models as the standard CAPM single index
model, a multi-index model incorporating national and global equity and fixed in-
come benchmarks, the quadratic Treynor & Mazuy (1966) model as well as the op-
tions model of Henriksson & Merton (1981), we have analysed the selection and
timing abilities of these funds.


The general conclusion is that we have not been able to identify significant per-
formance that is, Danish mutual funds have not possessed selection ability. In most
cases, their performance have been neutral and in a few cases we find even signifi-
cantly negative performance. Concerning timing ability the general conclusion is
also negative. Only in few cases we find evidence in favour of significant timing
ability, but most of the funds that have been successful market timers show signifi-
cantly negative selection ability. Finally, we have analysed whether fund perform-
ance is related to the expense ratios, but we find no evidence whatsoever to sus-
tain this hypothesis.


Although we cannot confirm the US evidence on the relationship between expense
ratios and fund performance, our results on performance concerning selection and
timing abilities are very much in line with the US evidence. We cannot reject that
Danish mutual funds net of expenses have performed neutrally, which is consistent
to the Grossman & Stiglitz theory of informationally efficient markets.




                                         20
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                                         22
Footnotes
  *   I am grateful to Lars Rasmussen for research assistance, comments from Jan
      Bartholdy, Tom Engsted and Asger Lunde and to Jesper Kirstein for supplying
      his expense data. The usual disclaimer applies.

  1. Whereas, the original Sharpe ratio measures the excess return over the risk-
     free rate, Sharpe (1994), however, extents the ratio to account for the ex-
     cess return over a benchmark rate.

  2. Danish mutual funds are not allowed to invest these 25% in the money mar-
     ket, which means that they must invest in either equities and/or bonds.
     Most funds will, however, choose to invest solely in equities or solely in
     bonds, since otherwise the fund will be classified as a mixed fund, changing
     how the fund is taxed.




                                       23
                     Table 1. Mutual Fund Categorisation
              Mutual funds                                Benchmarks
   Danish equities                   5     KFX
   Foreign equities
      - Europe                       5     MSCI Europe
      - Eastern Europe               1     MSCI EM Eastern Europe
      - World                        6     MSCI World
      - Pacific                      5     MSCI AC Pacific Free ex. Japan
      - Japan                        1     MSCI Japan
      - North America                1     S&P 500
      - Latin America                3     MSCI EMF Latin America
   Danish fixed income              14     J.P. Morgan Government Bond Denmark
   Foreign fixed income              3     J.P. Morgan Global Broad ex. Denmark

   Total                            44

                           Table 2. Summary statistics
                    No.    Excess   Standard      Aug.     Jarque-   Prob-      No.
                     of    return    devia-      Dickey     Bera     ability   passed
                   funds              tion       Fuller

Mutual funds
Danish equities     5        4.49    15.71       -4,77     13.07       0.03      1
Foreign equities
- Europe            5       6.38     18.34       -5,15     11.60       0.01      0
- Eastern Europe    1      -0.92     32.78       -5,24     244.76      0.00      0
- World             6       6.42     18.19       -5,93      7.48       0.11      4
- Pacific           5      -1.04     27.77       -4,95      1.41       0.62      5
- Japan             1      -6.76     23.57       -4.44      1.65       0.44      1
- North America     1       6.99     16.78       -6,69      2.61       0.27      1
- Latin America     3      -2.93     37.08       -5,93     60.61       0.00      0

Danish bonds        14       1.97        2.93    -5,23     35.09       0.24      8
Foreign bonds       3        3.21        5.39    -5,42      7.12       0.45      2

Benchmarks
KFX                        10.85     17.35       -5,55      2.62       0.27     Yes
MSCI Europe                 9.63     16.18       -5,39     10.83       0.00     No
MSCI East. Euro            -1.31     43.48       -5,06     139.04      0.00     No
MSCI World                  8.41     17.43       -5,64      6.11       0.05     No
MSCI Pacific               -3.73     26.45       -4.92      4.45       0.50     Yes
MSCI Japan                 -8.56     23.12       -8,58      2.33       0.31     Yes
MSCI Latin Am.              0.88     35.07       -5,74     48.05       0.00     No
S & P 500                  14.74     19.34       -5,72      5.73       0.06     Yes
JPM Denmark                 4.04      3.51       -4,91      3.16       0.21     Yes
JPM Global                  5.23      6.13       -6,00      0.68       0.71     Yes




                                          24
                  Table 3. Summary statistics of the Jensen measure
                                               The single-index model

   Mutual funds                No. of            α        t-stat.      No. sig-         β       R2-adj.
                               funds                                   nificant

 Danish equities                   5       -0.0033        -1.57                        0.78      0.82
 Foreign equities
 - Europe                          5       -0.0027        -1.28           (1)          1.00      0.86
 - Eastern Europe                  1        0.0006         0.15                        0.69      0.81
 - World                           6       -0.0011        -0.71           (1)          0.92      0.89
 - Pacific                         5        0.0021         1,14                        0.96      0.94
 - Japan                           1        0.0015         0.95                        1.00      0.96
 - North America                   1       -0.0031        -1.27                        0.73      0.70
 - Latin America                   3       -0.0031        -1.21                        0.96      0.89

 Danish bonds                      14      -0.0004        -1.12           (3)          0.59      0.79
 Foreign bonds                      3      -0.0002        -0.22                        0.66      0.71

 All funds                         44      -0.0010                        (5)          0.83      0.84
In the fifth column we indicate the number of funds in each category, which have a significantly
positive (negative) α at the 5% level.




               Figure 1. Frequency distribution of estimated Jensen α’s



                              25
                              20
                  Frequency




                              15
                              10
                               5
                               0
                                  1

                                           9

                                                   7

                                                           5

                                                                   3

                                                                           1




                                                                                                 05

                                                                                                         07
                                                                                  01

                                                                                         03
                                ,1

                                         ,0

                                                 ,0

                                                         ,0

                                                                 ,0

                                                                         ,0

                                                                                0,

                                                                                       0,

                                                                                              0,

                                                                                                      0,
                              -0

                                       -0

                                               -0

                                                       -0

                                                               -0

                                                                       -0




                                                                 Jensen's alpha




                                                           25
                  Table 4. Summary statistics of the Jensen measure
                                   The multi-index model

   Mutual funds        No. of        α        t-stat.     No. sig-      R2-adj.
                       funds                              nificant

 Danish equities          5       -0.0026      -1.32                        0.82
 Foreign equities
 - Europe                 5       -0.0010      -0.48                        0.87
 - Eastern Europe         1       -0.0014      -0.35                        0.82
 - World                  6        0.0001       0.07                        0.90
 - Pacific                5        0.0042       2,08                        0.95
 - Japan                  1        0.0009       0.44                        0.96
 - North America          1        0.0008       0.27                        0.73
 - Latin America          3       -0.0014      -0.56                        0.89

 Danish bonds             14      -0.0005      -1.67           (4)          0.83
 Foreign bonds             3      -0.0008      -0.98                        0.75

 All funds                44      -0.0002                      (4)          0.85
In the fifth column we indicate the number of funds in each category, which have a
significantly positive (negative) α at the 5% level.




             Table 5. Summary statistics of selection and timing ability
                         The Treynor & Mazuy quadratic model

   Mutual funds        No. of        α          t-      No. sig-      bi1      t-stat.     No. sig-
                       funds                  stat.     nificant                           nificant

 Danish equities          5       0.0001      0.05                   -1.33         -1.86     (2)
 Foreign equities
 - Europe                 5       -0.0007    -0.35                   -0.81         -1.35
 - Eastern Europe         1        0.0032     0.80                   -0.17         -1.14
 - World                  6        0.0005     0.22        (1)        -0.60         -1.26     (1)
 - Pacific                5       -0.0010    -0.50                    0.55          3.17      2
 - Japan                  1        0.0038     1.97         1         -0.52         -2.34     (1)
 - North America          1       -0.0023    -0.64                   -0.25         -0.40
 - Latin America          3       -0.0026    -0.97                   -0.05         -0.45     (1)

 Danish bonds             14      -0.0010    -2.03        (4)        5.72          2.55       5
 Foreign bonds             3      -0.0005    -0.46                   1.05          0.56

 All funds                44      -0.0001                1 (5)                              7 (5)
In the fifth and eight column we indicate the number of funds in each category, which have a sig-
nificantly positive (negative) αi and bi1 at the 5% level, respectively.




                                                26
             Table 6. Summary statistics of selection and timing ability
                         The Henriksson & Merton option model

   Mutual funds        No. of        α         t-    No. sig-         γ        t-stat.   No. sig-
                       funds                 stat.   nificant                            nificant

 Danish equities          5       0.0020     0.65                   -0.27      -1.86       (2)
 Foreign equities
 - Europe                 5       0.0001      0,03                  -0.15      -0.95
 - Eastern Europe         1       0.0018      0.25                  -0.03      -0.15
 - World                  6       0.0015      0.47                  -0.13      -0.99
 - Pacific                5      -0.0042     -1.55      (2)          0.22       2.71        2
 - Japan                  1       0.0054      2.21       1          -0.14      -2.16       (1)
 - North America          1       0.0028      0.48                  -0.16      -0.67
 - Latin America          3      -0.0044     -1.06                   0.03       0.32

 Danish bonds            14      -0.0013     -1.91      (4)         0.22        1.98        3
 Foreign bonds            3      -0.0010     -0.71                  0.12        0.82

 All funds               44       0.0003               1 (6)        -0.03                 5 (3)
In the fifth and eight column we indicate the number of funds in each category, which have a sig-
nificantly positive (negative) αi and γ1 at the 5% level, respectively.




          Table 7. Summary statistics of performance and expense ratios

                       Models                 ci1         t-stat.           R2-adj.

               Single index                  -2.63         -1.53             0.06

               Multi-index                   -0.67         -0.45            -0.02

               Treynor & Mazuy               -0.30         -0.12            -0.02

               Henriksson & Merton           -0.02         -0.01            -0.02




                                               27

				
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