The efficiency evaluation of mutual fund managers based on Dara, Cara, Iara.
Baghdadabad, Mohammad Reza Tavakoli ; Tanha, Farid Habibi ; Halid, Noreha 等
1. Introduction
Mutual fund performance can be evaluated by either the parametric
approach or nonparametric approach. The first approach has been
frequently studied in the literature, while the second approach has been
poorly considered in the performance evaluation models until now.
Earlier studies of fund performance evaluation are started with the
models based on Jensen's alpha (i.e., Jensen 1968), and are then
extended by adding more variables as explanatory factors (i.e., Carhart
1997) to improve the models. Most models are grounded on parametric
models, in which they require a strong theoretical model and a benchmark
to compute the outcome. Moreover, they only evaluate the funds
performance in terms of the relationship between risk premium and
return, without realizing the amount of resources that has been spent.
Data envelopment analysis is a non-parametric method used to
evaluate the relative efficiency of decision-making units (DMU), which
is first introduced by Charnes et al. (1978). DEA is employed for
relative efficiency appraisal of DMUs. The efficiency evaluation of
mutual fund managers in the DEA framework provides several advantages.
First, unlike the parametric models, there is no necessity to run a
theoretical model. Second, since the model evaluates the relative
performance of funds, there is no need to assign a benchmark as well.
Third, DEA does not require the assumptions of function forms relating
inputs to outputs. Finally, DEA can incorporate factors needed into the
model. Banker and Maindiratta (1986) explain that integral outcome of
the DEA analysis is a set of inefficiency measure that identifies the
source of the inefficiency and indicates the extent to which the various
inputs need to be reduced or outputs need to be increased for making the
inefficient DMUs efficient. The marginal share of each input or output
can be clarified by this information.
Since research on the efficiency of mutual funds is scarce and only
a few studies focus on this field (i.e., Murthi et al. 1997), we fill
several gaps in the literature. First, we evaluate the full universe of
more than 17,000 mutual funds in the Bloomberg database over the period
2005 to 2010. This large sample provides the possibility of overcoming
the small-sample problems that plagued prior studies concerning the
efficiency evaluation of mutual fund. Second, unlike earlier studies, we
evaluate funds managers' efficiency in terms of management style.
Third, we propose an optimal choice pattern to make decisions in
selecting the funds by investors; in addition, we prioritize efficient
managers in terms of their own efficiency scores. Fourth, unlike earlier
studies that only evaluate the funds in terms of relative (technical)
efficiency, we calculate two other efficiency measures, namely,
management and scale efficiency. Fifth, we propose three models of DEA
in the Decreasing Absolute Risk Aversion (DARA), Constant Absolute Risk
Aversion (CARA), and Increasing Absolute Risk Aversion (IARA) framework
to identify the best model in evaluating the efficiency of fund
managers.
2. Background
2.1. Fund performance measurement
Fund managers use many techniques to know how funds would perform.
The performance measures enable managers to distinguish funds in terms
of their performance. Although there are some performance measures, none
of them can accurately predict the fund performance. The existing
methods are simplistic and based on two variables, return and risk. They
often disregard the amount of resource consumption for increasing one
unit of return. Due to the fact that researchers are trying to propose a
top model for the comprehensive evaluation of the performance of funds,
they have extended the models in the framework of two parametric (i.e.,
Carhart 1997) and non-parametric approaches.
2.2. Non-parametric approach
Non-parametric approaches try to assess the efficacy of DMUs with
multiple input and output. DEA is one of the non-parametric methods that
can be used to evaluate a relative efficiency of DMUs (Charnes et al.
1978). It enables one to evaluate the relative efficiency of units as
well as being able to overcome certain shortcomings of parametric
approaches for performance evaluation. However, while several authors
have studied the efficiency of funds, they have entirely excluded the
evaluation of funds in terms of management and scale efficiency. The
first study that uses the DEA methodology in measuring fund performances
is related to Murthi et al. (1997). They evaluate the relative
efficiency of funds, and then compare them with other parametric
performance measures.
Basso and Funari (2001) consider two cost components and two risk
components as DEA inputs. They evaluate the performance of 48 funds in
three separate groups with the input-oriented DEA model supposing the
constant return to scale and analyze them with and without a stochastic
dominance index. Wilkens and Zhu (2001) propose characteristics of the
returns' distribution as output. Along with returns, they add
skewness and minimum return to more accurately assess fund performance.
Similarly, Joro and Na (2002) extend this line of research and add
distribution's third moment as output in their DEA model. The third
order approximation based on a Taylor's series' expansion of a
generalized utility function around the mean of the portfolio returns
exhibits three desirable properties for utility functions, as proposed
by Arrow and Pratt, namely, (i) positive marginal utility for wealth,
(ii) decreasing marginal utility for wealth, and (iii) non-increasing
absolute risk aversion.
Basso and Funari (2003) also use the multi-criteria capability of
DEA to measure the ethical mutual funds' performance. They consider
the ethical component, subscription, redemption fees and risk as DEA
inputs, and also the expected return of funds as output. Lozano and
Gutierrez (2008) find that most evaluation models overestimate risk, as
in conventional models (such as DEA) the risk of mutual funds is
calculated as a linear combination of fund assets. In other words, the
effect of diversification is neglected. Thus, they propose a new DEA
model combined with second-order stochastic dominance to compute the
relative efficiency of mutual funds. The main advantage of this method
is to consider the effects of portfolio diversification, which is
neglected by the conventional DEA approaches.
However, the literature does not consider (i) an enough number of
funds to minimize survivorship bias, (ii) management style as surveying
class, (iii) management and scale efficiency scores as evaluation
measures, and (iv) other stochastic dominance measures in the DEA model.
3. Methodology and data
The DEA measures the relative performance of DMUs and the score of
relative efficiencies with multiple outputs and inputs as:
Efficiency = Weighted Sum of Outputs/Weighted Sum of Inputs. (1)
This is a linear programming model (LPM) that was developed by
Charnes et al. (1978). They first proposed an input-oriented model with
respect to constant returns to scale (CRS), and then extended the model
based on a variable returns to scale (VRS) model.
3.1. The constant return to scale (CRS)
To describe the CRS model, assume that there is a set of n DMUs and
each unit defines s output and m input. The relative performance score
of each unit is defined as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2)
where r = 1 to s, i = 1 to m, j = 1 to n, [Y.sub.rj] = amount of
output r produced by [DMU.sub.i], [X.sub.ij] = amount of input j used by
[DMU.sub.i], [U.sub.r] = weight given to output r, [V.sub.i] = weight
given to input i. Since it is a nonlinear model, we change it to a
linear model with assumption of [m.summation over (i=1)]
[V.sub.i][X.sub.ij] = 1:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)
This model is run n times (the number of DMUs) to compute the
scores of relative performance of each DMU. This score is derived from
the input and output weights of each DMU. Those DMUs getting a score of
1 are efficient and the others getting a score of less than 1 are
inefficient.
3.2. The variable return to scale (VRS)
The CRS assumption is only appropriate when all management styles
are operating at an optimal scale. Many factors may cause a style to not
operate at the optimal scale. Banker et al. (1984) propose a CRS DEA
model to compute the VRS. The using of the CRS specification when not
all styles are operating at the optimal scale will lead to technical
(relative) efficiency, which is confounded by scale efficiency. Thus,
the CRS can be easily justified to compute VRS by adding
NI'[lambda] = 1 to Eq. (4):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4)
where NI' is an N x 1 vector of one. The calculated technical
efficiency by the VRS model is divided into management and scale
efficiency.
To compute the scale efficiency, many studies decompose the
technical efficiency scores obtained from a CRS DEA into two
components--scale inefficiency and pure technical inefficiency. This may
occur by computing both a CRS and a VRS on the same data. If there is a
difference in the two technical efficiency scores for a given DMU, this
shows that the DMU has scale inefficiency, and that the scale
inefficiency can be computed by the difference between the VRS and CRS
technical scores. One shortage of scale efficiency measure is that the
value does not detect whether the DMU is operating in an area of the
increasing or decreasing returns to scale. This may be determined by
running an additional DEA problem with non-increasing return to scale
(NIRS) imposed. This can be done by altering the DEA model in Eq. (4) by
substituting the NI'[lambda] = 1 restriction with NI'[lambda]
[less than or equal to] 1, as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)
3.3. Prioritization of efficient DMUs
Since it is possible to have more than one efficient style, the DEA
ranks the efficient and reference styles in terms of the number of
references and the weighted method as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)
where [D.sub.k] is inverse efficiency of kth DMU, [[lambda].sub.t]
is decision variable, [Y.sub.1j] is jth output for first DMU, and
[X.sub.1i] is ith input for the first DMU (style).
It is clear that if the kth value of [[lambda].sub.j] equals zero,
calculating the efficiency of kth DMU means that the jth DMU is not a
reference for that DMU. The positive values show that the DMU is
efficient. Moreover, the sum of the values of [[lambda].sub.j] after
solving LPM is captured as DMU weight, which can be used for ranking.
3.4. Data
This study goes through the effect of different kinds of management
styles on mutual funds performance. The 20 styles are considered based
on the classification of the Bloomberg Database. Hence, the monthly data
of 17,686 US mutual funds for a fiveyear period, 2005-2010, through the
Database are collected to minimize the impact of survivorship bias.
4. Research inputs and outputs
Each DMU refers to a given management style category of mutual
funds. We tend to evaluate the managers' efficiency of these
categories. Hence, we use several criteria, comprising one measure of
return and one measure of stochastic dominance in three different forms
for the outputs [Y.sub.ij] to be taken into account. As well as both the
costs and risk criteria of mutual funds for the outputs [X.sub.ij] to be
considered. Due to the fact that inputs represent management activity
instead of costly raw materials and because the ultimate purpose of the
study is outputs produced by that activity, we use the output-oriented
approach for two approaches of CRS and VRS.
Since the role of inputs and outputs are critical in the DEA, they
are introduced as follows:
4.1. Inputs
Murthi et al. (1997) use the fund costs and fees, including
subscription fees, redemption fees, operational expenses, purchase and
sale costs and management fees, as one of the DEA inputs. They state
that all costs are integrated in the expense ratio, loads and turnover
indicators. Similarly, Daraio and Simar (2006) find that costs and fees
are made by expense ratio, loads and turnover ratio, so they consider
these costs as inputs. In this research these three indicators are used
as our model inputs to reflect the role of fund costs and fees in the
mutual fund performance.
Expense Ratio--the interactions between the performance of mutual
funds and fund costs and fees report in many studies. Addressing the
expense issue, the early work of Elton et al. (1993) and Carhart (1997)
find a high relationship between fund performance and expenses. They
explain that funds with high expense ratios have under performed. Mutual
funds have a good financial status benefit from the increased size of
their funds not by raising their costs and fees.
Turnover--the study of Friend and Blume (1970), in this thread show
that a positive relationship between fund turnover and fund performance
exists. The study of Daraio and Simar (2006) also shows that turnover
gives an indication of trading activity: funds with higher turnover
incur greater brokerage fees for affecting the trades.
Load Cost--Ippolito (1989) presents that load funds generally earn
sufficiently higher rates of return compared with no-load funds to pay
for the extra charges. Sirri and Tufano (1998) also find a relationship
between fund performance and load cost.
Beta Coefficient--the second kind of input indicator for our DEA
model is the Beta index. Although researchers are not unanimous in
employing a similar risk measure, all believe that the risk parameter is
one of the most important variables affecting the performance of funds.
There are many techniques to quantify risk employed in different
performance measures. However, most methods use standard deviation and
beta. Murthi et al. (1997) employ the standard deviation as the measure
of risk. Basso and Funari (2001) capture standard deviation and Beta as
DEA inputs. They justify the selection of two risk measures and explain
that the standard deviation of the returns is a proper risk measure for
the investors who only hold one risky asset, and the Beta coefficient is
for the investors who diversify their investments. Chang (2004) also
applies standard deviation and beta coefficient as inputs. Thus, we use
Beta coefficient as a risk measure because it quantifies a fund's
volatility relative to the market as well as the beta measure commonly
used for portfolio performance.
4.2. Outputs
Return--the most fundamental factor for mutual fund appraisal in
both the parametric and non-parametric methods is the return on mutual
fund, since all investors require a maximum return with minimum risk. To
measure the return of portfolio and assets, some methods employ mean
return, while others use excess return. Basso and Funari (2001) and
Chang (2004) apply the mean return and Murthi et al. (1997) use the
excess return as the output. Excess return is also used by Chen and Lin
(2006), and Hsu and Lin (2007) as the output of the DEA model. This
paper applies the mean return as one of the DEA model outputs as the
mean return leads to a decrease in the presence of negative values.
DARA, CARA, and IARA--so far, some papers use a stochastic
dominance indicator as output of the DEA model. Two concepts of the time
occurrence of the returns and the investors' preference structure
are reflected by the stochastic dominance index. This index is analyzed
by giving a higher rank to the funds not dominated by other funds.
To apply stochastic dominance, although there is no need for any
assumptions regarding the functional shape of the return distribution,
it employs any function that is able to characterize a cumulative
probability distribution. These stochastic dominance characteristics are
consistent with the features of nonparametric methods, especially the
DEA model. Basso and Funari (2001) are the first researchers who use a
stochastic dominance indicator as one of the outputs of DEA model. They
clearly state that a highly desirable property for a mutual fund is that
it is not dominated by other funds--only the non-dominated portfolios
can be considered efficient--which can easily happen in an analysis of
fund returns over a long period. A fund turns out to be dominated in
some years but not in others. Hence, a stochastic dominance indicator
can be defined by determining in how many periods of a fund is efficient
according to a given stochastic dominance criterion. Basso and Funari
(2003) use the DARA as output, and Chen and Lin (2006) and Lozano and
Gutierrez (2008) apply the DARA index in the DEA model.
Thus, we propose a stochastic dominance criterion to each fund
corresponding to investors' preferences (manager') and
investors' attitude (managers') towards risk. There are three
kinds of attitude towards risk:
--Risk averse: If an investor is presented with two kinds of
investments, and he prefers the investment with the lower risk, then he
is called risk averse (U" (0) <0).
--Risk Neutral: If an investor is only concerned with an
investment's return and overlooks risk, then he is called risk
neutral (U" (0) = 0).
--Risk Seeking: if an investor prefers to take big risks to raise
the potential return on investments, then he is called risk seeking
(U" (0) > 0).
U" is the second derivative of the utility function of wealth.
The utility functions are models that describe an investor's
attitude toward risk. Fig. 1 shows the relationship between
investor' wealth and utility from three different aspects.
[FIGURE 1 OMITTED]
It is the second derivative of the utility function of wealth. The
utility function is able to describe the effect of fluctuations in
wealth on investor's preferences. An investor exhibits three kinds
of absolute risk aversion as follows:
--DARA: If investor's demand for investing in risky assets
increases as his wealth rises.
--CARA: If investor's demand for investing in risky assets
keeps the same as his wealth changes.
--IARA: If investor's demand for investing in risky assets
decreases as his wealth rises.
Arrow (1971) and Pratt (1964) design a coefficient of absolute risk
aversion to show how investors behave. This coefficient of absolute risk
aversion is defined as:
A(w) = u"(w)/u'(w). (7)
The first derivative of A(w) at wealth is an index of how absolute
risk aversion behaves with changes in wealth. Now, absolute risk
aversion can be calculated with respect to A'(w) as:
--Increasing absolute risk aversion A'(w) > 0;
--Constant absolute risk aversion A'(w) = 0;
--Decreasing absolute risk aversion A'(w) < 0.
The functions of absolute risk aversion are unique. Thus, the first
and second derivatives of utility functions at wealth can be easily
computed. In the next stage, the coefficients of absolute risk aversion
with Arrow and Pratt's measure are obtained. Finally, by the first
derivative of the absolute risk aversion coefficient, three measures of
DARA, CARA and IARA are computed and presented in Table 1.
We use Net Asset Value (NAV) of mutual funds instead of the wealth
(w) variable. There are three different Stochastic Dominance measures
used. Thus, the DEA is run three times, and each time one of the
measures of absolute risk aversion is considered as the output of our
DEA model.
5. Empirical evidence
We test the DEA model for different classes of fund strategies
discussed in the prior sections on all data available for the US market.
We consider the annual returns of 17,555 funds. These funds belong to a
variety of fund strategies as reported in Table 2. The DEA evaluates the
managers' efficiency of each class. We use the mean return along
with three dominance relations of CARA, DARA, and IARA as the output of
funds, besides the four inputs--fund turnover, expense ratio, load cost,
and variance. Then 17 distinguished LPMs are run to compute the
efficiency scores of technical, management, and scale based on the
inputs and outputs. In order to compare the fund strategies performance
with the behavior of a market benchmark, we use three measures--Sharpe,
Treynor, and Jensen's alpha--to investigate the correlation between
each of the three efficiencies with the market benchmark indexes.
Table 2 reports the descriptive statistics for the 17 categories of
US funds strategies. It shows an average of net asset value for each
fund strategy in the range of 1 to 27.72 billion dollars. The second
column shows the five-year average of fund turnover, which represents
substantial trading activity in fund strategies. The next column
represents the 5-year mean and standard deviation of any strategy for
the two measures of Sharpe and Treynor, respectively. The last column
also shows the number of funds in each of the fund strategies.
Then we use the CRS model assuming constant returns to scale for
computing technical efficiency, and the VRS model assuming variable
returns to scale to compute management and scale efficiency. Considering
the two models, first the DEA model is computed in the DARA form with
the two outputs of DARA and return and four input variables of fund
turnover, expense ratio, load cost, and variance. Table 3 shows the
results of the DARA model, in which six fund strategies--Contrarian,
First Tier, Government & Agency, Growth & Income, Index Fund,
and Principal Preservation--have the highest technical, management, and
scale efficiency scores compared to the others. The average value of the
three efficiencies is 0.81, 0.921, and 0.874, which means that assuming
all other conditions are fixed, the three efficiencies of relative,
management, and scale have empty capacity equal to 0.19, 0.079, and
0.126, respectively.
One of the most important objectives of the DEA model is to
determine the reference fund strategies for inefficient strategies.
Thus, we identify the reference strategies based on the results of
software Deap2, as shown in Table 4.
Table 4 represents the reference strategies for each fund
separately (i.e., three strategies for Contrarian, Growth & Income,
and Index Fund, which, respectively, are a reference strategy for
Blend). These interpretations can be followed for other fund strategies,
meaning that each fund strategy manager must consider the performance of
its own preference strategies as a pattern to achieve a level of
efficiency in future. As shown in Table 4, the efficient strategies
(i.e., Contrarian and Government & Agency) are self-referenced.
Since the six strategies in our study are determined as reference,
the DEA model is able to rank each reference strategy. Using two
methods--"number of referencing" and "weighted number of
references"--we rank each reference strategy. To rank the
strategies based on the first method, the number of referencing is
computed for each efficient strategy after running the 17 LPMs. A
strategy will become a reference unit, if its efficiency score has the
highest frequency. To rank the strategies based on the second method,
the weighted average is calculated for each efficient strategy and their
ranking is done according to the weighted average of each efficient unit
when they are preferred (Table 5).
Then, the two methods of CRS and VRS are again used to compute the
DEA model in the CARA form while considering two outputs of CARA and
return and four input variables of fund turnover, expense ratio, load
cost, and variance. Table 6 shows that three strategies of Contrarian,
First Tier, and Government & Agency have the highest technical,
management, and scale efficiency scores compared to the others. The
average values of the three efficiencies are 0.463, 1, and 0.463,
respectively, which means that assuming all other conditions being
fixed, the two efficiencies of relative and scale, similarly, have
considerable empty capacity equal to 0.537.
To determine the preference strategies, the evidence of Table 6
shows that the six strategies are referred to each other. Moreover, the
ranking of preference of strategies based on the method of the number of
referencing shows that three fund strategies--Principal Preservation,
Contrarian, and Current Income--have the highest rankings, while the
ranking of the method of the weighted number of references detects that
three strategies --Principal Preservation, Current Income, and
Contrarian--have the highest rankings, respectively.
Similarly, the DEA model is computed in the IARA form while
considering the two outputs of IARA and return and four research input
variables. Table 7 shows the results of the model IARA, where similar to
the CARA model, three fund strategies--Contrarian, First Tier, and
Government & Agency--have the highest technical, management, and
scale efficiency scores compared to the others. The average values of
the three efficiencies are 0.286, 0.611, and 0.383, respectively, which
means that assuming all other conditions being fixed, three efficiencies
have considerable empty capacity equal to 0.714, 0.389, and 0.617, which
is worse than the CARA model.
To determine the preference strategies, Table 6 shows that ten
strategies are referenced for the surveying strategies. Moreover, the
ranking of preference strategies based on the method of the number of
referencing and the weighted number of references shows that three
strategies--Government & Agency, Contrarian, and First Tier--are
respectively ranked among the surveying strategies.
In order to compare the fund strategies performance with the
behavior of a market benchmark (Table 8), we make the correlation
between three measures of Sharpe, Treynor, and Jensen's alpha with
each of the three efficiencies computed by the DEA involving technical,
management, and scale efficiency. Table 9 represents the correlation
coefficients between three DEA models--CARA, DARA, and IARA--with three
conventional measures--Sharpe, Treynor, and Jensen's alpha. The
evidence shows that the average correlation coefficient of the DARA for
technical efficiency is 0.54, while the average of this coefficient for
the CARA and IARA models is 0.61 and 0.57, respectively. For management
efficiency, the average of the correlation coefficients for DARA, CARA,
and IARA are 0.48, 0.62, and 0.56, respectively. Similarly the results
of the correlation coefficients of IARA are almost similar to DARA, in
which the three models of DARA, CARA, and IARA have values equal to
0.52, 0.61, and 0.56, respectively. Moreover, the values of the Sharpe,
Treynor, and Jensen's alpha measures have been normalized by
dividing the values by the highest value.
6. Conclusions
We propose a replacement approach to evaluate the performance of
mutual fund managers. We combine the DEA model and stochastic dominance
criteria and propose two new measures in the form of CARA and IARA
dominance relations along with the DARA model previously suggested by
Basso and Funari (2001) to evaluate the technical, management, and scale
efficiency of 17,555 US funds. The three DEA models being proposed in
the form of CARA, DARA, and IARA are used to evaluate the managers'
efficiency of management styles. Unlike prior studies that compute the
technical efficiency on the funds, we extend the analysis on two other
efficiencies, namely, management and scale. Moreover, we compute the
efficiency scores and determine the preference strategy for each fund
strategy.
The evidence shows that the scores of technical, management, and
scale efficiency are respectively 0.81, 0.921, and 0.874 for the DARA,
while the efficiency scores of the CARA and IARA are negligible.
Moreover, each strategy in any model is ranked based on two methods--the
number of referencing and the weighted number of references--so that the
managers of inefficient strategy must pattern the managers' ability
of reference (efficient) strategies to improve its efficiency on the
fund market in future.
Finally, since the average correlation coefficient between the IARA
model and the three measures--Sharpe, Treynor, and Jensen's
alpha--is higher than the two other models, it is able to provide a
better explanation of the DEA model than the others.
As a proposition for future studies, the three DEA models in the
DARA, CARA, and IARA form can be studied in terms of the cross DEA model
to improve the results.
doi: 10.3846/16111699.2011.651625
Caption: Fig. 1: The utility function of investors
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Mohammad Reza Tavakoli Baghdadabad (1), Farid Habibi Tanha (2),
Noreha Halid (3)
Graduate School of Business, University Kebangsaan Malaysia, JALAN
REKO, Universiti Kebangsaan Malaysia, Bangi, Selangor, Malaysia
E-mails: (1) Mr_tavakkoli@yahoo.com (corresponding author); (2)
farid_ht@yahoo.com; (3) noreha@ukm.my
Received 10 August 2011; accepted 17 December 2011
Mohammad Reza TAVAKOLI BAGHDADABAD was born in Iran in 1976. His
studies' background is in the field of Management and Finance.
Farid Habibi TANHA was born in Iran in 1983. He is DBA student of
the Graduate School of Business, National University of Malaysia-UKM
since 2010.
Noreha HALID was born in Malaysia in 1963. She is Associate
Professor of Graduate School of Business, National University of
Malaysia, UKM since 2009.
Table 1. The relation of DARA, CARA and IARA
Absolute U (w) U (w) U (w)
Risk
Aversion
DARA log(w) 1/c ln(10) 1/[c.sup.2]
ln(10)
CARA 1 - [alpha] -[[alpha].sup.2]
[e.sup.-[alpha]w] [e.sup.-[alpha]w] [e.sup.-[alpha]w]
IARA c - 1-2[alpha]w -2[alpha]
[alpha][w.sup.2]
Absolute A(c) = -U'(w)/U"(w) A (w)
Risk
Aversion
DARA 1/w -1/w
CARA [alpha] 0
IARA 2[alpha]/[1 - 2[alpha]w] 2[alpha]/[(1 - 2
[alpha]w).sup.2]
Table 2. The descriptive statistics of funds strategy
Fund Strategy NAV Turnover Sharp Index
Mean Standard
Deviation
Blend 18.05 89.9 0.13 0.12
Contrarian 12.57 367.4 -0.41 0.26
Current Income 10.2 86.81 0.43 0.42
Emerging Market 20.72 77.17 0.39 0.09
Equity Income 15.05 56.39 0.16 0.14
First Tier 1.03 1.29 -8.58 99.2
Geographically Focused 17.54 74.29 0.13 0.16
Government & Agency 1 0.08 -7.33 47.9
Growth 20.3 97.15 0.17 0.12
Growth & Income 21.97 55.5 0.1 0.11
Index Fund 27.72 127.7 0.12 0.13
Long-Short 12.72 259.8 -0.1 0.21
Market Neutral 12.01 326.4 -0.17 0.48
Principal Preservation 14.6 18.31 1.31 3.49
Sector Fund 21.39 114.8 0.19 0.2
Total Return 11.15 196 0.68 0.4
Value 17.83 62.71 0.12 0.13
Fund Strategy Treynor Index N
Mean Standard
Deviation
Blend 0.05 0.05 1312
Contrarian 0.08 0.09 82
Current Income 0.16 0.73 3454
Emerging Market 0.17 0.05 396
Equity Income 0.10 0.6 294
First Tier 57.8 83.9 534
Geographically Focused 0.07 0.38 2304
Government & Agency 6.76 5.42 332
Growth 0.07 0.14 2457
Growth & Income 0.04 0.02 195
Index Fund 0.05 0.05 630
Long-Short -0.08 0.17 129
Market Neutral -3.05 3.39 83
Principal Preservation -1.12 1.97 21
Sector Fund 0.1 0.09 1320
Total Return 0.12 0.21 1805
Value 0.06 0.14 2207
Table 3. The results of different efficiencies with DARA
Fund Strategy Technical Management Scale Scale
Efficiency Efficiency Efficiency Type
Blend 0.83 0.91 0.91 Drs
Contrarian 1 1 1 --
Current Income 0.84 0.84 0.99 Irs
Emerging Market 0.73 0.98 0.74 Drs
Equity Income 0.78 0.88 0.89 Drs
First Tier 1 1 1 --
Geographically Focused 0.70 0.91 0.77 Drs
Government & Agency 1 1 1 --
Growth 0.77 0.93 0.82 Drs
Growth & Income 1 1 1 --
Index Fund 1 1 1 --
Long-Short 0.46 0.81 0.57 Drs
Market Neutral 0.44 0.79 0.55 Drs
Principal Preservation 1 1 1 --
Sector Fund 0.67 0.93 0.72 Drs
Total Return 0.73 0.73 0.99 Drs
Value 0.79 0.92 0.85 Drs
Mean 0.81 0.92 0.87
Table 4. The results of determining the preference strategies
Fund Strategy Strategy 1 Strategy 2
Blend Contrarian Growth & Income
Contrarian Contrarian --
Current Income Contrarian First Tier
Emerging Market Contrarian Growth & Income
Equity Income Growth & Income Index Fund
First Tier First Tier --
Geographically Focused Index Fund Growth & Income
Government & Agency Government & Agency --
Growth Growth & Income Index Fund
Growth & Income Growth & Income --
Index Fund Index Fund --
Long-Short Index Fund Contrarian
Market Neutral Contrarian Index Fund
Principal Preservation Principal Preservation --
Sector Fund Index Fund Growth & Income
Total Return Index Fund Contrarian
Value Index Fund Growth &
Income
Fund Strategy Strategy 3 Strategy 4
Blend Index Fund --
Contrarian -- --
Current Income Growth & Income Index Fund
Emerging Market Index Fund --
Equity Income -- --
First Tier -- --
Geographically Focused -- --
Government & Agency -- --
Growth -- --
Growth & Income -- --
Index Fund -- --
Long-Short -- --
Market Neutral -- --
Principal Preservation -- --
Sector Fund -- --
Total Return -- --
Value -- --
Table 5. The ranking of funds strategy
Fund Strategy Index Fund Growth & Income Contrarian
Number of 12 9 7
Referencing
Weighted Number 7.06 6.13 1.58
of References
Fund Strategy First Tier Government Principal
& Agency Preservation
Number of 2 1 1
Referencing
Weighted Number 1.12 1 1
of References
Table 6. The results of DEA with CARA
Efficiency
Fund
Strategy Technical Management Scale
Blend 0.36 1 0.36
Contrarian 1 1 1
Current Income 0.44 1 0.44
Emerging Market 0.26 1 0.26
Equity Income 0.33 1 0.33
First Tier 1 1 1
Geographically 0.29 1 0.29
Focused
Government & 1 1 1
Agency
Growth 0.32 1 0.32
Growth & Income 0.39 1 0.39
Index Fund 0.49 1 0.49
Long-Short 0.23 1 0.23
Market Neutral 0.26 1 0.26
Principal 0.36 1 0.36
Preservation
Sector Fund 0.29 1 0.29
Total Return 0.47 1 0.47
Value 0.30 1 0.30
Mean 0.46 1 0.46
Scale Type Preference Strategy
Fund
Strategy 1 2
Blend drs Contrarian Principal
Preservation
Contrarian Constant Contrarian --
Current Income drs Current Income --
Emerging Market drs Emerging Market --
Equity Income drs Principal Current Income
Preservation
First Tier Constant First Tier --
Geographically drs Emerging Market Principal
Focused Preservation
Government & Constant Government & --
Agency Agency
Growth drs Contrarian Principal
Preservation
Growth & Income drs Current Income --
Index Fund drs Index Fund --
Long-Short drs Long-Short --
Market Neutral drs Geographically Contrarian
Focused
Principal drs Principal --
Preservation Preservation
Sector Fund drs -- Contrarian
Total Return drs --
Value drs Long-Short Equity Income
Mean
Preference Strategy
Fund
Strategy 3 4
Blend Geographically Index Fund
Focused
Contrarian -- --
Current Income -- --
Emerging Market -- --
Equity Income -- --
First Tier -- --
Geographically -- Contrarian
Focused
Government & -- --
Agency
Growth Geographically --
Focused
Growth & Income Principal Geographically
Preservation Focused
Index Fund -- --
Long-Short -- --
Market Neutral -- --
Principal -- --
Preservation
Sector Fund -- --
Total Return -- --
Value Principal --
Mean Preservation
Priority of References
Fund
Strategy number of weighted
referencing number
Blend -- --
Contrarian 6 1.54
Current Income 3 1.99
Emerging Market 2 1.40
Equity Income 1 0.38
First Tier 1 1
Geographically 4 1.19
Focused
Government & 1 1
Agency
Growth -- --
Growth & Income -- --
Index Fund 2 1.26
Long-Short 2 1.12
Market Neutral -- --
Principal 7 3.81
Preservation
Sector Fund -- --
Total Return -- --
Value -- --
Mean
Table 7. The result of DEA model 3 using IARA
Fund Strategy Efficiency Scale Type
Technical Management Scale
Blend 0.116 0.51 0.227 drs
Contrarian 1 1 1 Constant
Current Income 0.184 0.54 0.34 drs
Emerging Market 0.102 0.518 0.197 drs
Equity Income 0.111 0.52 0.213 drs
First Tier 1 1 1 Constant
Geographically 0.096 0.51 0.188 drs
Focused
Government & 1 1 1 Constant
Agency
Growth 0.106 0.51 0.207 drs
Growth & Income 0.126 0.51 0.248 drs
Index Fund 0.157 0.51 0.309 drs
Long-Short 0.196 0.604 0.325 drs
Market Neutral 0.204 0.597 0.342 drs
Principal 0.121 0.52 0.232 drs
Preservation
Sector Fund 0.094 0.51 0.185 drs
Total Return 0.157 0.52 0.302 drs
Value 0.097 0.51 0.19 drs
Mean 0.286 0.611 0.383 --
Fund Strategy Preference Strategy
1 2 3 4
Blend Government & Agency -- -- --
Contrarian Contrarian -- -- --
Current Income Contrarian Government & -- --
Agency
Emerging Market Government & Agency Contrarian -- --
Equity Income Government & Agency -- -- --
First Tier First Tier -- -- --
Geographically Government & Agency -- -- --
Focused
Government & Government & Agency -- -- --
Agency
Growth Government & Agency -- -- --
Growth & Income Government & Agency -- -- --
Index Fund Government & Agency -- -- --
Long-Short Government & Agency Contrarian -- --
Market Neutral Government & Agency Contrarian -- --
Principal Government & Agency -- -- --
Preservation
Sector Fund Government & Agency -- -- --
Total Return Government & Agency -- -- --
Value Government & Agency -- -- --
Mean -- -- -- --
Fund Strategy Priority of References
number of weighted
referencing number
Blend -- --
Contrarian 5 1.63
Current Income -- --
Emerging Market -- --
Equity Income -- --
First Tier 1 1
Geographically -- --
Focused
Government & 15 14.37
Agency
Growth -- --
Growth & Income -- --
Index Fund -- --
Long-Short -- --
Market Neutral -- --
Principal -- --
Preservation
Sector Fund -- --
Total Return -- --
Value -- --
Mean -- --
Table 8. Conventional measures versus DARA, CARA, and IARA
Fund Strategy Measures
Sharpe Treynor Jensen's Alpha
Blend 0.09 0.001 -0.18
Contrarian -0.31 0.001 -3.50
Current Income 0.32 0.003 0.19
Emerging Market 0.30 0.003 -0.50
Equity Income 0.12 0.002 0.63
First Tier -6.51 1 0.03
Geographically Focused 0.10 0.001 0.04
Government & Agency -5.55 0.117 0.28
Growth 0.13 0.001 -0.01
Growth & Income 0.08 0.001 -0.12
Index Fund 0.09 0.001 -0.42
Long-Short -0.07 -0.001 -1.62
Market Neutral -0.13 -0.053 -1.08
Principal Preservation 1 -0.019 -0.03
Sector Fund 0.14 0.002 1
Total Return 0.51 0.002 0.09
Value 0.09 0.001 0.09
Fund Strategy Technical Efficiency
DARA CARA IARA
Blend 0.82 0.36 0.11
Contrarian 1 1 1
Current Income 0.8 0.44 0.18
Emerging Market 0.73 0.26 0.10
Equity Income 0.78 0.33 0.11
First Tier 1 1 1
Geographically Focused 0.70 0.29 0.09
Government & Agency 1 1 1
Growth 0.77 0.32 0.10
Growth & Income 1 0.39 0.12
Index Fund 1 0.49 0.15
Long-Short 0.46 0.23 0.19
Market Neutral 0.44 0.26 0.20
Principal Preservation 1 0.36 0.12
Sector Fund 0.67 0.29 0.09
Total Return 0.73 0.47 0.15
Value 0.79 0.30 0.09
Table 9. The correlation coefficients
DARA
DARA SHARPE TREYNOR JENSEN
Technical DARA 1
Efficiency SHARPE 0.14 1
TREYNOR 0.64 0.35 1
JENSEN 0.37 -0.04 0.33 1
Mean 0.54 0.36 0.58 0.41
Management DARA 1
Efficiency SHARPE 0.17 1
TREYNOR 0.52 0.35 1
JENSEN 0.21 -0.04 0.33 1
Mean 0.48 0.37 0.55 0.37
Scale DARA 1
Efficiency SHARPE 0.07 1
TREYNOR 0.59 0.35 1
JENSEN 0.42 -0.04 0.33 1
Mean 0.52 0.34 0.57 0.43
CARA
CARA SHARPE TREYNOR JENSEN
Technical 1
Efficiency 0.65 1
0.57 0.35 1
0.21 -0.04 0.33 1
Mean 0.61 0.49 0.56 0.37
Management 1
Efficiency 0.66 1
0.58 0.35 1
0.23 -0.04 0.33 1
Mean 0.62 0.44 0.56 0.43
Scale 1
Efficiency 0.65 1
0.57 0.35 1
0.21 -0.04 0.33 1
Mean 0.61 0.49 0.56 0.37
IARA
IARA SHARPE TREYNOR JENSEN
Technical 1
Efficiency 0.71 1
0.48 0.35 1
0.11 -0.04 0.33 1
Mean 0.57 0.5 0.54 0.35
Management 1
Efficiency 0.72 1
0.44 0.35 1
0.07 -0.04 0.33 1
Mean 0.56 0.51 0.53 0.34
Scale 1
Efficiency 0.71 1
0.45 0.35 1
0.08 -0.04 0.33 1
Mean 0.56 0.5 0.53 0.34
