1.
The new assessment of US mutual fund returns through a
multiscaling approach
Francis Ina* and Sangbae Kimb
a
Department of Accounting and Finance, Monash University, Clayton, Victoria, 3168,
Australia
b
School of Business Administration, Kyungpook National University, Puk-ku, Daegu,
702-701, Republic of Korea
Abstract
This paper uses for the first time the multiscaling approach to evaluate the performance of
US mutual funds, namely Institutional, Active and Index funds, adopting a performance
measure (the Jensen’s alpha). Empirical results show that none of the funds are dominant
over all time-scales, while depending on the specific time scale, indicating that evaluating
the performance of the mutual funds depends on the investment horizons. In terms of the
premiums on the SMB, HML, and momentum factor, our results generally support that the
three risk factors do not prove to be a strong indicator of US mutual funds at the short
scales, while at the long scales, turn to be a strong indicator.
Keywords: Performance Measure; Mutual Fund; Multiscaling; Jensen’s alpha.
This paper (first draft) is prepared for seminar presentation on February 10 th 2006
“Workshop for Funds Management Research: Emerging Issues” by The Melbourne
Centre for Financial Studies
*
Corresponding author. Tel.: +61 3 9905 1561; fax: +61 3 9905 5475
E-mail address: Francis.In@BusEco.monash.edu.au
2.
1. Introduction
Since the advent of the Capital Asset Pricing Model (CAPM) by Sharpe (1964) and
Lintner (1965), a crucial extension of the model has been made into the issue of the
assessment of risk-adjusted performance of portfolios to evaluate the ability of portfolio
managers to realize returns in excess of a benchmark portfolio with similar risk. Along
with this aspect, four measures (the Sharpe (1964) ratio, Treynor and Black (1973)
appraisal ratio, Treynor (1966) ratio, and Jensen’s (1968) alpha) are proposed and
adopted by the academics and the practitioners. However, the standard measure of fund
performance, ‘‘alpha,’’ is typically not estimated with much precision. We would argue
that another important source of information about fund performance has been
overlooked up to now, both in traditional studies and in the more recent analyses. The
neglected information is the holding period1 effect of fund. Ignoring this effect might
cause a biased performance measure if the fund manager uses a time horizon shorter
than the ‘true’ time horizon, defined as the relevant time horizon implicit in the
decision-making process of investors. For example, consider an investment company
with a large number of investors and money managers. Clearly, the investors and the
money managers make decisions over different time scales. Suppose, for simplicity, that
the investment horizon of an investor is one year and that the investment company
reviews the performance of the money manager every quarter, using the Sharpe ratio.
1
The holding period that is relevant for portfolio allocation is the length of time investors hold any stocks
or bonds, no matter how many changes are made among the individual issues in their portfolio (Siegel,
1998, p29). In other words, the investment horizon sensitivity is very important to evaluate the
performance of one or more portfolios. An investor might not be interested in short-term performance of
portfolios at all. Institutional investors like pension funds have a very long investment horizon. Therefore,
it is interesting to examine the long-term performance of the investments when the investment horizon
increases.
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3.
The money manager will therefore focus on the three-month performance of a portfolio,
while the investor will concentrate on the one-year performance. Thus, for this investor,
the money manager may not provide the best service. To provide the best service for
diverse investors, the performance measure needs to be constructed over different
investment horizons.
While it is recognized that the investment horizon is important, the previous literatures
are silent empirically on this research topic. One exception is Hodges et al. (1997). They
examine the multiperiod Sharpe ratio using randomized historical data from 1926 to
1993. Siegel (1999) summarizes three problems in Hodges et al. (1997): a key feature of
long-term stock data – mean reversion of equity returns, the real return should be
considered, not nominal return, an assumption about the properties of returns of stocks
and bonds. To overcome this problem, Kim and In (2005a) examine the multiperiod
Sharpe ratio using wavelet analysis. The advantages of wavelet analysis in examining
the multiperiod Sharpe ratio are two-fold: the first and more serious problem concerns
the assumption of the distribution. The previous studies show that the real return on the
stock has a property of mean reversion, which indicates that the stock return is not
independently and identically distributed. The second problem concerns the
construction of an n-period return and variance. The construction of a long-period return
leaves us a handful observation. Therefore, this may result in a biased estimator.
Our paper aims to contribute to the literature on the study of the performance measures
of mutual fund returns using multiscaling approach: wavelet analysis. To the best of our
knowledge, no previous study has investigated the multihorizon performance measure
of mutual fund using multiscaling approach. Adopting wavelet analysis does not require
3
4.
any assumption on the distribution of returns, because wavelet analysis is a
nonparametric estimation and decomposes the unconditional variance into different time
scales.
The main advantage of wavelet analysis is the ability to decompose the data into several
time scales.2 Consider the large number of investors who trade in the security markets
and make decisions over different time scales. One can visualize investors operating
minute-by-minute, hour-by-hour, day-by-day, month-by-month, or year-by-year. In fact,
due to the different decision-making time scales among investors, the true dynamic
structure of the relationship between variables will vary over different time scales
associated with those different horizons. Economists and financial analysts have long
recognized the idea of several time periods in decision making, while economic and
financial analyses have been restricted to at most two time scales (the short-run and the
long-run), due to the lack of analytical tools to decompose data into more than two time
scales (In and Kim, 2006).
Several applications of wavelet analysis to economics and finance have been
documented in recent literature. To the best of our knowledge, applications in these
fields include examination of foreign exchange data using waveform dictionaries
(Ramsey and Zhang, 1997), decomposition of economic relationships of expenditure
and income (Ramsey and Lampart, 1998), the multiscale Sharpe ratio (Kim and In,
2005a), the multiscale relationship between stock returns and inflation (Kim and In,
2
The key distinctive features of wavelet analysis are that wavelets possess not only the ability to perform
nonparametric estimations of highly complex structures without knowledge of the underlying functional
form, but also are able to accurately locate discontinuity and high frequency bursts in dynamic systems.
In short, the major aspects of wavelet analysis are the ability to handle nonstationary data, localization in
time, and the resolution of the signal in terms of the time scale of analysis. Among these aspects, the most
important property of wavelet analysis is decomposition by time scale (Ramsey, 1999).
4
5.
2005b), systematic risk in a capital asset pricing model (Gençay et al., 2003 and 2005)
and the multiscale hedge ratio (In and Kim, 2006) among others.
Our results show that one fund are not dominant over all time scales, while depending
on the specific time scale, indicating that evaluating the performance of the mutual
funds depends on the investment horizons. In terms of the premiums on the SMB (small
minus big, a factor that is related to size), HML (high minus low, a factor related to
book-to-market equity), and Mom (momentum factor), our empirical results show that
the three risk factors do not prove to be a strong indicator of mutual funds at the short
scales, while at the long scales, turn to be a strong indicator.
The remainder of the paper is organized as follows. Section 2 discusses the
performance measure models. Section 3 describes the econometric methodology for the
wavelet analysis. The data and the empirical results are discussed in Section 4. Section 5
presents the summary and concluding remarks.
2. Performance measure models
This section briefly describes our empirical models. Currently, most performance
studies of multi-index asst pricing models use Jensen’ (1968) alpha. Its interpretation as
the risk-adjusted abnormal return of a portfolio makes it flexible enough to be used in
most asset pricing specifications. Kothari and Warner (2001) consider only this measure
for multi-index asset pricing models in their empirical comparison of mutual fund
performance measures.
5
6.
There are various versions of Jensen’s alpha3, corresponding to different asset pricing
models. For comparison purposes, our study of US mutual funds performance starts
with the CAPM. The basic multi-factor specifications are the Fama and French (1993)
three-factor model, and the Carhart (1997) model because they are not dominated by
any other model in the mutual funds performance literature. Following Bams and Otten
(2002), the Sharpe’s (1992) asset class model is not considered in our study because it is
an asset allocation model and not an asset evaluation model.
2.1 The capital asset pricing model
We start with a single index model based on the classical CAPM, which is developed by
Sharpe (1964) and Lintner (1965). In this framework, the Jensen’s alpha is estimated as
follows:
R pt − R ft = α p + β p ( RMt − R ft ) + ε pt (1)
where R pt is the return of fund p, R ft is the risk-free return, RMt is the return of market
portfolio, M, ε pt is the error term in calendar month t. In addition, α p and β p are the
Jensen’s alpha and the beta of the portfolio with respect to market portfolio,
respectively. If market portfolio M is efficient, then the true alpha of every security and
3
Jensen’s alpha is defined simply as the constant of regression analysis of CAPM model. Consequently,
its properties can be deduced by standard econometric techniques and are well known. Fabozzi and
Francis (1977) consider Jensen’s alpha in a log-linear CAPM model, but conclude that the change in
model specification did not change the estimate of α p significantly. Connor and Korajczyk (1986)
ˆ
examine the econometric properties of Jensen’s Alpha when the underlying model is a general APT, while
Chen, Copeland, and Mayers (1987) report significant differences between Jensen’s Alpha from the
CAPM and one calculated from an APT that includes firm size among its factors.
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7.
every portfolio will be zero, although an estimated alpha may be different from zero
because of estimation error. However, alpha can be calculated and be given a precise
interpretation in terms of portfolio optimization even if the benchmark is not efficient.
If α > 0 , then the investor can increase his expected utility by investing at least a small
amount in the fund. Of course, if α < 0 , then he can achieve the same effect by short-
selling the fund, if this is possible.4
2.2 The three-factor model of Fama and French (1993)
One extension for calculating the Jensen’s alpha is to use the three-factor model,
proposed by Fama and French (1993). This model has been well known for its
explanatory power of mutual fund returns (Do et al., 2005). The Fama and French three
factor model is estimated from an expected form of the CAPM. It utilizes the size and
book-to-market ratio of the firms into account. In this framework, we estimated the
Jensen’s alpha as follows:
R pt − R ft = α p + β p ( RMt − R ft ) + β p SMBt + β p HMLt + ε pt
M SMB HML
(2)
where SMBt and HMLt is the return on a portfolio of small stocks minus the return on
a portfolio of large stocks , the return on a portfolio of stocks with high book-to-market
ratios minus the return on a portfolio of stocks with low book-to-market ratios in
4
Nielsen and Vassalou (2004) show that Jensen’s alpha is proportional to the first derivative of the overall
Sharpe ratio with respect to the proportion invested in the active fund.
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8.
calendar month t. These factors are used to isolate the firm-specific components of
returns5.
2.3 The four-factor model of Carhart (1997)
The final model to estimate the Jensen’s alpha is the four-factor model of Carhart
(1997), which can be considered as an extended version of the Fama and French (1993)
three-factor model. It takes into account not only the size and book-to-market ratio, but
also an additional factor for the momentum effect6. The momentum factor has been
incorporated by Carhart (1997) because the three-factor model is lack of an ability to
explain cross-sectional variation in momentum-sorted portfolio returns (Fama and
French, 1996).
R pt − R ft = α p + β p ( RMt − R ft ) + β p SMBt + β p HMLt + β p Momt + ε pt
M SMB HML Mom
(3)
where Momt is the average return on the two high prior return portfolios minus the
average return on the two low prior return portfolios in calendar month t. According to
Carhart (1997), the four-factor model can be interpreted as a performance attribution
model, which the coefficients and premia on the factor-mimicking portfolios indicate
the proportion of mean return attributable to four elementary strategies.
5
Fama and French (1993, 1996) assert that the high return of value strategies is compensation for risk.
They suggest a three-factor (the market, SMB and HML) model supposed to capture this risk. Fama and
French (1993) argue that SMB and HML proxy for financial distress and they are state variables in an
intertemporal asset pricing model.
6
Grinblatt et al. (1995) define the momentum effect as buying stocks that were past winners and selling
past losers.
8
9.
3. Multiscaling approach: Wavelet analysis
A major innovation of this paper is the introduction of a new approach into studying the
performance measurement of a portfolio, as it allows us to investigate the multiperiod
performance measurement. The investors construct their portfolio all with different time
scales when they come to making an investment. Wavelet analysis is a natural tool used
to investigate our purpose, as it enables us to decompose the data on a scale-by-scale
basis. In this section, we summarize the discrete wavelet transform (DWT). The discrete
wavelet transform (DWT) is a kind of discretization of continuous wavelet transform.
Basic wavelets are characterized into father and mother wavelets, φ(t) and ψ(t),
respectively. These wavelets are functions of time only. A father wavelet (scaling
function) represents the smooth baseline trend, while mother wavelets (wavelet
function) are used to describe all deviations from trends. Consider a time series, f(t),
which we want to decompose into various wavelet scales. Given the father wavelet φ
such that its dilates and translates constitute orthonormal bases for all the Vj subspaces
that are scaled versions of the subspace V0 to which φ belongs, we can form a
Multiresolution Analysis (MRA) for a given time series (See Burrus et al., 1998 for
details).
With DWT, we are basically constructing a map from the signal domain to the wavelet
coefficients domain. In other words, we apply the transform w = Wf. The important
features of time series can better be captured by defining a slightly different set of
functions ψ (t ) , mother wavelets, which span the differences between two adjacent
spaces. Combining the orthogonality, we can describe L2 as follows:
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10.
L2 = V0 ⊕ W1 ⊕ W2 ⊕ W3 ⊕ (4)
where ⊕ denotes the orthogonal sum. In equation (4), the relationship of V0 to the
wavelet spaces can be described as V0 = W−∞ ⊕ ⊕ W−1 . This relationship shows that
the key idea of MRA consists in studying a time series by examining its increasingly
coarser approximations as more and more details are canceled from the data (Abry et
al., 1998). Based on this relationship, the mother wavelet ψ (t ) has the following form7:
j
−
2 ψ (2 − j t
−j
2ψ
t −2jk
ψ j ,k (t ) = 2 − k) = 2 (5)
2j
According to equation (5), any time series f (t ) ∈ L2 could be written as a series
expansion in terms of the scaling function and wavelets.
∞ ∞ ∞
f (t ) = ∑
k = −∞
s k φ k (t ) + ∑ ∑d
j =0 k = −∞
j , kψ j ,k (t ) (6)
As can be seen in equation (6), the DWT algorithm has an ability to produce the wavelet
coefficients for fine (coarse) scales, thus capturing high (low) frequency information.
7
Intuitively, a small j or a low resolution level can capture smooth components of the signal, while a large
j or a high resolution level can capture variable components of the signal (Lee and Hong, 2001).
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11.
Therefore, a series of smoothed data, captured by s k , and a series of details ( d j ,k ) not
previously accounted for, which give information at finer resolution levels, are obtained.
Our analysis adopts the MODWT instead of DWT. It provides basically all functions of
the DWT, such as MRA decomposition8 and analysis of variance.
Given the three models (equations (1) to (3)) and these wavelet coefficients at each
scale, the Jensen’s alpha at various time scales can be estimated as follows:
R pt (λ j ) − R ft (λ j ) = α p (λ j ) + β p (λ j )( RMt (λ j ) − R ft (λ j )) + ε pt (λ j ) (7)
R pt (λ j ) − R ft (λ j ) = α p (λ j ) + β p (λ j )( RMt (λ j ) − R ft (λ j ))
M
+ β p (λ j ) SMBt (λ j ) + β p (λ j ) HMLt (λ j ) + ε pt (λ j ) (8)
SMB HML
R pt (λ j ) − R ft (λ j ) = α p (λ j ) + β p (λ j )( RMt (λ j ) − R ft (λ j ))
M
+ β p (λ j ) SMBt (λ j ) + β p (λ j ) HMLt (λ j ) + β p (λ j ) Momt (λ j ) + ε pt (λ j )
SMB HML Mom
(9)
where R pt (λ j ) is the return on portfolio i in calendar month t at scale λj; RMt (λ j ) is the
CRSP value-weighted market index return in calendar month t at scale λj; R ft (λ j ) is
the risk free return (one-month Treasury bill) in calendar month t at scale λj; SMB(λj) is
the return on a portfolio of small stock minus the return on a portfolio large stocks at
scale λj, HML(λj) is the return on a portfolio of stocks with high book-to-market ratios
minus the return on a portfolio of stocks with low book-to-market ratios at scale λj,
Mom(λj) is the average return on the two high prior return portfolios minus the average
8
Note that this version of MRA provides an important feature, which is not available to the original
DWT. For more detail, see Percival and Walden (2000) and In and Kim (2006)
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12.
return on the two low prior return portfolios in calendar time t at scale λj, intercept
α p (λ j ) is the Jensen’s alpha of portfolio p at scale λj, β p (λ j ) is the beta of the
M
portfolio p with respect to market portfolio, β p (λ j ) is the assigned loadings on the
SMB
market size (SMB) at scale λj, β p (λ j ) is the assigned loadings on the value factor
HML
(HML) at scale λj,, and β p (λ j ) is the assigned loadings on the momentum factor at
Mom
scale λj; respectively. In this specification, α p (λ j ) indicates the wavelet multiscale
Jensen’s alpha of the portfolio, which can be varying depending on the wavelet scales
(i.e., investment horizons).
4. Data and empirical results
We use monthly nominal mutual fund returns (index fund, institutional, and active
funds) for the US in the period January 1991 to December 2002. Data were collected
from CRSP. To construct the returns of each fund, we simply use the equally weighted
average returns. More specifically, index fund returns are calculated by simply
averaging 12 index fund returns. Similarly, institutional fund and active fund returns are
calculated from 35 institutional funds and 346 active funds, respectively. For the the
CRSP value-weighted market index return, the risk free return (one-month Treasury
bill), SMB, and HML, Momentum factor (Mom) for the US in the period January 1991
to December 2002, obtained from the Kenneth French homepage.
Institutional funds are defined as a mutual fund that targets pension funds, endowments,
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and other high net worth entities and individuals. Institutional funds usually have lower
operating costs and higher minimum investments than retail funds. The main objective
of institutional funds is to reduce risk by investing in hundreds of different securities.
The objective of active funds is to outperform the market average by actively seeking
out stocks that will provide superior total return. In contrast, index funds are a form of
passive investment. Index funds are a mutual fund whose portfolio matches that of a
market index such as the S&P 500 Index. Therefore, their performance mirrors the
market as a whole.
Table 1 presents several summary statistics for the monthly data of three mutual fund
(Institutional, Active and Index funds) returns and four factor mimicking portfolios
(MKT, SMB, HML and Mom). As shown in Panel A of Table 1, all sample means
ranges from 0.201 (SMB) to 1.106 (Mom). Among three fund returns, Index fund has
the highest mean return and the highest standard deviation, implying that the high risk
has been compensated by the high return.
Among seven variables, first-order autocorrelation of monthly data ranges from –0.086
(Index) to 0.079 (HML), implying that the HML factor is more persistent than three
mutual funds and other factor mimicking portfolios. The Ljung-Box statistics indicate
the persistence of linear dependency of each set of data and the Ljung-Box statistics for
the squared data show strong evidence of non-linear dependency in all data except
Institutional and Active funds. The measures for skewness and kurtosis are also reported
to check whether monthly data are normally distributed. These statistics indicate that all
data are not normally distributed.
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We report the unconditional contemporaneous correlation coefficients between three
mutual funds and four factor-mimicking portfolios – Institutional, Active and Index
funds’ returns, MKT, SMB, HML and Mom – in Panel B of Table 1. The striking feature
is that the correlation with the HML is very low, while the correlations between the
MKT and three fund returns are very high.
The purpose of this paper is to examine the performance of the mutual funds
multihorizontally. To do so, we use a multihorizon performance measure, namely
Jensen’s alpha, using the wavelet multiscaling approach. For comparison reason, three
different specification has been adopted: the CAPM, Fama and French three-factor
model, and four-factor model by Carhart (1997). Considering the balance between the
sample size and the length of the wavelet filter, we settle with the Daubechies extremal
phase wavelet filter of length 4 (D(4)), while we decompose our data up to scale 5.
Since we use monthly data, scale 1 represents 2-4 month period dynamics. Equivalently,
scale 2, 3, 4, and 5 correspond to 4-8, 8-16, 16-32, and 32-64 month period dynamics,
respectively.
We examine the variances of three mutual returns against various time scales. An
important characteristic of the wavelet transform is its ability to decompose (analyze)
the variance of the stochastic process. Figure 1 illustrates the wavelet variance, taken by
logarithm, of three series against the wavelet scales. The variances of three mutual fund
returns decrease as the wavelet scale increases. Note that the variance-versus-wavelet
scale curves show a broad peak at the lowest scale (scale 1) in all mutual funds. This
result implies that an investor with a short investment horizon has to respond to every
fluctuation in the realized returns, while for an investor with a much longer horizon, the
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long-run risk is significantly less (Kim and In, 2005a). Interestingly, Index funds are
more volatile than the other funds at scale 1 and scale 5, while show lower volatility at
intermediate scales. This indicates the importance of considering the investment
horizon, when evaluating the performance of the funds.
To get a sense of the performance of three US mutual funds using the three empirical
specifications, we present empirical results from multiscaling tests of the CAPM, Fama-
French, and Carhart model.
Firstly, we examine the multihorizon Jensen’s alpha utilizing the CAPM. Table 2
illustrates the multihorizon Jensen’s alpha. Note that if α > 0 , then the investor can
increase his expected utility by investing at least a small amount in the fund. Of course,
if α < 0 , then he can achieve the same effect by short-selling the fund. The results show
that overall the Jensen’s alphas have positive values at the original data set. However,
most funds show negative value up to scale 2, equivalent to 2-8 month period except
Index fund at scale 1. However, after scale 3, three funds have positive Jensen’s alpha
except Institutional and Index funds at scale 5. This implies that the investor can
increase his expected utility by short-selling the fund in the short-run, while by
investing a small amount in the present fund in the long-run. In terms of their
performance, Index fund outperforms the institutional and active funds except scale 2
and 4, suggesting that the performance of mutual fund depends on the investment
horizon.
The betas estimated in Table 2 are very high in all funds regardless of the time scales.
However, in the case of Index fund, the beta values are decreasing as time scale
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increases. This implies that the evaluating the systematic risk is depending on the
investment horizon. In other words, for proper evaluation of the performance, the
multiscaling approach is considered as an appropriate tool.
Second, the Jensen’s alpha is estimated using the Fama and French three-factor model.
Generally, the adjusted R 2 is very high for all funds regardless of time scales with
average value of 0.982. Overall, 56% of Jensen’s alpha for three funds have positive
value. As can be seen in Table 3, the premium on the SMB factor is less explanatory
power at the short scales (up to scale 2) while after scale 3, the premium is significant at
5 percent level, except Index fund. For Institutional and Active funds, the premiums on
the SMB factor have a positive impact on fund returns in the long-run. In the case of
HML, the premiums are significant in the most scales for Institutional and Active funds,
while they are significant at the medium scales for Index funds. All significant
premiums for the HML factor are a positive impact on the fund returns. Interestingly,
even though the premiums for the SMB and HML factors are insignificant in the
original data set, the decomposed series show the significant premiums at longer time
scales. This means that the explanatory power of the risk factors depends on the time
scales. In terms of their performance of three mutual funds, Index fund outperforms the
institutional and active funds except scale 2 and 4, consistent with the results of Table 2.
Finally, we evaluate the performance of three funds using the four-factor model,
proposed by Carhart (1997). Overall, the results are similar to those of the three-factor
model. Therefore, we focus on the premiums on the momentum factor (Mom). For
Institutional and Active funds, the premiums are not significant at lower scales,
including the original data, while they are significant at the medium scales. However,
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17.
Index fund show the significant premiums on the momentum factor at all time scales,
except for scale 2. This implies that the momentum factor does not prove to be a strong
indicator of mutual funds at the short scales, while at the long scales, turn to be a strong
indicator. The performance of each fund has same results with the three-factor model.
In sum, our results show that the evaluation of performance for funds depends on the
time scales (or investment horizons). Focusing on our data set, using the Jensen’s alpha,
which is calculated by three different specifications, it can be concluded that Index fund
are dominant over all time scales, except scale 2 and 4.
5. Summary and Concluding Remarks
In literature, despite its importance in modern financial analysis, the evaluation of
mutual fund performance have not been accompanied by examination of the investment
horizon, an important factor for investments. This paper uses for the first time the
multihorizon performance measure (the Jensen;s alpha) to evaluate the performance of
three US mutual funds (Institutional, Active and Index funds) over various time scales.
The wavelet multiscaling approach has the advantage of being able to decompose the
time series over the various time scales. This advantage allows us to investigate the
behavior of our data over multiple horizons.
In terms of the performance measures of the three mutual funds, our empirical results
indicate that none of the funds are dominant over all time scales, while depending on
the specific time scale, they show that evaluating the performance of the mutual funds
depends on the investment horizons. Since risk and value (performannce) are
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timescale-dependent concepts, any attempt to measure performance, such as a popular
performance measure Jensen’s alpha, must take into account the investment horizon
effect. In terms of the premiums on the SMB, HML, and Mom, our results generally
support that even though they are not significant at the short scales, the premiums are
significant at the longer time scales. This implies that the three risk factors do not prove
to be a strong indicator of mutual funds at the short scales, while at the long scales, turn
to be a strong indicator.
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22.
Table 1. Descriptive statistics
Panel A. Basic statistics
Institutional Active Index MKT SMB HML Mom
mean 0.557 0.544 0.567 0.573 0.201 0.420 1.106
std.dev. 4.254 4.217 4.314 4.409 4.103 3.867 5.210
skewness -0.688 -0.684 -0.517 -0.700 0.878 0.030 -0.698
kurtosis 1.258 1.056 0.627 0.894 6.961 2.014 5.391
LB(5) 1.058 0.883 4.007 0.926 7.460* 4.936 5.580
(0.589) (0.643) (0.135) (0.629) (0.024) (0.085) (0.061)
LB(10) 8.207 7.843 11.912 9.683 10.832 5.673 14.983*
(0.315) (0.347) (0.104) (0.207) (0.146) (0.578) (0.036)
LB2(5) 2.694 4.061 9.449* 7.936* 50.655* 91.708* 17.342*
(0.260) (0.131) (0.009) (0.019) (0.000) (0.000) (0.000)
LB2(10) 9.193 11.469 15.280* 17.037* 50.737* 128.821* 47.361*
(0.239) (0.119) (0.033) (0.017) (0.000) (0.000) (0.000)
rho(t,t-1) -0.008 -0.015 -0.086 -0.021 -0.065 0.079 -0.085
Notes: Institutional funds are a mutual fund that targets pension funds, endowments, and other high net
worth entities and individuals. The objective of active funds is to outperform the market average by
actively seeking out stocks that will provide superior total return. Index funds are a mutual fund whose
portfolio matches that of a market index such as the S&P 500 Index. MKT is the difference between the
return of market portfolio and the risk-free return. SMB is the return on a portfolio of small stocks minus
the return on a portfolio of large stocks, HML is the return on a portfolio of stocks with high book-to-
market ratios minus the return on a portfolio of stocks with low book-to-market ratios in calendar month
t. Mom is the average return on the two high prior return portfolios minus the average return on the two
low prior return portfolios in calendar month t. * indicates significance at 5% level. LB(k) and LB2(k)
denotes the Ljung-Box test of significance of autocorrelations of k lags for returns and squared returns,
respectively. ρ is the first order autocorrelation coefficient. Skewness and kurtosis are defined as
E [ ( Rt − µ )] 3 and E [ ( Rt − µ )] 4 , respectively, where µ is the sample mean.
Panel B. Correlation matrix
MKT SMB HML Mom
Institutional 0.987 0.133 -0.492 -0.186
Active 0.992 0.156 -0.512 -0.165
Index 0.973 -0.048 -0.444 -0.230
Notes: The unconditional correlation coefficients have been calculated to check the relationship with four
factors. The striking feature is that the correlation with the HML is very low, while the correlations
between the MKT and three fund returns are very high. In addition, the HML and the Mom show the
negative correlation with three mutual fund returns.
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Table 2. the Mutual fund performance using the excess market return
Portfolio original Scale 1 Scale 2 Scale 3 Scale 4 Scale 5
Institutional alpha 0.011 -0.003 -0.001 0.000 0.023 -0.063
(0.069) (0.014) (0.022) (0.035) (0.035) (0.039)
M
βp 0.953* 0.948* 0.979* 0.961* 0.944* 0.956*
(0.018) (0.017) (0.018) (0.026) (0.042) (0.082)
R2 0.975 0.980 0.972 0.964 0.937 0.916
Active alpha 0.000 -0.004 -0.002 0.000 0.022 -0.052
(0.059) (0.011) (0.018) (0.028) (0.026) (0.035)
M
βp 0.949* 0.944* 0.955* 0.959* 0.945* 0.928*
(0.015) (0.013) (0.014) (0.021) (0.032) (0.059)
R2 0.983 0.987 0.983 0.976 0.965 0.937
Index alpha 0.021 0.004 -0.002 0.009 0.002 0.113*
(0.074) (0.025) (0.030) (0.038) (0.039) (0.024)
M
βp 0.952* 0.992* 0.883* 0.851* 0.818* 0.759*
(0.022) (0.024) (0.027) (0.037) (0.040) (0.048)
R2 0.946 0.952 0.930 0.933 0.896 0.921
Note: Excess returns are used to calculate the multiscale beta utilizing the following equation:
R pt (λ j ) − R ft (λ j ) = α p (λ j ) + β p (λ j )( R Mt (λ j ) − R ft (λ j )) + ε pt (λ j )
To calculate the multihorizon Jensen’s alpha at scale λj, we decompose each time series up to level 5,
using the Daubechies extremal phase wavelet filter of length 4 (D(4)). Scale 1, 2, 3, 4, and 5 represent
2-4, 4-8, 8-16, 16-32, and 32-64 month period dynamics, respectively.
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24.
Table 3. Mutual fund performance in Fama and French three factor model
Portfolio original Scale 1 Scale 2 Scale 3 Scale 4 Scale 5
Institutional alpha -0.052 -0.003 0.001 -0.004 0.015 -0.008
(0.053) (0.014) (0.022) (0.030) (0.028) (0.008)
M
βp 0.994* 0.977* 1.029* 0.983* 1.028* 0.915*
(0.013) (0.020) (0.016) (0.019) (0.040) (0.022)
SMB
βp 0.010 -0.003 -0.037 0.064* -0.008 0.133*
(0.022) (0.021) (0.031) (0.030) (0.039) (0.016)
HML
βp 0.089* 0.061 0.082* 0.098* 0.104* 0.057*
(0.024) (0.032) (0.031) (0.040) (0.028) (0.014)
R2 0.978 0.981 0.979 0.970 0.959 0.989
Active alpha -0.053 -0.003 -0.001 -0.004 0.017 0.025*
(0.046) (0.011) (0.017) (0.024) (0.022) (0.004)
M
βp 0.979* 0.974* 0.987* 0.971* 1.003* 0.812*
(0.008) (0.014) (0.011) (0.013) (0.026) (0.010)
SMB
βp 0.027 0.023 -0.004 0.058* -0.006 0.164*
(0.017) (0.016) (0.023) (0.025) (0.025) (0.010)
HML
βp 0.072* 0.063* 0.068* 0.075* 0.071* -0.004
(0.020) (0.029) (0.025) (0.028) (0.021) (0.006)
R2 0.985 0.988 0.986 0.980 0.976 0.995
Index alpha 0.035 0.002 -0.002 0.004 0.002 0.047*
(0.037) (0.011) (0.015) (0.020) (0.012) (0.013)
M
βp 0.991* 0.984* 0.975* 0.982* 0.951* 0.877*
(0.014) (0.017) (0.020) (0.014) (0.017) (0.025)
SMB
βp -0.212* -0.216* -0.215* -0.209* -0.221* -0.135*
(0.014) (0.019) (0.017) (0.021) (0.012) (0.017)
HML
βp 0.016 -0.014 0.028* 0.072* 0.006 0.029
(0.013) (0.022) (0.013) (0.018) (0.011) (0.016)
R2 0.989 0.991 0.986 0.982 0.990 0.972
Note: The three-factor model is used to calculate the multiscale Jensen’s alpha utilizing following
M
equation: R pt (λ j ) − R ft (λ j ) = α p (λ j ) + β p (λ j )( R Mt (λ j ) − R ft (λ j ))
SMB HML
+ β p (λ j ) SMBt (λ j ) + β p (λ j ) HMLt (λ j ) + ε pt (λ j )
To calculate the multihorizon Jensen’s alpha at scale λj, we decompose each time series up to level 5,
using the Daubechies extremal phase wavelet filter of length 4 (D(4)). Scale1, 2, 3, 4, and 5 represent 2-4,
4-8, 8-16, 16-32, and 32-64 month period dynamics, respectively.
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25.
Table 4. Mutual fund performance in four factor model of Carhart (1997)
Portfolio original Scale 1 Scale 2 Scale 3 Scale 4 Scale 5
Institutional alpha -0.033 -0.003 0.001 -0.008 0.015 0.005
(0.064) (0.014) (0.020) (0.026) (0.028) (0.006)
M
βp 0.989* 0.977* 1.010* 0.968* 1.032* 0.872*
(0.014) (0.020) (0.015) (0.015) (0.036) (0.013)
SMB
βp 0.012 -0.003 -0.027 0.066* -0.006 0.135*
(0.019) (0.021) (0.023) (0.023) (0.038) (0.014)
HML
βp 0.085* 0.061 0.073* 0.044 0.084* 0.068*
(0.025) (0.032) (0.027) (0.027) (0.028) (0.010)
Mom
βp -0.014 0.000 -0.041* -0.079* -0.037 0.072*
(0.018) (0.022) (0.018) (0.023) (0.022) (0.015)
R2 0.978 0.981 0.981 0.977 0.961 0.993
Active alpha -0.052 -0.004 -0.001 -0.006 0.017 0.023*
(0.057) (0.011) (0.017) (0.021) (0.022) (0.004)
M
βp 0.979* 0.975* 0.977* 0.960* 1.004* 0.819*
(0.011) (0.013) (0.012) (0.012) (0.025) (0.008)
SMB
βp 0.027 0.017 0.002 0.059* -0.006 0.163*
(0.016) (0.016) (0.018) (0.021) (0.025) (0.010)
HML
βp 0.072* 0.060* 0.063* 0.036* 0.067* -0.006
(0.021) (0.023) (0.023) (0.018) (0.021) (0.007)
Mom
βp 0.000 0.019 -0.022 -0.056* -0.008 -0.011
(0.017) (0.019) (0.019) (0.017) (0.017) (0.013)
R2 0.985 0.989 0.986 0.984 0.976 0.995
Note: The four-factor model is used to calculate the multiscale Jensen’s alpha utilizing the following
M
equation: R pt (λ j ) − R ft (λ j ) = α p (λ j ) + β p (λ j )( R Mt (λ j ) − R ft (λ j ))
SMB HML Mom
+ β p (λ j ) SMBt (λ j ) + β p (λ j ) HMLt (λ j ) + β p (λ j ) Momt (λ j ) + ε pt (λ j )
To calculate the multihorizon Jensen’s alpha at scale λj, we decompose each time series up to level 5,
using the Daubechies extremal phase wavelet filter of length 4 (D(4)). Scale1, 2, 3, 4, and 5 represent 2-4,
4-8, 8-16, 16-32, and 32-64 month period dynamics, respectively.
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26.
Table 4. (cont’d) Mutual fund performance in four factor model of Carhart (1997)
Index alpha 0.078* 0.002 -0.002 0.001 0.002 0.021*
(0.033) (0.011) (0.014) (0.017) (0.009) (0.005)
M
βp 0.981* 0.983* 0.966* 0.970* 0.955* 0.959*
(0.014) (0.015) (0.023) (0.013) (0.013) (0.010)
SMB
βp -0.208* -0.209* -0.210* -0.207* -0.218* -0.140*
(0.012) (0.017) (0.016) (0.015) (0.011) (0.006)
HML
βp 0.008 -0.011 0.023 0.029* -0.013 0.007
(0.013) (0.017) (0.014) (0.014) (0.009) (0.004)
Mom
βp -0.032* -0.024* -0.019 -0.062* -0.036* -0.135*
(0.009) (0.010) (0.014) (0.019) (0.006) (0.011)
R2 0.990 0.991 0.986 0.987 0.993 0.993
Note: The four-factor model is used to calculate the multiscale Jensen’s alpha utilizing the following
M
equation: R pt (λ j ) − R ft (λ j ) = α p (λ j ) + β p (λ j )( R Mt (λ j ) − R ft (λ j ))
SMB HML Mom
+ β p (λ j ) SMBt (λ j ) + β p (λ j ) HMLt (λ j ) + β p (λ j ) Momt (λ j ) + ε pt (λ j )
To calculate the multihorizon Jensen’s alpha at scale λj, we decompose each time series up to level 5,
using the Daubechies extremal phase wavelet filter of length 4 (D(4)). Scale1, 2, 3, 4, and 5 represent 2-4,
4-8, 8-16, 16-32, and 32-64 month period dynamics, respectively.
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27.
Figure 1. Estimated wavelet variance
3.0
2.5
2.0
1.5
1.0
0.5
0.0
1 2 4 8 16
-0.5
-1.0
-1.5
-2.0
-2.5
wavelet scale
Institutional fund Active fund Index fund
Notes: The y-axis indicates the log wavelet variance and the x-axis indicates the wavelet time scale. To
calculate the wavelet variance at scale λj, we decompose each time series up to level 5, using the
Daubechies extremal phase wavelet filter of length 4 (D(4)). Scale1, 2, 3, 4, and 5 represent 2-4, 4-8,
8-16, 16-32, and 32-64 month period dynamics, respectively.
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