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Bootstrap Methods with application in
Econometrics and Finance*
WEI Zhen+
Department of Statistics and Probability
School of Mathematical Sciences
Peking University
People’s Republic of China
Email: [email protected]
Abstract
This paper studies both the traditional and the state of art aspects of Bootstrap
Resampling Procedures. The theoretical asymptotic performances of such procedures are
explored by an intuitive way of Edgeworth Expansion. Several refinements of standard
bootstrap methods are advanced by a brief introduction, with suggestions for further
research frontiers. Further more, empirical experiments are carried out in terms of
bootstrap confidence intervals and hypothesis tests to evaluation the managerial
performance of several representative funds in the financial market of People’s Republic
China. Another contribution of this paper is that it develops an integrated numerical
method to analyze the overall performance of each fund instead of relying on the traditional
four types of performance measures respectively. Concluding remarks are forwarded in an
attempt to explicit that any experimental analysis should be carried out with great caution
because a minor intrusion of theoretical assumptions may result in catastrophic failure in
understanding the intrinsic properties of the data.
Keywords bootstrap, resampling, jackknife, block bootstrap, circular bootstrap, wild
bootstrap, paired bootstrap, econometrics, financial time series, Edgeworth expansion,
cross score, performance measure, computer intensive methods
* This paper is presented as the undergraduate thesis work of Mr. WEI Zhen in the Department of Statistics and Probability, School of Mathematical Sciences, Peking University, PRC. + The author is grateful to Doctor QIN Huaizhen for his helpful suggestions and comments, to Doctor DAI Ming, HUANG Hai and professor YANG jingping for their gracious support in obtaining the financial data presented in this work.
Contents
1 Introduction 1
2 Bootstrap approach to Confidence Intervals and Hypothesis Tests 2 2.1 Bootstrap Confidence Intervals 3 2.2 Bootstrap Hypothesis Tests 4 2.3 Refinements of standard bootstrap techniques 5 2.3.1 Serial Correlation 5 2.3.2 Conditional Heteroskedasticity 6 2.3.3 Structural Equation Models 8 2.3.4 Non-stationary time series
3 The asymptotic theories of Bootstrap Statistics 9
4 Experimental Bootstrap with Financial Data 13 4.1 Bootstrapping the confidence intervals and hypothesis tests 13 4.2 Cross Scoring method for ranking the performance of funds 13 4.3 The Experimental Data 14 4.4 Experimental Results 15
5 Conclusion 17
Reference 25
Figures
Figure 1 Serial correlation … 6 Figure 2 Conditional Heteroskedasticity… 7 Figure 3 Co-integration of fund index… 9 Figure 4 Histograms and normal quantile-comparison plots… 18 Figure 5 Confidence ellipses of bootstrap samples… 19
Tables
Table 1 Ranking 33 representative close funds… 20 Table 2 Cross Scoring of 33 representative funds… 21 Table 3 Bootstrapping the 95% Confidence Intervals… 22 Table 4 The 95% Confidence Intervals of the transformed difference… 23 Table 5 Bootstrapping the 95% Confidence Intervals of the difference… 24
1. Introduction
The past several decades have experienced an explosive development in Computer Science
and Technology. One tremendous advance among them is the dramatic improvement in
Hardware capacity and Computational efficiency, which allows otherwise impossible access
into a new era of Statistical and Computational research frontier. That is what we often call
Computer Intensive Methods (CIM)1 with their wide applications in various disciplines and
areas such as Biological Computation, Signal Processing, Financial Analysis, Actuarial
Science etc. For recent references, see, for example, James MacKinnon (2002), Krzysztof
Ostaszewski and Grzegorz Rempala (2000) and Ansgar Heinrich Ludolf West (2000). Among
many versions of CIMs, the Bootstrap Methods receive particular recognition partly due to
its underlying seemingly simple but actually profound idea and its easy implementation
using computer simulation either through software packages that are readily available (e.g.
The boot library in R or S-plus) or through recoding according to user-defined algorithms
borrowing new features.
The term “Bootstrap” (or Bootstrapping) which is due to Efron (1979) is an allusion to
the expression "pulling oneself up by one's bootstraps" meaning doing the impossible [see
also, e.g. Efron and Tibshirani (1993)]. As it turns out, through a decade’s practices and
advances both in theoretical framework and simulation techniques, it does provide a
promising tool to practitioners to solve the traditionally unsolvable problems. On the other
hand, I should also note that there is no universal rule in the quantitative world which
confronts us in that its intrinsic complexity may well be far beyond what the data can tell us.
Then, it is natural to see that there are cases when standard bootstrap performs not as well
as the traditional asymptotic theories or simple Markov Chain Monte Carlo (MCMC)
Methods. However, to be generally speaking, as will be shown later in this paper, under
certain probabilistic conditions (which is often true in reality), bootstrap usually produces
more accurate and reliable results than traditional methods which are either based upon
the large number distribution properties of the observations [for example, see Pakes (1986),
Rust (1987) and Stern (1997)] or the prior beliefs of the authors [see, Albert and Chib (1993),
Geweke (1999) and Elerian, Chib, and Shephard (2001), among many others], especially for
inferences based upon small samples.
Furthermore, in cases when standard bootstrap procedures do not produce unbiased or
consistent estimate of statistics of interest, refinements are suggested to handle this
incompatibility. Specifically, Künsch (1989), Politis and Romano (1992), Liu (1988) and
Mammen (1993) and Freedman (1981) proposed different novel bootstrap methods to solve
problems in empirical econometrical and financial analysis. For a recent summarization of
these methods and their recent developments, see for example, Emmanuel Flachaire (2003)
and James MacKinnon (2002). For empirical study on the numerical performance of Fast
Bootstrap Procedures, see the work of Francois Lamarche (2003) as a good reference.
The remaining part of this paper is organized as follows. Section 2 introduces elements 1 Including various statistical and computational techniques such as Delta-methods, Cross validation, Jackknife, and AI Networks related algorithms.
1
of employing basic Bootstrap resampling methods to construct Confidence Intervals and
perform Hypothesis Tests, which ends with a discussion of some situations when standard
bootstrap resampling may not produce satisfying results and suggests several techniques
that are useful to overcome these difficulties. Asymptotic superiorities of general Bootstrap
methods over the traditional simulation methods are the main focus of Section 3, where a
fresh probabilistic interpretation of Edgeworth expansion is put forward to prove analytical
results. Several numerical experiments are carried out in Section 4 to give the reader an
intuitive understanding of the performance of Bootstrap Methods and its promising
application in real practices. Lastly, Section 5 supplements a short conclusion of this paper
and provides suggestions for further research directions.
2. Bootstrap approach to Confidence Intervals and Hypo- thesis Tests
There are two major resampling methods in the family of Bootstrap: parametric and
nonparametric. The former does assume any prior knowledge of the distribution of the
studied data set and samples directly from the original data (or the empirical distribution
function implied by the data) with replacement, hence is robust and more favored by
researchers (because therein lies its strength), while the latter makes assumptions about the
underlying distribution of the data, estimates its parameters (using maximum likelihood,
least square methods or other numerical procedures) and then resamples from the
estimated parametric distribution. Here, I focus on how to use nonparametric bootstrap to
construct confidence intervals and perform hypothesis test while assuming the reader can
easily carry out the parametric case following the steps described in this section.
Suppose we draw a sample { }1 2= , , ,… nX X XS from a population { }1 2= , , , Nx x xP … .
We proceed to draw a sample of size n from among the elements of , sampling with
replacement. The result is called bootstrap sample
S
{ }* * *1 2= , , , nX X X*S … . The key
bootstrap analogy is as follows:
The population is to the sample as the sample is to the bootstrap samples
That is
*v. s. v. s.∼P S S S
Following this conception, we could use the average of the bootstrapped statistics,
*** * 1( ) == = ∑
R
bbT
T E TR
(1)
to estimate the expectation of the bootstrapped statistics. Similarly, the estimated bootstrap
variance of *T
** 2* * 1
( )( )
1=
−=
−∑R
bbT T
V TR
(2)
2
estimates the sampling variance of T, and SE T V T=*( ) ( )* *
* is the bootstrap estimate of
the standard error of T.
2.1 Bootstrap Confidence Intervals
Standard (first suggested by Efron (1979) in his pioneering paper), studentized,
normal-theory, percentile, bias-corrected accelerated (or BCa) percentile and double
bootstrap confidence intervals are widely used. In Section 4 where numerical experiments
are carried out, I only stress on normal-theory, percentile and bias-corrected accelerated
CIs. Following is a brief introduction of how to construct these three types of confident
intervals. The theoretical deductions of these intervals are omitted because they can be
constructed through either simple statistical intuition or mathematical procedure that is
well described in various text books.
To construct a α( - )100 1 percent confidence interval, following the results in equations
(1) and (2), we have
1 2T B z SE T SE T V Tαθ −= − ± =* * * *
* */( ) ( ) where ( ) ( ) * (3)
* *Where B T T= − is the bootstrap estimate of the bias of the statistic T.
The so-called bootstrap percentile interval is to use the empirical quantiles of to
form a confidence interval for θ:
*bT
R RT Tα θ+ +< <* *[( ) / ] [( )( / )]1 2 1 1 2α− (4)
where are the ordered bootstrap replicates of the statistic, and the
operator nT T T…* * *
( ) ( ) ( ), , ,1 2
[ ]i indicates rounding to the nearest integer.
A more accurate approach which is recommended by many literatures is the so-called
bias-corrected, accelerated percentile (BCa) intervals. This type of interval can be calculated
from observations through the following steps:
1. Let correction factor 1
R
bb
I T Tz
R− =
⎡ ⎤≤⎢ ⎥
⎢ ⎥=+⎢ ⎥
⎢ ⎥⎣ ⎦
∑Φ
*( )1 1
1 (5)
where is the inverse of standard normal cumulative distribution function, is
an indicator function satisfying:
( )Φ i-1 ( )I i
( )I A⎧
= ⎨⎩
if A is true
if A is false
10
(6)
3
2. Let be the jackknife value( )iT −2 of T , and T be the average of . Then
correction factor 2 ( )iT −
( )( )( )( )
n
ii
n
ii
T Ta
T T
−=
−=
−=
⎡ ⎤−⎢ ⎥⎣ ⎦
∑
∑/
3
13 22
16
(7)
3. Having the two correction factors at hand, we compute
( )
( )
z za z
z z
z za z
z z
α
α
α
α
α
α
−
−
−
−
⎡ ⎤−⎢ ⎥= +
− −⎢ ⎥⎣ ⎦⎡ ⎤+⎢ ⎥= +
− −⎢ ⎥⎣ ⎦
Φ
Φ
/
/
/
/
1 21
1 2
1 22
1 2
1
1
(8)
where ( )Φ i is the standard normal cumulative distribution function.
4. The corrected percentile confidence interval is as follows
[ ] [ ]RaT Tθ< <*
1 2Ra* (9)
where the operator [ ]i indicates rounding to the nearest integer as before. It should be
noted that when , BCa confidence interval reduces to the percentile interval stated
previously.
a z= = 0
2.2 Bootstrap Hypothesis Tests
Although there is a natural correspondence between constructing confidence intervals and
performing hypothesis tests, it is rather important to stress out some important aspects of
bootstrap hypothesis tests for at least two reasons:
1. There so many ways to construct Bootstrap confidence intervals that we occasionally get
contradictory or inconsistent results. Bootstrap Hypothesis Testing, at this time, may be
more informative and reliable than a set of confidence intervals
2. In the fields such as Financial Analysis, innovative ideas often allows us to perform
original hypothesis tests which are beyond the scope of traditional confidence intervals.
Standard bootstrap hypothesis test can be explained as follows. Suppose the statistics
of interest isθ and the null hypothesis we want to test is ( )θ θ θ θ≤ =0 0 for someθ0 , the
alternative hypothesis isθ θ> 0 . Then, a pivotal statisticsV which is a function ofθ andθ0 is
constructed (e.g. the t statistics usually used in regression model is ( ) ( )Varθ θ θ− 0 ).
From the sample data, we can compute the value of such statistics d. Further, we employ
the standard bootstrap resampling procedure to calculate the bootstrap sample value of , V
2 Jackknife value refers to the value calculated from the sample when the i-th observation is
removed. ( )iT−
4
denoted as . Thereafter, we calculate the frequency of the event that is
greater than :
* ( , , ,id i n= 1 2 … ) *id
d
(* *n
d ni
)P I d dn =
= ≤∑1
1
which is the bootstrap P-value of the original hypothesis tests. Suppose *dP α< at a given
significant levelα , we have sufficient reason to discredit the null hypothesis.
While the standard form of bootstrap hypothesis test is appealing as it stands, here, I
instead focus on a particular case where Bootstrap hypothesis test is applied to a problem in
Finance and a novel approach is advanced and further experiments and discussion will be
carried out in Section 4 of this paper.
The Sharpe (1966) and Treynor (1965) performance measures are widely accepted in
both academia and industry to assess the Risk-adjusted value of a particular portfolio. It
can be shown, after some mathematical treatment, that the Sharpe performance measure is
useful when the portfolio of interest represents all of the investor’s investment, while
Treynor’s measure is preferred when the portfolio under discussion is only a portion of the
whole investment package. Previous efforts of the estimation of the two measures are based
on the assumption of the asymptotic normality of the data. For example, see the pioneering
paper of Jobson and Korkie (1981), in which a Z statistics is constructed to facilitate
Hypothesis Testing and construct Confident Intervals. However, practice tells us that under
some circumstances, which are not rare, such assumption can be seriously cast into doubt.
Besides, the data transformation process used to construct the Z statistics is also
questionable because certain regularity condition in the transformation is not necessarily
satisfied. Vinod and Morey (1999) suggested a Bootstrapping Methodology to reconcile
these problems and to produce more reliable hypothesis tests. Another two main stream
performance measures are the Jensen’s measure and Appraisal Ratio, which are also
employed by financial and rating system. In section 4, I apply both the JK (Jobson and
Korkie) Method and the Bootstrap Method (suggested by Vinod and Morey) to the data of
several funds in China, and comment on their relative strength.
2.3 Refinements of standard Bootstrap techniques
As every coin has its two sides, there are also cases where standard bootstrap procedures
may fail to do the right job. Thus we should exercise extreme caution when applying
bootstrap methods to perform real analysis and draw a conclusion.
2.3.1 Serial Correlation
It is not difficult to show that the standard Bootstrap resampling procedure may fail to
generate unbiased and consistent estimation of a statistic when the observed data is serially
correlated. Unfortunately, in industrial and financial practices, the retrieved data may well
violate the independent assumption of many statistical frameworks. Time Series Analysis
mainly focuses on the short-term and long-term interdependence and correlation of the
5
data, in which AC (Auto-Correlation) and PAC (Partial Auto-Correlation) Statistics are used
to describe the serial dependence. One particular case can be shown by Figure 1, which
describes the AC and PAC of the time series of a GDP data set.
To solve this problem, many refinements are suggested by academia. Particularly,
Künsch (1989) considered a moving block bootstrap procedure which resamples a block of
observations each time instead of only one observation. This approach does not generate an
equal sampling because the middle region of the data seems to have a greater chance to be
sampled. Politis and Romano (1992) thus forwarded a circular bootstrap, which takes the
data series as a circle and resamples the moving block circularly from the data. Empirical
evidence shows that both block bootstrap and circular bootstrap can capture the short-term
correlation of the data and outperform the standard bootstrap resampling method.
2.3.2 Conditional Heteroskedasticity
Financial and Economic data often exhibits some form of unknown volatility that beyond
the explanation of traditional Econometric Models. For example, the GARCH-related
models (such as ARCH, TARCH, EGARCH, GJR-GARCH, etc.) are particularly designed to
solve the problem of conditional heteroskedasticity that is produced by the error term of
traditional least square regression. This phenomenon is often called Volatility Clustering of
financial and macroeconomic time series, which has brought recognition in industry long
Figure 1 Serial correlation of Gross Domestic Productivity from January 1952 to
March 1996. The data is from the example file of the software package
Eviews 4.1
6
before any theoretical model was established to give explanation and prediction. As an
illustration, I regression the daily return of Fund index by that of Market Index of the
Financial Market of China3, producing error terms whose variances are AR(1) correlated.
The result is demonstrated in Figure 2.Under such circumstances, standard bootstrap also
can not make satisfying results.
Basically, there are two ways to alleviate this drawback. They are namely, Wild
Bootstrap and Paired Bootstrap (or Pairwise Bootstrap). The wild bootstrap which was
suggested by Liu (1988) and Mammen (1993) considers a bootstrap data generation process
(DGP)
( )tt ty fβ ε= +X tv (10)
where β is the OLS estimates of the standard regression model, tε is the corresponding
residual term and ( )f i is a transformation probability function, usually takes on binary
distribution having equal probability on -1 and 1. Bose (1988), Kreiss (1997) and He and
Teräsvirta (1999) developed and experimented many new version of the Wild Bootstrap
application, such as Fixed-Design and Recursive-Design Wild Bootstrap Methods.
-2
-1
0
1
2
3
-4
-2
0
2
4
6
50 100 150 200 250
ErrorReturn of Fund IndexFitted value
Figure 2 Conditional Heteroskedasticity of Regression Model of Financial data
The Paired Bootstrap was first proposed by Freedman (1981), which resamples the
dependent and independent variables in pairs in the regression or auto-regression models.
Silvia Goncalves and Lutz Kilian (2002) shows that, if certain regularity conditions are
3 The data are daily index extracted from January, 3, 2003 to February, 12, 2004.
7
satisfied, for the fixed-design Wild Bootstrap and the Paired Bootstrap
( ) ( )*
P* * * *supx
P t x P t xθθ∈
≤ − ≤ ⎯⎯→R
0 (11)
where **tθ
and *tθ are the bootstrap-based and ordinary t statistics for the regression model,
respectively. Furthermore, if more constraints are added, it may be shown that equation
(11) also holds for Recursive-Design Wild Bootstrap.
2.3.3 Structural Equation Models
Macroeconomic models usually deals with various equilibrium of the general Market. A
classical Supply-Demand Model that can be found in many texts can be written as
St t t
Dt t t
S Dt t
Q a a P a P
Q b b P b P
Q Q
t
t
ε
ε−
−
⎧ = + + +⎪⎪ = + + +⎨⎪ =⎪⎩
0 1 2 1 1
0 1 2 1 2 (12)
where StQ and D
tQ denote the amount of supply and demand at time respectively. t
tP and ty represent the price and income at time t .
In such cases, we should also be cautious when applying bootstrap resampling to the
standard techniques in SEM, such as TSLS (Two-Stage Least Squares), SUR (Seemingly
Unrelated Regression) and 3SLS (Three Stage Least Squares). For detailed discussion of
such events, I recommend James MacKinnon (2002) for reference. To see more topics on
residual resampling dealing with more generalized cases, see, for example, Urban Hjorth
(1994).
2.3.4 Non-stationary time series
It also can be shown that for models dealing with times series that are unstationary,
standard bootstrap procedure also fails. In Financial practices, we often need to extract the
underlying stationary or equilibrium characteristics from several un-stationary time series.
This is often modeled by Co-integration, which claims that a certain linear combination of
I(di) process4 (where di are positive integers) may yield a stationary outcome. The result
provides evidence for long-term equilibrium of many macroeconomic factors which are
themselves exposed to uncertainties of unknown form. Seeking out an efficient way to
bootstrap such processes still remains one of the cutting edge frontiers of theoretical and
empirical Bootstrap researches.
An example of co-integration of two I(1) sequences are shown in Figure 3, where Fund
Index and Market Index of Chinese Market are both I(1) time series5 and the residual
produced by the regression of the two series is a stationary process, which reflects the
long-term equilibrium relationship between the two macroeconomic indicators.
4 I(d) process refers the a series whose difference of d-th order is stationary. 5 This can be validated through uniroot test by Augmented Dickey-Fuller Test.
8
Figu
3. T
Th
Edgew
of othe
Su
which
for som
order m
Fo
mome
where
-40
-20
0
20
40
60
880
920
960
1000
1040
1080
50 100 150 200 250
Error Fund Index Fitted value
re 3 Co-integration of Fund Index and Market Index in the financial market of
China. The data are the daily close index extracted from January 3, 2003 to
February 12, 2004.
he asymptotic theories of Bootstrap Statistics
e asymptotic behaviors of Bootstrapping Methods are essentially examined by
orth Expansion, although it is frequently addressed by researchers that the omission
r potential aspects of bootstrap methodologies may blind us from the large picture.
ppose random variable has finite moments of all orders, one particular case of
is that decreases in tails at least in a exponential rate, or
X
X
( ) ( )Pr expX x c c x> < −1 2 (13)
e positive real number c and . In the following paper, I denote the j-th centered
oment of by1 c2
X ju , i.e.
( )jju E X= (14)
llowing probability theory, there is a one to one relationship between and its
nt generating function
X
( )XM t , which is defined as
( ) ( ) ( )dtX txXM t E e e F x= = ∫ (15)
is the cumulative distribution function of . Furthermore, if follows standard ( )F i X X
9
normal distribution, we have
( ) ( )dtxXM t e x e= ∫ Φ 2
t
=2
(16)
Define the culmulant generating function ( )XK t as
( ) ( )log!
r
X Xr
tK t M t
rκ
∞
=
= =∑1
r (17)
where is called the r-th cumulant ofrκ X ( ), ,r = 1 2 …
Theorem 1 Under the condition that has finite moments of all orders and mild
regularity constraints, we have
X
( ) ( ) !!
!!
i
i
rsr
r pr s
k i ii
i
u rk
pr
κ∆
= =
=
⎛ ⎞= − − ⎜ ⎟
⎝ ⎠∑ ∑ ∏
∏1 1
1
1 1 (18)
where the sum extends over all possible combinations of positive integers
and , such that ∆Σ sp p< <1
, sr1 r
s
i ii
p r p=
=∑1
and s
ii
r r=
=∑1
Proof We expand ( )log XM t using Taylor’s formula round t = 0
( ) ( ) ( )dlog
!d
r rX
X X rr t
K t tK t M t
rt
∞
= =
= =∑1 0
(19)
Comparing (18) and (19) yields
( ) ( ) 'd d log d
d d d
r r rX X X
r r r rX tt t
K t M t M
Mt t tκ −
== =
⎛ ⎞= = = ⎜ ⎟
⎝ ⎠1
00 0
(20)
On the other hand, under mild regularity conditions
( ) ( ) ( ) (d dd d
d d
r rtx r tx r tX
Xr rM t e F x t e F x E X e
t t= = =∫ ∫ ) (21)
Then, it is easy to see
( ) ( )d
d
rr
Xr
t
M t E X ut
=
r= =0
(22)
From (20) and (22), through a brief mathematical deduction, we get (18). ■
From formula (18), under the condition that ( )u E X= =1 0 , we can easily calculate
10
that: for , and so on. r uκ = r 2, or r = 1 2 3 u uκ = − 24 4 3
Let be independently identically distributed variables satisfying
conditions described in Theorem 1. If we denote their population mean and standard
error by
, ,X X1 2 …
µ andσ respectively, from Lindeberg-Levy Theorem, we know that
( ) ( )/ dnZ n X Nµ σ= − ⎯⎯→ 0 1, (23)
when n approximates infinity, and ( ) t2
nZM t e→ 2 equivalently.
From theorem 1, it is easy to show that
( ) ( )/n
n
Z X uM t M t nσ−⎡ ⎤= ⎣ ⎦ (24)
It follows that
( ) ( )( ) ( )
log /
nZ X uM t nK t n
uutt t
nn
σ
σ
σσ
−=
−= + + +
4243 43
43
32 246
o n (25)
Then, we know that
( ) ( ) ( )t
nZM t e c t c t c t o nnn
⎡ ⎤= + + + +⎢
⎣ ⎦
22 3 4 6
1 2 31 11 ⎥ (26)
where ( )
, ,uu u
c c cσ
σ σ σ
−= = =
243 3
1 2 33 4
36 24 7 62
.
Theorem 2 ( ) ttX rrE H X e e t⎡ ⎤ =⎣ ⎦
22 , where conforms standard normal
distribution, and
X
( )rH i refers to the Hermitian polynomials function, which is
recursively defined as:
( ) ( ),H x H x= =0 11 x and ( ) ( ) ( ) ( )r r rH x xH x r H x− −= − −1 21
Proof It is easy to see that
( ) ( )
( ) ( )
d
d
x
t
tX txr r
r
E H X e H x e e x
e H t x x
π
+∞ −
−∞
+∞
−∞
⎡ ⎤ =⎣ ⎦
= +
∫
∫ Φ
22
22
12 (27)
Following the definition of ( )rH x , we can employ the routine of Mathematical Deduction
again to prove that
( ) ( )d rrH t x x t
+∞
−∞+ =∫ Φ (28)
Combining equations (27) and (28) justifies the claim of Theorem 2. ■
11
Particularly, substituting r with 0 yields ( ) t
XM t e=22 . So Theorem 2 can be seen as a
generalization of Moment of generation function.
From Theorem 2, it can be easily show that the probability density function of has
the form: nZ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )nZ
x xf x x c H x c H x c H x o n
nn
φ φφ −⎡ ⎤= + + + +⎣ ⎦
11 3 2 4 3 6 (29)
where ( )φ i is the density function of standard normal distribution. Moreover, from the
notion that ( ) ( ) ( )dr rH x x x H xφ −= −∫ 1 , we can integration equation (29) to get the form
of the cumulative distribution function of nZ
( ) ( ) ( ) ( ) ( ) ( ) ( ) (nZ
x x )F x x c H x c H x c H x o nnn
φ φ −⎡ ⎤= − − + +⎣ ⎦Φ 11 2 2 3 3 5 (30)
From Theorem 1 and Theorem 2, an intuitive understanding of the asymptotic
behavior of various bootstrap resampling procedures can be explained as follows. Suppose
we computed the bootstrap estimate of a statistics ( )Xθ as ( )* Xθ , if ( )Xθ is pivotal, then
its cumulative distribution function (CDF) of the traditional normal theory estimateθ can
be expanded as
( ) ( ) ( ) ( ) ( ) ( ) (x x )F x F x G x G x o nnn
θθ
φ φ −= − − + 11 2 (31)
where ( )F xθ is the CDF of the true parameterθ . Similarly, the CDF of ( )* Xθ can also be
expanded as
( ) ( ) ( ) ( ) ( ) ( ) (*
x x )F x F x G x G x o nnn
θθ
φ φ −= − − + 11 2 (32)
According to the bootstrapping estimate procedure, we know that can be seen as
an asymptotic approximation of
( )G1 i( )G ≡ 0i . Following the traditional asymptotic theory, we
get
( ) ( ) ( )/G o n o n−= + 1 2 1 21 0i ∼ /− (33)
Comparing equations (31)-(33), we can write
( ) ( ) ( )*F x F x o nθθ−= + 1 (34)
whereas ( ) ( ) ( /F x F x o nθθ−= + 1 2 ) , which consolidates our former claim of the asymptotic
superiority of bootstrap procedures over the traditional ones.
From the steps above, it is not difficult to show that when the statistic ( )Xθ is not
pivotal, bootstrap can also provide consistent estimates while traditional approach can not,
but the asymptotic order is reduced to ( )/o n−1 2 .
12
4. Experimenting Bootstrap with Financial Data
In this section, I employ the above-mentioned bootstrap methods to construct confidence
intervals and perform hypothesis tests on the data of several representative funds in the
financial market of China. In the context of hypothesis testing, I compare the result
provides by JK’s method and bootstrap methods. From the result of the experiments, it
seems evident that bootstrapping hypothesis tests generally provide more accurate results
and more convincing conclusion. Furthermore, in order to provide an objective and
integrated evaluation of the collective performance of each fund, I proposed a Cross
Scoring method to rank these funds. The ultimate rank is summarized in Table 2.
4.1 Bootstrapping the confidence intervals and hypothesis tests
In the paper of Jobson and Korkie (1981), they used the following transformation
( ) ( )
( )
kn n k k n
nm kmkn k n
m m
E G ShT T
E G Tr
σ µ σ µ
σ σµ µ
σ σ
⎛ ⎞⎡ ⎤ = − − +⎜ ⎟⎣ ⎦ ⎝ ⎠⎛ ⎞⎡ ⎤ = −⎜ ⎟⎣ ⎦ ⎝ ⎠
2
2 2
1 114 32
(35)
to perform hypothesis tests on the difference of the performances of two funds,
particularly in terms of Sharpe and Treynor performance measures, where is the
transformation function, n and k represent two arbitrarily selected funds and T is the
sample size at hand. Essentially, the subsequent Z statistic is constructed from equation
(35). JK (1981) found that the Z statistics of Sharpe measure performs well for small
sample size, but the Treynor measure behaves not as well in some cases.
( )G i
4.2 Cross Scoring method for ranking the performance of funds
Having all the point estimates of the four performance measures, I suggest employing the
integrated rule to rank the performance of the funds. I define the Cross Scoring (CS) as
the following form
( ) ( )ω signk ij i ii j k
Score Perf k Perf j= ≠
⎡ ⎤= −⎣ ⎦∑∑4
1 (36)
where i represents the four types of performance measures, N is the number of funds to be
ranked, ( )sign x is a function taking values in { }, ,−1 0 1 when , ,x < = > 0 respectively
representing individual penalty (award) in each performance measures and ijω is a
weighting parameter satisfying
ωiji j k= ≠
=∑∑4
11 (37)
In the experiment, I fix ijω to ( )N −1 4 1 while more realistic approach is to adjust the
13
distribution of ijω s according to the specific property of the investor’s portfolio. For
example, when assessing the portfolio which represents the entire investment of the
investor, we can justifiably define:
when ω
otherwiseij
iN
⎧ =⎪= −⎨⎪⎩
1 11
0 (38)
This corresponds to using Sharpe’s Measure alone.
One natural explanation of using formula (36) to evaluate the performance of any
fund particularly or any financial institution in general is that we can adjust the
parameters freely to fit the need of individual interest. As has been formerly stated, the
Sharpe, Treynor measures and Appraisal Ratio have their respective importance in the
evaluation. However, in the real financial practices, our portfolio at hand does not
necessarily satisfy the conditions when a specific type of performance measure works
extremely well. So, it is realistic and promising to employ this hybrid rule. For instance, if
the portfolio of interest consists mainly of one stock and a combined investment on
another stock and the fund index, none of the four performance measures will work well in
this case alone. Following our idea of integration of these measures, one possible solution
is to take the weighted average of Appraisal Ratio and Sharpe (Treynor) measure. The
weights are proportional to the net value of each type of investment respectively. This idea
can be applied to more complicated situations in more or less the same manner.
In the experiments, using the results of section 4.1, we can easily calculate the Cross
Scores of all the funds of interest from equation (36). Here, I use the equal weights to
produce the rank according to the cross score. The results are shown in Table 4.
4.3 The Experimental Data
Due to the huge amount of the experimental data that have to be employed in order to give
a comprehensive rank of all the representative funds in Chinese Market, I use SAS system
to manipulate the daily returns of all the funds. Of 54 funds6 that currently exist in Chinese
Market, only 33 funds functions smoothly more than 3 years. Since it is uninformative to
explore the less-than-one-year daily returns of the remaining 21 funds, I only ranked the
33 representative funds using the scoring method described above. The experimental data
then is reduced to the daily returns of these representative funds from Jan. 3, 2001 to Dec.
31, 2003 (717 observations).
Then, the experiments in this section can be described as the following steps:
1. Rank the selected funds using the traditional four types of performance measure
respectively.
2. Rank the selected funds using the cross scoring method while assuming the investor
6 We only discuss close fund here since the inaccuracy of the information gained from the daily prices of open funds in China lends little insights into what we are interested in.
14
takes equal weights on each measures.
3. Comparison of the results in steps 1 and 2 are stated, and the top 3 and bottom 3 funds
according to step 2 are selected to construct confidence intervals and perform
hypothesis tests following the way described below.
4. Bootstrapping the confidence intervals of all the performance measures of the selected
funds in step3 and compare the results with their point estimates in step 1.
5. Perform hypothesis tests using JK’s method and Bootstrap method to access the
difference of the performance measures of the top 2 and bottom 2 funds according to
step 2 and comparison is stated to evaluate the accuracy of these tests.
4.4 Experimental Results
Table 1 reports the rank of the 33 representative close funds in the financial market of
China. The result is ordered by the four types of performance measures: Sharpe’s measure,
Treynor’s measure, Jensen’s measure and Appraisal Ratio in each category. Table 2
represents the cross rank of the 33 representative close funds using the method describe in
Section 4.2. Comparing the two tables, we can see that they generally produce similar
results with only a little deviation from each other which reflects the intrinsic differences
in the nature of these measures. Table 2 generates an objective rank of the funds due to the
reason that has been formerly stated. However, if the investor has a particular taste of
portfolio management, for instance, if he or she would like to risk all his investment in an
individual fund, then it may not be wise to use the rank in Table 2. Under such
circumstances, it is more reasonable to refer to the rank of the Sharpe performance
measure in Table 1.
Table 3 summarizes the point estimates and 95% bootstrap confidence intervals
(lower and upper bounds) of four performance measures of the top 3 and bottom 3
representative funds in Chinese Market which are selected according to the cross rank in
produced in Table 2. It can be seen from the numerical results that the bootstrap estimates
of confidence intervals of the two groups of funds 1-3 and 31-33 exhibit a sort of in-group
similarity, producing nearly same interval width, but the confidence intervals between the
two groups are somewhat different from each other. Then, we may further ask that is there
significant evidence to discredit the hypothesis that all funds in the Chinese Market are
managed the same? If this is true, it is useless to discriminate which portfolio of funds is
the best choice, just throw your money into a random bundle of them. This question will
be answered through the hypothesis techniques using JK’s method (see Section 4.1) and
bootstrap hypothesis test.
In order to explicit that the bootstrap data generating process does conform to the
theoretical assumption of creating asymptotically normally distributed resamples, I plot
both the histograms and the quantile comparison graphs of the bootstrap distribution of
the samples (the sample size here is fixed at 3999) of four performance measures given in
Table 3. To see the interactive effects of the performances on these funds, I plot the 5%, 95% and 99% confidence ellipses of the bootstrap samples of these measures either from
15
the same fund or from different funds. From figure 5, the Sharpe and Treynor
performance measures are strictly positively correlated, as can be seen from (a1) where the
three confidence ellipses reflects the close positive correlation between the two
performance measures within a single fund. This phenomenon can be explained by the
similarity between the definitions of the two measures. From (a2), we can see that the
Sharpe measure and Jensen measure for a single fund are also positive correlated, but the
correlation is reduced and not so evident. Also, it is not difficult to show that Jensen
measure and the Appraisal Ratio are linearly correlated as can be seen from (a3). Further
more, if we want to look at the interactive relationship between the measures of different
funds, we could also follow this graphical way. For example, (b1) tells us that the Sharpe
measure of different funds are also positively correlated, but comparing with (a1), we
conclude intuitively that this relationship is not as strong as that of Sharpe and Treynor
measures within one fund. (b2) and (b3) also show some positive correlation between
different measures of different funds.
Economic instinct tells us that the performances of all funds are largely subjected to
the macroeconomic factors such as the functioning of the national economy, the influence
of the international politics and trading, the wholesomeness of the financial and
managerial system and so on. This partly explains the positive correlation of different
types of performance measures. It also provides evidence of the consistency of employing
these measures to evaluate the performance of individual fund or the fund index of a
regional or national economy.
Table 4 provides the results of hypothesis testing using JK’s method. According to the
correspondence of confidence interval and hypothesis testing, if the 95% confidence
interval of the transformed difference of performance measure does not contain 0, then we
have sufficient reason to belief that there is a significant difference between the
performances of the two funds. The same concept can also be applied to the result of Table
3, where bootstrapping confidence intervals of the original differences (not the
transformed ones) of the performance measures are calculated for each combination of
two funds. From the results we conclude that there is no significant difference in the
performance of the funds under investigation, confirming our former intuition about the
analogous bootstrap CIs of the performance measures of the selected funds.
Accordingly, Table 5 describes the 95% bootstrap Confidence Intervals of the
difference in the four performance measures of top 2 and bottom 2 funds selected from
Table 2. The Confidence Intervals are constructed through three popular bootstrapping
methods, namely, the normal-theory CI, the percentile CI and the BCa CIs, respectively.
The bolded values in the following table indicate that the specified confidence interval
does not contain 0, so we have sufficient evidence to believe that there is significant
difference between the corresponding measures. Comparing the result in Table 4 and
Table 5, we can see that there are several discrepancies as are indicated by the bold values
of the confidence intervals. For example, the strict positive values in the three types of
confidence intervals of the difference in the Sharpe measure of Fund 1 (184695) and Fund
33 (500015) indicate that we have sufficient information to believe that investing your total
16
portfolio in Fund 1 is more promising than in Fund 33. Other results can be explained in
the similar way. These results are quite different from those in Table 4, which states that
all the 33 funds are managed similarly (since the top and bottom funds have similar
managerial performances). Intuition and real evidence both discredit this allegation,
which in turn consolidates that accuracy of employing bootstrap hypothesis tests to
generate results and gain useful information for decision making.
5. Concluding Remarks
This paper studied the theoretical and numerical aspects of various bootstrap
methodologies, especially on how to employing bootstrap ideas to construct confidence
intervals and perform hypothesis tests. Theoretical findings based upon some mild
regularity conditions are proved in an attempt to provide a straightforward explanation of
the asymptotic superiority of bootstrap procedures over the traditional ones.
Numerical experiments are carried out to evaluate the managerial performances of 33
representative funds in the financial market of China. The result is forwarded in a
graphical and tabular way to give the reader an intuitive understanding of how bootstrap
works and excels. Furthermore, this paper also advanced a cross scoring methods to assess
the various measures in a synthesizing form. The resulting rank of the 33 representative
funds in the financial market of China provides a pioneering step to evaluate their overall
performance and suggests an objective standard for the investors to resort to. Further
more, the comparison between the accuracy of employing JK’s method and bootstrap
method to perform hypothesis test reveals the fact that the JK’s asymptotic statistics can
not provide a coherent way to test the managerial performance of financial products.
To conclude this paper, I should emphasize again that the utilization of any new
theoretical methodology should be exercised with great care. The real world is not as
perfect as that is assumed in theories, so it is natural to see that some constraints in our
theory are too demanding to be satisfied. However, every problem has its solution as long
as you can realize it. The various refinements of standard bootstrap procedures can be
seen as pioneering steps to reconcile theory and reality. As has been formerly stated, it is
unrealistic to make a tool omnipotent. So, there are still tasks left to us to finish, one of
which mentioned in Section 2 is to bootstrap the models involving unstationary process. In
summary, to unravel the mysteries of the universe is to experience a joyous trip of life and
to contribute what we can to the present world and the future.
17
(a)
Histogram of t
t*
Den
sity
-1.1 -1.0 -0.9 -0.8 -0.7 -0.6
02
46
-3 -2 -1 0 1 2 3
-1.0
-0.9
-0.8
-0.7
-0.6
Quantiles of Standard Normal
t*
(b)
Histogram of t
t*
Den
sity
-2.6 -2.4 -2.2 -2.0 -1.8 -1.6 -1.4
0.0
0.5
1.0
1.5
2.0
2.5
-3 -2 -1 0 1 2 3
-2.6
-2.4
-2.2
-2.0
-1.8
-1.6
Quantiles of Standard Normal
t*
(c)
Histogram of t
t*
Den
sity
-0.016 -0.015 -0.014 -0.013 -0.012
020
040
060
080
0
-3 -2 -1 0 1 2 3
-0.0
16-0
.015
-0.0
14-0
.013
-0.0
12
Quantiles of Standard Normal
t*
(d)
Histogram of t
t*
Den
sity
-1.4 -1.2 -1.0 -0.8
01
23
45
6
-3 -2 -1 0 1 2 3
-1.4
-1.3
-1.2
-1.1
-1.0
-0.9
-0.8
-0.7
Quantiles of Standard Normal
t*
Figure 4 Histograms and normal quantile-comparison plots for the bootstrap
replicates of performance measures of Fund JingBao (184695), where (a) Sharpe Measure,
(b) Treynor Measure, (c) Jensen Measure and (d) Appraisal Ratio. From the graphics, we
can see that the bootstrap samples of all measures are approximately normally distributed.
18
-1.0 -0.9 -0.8 -0.7 -0.6
-2.6
-2.4
-2.2
-2.0
-1.8
-1.6
Sharpe Measure 1
Trey
nor M
easu
re 1
(a1)
-1.0 -0.9 -0.8 -0.7 -0.6
-1.5
-1.4
-1.3
-1.2
-1.1
-1.0
-0.9
-0.8
Sharpe Measure 1
Sha
rpe
Mea
sure
2
(b1)
-1.0 -0.9 -0.8 -0.7 -0.6
-0.0
16-0
.015
-0.0
14-0
.013
-0.0
12
Sharpe Measure 1
Jens
en M
easu
re 1
(a2)
-1.0 -0.9 -0.8 -0.7 -0.6
-2.8
-2.6
-2.4
-2.2
-2.0
-1.8
Sharpe Measure 1
Trey
nor M
easu
re 2
(b2)
-1.4 -1.3 -1.2 -1.1 -1.0 -0.9 -0.8 -0.7
-0.0
16-0
.015
-0.0
14-0
.013
-0.0
12
Appraisal Ratio 1
Jens
en M
easu
re 1
(a3)
-1.0 -0.9 -0.8 -0.7 -0.6
-0.0
155
-0.0
150
-0.0
145
-0.0
140
-0.0
135
-0.0
130
Sharpe Measure 1
Jens
en M
easu
re 2
(b3)
Figure 5 Confidence ellipses of bootstrap samples of performance measures representing the interactive dependence of individual funds and the measures used to evaluate them. (a1) Sharpe measure v.s. Treynor measure, (a2) Sharpe measure v.s. Jensen measure, (a3) Appraisal Ratio v.s. Jensen measure of the same fund. (b1) Sharpe measures of two funds, (b2) Sharpe measure v.s. Treynor measure of another fund and (b3) Sharpe measure v.s. Jensen measure of another fund.
19
Table 1 Ranking 33 representative close funds in the financial market of China. The
result is ordered by the four types of performance measures: Sharpe’s measure, Treynor’s
measure, Jensen’s measure and Appraisal Ratio in each category, using the daily returns of
these representative funds from Jan. 3, 2001 to Dec. 31, 2003.
Sharpe Measure Treynor Measure Jensen Measure Apprasial Ratio
Fund Code value Fund Code value Fund Code value Fund Code value hand 500025 -0.755 taih 500002 -1.862 xingh 500008 -0.014 hand 500025 -0.878 jingb 184695 -0.803 jind 500021 -1.877 xingk 184708 -0.014 jingb 184695 -0.957 tongy 184690 -0.856 jiny 500010 -1.883 taih 500002 -0.014 anx 500003 -0.966 kaiy 184688 -0.858 hand 500025 -1.928 xinghe 500018 -0.014 tongy 184690 -0.977 anx 500003 -0.867 jingb 184695 -1.933 yuz 184705 -0.014 kaiy 184688 -0.983 yuy 500006 -0.873 hanbo 500035 -1.934 jins 184703 -0.014 yuy 500006 -1.023 huip 184689 -0.888 yuz 184705 -1.954 xingan 184718 -0.014 huip 184689 -1.031 taih 500002 -0.897 xingan 184718 -1.964 yuyuan 500016 -0.014 jingy 500007 -1.061 jingy 500007 -0.922 jins 184703 -1.973 yuh 184696 -0.014 jint 500001 -1.093 jingh 184691 -0.925 longy 184710 -1.977 tiany 184698 -0.014 yuyuan 500016 -1.135 jint 500001 -0.950 jingf 184701 -2.070 jingy 500007 -0.014 xingh 500008 -1.139 yul 184692 -0.950 jingh 184691 -2.072 jingb 184695 -0.014 jingh 184691 -1.149 jiny 500010 -0.950 xingk 184708 -2.091 jingf 184701 -0.014 taih 500002 -1.169
yuyuan 500016 -0.960 hans 500005 -2.095 jiny 500010 -0.014 yul 184692 -1.183 xingh 500008 -0.960 yul 184692 -2.118 jinx 500011 -0.014 hans 500005 -1.222 hanbo 500035 -0.963 tiany 184698 -2.139 tongs 184699 -0.014 huif 184693 -1.240 hans 500005 -0.965 tongz 184702 -2.141 yul 184692 -0.014 ans 500009 -1.256 jind 500021 -0.978 hanx 500015 -2.197 jind 500021 -0.014 tiany 184698 -1.285 yuz 184705 -0.992 yuy 500006 -2.200 longy 184710 -0.014 xinghe 500018 -1.285 jins 184703 -1.000 jinx 500011 -2.253 ans 500009 -0.014 jiny 500010 -1.289
tiany 184698 -1.003 yuh 184696 -2.257 hanbo 500035 -0.014 hanbo 500035 -1.292 xingan 184718 -1.003 xinghe 500018 -2.272 hans 500005 -0.014 tongz 184702 -1.325
huif 184693 -1.011 kaiy 184688 -2.290 tongy 184690 -0.014 tongs 184699 -1.342 longy 184710 -1.017 huip 184689 -2.292 hanx 500015 -0.014 yuz 184705 -1.354 tongz 184702 -1.021 tongs 184699 -2.312 huif 184693 -0.014 jins 184703 -1.362
xinghe 500018 -1.026 tongy 184690 -2.313 tongz 184702 -0.014 jind 500021 -1.367 jingf 184701 -1.036 huif 184693 -2.316 huip 184689 -0.014 xingan 184718 -1.376 ans 500009 -1.048 xingh 500008 -2.352 yuy 500006 -0.014 jinx 500011 -1.391
xingk 184708 -1.061 yuyuan 500016 -2.374 jingh 184691 -0.014 jingf 184701 -1.396 tongs 184699 -1.061 jingy 500007 -2.446 kaiy 184688 -0.014 longy 184710 -1.405 jinx 500011 -1.072 jint 500001 -2.517 jint 500001 -0.014 xingk 184708 -1.448
hanx 500015 -1.122 ans 500009 -2.522 hand 500025 -0.015 yuh 184696 -1.539 yuh 184696 -1.135 anx 500003 -2.538 anx 500003 -0.015 hanx 500015 -1.542
20
Table 2 Cross Scoring of 33 representative funds in the financial market of China. The
result is calculated from the cross scoring algorithm using the daily returns of these
representative funds from Jan. 3, 2001 to Dec. 31, 2003. The rank corresponds to both the
name (abbreviation of Chinese character) and the trading code in the security exchange of
China.
Rank Fund Code Sharpe Treynor Jensen ApRatio Cross score
1 jingb 184695 -0.803 -1.933 -0.014 -0.957 0.734 2 taih 500002 -0.897 -1.862 -0.014 -1.169 0.672 3 hand 500025 -0.755 -1.928 -0.015 -0.878 0.469 4 jiny 500010 -0.950 -1.883 -0.014 -1.289 0.281 5 yuz 184705 -0.992 -1.954 -0.014 -1.354 0.203 6 xingh 500008 -0.960 -2.352 -0.014 -1.139 0.203 7 tongy 184690 -0.856 -2.313 -0.014 -0.977 0.188 8 yul 184692 -0.950 -2.118 -0.014 -1.183 0.156 9 jingy 500007 -0.922 -2.446 -0.014 -1.061 0.156
10 yuy 500006 -0.873 -2.200 -0.014 -1.023 0.141 11 jins 184703 -1.000 -1.973 -0.014 -1.362 0.125 12 yuyuan 500016 -0.960 -2.374 -0.014 -1.135 0.109 13 kaiy 184688 -0.858 -2.290 -0.014 -0.983 0.094 14 jingh 184691 -0.925 -2.072 -0.014 -1.149 0.078 15 xingan 184718 -1.003 -1.964 -0.014 -1.376 0.063 16 jind 500021 -0.978 -1.877 -0.014 -1.367 0.063 17 hanbo 500035 -0.963 -1.934 -0.014 -1.292 0.063 18 huip 184689 -0.888 -2.292 -0.014 -1.031 0.047 19 tiany 184698 -1.003 -2.139 -0.014 -1.285 0.047 20 hans 500005 -0.965 -2.095 -0.014 -1.222 0.000 21 xinghe 500018 -1.026 -2.272 -0.014 -1.285 -0.047 22 anx 500003 -0.867 -2.538 -0.015 -0.966 -0.094 23 xingk 184708 -1.061 -2.091 -0.014 -1.448 -0.109 24 jingf 184701 -1.036 -2.070 -0.014 -1.396 -0.188 25 jint 500001 -0.950 -2.517 -0.014 -1.093 -0.219 26 longy 184710 -1.017 -1.977 -0.014 -1.405 -0.234 27 tongz 184702 -1.021 -2.141 -0.014 -1.325 -0.344 28 huif 184693 -1.011 -2.316 -0.014 -1.240 -0.359 29 tongs 184699 -1.061 -2.312 -0.014 -1.342 -0.406 30 jinx 500011 -1.072 -2.253 -0.014 -1.391 -0.406 31 yuh 184696 -1.135 -2.257 -0.014 -1.539 -0.422 32 ans 500009 -1.048 -2.522 -0.014 -1.256 -0.453 33 hanx 500015 -1.122 -2.197 -0.014 -1.542 -0.609
21
Table 3 Bootstrapping the 95% Confidence Intervals of performance measures of
several funds selected from Chinese Market. The Confidence Intervals are constructed
through three popular bootstrapping methods, namely, the normal-theory CI, the
percentile CI and the BCa CIs, respectively. The funds are ranked according to the
performance measures in each category. The rank corresponds to both the name
(abbreviation of Chinese character) and the trading code in the security exchange of China.
Norm-theory
CIs Percentile CIs BCa CIs
Fund Name
Code Point
Estimate Lower Upper Lower Upper Lower Upper
hand 500025 -0.755 -0.876 -0.626 -0.894 -0.642 -0.881 -0.631 jingb 184695 -0.803 -0.914 -0.683 -0.928 -0.698 -0.915 -0.686 taih 500002 -0.897 -1.078 -0.676 -1.139 -0.738 -1.082 -0.695 ans 500009 -1.048 -1.314 -0.712 -1.400 -0.828 -1.317 -0.749 hanx 500015 -1.122 -1.307 -0.912 -1.348 -0.954 -1.307 -0.928
Sharpe
Measure
yuh 184696 -1.135 -1.297 -0.958 -1.319 -0.976 -1.287 -0.936 taih 500002 -1.862 -2.132 -1.563 -2.194 -1.619 -2.156 -1.596 hand 500025 -1.928 -2.274 -1.548 -2.359 -1.633 -2.319 -1.621 jingb 184695 -1.933 -2.228 -1.606 -2.303 -1.665 -2.272 -1.651 hanx 500015 -2.197 -2.501 -1.859 -2.564 -1.923 -2.502 -1.881 yuh 184696 -2.257 -2.564 -1.909 -2.623 -1.975 -2.558 -1.913
Treynor
Measure
ans 500009 -2.522 -2.929 -2.064 -3.002 -2.150 -2.933 -2.076 taih 500002 -0.014 -0.015 -0.013 -0.015 -0.013 -0.015 -0.013 yuh 184696 -0.014 -0.015 -0.013 -0.015 -0.013 -0.015 -0.013 jingb 184695 -0.014 -0.015 -0.013 -0.015 -0.013 -0.015 -0.013 ans 500009 -0.014 -0.015 -0.013 -0.015 -0.014 -0.015 -0.014 hanx 500015 -0.014 -0.015 -0.014 -0.015 -0.014 -0.015 -0.014
Jensen
Measure
hand 500025 -0.015 -0.016 -0.013 -0.016 -0.013 -0.016 -0.013 hand 500025 -0.878 -1.016 -0.724 -1.045 -0.750 -1.019 -0.734 jingb 184695 -0.957 -1.094 -0.805 -1.118 -0.830 -1.099 -0.812 taih 500002 -1.169 -1.488 -0.743 -1.629 -0.920 -1.548 -0.822 ans 500009 -1.256 -1.701 -0.629 -1.864 -0.923 -1.735 -0.794 yuh 184696 -1.539 -1.698 -1.363 -1.726 -1.394 -1.700 -1.371
Appraisal
Ratio
hanx 500015 -1.542 -1.829 -1.190 -1.866 -1.271 -1.792 -1.147
22
Table 4 The 95% Confidence Intervals of the transformed difference in four
performance measures of several funds selected from Chinese Market using JK’s method,
where the numbers in the column “Diff” represents the Cross Rank generated in Table 2.
Transformed Sharpe
Measure
Transformed Treynor
Measure Diff
Estiamte Lower Upper Estimate Lower Upper
1-2 2.70E-05 -5.15E-04 5.69E-04 -4.07E-06 -1.83E-03 1.83E-03 1-3 -1.67E-05 -4.87E-04 4.54E-04 -2.61E-07 -1.85E-03 1.85E-03
1-31 7.56E-05 -4.76E-04 6.27E-04 1.54E-05 -1.76E-03 1.79E-03 1-32 6.11E-05 -3.96E-04 5.18E-04 2.53E-05 -1.73E-03 1.78E-03 1-33 7.46E-05 -4.56E-04 6.05E-04 1.31E-05 -1.79E-03 1.81E-03 2-3 -4.42E-05 -5.33E-04 4.44E-04 3.90E-06 -1.85E-03 1.85E-03
2-31 4.80E-05 -5.20E-04 6.16E-04 1.93E-05 -1.76E-03 1.80E-03 2-32 3.34E-05 -4.87E-04 5.54E-04 2.91E-05 -1.73E-03 1.79E-03 2-33 4.66E-05 -5.32E-04 6.25E-04 1.71E-05 -1.78E-03 1.82E-03 3-31 9.40E-05 -4.65E-04 6.53E-04 1.60E-05 -1.78E-03 1.81E-03 3-32 7.93E-05 -3.84E-04 5.42E-04 2.62E-05 -1.75E-03 1.80E-03 3-33 9.32E-05 -4.35E-04 6.21E-04 1.37E-05 -1.81E-03 1.83E-03
31-32 -1.51E-05 -6.05E-04 5.74E-04 9.69E-06 -1.71E-03 1.73E-03 31-33 -2.20E-06 -6.26E-04 6.22E-04 -2.51E-06 -1.76E-03 1.75E-03
Jobson
Korkie
Method
32-33 1.31E-05 -6.32E-04 6.59E-04 -1.24E-05 -1.75E-03 1.72E-03
23
Table 5 Bootstrapping the 95% Confidence Intervals of the difference in four
performance measures of top 2 and bottom 2 funds selected from Chinese Market. The
Confidence Intervals are constructed through three popular bootstrapping methods,
namely, the normal-theory CI, the percentile CI and the BCa CIs, respectively. The bolded
values in the following table indicate that the specified confidence interval does not
contain 0, so we have sufficient evidence to believe that there is significant difference
between the corresponding measures.
Norm-theory CIs Percentile CIs BCa CIs
Measure Diff
Point Estimate Lower Upper Lower Upper Lower Upper
1-2 0.094 -0.108 0.265 -0.066 0.296 -0.119 0.255 1-32 0.246 -0.069 0.504 0.025 0.558 -0.063 0.505 1-33 0.319 0.154 0.471 0.172 0.488 0.154 0.472 2-32 0.152 -0.190 0.467 -0.146 0.486 -0.199 0.445 2-33 0.225 0.023 0.444 0.007 0.428 0.038 0.468
Sharpe
Measure
32-33 0.073 -0.103 0.293 -0.151 0.223 -0.116 0.254 1-2 -0.071 -0.321 0.173 -0.324 0.181 -0.325 0.180 1-32 0.589 0.219 0.933 0.266 0.975 0.253 0.959 1-33 0.264 -0.019 0.534 0.001 0.555 -0.031 0.536 2-32 0.660 0.321 0.980 0.373 1.033 0.372 1.032 2-33 0.335 0.093 0.570 0.109 0.591 0.116 0.599
Treynor
Measure
32-33 -0.325 -0.611 -0.027 -0.649 -0.064 -0.652 -0.064 1-2 0.000 -0.001 0.001 -0.001 0.001 -0.001 0.001 1-32 0.000 -0.001 0.001 -0.001 0.001 -0.001 0.001 1-33 0.000 -0.001 0.001 -0.001 0.001 -0.001 0.001 2-32 0.000 -0.001 0.001 -0.001 0.001 -0.001 0.001 2-33 0.000 -0.001 0.001 -0.001 0.001 -0.001 0.001
Jensen
Measure
32-33 0.000 -0.001 0.001 -0.001 0.001 -0.001 0.001 1-2 0.212 -0.230 0.562 -0.085 0.684 -0.186 0.598 1-32 0.299 -0.324 0.766 -0.056 0.908 -0.188 0.808 1-33 0.585 0.236 0.895 0.285 0.910 0.189 0.847 2-32 0.087 -0.605 0.715 -0.500 0.764 -0.567 0.712 2-33 0.373 -0.090 0.889 -0.164 0.789 -0.120 0.828
Appraisal
Ratio
32-33 0.286 0.047 0.643 -0.144 0.428 -0.069 0.465
24
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