PAPER:
Determining
a stable relationship between hedge fund index HFRIEquity and S&P 500
behaviour, using filtering and maximum likelihood
Paolo Capelli, Francesca Mariani, Maria Cristina Recchioni,
Fabio Spinelli, Francesco Zirilli
1.
Abstract
2.
Introduction
3. The
calibration procedure and the formulae used to forecast the logreturns
4. Analysis
of the S&P 500 and HRFI Equity indices (digital
movies)
5. References
Warning: If your browser does not allow you to see the content of this website click here to download a word file that
reproduces the material contained here.
We
test the ability of a stochastic differential model proposed in [5] of forecasting the returns of a longshort equity
hedge fund index and of a market index, that is of the HFRIEquity index and of
the S&P 500 index respectively. The model is based on the assumptions that
the value of the variation of the logreturn of the hedge fund index (HFRI Equity)
is proportional up to an additive stochastic error to the value of the
variation of the logreturn of a market index (S&P 500) and that the
logreturn of the market index can be satisfactorily modeled using the Heston
stochastic volatility model. The model consists in a system of three stochastic
differential equations, two of them are the Heston stochastic volatility model
and the third one is the equation that models the behaviour of the hedge fund
index and its relation with the market index. The model is calibrated on
observed data using a method based on filtering and maximum likelihood proposed
in [9] and further developed in [4], [5], [6]. The data observed and analyzed go from January
1990 to June 2007, and are monthly data. For each observation time they consist
in the value at the observation time of the logreturns of the HFRIEquity and
of the S&P 500 indices. The calibration procedure uses appropriate subsets
of the data, that is the data observed in a six months time period. The values
of the HFRIEquity and of the S&P 500 indices logreturns forecasted by the
calibrated models are compared to the values of the observed indices logreturns.
The result of the comparison is very satisfactory. This website contains some
auxiliary material that helps the understanding of [4].
A more general reference to the work of some of the authors and of their
coauthors in mathematical finance is the website: http://www.econ.univpm.it/recchioni/finance.
We test on real data the model proposed in [4],
[5] that describes the dynamics of the index of
some classes of hedge funds and of a market index. Let us recall the model
proposed. We denote with R and R^{+} the set of real numbers and of
positive real numbers respectively, with t the time variable, and with (x_{t},z_{t},v_{t}),
t > 0, a stochastic process, we will interpret x_{t},
t > 0, as the logreturn of a market index, z_{t},
t > 0, as the logreturn of the index of a class of
hedge funds and v_{t}, t > 0, as the stochastic variance of the
market index. We consider the S&P 500 index as market index and the
HFRIEquity hedge fund index as index of a class of hedge funds, that is the
class of ``longshort equity" hedge funds. The model proposed in [4], [5] can be used with
several other choices for the meaning of the indices. In [4],
[5] the
dynamics of the stochastic process (x_{t},z_{t},v_{t}),
t > 0, is modeled by the following system of
stochastic differential equations:
_{}
where _{}, b, g, c, q, e are
constants, _{}, _{}, _{}, t ³ 0, are standard Wiener processes such that W_{0}^{1}=W_{0}^{2}= W_{0}^{3
}=0, and d_{}, d_{}, d_{}, t>0, are their stochastic differentials. Moreover we assume that:


where
< · > denotes the expected value of ·, and
the quantities r_{1,2}, r_{1,3}, r_{2,3} Î [1,1] are constants known as
correlation coefficients. We note that the autocorrelation coefficients of dW^{i}_{t},
t>0, i = 1,2,3 are equal to one.
Equations
(1), (3) are the well known Heston stochastic volatility model [8]. This model was introduced with the aim of overcoming
some limitations of the Black Scholes model [3] that
are pointed out by market data such as, for example, the assumption of constant
volatility. Let S_{t}, t > 0, be the price at time t of an
asset, the Heston model (1), (3) describes satisfactorily the dynamics of the logreturn
x_{t} = log(S_{t}/S_{0}), t > 0, of the asset and of its stochastic variance v_{t}, t > 0. Note that
x_{0} = log1 = 0 =_{} and that since z_{t}, t>0, is a logreturn a
similar statement holds for z_{0}, that is z_{0} = 0 =_{} . A systematic data analysis has shown that the Heston stochastic
volatility model is well suited to model the behaviour of several market indices
such as, for example, the S&P 500 index, the DowJones Industrials index
and the Nasdaq Composite index (see [7]). We note
that equation (3) is known as equation of the meanreverting process with speed c and parameters q and e. The initial stochastic variance _{} is a random variable
that we assume to be concentrated in a point, that we continue to denote
with _{}, with probability one.
We
note that the model (1), (2), (3) introduced
in [5] and analyzed on real data in [4]
is obtained adding to the Heston stochastic volatility model (1), (3)
a third equation, that is equation (2), that describes the behaviour of
the logreturn of the index of some
classes of hedge funds. In fact statistical studies, such as [10], of the
time series of the data relative to the
logreturns of the indices of several classes of hedge funds have shown
that the logreturns of the indices of some classes of hedge funds are related
to the logreturns of the S&P 500 index. In particular in [10] some discrete time models are proposed to
study nine different classes of hedge funds.
The model proposed in [10] to describe the dynamics of the logreturn of the index of the
“longshort equity” class of hedge funds is based on the assumption that up to
a stochastic additive error there exists a kind of direct proportionality
between the behaviour of the variation of the logreturn of the index of the “longshort equity"
class of hedge funds and the variation
of the logreturn of the S&P500 index. In [10]
this assumption is supported by convincing empirical evidence. So that
interpreting z_{t}, t>0, as the logreturn at time t of an index
of the “longshort equity" class of
hedge funds, that is the HFRIEquity
index, x_{t}, t>0, as the logreturn at time t of the S& P500
index and v_{t}, t>0, as the stochastic variance of x_{t}, t>0, the
system of stochastic differential equations (1), (2), (3) obtained
coupling the Heston model (1), (3) with
equation (2) can be seen as a reasonable translation in continuous time of the model proposed in [10]. In fact equation (2) states that dz_{t}
is given by the sum of b dx_{t} and a random disturbance given
by g_{}dW_{t}^{3}.
The calibration
problem of model (1), (2), (3) can be stated as follows: given a discrete
set of time values t=t_{i}, i=0,1,…,n, such that t_{0}=0, t_{i}<t_{i+1},
i=0,1,…,n1, and the observation _{} of the logreturns of
the market and of the hedge fund
indices, that is the observation of (x_{t},z_{t}), at time t=t_{i},
i=0,1,…,n, determine the values of the parameters appearing in (1), (2), (3), of the correlation
coefficients r_{1,2}, r_{1,3}, r_{2,3
}and of the
initial stochastic variance _{}. That is the
parameters (including the correlation coefficients) and the unknown initial
condition component of model (1), (2),
(3) that we want to estimate starting from the observations are: _{}, q, c, e, g, b, r_{1,2, }r_{1,3, }r_{2,3 } and _{}.
Note
that the choice t_{0}=0 corresponds to choosing the origin of the time
axis in the first observation time where we choose _{}=(0,0). Note that the origin of the time axis can be moved at our convenience in the time
series of the data without changing substantially the problem considered.
In
order to study time series of real data the calibration problem
of model (1), (2), (3) stated
previously has been solved using the approach suggested in [5] based on the methods of filtering and maximum likelihood. This approach was introduced in the context of mathematical finance in [9] and
further developed in [5], [6].
Going into details, we suppose that at discrete times 0=t_{0}<t_{1}<t_{2}<…<t_{n}<t_{n+1
}=+¥,
the logreturn of the
S&P500 index and the
logreturn of the HFRI Equity index are
observed and let F_{t}=
{(_{},_{}) : t_{i }£t}, t>0, be the set of the observations
available at time t>0 of the
logreturns of the market index x_{t} and of the logreturns of the
index of the hedge funds z_{t}. We assume that the observations are
error free, that is we assume that _{}=_{}, _{}=_{}, i=0,1,…,n. Let _{} be the vector of the
model parameters (including the correlation coefficients) and of the initial
stochastic variance, where the superscript ^{T} denotes the transposed
operator. We use the notation t_{0} =0 and F_{0}={(_{})}to simplify some of the formulae that follow.
We
determine the joint probability density function of having x_{t}=x, z_{t}=z
and v_{t} =v at time t>0 conditioned to the observations contained in F_{t}, t>0, and to the initial condition (6). Note that the initial conditions (4), (5)
are already contained in F_{t}, t>0. This joint probability
density function is determined solving a filtering problem that has been
presented in [9], [5], [6].
In [5] the calibration problem is translated in a maximum
likelihood optimization problem (see Section 3). We apply
these methods to analyze real data. In particular we consider the time series of monthly data of the S&P
500 and of the HFRIEquity indices covering a period of 210 months going from
January 31, 1990 to June 30, 2007.
The observation times will be denoted with t=_{}, i=1,2,…,210, and we introduce t=_{} corresponding to December 31, 1989. From these data we
derive the corresponding time series of the logreturns. We have applied the calibration procedure
described above using as data the data contained in a window of six consecutive observation times (that is a window covering the data relative to a period of six
months) corresponding to twelve data,
that is the window corresponding to the data
(_{},_{}) observed at time t=_{}, i=k, k+1,…,k+5 for some k. Note that in the calibration
problems derived from the data time series the origin of the time axis is
translated to the first observation time contained in the data window
considered. The numerical results
obtained considering the data windows
associated to the choices k=0,1,…,205 are presented in Section
4 and they show that the solutions of the calibration problems considered are
really associated to the data time series, that is they are “stable” when the time window of the data used in the
calibration is shifted. In fact as shown
in Figures 1, 2, 3 the solution of the calibration problem as a function of the time window of the data
used in the calibration can be grouped
into three sets associated to the data
of three non overlapping time periods and in
these three sets the parameters
(including the correlation coefficients)
and the initial stochastic variance found
are approximately constants (see Section 4 for further details). After calibrating the model using the data belonging to
a six month window we use the resulting estimate of the parameters and
of the initial stochastic variance to
forecast the market and the hedge fund
indices one, three and six months in the future counting as future the time after the last observation time
contained in the data window used in the calibration. The forecasted values of the market and of the
hedge fund indices are obtained using some formulae derived in [5] that
translate to the case of model (1), (2), (3)
standard formulae of filtering theory.
We perform this forecasting exercise moving the data window along the data time
series step by step, at each step
we discard the observations relative to the first observation time of the
window and we insert the observations relative to the next observation time
after the window, that is in the previous notation we consider the data windows
associated to k=0,1,...,205.The quality of these forecasts is established a
priori using filtering theory and a posteriori comparing the forecasted
values with the historical data. The results obtained suggest that the model proposed
describes satisfactorily the
data, and that it is able to produce high quality forecasts of the value of the hedge fund index logreturn
several months in the future. Forecasts of approximately the same quality are
obtained for the logreturns of the market index. We remark that the forecasts
that are expected to be good a priori on the basis of filtering theory are a
posteriori actually better than the average forecast.
Let us
formulate the calibration problem and let us give some formulae used to
forecast the logreturns x_{t}, z_{t} and the stochastic
variance v_{t}, t > 0, solution of problem (1), (2), (3),
(4), (5), (6). Moreover we give some
formulae that can be used to evaluate “a priori” the quality of the forecasted
values of the logreturns (and of the stochastic variance) as explained in Section 4.
Let us consider the joint
probability density function _{} =_{}(x,z,v,tF_{t}, Q), (x,z,v) Î R×R×R^{+}, t > 0, of having x_{t} = x, z_{t}
= z, v_{t} = v given F_{t}, t > 0, and Q. Remind that _{}=_{} = 0. The joint probability density function _{} is the solution of a
filtering problem, and in [4] , [5] we have shown that the function _{} is given by:
_{} _{} i=0,1,…,n, (10)
where G(x,z,v,t,x¢,z¢,v¢,t¢ Q), (x,z,v), (x¢,z¢,v¢) Î R×R×R^{+}, t, t¢ > 0, tt¢ > 0, is the fundamental solution of
the Fokker Planck equation associated to the system of stochastic differential
equations (1), (2), (3), and we have
for: i = 0:





_{} (12)
where d(^{.}) is the Dirac’s delta and _{} denotes the left limit t that goes to t_{i}.
In order to determine the
vector Q we solve the calibration problem, that is we
solve the optimization problem:


where the
loglikelihood function F( Q) is given by:
_{} (14)
and the set of the admissible vectors M
is given by:
_{}. (15)
We
maximize the function (14) using as optimization method a variable metric
steepest ascent method. This method is a kind of steepest ascent method based
on an iterative procedure that searches the maximum likelihood estimate Q^{*},
solution of (13), beginning from an initial guess Q^{0} Î M and that for k = 1,2,¼ generates at step k a feasible point Q^{k} Î M satisfying the inequality F(Q^{k}) > F( Q^{k1}), that is the objective function is
monotonically increasing along the sequence {Q^{k}}, k = 0,1,¼. In the numerical experience presented in Section
_{}
where e_{tol}, k_{max} are
positive constants that will be chosen later.
We note that the loglikelihood function (14)
is only one possible choice between many
other possibilities and that the
constraints contained in (15) that define M express some elementary properties satisfied by model (1), (2), (3).
Given
the joint probability density function _{}(x,z,v,tF_{t}, Q), (x,z,v) Î R×R×R^{+}, t ³ 0, we can forecast the values of
the market index logreturn, of the hedge fund index logreturn x_{t},
z_{t}, t > 0, t ¹ t_{i},
i = 0,1,¼,n, and of the stochastic variance v_{t}, t > 0, using respectively the mean values _{}_{t}_{}_{ }_{Q}, _{}_{t}_{}_{ }_{Q}, _{}_{t}_{}_{ }_{Q}, t > 0, conditioned to the observations contained in F_{t}, t > 0, of the random variables x_{t}, z_{t}, v_{t},
t > 0, that is:
_{} (17)
_{}
(18)
_{}_{}
(19)
Note that in Section 4 we
are interested in forecasting x_{t}, z_{t}, v_{t } when t>t_{n} and that this
corresponds to the genuine meaning of the word “forecast”.
As shown in
[5] from (10), (12), (17), (18), (19), we have:
_{}
_{}
_{}
Remind
that we use the notation t_{n+1} = +¥. Note that it
is easy to see that, as it should be, we have: _{} , _{} , i = 0,1,¼,n. The quality of the forecasted
values _{}depends from the variance of the random variables x_{t},
z_{t}, v_{t}, t>0, conditioned
to the observations, that is we can estimate the quality of the estimates (20),
(21), (22) computing respectively the quantities:
_{}
(23)
_{}
(24)
_{}
(25)
The estimates (20), (21), (22) are expected to
be good, when the variances (23), (24), (25) are small. In [5] some formulae similar to (20), (21), (22) are
derived from (23), (24), (25).
Let us present the results obtained applying
the procedure described in the previous Sections to the historical series of
210 monthly data relative to the variation DI_{x,i}
of the S&P 500 index I_{x,i}, at time t=_{}, i = 1,2,¼,210, and to the variation DI_{z,i} of the HFRIEquity index I_{z,i}, at time t=_{}, i=1,2,…,210 (see formulae (30), (31)). The data cover the
period of 210 months going from January 31, 1990 to June 30, 2007. The
observation dates t =_{}, i = 1,2,¼, 210 are the last day of the month
(i.e. January 31, February 28, March 31, April 30, May 31, June 30, July 31,
August 31, September 30, October 31, November 30, December 31) of the period
January 1990, June 2007. We have added to these data the observation at date t
=_{} corresponding to December 31, 1989. We choose I_{x,0}=I_{z,0}=1 and _{}, _{}. First of all we have manipulated the data DI_{x,i}, DI_{z,i}, i = 1,2,¼,
We
have used the following recursive formulae to construct the logreturns (i.e.
the last two columns of Table 1):
_{}
_{}
where as mentioned above _{}, _{} correspond to the logreturns at time t =_{} (December 31, 1989) that have been chosen equal to zero.
The
indices I_{x,0}, I_{z,0 } have been assumed to be equal one and I_{x,i},
I_{z,i}, i = 1,2,¼, 210 are related to the monthly
logreturns _{}, _{}, i = 1,2,¼,210, defined in (26), (27) by the
following formulae:

(28) 

(29) 

(30) 

(31) 
We consider a window of six consecutive observation
times (that is a window covering a time period of six months) corresponding to
twelve data and we move from a window to the next one removing the data
relative to the first observation time of the window and adding the data
corresponding to the observation time that follows the last observation time of
the window. That is the jth window contains the data (_{} , _{}), i = 1,2,¼,6, j = 1,2,¼,206. For each data window we solve the calibration problem (13). We
have solved 206 calibration problems. Remind that in the solution of each
calibration problem, coherently with the statement of the problem given in the
Introduction, we translate appropriately the origin of the time axis.
The
data analysis presented investigates the following two problems.
Problem 1: understand
if the calibration procedure described in Section 2 to determine the value of
the vector Q, that is the maximum likelihood
problem (13), gives values of Q that are really associated to the
data time series, that is values of Q that are approximately constant
when we change the data window used in the calibration. If this is the case the
values of Q determined by the calibration
procedure are not an artifact of the computational procedure used to determine
them. Note that despite the fact that the stochastic variance is the solution
of a stochastic differential equation it is reasonable to assume that the
stochastic variance is approximately constant over long periods of time and
that it changes abruptly from time to time.
Problem 2: evaluate
the capability of the models corresponding to the values of Q determined with the calibration procedure of forecasting the values
of x_{t}, z_{t} in the future, that is of forecasting using the
formulae (20), (21) the values x_{t},
z_{t} relative to observation times posterior to the observation times
contained in the data window used to estimate the vector Q. This capability is evaluated a
priori using formulae (23), (24) and established a posteriori comparing the
forecasted values with the observations actually made. Note that Problem 2 is considered
since Problem 1 is solved positively.
Investigation
of Problem 1
As said previously
we have solved 206 calibration problems relative to the 206 data windows
described above determining 206 values of the vector Q. In the solution of the 206 maximization problems (13) we have chosen
the initial guess of the maximization procedure to be always the same vector
and in the stopping criterion (16) we have chosen e_{tol}= 5·10^{4} and k_{max}
= 10000.
Figures
1, 2, 3 show (in ordinate) the
components of the vector Q obtained solving the maximum
likelihood problem (13) as a function of i (in abscissa), where i is the index
value of the last observation time _{} of the data window
considered. That is the vector Q obtained using a given data window
is associated to the index of the last observation time of the data contained
in the window.
Figure 1: Reconstruction of the parameters e, q, c, g (in ordinate)
of the model (1), (2), (3) as a function of the index value of the last
observation time contained in the data window considered (in abscissa)
Figure 2: Reconstruction of the initial stochastic
variance _{} and of the parameters b, m (in ordinate) of the model (1), (2), (3) as a
function of the index value of the last observation time contained in the data
window considered (in abscissa)
Figure 3:
Reconstruction of the correlation coefficients r_{1,2}, r_{1,3}, r_{2,3} (in ordinate) of the model (1), (2),
(3) as a function of the index value of the last observation time contained in
the data window considered (in abscissa)
Figures
1, 2 and 3 show that the parameters e, g, the initial stochastic variance _{} and the correlation
coefficients r_{1,2} and r_{2,3} as a function of i are approximately given by
piecewise constant functions. In particular two observation times (marked with the
red bars in the Figures) where the piecewise constant functions jump are
evidenced, the first one is located approximately after the first year of
observation (October 31, 1990) and the second one is located approximately after
ten years of observations (October 31, 2000). The data analysis carried out and
illustrated in Figures 1, 2, 3 shows that the data can be divided into three
periods:
That
is the data analysis carried out solving the 206 calibration problems
considered shows that the vectors Q determined by the calibration
procedure can be considered approximately constants in each one of the three
Periods mentioned above.
Moreover
in Figure 3 we can see that the correlation coefficient r_{1,3} that
measures the correlation between two of the stochastic differentials present in
the equations of the logreturn of the S&P 500 index and of the logreturn
of the hedge fund HFRIEquity index, is almost constant and approximately equal
to one over the entire observation period. This fact seems to confirm the
validity of model (1), (2), (3) that assumes that the behaviour of the hedge
fund HFRIEquity index depends strongly from the behaviour of the S&P 500
index. Furthermore the fact that the parameter b is
substantially constant (exception made when we go from Period 1 to Period 2
where b goes approximately from 0.5 to 0.7) confirms
the assumption of the existence of a kind of
“proportionality” between the variations of the HFRIEquity hedge
fund index and of the S&P 500 index logreturns.
We
can conclude that the data analysis presented shows a convincing evidence of
the fact that the values of the vector Q determined by the maximum
likelihood procedure (13) are really associated to the data time series and
supports some of the assumptions made to build model (1), (2), (3).
Investigation
of Problem 2
The
second problem addressed in the data analysis is the investigation of the
quality of the forecasted values of the logreturns of the indices obtained
using model (1), (2), (3) and the values
of Q shown in
Figures 1, 2, 3 determined solving the calibration problems. That is, we use
the values of the vectors Q determined solving the maximum
likelihood problems and formulae (20), (21) to forecast the values of the
returns x_{t} and z_{t} and of the increments of the
HFRIEquity index_{}= _{}, i=5,6,…,210m, m = 1,2,¼,6, that is one month, two months,... up to six months in
the future. Note that here “future” means the time that follows the last
observation contained in the data window used to estimate the vector Q.
Let
N_{tot} be a positive integer that denotes the number of forecasted values m months in the future of the returns of the S&P 500 index and of
the HFRIEquity index used in this study. Note that below we will choose N_{tot}
independent of m, m=1,2,…,6. The forecasted values are compared with the
historical data and the following quantities have been defined to measure the
accuracy of the forecasted values:
where the
quantities _{}, _{} are given by:
_{}
where_{}, _{} and _{}, i = 1,2,¼,N_{tot} , m=1,2,…,6, are the observed values (real data) and _{} _{}, _{} are the corresponding
forecasted values m months in the future, m=1,2,…,6 and_{}, i=1,2,…,N_{tot}, j=1,2,…m, are the variations of
the index I_{z,4+i+j1 }computed using the forecasted values _{}, i=1,2,…, N_{tot}, m=1,2,…,6. Remind that t =_{}, i = 1,2,¼,210 are the observation times where
the historical data are given. Finally we have computed the mean value of the
quantities _{}, i = 1,2,¼,N_{tot}, that is s_{m}
= _{} , m=1,2,…,6.
In
Table 2 we show the values of the quantities e_{x,m}, e_{z,m}, e_{var,m} and
s_{m} as a function of the forecasting period m (one, two,..., six
months in the future) and we consider N_{tot} = 200 observation times.
The choice N_{tot} = 200 depends from the fact that when we forecast
the values of the two indices up to six months in the future using data windows
of six months and a time series of 211 monthly observation times (remind that
we have also the couple (_{},_{}) at time t=_{}), we cannot forecast values corresponding to the first six
observation times and we cannot check the quality comparing with the historical
data of the forecasted values obtained using as data windows containing data
corresponding to the last six observation times so that we can consider the
last observation time of the windows indexed by i=5,6,…,204, that is we can
consider N_{tot} = 2045+1 = 200
dates (i.e. t=_{}, _{}, …., _{}) such that “starting”
from them we can compute forecasted values six months in the future that
can be compared with the historical data.
We
note that the quantities e_{x,m}, e_{z,m} are the mean values of the relative errors
committed on the forecasted values of the logreturns x_{t}, z_{t}
of the S& P 500 index and of the HFRIEquity index respectively and that
for m=1,2,…,6, e_{var,m} is the mean value of the absolute errors
committed on the forecasted values of the HFRIEquity index I_{z,t} = _{} m months in the
future, that is one month, two months, ..., up to six months in the future.
Table 2: Quality indices of the forecasted values
Forecasting period m 
_{} 
_{} 
_{} 
_{} 
1 month 
0.1271 
0.0214 
0.0202 
0.0217 
2 months 
0.1739 
0.0331 
0.0327 
0.0365 
3 months 
0.1976 
0.0397 
0.0435 
0.0499 
4 months 
0.2226 
0.0478 
0.0530 
0.0631 
5 months 
0.2425 
0.0577 
0.0633 
0.0770 
6 months 
0.2967 
0.0616 
0.0715 
0.0907 
Finally we show the
histograms of the absolute errors committed on the quantities a_{t,m}, that
is the histograms of the values taken by the function n_{m}(t) =_{}, t =_{}, i = 1,2,¼,N_{tot}, m = 1,3,6 (see
Figures 4, 5, 6).
Figure 4: Histogram of n_{1}(t)
Figure 5:
Histogram of n_{3}(t)
Figure 6:
Histogram of n_{6}(t)
Table 3 shows the observations and the forecasted
values. Note that in Table 3 in the first 46 observation times some
values are evidenced in blue colour. These values are some of the most
satisfactory forecasted values of a_{t,m},
m = 1,6 one month and six months in the future. We have computed the
conditioned variance of the state variable z_{t} (formula (24))
corresponding to these forecasted values of the HFRIEquity index. We observe that the
conditioned variances relative to the forecasted values of the returns one
month or six months in the future marked in blue are significantly smaller than
the remaining ones. For example, while the mean value of the conditioned
variances of the forecasted values of the returns of the HFRIEquity index one
month in the future is 1.27·10^{2} the value of the variances of the
forecasted values marked in blue (in time order) are 7.4·10^{3},
4.4·10^{3}, 6.1·10^{3}, that is the conditioned variance of
the blue forecasts is reduced of approximately a factor 0.5 with respect to its
mean value. The same happens in the case of the forecasted values six months in
the future where the mean value of the conditioned variance is 2.5·10^{2}
and the conditioned variances of the forecasted values marked in blue (in time
order) are 9.94·10^{3}, 9.96·10^{3}, 1.18·10^{2} and
1.23·10^{2}. That is also in this last case the conditioned variances
of the forecasted values marked in blue are reduced of approximately a factor
0.5 with respect to its mean value. This
fact observed in the high quality forecasts contained in the first 46
observation times is confirmed when we look to the entire data time series. We
have limited the study of the high quality forecasts to the first 46
observation times for simplicity. Similar results are obtained studying the
forecasted values of the S&P 500 index.
This
analysis shows that the conditioned variance allows us to assign a priori a
degree of reliability to the corresponding forecasted value, that is: a small
conditioned variance actually corresponds to a great degree of reliability of
the corresponding forecasted value. This fact may be of great relevance in
practical situations since using the conditioned variances we can evaluate a
priori the quality of the forecasts of
the HFRIEquity index.
We conclude showing two
digital movies concerning the forecasting problem.
The
first movie shows
the forecasted values one month , three months and six months in the
future of the SP&500 index and of
the HFRI Equity hedge fund index in comparison with the observed values, that
is the point _{} one, three, six months
in the future is shown in the cartesian plane and compared with the
corresponding observed points when the
time t ranges in the observation period that goes from May 31, 1990 to December 31,
2006.
Click here to see the first movie
The
second movie shows how the conditioned
variances of the indices (see formulae (23), (24) ) can be used as a priori
estimates of the quality of the forecasted values. In the movie we show the forecasted values _{} of the variations of
the logreturn of the HFRI hedge fund index, the corresponding historical
variations _{}, i=1,2,…,N_{tot} ,m=1,3,6, and the corresponding
conditioned variances. Note that when the conditioned variance of z_{t}
is small the quality of the forecasted
value is high.
Click here to see the second movie
[1] Y. AitSahalia, R. Kimmel, Maximum
likelihood estimation of stochastic volatility models, Journal of Financial
Economics, 83 (2007), 413452.
[2] D.S. Bates, Maximum
likelihood estimation of latent affine processes, The Review of Financial
Studies, 19 (2006), 909965.
[3] F. Black, M. Scholes, The
pricing of options and corporate liabilities, Journal of Political Economy,
81 (1973), 637659.
[4] P. Capelli, F. Mariani, M.C.
Recchioni, F. Spinelli, F.Zirilli, Determining a stable relationship between
hedge fund index HFRIEquity and S&P 500 behaviour, using filtering and maximum
likelihood,Inverse Problems in Science and Engineering 18 (2010),83109.
[5] L.Fatone,
F. Mariani, M.C.Recchioni, F.Zirilli, Maximum likelihood estimation of the
parameters of a system of stochastic differential equations that models the returns
of the index of some classes of hedge funds, Journal of Inverse and
IllPosed Problems, 15 (2007), 329362, http://www.econ.univpm.it/recchioni/finance/w5.
[6] L. Fatone, F. Mariani, M.C. Recchioni,
F.Zirilli, The calibration of the Heston stochastic volatility model using
filtering and maximum likelihood methods, in Proceedings of Dynamic Systems and Applications,
G.S.Ladde, N.G.Medhin, Chuang Peng, M.Sambandham Editors, Dynamic Publishers, Atlanta, USA, 5 (2008), 170181, http://www.econ.univpm.it/recchioni/finance/w6.
[7] A. Harvey, R. Whaley, Market volatility prediction and the efficiency of the S&P 500
Index Option Market, Journal of Financial Economics, 31 (1992), 4374.
[8] S.L. Heston, A closedform solution for
options with stochastic volatility with applications to bond and currency
options, The Review of Financial Studies, 6 (1993), 327343.
[9] F. Mariani, G.
Pacelli, F. Zirilli, Maximum likelihood
estimation of the Heston stochastic volatility model using asset and option
prices: an application of nonlinear filtering theory, Optimization Letters, 2 (2008), 177222, http://www.econ.univpm.it/pacelli/mariani/w1.
[10] P. Pillonel, L. Solanet, Predictability
in hedge fund index returns and its application in fund of hedge funds style
allocation, Master's Thesis in Banking and Finance at Université de
Lausanne, Hautes Etudes Commerciales (HEC), 2006. (downloadable from the website:
http://www.hec.unil.ch/cms_mbf/master_thesis/0403.pdf).
TABLE 1: Historical data
i 
date 
variation HFRI
Equity Hedge Index 
variation S&P
500 
HFRI Equity
Hedge Index _{} 
S&P
500 _{} 
HFRI Equity Hedge return _{} 
S&P 500 return _{} 
0 
31/12/1989 
 
 
 
 
0.00000 
0.00000 
1 
31/01/1990 
3.34% 
6.88% 
3.340E02 
6.880E02 
0.03397 
0.07128 
2 
28/02/1990 
2.85% 
0.85% 
2.850E02 
8.539E03 
0.00587 
0.06278 
3 
31/03/1990 
5.67% 
2.43% 
5.670E02 
2.426E02 
0.04928 
0.03881 
4 
30/04/1990 
0.87% 
2.69% 
8.700E03 
2.689E02 
0.04054 
0.06607 
5 
31/05/1990 
5.92% 
9.20% 
5.920E02 
9.199E02 
0.09806 
0.02194 
6 
30/06/1990 
2.52% 
0.89% 
2.520E02 
8.886E03 
0.12295 
0.01301 
7 
31/07/1990 
2.00% 
0.52% 
2.000E02 
5.223E03 
0.14275 
0.00777 
8 
31/08/1990 
1.88% 
9.43% 
1.880E02 
9.431E02 
0.12377 
0.09129 
9 
30/09/1990 
1.65% 
5.12% 
1.650E02 
5.118E02 
0.14013 
0.14382 
10 
31/10/1990 
0.77% 
0.67% 
7.700E03 
6.698E03 
0.14780 
0.15054 
11 
30/11/1990 
2.29% 
5.99% 
2.290E02 
5.993E02 
0.12464 
0.09234 
12 
31/12/1990 
1.02% 
2.48% 
1.020E02 
2.483E02 
0.13479 
0.06781 
13 
31/01/1991 
4.90% 
4.15% 
4.900E02 
4.152E02 
0.18262 
0.02713 
14 
28/02/1991 
5.20% 
6.73% 
5.200E02 
6.728E02 
0.23332 
0.03798 
15 
31/03/1991 
7.22% 
2.22% 
7.220E02 
2.220E02 
0.30303 
0.05994 
16 
30/04/1991 
0.47% 
0.03% 
4.700E03 
3.198E04 
0.30772 
0.06026 
17 
31/05/1991 
3.20% 
3.86% 
3.200E02 
3.860E02 
0.33922 
0.09813 
18 
30/06/1991 
0.59% 
4.79% 
5.900E03 
4.789E02 
0.34510 
0.04906 
19 
31/07/1991 
1.41% 
4.49% 
1.410E02 
4.486E02 
0.35910 
0.09294 
20 
31/08/1991 
2.17% 
1.96% 
2.170E02 
1.965E02 
0.38057 
0.11240 
21 
30/09/1991 
4.30% 
1.91% 
4.300E02 
1.914E02 
0.42267 
0.09308 
22 
31/10/1991 
1.16% 
1.18% 
1.160E02 
1.183E02 
0.43420 
0.10484 
23 
30/11/1991 
1.08% 
4.39% 
1.080E02 
4.390E02 
0.42335 
0.05994 
24 
31/12/1991 
5.02% 
11.16% 
5.020E02 
1.116E01 
0.47233 
0.16574 
25 
31/01/1992 
2.49% 
1.99% 
2.490E02 
1.990E02 
0.49692 
0.14564 
26 
29/02/1992 
2.90% 
0.96% 
2.900E02 
9.565E03 
0.52551 
0.15516 
27 
31/03/1992 
0.28% 
2.18% 
2.800E03 
2.183E02 
0.52270 
0.13309 
28 
30/04/1992 
0.27% 
2.79% 
2.700E03 
2.789E02 
0.52540 
0.16060 
29 
31/05/1992 
0.85% 
0.10% 
8.500E03 
9.640E04 
0.53386 
0.16156 
30 
30/06/1992 
0.92% 
1.74% 
9.200E03 
1.736E02 
0.52462 
0.14405 
31 
31/07/1992 
2.76% 
3.94% 
2.760E02 
3.940E02 
0.55185 
0.18269 
32 
31/08/1992 
0.85% 
2.40% 
8.500E03 
2.402E02 
0.54331 
0.15838 
33 
30/09/1992 
2.51% 
0.91% 
2.510E02 
9.106E03 
0.56810 
0.16744 
34 
31/10/1992 
2.03% 
0.21% 
2.030E02 
2.106E03 
0.58820 
0.16955 
35 
30/11/1992 
4.51% 
3.03% 
4.510E02 
3.026E02 
0.63231 
0.19936 
36 
31/12/1992 
3.38% 
1.01% 
3.380E02 
1.011E02 
0.66555 
0.20942 
37 
31/01/1993 
2.09% 
0.70% 
2.090E02 
7.046E03 
0.68624 
0.21644 
38 
28/02/1993 
0.57% 
1.05% 
5.700E03 
1.048E02 
0.68052 
0.22687 
39 
31/03/1993 
3.26% 
1.87% 
3.260E02 
1.870E02 
0.71260 
0.24539 
40 
30/04/1993 
1.30% 
2.54% 
1.300E02 
2.542E02 
0.72552 
0.21964 
41 
31/05/1993 
2.72% 
2.27% 
2.720E02 
2.272E02 
0.75235 
0.24211 
42 
30/06/1993 
3.01% 
0.08% 
3.010E02 
7.552E04 
0.78201 
0.24287 
43 
31/07/1993 
2.12% 
0.53% 
2.120E02 
5.327E03 
0.80299 
0.23752 
44 
31/08/1993 
3.84% 
3.44% 
3.840E02 
3.443E02 
0.84067 
0.27137 
45 
30/09/1993 
2.52% 
1.00% 
2.520E02 
9.988E03 
0.86556 
0.26134 
46 
31/10/1993 
3.11% 
1.94% 
3.110E02 
1.939E02 
0.89618 
0.28054 
47 
30/11/1993 
1.93% 
1.29% 
1.930E02 
1.291E02 
0.87669 
0.26755 
48 
31/12/1993 
3.59% 
1.01% 
3.590E02 
1.009E02 
0.91197 
0.27759 
49 
31/01/1994 
2.35% 
3.25% 
2.350E02 
3.250E02 
0.93519 
0.30957 
50 
28/02/1994 
0.40% 
3.00% 
4.000E03 
3.005E02 
0.93119 
0.27906 
51 
31/03/1994 
2.08% 
4.57% 
2.080E02 
4.575E02 
0.91017 
0.23223 
52 
30/04/1994 
0.37% 
1.15% 
3.700E03 
1.153E02 
0.90646 
0.24369 
53 
31/05/1994 
0.41% 
1.24% 
4.100E03 
1.242E02 
0.91055 
0.25604 
54 
30/06/1994 
0.41% 
2.68% 
4.100E03 
2.681E02 
0.90644 
0.22886 
55 
31/07/1994 
0.91% 
3.15% 
9.100E03 
3.149E02 
0.91550 
0.25986 
56 
31/08/1994 
1.27% 
3.76% 
1.270E02 
3.762E02 
0.92812 
0.29679 
57 
30/09/1994 
1.32% 
2.69% 
1.320E02 
2.690E02 
0.94123 
0.26953 
58 
31/10/1994 
0.40% 
2.08% 
4.000E03 
2.083E02 
0.94523 
0.29014 
59 
30/11/1994 
1.48% 
3.95% 
1.480E02 
3.950E02 
0.93032 
0.24984 
60 
31/12/1994 
0.74% 
1.23% 
7.400E03 
1.230E02 
0.93769 
0.26207 
61 
31/01/1995 
0.30% 
2.43% 
3.000E03 
2.428E02 
0.94068 
0.28606 
62 
28/02/1995 
1.68% 
3.61% 
1.680E02 
3.607E02 
0.95734 
0.32149 
63 
31/03/1995 
2.09% 
2.73% 
2.090E02 
2.733E02 
0.97803 
0.34845 
64 
30/04/1995 
2.64% 
2.80% 
2.640E02 
2.796E02 
1.00409 
0.37603 
65 
31/05/1995 
1.22% 
3.63% 
1.220E02 
3.631E02 
1.01621 
0.41170 
66 
30/06/1995 
4.73% 
2.13% 
4.730E02 
2.128E02 
1.06243 
0.43275 
67 
31/07/1995 
4.46% 
3.18% 
4.460E02 
3.178E02 
1.10606 
0.46404 
68 
31/08/1995 
2.93% 
0.03% 
2.930E02 
3.203E04 
1.13494 
0.46372 
69 
30/09/1995 
2.90% 
4.01% 
2.900E02 
4.010E02 
1.16353 
0.50303 
70 
31/10/1995 
1.44% 
0.50% 
1.440E02 
4.979E03 
1.14902 
0.49804 
71 
30/11/1995 
3.43% 
4.10% 
3.430E02 
4.105E02 
1.18275 
0.53827 
72 
31/12/1995 
2.56% 
1.74% 
2.560E02 
1.744E02 
1.20803 
0.55556 
73 
31/01/1996 
1.06% 
3.26% 
1.060E02 
3.262E02 
1.21857 
0.58766 
74 
29/02/1996 
2.82% 
0.69% 
2.820E02 
6.934E03 
1.24638 
0.59457 
75 
31/03/1996 
1.90% 
0.79% 
1.900E02 
7.917E03 
1.26520 
0.60246 
76 
30/04/1996 
5.34% 
1.34% 
5.340E02 
1.343E02 
1.31723 
0.61580 
77 
31/05/1996 
3.70% 
2.29% 
3.700E02 
2.285E02 
1.35356 
0.63839 
78 
30/06/1996 
0.73% 
0.23% 
7.300E03 
2.257E03 
1.34623 
0.64065 
79 
31/07/1996 
2.87% 
4.57% 
2.870E02 
4.575E02 
1.31711 
0.59382 
80 
31/08/1996 
2.63% 
1.88% 
2.630E02 
1.881E02 
1.34307 
0.61245 
81 
30/09/1996 
2.18% 
5.42% 
2.180E02 
5.417E02 
1.36464 
0.66521 
82 
31/10/1996 
1.56% 
2.61% 
1.560E02 
2.613E02 
1.38012 
0.69100 
83 
30/11/1996 
1.66% 
7.34% 
1.660E02 
7.338E02 
1.39658 
0.76181 
84 
31/12/1996 
0.83% 
2.15% 
8.300E03 
2.151E02 
1.40485 
0.74007 
85 
31/01/1997 
2.78% 
6.13% 
2.780E02 
6.132E02 
1.43227 
0.79958 
86 
28/02/1997 
0.24% 
0.59% 
2.400E03 
5.928E03 
1.42986 
0.80549 
87 
31/03/1997 
0.73% 
4.26% 
7.300E03 
4.261E02 
1.42254 
0.76195 
88 
30/04/1997 
0.27% 
5.84% 
2.700E03 
5.841E02 
1.41983 
0.81871 
89 
31/05/1997 
5.04% 
5.86% 
5.040E02 
5.858E02 
1.46900 
0.87564 
90 
30/06/1997 
1.97% 
4.35% 
1.970E02 
4.345E02 
1.48851 
0.91818 
91 
31/07/1997 
5.05% 
7.81% 
5.050E02 
7.812E02 
1.53778 
0.99339 
92 
31/08/1997 
1.35% 
5.74% 
1.350E02 
5.745E02 
1.55119 
0.93423 
93 
30/09/1997 
5.69% 
5.32% 
5.690E02 
5.315E02 
1.60653 
0.98601 
94 
31/10/1997 
0.39% 
3.45% 
3.900E03 
3.448E02 
1.61042 
0.95093 
95 
30/11/1997 
0.93% 
4.46% 
9.300E03 
4.459E02 
1.60108 
0.99455 
96 
31/12/1997 
1.42% 
1.57% 
1.420E02 
1.573E02 
1.61518 
1.01016 
97 
31/01/1998 
0.16% 
1.02% 
1.600E03 
1.015E02 
1.61358 
1.02026 
98 
28/02/1998 
4.09% 
7.04% 
4.090E02 
7.045E02 
1.65366 
1.08834 
99 
31/03/1998 
4.54% 
4.99% 
4.540E02 
4.995E02 
1.69806 
1.13708 
100 
30/04/1998 
1.39% 
0.91% 
1.390E02 
9.076E03 
1.71187 
1.14611 
101 
31/05/1998 
1.27% 
1.88% 
1.270E02 
1.883E02 
1.69908 
1.12710 
102 
30/06/1998 
0.50% 
3.94% 
5.000E03 
3.944E02 
1.70407 
1.16579 
103 
31/07/1998 
0.67% 
1.16% 
6.700E03 
1.162E02 
1.69735 
1.15410 
104 
31/08/1998 
7.65% 
14.58% 
7.650E02 
1.458E01 
1.61777 
0.99651 
105 
30/09/1998 
3.16% 
6.24% 
3.160E02 
6.240E02 
1.64888 
1.05704 
106 
31/10/1998 
2.47% 
8.03% 
2.470E02 
8.029E02 
1.67328 
1.13427 
107 
30/11/1998 
3.84% 
5.91% 
3.840E02 
5.913E02 
1.71096 
1.19172 
108 
31/12/1998 
5.39% 
5.64% 
5.390E02 
5.638E02 
1.76345 
1.24656 
109 
31/01/1999 
4.98% 
4.10% 
4.980E02 
4.101E02 
1.81205 
1.28675 
110 
28/02/1999 
2.41% 
3.23% 
2.410E02 
3.228E02 
1.78766 
1.25394 
111 
31/03/1999 
4.05% 
3.88% 
4.050E02 
3.879E02 
1.82736 
1.29200 
112 
30/04/1999 
5.25% 
3.79% 
5.250E02 
3.794E02 
1.87853 
1.32924 
113 
31/05/1999 
1.22% 
2.50% 
1.220E02 
2.497E02 
1.89066 
1.30395 
114 
30/06/1999 
3.80% 
5.44% 
3.800E02 
5.444E02 
1.92795 
1.35696 
115 
31/07/1999 
0.61% 
3.20% 
6.100E03 
3.205E02 
1.93403 
1.32438 
116 
31/08/1999 
0.04% 
0.63% 
4.000E04 
6.254E03 
1.93443 
1.31811 
117 
30/09/1999 
0.35% 
2.86% 
3.500E03 
2.855E02 
1.93793 
1.28915 
118 
31/10/1999 
2.33% 
6.25% 
2.330E02 
6.254E02 
1.96096 
1.34981 
119 
30/11/1999 
6.76% 
1.91% 
6.760E02 
1.906E02 
2.02637 
1.36869 
120 
31/12/1999 
10.88% 
5.78% 
1.088E01 
5.784E02 
2.12965 
1.42492 
121 
31/01/2000 
0.25% 
5.09% 
2.500E03 
5.090E02 
2.13215 
1.37268 
122 
29/02/2000 
10.00% 
2.01% 
1.000E01 
2.011E02 
2.22746 
1.35236 
123 
31/03/2000 
1.73% 
9.67% 
1.730E02 
9.672E02 
2.24461 
1.44469 
124 
30/04/2000 
4.19% 
3.08% 
4.190E02 
3.080E02 
2.20181 
1.41340 
125 
31/05/2000 
2.44% 
2.19% 
2.440E02 
2.191E02 
2.17710 
1.39125 
126 
30/06/2000 
4.85% 
2.39% 
4.850E02 
2.393E02 
2.22446 
1.41490 
127 
31/07/2000 
1.58% 
1.63% 
1.580E02 
1.634E02 
2.20854 
1.39842 
128 
31/08/2000 
5.35% 
6.07% 
5.350E02 
6.070E02 
2.26066 
1.45735 
129 
30/09/2000 
1.08% 
5.35% 
1.080E02 
5.348E02 
2.24980 
1.40239 
130 
31/10/2000 
2.01% 
0.49% 
2.010E02 
4.949E03 
2.22949 
1.39743 
131 
30/11/2000 
4.30% 
8.01% 
4.300E02 
8.007E02 
2.18554 
1.31397 
132 
31/12/2000 
3.16% 
0.41% 
3.160E02 
4.053E03 
2.21665 
1.31801 
133 
31/01/2001 
2.88% 
3.46% 
2.880E02 
3.464E02 
2.24504 
1.35207 
134 
28/02/2001 
2.56% 
9.23% 
2.560E02 
9.229E02 
2.21911 
1.25524 
135 
31/03/2001 
2.30% 
6.42% 
2.300E02 
6.420E02 
2.19584 
1.18888 
136 
30/04/2001 
2.27% 
7.68% 
2.270E02 
7.681E02 
2.21829 
1.26289 
137 
31/05/2001 
0.90% 
0.51% 
9.000E03 
5.090E03 
2.22725 
1.26796 
138 
30/06/2001 
0.32% 
2.50% 
3.200E03 
2.500E02 
2.22404 
1.24264 
139 
31/07/2001 
1.06% 
1.08% 
1.060E02 
1.077E02 
2.21339 
1.23182 
140 
31/08/2001 
1.22% 
6.41% 
1.220E02 
6.411E02 
2.20111 
1.16556 
141 
30/09/2001 
3.73% 
8.17% 
3.730E02 
8.172E02 
2.16310 
1.08031 
142 
31/10/2001 
1.85% 
1.81% 
1.850E02 
1.810E02 
2.18143 
1.09824 
143 
30/11/2001 
1.97% 
7.52% 
1.970E02 
7.518E02 
2.20094 
1.17073 
144 
31/12/2001 
1.99% 
0.76% 
1.990E02 
7.574E03 
2.22064 
1.17828 
145 
31/01/2002 
0.22% 
1.56% 
2.200E03 
1.557E02 
2.22284 
1.16258 
146 
28/02/2002 
0.89% 
2.08% 
8.900E03 
2.077E02 
2.21390 
1.14160 
147 
31/03/2002 
2.03% 
3.67% 
2.030E02 
3.674E02 
2.23400 
1.17768 
148 
30/04/2002 
0.17% 
6.14% 
1.700E03 
6.142E02 
2.23570 
1.11429 
149 
31/05/2002 
0.00% 
0.91% 
0.000E+00 
9.081E03 
2.23570 
1.10517 
150 
30/06/2002 
2.63% 
7.25% 
2.630E02 
7.246E02 
2.20904 
1.02995 
151 
31/07/2002 
3.93% 
7.90% 
3.930E02 
7.900E02 
2.16895 
0.94765 
152 
31/08/2002 
0.28% 
0.49% 
2.800E03 
4.881E03 
2.17175 
0.95252 
153 
30/09/2002 
1.96% 
11.00% 
1.960E02 
1.100E01 
2.15195 
0.83599 
154 
31/10/2002 
0.56% 
8.64% 
5.600E03 
8.645E02 
2.15754 
0.91890 
155 
30/11/2002 
2.67% 
5.71% 
2.670E02 
5.707E02 
2.18389 
0.97440 
156 
31/12/2002 
1.14% 
6.03% 
1.140E02 
6.033E02 
2.17242 
0.91218 
157 
31/01/2003 
0.01% 
2.74% 
1.000E04 
2.741E02 
2.17232 
0.88439 
158 
28/02/2003 
0.78% 
1.70% 
7.800E03 
1.700E02 
2.16449 
0.86724 
159 
31/03/2003 
0.07% 
0.84% 
7.000E04 
8.358E03 
2.16379 
0.87556 
160 
30/04/2003 
2.43% 
8.10% 
2.430E02 
8.104E02 
2.18780 
0.95349 
161 
31/05/2003 
4.08% 
5.09% 
4.080E02 
5.090E02 
2.22779 
1.00313 
162 
30/06/2003 
1.52% 
1.13% 
1.520E02 
1.132E02 
2.24287 
1.01439 
163 
31/07/2003 
2.41% 
1.62% 
2.410E02 
1.622E02 
2.26669 
1.03048 
164 
31/08/2003 
2.38% 
1.79% 
2.380E02 
1.787E02 
2.29021 
1.04819 
165 
30/09/2003 
0.78% 
1.19% 
7.800E03 
1.194E02 
2.29798 
1.03618 
166 
31/10/2003 
3.12% 
5.50% 
3.120E02 
5.496E02 
2.32870 
1.08968 
167 
30/11/2003 
1.14% 
0.71% 
1.140E02 
7.129E03 
2.34004 
1.09679 
168 
31/12/2003 
1.93% 
5.08% 
1.930E02 
5.077E02 
2.35915 
1.14631 
169 
31/01/2004 
1.95% 
1.73% 
1.950E02 
1.728E02 
2.37847 
1.16344 
170 
29/02/2004 
1.11% 
1.22% 
1.110E02 
1.221E02 
2.38951 
1.17558 
171 
31/03/2004 
0.36% 
1.64% 
3.600E03 
1.636E02 
2.39310 
1.15908 
172 
30/04/2004 
2.08% 
1.68% 
2.080E02 
1.679E02 
2.37208 
1.14215 
173 
31/05/2004 
0.19% 
1.21% 
1.900E03 
1.208E02 
2.37018 
1.15416 
174 
30/06/2004 
1.07% 
1.80% 
1.070E02 
1.799E02 
2.38082 
1.17199 
175 
31/07/2004 
1.88% 
3.43% 
1.880E02 
3.429E02 
2.36184 
1.13710 
176 
31/08/2004 
0.37% 
0.23% 
3.700E03 
2.287E03 
2.35814 
1.13938 
177 
30/09/2004 
1.99% 
0.94% 
1.990E02 
9.364E03 
2.37784 
1.14870 
178 
31/10/2004 
0.48% 
1.40% 
4.800E03 
1.401E02 
2.38263 
1.16261 
179 
30/11/2004 
3.37% 
3.86% 
3.370E02 
3.859E02 
2.41577 
1.20048 
180 
31/12/2004 
1.76% 
3.25% 
1.760E02 
3.246E02 
2.43322 
1.23242 
181 
31/01/2005 
0.58% 
2.53% 
5.800E03 
2.529E02 
2.42740 
1.20681 
182 
28/02/2005 
2.13% 
1.89% 
2.130E02 
1.890E02 
2.44848 
1.22553 
183 
31/03/2005 
1.05% 
1.91% 
1.050E02 
1.912E02 
2.43792 
1.20623 
184 
30/04/2005 
2.23% 
2.01% 
2.230E02 
2.011E02 
2.41537 
1.18591 
185 
31/05/2005 
1.55% 
3.00% 
1.550E02 
2.995E02 
2.43075 
1.21542 
186 
30/06/2005 
1.96% 
0.01% 
1.960E02 
1.427E04 
2.45016 
1.21528 
187 
31/07/2005 
2.95% 
3.60% 
2.950E02 
3.597E02 
2.47924 
1.25062 
188 
31/08/2005 
0.74% 
1.12% 
7.400E03 
1.122E02 
2.48661 
1.23933 
189 
30/09/2005 
2.25% 
0.69% 
2.250E02 
6.949E03 
2.50886 
1.24626 
190 
31/10/2005 
1.87% 
1.77% 
1.870E02 
1.774E02 
2.48998 
1.22836 
191 
30/11/2005 
2.14% 
3.52% 
2.140E02 
3.519E02 
2.51116 
1.26294 
192 
31/12/2005 
2.32% 
0.10% 
2.320E02 
9.524E04 
2.53409 
1.26199 
193 
31/01/2006 
3.95% 
2.55% 
3.950E02 
2.547E02 
2.57283 
1.28714 
194 
28/02/2006 
0.02% 
0.05% 
2.000E04 
4.531E04 
2.57303 
1.28759 
195 
31/03/2006 
2.55% 
1.11% 
2.550E02 
1.106E02 
2.59821 
1.29859 
196 
30/04/2006 
1.76% 
1.22% 
1.760E02 
1.219E02 
2.61566 
1.31071 
197 
31/05/2006 
2.32% 
3.09% 
2.320E02 
3.092E02 
2.59219 
1.27930 
198 
30/06/2006 
0.54% 
0.01% 
5.400E03 
8.661E05 
2.58677 
1.27939 
199 
31/07/2006 
0.54% 
0.51% 
5.400E03 
5.086E03 
2.58136 
1.28446 
200 
31/08/2006 
1.03% 
2.13% 
1.030E02 
2.127E02 
2.59160 
1.30551 
201 
30/09/2006 
0.16% 
2.46% 
1.600E03 
2.457E02 
2.59320 
1.32978 
202 
31/10/2006 
1.86% 
3.15% 
1.860E02 
3.151E02 
2.61163 
1.36081 
203 
30/11/2006 
2.00% 
1.65% 
2.000E02 
1.647E02 
2.63143 
1.37714 
204 
31/12/2006 
1.35% 
1.26% 
1.350E02 
1.262E02 
2.64484 
1.38968 
205 
31/01/2007 
1.16% 
1.41% 
1.160E02 
1.406E02 
2.65638 
1.40364 
206 
28/02/2007 
0.63% 
2.18% 
6.300E03 
2.185E02 
2.66266 
1.38155 
207 
31/03/2007 
1.02% 
1.00% 
1.020E02 
9.980E03 
2.67281 
1.39148 
208 
30/04/2007 
1.95% 
4.33% 
1.950E02 
4.329E02 
2.69212 
1.43386 
209 
31/05/2007 
2.29% 
3.25% 
2.290E02 
3.255E02 
2.71476 
1.46589 
210 
30/06/2007 
1.09% 
1.78% 
1.090E02 
1.782E02 
2.72560 
1.44791 
TABLE 3: Forecasted values of the logreturns
i 
date 
one month variation (in per cent) of the S&P 500
index _{} 
one month variation (in per cent) of the HFRI Equity hedge fund index _{} 
observed one month return _{} 
forecasted
one month return _{} 
observed six months return _{} 
forecasted
six months return _{} 
1 
31/01/1990 
6.88% 
3.34% 




2 
28/02/1990 
0.85% 
2.85% 




3 
31/03/1990 
2.43% 
5.67% 




4 
30/04/1990 
2.69% 
0.87% 




5 
31/05/1990 
9.20% 
5.92% 




6 
30/06/1990 
0.89% 
2.52% 
2.52E02 
6.55E03 


7 
31/07/1990 
0.52% 
2.00% 
2.00E02 
1.18E02 


8 
31/08/1990 
9.43% 
1.88% 
1.88E02 
1.10E02 


9 
30/09/1990 
5.12% 
1.65% 
1.65E02 
4.48E03 


10 
31/10/1990 
0.67% 
0.77% 
7.70E03 
6.66E03 


11 
30/11/1990 
5.99% 
2.29% 
2.29E02 
2.33E03 
2.694E02 
3.993E02 
12 
31/12/1990 
2.48% 
1.02% 
1.02E02 
1.63E03 
1.191E02 
7.308E02 
13 
31/01/1991 
4.15% 
4.90% 
4.90E02 
2.97E03 
4.068E02 
6.786E02 
14 
28/02/1991 
6.73% 
5.20% 
5.20E02 
3.42E03 
1.158E01 
2.717E02 
15 
31/03/1991 
2.22% 
7.22% 
7.22E02 
6.43E03 
1.769E01 
4.062E02 
16 
30/04/1991 
0.03% 
0.47% 
4.70E03 
2.00E02 
1.734E01 
1.409E02 
17 
31/05/1991 
3.86% 
3.20% 
3.20E02 
1.53E02 
2.393E01 
9.814E03 
18 
30/06/1991 
4.79% 
0.59% 
5.90E03 
1.64E02 
2.341E01 
1.766E02 
19 
31/07/1991 
4.49% 
1.41% 
1.41E02 
2.28E02 
1.930E01 
2.071E02 
20 
31/08/1991 
1.96% 
2.17% 
2.17E02 
5.36E03 
1.586E01 
3.921E02 
21 
30/09/1991 
1.91% 
4.30% 
4.30E02 
6.54E03 
1.271E01 
1.262E01 
22 
31/10/1991 
1.18% 
1.16% 
1.16E02 
1.34E02 
1.348E01 
9.517E02 
23 
30/11/1991 
4.39% 
1.08% 
1.08E02 
1.18E02 
8.777E02 
1.026E01 
24 
31/12/1991 
11.16% 
5.02% 
5.02E02 
1.00E02 
1.357E01 
1.445E01 
25 
31/01/1992 
1.99% 
2.49% 
2.49E02 
1.48E02 
1.478E01 
3.261E02 
26 
29/02/1992 
0.96% 
2.90% 
2.90E02 
1.72E02 
1.560E01 
3.990E02 
27 
31/03/1992 
2.18% 
0.28% 
2.80E03 
1.35E02 
1.052E01 
8.335E02 
28 
30/04/1992 
2.79% 
0.27% 
2.70E03 
1.06E02 
9.548E02 
7.318E02 
29 
31/05/1992 
0.10% 
0.85% 
8.50E03 
1.59E02 
1.169E01 
6.176E02 
30 
30/06/1992 
1.74% 
0.92% 
9.20E03 
8.66E03 
5.369E02 
9.231E02 
31 
31/07/1992 
3.94% 
2.76% 
2.76E02 
4.94E03 
5.646E02 
1.079E01 
32 
31/08/1992 
2.40% 
0.85% 
8.50E03 
4.88E03 
1.796E02 
8.346E02 
33 
30/09/1992 
0.91% 
2.51% 
2.51E02 
4.50E03 
4.644E02 
6.531E02 
34 
31/10/1992 
0.21% 
2.03% 
2.03E02 
4.80E03 
6.481E02 
9.928E02 
35 
30/11/1992 
3.03% 
4.51% 
4.51E02 
4.37E03 
1.035E01 
5.308E02 
36 
31/12/1992 
1.01% 
3.38% 
3.38E02 
1.22E02 
1.513E01 
3.001E02 
37 
31/01/1993 
0.70% 
2.09% 
2.09E02 
1.30E02 
1.438E01 
2.966E02 
38 
28/02/1993 
1.05% 
0.57% 
5.70E03 
1.90E02 
1.471E01 
2.731E02 
39 
31/03/1993 
1.87% 
3.26% 
3.26E02 
1.22E02 
1.555E01 
2.913E02 
40 
30/04/1993 
2.54% 
1.30% 
1.30E02 
1.62E02 
1.472E01 
2.654E02 
41 
31/05/1993 
2.27% 
2.72% 
2.72E02 
1.18E02 
1.275E01 
7.539E02 
42 
30/06/1993 
0.08% 
3.01% 
3.01E02 
1.04E02 
1.235E01 
8.066E02 
43 
31/07/1993 
0.53% 
2.12% 
2.12E02 
1.36E02 
1.238E01 
1.195E01 
44 
31/08/1993 
3.44% 
3.84% 
3.84E02 
1.54E02 
1.737E01 
7.566E02 
45 
30/09/1993 
1.00% 
2.52% 
2.52E02 
1.75E02 
1.653E01 
1.015E01 
46 
31/10/1993 
1.94% 
3.11% 
3.11E02 
1.84E02 
1.861E01 
7.300E02 
47 
30/11/1993 
1.29% 
1.93% 
1.93E02 
1.91E02 
1.324E01 
6.430E02 
48 
31/12/1993 
1.01% 
3.59% 
3.59E02 
1.23E02 
1.388E01 
8.470E02 
49 
31/01/1994 
3.25% 
2.35% 
2.35E02 
1.28E02 
1.413E01 
9.576E02 
50 
28/02/1994 
3.00% 
0.40% 
4.00E03 
1.20E02 
9.474E02 
1.095E01 
51 
31/03/1994 
4.57% 
2.08% 
2.08E02 
7.95E03 
4.562E02 
1.158E01 
52 
30/04/1994 
1.15% 
0.37% 
3.70E03 
4.35E03 
1.033E02 
1.202E01 
53 
31/05/1994 
1.24% 
0.41% 
4.10E03 
3.93E03 
3.444E02 
7.632E02 
54 
30/06/1994 
2.68% 
0.41% 
4.10E03 
1.39E03 
5.508E03 
7.920E02 
55 
31/07/1994 
3.15% 
0.91% 
9.10E03 
5.35E03 
1.950E02 
7.448E02 
56 
31/08/1994 
3.76% 
1.27% 
1.27E02 
3.47E03 
3.059E03 
4.868E02 
57 
30/09/1994 
2.69% 
1.32% 
1.32E02 
4.48E03 
3.156E02 
2.639E02 
58 
31/10/1994 
2.08% 
0.40% 
4.00E03 
4.21E03 
3.953E02 
2.381E02 
59 
30/11/1994 
3.95% 
1.48% 
1.48E02 
2.82E03 
1.996E02 
8.331E03 
60 
31/12/1994 
1.23% 
0.74% 
7.40E03 
2.45E03 
3.174E02 
3.169E02 
61 
31/01/1995 
2.43% 
0.30% 
3.00E03 
2.54E03 
2.550E02 
2.064E02 
62 
28/02/1995 
3.61% 
1.68% 
1.68E02 
4.04E03 
2.965E02 
2.718E02 
63 
31/03/1995 
2.73% 
2.09% 
2.09E02 
2.54E03 
3.748E02 
2.555E02 
64 
30/04/1995 
2.80% 
2.64% 
2.64E02 
4.70E03 
6.063E02 
1.705E02 
65 
31/05/1995 
3.63% 
1.22% 
1.22E02 
9.33E03 
8.969E02 
1.477E02 
66 
30/06/1995 
2.13% 
4.73% 
4.73E02 
9.14E03 
1.329E01 
1.536E02 
67 
31/07/1995 
3.18% 
4.46% 
4.46E02 
1.59E02 
1.798E01 
2.448E02 
68 
31/08/1995 
0.03% 
2.93% 
2.93E02 
1.89E02 
1.943E01 
1.535E02 
69 
30/09/1995 
4.01% 
2.90% 
2.90E02 
2.14E02 
2.038E01 
2.852E02 
70 
31/10/1995 
0.50% 
1.44% 
1.44E02 
1.92E02 
1.560E01 
5.730E02 
71 
30/11/1995 
4.10% 
3.43% 
3.43E02 
1.72E02 
1.812E01 
5.608E02 
72 
31/12/1995 
1.74% 
2.56% 
2.56E02 
1.57E02 
1.567E01 
9.919E02 
73 
31/01/1996 
3.26% 
1.06% 
1.06E02 
1.32E02 
1.191E01 
1.188E01 
74 
29/02/1996 
0.69% 
2.82% 
2.82E02 
1.06E02 
1.179E01 
1.352E01 
75 
31/03/1996 
0.79% 
1.90% 
1.90E02 
1.07E02 
1.070E01 
1.211E01 
76 
30/04/1996 
1.34% 
5.34% 
5.34E02 
1.56E02 
1.832E01 
1.078E01 
77 
31/05/1996 
2.29% 
3.70% 
3.70E02 
1.85E02 
1.863E01 
9.766E02 
78 
30/06/1996 
0.23% 
0.73% 
7.30E03 
2.00E02 
1.482E01 
8.200E02 
79 
31/07/1996 
4.57% 
2.87% 
2.87E02 
1.66E02 
1.036E01 
6.507E02 
80 
31/08/1996 
1.88% 
2.63% 
2.63E02 
8.76E03 
1.015E01 
6.613E02 
81 
30/09/1996 
5.42% 
2.18% 
2.18E02 
9.14E03 
1.045E01 
9.762E02 
82 
31/10/1996 
2.61% 
1.56% 
1.56E02 
5.59E03 
6.491E02 
1.163E01 
83 
30/11/1996 
7.34% 
1.66% 
1.66E02 
4.54E03 
4.396E02 
1.263E01 
84 
31/12/1996 
2.15% 
0.83% 
8.30E03 
6.47E03 
6.037E02 
1.038E01 
85 
31/01/1997 
6.13% 
2.78% 
2.78E02 
1.19E02 
1.220E01 
5.374E02 
86 
28/02/1997 
0.59% 
0.24% 
2.40E03 
1.14E02 
9.067E02 
5.610E02 
87 
31/03/1997 
4.26% 
0.73% 
7.30E03 
8.02E03 
5.961E02 
3.398E02 
88 
30/04/1997 
5.84% 
0.27% 
2.70E03 
4.68E03 
4.052E02 
2.753E02 
89 
31/05/1997 
5.86% 
5.04% 
5.04E02 
4.58E03 
7.511E02 
3.942E02 
90 
30/06/1997 
4.35% 
1.97% 
1.97E02 
7.34E03 
8.727E02 
7.370E02 
91 
31/07/1997 
7.81% 
5.05% 
5.05E02 
6.88E03 
1.113E01 
7.040E02 
92 
31/08/1997 
5.74% 
1.35% 
1.35E02 
1.40E02 
1.290E01 
4.910E02 
93 
30/09/1997 
5.32% 
5.69% 
5.69E02 
1.86E02 
2.020E01 
2.842E02 
94 
31/10/1997 
3.45% 
0.39% 
3.90E03 
2.43E02 
2.100E01 
2.779E02 
95 
30/11/1997 
4.46% 
0.93% 
9.30E03 
1.84E02 
1.412E01 
4.482E02 
96 
31/12/1997 
1.57% 
1.42% 
1.42E02 
9.80E03 
1.350E01 
4.199E02 
97 
31/01/1998 
1.02% 
0.16% 
1.60E03 
6.82E03 
7.874E02 
8.665E02 
98 
28/02/1998 
7.04% 
4.09% 
4.09E02 
6.72E03 
1.079E01 
1.172E01 
99 
31/03/1998 
4.99% 
4.54% 
4.54E02 
4.78E03 
9.585E02 
1.551E01 
100 
30/04/1998 
0.91% 
1.39% 
1.39E02 
9.83E03 
1.068E01 
1.154E01 
101 
31/05/1998 
1.88% 
1.27% 
1.27E02 
1.26E02 
1.030E01 
6.026E02 
102 
30/06/1998 
3.94% 
0.50% 
5.00E03 
1.06E02 
9.297E02 
4.161E02 
103 
31/07/1998 
1.16% 
0.67% 
6.70E03 
1.10E02 
8.738E02 
4.100E02 
104 
31/08/1998 
14.58% 
7.65% 
7.65E02 
4.39E03 
3.526E02 
2.904E02 
105 
30/09/1998 
6.24% 
3.16% 
3.16E02 
8.70E03 
4.800E02 
6.042E02 
106 
31/10/1998 
8.03% 
2.47% 
2.47E02 
9.49E03 
3.785E02 
7.819E02 
107 
30/11/1998 
5.91% 
3.84% 
3.84E02 
4.50E03 
1.194E02 
6.498E02 
108 
31/12/1998 
5.64% 
5.39% 
5.39E02 
4.29E03 
6.118E02 
6.790E02 
109 
31/01/1999 
4.10% 
4.98% 
4.98E02 
5.03E03 
1.215E01 
2.662E02 
110 
28/02/1999 
3.23% 
2.41% 
2.41E02 
2.63E02 
1.852E01 
5.108E02 
111 
31/03/1999 
3.88% 
4.05% 
4.05E02 
1.56E02 
1.954E01 
5.558E02 
112 
30/04/1999 
3.79% 
5.25% 
5.25E02 
1.37E02 
2.278E01 
2.667E02 
113 
31/05/1999 
2.50% 
1.22% 
1.22E02 
1.95E02 
1.969E01 
2.602E02 
114 
30/06/1999 
5.44% 
3.80% 
3.80E02 
2.72E02 
1.788E01 
3.055E02 
115 
31/07/1999 
3.20% 
0.61% 
6.10E03 
1.60E02 
1.297E01 
1.686E01 
116 
31/08/1999 
0.63% 
0.04% 
4.00E04 
1.99E02 
1.581E01 
9.733E02 
117 
30/09/1999 
2.86% 
0.35% 
3.50E03 
1.55E02 
1.169E01 
8.488E02 
118 
31/10/1999 
6.25% 
2.33% 
2.33E02 
7.13E03 
8.592E02 
1.228E01 
119 
30/11/1999 
1.91% 
6.76% 
6.76E02 
7.55E03 
1.454E01 
1.746E01 
120 
31/12/1999 
5.78% 
10.88% 
1.09E01 
1.56E02 
2.235E01 
9.959E02 
121 
31/01/2000 
5.09% 
0.25% 
2.50E03 
2.53E02 
2.191E01 
1.253E01 
122 
29/02/2000 
2.01% 
10.00% 
1.00E01 
2.19E02 
3.405E01 
9.659E02 
123 
31/03/2000 
9.67% 
1.73% 
1.73E02 
3.03E02 
3.589E01 
4.357E02 
124 
30/04/2000 
3.08% 
4.19% 
4.19E02 
3.70E02 
2.723E01 
4.614E02 
125 
31/05/2000 
2.19% 
2.44% 
2.44E02 
2.26E02 
1.627E01 
9.711E02 
126 
30/06/2000 
2.39% 
4.85% 
4.85E02 
5.99E03 
9.945E02 
1.614E01 
127 
31/07/2000 
1.63% 
1.58% 
1.58E02 
1.24E02 
7.938E02 
1.391E01 
128 
31/08/2000 
6.07% 
5.35% 
5.35E02 
3.54E03 
3.376E02 
1.964E01 
129 
30/09/2000 
5.35% 
1.08% 
1.08E02 
4.74E03 
5.201E03 
2.432E01 
130 
31/10/2000 
0.49% 
2.01% 
2.01E02 
6.00E03 
2.807E02 
1.436E01 
131 
30/11/2000 
8.01% 
4.30% 
4.30E02 
5.76E03 
8.473E03 
3.648E02 
132 
31/12/2000 
0.41% 
3.16% 
3.16E02 
7.03E03 
7.782E03 
7.690E02 
133 
31/01/2001 
3.46% 
2.88% 
2.88E02 
4.59E03 
3.718E02 
2.107E02 
134 
28/02/2001 
9.23% 
2.56% 
2.56E02 
3.07E03 
4.069E02 
2.877E02 
135 
31/03/2001 
6.42% 
2.30% 
2.30E02 
6.21E03 
5.252E02 
3.656E02 
136 
30/04/2001 
7.68% 
2.27% 
2.27E02 
6.29E03 
1.114E02 
3.509E02 
137 
31/05/2001 
0.51% 
0.90% 
9.00E03 
4.25E03 
4.259E02 
4.145E02 
138 
30/06/2001 
2.50% 
0.32% 
3.20E03 
4.23E03 
7.419E03 
2.786E02 
139 
31/07/2001 
1.08% 
1.06% 
1.06E02 
4.85E03 
3.116E02 
1.829E02 
140 
31/08/2001 
6.41% 
1.22% 
1.22E02 
2.40E03 
1.784E02 
3.670E02 
141 
30/09/2001 
8.17% 
3.73% 
3.73E02 
3.70E03 
3.221E02 
3.718E02 
142 
31/10/2001 
1.81% 
1.85% 
1.85E02 
7.58E03 
3.619E02 
2.577E02 
143 
30/11/2001 
7.52% 
1.97% 
1.97E02 
7.25E03 
2.597E02 
2.566E02 
144 
31/12/2001 
0.76% 
1.99% 
1.99E02 
3.83E03 
3.395E03 
2.873E02 
145 
31/01/2002 
1.56% 
0.22% 
2.20E03 
3.94E03 
9.498E03 
1.429E02 
146 
28/02/2002 
2.08% 
0.89% 
8.90E03 
3.89E03 
1.287E02 
2.241E02 
147 
31/03/2002 
3.67% 
2.03% 
2.03E02 
5.07E03 
7.347E02 
4.462E02 
148 
30/04/2002 
6.14% 
0.17% 
1.70E03 
5.29E03 
5.577E02 
4.273E02 
149 
31/05/2002 
0.91% 
0.00% 
2.13E18 
3.90E03 
3.537E02 
2.273E02 
150 
30/06/2002 
7.25% 
2.63% 
2.63E02 
3.47E03 
1.153E02 
2.390E02 
151 
31/07/2002 
7.90% 
3.93% 
3.93E02 
2.57E03 
5.246E02 
2.360E02 
152 
31/08/2002 
0.49% 
0.28% 
2.80E03 
7.13E03 
4.128E02 
3.082E02 
153 
30/09/2002 
11.00% 
1.96% 
1.96E02 
9.70E03 
7.877E02 
3.213E02 
154 
31/10/2002 
8.64% 
0.56% 
5.60E03 
1.28E02 
7.518E02 
2.366E02 
155 
30/11/2002 
5.71% 
2.67% 
2.67E02 
1.10E02 
5.049E02 
2.099E02 
156 
31/12/2002 
6.03% 
1.14% 
1.14E02 
5.42E03 
3.596E02 
1.532E02 
157 
31/01/2003 
2.74% 
0.01% 
1.00E04 
3.66E03 
3.376E03 
4.204E02 
158 
28/02/2003 
1.70% 
0.78% 
7.80E03 
3.44E03 
7.230E03 
5.680E02 
159 
31/03/2003 
0.84% 
0.07% 
7.00E04 
3.95E03 
1.191E02 
7.432E02 
160 
30/04/2003 
8.10% 
2.43% 
2.43E02 
3.83E03 
3.073E02 
6.428E02 
161 
31/05/2003 
5.09% 
4.08% 
4.08E02 
3.65E03 
4.488E02 
3.206E02 
162 
30/06/2003 
1.13% 
1.52% 
1.52E02 
5.84E03 
7.300E02 
2.218E02 
163 
31/07/2003 
1.62% 
2.41% 
2.41E02 
7.43E03 
9.896E02 
2.083E02 
164 
31/08/2003 
1.79% 
2.38% 
2.38E02 
1.29E02 
1.340E01 
2.393E02 
165 
30/09/2003 
1.19% 
0.78% 
7.80E03 
1.81E02 
1.436E01 
2.322E02 
166 
31/10/2003 
5.50% 
3.12% 
3.12E02 
1.27E02 
1.513E01 
2.211E02 
167 
30/11/2003 
0.71% 
1.14% 
1.14E02 
1.06E02 
1.188E01 
3.554E02 
168 
31/12/2003 