PAPER:

 

Determining a stable relationship between hedge fund index HFRI-Equity 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 log-returns

4.     Analysis of the S&P 500 and HRFI Equity indices (digital movies)

5.     References

 

 

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1. Abstract

We test the ability of a stochastic differential model proposed in [5] of forecasting the returns of a long-short equity hedge fund index and of a market index, that is of the HFRI-Equity index and of the S&P 500 index respectively. The model is based on the assumptions that the value of the variation of the log-return of the hedge fund index (HFRI Equity) is proportional up to an additive stochastic error to the value of the variation of the log-return of a market index (S&P 500) and that the log-return 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 log-returns of the HFRI-Equity 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 HFRI-Equity and of the S&P 500 indices log-returns forecasted by the calibrated models are compared to the values of the observed indices log-returns. 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.

 

 

 

 

2.  Introduction

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 (xt,zt,vt), t > 0, a stochastic process, we will interpret xt, t > 0, as the log-return of a market index, zt, t > 0, as the log-return of the index of a class of hedge funds and vt, t > 0, as the stochastic variance of the market index. We consider the S&P 500 index as market index and the HFRI-Equity hedge fund index as index of a class of hedge funds, that is the class of ``long-short 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 (xt,zt,vt), t > 0, is modeled by the following system of stochastic differential equations:

 

with initial condition:

 

 

 

 

 

 

 

 

where , b, g, c, q, e are constants,  ,  , , t ³ 0, are standard Wiener processes such   that W01=W02= W03 =0,   and  d, d, d, t>0, are their stochastic differentials. Moreover we assume that:

 

< dW1t dW2t >

=

r1,2 dt,  t > 0,

(7)

< dW1t dW3t >

=

r1,3 dt,  t > 0,

(8)

< dW2t dW3t >

=

r2,3 dt,  t > 0,

(9)

 

 

where < · > denotes the expected value of ·, and the quantities r1,2r1,3r2,3 Î [-1,1] are constants known as correlation coefficients. We note that the autocorrelation coefficients of dWit, 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 St, t > 0, be the price at time t of an asset, the Heston model (1), (3) describes satisfactorily the dynamics of the log-return xt = log(St/S0), t > 0, of the asset and of its stochastic variance vt, t > 0. Note that                        x0 = log1 = 0 = and that since zt, t>0, is a log-return a similar statement holds for z0, that is              z0 = 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 Dow-Jones Industrials index and the Nasdaq Composite index (see [7]). We note that equation (3) is known as equation of the mean-reverting 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 log-return 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  log-returns of the indices of several classes of hedge funds have shown that the log-returns of the indices of some classes of hedge funds are related to the log-returns 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 log-return of the index of the “long-short 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 log-return  of the index of the “long-short equity" class of hedge funds  and the variation of the log-return of the S&P500 index. In [10] this assumption is supported by convincing empirical evidence. So that interpreting zt, t>0, as the log-return at time t of an index of  the “long-short equity" class of hedge funds, that is  the HFRI-Equity index, xt, t>0, as the log-return at time t of the S& P500 index and vt, t>0, as the stochastic variance of  xt,  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 dzt is given by the sum of  b dxt and a random disturbance given by gdWt3.

The calibration problem of  model (1), (2), (3)   can be stated as follows: given a discrete set of time values t=ti, i=0,1,…,n, such that t0=0, ti<ti+1, i=0,1,…,n-1, and the observation  of the log-returns of the market  and of the hedge fund indices, that is the observation of (xt,zt), at time t=ti, i=0,1,…,n,  determine the  values of the parameters appearing in  (1), (2), (3), of the correlation coefficients r1,2r1,3r2,3  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, r1,2, r1,3, r2,3  and .

Note that the choice t0=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=t0<t1<t2<…<tn<tn+1 =+¥,  the log-return of the  S&P500  index and the log-return of  the HFRI Equity index are observed and let  Ft= {(,) : ti  £t}, t>0, be the set of the observations available at  time t>0 of the log-returns of the market index xt and of the log-returns of the index of the hedge funds zt. 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 t0 =0 and F0={()}to simplify some of the formulae that follow.

We determine the joint probability density function of having xt=x, zt=z and vt =v at time t>0 conditioned to the observations contained  in Ft, t>0,  and to the initial condition  (6). Note that the initial conditions (4), (5) are already contained in Ft, 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 HFRI-Equity 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 log-returns.  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 log-return several months in the future. Forecasts of approximately the same quality are obtained for the log-returns 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.

 

3.  The calibration procedure and the formulae used to forecast the log-returns

Let us formulate the calibration problem and let us give some formulae used to forecast the log-returns xt, zt and the stochastic variance vt, 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 log-returns (and of the stochastic variance) as explained in  Section 4.

Let us consider the joint probability density function  =(x,z,v,t|Ft, Q), (x,z,v) Î R×R×R+, t > 0, of having xt = x, zt = z, vt = v given Ft, 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, t-t¢ > 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:

 

 

                                fi(v; Q) = d,

 

 

   v Î R+,

    (11)

 

 

 

 

for i = 1,2,...,n:

                   (12)

where d(.) is the Dirac’s delta and  denotes the  left limit t  that goes to  ti.

In order to determine the vector Q we solve the calibration problem, that is we solve the optimization problem:

 

 


max
Q Î M 

F( Q),

(13)

 

 

where the log-likelihood 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 Q0 Î M and that for k = 1,2,¼ generates at step k a feasible point Qk Î M satisfying the inequality F(Qk) > F( Qk-1), that is the objective function is monotonically increasing along the sequence {Qk}, k = 0,1,¼. In the numerical experience presented in Section 4 in the optimization procedure we use as stopping rule the condition:

where etol, kmax are positive constants that will be chosen later.

We note that the log-likelihood 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,t|Ft, Q), (x,z,v) Î R×R×R+, t ³ 0, we can forecast the values of the market index log-return, of the hedge fund index log-return xt, zt, t > 0, t ¹ ti,                i = 0,1,¼,n, and of the stochastic variance vt, t > 0, using respectively the mean values t| Q, t| Q, t| Q, t > 0, conditioned to the observations contained in Ft, t > 0, of the random variables xt, zt, vt, t > 0, that is:

                      (17)

                             (18)

                       (19)

Note that in Section 4 we are interested in forecasting xt, zt, vt  when t>tn 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 tn+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 xt, zt,  vt, 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).

 

 

 

4. Analysis of the S&P 500 and HRFI Equity indices

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 DIx,i of the S&P 500 index Ix,i,                    at time t=, i = 1,2,¼,210, and to the variation DIz,i of the HFRI-Equity index Iz,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  Ix,0=Iz,0=1 and  , . First of all we have manipulated the data DIx,i, DIz,i, i = 1,2,¼, 210, in order to get the time series of the observed log-return , i = 1,¼, 210, of the S&P 500 index and of the observed log-return  , i = 1,¼,210, of the HFRI-Equity index. To visualize and eventually download  the data click here Table 1 .

We have used the following recursive formulae to construct the log-returns (i.e. the last two columns of Table 1):

 

where as mentioned above ,   correspond to the log-returns at time t = (December 31, 1989) that have been chosen equal to zero.

The indices Ix,0, Iz,0  have been assumed to be equal one and Ix,i, Iz,i, i = 1,2,¼, 210 are related to the monthly log-returns , , i = 1,2,¼,210, defined in (26), (27) by the following formulae:

 



 



= logIx,i = log(Ix,i-1(1+DIx,i)),   i = 1,2,¼,210,

(28)

 

 



 


 

= logIz,i = log(Iz,i-1(1+DIz,i)),   i = 1,2,¼,210,

(29)

that is we have:

Ix,i = Ix,i-1(1+DIx,i),   i = 1,2,¼,210,     Ix,0 = 1,

(30)

 

Iz,i = Iz,i-1(1+DIz,i),   i = 1,2,¼,210,     Iz,0 = 1.

(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 j-th 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 xt, zt in the future, that is of forecasting using the formulae (20), (21)  the values xt, zt 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  etol= 5·10-4 and kmax = 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 r1,2, r1,3, r2,3 (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 r1,2 and r2,3 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 r1,3 that measures the correlation between two of the stochastic differentials present in the equations of the log-return of the S&P 500 index and of the log-return of the hedge fund HFRI-Equity 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 HFRI-Equity 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 HFRI-Equity hedge fund index and of the S&P 500 index log-returns.

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 log-returns 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 xt and zt and of the increments of the HFRI-Equity index= ,          i=5,6,…,210-m, 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 Ntot 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 HFRI-Equity index used in this study. Note that below we will choose Ntot 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,Ntot , 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,…,Ntot, j=1,2,…m, are the variations of the index Iz,4+i+j-1 computed using the forecasted values , i=1,2,…, Ntot, 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,Ntot, that is sm =  , m=1,2,…,6.

In Table 2 we show the values of the quantities ex,m, ez,m, evar,m and sm as a function of the forecasting period m (one, two,..., six months in the future) and we consider Ntot = 200 observation times. The choice Ntot = 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  Ntot = 204-5+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 ex,m, ez,m are the mean values of the relative errors committed on the forecasted values of the log-returns xt, zt of the S& P 500 index and of the HFRI-Equity index respectively and that for m=1,2,…,6, evar,m is the mean value of the absolute errors committed on the forecasted values of the HFRI-Equity index Iz,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 at,m, that is the histograms of the values taken by the function nm(t) =, t =, i = 1,2,¼,Ntot, m = 1,3,6 (see Figures 4, 5, 6).

Figure 4: Histogram of n1(t)

 

Figure 5: Histogram of n3(t)

 

 

Figure 6: Histogram of n6(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  at,m, m = 1,6 one month and six months in the future. We have computed the conditioned variance of the state variable zt (formula (24)) corresponding to these forecasted values of the  HFRI-Equity 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 HFRI-Equity 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 HFRI-Equity 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 log-return of the HFRI hedge fund index, the corresponding historical variations , i=1,2,…,Ntot ,m=1,3,6, and the corresponding conditioned variances. Note that when the conditioned variance of zt is small the quality of  the forecasted value is high.

Click here to see the second movie

 

5. References

 [1]  Y. Ait-Sahalia, R. Kimmel, Maximum likelihood estimation of stochastic volatility models, Journal of Financial Economics, 83 (2007), 413-452.

 

[2]  D.S. Bates, Maximum likelihood estimation of latent affine processes, The Review of Financial Studies, 19 (2006), 909-965.

 

[3]  F. Black, M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy, 81 (1973), 637-659.

 

[4] P. Capelli, F. Mariani, M.C. Recchioni, F. Spinelli, F.Zirilli, Determining a stable relationship between hedge fund index HFRI-Equity and S&P 500 behaviour, using filtering and maximum likelihood,Inverse Problems in Science and Engineering 18 (2010),83-109.

 

[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 Ill-Posed Problems, 15 (2007), 329-362, 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), 170-181, 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), 43-74.

 

 

 [8]  S.L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, The Review of Financial Studies, 6 (1993), 327-343.

 

 [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), 177-222, 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%

-3.340E-02

-6.880E-02

-0.03397

-0.07128

2

28/02/1990

2.85%

0.85%

2.850E-02

8.539E-03

-0.00587

-0.06278

3

31/03/1990

5.67%

2.43%

5.670E-02

2.426E-02

0.04928

-0.03881

4

30/04/1990

-0.87%

-2.69%

-8.700E-03

-2.689E-02

0.04054

-0.06607

5

31/05/1990

5.92%

9.20%

5.920E-02

9.199E-02

0.09806

0.02194

6

30/06/1990

2.52%

-0.89%

2.520E-02

-8.886E-03

0.12295

0.01301

7

31/07/1990

2.00%

-0.52%

2.000E-02

-5.223E-03

0.14275

0.00777

8

31/08/1990

-1.88%

-9.43%

-1.880E-02

-9.431E-02

0.12377

-0.09129

9

30/09/1990

1.65%

-5.12%

1.650E-02

-5.118E-02

0.14013

-0.14382

10

31/10/1990

0.77%

-0.67%

7.700E-03

-6.698E-03

0.14780

-0.15054

11

30/11/1990

-2.29%

5.99%

-2.290E-02

5.993E-02

0.12464

-0.09234

12

31/12/1990

1.02%

2.48%

1.020E-02

2.483E-02

0.13479

-0.06781

13

31/01/1991

4.90%

4.15%

4.900E-02

4.152E-02

0.18262

-0.02713

14

28/02/1991

5.20%

6.73%

5.200E-02

6.728E-02

0.23332

0.03798

15

31/03/1991

7.22%

2.22%

7.220E-02

2.220E-02

0.30303

0.05994

16

30/04/1991

0.47%

0.03%

4.700E-03

3.198E-04

0.30772

0.06026

17

31/05/1991

3.20%

3.86%

3.200E-02

3.860E-02

0.33922

0.09813

18

30/06/1991

0.59%

-4.79%

5.900E-03

-4.789E-02

0.34510

0.04906

19

31/07/1991

1.41%

4.49%

1.410E-02

4.486E-02

0.35910

0.09294

20

31/08/1991

2.17%

1.96%

2.170E-02

1.965E-02

0.38057

0.11240

21

30/09/1991

4.30%

-1.91%

4.300E-02

-1.914E-02

0.42267

0.09308

22

31/10/1991

1.16%

1.18%

1.160E-02

1.183E-02

0.43420

0.10484

23

30/11/1991

-1.08%

-4.39%

-1.080E-02

-4.390E-02

0.42335

0.05994

24

31/12/1991

5.02%

11.16%

5.020E-02

1.116E-01

0.47233

0.16574

25

31/01/1992

2.49%

-1.99%

2.490E-02

-1.990E-02

0.49692

0.14564

26

29/02/1992

2.90%

0.96%

2.900E-02

9.565E-03

0.52551

0.15516

27

31/03/1992

-0.28%

-2.18%

-2.800E-03

-2.183E-02

0.52270

0.13309

28

30/04/1992

0.27%

2.79%

2.700E-03

2.789E-02

0.52540

0.16060

29

31/05/1992

0.85%

0.10%

8.500E-03

9.640E-04

0.53386

0.16156

30

30/06/1992

-0.92%

-1.74%

-9.200E-03

-1.736E-02

0.52462

0.14405

31

31/07/1992

2.76%

3.94%

2.760E-02

3.940E-02

0.55185

0.18269

32

31/08/1992

-0.85%

-2.40%

-8.500E-03

-2.402E-02

0.54331

0.15838

33

30/09/1992

2.51%

0.91%

2.510E-02

9.106E-03

0.56810

0.16744

34

31/10/1992

2.03%

0.21%

2.030E-02

2.106E-03

0.58820

0.16955

35

30/11/1992

4.51%

3.03%

4.510E-02

3.026E-02

0.63231

0.19936

36

31/12/1992

3.38%

1.01%

3.380E-02

1.011E-02

0.66555

0.20942

37

31/01/1993

2.09%

0.70%

2.090E-02

7.046E-03

0.68624

0.21644

38

28/02/1993

-0.57%

1.05%

-5.700E-03

1.048E-02

0.68052

0.22687

39

31/03/1993

3.26%

1.87%

3.260E-02

1.870E-02

0.71260

0.24539

40

30/04/1993

1.30%

-2.54%

1.300E-02

-2.542E-02

0.72552

0.21964

41

31/05/1993

2.72%

2.27%

2.720E-02

2.272E-02

0.75235

0.24211

42

30/06/1993

3.01%

0.08%

3.010E-02

7.552E-04

0.78201

0.24287

43

31/07/1993

2.12%

-0.53%

2.120E-02

-5.327E-03

0.80299

0.23752

44

31/08/1993

3.84%

3.44%

3.840E-02

3.443E-02

0.84067

0.27137

45

30/09/1993

2.52%

-1.00%

2.520E-02

-9.988E-03

0.86556

0.26134

46

31/10/1993

3.11%

1.94%

3.110E-02

1.939E-02

0.89618

0.28054

47

30/11/1993

-1.93%

-1.29%

-1.930E-02

-1.291E-02

0.87669

0.26755

48

31/12/1993

3.59%

1.01%

3.590E-02

1.009E-02

0.91197

0.27759

49

31/01/1994

2.35%

3.25%

2.350E-02

3.250E-02

0.93519

0.30957

50

28/02/1994

-0.40%

-3.00%

-4.000E-03

-3.005E-02

0.93119

0.27906

51

31/03/1994

-2.08%

-4.57%

-2.080E-02

-4.575E-02

0.91017

0.23223

52

30/04/1994

-0.37%

1.15%

-3.700E-03

1.153E-02

0.90646

0.24369

53

31/05/1994

0.41%

1.24%

4.100E-03

1.242E-02

0.91055

0.25604

54

30/06/1994

-0.41%

-2.68%

-4.100E-03

-2.681E-02

0.90644

0.22886

55

31/07/1994

0.91%

3.15%

9.100E-03

3.149E-02

0.91550

0.25986

56

31/08/1994

1.27%

3.76%

1.270E-02

3.762E-02

0.92812

0.29679

57

30/09/1994

1.32%

-2.69%

1.320E-02

-2.690E-02

0.94123

0.26953

58

31/10/1994

0.40%

2.08%

4.000E-03

2.083E-02

0.94523

0.29014

59

30/11/1994

-1.48%

-3.95%

-1.480E-02

-3.950E-02

0.93032

0.24984

60

31/12/1994

0.74%

1.23%

7.400E-03

1.230E-02

0.93769

0.26207

61

31/01/1995

0.30%

2.43%

3.000E-03

2.428E-02

0.94068

0.28606

62

28/02/1995

1.68%

3.61%

1.680E-02

3.607E-02

0.95734

0.32149

63

31/03/1995

2.09%

2.73%

2.090E-02

2.733E-02

0.97803

0.34845

64

30/04/1995

2.64%

2.80%

2.640E-02

2.796E-02

1.00409

0.37603

65

31/05/1995

1.22%

3.63%

1.220E-02

3.631E-02

1.01621

0.41170

66

30/06/1995

4.73%

2.13%

4.730E-02

2.128E-02

1.06243

0.43275

67

31/07/1995

4.46%

3.18%

4.460E-02

3.178E-02

1.10606

0.46404

68

31/08/1995

2.93%

-0.03%

2.930E-02

-3.203E-04

1.13494

0.46372

69

30/09/1995

2.90%

4.01%

2.900E-02

4.010E-02

1.16353

0.50303

70

31/10/1995

-1.44%

-0.50%

-1.440E-02

-4.979E-03

1.14902

0.49804

71

30/11/1995

3.43%

4.10%

3.430E-02

4.105E-02

1.18275

0.53827

72

31/12/1995

2.56%

1.74%

2.560E-02

1.744E-02

1.20803

0.55556

73

31/01/1996

1.06%

3.26%

1.060E-02

3.262E-02

1.21857

0.58766

74

29/02/1996

2.82%

0.69%

2.820E-02

6.934E-03

1.24638

0.59457

75

31/03/1996

1.90%

0.79%

1.900E-02

7.917E-03

1.26520

0.60246

76

30/04/1996

5.34%

1.34%

5.340E-02

1.343E-02

1.31723

0.61580

77

31/05/1996

3.70%

2.29%

3.700E-02

2.285E-02

1.35356

0.63839

78

30/06/1996

-0.73%

0.23%

-7.300E-03

2.257E-03

1.34623

0.64065

79

31/07/1996

-2.87%

-4.57%

-2.870E-02

-4.575E-02

1.31711

0.59382

80

31/08/1996

2.63%

1.88%

2.630E-02

1.881E-02

1.34307

0.61245

81

30/09/1996

2.18%

5.42%

2.180E-02

5.417E-02

1.36464

0.66521

82

31/10/1996

1.56%

2.61%

1.560E-02

2.613E-02

1.38012

0.69100

83

30/11/1996

1.66%

7.34%

1.660E-02

7.338E-02

1.39658

0.76181

84

31/12/1996

0.83%

-2.15%

8.300E-03

-2.151E-02

1.40485

0.74007

85

31/01/1997

2.78%

6.13%

2.780E-02

6.132E-02

1.43227

0.79958

86

28/02/1997

-0.24%

0.59%

-2.400E-03

5.928E-03

1.42986

0.80549

87

31/03/1997

-0.73%

-4.26%

-7.300E-03

-4.261E-02

1.42254

0.76195

88

30/04/1997

-0.27%

5.84%

-2.700E-03

5.841E-02

1.41983

0.81871

89

31/05/1997

5.04%

5.86%

5.040E-02

5.858E-02

1.46900

0.87564

90

30/06/1997

1.97%

4.35%

1.970E-02

4.345E-02

1.48851

0.91818

91

31/07/1997

5.05%

7.81%

5.050E-02

7.812E-02

1.53778

0.99339

92

31/08/1997

1.35%

-5.74%

1.350E-02

-5.745E-02

1.55119

0.93423

93

30/09/1997

5.69%

5.32%

5.690E-02

5.315E-02

1.60653

0.98601

94

31/10/1997

0.39%

-3.45%

3.900E-03

-3.448E-02

1.61042

0.95093

95

30/11/1997

-0.93%

4.46%

-9.300E-03

4.459E-02

1.60108

0.99455

96

31/12/1997

1.42%

1.57%

1.420E-02

1.573E-02

1.61518

1.01016

97

31/01/1998

-0.16%

1.02%

-1.600E-03

1.015E-02

1.61358

1.02026

98

28/02/1998

4.09%

7.04%

4.090E-02

7.045E-02

1.65366

1.08834

99

31/03/1998

4.54%

4.99%

4.540E-02

4.995E-02

1.69806

1.13708

100

30/04/1998

1.39%

0.91%

1.390E-02

9.076E-03

1.71187

1.14611

101

31/05/1998

-1.27%

-1.88%

-1.270E-02

-1.883E-02

1.69908

1.12710

102

30/06/1998

0.50%

3.94%

5.000E-03

3.944E-02

1.70407

1.16579

103

31/07/1998

-0.67%

-1.16%

-6.700E-03

-1.162E-02

1.69735

1.15410

104

31/08/1998

-7.65%

-14.58%

-7.650E-02

-1.458E-01

1.61777

0.99651

105

30/09/1998

3.16%

6.24%

3.160E-02

6.240E-02

1.64888

1.05704

106

31/10/1998

2.47%

8.03%

2.470E-02

8.029E-02

1.67328

1.13427

107

30/11/1998

3.84%

5.91%

3.840E-02

5.913E-02

1.71096

1.19172

108

31/12/1998

5.39%

5.64%

5.390E-02

5.638E-02

1.76345

1.24656

109

31/01/1999

4.98%

4.10%

4.980E-02

4.101E-02

1.81205

1.28675

110

28/02/1999

-2.41%

-3.23%

-2.410E-02

-3.228E-02

1.78766

1.25394

111

31/03/1999

4.05%

3.88%

4.050E-02

3.879E-02

1.82736

1.29200

112

30/04/1999

5.25%

3.79%

5.250E-02

3.794E-02

1.87853

1.32924

113

31/05/1999

1.22%

-2.50%

1.220E-02

-2.497E-02

1.89066

1.30395

114

30/06/1999

3.80%

5.44%

3.800E-02

5.444E-02

1.92795

1.35696

115

31/07/1999

0.61%

-3.20%

6.100E-03

-3.205E-02

1.93403

1.32438

116

31/08/1999

0.04%

-0.63%

4.000E-04

-6.254E-03

1.93443

1.31811

117

30/09/1999

0.35%

-2.86%

3.500E-03

-2.855E-02

1.93793

1.28915

118

31/10/1999

2.33%

6.25%

2.330E-02

6.254E-02

1.96096

1.34981

119

30/11/1999

6.76%

1.91%

6.760E-02

1.906E-02

2.02637

1.36869

120

31/12/1999

10.88%

5.78%

1.088E-01

5.784E-02

2.12965

1.42492

121

31/01/2000

0.25%

-5.09%

2.500E-03

-5.090E-02

2.13215

1.37268

122

29/02/2000

10.00%

-2.01%

1.000E-01

-2.011E-02

2.22746

1.35236

123

31/03/2000

1.73%

9.67%

1.730E-02

9.672E-02

2.24461

1.44469

124

30/04/2000

-4.19%

-3.08%

-4.190E-02

-3.080E-02

2.20181

1.41340

125

31/05/2000

-2.44%

-2.19%

-2.440E-02

-2.191E-02

2.17710

1.39125

126

30/06/2000

4.85%

2.39%

4.850E-02

2.393E-02

2.22446

1.41490

127

31/07/2000

-1.58%

-1.63%

-1.580E-02

-1.634E-02

2.20854

1.39842

128

31/08/2000

5.35%

6.07%

5.350E-02

6.070E-02

2.26066

1.45735

129

30/09/2000

-1.08%

-5.35%

-1.080E-02

-5.348E-02

2.24980

1.40239

130

31/10/2000

-2.01%

-0.49%

-2.010E-02

-4.949E-03

2.22949

1.39743

131

30/11/2000

-4.30%

-8.01%

-4.300E-02

-8.007E-02

2.18554

1.31397

132

31/12/2000

3.16%

0.41%

3.160E-02

4.053E-03

2.21665

1.31801

133

31/01/2001

2.88%

3.46%

2.880E-02

3.464E-02

2.24504

1.35207

134

28/02/2001

-2.56%

-9.23%

-2.560E-02

-9.229E-02

2.21911

1.25524

135

31/03/2001

-2.30%

-6.42%

-2.300E-02

-6.420E-02

2.19584

1.18888

136

30/04/2001

2.27%

7.68%

2.270E-02

7.681E-02

2.21829

1.26289

137

31/05/2001

0.90%

0.51%

9.000E-03

5.090E-03

2.22725

1.26796

138

30/06/2001

-0.32%

-2.50%

-3.200E-03

-2.500E-02

2.22404

1.24264

139

31/07/2001

-1.06%

-1.08%

-1.060E-02

-1.077E-02

2.21339

1.23182

140

31/08/2001

-1.22%

-6.41%

-1.220E-02

-6.411E-02

2.20111

1.16556

141

30/09/2001

-3.73%

-8.17%

-3.730E-02

-8.172E-02

2.16310

1.08031

142

31/10/2001

1.85%

1.81%

1.850E-02

1.810E-02

2.18143

1.09824

143

30/11/2001

1.97%

7.52%

1.970E-02

7.518E-02

2.20094

1.17073

144

31/12/2001

1.99%

0.76%

1.990E-02

7.574E-03

2.22064

1.17828

145

31/01/2002

0.22%

-1.56%

2.200E-03

-1.557E-02

2.22284

1.16258

146

28/02/2002

-0.89%

-2.08%

-8.900E-03

-2.077E-02

2.21390

1.14160

147

31/03/2002

2.03%

3.67%

2.030E-02

3.674E-02

2.23400

1.17768

148

30/04/2002

0.17%

-6.14%

1.700E-03

-6.142E-02

2.23570

1.11429

149

31/05/2002

0.00%

-0.91%

0.000E+00

-9.081E-03

2.23570

1.10517

150

30/06/2002

-2.63%

-7.25%

-2.630E-02

-7.246E-02

2.20904

1.02995

151

31/07/2002

-3.93%

-7.90%

-3.930E-02

-7.900E-02

2.16895

0.94765

152

31/08/2002

0.28%

0.49%

2.800E-03

4.881E-03

2.17175

0.95252

153

30/09/2002

-1.96%

-11.00%

-1.960E-02

-1.100E-01

2.15195

0.83599

154

31/10/2002

0.56%

8.64%

5.600E-03

8.645E-02

2.15754

0.91890

155

30/11/2002

2.67%

5.71%

2.670E-02

5.707E-02

2.18389

0.97440

156

31/12/2002

-1.14%

-6.03%

-1.140E-02

-6.033E-02

2.17242

0.91218

157

31/01/2003

-0.01%

-2.74%

-1.000E-04

-2.741E-02

2.17232

0.88439

158

28/02/2003

-0.78%

-1.70%

-7.800E-03

-1.700E-02

2.16449

0.86724

159

31/03/2003

-0.07%

0.84%

-7.000E-04

8.358E-03

2.16379

0.87556

160

30/04/2003

2.43%

8.10%

2.430E-02

8.104E-02

2.18780

0.95349

161

31/05/2003

4.08%

5.09%

4.080E-02

5.090E-02

2.22779

1.00313

162

30/06/2003

1.52%

1.13%

1.520E-02

1.132E-02

2.24287

1.01439

163

31/07/2003

2.41%

1.62%

2.410E-02

1.622E-02

2.26669

1.03048

164

31/08/2003

2.38%

1.79%

2.380E-02

1.787E-02

2.29021

1.04819

165

30/09/2003

0.78%

-1.19%

7.800E-03

-1.194E-02

2.29798

1.03618

166

31/10/2003

3.12%

5.50%

3.120E-02

5.496E-02

2.32870

1.08968

167

30/11/2003

1.14%

0.71%

1.140E-02

7.129E-03

2.34004

1.09679

168

31/12/2003

1.93%

5.08%

1.930E-02

5.077E-02

2.35915

1.14631

169

31/01/2004

1.95%

1.73%

1.950E-02

1.728E-02

2.37847

1.16344

170

29/02/2004

1.11%

1.22%

1.110E-02

1.221E-02

2.38951

1.17558

171

31/03/2004

0.36%

-1.64%

3.600E-03

-1.636E-02

2.39310

1.15908

172

30/04/2004

-2.08%

-1.68%

-2.080E-02

-1.679E-02

2.37208

1.14215

173

31/05/2004

-0.19%

1.21%

-1.900E-03

1.208E-02

2.37018

1.15416

174

30/06/2004

1.07%

1.80%

1.070E-02

1.799E-02

2.38082

1.17199

175

31/07/2004

-1.88%

-3.43%

-1.880E-02

-3.429E-02

2.36184

1.13710

176

31/08/2004

-0.37%

0.23%

-3.700E-03

2.287E-03

2.35814

1.13938

177

30/09/2004

1.99%

0.94%

1.990E-02

9.364E-03

2.37784

1.14870

178

31/10/2004

0.48%

1.40%

4.800E-03

1.401E-02

2.38263

1.16261

179

30/11/2004

3.37%

3.86%

3.370E-02

3.859E-02

2.41577

1.20048

180

31/12/2004

1.76%

3.25%

1.760E-02

3.246E-02

2.43322

1.23242

181

31/01/2005

-0.58%

-2.53%

-5.800E-03

-2.529E-02

2.42740

1.20681

182

28/02/2005

2.13%

1.89%

2.130E-02

1.890E-02

2.44848

1.22553

183

31/03/2005

-1.05%

-1.91%

-1.050E-02

-1.912E-02

2.43792

1.20623

184

30/04/2005

-2.23%

-2.01%

-2.230E-02

-2.011E-02

2.41537

1.18591

185

31/05/2005

1.55%

3.00%

1.550E-02

2.995E-02

2.43075

1.21542

186

30/06/2005

1.96%

-0.01%

1.960E-02

-1.427E-04

2.45016

1.21528

187

31/07/2005

2.95%

3.60%

2.950E-02

3.597E-02

2.47924

1.25062

188

31/08/2005

0.74%

-1.12%

7.400E-03

-1.122E-02

2.48661

1.23933

189

30/09/2005

2.25%

0.69%

2.250E-02

6.949E-03

2.50886

1.24626

190

31/10/2005

-1.87%

-1.77%

-1.870E-02

-1.774E-02

2.48998

1.22836

191

30/11/2005

2.14%

3.52%

2.140E-02

3.519E-02

2.51116

1.26294

192

31/12/2005

2.32%

-0.10%

2.320E-02

-9.524E-04

2.53409

1.26199

193

31/01/2006

3.95%

2.55%

3.950E-02

2.547E-02

2.57283

1.28714

194

28/02/2006

0.02%

0.05%

2.000E-04

4.531E-04

2.57303

1.28759

195

31/03/2006

2.55%

1.11%

2.550E-02

1.106E-02

2.59821

1.29859

196

30/04/2006

1.76%

1.22%

1.760E-02

1.219E-02

2.61566

1.31071

197

31/05/2006

-2.32%

-3.09%

-2.320E-02

-3.092E-02

2.59219

1.27930

198

30/06/2006

-0.54%

0.01%

-5.400E-03

8.661E-05

2.58677

1.27939

199

31/07/2006

-0.54%

0.51%

-5.400E-03

5.086E-03

2.58136

1.28446

200

31/08/2006

1.03%

2.13%

1.030E-02

2.127E-02

2.59160

1.30551

201

30/09/2006

0.16%

2.46%

1.600E-03

2.457E-02

2.59320

1.32978

202

31/10/2006

1.86%

3.15%

1.860E-02

3.151E-02

2.61163

1.36081

203

30/11/2006

2.00%

1.65%

2.000E-02

1.647E-02

2.63143

1.37714

204

31/12/2006

1.35%

1.26%

1.350E-02

1.262E-02

2.64484

1.38968

205

31/01/2007

1.16%

1.41%

1.160E-02

1.406E-02

2.65638

1.40364

206

28/02/2007

0.63%

-2.18%

6.300E-03

-2.185E-02

2.66266

1.38155

207

31/03/2007

1.02%

1.00%

1.020E-02

9.980E-03

2.67281

1.39148

208

30/04/2007

1.95%

4.33%

1.950E-02

4.329E-02

2.69212

1.43386

209

31/05/2007

2.29%

3.25%

2.290E-02

3.255E-02

2.71476

1.46589

210

30/06/2007

1.09%

-1.78%

1.090E-02

-1.782E-02

2.72560

1.44791

 

 

 

TABLE 3: Forecasted values of the log-returns

 

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%

2.52E-02

6.55E-03

 

 

7

31/07/1990

-0.52%

2.00%

2.00E-02

1.18E-02

 

 

8

31/08/1990

-9.43%

-1.88%

-1.88E-02

1.10E-02

 

 

9

30/09/1990

-5.12%

1.65%

1.65E-02

4.48E-03

 

 

10

31/10/1990

-0.67%

0.77%

7.70E-03

6.66E-03

 

 

11

30/11/1990

5.99%

-2.29%

-2.29E-02

2.33E-03

2.694E-02

3.993E-02

12

31/12/1990

2.48%

1.02%

1.02E-02

1.63E-03

1.191E-02

7.308E-02

13

31/01/1991

4.15%

4.90%

4.90E-02

-2.97E-03

4.068E-02

6.786E-02

14

28/02/1991

6.73%

5.20%

5.20E-02

3.42E-03

1.158E-01

2.717E-02

15

31/03/1991

2.22%

7.22%

7.22E-02

6.43E-03

1.769E-01

4.062E-02

16

30/04/1991

0.03%

0.47%

4.70E-03

2.00E-02

1.734E-01

1.409E-02

17

31/05/1991

3.86%

3.20%

3.20E-02

1.53E-02

2.393E-01

9.814E-03

18

30/06/1991

-4.79%

0.59%

5.90E-03

1.64E-02

2.341E-01

-1.766E-02

19

31/07/1991

4.49%

1.41%

1.41E-02

2.28E-02

1.930E-01

2.071E-02

20

31/08/1991

1.96%

2.17%

2.17E-02

5.36E-03

1.586E-01

3.921E-02

21

30/09/1991

-1.91%

4.30%

4.30E-02

6.54E-03

1.271E-01

1.262E-01

22

31/10/1991

1.18%

1.16%

1.16E-02

1.34E-02

1.348E-01

9.517E-02

23

30/11/1991

-4.39%

-1.08%

-1.08E-02

1.18E-02

8.777E-02

1.026E-01

24

31/12/1991

11.16%

5.02%

5.02E-02

1.00E-02

1.357E-01

1.445E-01

25

31/01/1992

-1.99%

2.49%

2.49E-02

1.48E-02

1.478E-01

3.261E-02

26

29/02/1992

0.96%

2.90%

2.90E-02

1.72E-02

1.560E-01

3.990E-02

27

31/03/1992

-2.18%

-0.28%

-2.80E-03

1.35E-02

1.052E-01

8.335E-02

28

30/04/1992

2.79%

0.27%

2.70E-03

1.06E-02

9.548E-02

7.318E-02

29

31/05/1992

0.10%

0.85%

8.50E-03

1.59E-02

1.169E-01

6.176E-02

30

30/06/1992

-1.74%

-0.92%

-9.20E-03

8.66E-03

5.369E-02

9.231E-02

31

31/07/1992

3.94%

2.76%

2.76E-02

4.94E-03

5.646E-02

1.079E-01

32

31/08/1992

-2.40%

-0.85%

-8.50E-03

4.88E-03

1.796E-02

8.346E-02

33

30/09/1992

0.91%

2.51%

2.51E-02

4.50E-03

4.644E-02

6.531E-02

34

31/10/1992

0.21%

2.03%

2.03E-02

4.80E-03

6.481E-02

9.928E-02

35

30/11/1992

3.03%

4.51%

4.51E-02

4.37E-03

1.035E-01

5.308E-02

36

31/12/1992

1.01%

3.38%

3.38E-02

1.22E-02

1.513E-01

3.001E-02

37

31/01/1993

0.70%

2.09%

2.09E-02

1.30E-02

1.438E-01

2.966E-02

38

28/02/1993

1.05%

-0.57%

-5.70E-03

1.90E-02

1.471E-01

2.731E-02

39

31/03/1993

1.87%

3.26%

3.26E-02

1.22E-02

1.555E-01

2.913E-02

40

30/04/1993

-2.54%

1.30%

1.30E-02

1.62E-02

1.472E-01

2.654E-02

41

31/05/1993

2.27%

2.72%

2.72E-02

1.18E-02

1.275E-01

7.539E-02

42

30/06/1993

0.08%

3.01%

3.01E-02

1.04E-02

1.235E-01

8.066E-02

43

31/07/1993

-0.53%

2.12%

2.12E-02

1.36E-02

1.238E-01

1.195E-01

44

31/08/1993

3.44%

3.84%

3.84E-02

1.54E-02

1.737E-01

7.566E-02

45

30/09/1993

-1.00%

2.52%

2.52E-02

1.75E-02

1.653E-01

1.015E-01

46

31/10/1993

1.94%

3.11%

3.11E-02

1.84E-02

1.861E-01

7.300E-02

47

30/11/1993

-1.29%

-1.93%

-1.93E-02

1.91E-02

1.324E-01

6.430E-02

48

31/12/1993

1.01%

3.59%

3.59E-02

1.23E-02

1.388E-01

8.470E-02

49

31/01/1994

3.25%

2.35%

2.35E-02

1.28E-02

1.413E-01

9.576E-02

50

28/02/1994

-3.00%

-0.40%

-4.00E-03

1.20E-02

9.474E-02

1.095E-01

51

31/03/1994

-4.57%

-2.08%

-2.08E-02

7.95E-03

4.562E-02

1.158E-01

52

30/04/1994

1.15%

-0.37%

-3.70E-03

4.35E-03

1.033E-02

1.202E-01

53

31/05/1994

1.24%

0.41%

4.10E-03

3.93E-03

3.444E-02

7.632E-02

54

30/06/1994

-2.68%

-0.41%

-4.10E-03

-1.39E-03

-5.508E-03

7.920E-02

55

31/07/1994

3.15%

0.91%

9.10E-03

-5.35E-03

-1.950E-02

7.448E-02

56

31/08/1994

3.76%

1.27%

1.27E-02

-3.47E-03

-3.059E-03

4.868E-02

57

30/09/1994

-2.69%

1.32%

1.32E-02

4.48E-03

3.156E-02

2.639E-02

58

31/10/1994

2.08%

0.40%

4.00E-03

4.21E-03

3.953E-02

2.381E-02

59

30/11/1994

-3.95%

-1.48%

-1.48E-02

2.82E-03

1.996E-02

-8.331E-03

60

31/12/1994

1.23%

0.74%

7.40E-03

2.45E-03

3.174E-02

-3.169E-02

61

31/01/1995

2.43%

0.30%

3.00E-03

2.54E-03

2.550E-02

-2.064E-02

62

28/02/1995

3.61%

1.68%

1.68E-02

4.04E-03

2.965E-02

2.718E-02

63

31/03/1995

2.73%

2.09%

2.09E-02

2.54E-03

3.748E-02

2.555E-02

64

30/04/1995

2.80%

2.64%

2.64E-02

4.70E-03

6.063E-02

1.705E-02

65

31/05/1995

3.63%

1.22%

1.22E-02

9.33E-03

8.969E-02

1.477E-02

66

30/06/1995

2.13%

4.73%

4.73E-02

9.14E-03

1.329E-01

1.536E-02

67

31/07/1995

3.18%

4.46%

4.46E-02

1.59E-02

1.798E-01

2.448E-02

68

31/08/1995

-0.03%

2.93%

2.93E-02

1.89E-02

1.943E-01

1.535E-02

69

30/09/1995

4.01%

2.90%

2.90E-02

2.14E-02

2.038E-01

2.852E-02

70

31/10/1995

-0.50%

-1.44%

-1.44E-02

1.92E-02

1.560E-01

5.730E-02

71

30/11/1995

4.10%

3.43%

3.43E-02

1.72E-02

1.812E-01

5.608E-02

72

31/12/1995

1.74%

2.56%

2.56E-02

1.57E-02

1.567E-01

9.919E-02

73

31/01/1996

3.26%

1.06%

1.06E-02

1.32E-02

1.191E-01

1.188E-01

74

29/02/1996

0.69%

2.82%

2.82E-02

1.06E-02

1.179E-01

1.352E-01

75

31/03/1996

0.79%

1.90%

1.90E-02

1.07E-02

1.070E-01

1.211E-01

76

30/04/1996

1.34%

5.34%

5.34E-02

1.56E-02

1.832E-01

1.078E-01

77

31/05/1996

2.29%

3.70%

3.70E-02

1.85E-02

1.863E-01

9.766E-02

78

30/06/1996

0.23%

-0.73%

-7.30E-03

2.00E-02

1.482E-01

8.200E-02

79

31/07/1996

-4.57%

-2.87%

-2.87E-02

1.66E-02

1.036E-01

6.507E-02

80

31/08/1996

1.88%

2.63%

2.63E-02

8.76E-03

1.015E-01

6.613E-02

81

30/09/1996

5.42%

2.18%

2.18E-02

9.14E-03

1.045E-01

9.762E-02

82

31/10/1996

2.61%

1.56%

1.56E-02

5.59E-03

6.491E-02

1.163E-01

83

30/11/1996

7.34%

1.66%

1.66E-02

4.54E-03

4.396E-02

1.263E-01

84

31/12/1996

-2.15%

0.83%

8.30E-03

6.47E-03

6.037E-02

1.038E-01

85

31/01/1997

6.13%

2.78%

2.78E-02

1.19E-02

1.220E-01

5.374E-02

86

28/02/1997

0.59%

-0.24%

-2.40E-03

1.14E-02

9.067E-02

5.610E-02

87

31/03/1997

-4.26%

-0.73%

-7.30E-03

8.02E-03

5.961E-02

3.398E-02

88

30/04/1997

5.84%

-0.27%

-2.70E-03

4.68E-03

4.052E-02

2.753E-02

89

31/05/1997

5.86%

5.04%

5.04E-02

4.58E-03

7.511E-02

3.942E-02

90

30/06/1997

4.35%

1.97%

1.97E-02

7.34E-03

8.727E-02

7.370E-02

91

31/07/1997

7.81%

5.05%

5.05E-02

6.88E-03

1.113E-01

7.040E-02

92

31/08/1997

-5.74%

1.35%

1.35E-02

1.40E-02

1.290E-01

4.910E-02

93

30/09/1997

5.32%

5.69%

5.69E-02

1.86E-02

2.020E-01

2.842E-02

94

31/10/1997

-3.45%

0.39%

3.90E-03

2.43E-02

2.100E-01

2.779E-02

95

30/11/1997

4.46%

-0.93%

-9.30E-03

1.84E-02

1.412E-01

4.482E-02

96

31/12/1997

1.57%

1.42%

1.42E-02

9.80E-03

1.350E-01

4.199E-02

97

31/01/1998

1.02%

-0.16%

-1.60E-03

6.82E-03

7.874E-02

8.665E-02

98

28/02/1998

7.04%

4.09%

4.09E-02

6.72E-03

1.079E-01

1.172E-01

99

31/03/1998

4.99%

4.54%

4.54E-02

4.78E-03

9.585E-02

1.551E-01

100

30/04/1998

0.91%

1.39%

1.39E-02

9.83E-03

1.068E-01

1.154E-01

101

31/05/1998

-1.88%

-1.27%

-1.27E-02

1.26E-02

1.030E-01

6.026E-02

102

30/06/1998

3.94%

0.50%

5.00E-03

1.06E-02

9.297E-02

4.161E-02

103

31/07/1998

-1.16%

-0.67%

-6.70E-03

1.10E-02

8.738E-02

4.100E-02

104

31/08/1998

-14.58%

-7.65%

-7.65E-02

4.39E-03

-3.526E-02

2.904E-02

105

30/09/1998

6.24%

3.16%

3.16E-02

-8.70E-03

-4.800E-02

6.042E-02

106

31/10/1998

8.03%

2.47%

2.47E-02

-9.49E-03

-3.785E-02

7.819E-02

107

30/11/1998

5.91%

3.84%

3.84E-02

-4.50E-03

1.194E-02

6.498E-02

108

31/12/1998

5.64%

5.39%

5.39E-02

4.29E-03

6.118E-02

6.790E-02

109

31/01/1999

4.10%

4.98%

4.98E-02

5.03E-03

1.215E-01

2.662E-02

110

28/02/1999

-3.23%

-2.41%

-2.41E-02

2.63E-02

1.852E-01

-5.108E-02

111

31/03/1999

3.88%

4.05%

4.05E-02

1.56E-02

1.954E-01

-5.558E-02

112

30/04/1999

3.79%

5.25%

5.25E-02

1.37E-02

2.278E-01

-2.667E-02

113

31/05/1999

-2.50%

1.22%

1.22E-02

1.95E-02

1.969E-01

2.602E-02

114

30/06/1999

5.44%

3.80%

3.80E-02

2.72E-02

1.788E-01

3.055E-02

115

31/07/1999

-3.20%

0.61%

6.10E-03

1.60E-02

1.297E-01

1.686E-01

116

31/08/1999

-0.63%

0.04%

4.00E-04

1.99E-02

1.581E-01

9.733E-02

117

30/09/1999

-2.86%

0.35%

3.50E-03

1.55E-02

1.169E-01

8.488E-02

118

31/10/1999

6.25%

2.33%

2.33E-02

7.13E-03

8.592E-02

1.228E-01

119

30/11/1999

1.91%

6.76%

6.76E-02

7.55E-03

1.454E-01

1.746E-01

120

31/12/1999

5.78%

10.88%

1.09E-01

1.56E-02

2.235E-01

9.959E-02

121

31/01/2000

-5.09%

0.25%

2.50E-03

2.53E-02

2.191E-01

1.253E-01

122

29/02/2000

-2.01%

10.00%

1.00E-01

2.19E-02

3.405E-01

9.659E-02

123

31/03/2000

9.67%

1.73%

1.73E-02

3.03E-02

3.589E-01

4.357E-02

124

30/04/2000

-3.08%

-4.19%

-4.19E-02

3.70E-02

2.723E-01

4.614E-02

125

31/05/2000

-2.19%

-2.44%

-2.44E-02

2.26E-02

1.627E-01

9.711E-02

126

30/06/2000

2.39%

4.85%

4.85E-02

5.99E-03

9.945E-02

1.614E-01

127

31/07/2000

-1.63%

-1.58%

-1.58E-02

1.24E-02

7.938E-02

1.391E-01

128

31/08/2000

6.07%

5.35%

5.35E-02

-3.54E-03

3.376E-02

1.964E-01

129

30/09/2000

-5.35%

-1.08%

-1.08E-02

4.74E-03

5.201E-03

2.432E-01

130

31/10/2000

-0.49%

-2.01%

-2.01E-02

6.00E-03

2.807E-02

1.436E-01

131

30/11/2000

-8.01%

-4.30%

-4.30E-02

5.76E-03

8.473E-03

3.648E-02

132

31/12/2000

0.41%

3.16%

3.16E-02

-7.03E-03

-7.782E-03

7.690E-02

133

31/01/2001

3.46%

2.88%

2.88E-02

4.59E-03

3.718E-02

-2.107E-02

134

28/02/2001

-9.23%

-2.56%

-2.56E-02

-3.07E-03

-4.069E-02

2.877E-02

135

31/03/2001

-6.42%

-2.30%

-2.30E-02

-6.21E-03

-5.252E-02

3.656E-02

136

30/04/2001

7.68%

2.27%

2.27E-02

-6.29E-03

-1.114E-02

3.509E-02

137

31/05/2001

0.51%

0.90%

9.00E-03

4.25E-03

4.259E-02

-4.145E-02

138

30/06/2001

-2.50%

-0.32%

-3.20E-03

4.23E-03

7.419E-03

2.786E-02

139

31/07/2001

-1.08%

-1.06%

-1.06E-02

-4.85E-03

-3.116E-02

-1.829E-02

140

31/08/2001

-6.41%

-1.22%

-1.22E-02

-2.40E-03

-1.784E-02

-3.670E-02

141

30/09/2001

-8.17%

-3.73%

-3.73E-02

3.70E-03

-3.221E-02

-3.718E-02

142

31/10/2001

1.81%

1.85%

1.85E-02

-7.58E-03

-3.619E-02

2.577E-02

143

30/11/2001

7.52%

1.97%

1.97E-02

-7.25E-03

-2.597E-02

2.566E-02

144

31/12/2001

0.76%

1.99%

1.99E-02

-3.83E-03

-3.395E-03

-2.873E-02

145

31/01/2002

-1.56%

0.22%

2.20E-03

3.94E-03

9.498E-03

-1.429E-02

146

28/02/2002

-2.08%

-0.89%

-8.90E-03

3.89E-03

1.287E-02

2.241E-02

147

31/03/2002

3.67%

2.03%

2.03E-02

5.07E-03

7.347E-02

-4.462E-02

148

30/04/2002

-6.14%

0.17%

1.70E-03

5.29E-03

5.577E-02

-4.273E-02

149

31/05/2002

-0.91%

0.00%

-2.13E-18

3.90E-03

3.537E-02

-2.273E-02

150

30/06/2002

-7.25%

-2.63%

-2.63E-02

3.47E-03

-1.153E-02

2.390E-02

151

31/07/2002

-7.90%

-3.93%

-3.93E-02

-2.57E-03

-5.246E-02

2.360E-02

152

31/08/2002

0.49%

0.28%

2.80E-03

-7.13E-03

-4.128E-02

3.082E-02

153

30/09/2002

-11.00%

-1.96%

-1.96E-02

-9.70E-03

-7.877E-02

3.213E-02

154

31/10/2002

8.64%

0.56%

5.60E-03

-1.28E-02

-7.518E-02

2.366E-02

155

30/11/2002

5.71%

2.67%

2.67E-02

-1.10E-02

-5.049E-02

2.099E-02

156

31/12/2002

-6.03%

-1.14%

-1.14E-02

-5.42E-03

-3.596E-02

-1.532E-02

157

31/01/2003

-2.74%

-0.01%

-1.00E-04

3.66E-03

3.376E-03

-4.204E-02

158

28/02/2003

-1.70%

-0.78%

-7.80E-03

3.44E-03

-7.230E-03

-5.680E-02

159

31/03/2003

0.84%

-0.07%

-7.00E-04

3.95E-03

1.191E-02

-7.432E-02

160

30/04/2003

8.10%

2.43%

2.43E-02

3.83E-03

3.073E-02

-6.428E-02

161

31/05/2003

5.09%

4.08%

4.08E-02

3.65E-03

4.488E-02

-3.206E-02

162

30/06/2003

1.13%

1.52%

1.52E-02

5.84E-03

7.300E-02

2.218E-02

163

31/07/2003

1.62%

2.41%

2.41E-02

7.43E-03

9.896E-02

2.083E-02

164

31/08/2003

1.79%

2.38%

2.38E-02

1.29E-02

1.340E-01

2.393E-02

165

30/09/2003

-1.19%

0.78%

7.80E-03

1.81E-02

1.436E-01

2.322E-02

166

31/10/2003

5.50%

3.12%

3.12E-02

1.27E-02

1.513E-01

2.211E-02

167

30/11/2003

0.71%

1.14%

1.14E-02

1.06E-02

1.188E-01

3.554E-02

168

31/12/2003