Andre Santos

Joao Tusi

Newton Da Costa Jr

Sergio Da Silva

English

Research Papers in Economics

Evaluating Brazilian Mutual 
Funds with Stochastic Frontiers 
 
 
Andre Santos 
Department of Economics, Federal University of Santa Catarina 
 
Joao Tusi 
Department of Economics, Federal University of Santa Catarina 
 
Newton Da Costa, Jr 
Department of Economics, Federal University of Santa Catarina 
 
Sergio Da Silva 
Department of Economics, Federal University of Santa Catarina 
 
 
 
Abstract 
 
We evaluate the performance of 307 Brazilian stock mutual funds employing 
stochastic frontiers.  We list the top ten actively managed funds and the bottom ten for 
the period April 2001−July 2003, and show that a fund’s efficiency increases with 
management skill to beat the market.  We also find that portfolios with low volatility 
tend to be more efficient.  Yet we find no relationship between fund size and 
performance, though this might be blurred by a survivorship bias. 
 
 
Keywords: Stochastic frontiers, mutual funds 
 
JEL classification: M21, G11, G23 1.  Introduction 
 
Actively managed mutual funds by professional managers are generally viewed as 
unable to beat on average either a benchmark (the market) or passively managed funds 
(e.g. Annaert, Van De Broeck, and Vennet 2003).  This is in line with the efficient 
market hypothesis.  Active funds can be perceived to beat the benchmark only due to 
large sampling noise in standard performance measures; and this should not be mistaken 
with superior management (Kothari and Warner 2001). 
  So active management of funds can at best match the performance of an efficient 
mix of passive portfolios.  Investors will not chase greater profitability; rather, they can 
do better by looking for relief from the burden of managing a diversified portfolio 
(Annaert, Van De Broeck, and Vennet 2003). Managing funds this way reaches the 
efficient frontier, where expected returns are maximized given the risk (variance of 
portfolio returns). 
  Departures from the efficient frontier can be imagined if its nature is seen as 
stochastic (Aigner, Lovell, and Schmidt 1977, Battese and Corra 1977, Meeusen and 
Van De Broeck 1977).  To get an efficiency measure that is not blurred with noise one 
can take (1) a one-sided distribution (such as a seminormal distribution) to model the 
efficiency term, and (2) a normal distribution to model the measurement noise. A clear-
cut measure then emerges, which can relate efficiency to particular fund characteristics, 
such as size. When deviations from this frontier are unequivocally attributable to a 
fund’s inefficiency, the frontier is called deterministic. 
  We employ the stochastic frontier approach to track the leaders of Brazilian 
stock mutual funds from April 2001 to July 2003.  Our technique will identify the 
funds’ features that can be unambiguously attributable to efficiency. Section 2 
elaborates further on the methods of evaluating mutual fund performance. Section 3 
describes data.  Section 4 presents results and discusses their implications. Section 6 
concludes. 
 
2.  Alternative evaluation methods 
 
To rank funds according to their performance, returns should be corrected for risk.   
Sharpe (1966) suggests dividing a fund’s excess return by the standard deviation of its 
return, thereby correcting for total risk. Here the excess return is given by the difference 
between actual return and a risk-free rate, such as that of government bonds. 
As people diversify their investments, correcting for nondiversifiable risk (rather 
than total risk) stands as more appropriate.  Here a fund’s CAPM beta can be employed 
(Treynor 1965).  Alternatively one can take the difference between actual and expected 
excess returns (Jensen 1968).  Jensen alpha is 
() pp p RE R α= −                                                                                                     (1) 
where  p R  is average actual excess return of fund  p , and  () p ER  is expected excess 
return.  In a time-series setting, for the basic CAPM 
() pppm RE R αβ =−                                                                                              (2) 
where  () m ER  is expected excess return on the market portfolio. 
Performance as measured by Jensen alpha might be plagued by sampling noise.  
Some funds can be found to outperform the benchmark, even when their managers have 
no real ability to outperform it.  So equation (2) needs to be extended to take account of 
this. One way of doing that is including a compound error component, following the stochastic frontier approach.  One term tracks efficiency ( p ξ ), and one captures the 
measurement error ( p ν ), i.e. 
ˆ ˆ ppp m p p RR v γβ ξ =+ −+                                                                                         (3) 
where  p R  is the fund’s excess return,  m R  is the excess return on market portfolio, and 
(nonnegative)  p ξ  and (zero-mean)  p ν  are assumed to be IID both between one another 
and across the funds.  Nonnegativity of  p ξ  follows from the evidence that most actively 
managed funds cannot systematically outperform the benchmark.  From this it follows 
that the stochastic upper bound is at  ˆ ˆpp m p Rv γβ ++ .  The efficiency term captures the 
ability in managing a fund whereas the measurement error tracks out-of-control shocks 
interfering with efficiency.  A useful efficiency index is 
ˆˆ ˆˆ () / ( ) pp m p ppp m p Rv R v γβ ξ γβ +− + ++ , which ranges from zero to unity (full 
efficiency). 
Following Battese and Coeli (1992), here we estimate technical efficiency 
assuming a seminormal distribution and defining 
()
22
01 2
11
1
log ( , , , ) ln2 ln ln
21 2
NN
p
p
ii
e N
Le
δ
γγσ δ π σ φ
σδ σ ==
 
=− + + − −    −   ∑∑      (4) 
where  log ( ) L ⋅  is the maximum log-likelihood function, 
2
ξ σ  and 
2
v σ  are variance of ξ  
and  v respectively, 
22
ξ δσσ =  (where 
222
v ξ σσσ =+ and  [] 0,1 δ∈ ), 
() ˆ ˆ pP pP m eR R γβ =−+  is compound error, and φ  is the cumulative normal density 
function.  (An unbalanced panel data is also employed.) 
 
3. Data 
 
We take 307 actively managed Brazilian stock funds, and get their monthly return series 
net of management fees, dividends, bonuses, and splits.  Time period ranges from April 
2001 to July 2003, and we consider the Bovespa stockmarket index as the benchmark.  
The funds are evaluated with and without leverage.  Excess returns are calculated using 
the Brazilian CDI OVER rate.  And the LIMDEP software (www.limdep.com) is 
employed to estimate the stochastic frontiers through maximum likelihood. 
Table 1 shows cross-section statistics for equation (3).  Jansen alpha averages 0.05 
percent, showing no evidence of the funds to outperform the benchmark (for the sample 
as a whole).  Monthly fund returns vary 0.19 percent on average, from –45.09 to 37.42 
percent.  And the Bovespa index return (not shown) averages 0.18 percent (standard 
deviation of 9.06 percent).  This reinforces the fact that the funds cannot outperform the 
passive benchmark.  The betas are mostly near unity, thereby justifying the use of the 
Bovespa index for benchmark. 
 
4. Results 
 
One cannot take the efficient market hypothesis for granted, however.  Indeed Table 2 
displays variances for the compound error.  Both errors are significant at the one percent 
level; and this justifies employing the stochastic frontier.  So departures from efficiency 
can occur and funds can be ranked according to our efficiency index. Table 3 ranks the top ten funds as far as efficiency is concerned.  Variables are 
estimated using equation (3).  As can be seen, no fund reaches one (as expected from 
non-deterministic frontiers).  The top ten funds all outperform the benchmark.  The 
winning fund over the period is Hedging Griffo Skopos HG with an index of 0.923.  
Jensen alphas are all positive (there is a positive difference between the funds’ returns 
and market return weighted by the beta), though some fail to be statistically significant. 
By contrast, the least efficient funds in Table 4 cannot beat the benchmark.  These 
also present greater volatility than the winning funds (standard deviation of 10.39 and 
8.21 percent respectively). 
Table 5 shows Pearson correlation matrix of the variables for the entire sample.  
The efficiency index is strongly correlated with Jansen alpha (as one might expect) and 
negatively related to standard deviation.  So both superior management skill to beat the 
benchmark and low volatility boost a fund’s efficiency.  Yet fund size is not related to 
performance.  This might probably occur thanks to a survivorship bias (Elton, Gruber, 
and Blake 1996).  This bias is due to the fact that smaller firms face a higher probability 
to disappear.  Indeed some funds in our sample have appeared while others disappeared 
over the period. 
 
5. Conclusion 
 
We list top and bottom ten Brazilian mutual funds for the period April 2001−July 2003.  
This is done to a sample of 307 stock mutual funds with the help of a stochastic frontier 
approach.  The estimated variances of the efficiency and measurement errors are both 
significant, a fact that justifies the use of stochastic frontiers. 
We thus are able to rank funds according to an efficiency index.  Doing so we find 
that the top ten funds outperform the benchmark and thus violate the efficiency market 
hypothesis.  Performance is dependent on superior management skills.  We also find 
that portfolios with low volatility tend to be more efficient.  Yet we find no relationship 
between fund size and performance, though this might be blurred by a survivorship bias. 
  Table 1.  Cross-section statistics for 307 Brazilian mutual funds, April 2001−July 2003 
 
 Mean  Median  Standard 
Deviation  Minimum Maximum 
Monthly Return (%)  0.19  −0.38  8.37  −45.09  37.42 
Alpha (%)  0.05 0.11  1.19  −10,22  7.80 
Beta  0.88 0.92  0.31  −1.04  2.91 
R Squared  0.87 0.94  0.20  0.0002  0.997 
 
 
 
Table 2.  Compound error’s estimated variance parameters 
 
Parameter Coefficient  Standard  Error   t Statistics 
) (
2 ξ σ   0.0745  −  − 
) (
2 v σ   0.1002 0.05145  194.705
* 
) ( / ) (
2 2 v σ ξ σ   0.0042 0.00121 3.521
* 
 
* Significant at one percent 
 
 
Table 3.  Top ten stock mutual funds 
 
Fund Time  Period  Return 
(%) 
Standard 
Deviation 
(%) 
Alpha 
(%)  Beta 
Efficiency 
Index 
HEDGING GRIFFO SKOPOS HG  May 2001−Mar 2003  118.18 8.65  3.56
**  0.55
*  0.923 
HEDGING GRIFFO CAMINO FIA  Apr 2001−Jul 2003  110.90 3.76  1.30
***  0.06
N  0.905 
COINVALORES COIN FATOR 
FIA  Apr 2001−Jul 2003  69.99 9.35  2.07
*  0.94
*  0.897 
FATOR PLURAL JAGUAR FIA  Apr 2001−Jul 2003  56.74 9.47  1.88
*  1.01
*  0.891 
HEDGING GRIFFO CSAM KITE 
FIA  Apr 2001−Sep 2001  13.18 4.77  1.24
N  0.52
N  0.881 
BCN ACTIVE FIA  Apr 2001−Jul 2003  38.00 9.55  1.43
*  1.02
*  0.879 
PROSPER ADVINVEST FIA  Apr 2001−Jul 2003  43.61 7.20  0.96
N  0.70
*  0.878 
ITAU INSTITUCIONAL ONIX FIA  Apr 2001−May 2003  26.37 9.66  1.05
N  0.74
*  0.877 
MELLON BRASCAN ARX FIA  Jul 2001−Jul 2003  34.28 9.51  1.37
**  0.94
*  0.876 
BRADESCO BRAM FIA 
IBOVESPA ALAVANCADO  Jun 2002−Jul2003  43.03 10.17  2.13
N  0.82
*  0.874 
 
*   Significant at one percent 
**   Significant at 5 percent 
***   Significant at 10 percent 
N   Not significant Table 4. Bottom ten stock mutual funds 
 
Fund Time  Period  Return 
(%) 
Standard 
Deviation 
(%) 
Alpha 
(%)  Beta 
Efficiency 
Index 
MULTISTOCK MAXIMA 
MARKET PORTFOLIO  May 2001−Jan 2003  −90.28  16.23  −10.22
**  0.02
N  0.315 
JP MORGAN BASE FIA  Dec 2001−Sep 2002  −60.15  15.14  −2.84
N  1.24
**  0.740 
MELLON BRASCAN ORYX FIA  Apr 2001−Jul 2003  −40.42  9.66  −1.56
**  0.99
*  0.744 
COMPACTA CL  Apr 2001−Jul 2001  −35.77  8.53  −7.16
N  1.04
N  0.763 
MELLON BRASCAN SANTA FÉ 
SCORPIUS FIA  Apr 2001−Jul 2003  −33.42  9.78  −1.15
***  1.01
*  0.769 
GERAÇÃO FUTURO BNL 
CHIUSO 157  Apr 2001−Jul 2003  −31.01  9.19  −1.17 
***  0.94
*  0.774 
PACTUAL ASSET FIA 
INSTITUCIONAL X  Feb 2002−Jul 2003  −23.49  9.9  −1.79
**  0.99
*  0.775 
QUALITY HAMBURG FIA  Jul 2001−Jul 2003  −27.74  8.91  −2.55
*  0.89
*  0.782 
LATINVEST FIA FC  Apr 2001−Jun 2003  −28.13  7.62  −1.28
**  0.78
*  0.783 
UNIBANCO QUALIFICADO FIA  Apr 2001−Jul 2003  −27.23  8.89  −0.97
*  0.96
*  0.784 
 
*   Significant at one percent 
**   Significant at 5 percent 
***   Significant at 10 percent 
N   Not significant 
 
 
 
Table 5.  Pearson correlation matrix 
 
 
Efficiency Index  Beta  Alpha  Standard 
Deviation  Fund Size 
Efficiency Index  −        
Beta  −0.012
N  −      
Alpha 0.702
*  0.351
*  −    
Standard Deviation  −0.286
*  0.537
*  0.097
N  −   
Fund Size  0.034
N  0.031
N  0.031
N 0.02
N  − 
 
*   Significant at one percent 
**   Significant at 5 percent 
***   Significant at 10 percent 
N   Not significant References 
 
Aigner, D, C. Lovell, and P. Schmidt (1977) “Formulation and estimation of stochastic 
frontier production function models” Journal of Econometrics 6, 21-37. 
 
Annaert, J., J. Van De Broeck, and R. Vennet (2003) “Determinants of mutual fund 
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Battese, G.E., and Coelli, T.J. (1992) “Frontier production functions, technical 
efficiency and panel data with application to paddy farmers in India” Journal of 
Productivity Analysis 3, 153-169. 
 
Battese, G.E., and G.S. Corra (1977) “Estimation of a production frontier model: with 
application to the pastoral zone of Eastern Australia” Australian Journal of Agricultural 
Economics 21, 169-179. 
 
Elton, E.J., M.J. Gruber, and C.R. Blake (1996) “Survivorship bias and mutual fund 
performance” Review of Financial Studies 9, 1097-1120. 
 
Jensen, M. (1968) “The performance of mutual funds in the period 1945-1964” Journal 
of Finance 23, 389-416. 
 
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Finance 56, 1985-2020. 
 
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De Broeck

Determinants of mutual fund underperformance: a Bayesian stochastic frontier approach”

Estimation of a production frontier model: with application to the pastoral zone of

Evaluating mutual fund performance”

Formulation and estimation of stochastic frontier production function models”

Frontier production functions, technical efficiency and panel data with application to paddy farmers in

How to rate management of investment funds”

Mutual fund performance”

Survivorship bias and mutual fund performance”

The performance of mutual funds in the period 1945-1964”

Evaluating Brazilian Stock Mutual Funds with Stochastic Frontiers

http://128.118.178.162/eps/fin/papers/0510/0510030.pdf

Evaluating Brazilian Stock Mutual Funds with Stochastic Frontiers

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Research Papers in Economics