Enhanced Stutzer Index Optimization Using Hybrid Genetic Algorithm and Sequential Quadratic Programming
This study presents a hybrid approach by associating the Genetic Algorithm (GA) and the Sequential Quadratic Programming (SQP) to improve the Stutzer Index optimization. The Stutzer Index is a well-recognized portfolio performance measure that provides unbiased estimates of risk-adjusted performance. However, the tasks in optimizing and determining a good starting point for the constrained optimization of Stutzer Index are challenging, especially with the additional constraint on the negative term θ. By integrating GA and SQP, this study anticipates the hybrid model to improve the efficiency and the performance of the optimization. The optimal indices obtained from both the SQP and the hybrid GA-SQP that used the initial guess recommended by Stutzer and the optimal index acquired via the hybrid GA-SQP with random starting point, for different period of data and number of assets respectively, are utilized for the comparative study. The results revealed that the hybrid model is superior in the Stutzer Index optimization, owing to the consistent capability of GA to locate the global optimum region and SQP to reach the optimal solution. The results also attested that the hybrid model enhanced the efficiency of the optimization as it does not required user-defined starting point and can sufficiently attained the optimal solution by utilizing a randomly generated starting point. In general, the hybrid model is competent in improving the efficiency and the performance of the Stutzer Index optimization, albeit the enhancement is not statistically significant in smaller number of observations.
Received: October 26, 2010;
Accepted: January 22, 2011;
Published: February 24, 2011
The most widely used approaches in portfolio construction and risk-adjusted
performance measurement are founded principally on the mean-variance framework
of Markowitz (1952) and the reward-to-variability (or
Sharpe Ratio) of Sharpe (1966). However, the necessity
for alternative performance measurements escalates as the facts of non-normality
in asset returns surface. In response to the inappropriate assumption of normality,
Stutzer (2000) proposed a portfolio performance index
that uses the quantifiable exponential decay rate of the likelihood that a portfolio
will underperform a benchmark, i.e. the Stutzer Index. The Stutzer Index, also
known as the Stutzers Portfolio Performance Index, is an alternative risk-adjusted
performance measurement that is sensitive to the shape and the higher moments
of the returns distribution. Stutzer Index rewards those portfolios that have
a lower probability of underperforming a specified benchmark on average, or
portfolios with returns distribution that is positively skewed. The indexis
established on the assumption that fund managers have an aversion towards excess
returns that underperformed a pre-designated benchmark and hence, will favor
portfolios with positive expected excess returns, i.e., portfolios with higher
probability of decay rates.
Stutzer Index is a consistent generalization of the Sharpe Ratio that provides
unbiased estimates of risk-adjusted performance, even with the presence of skewness
and kurtosis in the asset returns and it was formerly used by the Morningstar,
Inc. in its Star Ratings of mutual funds. Stutzer (2000)
has shown that the Stutzer Index can be estimated by:
where, θ is a negative value, N is the total number of observations over
the evaluation period, Rt is the excess returns of a portfolio over
the predetermined benchmark at time t. Thus, for n assets under consideration,
Stutzer (2000) has also attested that the optimal value
of θ and the optimal weights, that maximizes the Stutzer Index can be obtained
Notwithstanding the advantages demonstrated by the Stutzer Index, the optimization
of Stutzer Index is slightly more challenging than other portfolio performance
measures such as Sharpe Ratio and Sortino Ratio (Sortino
and Price, 1994), due to the extra constraint applied on the negative value
θ, in addition to the constraints in weights.
Optimization technique is generally used to find a set of optimal parameters
of some objective function and it might be subject to some parameter bounds,
equality and/or inequality constraints. A good starting point for the optimizer
is essential in ensuring the accomplishment of the optimization, especially
in a global optimization problem to avoid local optimum. Stutzer
(2000) solved the maximization of Stutzer Index by using numerical optimization
and he proposed that the estimated optimal portfolios weights of the Sharpe
Ratio and the -1 times the portfolios mean excess return divided by its variance,
is a good initial guess for the portfolio weights and θ, respectively in
optimizing the Stutzer Index. Following his work, Benson
et al. (2008) employed numerical optimization and the initial guess
recommended by him to solve the Stutzer Index optimization problem in their
study. In a more recent study, Lye and Ng (2010) also
followed Stutzers suggestion in the choice of starting point, although
they applied SQP to optimize the Stutzer Index. However, most of the standard
optimization methods such as numerical optimization and SQP are sensitive to
the starting point and their solutions are more likely to be trapped in a local
minimum or converge prematurely. Motivated by the study conducted by El-Mihoub
et al. (2006) which highlighted the merits of hybrid genetic algorithms,
and the fact that optimizing and determining a good starting point for the constrained
optimization of the Stutzer Index is a challenging task, therefore, this study
proposes to use the integrated genetic algorithm and the sequential quadratic
programming (GA-SQP) to:
||Enhance the efficiency in optimizing the Stutzer Index by
using randomly generated starting point, in which it diminished the difficulty
and the necessity to determine the best starting point preceding the optimization
||Improve the performance of the Stutzer Index optimization
by utilizing the advantages of GA andSQP in finding the global optimum and
reaching the optimal solution, respectively
By combining the GA (a well-known global search algorithm) with the SQP (an efficient local search method), it is needless to decide the best starting point for the hybrid GA-SQP and it is anticipated to improve the performance of the Stutzer Index optimization as well.
MATERIALS AND METHODS
This study was conducted from January 2010 to September 2010, in the Multimedia
University of Malaysia. Figure 1 displays the basic steps
in the hybrid GA-SQP. The first component of the hybrid model uses the GA to
search for the region in which the global optimum is located. Genetic algorithms
(Goldberg, 1989; Mitchell, 1998) apply
an evolutionary process via the genetic operators (selection, crossover, and
mutation) to perform global search in a solution space gradually, to discover
the best solution for the problem. Genetic algorithms are well-recognized effective
global optimization tools in solving both constrained and unconstrained optimization
problems because of their simplicity, derivative-free, inbuilt parallel processing
capability and most importantly, it can perform equally well even without user-defined
starting point. In this study, the search for the best solution is continued
by SQP in the second part of the hybrid model. Sequential Quadratic Programming
(Fletcher, 1987) is a well-known method in solving optimization
of nonlinear continuous objective function, in which it iteratively solves a
series of quadratic programming subproblems that are subject to some linear
constraints. Even though the SQP is proven in finding the local optimum of an
optimization problem, the starting point provided to the method is very crucial
in locating the desired optimal solution.
||The basic steps in the hybrid Genetic algorithm-sequential
Hybrid GA-SQP is applied in various fields and its competency to solve various
real-world problems is verified (Da-kuo et al., 2008;
He et al., 2008; Mansoornejad
et al., 2008; Nisar et al., 2008;
Rentizelas and Tatsiopoulos 2010; Wang
et al., 2006; Zeeshan et al., 2010).
By retaining the merits of GA and SQP, respectively, in locating the most promising
region of convergence and in obtaining the desired optimal solution, the hybrid
GA-SQP applied in this study aims to enhance the efficiency and the performance
of the optimization of the Stutzer Index. This study used the Genetic Algorithm
and Direct Search Toolbox and the Optimization Toolbox of the MATLAB for the
constrained optimizations. The optimization of the Stutzer Index is realized
by using the Eq. 2 and it is subject to the following constraints:
||The total weights of the portfolio for n assets under consideration
is equal to one
||The weight in the portfolio is non-negative (wi,
0), by assuming short selling is not permitted
||The value of θ is negative (θ<0)
The data employed for the empirical study are the adjusted daily returns of
the top 30 constituent stocks (by market capitalization as of 31st December
2008) of the Kuala Lumpur Composite Index (KLCI) in Bursa Malaysia, Malaysia,
dated from 2nd January 1996 to 31st December 2008, that are available throughout
the period of study. All the data, including the market benchmark index KLCI,
are retrieved from the DataStream. Table 1 presents the summary
statistics of the stock returns and it is verified from the Jarque-Bera test
that the adjusted daily returns are not normally distributed. The empirical
study in this paper is segmented into two main parts. The first part of the
study utilizes the entire period of data to maximize theStutzer Index for different
number of assets (10, 20 and 30). The optimal Stutzer Index acquired from the
SQP by using the starting point suggested by Stutzer (2000)
is compared with the optimal index attained by the hybrid GA-SQP. The study
on the sufficiency of randomly generated starting point in contrast to the initial
guess suggested by Stutzer (2000) is also carry out
via the hybrid GA-SQP and the S-GA-SQP (hybrid GA-SQP with initial guess proposed
by Stutzer) respectively. As for the latter part of the empirical study, the
data are divided into a series of subperiods (3-month and 6-month) in which
each corresponding portfolio is revised after every subperiod. The respective
optimal index acquired from the S-SQP, hybrid GA-SQP and the S-GA-SQP are consequently
utilized in the Mann-Whitney nonparametric test for the comparative study.
|| Descriptive statistics for stock daily returns and test for
|JB indicates the Jarque-Bera normality test. If the JB test
statistic is greater than the critical value 15.6372, the JB test rejects
the null hypothesis that the data are from a normal distribution at 1% significance
RESULTS AND DISCUSSION
The first part of the study employed the entire period of data and compared
the performance of the S-SQP with the hybrid GA-SQP. Table 2
presents the optimal Stutzer Index and the daily returns (%) for different number
of assets. The results revealed that the optimal Stutzer Index acquired from
the hybrid GA-SQP is higher than the optimal index obtained by the S-SQP and
it is consistent in all three different numbers of assets. The superiority of
the hybrid GA-SQP attested the importance of starting point in standard optimization
such as SQP and the capability of GA and its consistency in locating the global
optimum region. The findings are consistent with some of the earlier studies
conducted although the hybrid model was applied in other areas (Mansoornejad
et al., 2008; Nisar et al., 2008;
Rentizelas and Tatsiopoulos 2010; Wang
et al., 2006). In addition, the results also disclosed that the hybrid
GA-SQP has enhanced the efficiency of the optimization of Stutzer Index as it
does not required user-defined starting point. Furthermore, the identical optimal
indices yielded by the hybrid GA-SQP and the S-GA-SQP, as shown in Table
2, verified that a randomly generated starting point is sufficient for the
hybrid GA-SQP in optimizing the Stutzer Index. The competency shown by the hybrid
model in utilizing a random starting point is also highlighted in other applications
(He et al., 2008; Zeeshan
et al., 2010). Moreover, the optimal daily returns (the product of
the assets average daily return and its respective optimal weights) obtained
from the hybrid GA-SQP also outperformed the S-SQP. No significant difference
is evident between the daily returns acquired by the hybrid GA-SQP and the S-GA-SQP
and this further confirmed that the two methods have successfully reached the
The weights and the value of θ in the optimal portfolio for different
number of assets are disclosed in Table 3. The overall weights
allocated in each stock are nearly comparable in all the three methods. On the
other hand, the divergence in the optimal value of θ is noticeable between
the S-SQP and the other two approaches.
|| The optimal Stutzer Index and daily returns (%) for different
number of assets
|| The weight (%) and the value θ in the optimal portfolio
for different number of assets
||The Mann-whitney nonparametric test on the optimal Stutzer
Index for a series of subperiods with null hypothesis: the median of GA-SQP
is not greater than the median of S-SQP
||The Mann-Whitney nonparametric test on the optimal Stutzer
Index for a series of subperiods with null hypothesis: the median of GA-SQP
is equal to the median of S-GA-SQP
With no significant variations in the weights between the three approaches,
the smaller value of θ in S-SQP ought to be the factor why its performance
is inferior to the hybrid GA-SQP and the S-GA-SQP. This further demonstrated
the importance of a good starting point in a standard optimizer, particularly
in the problem that involved wider search space such as the value of θ
θ < 0) in the Stutzer Index optimization.
In the second part of the study, the corresponding optimal Stutzer Index of
each subperiod is acquired via the S-SQP, hybrid GA-SQP and S-GA-SQP. Table
4 exhibits the hypothesis test summary obtained from the Mann-Whitney nonparametric
test. Even though the results revealed that the hybrid GA-SQPfaintly better
than the S-SQP, the superiority is not statistically significant in smaller
number of observations. However, the outperformance of the hybrid GA-SQP over
the S-SQP becomes perceptible when the number of observations under consideration
increased. This can be evident from the decreases in the p-value shown in the
Table 4, when the range of the subperiod widen from 3 to 6
month. The factual deficiency of SQP in solving large-scale problem is verifiable
by the inbuilt weakness of standard SQP algorithm, which was pointed out as
well in the research of Murray (1997). On the other
hand, the hybrid model attested its potential to overcome such limitations and
the prospect to solve optimization problems particularly in finance, business
and economics, which often have enormous number of data.
As for the comparative study between the hybrid GA-SQP and the S-GA-SQP, as shown in Table 5, the Mann-Whitney nonparametric test disclosed no significant difference between the optimal indices regardless of the number of observations. These outcomes further verified the sufficiency of the hybrid GA-SQP in optimizing the Stutzer Index by using random starting point and the capability of GA in conducting global search.
This study used a hybrid model that is incorporated by two well-recognized
optimization methods that complement each other perfectly: the genetic algorithm
and the sequential quadratic programming, to improve the optimization of the
Stutzer Index. The results revealed that the hybrid model successfully enhanced
the efficiency and the performance of the Stutzer Index optimization, in comparison
to the standard sequential quadratic programming that employed the initial guess
suggested by Stutzer (2000). The findings also attested
that the hybrid model does not required any user-defined starting point as it
can efficiently attained the optimal index by utilizing a randomly generated
starting point, i.e., consumed less time. Even though the improvement is not
statistically significant in smaller number of observations, the hybrid model,
in general, demonstrated competency in improving the efficiency and the performance
of the Stutzer Index optimization.
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