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The present study aimed to choose the optimum multi-period investment portfolio model with attitude toward lower partial moment as a measure of risk under transaction costs constrain. In this study, a new method was used to calculate the LPM model. To compare the proposed method with the conventional method, some qualitative performance measures such as Skewness, the Sharpe ratio, Sortino ratio, Treyner ratio and Jensen measure is used. The results show that the proposed method improved efficiency compared to the conventional method. As multi-period optimization problem is NP-hard, meta-heuristic algorithm NSGAII was used to solve it. Selecting optimal investment portfolio is one of the most important decision-making issues in the field of management and financial engineering so, that's why it has been considered by investors and researchers of this area. By introducing Markowitz Mean-Variance (MV) model (1952) new insights in the field of portfolio selection was created. But since variance deal with the positive and negative deviations in the same way, it was in conflict with investors’ perceptions of risk so, Markowitz (1959) introduced Mean-Semivarience (MSV) model. The objective function of investment portfolio optimization models is defined based on one of two types of to maximize returns’ value from investment or minimize the risk of investment. The constraints section also contains limitations which will apply by the investors or the market where the investment is made.
In the real world investment strategies usually defined as a multi-period, because investors need to review and revise their portfolio in each period. This has led to the emergence of transaction costs as a key factor in selecting portfolio model, which these costs may vary from one period to another.
Bawa (1975) was the first author introduced the lower partial moment (LPM) as a general family of downside risk measures. Fishburn (1977), Lindenberg and Bawa (1977) Showed that there is a strong relationship between different levels of LPM and stochastic dominance. Fishburn (1977) presented a general LPM model and indicated that, regardless of any distribution restrictions, utility function defined by LPM are as general as the utility functions defined by stochastic dominance analysis.
LPM allows the investors to choose a large number utility function from risk loving to risk averse behavior (Kahneman and Tversky 1979). This feature makes the LPM to be considered in the academic literature of management and financial engineering Estrada (2002), Galagedera (2007), Jarrow and Zhao (2006), Jin et al. (2006). Nawrocki (1992) by examining the LPM portfolio selection algorithm and comparison with the Mean-Variance model showed that the LPM portfolio has less number of shares in compared with Mean-Variance model. Unser (2000) studied to examine experimentally people’s risk perception in a financial context and correspondence of risk perception with specific LPMs. He indicated that the LPM which explains risk perception best is the LPM0 and the reference point (target) of individuals for defining losses is not distribution’s mean but the initial price in a time series of stock prices. Mehrjoo and Jasami (2012) tried to solve lower partial moment portfolio optimization problem. In comparison between the Mean-Variance model and LPM model which was solved by the genetic algorithm, the results showed lower partial moment was more efficient than Mean-Variance model. Numerous empirical and simulation studies showed that investors are more interest about downside risk measures such as LPM in compares to the Mean -Variance measure Leibowitz and Langetieg (1989)، Nawrocki and Staples (1989)، Nawrocki (1992), Unser(2000), Mehrjoo and Jasemi (2012). Jaaman et al (2011) compared the Mean-Variance measure and the LPM through the multi-objective programming. Their results showed that the optimal portfolios that based on lower partial moment give higher expected return and skewness than the optimal portfolios that based on MV at the same level of downside risk.
Downside risk measures are also very useful in asset pricing. Eghbal et al (2007) compared CAPM of Black (1972) models with the Model Mean Lower Partial Moment Capital Asset Pricing of Bawa and Lindenberg (1977) and Harlow and Rao (1989) model in the context of emerging markets. Galagedera (2007) compared three measures of downside beta with CAPM beta in emerging markets monthly returns and observed that the downside beta is a better explanatory variables than the CAPM beta.
Fulga(2015) by considering links between stochastic dominance criteria, conditional value at risk and lower partial moment of first order, offered a method to determine an optimal portfolio. Buslama and Vadra (2014) used different risk measures such as LPM, conditional value at risk and variance and GARCH variance to assess the robustness of the portfolio.
Solving multi-period investment portfolio Problem also has a significant history of research. Cui et al. (2013), Li and Ng (2000), Gulpinar and Rustem (2007) and Çelikyurt and Özekici(2007) offered optimal multi-period Mean-Variance portfolio model. Calafiore and Kharaman (2014), Yan (2009), Pinar (2007) and Huang and Qiao (2012) by replacing the semivariance instead of variance solved multi-period portfolio optimization model. Liu and Zhang (2015) and Zhang et al (2012) modeled multi-period fuzzy portfolio optimization problem. They formulated Mean-Semivariance portfolio selection model under minimum transaction lots, transaction costs and portfolio’s diversification degree constrains. Najafi and Mushakhian (2015) solved Mean-Semivariance multi-period portfolio optimization problem by using hybrid of genetic and particle swarm optimization algorithm.
Arnott and Wagner (1990) stated in their study that ignoring the transaction costs can lead to failure in obtaining the efficient portfolio. Therefore, Bertsimas and Pachamanova (2008) and Gulpınar et al. (2003) and Konno and Wijayanayake (2001) considered transaction costs in the multi-period problems. Wei and Ye (2007) and Zhu et al (2004) examined multi-period Mean-Variance portfolio selection model with bankruptcy control in stochastic market. Leippold et al (2004) offered geometric approach to discrete time multi-period mean variance portfolio optimization that largely simplifies the mathematical analysis and the economic interpretation.
However, Mean-Semivariance model were examined only returns less than the average and did not have variance problem, but still the quadratic objective function feature was a limitation for it. For this purpose Bawa (1975) by introducing a lower partial moment solved this problem. LPM measure proposed by Bawa (1975) defined as follows.
Eq(1)
Where (FR) is the cumulative distribution function and Rnec is the target parameter. The parameter α determines the weight that investor is allowed for deviations. Since the calculation LPM in continuous mode is very difficult to do this, tried to be calculated adequately in discrete mode.
Nawrocki (1991) and Huang et al. (2001) presented symmetric matrix method to calculate LPM as follows.
Eq(2)
where di is the semideviation which is the square root of the semivariance Si for security I and pij is correlation matrix between securities i and j.
There is a fundamental flaw in this method. When the return of an asset is a less (more) than Rnec but the average weighted of portfolio’s (other assets) is greater (less) than Rnec then risk function value will increase (decrease) while we have not (have) face with loss in that specific period, so a new method used in this study to calculate LPM. If the previous method was called stock-driven new method will called portfolio-driven. Portfolio-driven LPM calculation method is as follows.
Eq(3)
Where Rp (w1,w2 ,…,wn)is return of P(w1,w2,…,wn).
To estimate from equation (3) first of all has to be calculate. Here it is done by applying the concept of histograms. As you can see both return and operating frequency, are the main architectural elements of histogram. In this case, the ranges (ri) are portfolio returns while other element (fi) is frequency that each interval occurs during a given period. So the method is applicable to all distributions.
To calculate LMP (Rnec, α) by drawing a histogram, you must first convert it to a discrete equation.
Eq(4)
If rk <Rn <rk + 1 is. In this case, we have:
Eq(5)
If we assume that the time horizon of T and portfolio’s return P (w1,w2,…,wn) at tth time unit is calculated as follows.
t=1, 2,…,T Eq(6)
Where rit is return of security I at tth time unit. So to calculate LPM (Rnec,α) we have:
Eq(7)
To demonstrate the efficient performance of portfolio-driven approach towards stock-driven method, some of qualitative criteria such as the skewness, Sharpe ratio, Sortino ratio, Treyner ratio and Jensen measure was used. Impact of the new method on processing time was also examined.
The Sharpe Ratio is a measure for calculating risk-adjusted return. The Sharpe ratio is the average return earned in excess of the risk-free rate per unit of volatility or total risk. Higher Sharpe ratio showed a higher rate of return to risk and Investment attractiveness. In fact, the Sortino ratio is modified version of Sharpe ratio. Therefore, it only consider negative volatility. So the higher the Sortino ratio is also preferred. The Treynor ratio is a metric for returns that exceed those that might have been gained on a risk-less investment, per each unit of market risk. The Jensen's measure is a risk-adjusted performance measure that represents the average return on a portfolio or investment above or below that predicted by the CAPM given the portfolio's or investment's beta and the average market return. Investors prefer positive skewness and dislike negative skewness. So the higher amount of these criteria are our utility.
To compare criteria for both methods the optimum number of trials was extracted by software minitab17. Then, using t-student test to approve the claim that the portfolio-driven approach works better than a stock-driven method.
The results were tested by both methods.
Tabel 1: The results of the portfolio’s statistical tests
Amount
Portfolio size skewness Sharpe ratio Sortino ratio Treyner ratio Jensen Measure
P-value t-static P-value t-static P-value t-static P-value t-static P-value t-static
5- shares 0.097 1.26 0.227 0.76 0.86 -1.11 0.074 1.49 0.187 0.91
15-shares 0.017 2.21 0.006 2.64 0.999 -3.29 0.12 1.2 0.092 1.36
25-shares 0.027 1.99 0.024 2.05 0.999 -3.19 0.04 1.8 0.008 2.53
According to the results observed in the 0.95 confidence level all criteria for a portfolio of 25-shares and skewness and Sharpe ratio for a portfolio of 15-shares is significant. In the 0.9 confidence level Jensen measure for a portfolio of 15-shares and Treyner ratio and skewness for portfolio of 5 -shares is significant.
Modeling multi-period investment portfolio
Variables and parameters used in this study included the following:
πs: probability that scenario s occurs, thus
rsn,t :return percentage of asset n, in time period t under scenario s
Cbuy: transaction costs incurred in rebalancing assets at the beginning of time period t for buying assets
Csell: transaction costs incurred in rebalancing assets at the beginning of time period t for selling assets
rl :rate of lending
rb :rate of borrowing
W0: wealth in the beginning of time period 0
xsn,t : amount of money for asset n, under scenario s, in the beginning of time period t before rebalancing
ysn,t :amount of money for asset n, under scenario s, in the beginning of time period t after rebalancing
vsn,t :amount of money bought of asset n, under scenario s, in the beginning of time period t
usnt: amount of money sold of asset n, under scenario s, in the beginning of time period t
bst :amount of money borrowed in period t, under scenario s
Wst : wealth at the beginning of time period t, under scenario s
ys :deviation from mean of wealth under scenario s
LPMsT: amount of lower partial moment at the end of time period T, under scenario s
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
Constraint (1) states that first objective function to maximize the wealth. Constraint (2) expresses second objective function to minimize risk (LPM measure). Constraint (3) represents the amount of money invested in assets n, under scenario s, in the beginning of the time period t after rebalancing. Constraint (4) shows budget constraints at the time 0 that guarantees the total initial investment equals to the initial budget and amount of money borrowed at the beginning of period 0. Constraints (5) - (8) are cash flow restrictions in period t. Constraint (9) is the wealth accumulated at the end of tth period under scenario s before rebalancing.
NSGAII
Deb et al (2002) provided multi-objective evolutionary algorithm (MOEA) Non-dominated Sorting Genetic Algorithm II (NSGAII), which was the newer version of (NSGA). NSGA has drawbacks such as computational complexity, Lack of elitism and the need to identify the parameters of sharing. By addressing these problems improved version of the NSGA was presented. According to the results of several complex problem simulation, they found that NSGAII works better than the two other contemporary MOEAs: (PAES) and (SPEA) in finding the answer set, and in covering Pareto optimal set.
One of the fundamental issues in meta-heuristic algorithms is determining the optimal parameters affecting the performance of the algorithms such as mutation rate, crossover rate, the number of initial population and the optimal number of iteration. For this purpose, Taguchi experimental design approach was used to obtain optimal parameters.
Taguchi experimental design
One of the important part of the Taguchi experiment design is selection of control factors. For this purpose, a standard orthogonal matrix is created to meet this need that produce the number of tests for each level of optimal parameters. To determine the orthogonal matrix the number of factors and the number of levels which is taken for each factor should to be considered. In the genetic algorithm some factors such as initial population, number of iteration, the rate of mutation, crossover and mutation rate percentage could affect the results. In this study, 3 levels is considered for each parameter. Thus, according to an orthogonal matrix 27 tests is needed to perform.
In order to assess the relative importance of each factor by taking into basic effects of that on the performance of the algorithm, the S/N exchange rate is used. The S refers to optimal values and the N points out Undesirable values (Noise) that the aim is to maximize the rate.
Rate of S/N is calculated as follows:
Eq(8)
According to the results of calculations, optimization parameters are as follows.
Table 2 - The results of Taguchi experiment
parameters initial population number of iteration Crossover rate mutation rate percentage mutation rate
Optimal value 150 300 0.8 0.06 0.12
But the main feature that make portfolio-driven method more attractive than stock-driven method is reducing the solution time and its simplicity. Because it does no need to calculate the variance-covariance matrix and computational complexity is reduced. Table 4-10 show results of model’s computation processing time, for both methods at α = 2. The system is used in the calculations has the following specifications:
CPU:Pentium(R) Dual-Core 2.30GHz , RAM:3.00GB , MATLAB R2015b
Table 3 compares the results of calculation of different methods for investment portfolios
Method
Portfolio size Stock-driven Portfolio-driven
5-shares 100.694 minutes 86.557 minutes
15-shares 120.962 minutes 94.350 minutes
25-shares 165.038 minutes 100.528 minutes
According to the results, observed that the portfolio-driven reduces the process time dramatically. Since time is a key factor in the financial area thus, this method can be used in on-line trading systems and users benefit from its advantages.
Figure 2 shows the result of the NSGAII algorithm’s run and each point represents an investment strategic. For example, the point is shown has following risks and returns.
Results
This study aims to provide a model for selecting optimal investment portfolio with lower partial moment risk measure approach. For this purpose, three different investment portfolio (5 a share, 15 shares and 25 shares) was created by using data from the New York Stock Exchange. Then a new method for calculating LPM portfolio-driven was introduced and were compared with the conventional method (stock-driven). In order to evaluate the performance of portfolio-driven, skewness rate, the Sharpe ratio, Sortino ratio, Trayner ratio and Jensen ratio were used. According to the results observed in the 0.95 confidence level all the criteria for a portfolio of 25 shares and skewness and sharpe ratio for a portfolio of 15 shares is significant. In the 0.9 confidence level jensen ratio for a portfolio of 15shares and treyner and skewness rations for portfpio of 5 shares is significant. Then the multi-period portfolio optimization model by taking into transaction costs constrain was solved by meta-heuristic algorithm NSGAII. By comparing the portfolio-driven method and stock-driven method processing time observed that the portfolio-driven approach requires less time to reach the optimal solution.
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...utes 86.557 minutes 15-shares 120.962 minutes 94.350 minutes 25-shares 165.0...
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...nutes 15-shares 120.962 minutes 94.350 minutes 25-shares 165.038 minutes 100....
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...nutes 94.350 minutes 25-shares 165.038 minutes 100.528 minutes According to t...
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...utes 25-shares 165.038 minutes 100.528 minutes According to the results, obser...
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...wn has following risks and returns. Results This study aims to provide a mo...
^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Transition Words or Phrases used:
also, but, first, however, if, may, second, so, still, then, therefore, thus, while, as to, for example, in fact, such as, first of all, in the same way
Attributes: Values AverageValues Percentages(Values/AverageValues)% => Comments
Performance on Part of Speech:
To be verbs : 88.0 15.1003584229 583% => Less to be verbs wanted.
Auxiliary verbs: 15.0 9.8082437276 153% => OK
Conjunction : 98.0 13.8261648746 709% => Less conjunction wanted
Relative clauses : 44.0 11.0286738351 399% => Less relative clauses wanted (maybe 'which' is over used).
Pronoun: 81.0 43.0788530466 188% => OK
Preposition: 323.0 52.1666666667 619% => Less preposition wanted.
Nominalization: 128.0 8.0752688172 1585% => Less nominalization wanted.
Performance on vocabulary words:
No of characters: 14834.0 1977.66487455 750% => Less number of characters wanted.
No of words: 2701.0 407.700716846 662% => Less content wanted.
Chars per words: 5.49203998519 4.8611393121 113% => OK
Fourth root words length: 7.20910159734 4.48103885553 161% => OK
Word Length SD: 3.14010551217 2.67179642975 118% => OK
Unique words: 917.0 212.727598566 431% => Less unique words wanted.
Unique words percentage: 0.339503887449 0.524837075471 65% => More unique words wanted or less content wanted.
syllable_count: 4378.5 618.680645161 708% => syllable counts are too long.
avg_syllables_per_word: 1.6 1.51630824373 106% => OK
A sentence (or a clause, phrase) starts by:
Pronoun: 12.0 9.59856630824 125% => OK
Interrogative: 5.0 0.994623655914 503% => OK
Article: 23.0 3.08781362007 745% => Less articles wanted as sentence beginning.
Subordination: 10.0 3.51792114695 284% => Less adverbial clause wanted.
Conjunction: 4.0 1.86738351254 214% => Less conjunction wanted as sentence beginning.
Preposition: 43.0 4.94265232975 870% => Less preposition wanted as sentence beginnings.
Performance on sentences:
How many sentences: 117.0 20.6003584229 568% => Too many sentences.
Sentence length: 23.0 20.1344086022 114% => OK
Sentence length SD: 201.119032466 48.9658058833 411% => The lengths of sentences changed so frequently.
Chars per sentence: 126.786324786 100.406767564 126% => OK
Words per sentence: 23.0854700855 20.6045352989 112% => OK
Discourse Markers: 1.29914529915 5.45110844103 24% => More transition words/phrases wanted.
Paragraphs: 83.0 4.53405017921 1831% => Less paragraphs wanted.
Language errors: 64.0 5.5376344086 1156% => Less language errors wanted.
Sentences with positive sentiment : 50.0 11.8709677419 421% => Less positive sentences wanted.
Sentences with negative sentiment : 22.0 3.85842293907 570% => Less negative sentences wanted.
Sentences with neutral sentiment: 45.0 4.88709677419 921% => Less facts, knowledge or examples wanted.
What are sentences with positive/Negative/neutral sentiment?
Coherence and Cohesion:
Essay topic to essay body coherence: 0.0 0.236089414692 0% => The similarity between the topic and the content is low.
Sentence topic coherence: 0.0 0.076458572812 0% => Sentence topic similarity is low.
Sentence topic coherence SD: 0.0 0.0737576698707 0% => Sentences are similar to each other.
Paragraph topic coherence: 0.0 0.150856017488 0% => Maybe some paragraphs are off the topic.
Paragraph topic coherence SD: 0.0 0.0645574589148 0% => Paragraphs are similar to each other. Some content may get duplicated or it is not exactly right on the topic.
Essay readability:
automated_readability_index: 16.0 11.7677419355 136% => OK
flesch_reading_ease: 48.13 58.1214874552 83% => OK
smog_index: 8.8 6.10430107527 144% => OK
flesch_kincaid_grade: 12.3 10.1575268817 121% => OK
coleman_liau_index: 14.86 10.9000537634 136% => OK
dale_chall_readability_score: 7.86 8.01818996416 98% => OK
difficult_words: 528.0 86.8835125448 608% => Less difficult words wanted.
linsear_write_formula: 8.5 10.002688172 85% => OK
gunning_fog: 11.2 10.0537634409 111% => OK
text_standard: 9.0 10.247311828 88% => OK
What are above readability scores?
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Try to use less pronouns (like 'It, I, They, We, You...') as the subject of a sentence.
Write the essay in 30 minutes. We are expecting: No. of Words: 350 while No. of Different Words: 200
Maximum five paragraphs wanted.
It is not exactly right on the topic in the view of e-grader. Maybe there is a wrong essay topic.
Rates: 3.33333333333 out of 100
Scores by essay e-grader: 1.0 Out of 30
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Note: the e-grader does NOT examine the meaning of words and ideas. VIP users will receive further evaluations by advanced module of e-grader and human graders.