Higher moment portfolio management with downside risk

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1 AMERICAN JOURNAL OF SOCIAL AND MANAGEMEN SCIENCES ISSN Print: ISSN Online: doi:0.525/ajsms ScienceHuβ Higher moment portfolio management with downside risk Saiful Hafizah Jaaman Weng Hoe Lam and Zaidi Isa Center for Modelling and Data Analysis (DELA) School of Mathematical Sciences Universiti Kebangsaan Malaysia 4600 UKM Bangi Selangor Malaysia. *Phone : (60) Fax : (60) * shj@ukm.my ABSRAC Variance is the common risk measure that has been used in portfolio optimization since the introduction of the mean-variance model. However the mean-variance model penalizes not only the downside deviation but also the upside deviation. he upside deviation is desirable to the investors. he objective of this study is to compare the compositions and performances of optimal portfolios by replacing variance with lower partial moment. he lower partial moment is the downside risk that focuses on the deviation below the specified target rate of return which better matches investor s perception towards risk. Incorporating skewness this study employs the polynomial goal programming method. his method is flexible to incorporate different degree of investor s preference for mean return and skewness. Skewness is important because increasing skewness reduces the probability of getting negative rates of return. he empirical results demonstrate that the optimal portfolios compositions base on lower partial moment are different to the optimal portfolios compositions based on variance. Furthermore at the same level of downside risk optimal portfolios that based on lower partial moment are found to give higher expected return and skewness than the optimal portfolios that are based on variance. his implies that the optimal portfolios that are based on lower partial moment dominate those based on variance at the same level of downside risk. Lower partial moment is a more appropriate risk measure for the investors because it only penalizes the downside deviation. Keywords: variance skewness lower partial moment portfolio optimization polynomial goal programming INRODUCION he mean-variance model is commonly used in portfolio optimization since the seminal work of [2]. Mean is used as the expected return while variance as the risk measure. Variance measures the deviation above and below the mean return. he mean-variance model penalizes not only the downside deviation but also the upside deviation. However upside deviation is desirable to investors for it is the wish of the investors to maximize the upside deviation and minimize the downside deviation. [2 6] introduce the lower partial moment as risk measure. he lower partial moment is the downside risk that focuses on the deviation below the specified target rate of return which better matches investor s perception towards risk. he objective of this study is to compare the compositions of the optimal portfolios and their performances by replacing variance with lower partial moment. Besides this study also incorporates the third moment skewness into portfolio optimization. According to [78] higher moments such as skewness cannot be neglected unless the assets returns are normally distributed or the investors have the quadratic utility function. However many studies [559] demonstrate that the returns of assets are asymmetrically distributed. [6] argues that quadratic function is very unlikely because it implies increasing [0] absolute risk aversion. point out that the expected utility is a function that associated with higher moments of a probability distribution. his paper is organized as follows. he next section discusses the concept and formula of lower partial moment. Section describes the data and methodology. Section 4 discusses the empirical results of this study. he last section provides concluding remarks. [26] Lower Partial Moment: introduce a general definition of downside risk in the form of lower partial moment (LPM). he lower partial moment of order around τ is defined as:

2 Am. J. Soc. Mgmt. Sci. 20 2(2): LPM ( τ ; R ) = ( τ R ) τ df ( R ) = E{(max[ 0 τ R ]) where F(R) is the cumulative distribution function of the investment return R τ is the target return is the degree of the lower partial moment. It specifies risk in terms of probability-weighted functions of deviations below some target return. [8] defines the discrete lower partial moment of order with the expectation is defined as the average of historical portfolio returns R during period t as follows: pt } () LPM ( τ ; R) = (max[0τ Rpt ]) (2) t = where max is a maximization function that chooses the larger of two numbers 0 or τ R Semi-variance is a special case of the lower partial moment when is equal to 2 and τ is equal to E(R) []. [6] shows that = (which suits a risk-neutral investor) separates risk-seeking (0 < < ) from risk averse behavior ( > ) with regard to returns below the target Г. Many studies have used lower partial moment as risk measure in portfolio optimization [789420]. MAERIALS AND MEHODS he data comprises of 5 stocks monthly returns included in the Kuala Lumpur Composite Index (KLCI). hese data are selected for KLCI is used as a measure to indicate the performance of the Malaysia stock market. he period of this study is from July 2002 until December he polynomial goal programming (PGP) method is adopted by incorporating skewness. his method is first [2] introduced by to solve bank balance-sheet management problem. PGP is a multiple objectives approach to find the trade-off between the various conflicting objectives. Next [55] apply PGP method in portfolio management problem with skewness using variance as risk measure. According to [[ the important features of the PGP method are its flexibility of incorporating investor s preference for mean and skewness as well as the simplicity of implementation. -skewness model [] is as follows: Maximize Z = E( X ) Maximize R Z )] = E[ X ( R R pt X VX = X 0. () where X be the transposed of X X = x x... x x denote the percentage of wealth ( 2 n ) i invested in the ith risky asset R is the mean return of the assets R is the rate of return of the assets V is the covariance matrix of rates of return of the assets Z is the portfolio expected return Z is skewness. According to [] the portfolio X can be rescaled and restricted on the unit variance because the portfolio decision depends on the percentage of wealth invested in the assets. here are two objectives in model () they maximize portfolio mean return and skewness. PGP involves two steps procedure since it is highly impossible to solve the multiple-objectives simultaneously. Firstly the objectives Z and Z are solved respectively within the unit variance constraint to obtain the optimal values Z * and Z *. hen the optimal values Z * and Z * are substituted into the following model: Minimize Z + β = ( d ) ( d) E( X R) + d = Z E[ X ( R R )] + d = Z X VX = X 0 d d 0. (4) where d d are non-negative variables which represents the deviation of Z Z from the optimal values Z * = Max{ Z X VX = } and Z * = Max{ Z X VX = } respectively. and β are the non-negative investor-specific parameters representing the investor s degree of preferences on the portfolio mean and skewness respectively. he greater the and β imply the more important of mean and skewness to the investors. [4] proposes the new 22

3 Am. J. Soc. Mgmt. Sci. 20 2(2): model by replacing variance with lower partial moment using the same PGP approach. he model can be formulated as follows: Minimize Z + β = ( d ) ( d) E( X R) + d = Z E[ X ( R R )] + LPM (τ ; R p ) = L u X 0. d = Z able : Monthly optimal portfolio compositions based on variance (%) Parameters = β = 0 d d 0. (5) he optimal portfolios that are based on variance are constructed using model () and (4) with different combinations of and β. For comparison the lower partial moments of the optimal portfolios that are based on variance are measured. hese lower partial moments are substituted into model (5) to obtain the optimal portfolios based on lower partial moment. his study adopts = 2 and a target return of zero which is similar to [47]. Empirical Results able presents the monthly optimal portfolio compositions based on variance in percentages. skewness = β = = β = 2 = 2β = BA DIGI GAMUDA GUOCO IOICORP KULIM LINGUI MISC MMCCORP PEDAG PPB SHANG SHELL SIME SAR UMW As shown in table different combinations of and β result in different compositions of stocks. his result [] is in accordance to which states that the incorporation of skewness into portfolio decision change optimal portfolios compositions. PEDAG (20.96%) dominates other stocks for = β = 0 (mean-variance efficient portfolio). It implies that the investors should invest 20.96% of their fund in PEDAG. For = β = 2 (mean-variance-skewness efficient portfolio) GAMUDA (2.52%) is the largest component while KULIM (.42%) is the smallest component. For = β = and = 2 β = mean- 222

4 Am. J. Soc. Mgmt. Sci. 20 2(2): variance-skewness efficient portfolios the optimal portfolio compositions are very similar which are dominated by GAMUDA. GAMUDA GUOCO LINGUI MMCCORP and SAR are selected in all mean-variance-skewness efficient portfolios but are not included in the mean-variance efficient portfolio. able 2: Monthly optimal portfolio compositions based on lower partial moment (%) able 2 demonstrates the monthly optimal portfolio compositions base on lower partial moment in percentages. Parameters = β = 0 = β = = β = 2 = 2β = BA DIGI GUOCO IOICORP PEDAG PPB SHANG SHELL SIME and β based on lower partial moment. Optimal portfolio compositions based on lower partial moment are different with the optimal portfolio compositions based on variance. his result is also shown by [4] that the portfolio compositions will change by replacing variance with lower partial moment. able 2 indicates that different combinations of and β give different compositions of stocks based on lower partial moment. Shell (22.59%) has the largest component for = β = 0 while GUOCO (60.54%) is the most dominant stock for = β = 2 optimal portfolio. he compositions of the optimal portfolio are almost similar for another two portfolios with = β = and = 2 β =. BA PEDAG and PPB are chosen in all optimal portfolios for all combinations of able and table 4 display the monthly summary statistics of optimal portfolios based on variance and lower partial moment respectively. able : Monthly summary statistics of optimal portfolios based on variance Parameters = β = 0 = β = = β = 2 = 2β = Expected return Skewness Lower partial moment able 4: Monthly summary statistics of optimal portfolios based on lower partial moment Parameters = β = 0 = β = = β = 2 = 2β = Expected return Skewness Lower partial moment As reported in table the = β = 0 optimal portfolio gives the highest expected return (0.028) but the lowest skewness (0.082) among the various portfolios that based on variance. All skewness of the mean-variance-skewness efficient portfolios are higher than the mean-variance efficient portfolio. his implies that the investors will trade the expected return of the portfolio for skewness. However table 4 shows that the = β = optimal portfolio has the highest expected return (0.028) and the highest 22

5 Am. J. Soc. Mgmt. Sci. 20 2(2): skewness (.92) based on lower partial moment. As a whole the expected return and skewness of the optimal portfolios that based on lower partial moment are higher than the optimal portfolios that based on variance for all combinations of investor preferences. his indicates that the optimal portfolios that based on lower partial moment dominate that based on variance at the same level of downside risk. CONCLUSIONS his paper compares the optimal portfolios compositions and portfolios performances by replacing variance with lower partial moment. he PGP method is used by incorporating skewness into portfolio decision. he empirical evidences show that the optimal portfolios compositions based on lower partial moment are different with the optimal portfolios compositions based on variance. Besides that the optimal portfolios that based on lower partial moment give higher expected return and skewness than the optimal portfolios that based on variance at the same level of downside risk. he lower partial moment is a more appropriate risk measure for investors because it only penalizes the downside deviation. he future researches should include the fourth moment kurtosis for the extreme events. ACKNOWLEDGEMENS his study is supported by Universiti Kebangsaan Malaysia s Research Grant Code: UKM- S-06- FGRS REFERENCES Arditti F.D Risk and the Required Return on Equity. Journal of Finance 22 (): 9 6. Bawa V.S Optimal Rules for Ordering Uncertain Prospects. Journal of Financial Economics 2: Cheng P Comparing Downside-Risk and Mean- Variance Analysis Using Bootstrap Simulation. Journal of Real Estate Portfolio Management 7( ): Chen H. and Shia B Multinational Portfolio Construction Using Polynomial Goal Programming and Lower Partial Moments. Journal of the Chinese Statistical Association 45: 0 4. Chunhachinda P. Dandapani K. Hamid S. and Prakash A.J Portfolio Selection and Skewness: Evidence from International Stock Markets. Journal of Banking & Finance 2:4-67. Fishburn P.C Mean-Risk Analysis with Risk Associated with Below-arget Returns he American Economic Review 67: Gilmore C.G. McManus G.M. and ezel A Portfolio Allocations and the Emerging Equity Markets of Central Europe. Journal of Multinational Financial Management 5(): Grootveld H. and Hallerbach W Variance vs Downside Risk: Is there Really hat Much Difference? European Journal of Operational Research 4( 2): Harlow W. V.99. Asset allocation in a Downshle-Risk Framework. Financial Analysts Journal Jean W. 97. he Extension of Portfolio Analysis to hree or More Parameters. Journal of Financial and Quantitative Analysis 6: Lai.Y. 99. Portfolio selection with skewness: A Multiple-objective Approach. Review of Quantitative Finance and Accounting (): Markowitz H Portfolio Selection. Journal of Finance 7(): Markowitz H Portfolio selection: efficient diversification of investments. John Wiley & Sons New York. Nawrocki David N.99. Optimal Algorithms and Lower Partial Moment: Ex Post Results. Applied Economics 2(): Prakash A.J. Chang C. and Pactwa..E Selecting a Portfolio with Skewness: Recent Evidence from US European and Latin American Equity Markets. Journal of Banking and Finance 27: Pratt J Risk Aversion in the Small and in the Large. Econometrica 2(/2): Rubinstein M. 97. he Fundamental heorem of Parameter Preference and Security Valuation. Journal of Financial and Quantitative Analysis 8: Samuelson P he Fundamental Approximation of heorem of Portfolio Analysis in erms of Means Variances and Higher Moments. Review of Economic Studies 7: Simkowitz M. and Beedles W Diversification in a hree Moment World. Journal of Financial and Quantitative Analysis : Sing.F. and Ong S.E Asset allocation in a Downside Risk Framework. Journal of Real Estate Portfolio Management 6(): ayi G. and Leonard P Bank Balance-Sheet Management: An Alternative Multi-Objective Model. Journal of the Operational Research Society 9: