Strathmore UNIVERSITY A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi Stock Exchange. Strathmore UNIVERSITY Mwabaya Fahari Wasi A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi Stock Exchange. 068811 Mwabaya Fahari Wasi 068811 Submitted in partial fulfillment of the requirements for the Degree of Submitted in partial fulfillment of the requirements for the Degree of Bachelor in BusBiancheeslosr i nS Bucsiineesns Sccieen cAe Acctutauriaal rati Satrlat hamto rSe Utnrivaerstihty more University School of Finance and Applied Economics Strathmore University School of FinancNea iraobni, Kde nAyap plied Economics Strathm[Noovemrbeer ,U 201n5]i versity Nairobi, Kenya This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper acknowledgement. [November, 2015] This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper acknowledgement. DECLARATION I declare that this work has not been previously submitted and approved for the award of a degree by this or any other University. To the best of my knowledge and belief, the Research Proposal contains no material previously published or written by another person except where due reference is made in the Research Proposal itself. ©No part of this Research Proposal may be reproduced without the permission of the author and Strathmore University Strathmore UNIVERSITY Mwabaya Fahari Wasi A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi Stock Exchange. Strathmore UNIVERSITY 20111/2015 Mwabaya Fahari Wasi A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi Stock Exchange. 068811 Mwabaya Fahari Wasi 068811 Submitted in partial fulfillment of the requirements for the Degree of Submitted in partial fulfillment of the requirements for the Degree of Bachelor in BusBiancheeslosr i nS Bucsiineesns Sccieen cAe Acctutauriaal rati Satrlat hamto rSe Utnrivaerstihty more University School of Finance and Applied Economics Strathmore University School of FinancNea iraobni, Kde nAyap plied Economics This Research Proposal has been submitted for ex Strathm[Noovemrbeer ,U 2015 a] mination with my approval as the Supervisor. niversity Nairobi, Kenya This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper acknowledgement. [November, 2015] This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper MercyKano acknowledgement. 20/11/2015 School of Finance and Applied Economics Strathmore University ii Abstract. The main objectives of this paper were to measure and compare the portfolio performance of portfolios weighted through the mean variance and semi variance approach in the Kenyan context and to compare portfolio performance in terms of return between portfolios weighted using a Geometric Mean Variance Frontier Approach V s Semi Variance Approach. Equities used were from broad sectors of the NSE: Agricultural, Financial, Commercial and Services and Industrial. Based on the empirical results, there is no significant advantage in using semi variance as a risk Strathmore measure as compared to variance in optimizaUtioNnIV. TEhRiSs IiTs Yb ecause the equity returns in the Nairobi Stock Exchange follow a normal probability distribution. The geometric mean variance returns are also compared to the seAm Coim pvariasorni oafn Mceaen- Voaripantciem andi MStoza eatni-Soemni Variance Optimization on the Nairobi ck Exchang e.m ethods and results show they statistically Strathmore UNIVERSITY approximate each other. Key limitations for the study that NSE duration used may have been a Mwabaya Fahari Wasi A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi Stock Exchange. unique case for a normal distribution. Further 0r6e88s11e arch needs to be done to ascertain whether the Mwabaya Fahari Wasi 068811 geometric mean variance optimSubimzitatetdi ion pna rtaiapl fpulfrilolment of the requirements for the Degree of Submitted in partial fuxlfillimemnt of thae retquieremsents fotr the Degr ee sof emi variance approach. Bachelor in BusBiancheeslosr i nS Bucsiineesns Sccieen cAe Acctutauriaal rati Satrlat hamto rSe Utnrivaerstihty more University School of Finance and Applied Economics Strathmore University School of FinancNea iraobni, Kde nAyap plied Economics Strathm[Noovemrbeer ,U 201n5]i versity Key words: NSE, Semi variance, Mean Vari Nairobi, Kenya This Research Project is availablen for Lcibraery u se oOn the upndertstaindimng thati it zis coapyritghti on material and that no quotation from the Research Project may be published without proper acknowledgement. [November, 2015] This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper acknowledgement. iii Acknowledgments. First of all, I would like to thank God for giving me both the physical and mental strength to undertake this research project. I would also like to appreciate my family for the continuous support they have given me throughout my university education. A special thanks also goes out to my supervisor Mrs. Mercy Kana for guiding me through the research process and offering me critiques and advice that bettered my work. Strathmore UNIVERSITY A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi Stock Exchange. Strathmore UNIVERSITY Mwabaya Fahari Wasi A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi Stock Exchange. 068811 Mwabaya Fahari Wasi 068811 Submitted in partial fulfillment of the requirements for the Degree of Submitted in partial fulfillment of the requirements for the Degree of Bachelor in BusBiancheeslosr i nS Bucsiineesns Sccieen cAe Acctutauriaal rati Satrlat hamto rSe Utnrivaerstihty more University School of Finance and Applied Economics Strathmore University School of FinancNea iraobni, Kde nAyap plied Economics Strathm[Noovemrbeer ,U 201n5]i versity Nairobi, Kenya This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper acknowledgement. [November, 2015] This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper acknowledgement. iv Contents Abstract ............................................................................................................................................ iii Acknowledgments ............................................................................................................................ iv list of Abbreviations ........................................................................................................................ vii list of Figures .................................................................................................................................. viii list of Tables ................................................................................................................................... viii 1 Introduction ............................................................................................................................... 1 1.1 Background ................................................................................................................................... 1 1.2 Motivation for Study ..................................................................................................................... 3 Strathmore 1.3 Problem Statement ....................................................................................................................... 3 UNIVERSITY 1.4 Research Objectives ...................................................................................................................... 4 1.5 Research Questions ...................................................................................................................... 4 1.6 Significance of StAu Cdoym .p.a.r.is.o.n.. o.f. M..e.a..n.-V..a.r.i.a.n.c.e. a.n..d. M..e.a.n..-S..e.m.i. .V.a.r.i.a.n.c.e. O..p.t.i.m.i.z.a.t.io..n. o.n.. t.h.e. .N.a.i.r.o.b.i. ......................................... 4 Stock Exchange. 2 literature Review ........................................S.t.ra.t.hUNIVE. m R. o S.I. re T.Y. ..................................................................... 5 2.1 Introduction .................................................................................................................................. 5 Mwabaya Fahari Wasi A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi Stock Exchange. 2.2 Mean Variance Approach ............................................................................................................. 5 068811 Mwabaya Fahari Wasi 2.3 Downside Risk Measures .........................0.68.8.11. ................................................................................. 7 Submitted in partial fulfillment of the requirements for the Degree of Submitted in partial fulfillment of the requirements for the Degree of 1 Methodology ......................B..a.c.h.e.lo.r. i.n. .B.u.s.Bian.che.esl.osr. i nS. B.ucsii.nee.sns .Scc.iee.n cAe. A.cct.utau.riaa.l ra.ti .Satr.lat .ham.to. rSe. Utn.riv.aer.stiht.y m..o.r.e. U..n.i.v.e.rs.i.ty. ............................................... 11 School of Finance and Applied Economics 2.4 Introduction ............................ StrathmorS.c.h.o..o.l .o.f. F..in.a..n.cN.ea. irao.bn. e University i, K.de. nAy.ap ..p.li.e.d. .E.c.o.n.o.m..i.c.s .............................................................. 11 [November, 2015] 2.5 Research Design ...............................S.t.r.a.t.h.m..o.r.e. U..n.i.v.e.r.si.t.y. ...................................................................... 11 Nairobi, Kenya This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper 2.6 Sampling Design .....................................ac.kn.o.wl.ed.ge.me.nt.. ............................................................................. 11 [November, 2015] 2.7 Data ............................................................................................................................................. 11 2.8 Population and SaTmhisp Rleisneagrc h. P.r.o.j.e.c.t .i.s .a.v.a.i.la.b.l.e. f.o.r. L..ib.r.a.r.y. u.s.e. .o.n. .th.e. .u.n.d.e.r.s.ta.n.d.i.n.g. .th.a.t. .it. i.s. c.o.p.y.r.i.g.h.t. ......................................... 11 material and that no quotation from the Research Project may be published without proper 2.9 Conceptual Model ..............................a.c.k.n.o..w.l.e.d.g.e.m.e.n.t.. ......................................................................... 12 2.10 Selection of the securities to be used in constructing portfolios ............................................... 13 2.11 Mean Variance Optimization ...................................................................................................... 13 2.12 Mean Semi Variance Approach ................................................................................................... 16 2.13 Efficient Frontiers ........................................................................................................................ 21 3 Results and Analysis ................................................................................................................. 22 3.1 Portfolio Selection ....................................................................................................................... 22 3.2 Portfolio Analysis ........................................................................................................................ 22 3.3 Optimization ............................................................................................................................... 23 3.4 Scenario 1 .................................................................................................................................... 23 v 3.4.1 Mean Variance Optimization .............................................................................................. 23 3.4.2 Mean Semi Variance optimization ...................................................................................... 24 3.4.3 Geometric Mean Variance Optimization ............................................................................ 25 3.5 Scenario 2 .................................................................................................................................... 26 3.5.1 Mean Variance Optimization .............................................................................................. 27 3.5.2 Mean Semi variance optimization ...................................................................................... 28 3.5.3 Geometric mean variance yielded the following efficient frontier .................................... 29 3.6 Portfolio Performance Analysis .................................................................................................. 30 3.6.1 Scenario 1 ............................................................................................................................ 30 3.6.2 Scenario 2 .....................................S...t.r..a...t.h...m...o...r..e.. ................................................................ 31 3.7 Test for Normality ................................U...N..I..V..E...R..S...I.T...Y.. ................................................................ 32 4 Discussions and Conclusion .......................................................................................................3 4 4.1 Limitations of the Study .............................................................................................................. 34 A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi 5 Works Cited ............................................S.t.o.c.k. .E.x.c.h.a.n.g.e.. ....................................................................3 5 Strathmore UNIVERSITY 6 Appendix ..................................................................................................................................3 7 Mwabaya Fahari Wasi A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi Stock Exchange. 068811 Mwabaya Fahari Wasi 068811 Submitted in partial fulfillment of the requirements for the Degree of Submitted in partial fulfillment of the requirements for the Degree of Bachelor in BusBiancheeslosr i nS Bucsiineesns Sccieen cAe Acctutauriaal rati Satrlat hamto rSe Utnrivaerstihty more University School of Finance and Applied Economics Strathmore University School of FinancNea iraobni, Kde nAyap plied Economics Strathm[Noovemrbeer ,U 201n5]i versity Nairobi, Kenya This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper acknowledgement. [November, 2015] This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper acknowledgement. vi List of Abbreviations. E- Expected return V- Variance SVm- Below-mean semi variance SVt- Below-target semi variance SV- Semi variance Strathmore Vwretd-Value-weighted-return of the marketU NIVERSITY GMV -Geometric Mean Variance A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi Stock Exchange. SVP-Semi Variance Portfolio Strathmore UNIVERSITY Mwabaya Fahari Wasi MVP- Mean Variance Portfolio. A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi Stock Exchange. 068811 Mwabaya Fahari Wasi NSE-Nairobi Securities Exchan 068811 Subgmeitt ed in partial fulfillment of the requirements for the Degree of Submitted in partial fulfillment of the requirements for the Degree of Bachelor in BusBiancheeslosr i nS Bucsiineesns Sccieen cAe Acctutauriaal rati Satrlat hamto rSe Utnrivaerstihty more University School of Finance and Applied Economics Strathmore University School of FinancNea iraobni, Kde nAyap plied Economics Strathm[Noovemrbeer ,U 201n5]i versity Nairobi, Kenya This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper acknowledgement. [November, 2015] This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper acknowledgement. vii List of Figures Figure 1 ....................................................................................................................................................... 24 Figure 2 ....................................................................................................................................................... 25 Figure 3 ....................................................................................................................................................... 26 Figure 4 ....................................................................................................................................................... 27 Figure 5 ....................................................................................................................................................... 28 Figure 6 ....................................................................................................................................................... 29 Figure 7 ....................................................................................................................................................... 30 Figure 8 ....................................................................................................................................................... 30 Figure 9 ....................................................................................................................................................... 31 Figure 10 ..................................................................................................................................................... 32 Figure 11 ..............................................................S...t..r.a...t.h...m....o..r..e.. ................................................................ 33 Figure 12 ..................................................................................................................................................... 37 UNIVERSITY Figure 13 ..................................................................................................................................................... 37 A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi List of Tables Stock Exchange. Strathmore UNIVERSITY Table 1. .............................................................M..w..a.b.a.y.a. .F.a.h..a.ri. .W..a.s.i ...................................................................... 22 A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi Table 2 ....................................................................S.to.ck. E.xc.ha.n.ge.. ............................................................................. 22 068811 Table 3 ...................................................................M.w.ab.ay.a .Fa.h.ari. W.a.si ............................................................................ 23 068811 Table 4 .........................................S.u.b.m..i.tt.e.d. .in. .p.a.r.t.i.a.l .f.u.lf.i.ll.m..e.n.t. o.f. t.h.e. .r.e.q.u.i.r.e.m.e.n..ts. .f.o.r. t.h.e. .D.e.g.r.e.e. .o.f. ................................................ 24 Submitted in partial fulfillment of the requirements for the Degree of Table 5 ...........................................B.a.c.h.e.l.o.r. i.n. .B.u.s.Bi.anc.he.eslo.sr .i nS .Bu.csi.inee.sns. Sc.ciee.n c.Ae .Acc.tut.aur.iaa.l ra.ti S.at.rlat. ham.to. rSe. U.tnr.ivae.rst.ihty. m..o.r.e. .U.n.i.v.e.r.s.i.ty. ................................................... 25 School of Finance and Applied Economics Table 6 ...................................................................S.tra.th.m.or.e .Un.iv.er.sit.y ............................................................................ 26 School of FinancNea iraobni, Kde nAyap plied Economics Table 7 ..............................................................S.t.r.a.t.h.m[N.o.ove.mr.bee.r ,U .20.1n5.]i .v.e.r.s.it.y. ...................................................................... 28 Table 8 .................................................................N..a.i.robi, Kenya This Research Project is available for L.ib.rar.y .us.e o.n .the. u.nd.er.sta.nd.in.g .th.at .it .is .co.py.rig.ht. ............................................................. 29 material and that no quotation from the Research Project may be published without proper acknowledgement. [November, 2015] This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper acknowledgement. viii 1 Introduction 1.1 Background Portfolio Optimization (K. Liagkouras, 2013) is the process of choosing the assets and their proportions, so that it attains the maximum profitability for the risk undertaken .T here are many theories that attempt to solve the optimization problem ,popular of this is the mean variance optimization theory (Markowitz H. , 1952). The theory attempts to maximize portfolio exSptercatetdh mreoturren for a given amount of portfolio risk, or equivalently minimize risk for a given levUelN IoVfE eRxSpIeTcYte d return by selectively choosing the proportions of various assets. The concept uses diversification in investing so that we obtain a collection of assets with collectively lower risk than any individual asset. (Sullivan & Steven.M., A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi 2003). Stock Exchange. Strathmore UNIVERSITY In the theory (Markowitz, 1952) makes maMnwyab aayas Fsauhamri Wpatsii ons about the investor and the market. The A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi Stock Exchange. assumptions made are not entirely true and eac0h68 8h11a s the mean variance theory to some extent. Mwabaya Fahari Wasi 068811 These assumptions include: Submitted in partial fulfillment of the requirements for the Degree of Submitted in partial fulfillment of the requirements for the Degree of Bachelor in BusBiancheeslosr i nS Bucsiineesns Sccieen cAe Acctutauriaal rati Satrlat hamto rSe Utnrivaerstihty more University School of Finance and Applied Economics • investors are interested in maximiz Strathmore University School of Fininagnc Nera iraeobni, tKdeu nAyap rpnlie df Eocron oam icgs iven variance; Strathm[Noovemrbeer ,U 201n5]i versity • asset returns are normally distributeNda;ir ocbi,o Krernyeal ation between assets are fixed and constant This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper acknowledgement. forever; [November, 2015] • all investors are rational and risk averse; • all investors have tThhise R esseaarmch Pero jaecct ics aevasilsab lteo fo r iLnibrfarmy umse oan tthie ounnde rsatatn dtinhg eth ats iat ism coepy right material and that no quotation from the Research Project may be published without protpiemr e; there are no taxes or acknowledgement. • transaction costs; • Risk of an asset is known in advance and any investor can lend and borrow an unlimited • Amount at the risk free rate. Mean Variance has been a method of choice for many financial modelers in the 21 'st Century (Estrada J. , 2006).Unknown to many Markowitz favored another measure of risk: the Semi- variance of returns. In fact, Markowitz (1959) allocates the entire chapter IX to discuss semi- variance, where he argues that "analyses based on S [semi variance] tend to produce better 1 portfolios than those based on V [variance]" (see Markowitz, 1991, page 194). In the revised edition of his book (Markowitz H. , 1991 ), he goes further and claims that "semi variance is the more plausible measure of risk" (page 374). Later he claims that because "an investor worries about underperformance rather than over performance, semi deviation is a more appropriate measure of investor's risk than variance" (Markowitz, Todd, Xu, and Yamane, 1993, page 307). However Markowitz gives preference to the mean variance optimization theory due to what he defines as the convenience and cost effective use of variance for analysis in comparison to semi variance as it is less costly to use and more familiar to practitioners. The foundations of this theory however, are based on a set of strict assumptions with the result that the majority of models fail to Strathmore capture reality perfectly and exhibit significUanNtI VmEoRdSeIlT rYis k. Downside risk optimization, which models the efficient frontier using semi variance, has exhibited potential for providing better risk metrics. A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi In recent time however with improved technSotoclko Egxcyhange. Strathm aorne d computing power in computers, downside UNIVERSITY risk has been gaining increasing attention, and the many magnitudes that capture downside risk Mwabaya Fahari Wasi A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi are by now well-known and widely used. (EsSttocrk Eaxchadnge.a J. , 2006). Over the last decade, researchers, 068811 Mwabaya Fahari Wasi individual and institution investors have com0e688 11 with various methods to maximize yields and Submitted in partial fulfillment of the requirements for the Degree of Submitted in partial fulfillment of the requirements for the Degree of minimize risk to suit the differeBnacth etlyorp ine Bsus Biaoncheeslofsr i n S iBucnsiineesvns Sccieen cAes Accttutoauriaal rati sSatrla.t ha mtoO rSe Utnrinvaersti htyt mhoere Ubniavecrskityd rop of the financial crisis in the School of Finance and Applied Economics Strathmore University U.S and the Euro Crisis over the last Sc1h0oo l yofe Fainransc Nea i,raotbni,h Kde nAeyap rpelie dh Eacosno mbices en an increased interest by the players Strathm[Noovemrbeer ,U 201n5]i versity in the financial markets to the risk portfoliosN airnobdi, Kmenyaax imize returns over the long tenn. This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper acknowledgement. [November, 2015] Semi Variance optimizatTihoisn R esaeasrc h Paro jedct ois wavanila bles fiodr Leib ramry uesea ons tuher uend erostafnd irngi sthkat it ios cfofpyerirgsht such an alternative as it material and that no quotation from the Research Project may be published without proper incorporates a number of real world constraaciknnotwsle dsgeumcenht. as cardinality constraints, floor and ceiling constraints, non-negativity constraint and budget constraint and analyzes their effects on the efficient frontier formulation. (Markowitz, 201 0), also proposes the use of a mean variance approach using geometric mean return as an alternative to avoid the use of the use of a semi variance approach. While empirical tests has been investigated on foreign markets, studies relating to Kenya in particular are limited. This study aims investigate effectiveness of asset allocation and portfolio optimization using the mean-semi variance framework in the Kenyan context. 2 1.2 Motivation for Study A study by (Mutuku, 2012) noted that there is high volatility in the Kenyan stocks over the past years. This has been partly attributed to the post-election violence of 2007/2008 following the disputed presidential elections; the global financial crisis of2008/2009 and the steep depreciation of the Kenya shilling in 2011and 2015, which affected the financial asset prices significantly. Pension schemes which are significant investors in the financial assets have been greatly affected by this volatility. Strathmore UNIVERSITY 1.3 Problem Statement In Kenya financial performance of pension funds has been critical to their sustainability, enabling A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi them to meet their obligation to members.S toAck ExSt rakchtheanyge . more aspect has been how the fund's assets are UNIVERSITY managed in order to achieve the desired returns. According to (Mutuku, 2012) Pension funds in Mwabaya Fahari Wasi A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi Kenya have been significantly affected by mStoack Erxchkangee. t volatility over the years, with good periods 068811 Mwabaya Fahari Wasi showing significant positive growth and bad periods of negative performance. These swings are 068811 Submitted in partial fulfillment of the requirements for the Degree of Submitted in partial fulfillment of the requirements for the Degree of exacerbated by a significant neBagchaetloirv ine B uscBiancoheeslorsr i nrS Bucesiineelsns aSccieetn cAie Aocctutaunriaal ra ti Sabtrlat haemto rtSe Uwtnrivaersteihty meonre Uthnivee rsNity SE prices and interest rates on School of Finance and Applied Economics government securities which together con Strathmore University School of FsitnaintcuNea itraobeni, Kd e nAy7ap p0li%ed E coonofm ipcse nsion scheme assets (Mutuku, 2012) Strathm[Noovemrbeer ,U 201n5]i versity .Diversification reduces risk without compromising on expected return (Reilley & Brown, Nairobi, Kenya This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper acknowledgement. 20 12)The number of securities to invest in, their combination in a portfolio and the risk involved [November, 2015] are equally important considerations (Reilley & Brown, 2012). This Research Project is available for Library use on the understanding that it is copyright (Treynor & Black, 1973)m at,e riasl hanod twhat enod q uottahtioant f ropm other Rtfesoealrciho P ropjecet mrafyo ber mpublaisnhedc wei thocuat pnro pebr e improved by optimally acknowledgement. weighting a fund manager's stocks selection. (Mwangangi, 2006) , surveyed the application of Markowitz's mean variance portfolio optimization model in overall asset allocation decisions by pension fund managers in Kenya. He used a questionnaire and secondary data from Retirement Benefit Authority on funds allocation for three years from 2003 to 2005. The results of the study showed that 60% of the fund managers applied the Markowitz's mean variance optimization model in their allocation criteria. Markowitz (20 10 ) gives preference to the semi variance measure as compared to the mean variance method in producing optimized portfolios .There has however been no research done on the 3 application of mean semi variance optimization in Kenya. Should Kenyan fund managers switch to the mean semi-variance portfolio optimization approach? This paper aims to investigate the viability of asset allocations and portfolio optimization in a mean-semi variance framework in Kenya" 1.4 Research Objectives. 1. To measure and compare the portfolio performance of portfolios weighted through the mean variance and semi variance approach in the Kenyan Context. Strathmore 2. To compare portfolio performance in tem1s of return between portfolios weighted using a UNIVERSITY Geometric Mean Variance Frontier Approach V s Semi Variance Approach. 1.5 Research Questions A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi Stock Exchange. 1. Does the semi-variance portfolio optSitmrathmUNIVEiRz oare SITYt ion approach create portfolios that yield a higher return than mean variance optimized portfolios in Kenya? Mwabaya Fahari Wasi A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi Stock Exchange. 2. Can the Geometric Mean Variance Frontier Approach estimate the Semi Variance 068811 Mwabaya Fahari Wasi Approach in Kenya? 068811 Submitted in partial fulfillment of the requirements for the Degree of Submitted in partial fulfillment of the requirements for the Degree of Bachelor in BusBiancheeslosr i nS Bucsiineesns Sccieen cAe Acctutauriaal rati Satrlat hamto rSe Utnrivaerstihty more University School of Finance and Applied Economics Strathmore University School of FinancNea iraobni, Kde nAyap plied Economics 1.6 Significance of Study. Strathm [Noovemrbeer ,U 201n5]i versity Nairobi, Kenya This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper acknowledgement. This study contributes to the empirical evid[eNnovcemeb eor, 2n01 5t]h e determination of the optimal portfolio for investors within the Nairobi Securities exchange. This study has important implications for investors in making portfTohlisi Roe sesarechc Purojreictt iis available for Library use on the understanding that it imaterial and that no quoteatsio n sfreoml ethec Rteisoeanrch Paronjedct mfauy bne dpusbl ishaeld lwoithc s caoptyiright out propenr decisions in Kenya. This acknowledgement. study can inform future review of policy and regulatory guidelines for regulated institutional investors in Kenya. The study is of interest to researchers and financial analysts of the Kenyan economy. The study is of interest to portfolio and investment managers of insurance companies and retirement benefits schemes. 4 2 Literature Review 2.1 Introduction Risk has been with defined differently over the decades. Frank H. Knight (1921) argues that there is a difference between uncertainty and risk. According to Knight, risk is a combination of the likelihood of an occurrence of a hazardous event, meaning an event that could cause harm in terms of losses or undesirable outcome, and its magnitude. (Knight, 1921) , also proposes that it is possible to calculate the probability a risk, which makes it measureable. Uncetiainty on the other hand is characterized as the existence of more than one Strathmore possibility in the future, but unlike risk, uncertainty is not measureable. UNIVERSITY (Hubbard, 2007) further affirms Knights position by defining uncertainty as The lack of complete certainty, that is, the existence of more than one possibility i.e. the true outcome value is not A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi known. He also defines risk as a state of uncSteocrkt Eaxichange. Strathnmtoyre where some of the possibilities involve loss, UNIVERSITY catastrophe or other undesirable outcome. Mwabaya Fahari Wasi A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi Stock Exchange. This section will focus on risk and various methods of calculating risk that have been there over 068811 Mwabaya Fahari Wasi time and the empirical applications of the metho068d811 s in solving the optimization problem. I will then Submitted in partial fulfillment of the requirements for the Degree of Submitted in partial fulfillment of the requirements for the Degree of focus on the mean variance vs sBeacmheloir ivn aBursiBianacheeslnosr i nSc Bucseiineesn s Scacieen pcAe Accptutaurriaal orati Satarlat hacmto rhSe Utn rivaaerstihtsy m porre oUnpivoerssitey d by (Markowitz, 2010) School of Finance and Applied Economics Strathmore University School of FinancNea iraobni, Kde nAyap plied Economics Strathm[Noovemrbeer ,U 201n5]i versity Nairobi, Kenya This Research Project is available for Library use on the understanding that it is copyright 2.2 Mean Variance Approach material and that no quotation from the Research Project may be published without proper acknowledgement. [November, 2015] Modem Portfolio Theory, a finance theory pioneered by (Markowitz, 1952) is a tool that attempts maximize portfolio expectTehids R erseeatrcuh rPrnoj efcto isr a vaail agblei vfore Lnibr aarym useo oun tnhe tu noderfst pandoinrgt tfhaot ilt iiso co pryirsigkht by carefully allocating the material and that no quotation from the Research Project may be published without proper acknowledgement. proportions of various assets. Markowitz used mean returns, variances and covariance's to derive an efficient frontier where every portfolio on the frontier maximizes the expected return for a given variance or minimizes the variance for a given expected return. This called the EV (Expected Return and Variance) criterion. In selecting a portfolio of assets to invest in, an investor needs to make a tradeoff between risk and return. The investor's sensitivity to changing wealth and risk is known as a utility function. The utility function have been subjective so far and scholars have differed on the methods used to model utility. 5 (Boasson, Boasson, & Zhou, 2011), notes that, Markowitz's mean-variance approach, has two drawbacks. It assumes that the distribution of investment returns is jointly elliptically distributed. If the underlying return data is not normally distributed, the variance is likely to give misleading results. Studies have demonstrated that investment returns are not normally distributed (Fama, 1968). In the real world, security returns tend to be asymmetrically distributed, e.g. log normally distributed. The skewed distribution of investment returns makes the variance as an inefficient risk measure, because variance treats the favorable upside dispersion of investment return over the mean value of return as a part of risk and penalizes it as much as the unfavorable downside deviation from the mean returns. Strathmore If the retums are not normally distributed,U NinIvVeEsRtoSrsIT Yus ing variance or standard deviation to measure risk are more often than not likely to reach wrong asset allocation decisions. Skewness and kmiosis in real rates of return data with non-normal distributions can cause variance or A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi standard deviation to underestimate risk. Stock Exchange. STtrahthem omre ean-variance approach also overlooks the UNIVERSITY investor's risk aversion. Because variance is only a measure of dispersion of returns around a Mwabaya Fahari Wasi mean, it cannot be customized for iA nComdparisionv of Miedan-Vuarianace alnd Meiann-Semi Variance Optimization on the Nairobi Stock Exchavnge. estors' aversion ( (Boasson, Boasson, & Zhou, 068811 2011). Mwabaya Fahari Wasi 068811 Submitted in partial fulfillment of the requirements for the Degree of Submitted in partial fulfillment of the requirements for the Degree of Moreover, the real-world applBicacahetloiro in B uosBiancfheeslo srM i nS Bucsiinaeesns rSccieekn cAe Aocctutwauriaal ratii Sattrlat zhamto' rSe Utnmrivaerstihtye maorne -Uvnivaersiitay nce optimal pmifolio allocation School of Finance and Applied Economics Strathmore University has many pitfalls. The optimal portfoSclhiool ocfo FinasnctNear iraoubni, Kdce nAytap epldied iEnco naom imcs ean-variance framework may not lead Strathm[Noovemrbeer ,U 201n5]i versity to an optimal portfolio that optimizes exNpaierocbit, eKedny a retums while minimizing risk as required. This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper acknowledgement. (Michaud, 1989) , indicates that these re[Naolv emwbeor, 2r0l1d5] portfolio optimizers are essentially "error maximizers" because "optimizers" tend to treat the inputs as if they were exact quantities, while in reality they can only beThma is Rseseterialt ai amrch Prtoeject is availand that no dqu owtatioint h ble eforr Lriobrrar.y uTse hone th eo unpdetristmandainlg thfrom the Research Project may be publish ep ato it rist cod withofuo pyright t plroipoers constructed based on this acknowledgement. framework tend to suggest large bets on stocks with large estimation error in expected returns, often leading to poor-out of- sample performance. (Markowitz, 1970) , realized the limitations of variance. He showed that both the downside risk measurement and the variance measurement can produce the same correct results when retum distributions are normal. However, in situations where retum distributions are not nonnal, the downside risk measurement is more likely to produce a better solution. Because of this limitation in using variance as a risk measurement, various downside-risk measurements have been proposed and developed. One of the downside risk measurements is semi variance. By definition, a downside-risk measurement 6 measures only the returns below a certain threshold. This threshold captures the risk perspectives from investors to investors. Unlike standard deviation, downside risk accommodates different views of risk. Markowitz however affirms that variance as a measure of risk has an edge over other downside risk measures "with respect to cost, convenience, and familiarity. The difference in cost, (Markowitz, 1959) is given by the fact that efficient sets based on down side risk took, back then, two to four times as much computing time as those based on variance. The difference in convenience, in turn, is given by the fact that efficient sets based on variance require as inputs only means, variances, and covariance, where as those based on downside risk require the entire joint Strathmore distribution of returns (Estrada J. , 2006).WUiNthI VthEeR SimITpYro vement of technology and computing power of computers there has been increased interest in downside risk measures (Estrada, 2007) A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi Stock Exchange. Strathmore UNIVERSITY 2.3 Downside Risk Measures Mwabaya Fahari Wasi A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi Stock Exchange. 068811 Mwabaya Fahari Wasi The (Markowitz, 1959) defense of mean-variance optimization criterion was not unconditional. 068811 Submitted in partial fulfillment of the requirements for the Degree of Submitted in partial fulfillment of the requirements for the Degree of It asserts the existence of situatiBoacnheslo ri in B wusBianhcheesliosr ci nS Bhucsiinee snsr Sccieen cAte Aucctutaurriaanl rati Ssatrla t hamoto rSen Utnri vaetrstihty meo rpe Uonrivtefrsoityl io-as-whole are mostly confined School of Finance and Applied Economics to a range in which the investor's uStchioloilt oyf F inf Str auncn athmcoret University Nea iraobni, Kdie nAoyap npli edc Eaconno mbices approximated sufficiently well by a Strathm[Noovemrbeer ,U 201n5]i versity quadratic, and occasional departures from this range are "not too serious." As (Levy & Markowitz, Nairobi, Kenya This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper 1979) and others confirm, for many utility afcknuowlendgemcent.t ions and for distributions of returns, such as [November, 2015] historical returns of investment companies, functions of mean-variance supply robust approximations to expecteThdis Ruesteialricht yPr.o ject is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper acknowledgement. (Markowitz, 201 0) , assess the other alternatives available in the situations where the mean- variance approach is not applicable. Major alternatives include the following :( 1) . Use other measures of risk or return in a risk-return analysis (2.)Determine the investor's utility function explicitly and maximize its expected value (3.) Do not optimize; instead, use constraints and guidelines. We look at the first alternative that opts for the use of other risk measures in this case the downside risk measures. Roy (1952) was the first to look at other measures of risk. He proposed the safety first ratio. Roy states that an investor will prefer safety of principal first and will set some minimum 7 acceptable return that will conserve the principal. Roy called it the minimum acceptable return the disaster level and the resulting technique is the Roy safety first technique. Roy stated that the investor would prefer the investment with the smallest probability of going below the disaster level or target return. By maximizing a reward to variability ratio, (r- d)/s, the investor will choose the portfolio with the lowest probability of going below the disaster level, (d), given an expected mean return, (r), and a standard deviation (s). (Markowitz, 2010), advances Roy's safety principal first by recognizing Roy's important concept of downside risk measure. Markowitz observes that investors are interested in minimizing downside risk for two reasons: (1) only downside risk or safety first is relevant to an investor and Strathmore (2) security distributions may not be normaUllNyI VdEisRtrSibIuTtYe d. Therefore a downside risk measure would help investors make proper decisions when faced with non-normal security return distributions. A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi Stock Exchange. (Markowitz, 1959), provided two suggestionSstr atfhomorr e measuring downside risk: a semi vanance UNIVERSITY computed from the mean return or below-mean semi variance (SVm) and a semi variance Mwabaya Fahari Wasi A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi Stock Exchange. computed from a target return or below-target semi variance (SVt). The two measures compute a 068811 Mwabaya Fahari Wasi variance using only the returns below the mean 06r881e1 turn (SVm) or below a target return (SVt). Since Submitted in partial fulfillment of the requirements for the Degree of Submitted in partial fulfillment of the requirements for the Degree of only a subset of the return distrBiabchuelotri ion nBu sBiiancshee slosr ui nS Bucssiineesns Sdccieen ,cAe AccMtutauriaal raati Satrrlat hkamto roSe Utnwrivaerstihtiy mtzor ec Uanilvleresidty these measures partial or semi- School of Finance and Applied Economics Strathmore University variances. Markowitz however notes Stchhoaotl otfh Fiena ncNeao iraobnmi, Kde nAyap pulietda Etcioononmiscs of semi variances is rather tedious and Strathm[Noovemrbeer ,U 201n5]i versity would not add any major benefits at the timeN.a irobi, Kenya This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper acknowledgement. Fishburn (1877) and Harlow and Rao(198[9No)v emdbeer,v 20e15l]o ped the (a-t) model where a denotes the investors risk aversion while t represents the target return of investment or disaster level or This Research Project is available for Library use on the understanding that it is copyright proposed by Roy(1952).Fmiastehria l banud trhant n(ol q9uo9ta7tio)n afrolms tohe Rpesreoarcph Project may be published without proper acknowledgemoenst.e d the Mean-Lower Partial Moment model of which Harlow(1991) applied to portfolio selection. Harlow (1991) defined Lower Partial Model as: LPMn = LkpPp(T- Rp)n, where Pp is the probability that the return,Rp occurs. The type of moment, unspecified in the LPM equation captures an investors preference. For n = 0, the risk measure becomes a Oth-order moment (LPMO) which measures the probability of falling below the target rate. 8 However, for n = 1, LPM1 becomes the expected deviation of returns below the target. For n = 2, LPM2 is analogous to variance, in that it is a probability weighting of squared deviations. Thus, LPM2 can be referred to as a target semi variance. Harlow (1991) further explained that many popular notions of risk are special cases of the generalized LPM,n, measure. For example, with n = 0 and a target rate = 0%, LPMO is simply the probability of a loss. For n = 2 and a target rate= mean return, LPM2 becomes the traditional semi variance. Overall, LPM1 (target shortfall) and LPM2 (target semi variance) provide an intuitive set of risk definitions Strathmore that are more useful than traditional approachUeNs I(VHEaRrlSoIwT,Y 1 991). However, Harlow and Rao (1989) failed to consider the correlation of asset returns which is an important consideration for diversification of risk. This renders the low partial model only A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi effective for those assets whose returns areS tpocek Erxfchange. Stratehmcotrley or highly correlated. Lower Partial Model UNIVERSITY was however regarded as having more complexity in computation than does the variance Mwabaya Fahari Wasi A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi measurement. Stock Exchange. 068811 Mwabaya Fahari Wasi Foo& Eng (2000) made adjustments to Harlow06881 1 & Rao (1989) by incorporating the model with Submitted in partial fulfillment of the requirements for the Degree of Submitted in partial fulfillment of the requirements for the Degree of Bachelor in BusBiancheeslosr i nS Bucsiineesns Sccieen cAe Acctutauriaal rati Satrlat hamto rSe Utnriversity downside covariance of correlated asset returns. Hoganat h&mo reW Unaiverrrsietyn (1974) introduced the concept School of Finance and Applied Economics Strathmore University of co-lower-partial-variance, whichS chmool eofa FsinuanrcNeea iraosbni, Kde nAyapr pilsiekd Eyco noamiscss et and market portfolio. Bawa & Strathm[Noovemrbeer ,U 201n5]i versity Lindenberg (1977) further developeThid Nairs Res earcht Prhojecti is ava ilablec for oob-i,Library ul Kenya se on thwe understanrdin-g tphat ita is corpytrigiht al variance measure to an n-degree material and that no quotation from the Research Project may be published without proper acknowledgement. framework called generalized asymmetric[ Nocvoem-bLer,P 20M15] . However their methods were still more complex and computationally burdened and the variance measure was still regarded as the best This Research Project is available for Library use on the understanding that it is copyright method due to its simplicimtyate.r ial and that no quotation from the Research Project may be published without proper acknowledgement. (Estrada, 2007) , proposes the use of a heuristic approach in semi variance optimization. His method however does takes more time to calculate even using advance laptops to calculate. (Markowitz, 201 0) proposes a new a simple formula to calculating the optimal pmifolio, He proposes that to ease the calculation of downside risk using semi variance and finding the optimized portfolio we should combine the semi deviation as a measure of risk with the geometric mean as the measure of return, R since , 9 log(l + GM) = Elog(l + R) (Markowitz, 201 0) , reckons that it is much easier to compute a mean-variance efficient frontier than a OM-Semi variance efficient frontier. "So long as E log(l + R) can be estimated sufficiently well from E and V, the economical way to generate a GM-V frontier is to generate a mean-variance efficient frontier and then plot GM on the return axis. However, when distributions are too spread out for mean-variance approximations to be adequate, the extra expense is justified for deriving efficient portfolios on the GM-Sb frontier. This expense is not only computational, but also includes additional estimation requirements Sbtercaatuhsme Eo rleo g (1 + R) is not a function of first and second moments only." UNIVERSITY A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi Stock Exchange. Strathmore UNIVERSITY Mwabaya Fahari Wasi A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi Stock Exchange. 068811 Mwabaya Fahari Wasi 068811 Submitted in partial fulfillment of the requirements for the Degree of Submitted in partial fulfillment of the requirements for the Degree of Bachelor in BusBiancheeslosr i nS Bucsiineesns Sccieen cAe Acctutauriaal rati Satrlat hamto rSe Utnrivaerstihty more University School of Finance and Applied Economics Strathmore University School of FinancNea iraobni, Kde nAyap plied Economics Strathm[Noovemrbeer ,U 201n5]i versity Nairobi, Kenya This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper acknowledgement. [November, 2015] This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper acknowledgement. 10 3 Methodology 3.1 Introduction To achieve the objective of the study the research adopted the methodology proposed by (Markowitz, 201 0) and (Boasson, Boasson, & Zhou, 2011) in the calculation of the mean semi variance optimization. The research also employed the equations proposed by Markowitz 1952 in the calculation of the mean variance optimization frontier. 3.2 Research Design Strathmore This study is exploratory in nature as it seeksU toN IaVsEseRsSs ItThYe performance of portfolios created using the mean variance approach in comparison to the ones created using the mean semi variance approach. The exploratory design was also selected as previous researchers' ( (Estrada, 2007), A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi Stock Exchange. (Boasson, Boasson, & Zhou, 2011 ), similar topStiracthsm oarles o used the same approach. UNIVERSITY Mwabaya Fahari Wasi A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi Stock Exchange. 3.3 Sampling Design 068811 Mwabaya Fahari Wasi 068811 Submitted in partial fulfillment of the requirements for the Degree of Submitted in partial fulfillment of the requirements for the Degree of This study analyses portfolioB archeetlour irnn Bsus Bianchefesloosr i nS rBuc siineesnsv Sccieean cAer Accitutaouriaal urati Satsrlat ha mto raSe Utnsrivaesrstihtye mto rea Ulnlivoercsiatyt ions in the Nairobi Securities School of Finance and Applied Economics Strathmore University Exchange from between January 200Sc9ho otlo o f FNinaoncvNea ireaobni,m Kde nAyapb pelierd E2co0no1m4ics. This duration was chosen as it avoids Strathm[Noovemrbeer ,U 201n5]i versity the effect ofthe post-election violence that oNcaicroubi,r Kreenyda in 2007-2008. This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper acknowledgement. [November, 2015] 3.4 Data This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper acknowledgement. For the purposes of constructing efficient frontiers, we used data obtained from the Nairobi Stock Exchange on the daily closing prices of stocks over the period. This data was sourced from the Nairobi Stock Exchange. 3.5 Population and Sampling The population for the study comprised firms listed in the Nairobi Securities Exchange. The study used a census of all securities in the population which had complete information on prices for all 11 the months over the study period January 2009 to December 2014. Any firm that was delisted or suspended over the period was not considered. Portfolio sizes will the between 8 to 20 stocks as recommended by (Mbithi, 2013) in his research of the optimal sizes of portfolios he found that portfolio risk reduced by 40% with 8-securities portfolio, 46% with 20-securities portfolio and 47% with 30-ssecurities portfolio. The risk reduction achieved with 8-securities portfolio represents 85% of the risk reduction achievable with a 30-security portfolio. 3.6 Conceptual Model. Strathmore UNIVERSITY Calculate the Expected Return for Stocks and Standard Deviations and Correlations between Stocks To minimize the variance and semi variance of the portfolio for a target return, we need to first A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi calculate the expected return and varianceSt ocfko Exchange. Strra thmthore risky pmifolio from expected returns and UNIVERSITY variance of the securities comprising the portfolio. Mwabaya Fahari Wasi A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi Stock Exchange. Calculating Expected return of the risky portfolio 068811 Mwabaya Fahari Wasi 068811 Arithmetic Mean Method Submitted in partial fulfillment of the requirements for the Degree of Submitted in partial fulfillment of the requirements for the Degree of Bachelor in BusBiancheeslosr i nS Bucsiineesns Sccieen cAe Acctutauriaal rati Satrlat hamto rSe Utnrivaerstihty more University School of Finance and Applied Economics The expected return of the risky portfolio is Strsathimomre Univpersitly y the weighted average of expected returns of School of FinancNea iraobni, Kde nAyap plied Economics [November, 2015] the securities within the portfolio. Strathmore University Nairobi, Kenya This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper n acknowledgement. L [November, 2015] E[rv] = wiE[rd ( equation 1) i=1 This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper Or using matrices acknowledgement. (equation 2) 12 Calculation of Geometric Mean: In an investor's wealth perspective, the growth of the asset over the entire period [0, T] should be expressed as a geometric mean. The use of geometric mean has far better properties in terms of the interpretation of asset returns as compared to the arithmetic mean. In analyzing wealth over a longer period of time, the geometric mean conveys what the average financial rate of return would have been over the whole duration of the investment period. Then for n single periods returns, a sequence of asset returns defined as {Rt}nt=l based on the sequence of asset prices {S t }nt=O, the geometric mean return or the so-called "time-weighted rate of return," r(O,n), can be expressed as followsS: trathmore UNIVERSITY 1 A Comparison of Mean-Va(riaQnce an d Mea+n-S emi Varrianc-e Optimization on the Nairobi rco.n) = S(t1oc k Exchange. StrathRmo,r)e 1 (3) UNIVERSITY Mwabaya Fahari Wasi A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi Stock Exchange. 068811 Mwabaya Fahari Wasi 068811 Submitted in partial fulfillment of the requirements for the Degree of Submitted in partial fulfillment of the requirements for the Degree of 3. 7 Selection of the securitieBsac hteolor ibn Beu sBuiancheseslosre i nS Bducsiinee snsi Sccnieen cAe Accctutauroiaal ranti Satrlast hamtto rrSe Utunrivaercstihty mtionre gUn ipveorsirtyt folios. School of Finance and Applied Economics Strathmore University School of FinancNea iraobni, Kde nAyap plied Economics For the purposes of the research stocks wSitlralt hbm[Noovemrbe er ,U c201n5h]i voerssitey n on the basis of their average daily mean Nairobi, Kenya returns. The top 13 were selected toT his Research Projecmatecrial rande that no qtt ise ava ilabnle for Laibrarsy usse one the understanding that it is copyright uotation from the Research Project m ay bpe puoblishred twitfhout prolpeir os while giving considerations of the acknowledgement. conelation between the assets. [November, 2015] These assets were then to Tbhies R eusesarechd P rotjoec t cis raveaialatble foer fLifbiracryi uesen otn tfher ounndertsitaendri nfg othart it his ceo,py rnig,h t asset p01ifolio for both the material and that no quotation from the Research Project may be published without proper mean variance and semi variance optimizatiacoknnow mledgeemthenot. d. 3.8 Mean Variance Optimization Calculating Variance of the Risky Portfolio Defining Wi as the portfolio weight for security i and E[ri] as the expected return for security i . The variance of the risky portfolio can be derived as follows: 13 We can make use of a variance-covariance matrix to simplify the calculations. Thus Z is a symmetric matrix. The variance of the portfolio using matrices can be written as follows: Strathmore UNIVERSITY Constraints A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi Stock Exchange. Strathmore To minimize the variance of the portfolio we nUeNIeVEdR StITwY o constraints: I. The expected retum for the Mwabaya FaharA oComparistoni ofm Mean-iVazrianece adnd M eanp-Semoi Vrariati ncfWoasli e Optimiziation on thse Nahiroboi uld be equal to the one obtained by Stock Exchange. getting the sum of the weighted retums0 6i8n811 Mwabaya Fahar i Wtahsi e portfolio 068811 In S ubmitted in partial fulfillment of theN re quirements for the Degree of Submitted in partial fulfillment of the requirements for the Degree of Bachelor in BusBiancheeslosr i nS Bucsiineesns Scc~ieen ce Actuarial at Strathmore University wiE[ri] = r* ActIuarial awt SitrEath[mroirJe U n-ivrers*ity = 0 School of Finance and Applied Economics Strathmore University i=l School of FinancNea iraobni, Kdie nA=yap pll ied Economics Strathm[Noovemrbeer ,U 201n5]i versity Or written using matrices: Nairobi, Kenya This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper acknowledgement. [wNove.mRber, -20r15*] II. The portfolio weigThhis tRses esarhcho Pruojlecdt i s savuaimlabl eu fopr L itborar yo usne eon t(het hunede rsftuandliln g ithnatv it eiss cotpmyrigehnt t constraint).So as to meet material and that no quotation from the Research Project may be published without proper the assumption that the investor wanatcksn otwoled gienmevnte. st all his money in the risky portfolio. N N I wi = 1 ~ I wi - 1 = 0 i=l i=l Constraint optimization Using the Lagrangian method of optimization we can minimize the variance of the portfolio with respect to the two constraints and solve using the tools for the linear algebra. 14 This equation can be simplified using matrix notation as follows: Taking the first order condition of this lagrangian equation we find the partial derivative of a scalar with respect to a vector as follows: Strathmore UNIVERSITY a£ aw A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi Stock Exchange. Strathmore UNIVERSITY Mwabaya Fahari Wasi A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi Stock Exchange. 068811 Mwabaya Fahari Wasi 068811 Submitted in partial fulfillment of the requirements for the Degree of Submitted in partial fulfillment of the requirements for the Degree of Bachelor in BusBiancheeslosr i nS Bucsiineesns Sccieen cAe Acctutauriaal rati Satrlat hamto rSe Utnrivaerstihty more University School of Finance and Applied Economics Strathmore University School of FinancNea iraobni, Kde nAyap plied Economics Strathm[Noovemrbeer ,U 201n5]i versity ... + Nairobi, Kenya This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper acknowledgement. [November, 2015] + Wn -1) This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper L = + ... + + ... + acknowledg+em ent. wfa11 Wnw1a1n WnW1an1 ... + wJann + lt1 (w1£[r1 ] + ... + WnE[rn]) + Az(W1 + ... + Wn- 1) L = Ln Ln wiwjaij + A1(w1E[r1] + ... + WnE[rn]) + lt2 (w1 + ... + Wn -1) i=1 j=l 15 The partial derivative with respect to the other variables,it11 it2 , will be: aL aitl = W.R- r* aL aitz = W.l-1 Equating the partial derivatives for to zero wSillt reantahbmleo arse t o obtain the values of the it11 it2 i.e UNIVERSITY aL ait = W.R -r* = 0 A Comparison of Mean-Varian1c e and Mean-Semi Variance Optimization on the Nairobi Stock Exchange. Strathmore UNIVERSITY Mwabaya Fahari Wasi A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi Stock Exchange. The optimization process using the Lagrangi068Mwabaya Fn811 aha ri Woasi ptimization method is however tedious, time 068811 consuming and prone to mistSaubkmeittsed iinf p adrtioaln fuelfi llmenat of uthea rlelqyuir.e mTenhts ifosr thce aDneg rehe of Submitted in partial fulfillment of the requirements for the Degree of wever be simplified and done Bachelor in BusBiancheeslosr i nS Bucsiineesns Sccieen cAe Acctutauriaal rati Satrlat hamto rSe Utnrivaerstihty more University much faster using the solver add in on MicScrhoool ofs Finoancef antd A ppeliedx Ecocnomeics l or in Mat lab. It involves inputting the data Strathmore University School of FinancNea iraobni, Kde nAyap plied Economics required in excel. You then use the data toS tfraitnhm[dNoove mrbteer h,U 201ne5]i v emrsitey an returns of the securities and the variance Nairobi, Kenya covariance matrix. Setting up the weThiis Rgeseahrch Ptrosject, is material and that no quot avaatioyilable forn from thu Libr aryc use aon thne un dertstahndinge thant it is cofpyriight e Research Project may be published without propner d the expected return and variance of acknowledgement. [November, 2015] the portfolio. Finally using the Solver add-in you can minimize the variance of the expected return subject to the constraints that the weights of securities in the portfolio should be equal to one. The This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper other constraint, that the expected return sahckonouwlledge mbenet. equal to the weighted sum of the expected returns of the securities, is already satisfied by the method used in obtaining the expected returns before optimization in excel. 3.9 Mean Semi Variance Approach The underlying principle for semi variance model is the same as the variance model, in that investors are willing to bring downside risk as low as possible while keeping the rate of return above a certain level. The definition of semi variance below the mean value can be expressed by the following formula: 16 n 1 Semi - Variance = - n I ri) = School of FinancNea iraobni, Kde nya 1JtWj-ALpplied E c 2 1JonWomjic)s Pet)] (9) Strathm[Noovemrbeer ,U 201n5]i versity j=l j=l Nairobi, Kenya This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper acknowledgement. Where Pt denotes the probability. If we ass[Nigovnem btehr, e20 1s5]a me probability for all observations then we have Pt = 1/T This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper To further derive the simplified formula fora cksneowmledige mvenat.r iance model, we have to make an important assumption, which is the validity of Sharpe ( 1964) beta regression equation: It states that the random variable of the jth asset's return is related to the market portfolio return, return is related to the market portfolio return, wherea1 and /3j are constant, and Ej is a random error with zero covariance for (Ej , £;;) zero covariance for (Ej, fM). The market portfolio is the weighted sum of asset returns. In addition, we can obtain {31 by: 17 Where cov(1)", fm) is the covariance between the return ofthejth asset and the return of the market portfolio, O"m is the variance of the market return. Based on equation (7), we can get: Where 7Jand rm are expected return of the jth asset and market portfolio respectively.ej has zero mean value. Strathmore UNIVERSITY Considering all the assets in the portfolio and adding them up based on equation (13), A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi n I Stock Exchange. Strathmore CfJ-7J)Wj = e+ UNIVERSITY Crm- rm) I {JjWj (13) j Mwabaya Fahari Wasi A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi Stock Exchange. 068811 Mwabaya Fahari Wasi 068811 Submitted in partial fulfillment of the requirements for the Degree of Submitted in partial fulfillment of the requirements for the Degree of Based on (13), we can rewrite tBhacehe leorq inu BaustBianicheoeslosr i nS Buc siioneesns Sfccieen S cAe AcctVutauriaal( rati >Satrlat ha)mto rSea Utnrsivaer:stiht y more University School of Finance and Applied Economics Strathmore University School of FinancNea iraobni, Kde nAyap plied Economics Strathm[Noovemrbeer ,U 201n5]i versity Nairobi, Kenya This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper acknowledgement. [November, 2015] When the level of diversification goes to infinity, we can prove that: This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper acknowledgement. The definition of SV (>) and SV ( <) implies that: t n n svc <) + SV(>) = v = I[I 1JtWj-I 1J wjF Pet) (16) t=l j=l j=l Hence, we can get the expression ofSV (<)by subtracting SV (>)from V: 18 lim SV(<) = V- lim SV(>) (17) L->oo n->oo Note that the definition of the level of diversification is: a portfolio is considered to reach a level L of diversification if: ~ax Wj = 1/L and n/L = q Q > q > 1 and Lf=l Wj = 1, Wj > 0 for all J-l,.n j. Q is a constant that defines the bound of q. TShteran tthem hoigreh er the level of diversification is reached, UNIVERSITY i.e., the higher the value of L, the lower is the greatest weight. From equation (17), we can see the whole calculation is much simplified, and thus, all the data or parameter can thus be obAt aCoimnpeardiso. n of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi Stock Exchange. Strathmore UNIVERSITY Hogan & Warren (1972) presents the essential mathematical properties of mean-semi variance Mwabaya Fahari Wasi models, where they prove the convAe Coxmpairistony of M eaan-Vnariandce a nd Mdeani-Sfemfi Vaerianrcee Optnimiztatiion aon tbhe Nairobi Stock Exchange. ility of this model. Their contributions 068811 make the theoretical and computational viabiMlwiabtayya Fah aroi Wafsi mean-semi variance model guaranteed. 068811 Submitted in partial fulfillment of the requirements for the Degree of Submitted in partial fulfillment of the requirements for the Degree of Bachelor in BusBiancheeslosr i nS Bucsiineesns Sccieen cAe Acctutauriaal rati Satrlat hamto rSe Utnrivaerstihty more University School of Finance and Applied Economics Strathmore University School of FinancNea iraobni, Kenya 1. Calculation of Semi Variance Matrix d Applied Economics Strathm[Noovemrbeer ,U 201n5]i versity Nairobi, Kenya This Research Project is available for Library use on the understanding that it is copyright The first step is to derive the beta coematferiafl anid cthat ino equotnationt frsom theo Resefarch vProjeact mray ibe opubliusheds wit houst preoper acknowledgement. curities. We calculate the variance and [November, 2015] covariance matrix of different securities. The second step is to comTphius Rteese atrhch ePr omjecta isr akvaeilatb lem fore Laibnrar ya unse don tvhea unrdiearsntancdieng vthaat ilt uis ecospy raigsht follows: material and that no quotation from the Research Project may be published without proper acknowledgement. N Em = ! L vwretdt t=l N O"m = N ~ 1 L(vwretdt- Em) 2 t=l Where yrwretd is the value-weighted-return of the market. The covariance of different assets with then market return is calculated as follows: 19 n Cov(Xj,XM) =~I (Rjt- Rj)(vwretdt- Em) t=1 Then Beta is calculated according to equation (18): The third step is to calculate the market portfSoltiroa'sth smemoir vea riance above the mean return: UNIVERSITY LtT= l max(R- m,t- Em, 0) 2 A ComparisonV of Mmean(-V>arian)ce- an-d -Me-an-S-em-i V-arTianc-e O-p-tim-iz-ati-on- on the Nairobi Stock Exchange. Strathmore UNIVERSITY Mwabaya Fahari Wasi A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi Stock Exchange. The fourth step is to calculate the required semi variance matrix. Alternatively, the semi variance 068811 Mwabaya Fahari Wasi matrix can also be calculated as a result of the f0o6881l1 lowing matrix: Submitted in partial fulfillment of the requirements for the Degree of Submitted in partial fulfillment of the requirements for the Degree of Bachelor in BusBiancheeslosr i nS Bucsiineesns Sccieen cAe Acctutauriaal rati Satrlat hamto rSe Utnrivaerstihty more University School of Finance and Applied Economics Strathmore University School of FinancNea iraobni, Kde nAyap plied Economics Strathm[Noovemrbeer ,U 201n5]i versity Nairobi, Kenya This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper acknowledgement. [November, 2015] This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper acknowledgement. Optimization is then applied using mat lab, R, excel or a combination of all three. 20 3.10 Efficient Frontiers Efficient frontiers were drawn graphically after a simulation on the optimal weights for the risk measures. Once the various weights are obtained an analysis for the respective model will be applied. We compare the return using the perfonnance of the securities over the 7-year period and come up with a conclusion to meet the research's first objective. To meet the second objective, the research then uses the geometric mean to compute the mean variance optimized portfolio and compare the results and variances of the same using a statistical model. Strathmore UNIVERSITY To compare the effectiveness of the two optimization methods the research will first determine if there is a difference in the risk-adjusted return metrics produced by both optimization methods and secondly to statistically Ate Csotm ptahriseo n dofi Mfefaen-rVeanriacncee san.d TMewan-Semi Variance Optimization onStock Exoch atneges. ts will be use thde Ntaoir obai nalyze the output data, an Strathmore UNIVERSITY F -test, to test whether the output had equal variances or not. Mwabaya Fahari Wasi A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi Stock Exchange. 068811 Mwabaya Fahari Wasi 068811 Submitted in partial fulfillment of the requirements for the Degree of Submitted in partial fulfillment of the requirements for the Degree of Bachelor in BusBiancheeslosr i nS Bucsiineesns Sccieen cAe Acctutauriaal rati Satrlat hamto rSe Utnrivaerstihty more University School of Finance and Applied Economics Strathmore University School of FinancNea iraobni, Kde nAyap plied Economics Strathm[Noovemrbeer ,U 201n5]i versity Nairobi, Kenya This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper acknowledgement. [November, 2015] This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper acknowledgement. 21 4 Results and Analysis 4.1 Portfolio Selection The portfolio to be used in the research was derived from the Nairobi Securities Exchange data which contained 50 stocks. To select 13 portfolios. All stocks that had been delisted between 2008 and 2015 were filtered out. The research assumed that 2011 investment date any stock and data used to create the efficient frontier would therefore be a stock that had existed for at least two years any stock that was not in existence as from 2008 was also filtered out. For the remaining stocks the mean returns of the stocks over the two-year period was found and the loss making stocks were filtered out, since rational investor would not invest in a stock that has been losing value over a long period of time. The stocks that had positive returns on average were then ranked. From the top 20 stocks , 13S sttroactkhsm woerree selected. The selection criteria being the top two stocks from each available industUryN. IVTEheR SITYs 13th tock was chosen on the basis of which had the highest return over the time period from the remaining 8 stocks. This was to ensure that the portfolio universe was well diversified. A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi Stock Exchange. Strathmore UNIVERSITY 4.2 Portfolio Analysis Mwabaya Fahari Wasi A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi In order to achieve the first objective of the reSstocek Exachanrgec. h i.e. comparing the mean variance approach 068811 to optimization with the semi variance approaMwcabahya F,ah arit Whasi e mean monthly returns for the individual 068811 stocks were computed togetheSru bwmititetdh in tpharetiailr fu rlfiellsmpente ocf thiev requ ireimseknt sm fore thaes Duegree sof i.e. variance, semi variance. Submitted in partial fulfillment of the requirements for the Degree of Bachelor in BusBiancheeslosr i nS Bucsiineesns Sccieen cAe ActuariaSemi variance was computed using the mean recttuua l rati Satrlat hamto rSe Utnrivaerstihty rns for meoarec Uhni vsetrsoityc k as its benchmark return. School of Finance and Applied Economics Strathmore University The results were as follows: School of Financ Nea iraobni, Kde nAyap plied Economics Strathm[Noovemrbeer ,U 201n5]i versity Mean and Variance of the selected sto Nairobi, Kenya This Rceseakrch Psroject is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper acknowledgement. [November, 2015] This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper MMR 1.65% 3.08% 1.79% 2.46% 6.85% 6a.c1k1no%wle dgem4e.n8t.1 % 3.05% 3.45% 2.98% 4.07% 2.37% 0.31% Variance 1.39% 0.89% 1.14% 4.02% 7.06% 2.34% 1.97% 0.23% 1.78% 2.18% 0.75% 1.26% 0.74% Mean and Semi variance of selected stocks: MMR 1.65% 3.08% 1.79% 2.46% 6.85% 6.11% 4.81% 3.05% 3.45% 2.98% 4.07% 2.37% 0.31% 0.43% 0.40% 1.05% 0.59% 0.37% 0.67% 0.05% 0.39% 0.74% 0.27% 0.47% 0.23% 22 The second objective was to compare the geometric mean variance portfolio of returns and that of the created using the semi variance optimization methodology. The resultant geometric means for the individual stock returns over the 2-year period are as tabulated below. MMR 1.65% 3.08% 1.79% 2.46% 6.85% 6.11% 4.81% 3.05% 3.45% 2.98% 4.07% 2.37% 0.31% GMR 2.62% 1.24% 0.79% 4.40% 5.16% 3.91% 2.94% 2.69% 1.93% 3.71% 1.76% -0.04% 4.3 Optimization To obtain the set optimal portfolio returns peSr utrnaitt hrimsko trhee portfolio had to be optimized to give the highest return for a given set of risk. For tUhNe IpVuErRpoSsIeTsY o f the research and to eliminate bias from the research results optimization was done using three different scenarios. 1. An investor whose annualized rate of return is 55% with no restrictions on the weightings ofthe portfolio. A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi 2. An investor whose annualized rate oSfto rcek EtxuchranngeStrathmo ri.es 45% with restrictions on the weightings of UNIVERSITY the portfolio such that it must invest 50% on the first 6 stocks and 50% on the remaining 7 stocks. Mwabaya Fahari Wasi A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi Stock Exchange. The selection of scenarios was to see if the rese0688Mwabaya Fahr1arc1 i Wahsi results were consistent under a different set of investor objectives and to ensure that there w068a811s no bias in the research results. Submitted in partial fulfillment of the requirements for the Degree of Submitted in partial fulfillment of the requirements for the Degree of Bachelor in BusBiancheeslosr i nS Bucsiineesns Sccieen cAe Acctutauriaal rati Satrlat hamto rSe Utnrivaerstihty more University No short selling of assets was allowed in anSchoyol of Fionancef an dt Aphplied Ec onosmics enarios i.e. no negative weights. Strathmore University School of FinancNea iraobni, Kde nAyap plied Economics Strathm[Noovemrbeer ,U 201n5]i versity Nairobi, Kenya This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper acknowledgement. [November, 2015] 4.4 Scenario 1 This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper acknowledgement. 4.4.1 Mean Variance Optimization As per the set conditions in scenario one the efficient frontier generated using the financial toolbox was as follows. 23 Efficient Frontier with 0.8 •Jubilee ·Unga Strathmore UNIVERSITY 0.1 • PanAfrica oL~~-~~-·~~-~~··-·'··~~·~··-L·-~·~-~-·~-~·'~-·---L~-~~··d--~---··-L····-~--~ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 09 Standard Deviation of Returns (Annualized) A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi Stock Exchange. Strathmore UNIVERSITY The optimal portfolio for an annualized retMuwrabnay oa Ffah 5ar5i W%asi was as follows: A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi Stock Exchange. 068811 Mwabaya Fahari Wasi WTK 068811 10% Submitted in partial fulfillment of the requirements for the Degree of Submitted in partial fulfillment of the requirements for the Degree of KakuzBi achelor in BusBiancheeslosr i nS Bucsiineesns Sccieen cAe Acctutauriaal rati Satrlat hamto rSe Utnrivaerstihty more University 20% School of Finance and Applied Economics Scan Strathmore University 16% School of FinancNea iraobni, Kde nAyap plied Economics BAT Strathm[Noovemrbeer ,U 201n5]i versity 28% ARM Nairobi, Kenya This Research Project is available for Library use on the understanding that it is copyright 26% material and that no quotation from the Research Project may be published without proper acknowledgement. [November, 2015] This Research Project is available for Library use on the understanding that it is copyright BAT and ARM were accomratdereiald an dt hthaet noh qiugotahtioens frto mw the Riegseharcths P raojsec tt mhaye bye p uhblaisdhed hwithgouht proepetru rn compared to the risk acknowledgement. taken up as measured by standard deviation. The least weights were to WTK as it had a higher risk and higher return. 4.4.2 Mean Semi Variance optimization As per the set conditions in scenario one the efficient frontier generated using the financial toolbox was as follows. 24 Efficient Frontier with Targeted Portfolios 0.9 Vt/TK 0.8 TI' 0.7 (1) -~ (11 ::J 0.6 c ·Scan c <( ....._. 0.5 (:/) -,c_ ::J (1) 04 oc • Centum '0 Strathmore c 0.3 • Jubiloo 'Ung.a UNIVERSITY (11 (1) :2 • Safarioom 0.2 • CFC 0.1 A Comparison of Mean-Variance and Mean-Semi Variance Optimization on the Nairobi Stock Exchange. Strathmore • PanAfrica UNIVERSITY 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Mwabaya Fahari Wasi SemA Ciomdparieson vof Mieaan-Vtariianoce annd Msean -Semi Variance Optimization on the Nairobi Stock Exchange. 068811 Mwabaya Fahari Wasi 068811 Submitted in partial fulfillment of the requirements for the Degree of Submitted in partial fulfillment of the requirements for the Degree of The optimal portfolio for a targBeactheeldor irn eBtusuBiancrheeslnosr i n S Boucsiineesfns Sc ci5een cAe5 Acctut%auriaal rati Sactrlat haomto rmSe Utnrivaeprstihtyr miosree Udn ivoerfsit tyh e following stocks: School of Finance and Applied Economics Strathmore University School of FinancNea iraobni, Kde nAyap plied Economics Strathm[Noovemrbeer ,U 201n5]i versity WTK 21% Nairobi, Kenya This Research Project is available for Library use on the understanding that it is copyright Kakuzi material and that no quotation from the Research Project may be published without proper acknowledgement. 20% BAT [November, 2015] 45% ARM 15% This Research Project is available for Library use on the understanding that it is copyright material and that no quotation from the Research Project may be published without proper BAT was accorded the highest weight as it ahckanodw letdhgeeme nlto. west semi variance hence less risk. Scan group was eliminated as it had a higher semi deviance compared to standard deviation. 4.4.3 Geometric Mean Variance Optimization. Using the geometric mean variance optimization method under scenario 1 yielded the following as the constituents of the optimal portfolio. 25 Efficient Geometric Frontier with Targeted Portfolios 0.7 Ka.kuzi 0.6 ,......, -o • i/v'TK