JFBI Vol. 2, No.267

Needfor Sanitised Investment Culture.There is an apparent degradation of thequality oflife and investment in Nigeria.

Introduction

petroleum marketers over an 11 yearperiod. The optimisation processinvolves the introduction of the Elton­Gruber-Padberg approach which usesthesimple risk adjusted ranking devicesto determine,first, the limit of assets inaportfolio and theweightsyielding highreturnsdeterminedusing quantificationtools. Optimal portfolio is determinedto consistof assetsof UnionBank,Mobiland Dunlop using Elton-Gruber­Paderg approach. Multiple Analysis ofVarianceis used.Results show thatnon­linear objective functions andoptimisation techniques describe theinvestments getter and yield highreturns; thus discouraging a naive andqualitative investment approach.

* Dr (Mrs) C. Chinelo Ikoku is the Head, Deparmentof Finance and Banking, University of Port Harcourt.Mr Stephen B. Vilawa is an MBA graduate of theDepartment of Finance and Banking, University ofPort Harcourt.

Every rational investor will seek toincrease his wealth of assets throughhigh yield or return on investment andby minimising risk. The accuratequantificationof returnand risk involvesthe import of mathematical science andmanagement science tools as areemployed in this work using Simplexalgorithms, Gauss-Seidel iteration,Markowitz-Lagrangian approach,Cobb-Douglasfunction and QuadraticKuhn-TukaAlgorithms. Twentyassetsofcorporations are randomly selectedfrom an assortment of economicsectorsnamely: agriculture, banking,automobile and tyre, pharmaceuticaland animalfeeds, breweries, chemicaland paints, conglomerates, and

C.CHINELO IKOKU & S. B. VILAWA *

OPTIMISATION TECHNIQUES FOR EFFICIENTPORTFOLIO MANAGEMENT

7

ideas of optimisation techniques andtheir application to portfolio analysis,thus leading to efficient management.Investors with business acumen areprimarily concerned with how to getlarge positive returns on their portfolioof assets. The investors seek the securityin the capital or money market that isoften influenced by government policies,political process, legislative enactmentetc. Naive investors sometimes rely onthe overall marketing index that willinvoke the use of special decisionmaking tools. As the quality of ourfinancial market improves, portfolioevaluation and choice is being guidedwith the use of quantitative techniquesor deterministic models developed on alogical premise.

David and Starr (1965) define amodel as an abstraction or simplifiedrepresentation of reality intended toexplain the behaviour or some aspectof it.Deterministic models are the mostcommonly and extensively used modelsbecause there is in the discipline ofmathematics, an inherent rigour thatcompels the relationships that existamong the variables, and a powerful toolfor the manipulation of data that enablesderiving conclusions from givenpremises. Thus, many large marketsituations can be simulated and analysedwith the use of mathematical models.

Mathematically, linear programmingtechniques assume linearity between thevariables and the problems, while in non­linear optimisation techniques the

JFBI Vol. 2, No.2

Optimisation Techniques: TheNeedOne of the centra! goals of this work isto understand in-depth the fundamental

Unless there is a total sanitation of thewhole system, both politically andmorally, economic distress in anunimaginable scale is sure to wreck theinvestment institution, dimming thefuture further. A gloomy future can beaverted by restoring a higher priority toour investment institutions, improve onthe knowledge necessary to sustainhealth investment, the quality of ourfinancial market and the quantity of ourenvironment, thus, fulfilling our role inworld business affairs. Certainly, thereare no internationally accepted indexeson the health of Nigerian businesses.Many factors are responsible for this anddue to tremendous pressure forpolitically stimulated quick "fixes",government decided to go into businessacquiring large shares in big corporateinstitutions and installing managers whowould rather maintain personal loyaltiesto individuals in govenunent than thinkof how to improve the business theyshould be managing. There is no sputnikin the offing to induce the changenecessary. This reassessment of priorityis necessary, not because we are introuble today, but because we will haveno future as we move into the nextmillennium either internationally ornationally, if we let this deterioration ofthe productive capability of our financialcommunity decelerate further.

r­..

Chinelo Ikoku & s. B. Vilawa68

(

JFBl Vol. 2, No.2

return, the portfolio with theminimum risk is to be chosen.Conversely, where there areportfolio(s) with equal risk, theportfolio with the minimum rate ofreturn is to be selected.

The last qualitative guideline isknown as the dominance principle inportfolio theory. Some investors employthe risk premium as a measure of risk,while others adopt the variance orstandard deviation as the risk surrogate.To theoretically explain this paradox,utilitarianism is developed, which isborne out of the assumption that allinvestments (portfolios) are to be testedby their 'utility, especially that thegreatest investment returns on thesecurity or the portfolio of assets shouldbe the sole interest of ail investors orinvestments. The argument is furtheradvanced and complicated as to whenthe investor is a corporate organisation.Thus if we write E (u) = F(R8), whereE (u) the expected utility, is a singleindex ofperfonnance, R is the expectedreturn on the portfolio as a whole and 8is the associated risk and standarddeviation of the returns. Mathematicalimplication here is that E (u) is capableofmapping a two dimensional measureinto one dimension, thus giving anoptimum risk return combination. Afurther meaning to this is that, there willbe, in general, many porfolio(s) whichsatisfy the expected utility function anthese are known as Mean-Varianceefficient portfolio. Thus the target is to

Context of the ProblemInvestment experts are makingindefatigable attempts at evolvingportfolio theories. However, it isaxiomatic that many of these portfoliotheories have proved inadequate inevolving globally accepted optimaltechniques for efficient portfoliomanagement in the financial market andbusiness world. These obviousomissions or commissions arising fromeither over-constraining the objectivefunctions or over-relaxation of theobjective functions. The results cease tobe a panacea. Qualitative approaches toportfolio selection guidelines have nothelped matters either. For example,some of these qualitative guidelines stateas follows:

(i) The investment in the portfolioshould be diversified to maximisethe yield for a given level of risk.

(ii) The portfolio must be revieweddaily to decide which securitywould be liquidated and what newinvestment should be made.

(iii) Given portfolio(s) of equal rate of

objective function is expressed as apolynomial function of the variable, witheither non-linear or linear constraints.Most investment decisions are made inan environment known as the state ofrisk. A risk objective function requiresminimisation, whereas the returnfunction is always intended formaximisation.

69Optimisation Techniques Efficient Portfolio Management

n

Maximising R=LC.X.i=I I I

JFBJ Vol. 2. No.2

Literature Review

Technically,a portfolioencompassestheinvestors' entire set of assets, and in abroad term, the study of all aspects of aportfolio is known as portfoliomanagement. Prior to Markowitz(1950), and the introduction ofmathematical tools in managementscience, investors dealt loosely with theconcept of portfolio return and risk. Aformal framework introduced by

HypothesisAfter thorough research and the reviewof relevant literature on the subject ofhow best to evolve efficient portfoliousing a more simplified optimisationtechnique, the following hypothesiscame up. There is no significantdifference in the expected portfolioreturns using the followingoptimisationtechniques; Simplex algorithm, Gauss­Seidel iteration,QuadraticKuhn-Tuckerprogramming, Markowitz-Lagrangianmultiplier treatment, Cobb-Douglas­Lagrangian treatment and others.

e:

implies that most investors are in actualfact averse to risk. The objective then isto choose a portfolio of investment thatmaximises subjectto whatever financialand other constrains that may apply.

eliminate the efficient portfolio.Explaining this using the followingequations:

3E(u»0;(E(u) <0 -(-3R 33

Ii

II

Chinelo lkoku & S. B. Vilawa70

Markowitz quantified portfolio risk andhad limitations when the portfolio wereassets more than four.

Other portfolio theorists formulateddifferent models and William Sharpe(1966) introduced the use ofthe CapitalAssets Pricing Model which is used tocompute an index of performance inrelationship to the market. Treynor,working independently in 1966,introduced another model whichcomputes an index of performancedefined as the ratio or risk premium tothe portfolio beta (riskiness or volatilityof the portfolio) ; while Jensen (1968)introduced another approach altogetherwhich compares better than the Sharpeand Treynor indexes; the Jensen indexuses the linear relationship of the securitymarket line, as a benchmark to measureperformance.

King Wallingford, Gohen, Pogueet al (1967) constructed themulti-indexmodels,which capturesomenon-marketinfluences such as the industry factor.George Dantzig (1947) introduced andinvented the Simplex Algorithmwhichwas first used for optimisationmethodsin operational research. Gauss-Seidelalso introduced an optimisationmethodcloselyrelatedto the SimplexAlgorithmin that both are linear programmingtechniques, with intended objectivefunctions for maximisation and otherconstrainingfunctionsandvariables.Seethe following equations.

JFBI Vol. 2. No.2

Step!

1. A predetermined sample size of(20) securities are randomlyselected from a table of randomnumbers which have an equalchance of being selected.

Methodology

In management science techniques,source data are always quantitative andtherefore need special statisticsprocessing for effective use. Thereforethe following steps are adopted:

Where R&V are functions of x. This istheCobb-Douglasmodel.Elton-Gruber­Padberg (1978) formulated a newapproachtoportfolioconstructionwhichselects and weighs securities for aportfolio based on as a relatively simplerankingdevice.Theresultingportfolio(s)compares favourably with those ofquadratic programming as adopted byKuhn-Tucker and others.

j=1

n

I.X =1J

Maximising U = f (R,V)SUbjectto the following constraints:

8U>0; 8U<0 (7.3)DR 8V

Equation 7.2 is the Kuhn-Tuckermodeland Markowitz model when thecomponentn

I. C.X drops, then the objective will bej=i I I

that ofminimisation.

The Lagrangian functioncorrespondingto the equation 7.2 is

n

L (X,U)= C(X) - ~ = A (X)Uij=1

I. CX. + I. ~ c.kX.Xkj=1 I I j=I k=1 J J

n n n

I.Xj=1 J

Maximising I. I. a'kXXk < b (7.2)k=1 j=l J J J

Subject to the following constraintsn

n n

where i= 1,2,3, ... n, & j= 1,2,3, ...m.

A Simplex Tableau is formed fromequation 7.1, whereas the Gauss-Seideluses the same set of equations foriteration. Kuhn-Tucker, Cobb-Douglasand Harry Markowitz introduced thenonlinear optimisation methods. Theobj ective function is usually apolynominal function of variables. Allthe three methods introduced theLagrangian function. In particular,Kuhn-Tucker reasoned that investorswould normally select the efficientportfolio that is most appealing to themby furtivereferencesto itsutilityfunctionno doubt, or else would incorporatesome unsophisticated representation ofpreferences in an alternative function tothat of Lagrangianmultiplier treatment.Stating the three models below willexplain' out their various positions onoptimisation.

I. I. aX. <b.- (7.1)i=i j =1)1 J J

n n

subject to the constraints

71Optimisation Techniques for Efficient PortfolioManagement

~.

Portfolio Construction and CutoffRate

Elton-Gruber-Padbergformulatedanewapproachto portfolioconstructionwhichselects and weighs securities for aportfolio based on a relatively simpleranking device. The basic assumptionbehind the procedure is that the co­vanances among securities, oralternativelythebetas, canbe accuratelyforecasted. The resulting portfolios arethe same as that which the quadraticprogramming procedure would yield,assuming validity of the market modeland the CAMP. The EGP procedurerequires a forecast of the riskless rateandthevarianceof returnson themarketportfolio. Also, for each security underconsideration, it calls for the followingitems:expectedreturn,beta, andvariableof return and coefficient ofdetermination. The coefficient ofdetermination (r") IS the squarecorrelation of the coefficient of thesecurity's returns with market returns.A few ngorous computationalprocedures are required for EGP.

JFBI Vol. 2, No.2

11. All assets at the first convergenceto the critical rate constitute theoptimumportfolio.

111. Determine the relative weight i.e.the proportion ofthe fund to be onthe assets.

IV. Formulatedobjectivefunctionsandconstraints.

Step II

1. Determine feasibleassetsusing theEGPmethodsdescribedinequation7.4.

,.....

111. Regress a characteristic linewithout bias through the scatteredpoints formed using rate of returnof stock and rate of return on themarket (Rm) and determine thefollowing: the l3-coefficientof thestock,the intercepta.b is ameasureof the riskiness of the stock orportfolio, while a is the rate ofreturn when I\nis zero.

IV. Computethe regressioncoefficientand coefficientof determinationofeach stock rj& r2jrespectively.

11. Compute the rate of return RORfor eleven (11) years which is ameasure of the securities return.The computational process wasalso applicable for the marketindex. From the averageROR, thevariance of the stock, standarddeviation and co-variances aredetermined.

Chinelo Ikoku & s. B. Vilawa72

v. Solve problems using the fiveoptimisation methods - Simplexalgorithms, Gauss-Seidel iteration,Markowitz- Lagrangian method,Quadratic programming, Cobb­Douglas- Lagrangian approach -and obtain actual weights.

VI. Substitute weights into returnequation and obtain vanousportfolio returns.

JFBI Vol. 2, No.2

Weight Determination for OptimalPortfolio Assets

After using the cutoff rate to identify thesecurities that qualify for the optimalportfolio, the next task is to compute theappropriate weights for each security.These weights show how much of eachsecurity to buy to balance the portfolio

Where 0'2. = the variance of the returns1

on security r2 = the coefficient ofdetermination ofthe returns on securitywith the market portfolio.

Equation 7.4 looks complicated,butwith a few intermediate computationsand summations, its use is quitestraightforward.First, note that equation7.4 uses 0'2 .. This term is the variance

eJof the returns on the security notassociatedwith market returns. Itis alsocalled the variance of the error term inthe regression used to compute betacoefficient. It is computed as follows:

0'2 .=O'2=(1-rZ) (7.4)eJ I

Pi beta of the stock i

returns.

R, = risk free rate of return

0'2 variance of the marketm

the variance ofthe stock'smovementnot correlatedtomarket returns

R, expected return ofsecurity

Where C.= the cutoff rate for security iI

L = f32.J

] +0'2m

(3-1) (7.4)

C =~2i V m

n

Ci= i~1 (E (R) - Rf)

Pi

The first step in the EGP portfolio. construction process is to rank thesecurities, beginning with the highestrisk-adjusted excess return. The nextstep is the cutoff rate.

Since the objective is to find theoptimum portfolio, there is hence theneed to screen out stocks whoseexpected return is not sufficient to leadto this portfolio. We therefore need acutoff rate that represents the minimumrisk-adjusted excess return that we canaccept. Securities yielding less than thisreturn will not qualify for inclusion inthe portfolio. Of course, in perfectlyefficient markets, all securities wouldhave identical risk-adjusted excessreturn. In such case, there would be noneed for cutoff rate; all securities wouldbe in the portfolio. The value of thecutoff rate depends upon thecharacteristicsofthe optimumportfolio.To identifythe cutoff rate, it is necessary.to calculate its value for severaldifferentpossibilities of the optimum. Somevariables are required for computing thecutoff rate. The formula for the cutoffrate is as follows:

73Optimisation Techniques for Efficient Portfolio Management

JFBI Vol. 2, No.2

74 Chinelo lkoku & S.B. Vilawa

optimally. The following formular WhereW.= theweight for securityj (theJ

computes the weights: portion of funds in the portfolio to becommitted to securityj)

W. = Z. ................................. (7.6) = the number of securities in theI I nn optimal portfolio.L Z.j=l J R. = the optimal cutoff rate.

Z. = ~JE (R) - R{Cjl .............. (7.7) Z. = relative investment in eachIJ

O"ej~i security.

Table 7.1

Statistical and Parametric Requirement (%)

SIN Average Stand. Variance COY.of Inter a. Vol of Regress Determ. Syst. Unsyst.Securities Returns DeY. cr2j(%) Stock & Stock Coef, r" ofCoef (%) e2I

R.(%) cr2j(%) Mkt crm Stock(%) ~I r'. 1}2 cr2

J I m

AUTO-MOBILE&TYRE1. DUNLOP

NIG. LTD 66.9 4.89 23.9 3.4 5.3 0.67 0.40 0.16 6.5 17.4

BREWERIES2.NIGERIAN

,-. BREWERIES 112.7 28.3 800.0 39.7 0.72 3.78 0.50 0.25 204.4 596.3... 3. GUINNESS 37.0 5.05 25.5 7.3 11.2 0.83 0.62 0.38 9.8 15.8

CHEMICAL &PAINTS4. BERGER

PAINTS 49.3 8.28 68.5 4.8 12.2 1.53 0.70 0.49 33.5 34.95. HEAE-

MEYER 14.3 2.5 6.3 13.4 2.9 3.6 0.54 0.29 03.8 4.4

CONGLOME-RATES6. A.G.LEVENTIS 32.3 6.3 39.0 -0.9 32.2 -0.08 -0.5 0.0 0.1 39.77.K.CHELLA-RAMS 6.7 2.51 6.3 -0.9 13.8 -0.23 -.34 O.ll 0.7 5.5

BANKING8.' FIRST

BANK 33.5 29.6 81.5 4.8 26.1 0.24 .09 0.0 0.8 80.69. UBA 118.6 31.4 96.8 23.4 52.9 2.49 .32 0.10 99.4 897.510. UNION

BANK 33.3 2.59 73.9 81.0 12.3 0.67 .29 0.08 6.4 67.7

JFBI Vol. 2. No.2

intended to explain exactly the

market return in zero

Equation 7.6 simply scales weights oneach security so that they sum to 1 toensure full investment.

As follows in the Table 7.1 aboveR, = Average return on stock

computed monthly and yearlyfor each of the stock

O'j - Standard deviation yearly ofeach of the stock return foreleven years.

0'.2 = Variance of the stock when the1

aim = Covariance of the stock andmarket

uj = Return on the stock when themarket return in zero.

f3i = A measure of how volatile thestock is with respect to themarket.

ri = Regression coefficient

r2: =Coefficient of determinationI

.1 16.30.0.0.913.4 -0.10

o. 18.00.00733.1 0.08

3.9 21.90.15.3913.4 0.53

.27 3.8 50.60.Q726.5 0.52

1.2 31.90.03.1927.3 0.29

1.5 28.30.05.2329.1 033

3.3 15.75.2 10.56.0 32.72.0 18.3

0.170.320.140.10

.41

.57

.38

.32

31.8 0.4814.5 0.6022.9 0.6538.6 0.38

= 31.46%= 3.78%= 14.31%

·1.9

4.9

1.9

5.1

3.7

OJ

6.96.69.65.1

Market average return (Rm) or indexStandard deviation of the market returnVariance of the market return (aZj )

16.5

18.1

25.9

54.4

33.1

PHARMACEU·TiCAL &ANIMAl'.,FEED16.BEECHAM

LTD 36.5 5.7517.STERLING

PROD. 42.1 7.3918. PHARMA

DEKO LTD 29.9 5.0919. PFIXER

PRO. LTD 35.7 42.520. BOOTS CO.

NIG. LTD 10.1 4.06

29.8

18.915.838.820.4

PETROLEUMMARKETERS11.MOBILOIL 446.8 4.3412.A.P. LTD 33.4 3.97J3.AGlPLTD. 41.6 6.2214.TOTAL 50.5 4.5215.TEXACO

NIG LTD.· 39.3 5.45

~1

Determ.ofCoef

RegressCof. r"StockrJ

Vol ofStock

Inter aCov. ofStock 'B:Mkt lJm(%)

Stand. VarianceDev. 0'2j (%)0'2) (%)

AverageReturnsRJ%)

Syst.(%)

SINSecurities

Statistical and Parametric Requirement (%)

Table 7.1 (Cont'd)

75Optimisation Techniques for Efficient PortfolioManagement

-

that the efficient portfoliots) generatedby portfolio analysis are no better thanthe statistical inputs on which they arebased" the test of significance, the

(ratio

JFBI Vol.2. No.2

There are statistics - descriptivestatistics, which is the body oftechniquesfor effective organisation and communi­cation of data as has been used and which.leads to inferential statistics.appropriate inferential statisticstesting of hypothesis is MultipAnalysis of Variance (MAN01;lA).Francis and Archer (1971), pointed

<'_...,-.-,_._----_.SIN Securities E (R)-Rf % E(R)-Rf)~j p2, Rei Z W, ;

13 0'2 (%)I m

1. Union Bank 29.27 .1969 .663 2.5 ·.0762 ".029962. Mobil Oil 69.6 2.12 1467 25.4 .984 386973. Dunlop 79.6 2.06 2.579 37.4 1.635 .642994. Berger Paints 23.5 1.57 6.707 32.35. A.P. Ltd 33.3 1.14 3.42 15.86. Guinness 28.4 1.23 4,36 18.37. Total Nig. 97.6 .770 0.789 21.158. Nigeria 26.3 .623 2.396 21.6

Breweries9. Agip Nig, 43.4 .534 1.292 22.610. First Bank 83.8 .0598 0.74 22.7II. Pharma-Deko 31.1 3998 1.2826 23.1812. Texaco 78.5 .302 .384 2~.95613. Sterlin

Products 55.2 .294 ..534 24.5414. UBA 42.2 .2918 .6908 24.96

r: 15. K. Chelarams 28.7 .276 .9618 25.0816. Beecham 79.7 .21 .2636 25.5617. Pfizer 278.8 .099 .0355 25.8618. Hegemyer 2.2 .065 2.94 23.7619. Boots Co. 33.0 0.202 .0613 4.28620. A.G.

Leventis ·236.3 ·.038 .016 ·5.42,,-=>_.-. --,--._,..~~,._.."",",",','_'~- -'--.. --~"""".

security's return that explainedby the variance of return onthe index or market.

~?cr2m= Systematic risk component of

Table 7.2

Cutoff Rated Determination for Optimum Portfolio

ChineloIkoku& S. B. Vilawa76

the stock explained by thevariance of index,

e2j "" Unsystematic risk componentnot explained by the index.

JFBI Vol. 2, No.2

1. Row I is the objective function,which is intended formaximisation.

11. Row II is the risk measureconstraining the maximisation rowand the rate of return and, bycalculation, should not exceed197.8%.

iii. Row III is relative investmentthat isneeded for the securities and bycalculation forEGP is about250.(See table 7-3).

IV. Row IV is the risk adjusted rate ofreturn derived from the fact thatsuperfluous diversification canactually increase the beta (~) of theportfolio which we want to reduceas much as possible to yieldoptimum portfolio. Therefore, riskadjustedrateof returncorrespondingto a beta (~) of 1.978 is 70%.

* Computed from source dataR = 33.3xj + 46.8x2 + 66.9x367x1 + 48x2 + 67x3::;197.8OX1+ 9.8x2 + 6X3 ::;25 ( 7.8)26xl+22x2+25x3 ::;70 (7.9)XI'X2,X3~ 0 (7.10)

ri

Return 33.3 46.8 66.9Risk(13) 67 48 67 197.8 IIRelativeinvestment(Z) 98 160 250 III

Risk adjustedrate ofretum 25.5 22.1 25.4 70 IV

Union Bank Mobil Dunlop Limit Row(%) (%) (%) (%)

Summaryof Variables for Linear Techniques

Table 7.3

Modelling and FormulationRefer to Table (7.3),UnionBank,MobilOil and Dunlop Nig. Limited are thethree assetschosenunderEGPapproachand setting out format for formulation.

to carry out a test for homogeneousvarianceon the sampledata andthenusethe result of that test to decide if theanalysis of variance is legitimate. Inmanagementscience, andmore so in thiswork, this procedure is advocated.

77Optimisation Techniques for Efficient Portfolio Management

JFBI Vol. 2, No.2

Table 7.4

Simplex Tableau for Maximisation

Xl X2 X3 X4 Xs X6 Basis Row Operation

197.8 67 48 67 0 0 X4 Rl

25 0 9.8 16 0 1 0 Xs Rz

70 26 22 25 0 0 X6 R3

0- -33.3 -46.8 -66.9 0 0 0 R4

93.1 67 6.96 0 00 X4 Rl -67 R216

25 0 9.8 0 0 X3 _&_16 16 16 16

30.9 26 6.68 0 0 0 X6 R3 -25~

16

104.5 -33 5.8 0 0 66.9 0 RJ -66.9R2---

16 16

13.5 0 -10.25 0 0 0 X4 R, -67 R3

26r.o. 25 0 9.8 0 0 1 0 X4 R2

16 26 16

30.9 6.68 0 0 0 Xl Rz

26 26 26 26

143.7 0 14.27 0 0 66.9 R4+ 33R316 26

This implies that the optimum solution will give the following result:

Chinelolkoku & s. B. Vilawa78

JFBJ Vol. 2, No.2

IntroducingtheLagrangianfunctionandthe Kuhn-Tucker condition with the.setof partial derivatives and artificialvariables ul: u2 ; u3 ; & u4 and re­expressingtheconstraintscorrespondingto the partial derivatives so that theconstraint terms are on the RHS, thenwe shall arrive at the following set ofequations:171.8xj-100.2x2-38.8x3+y- t j+uj= 83.3-100.2xj+ 37.8x2+ 9x3+y-t2+ u2= 117-38.8xj + 9x2 - 47.8x3+ y - t3+U3=167.3

XI+ X2+ X3+ S + u4 ~ 2.5

F = 2.5[33.3xj+ 46.8x2+ 66.9x31-85.9x2- 18.9xj-23.9x/+100.2xjX2+ 38.8xjx 3-9x2X3

F = 83.3xj+117x2+167x3-85.9x2-18.9x2_23.9x2+100.2xjx2+38.8x,x3- 9X2X3

Quadratic Programming TECBNNBMMThe weight expresses the rate at whichvariance must increase to offset anincreasein expectedreturn.Alternatively(I1W) expresses the rate at whichexpected return must increase tocompensate for increased variance. Thelower thevalueofthe w, themore averseto risk the investor is.

Weare to suppose that not all of thebudget need be spent, but continue toassume that leveraged portfolios areprohibited.

For w = 2.5, the utility function

F = WR - V (7.11)

Intuitively, negative values of x, areto be discarded and we see the systemyielding results that maximise theobjectivefunction and alsomaintain thesecond equation of constraints.Therefore, for this condition, themaximum return stands at 39.56%.

Optimum Portfolio SecuritiesProportion to be invested in.Union.Bank = 0.536Proportion to be investedinMobil = 0.4463Proportion to be investedin Dunlop 0.0Optimum Return 39.56%

Gauss-SeidelThis technique is finding its way intoportfolio management for the first timethroughthisliterature.Accountingto thetheorem governing its use, aconvergence is necessary for effectiveanalytical results. The set of equations(7.8 to 7.10) inthis articledo not alwaysguarantee convergence. A little

Lrearrangementcan mduce convergence.For quick iteration with a little bit ofintuition,convergenceisbetter achievedat the fourth iteration.

Restating equations 7.8, 7.9& 7.10as shown below:

Maximise, R = 33.3xj + 446.8x2+66.9x3

Subject to the following constraints.

67xj + 48x2 + 67x3 .$ 197.8OXj + 9.8x2+ 6X3 .::;25.026xj + 22x2 + 25x3 .$ 70Xj'X2'X3~ 0

79Optimisation Techniques for Efficient Portfolio Management

u = f (R,V), where U representsexpected utility.

The function F can be interpreted as autility function.Ingeneral,the investor'sex-ante satisfaction with a portfoliodepends on both expected return andvariance and can, in principle at least,be written as a function of thesevariables. That is equation (7.11)

-F =WR-V.

Quadratic Programming Technique­KuhnTucker ConditionsThe quadratic programming algorithmiteratively minimises the quadraticobjective function, subject to differentvaluesof a linearconstraint.The startingpoint is the formulationof the efficiencyfrontier determination problem whichallowsthe incorporationof the investor'srisk-returnpreferencesso that the overalloptimum portfolio can be identifiedalong with the efficiency frontier itself.Thus, the objective is to maximise aweighted averageof expectedreturn andvariance.At thisjuncture, we shallrecallequation (7.11) which is restated here

JFBI Vol. 2, No.2

= 0.463

Optimum Portfolio SecuritiesProportion to be investedinUnion Bank = 0.536

Proportion to be investedinMobil

Objective FunctionTherefore, for this condition, themaximum return stands at 39.56%

Lagrangian Multiplier MethodFormulation ProblemIt is clearly known from table 7.3 thatonly three (3) securities are above thecutoff rate using the EGP approach.Hence in this section the problemformulationwill involve three securities.The problem of finding the set of themean-variancesefficientportfolio is oneto which the Lagrangian m.ultiplierapproach seems suited, since it is usedin constrained minimisation (ormaximisation). For each given value ofportfolio return, we required minimumvariance. The problem is thusformulated.

r:...

Rate of return R = (52.29%)

= (44.7%)Dunlop (x3)

Optimum Portfolio SecuritiesFrom the above results, the proportionsby weight of each security in theportfolio are as follows:

Union Bank (xl) = (20.4%)

Mobil (x2) = (34.5%)

Chinelo Ikoku & S. B. Vilawa80

Proportion to be investedin Dunlop = 0.0

Optimum Rate of Return = '3.956%

.ir». Vol, No. 2

many investment and management toolsfor that. It is the earnest desire thatquantitative techniques, if arduously andaptly applied, will save many investorsmuch of their hard-earned money andactually minimise risk. An optimal yieldportfolio theory lends support to theimport of traditional mathematicalscience in investment analysis, There isthe observation made that differentquantitative models when used tosimulate the same system yield differentresults,

Therefore, in an attempt to answerthe research questions, it is logical toexplain that the five quantitative modelsare merely part of the portfolio theoriesthat are not perfected. Quantitative

122 3,84

1.22 7,01

11.58 4,46

11.58 8.68

It is noteworthy that care should be takento avoid errors. There are two possibletypes of errors during statistical testingof a hypothesis: type I and II. The costinvolved for each kind will not be thesame. We can reject the null hypothesiswhen, in fact, it is correct (we call this atype I error), or we can accept the nullhypothesis when in fact it is incorrect (atype IIerror). It is important to note that,if we are making our decision on theevidence of some sample, we can neverbe absolutely sure, except in certaintrivial cases, that we have made thecorrect decision (in other words, theprobability of a type I error, or theprobability of a type II error, is neverzero).

I

r

,945 3.48

.945 5.09

CriticalValue

ComputedValue

CriticalValue

HypothesisTwo Way Analysis

HypothesisOne Way Analysis

F5 (4,10)

FI (4,10)

FS(A) (4,8)

Fl (A) (4,8)

P5(B) (2,8)

PI (B) (2,8)

ComputedValue

:F~testDegreeof Freedom

Level ofSignificance

---~'~,---'~'-~,--"--------------Summary Table for the Hypotheses with Their Degrees of Freedom

Table 7,5

81Optimisation Techniquesfor Efficient PortfolioManagement

--

References1. Baridam, M.D.~ Research in

Administrative Science, PortHarcourt, 1990.

2. Bell.C.E., Quantitative Methodsjar Administration,Homewood, IIIIrwin, 1977.

3. Brealey, Richard A. "How toCombine Active Management,with Index Funds" Journal ofPortfolio Management,12 (winter)pp 4-10,1986.

4. Charles P. Jones, InvestmentAnalysis and Management, 3rd edJohn Wiley & Sons Inc., NewYork, 1968, pp 626-662.

5, Clarke, G., Quantitative Analysis:Matrix-BasedMethodsfor BusinessUniversity of Port Harcourt, 1990.

6. David, W.M.and Starr, M. K. L.ExecutiveDecisionand OperationsResearch, 2nd Ed. (EnglewoodCliffs.) New Jersey: PrenticeHall,1965.

7. Dan W. French, Security andPortfolio Analysis, Concepts andManagement, Merrill PublishingCompany 1989, pp.151-171.

8. Donald E. Fischer & Ronald J.Jordan Security Analysis and

JFBf Vol. 2, No.2

Common share or equity investors areentitled to two kinds of income: cashdividend earning and earnings fromcapital [arising from price fluctuations].The rate of return (ROR) should be themeasure of yield for common shareinvestors. It is further recommended thatnon- linear models give optimumportfolio return and should be adoptedby any prospective investors or portfoliomanagers.

Quantitative techniques remain themost effective tool used in portfolioanalysis, evaluation, management,ranking and revision. When the assetsare many i.e. from four and above, itwill be appropriate to determine first the

r-...

Recommendations and Conclusions

financial system with precision. Moreso,each model involves a degree ofsensitivity, accuracy and errors. Themode of data collection and preparationgreatly determines what type of portfoliomodel to use.

Portfolio theorists use differentmethods in quantifying some basic inputvariables like the geometric mean, netpositive value, rate of return etc.

A further explanation to this,however, limitations in one model canbe complemented and augmented byanother order. As an example, theSimplex algorithm is mainly adaptablefor linear systems. However, the HarryMarkowitz integrates the Lagrangianmultiplier method which accommodatesboth linear and non-linear systems.

Chinelo lkoku & S.E. Vilawa82

number of assets in the optimal portfolioand the techniques used to apportion theweights.

JFBI Vol. 2, No.2

Englewood Cliffs, New Jersey,1971.

11 ll<:oku,C.c. FinancialAnalysis andInterpretation for Alert Investors,African Educational Services,Owerri. 1993.

12. Mchaugl;n,F.S. andPickhardt.R.CQuantitative TechniqueManagement DeCiSion, Boston:Houghton Mifflin Company 1979pp. 1-17.

13. Markowitz, H.M. PortfolioSelection:EfficientDiversificationof Investment, New York: JohnWiley, 1959.

Portfolio Management, 5thEdition. New Jersey: EnglewoodCliffs Prentice Hall 07632 1989,pp.637-665.

9. Edwin J. Elton, Martin .J Gruber,andManfredW.Padberg"OptimalPortfolio from Simple RankingDevices" in Peter L. Bernstein(Ed.), SecuritySelectionandActivePortfolioManagement,NewYork:Institutional Investors, 1978 pp.(219-232).

10. Francis J.C., and Archer, S.H,Portfolio Analysis, Prentice HallFoundation of Finance Series,

83Optimisation Techniques for Efficient Portfolio Management