The Modern Portfolio Theory As An Investment Decision Tool
Journal of Accounting and Taxation Vol. 4(2), pp. 19-28, March 2012Available online at http://www.academicjournals.org/JATDOI: 10.5897/JAT11.036ISSN 2141-6664 2012 Academic JournalsReviewThe modern portfolio theory as an investment decisiontoolIyiola Omisore1, Munirat Yusuf 2* and Nwufo Christopher .I.31Senator, Federal Republic of Nigeria.Department of Business Administration, University of Abuja, Abuja, Nigeria.3Department of Accounting, University of Abuja, Abuja, Nigeria.2Accepted 13 February, 2012This research paper is academic exposition into the modern portfolio theory (MPT) written with aprimary objective of showing how it aids an investor to classify, estimate, and control both the kind andthe amount of expected risk and return in an attempt to maximize portfolio expected return for a givenamount of portfolio risk, or equivalently minimize risk for a given level of expected return. Amethodology section is included which examined applicability of the theory to real time investmentdecisions relative to assumptions of the MPT. A fair critique of the MPT is carried out to determineinherent flaws of the theory while attempting to proffer areas of further improvement (for example, thepost-modern portfolio theory [PMPT]). The paper is summarised to give a compressed view of thediscourse upon which conclusions were drawn while referencing cited literature as employed in thecourse of the presentation.Key words: Assets, beta coefficient, diversification, expected returns, investment, portfolio, risk, uncertainty.INTRODUCTIONThis paper presentation is an assessment of the modernportfolio theory as an investment decision tool. In theinvestment world, there exist different motives for investment. The most prominent among all is to earn a returnon investment. However, selecting investments on thebasis of returns alone is not sufficient. The fact that mostinvestors invest their funds in more than one securitysuggests that there are other factors, besides return, andthey must be considered. The investors not only likereturn but also dislike risk.The financial market, despite the benefits and rewards,is a complexly volatile industry which requires criticalanalysis to adequately evaluate risks relative to returns toaid decisions as regards participation in the industry.Upon such premise, this research work is an academicinsight into some analytics of the financial market. Thepresentation is an attempt to create foundation*Corresponding author. E-mail: drmrsyusuf@yahoo.com.knowledge to understanding the workings of the financialmarket. Despite the span of the research, specificattention would be accorded to the Modern PortfolioTheory.In the course of this discourse, some historicalbackground to financial market analysis would beexamined, related literature (to Modern Portfolio Theory)reviewed, its applications, pros and cons of the theorywould equally be examined.It is intended that this write-up would add to the existingpool of knowledge on the concept being investigated.LITERATURE REVIEWInvestment portfolio theories guide the way an individualinvestor or financial planner allocates money and othercapital assets within an investing portfolio. An investingportfolio has long-term goals independent of a market'sday-to-day fluctuations; because of these goals,investment portfolio theories aim to aid investors or
20J. Account. Taxationfinancial planners with tools to estimate the expected riskand return associated with investments.Passive portfolio theories, on one hand, combine aninvestor's goals and temperament with financial actions.Passive theories propose minimal input from the investor;instead, passive strategies rely on diversification, buyingmany stocks in the same industry or market, to match theperformance of a market index. Passive theories usemarket data and other available information to forecastinvestment performance.Active Portfolio Theories come in three varieties. Activeportfolios can either be patient, aggressive or conservative. Patient portfolios invest in established, stablecompanies that pay dividends and earn revenue despiteeconomic conditions. Aggressive portfolios buy riskierstocks, those that are growing, in an attempt to maximizereturns; because of the volatility to which this type ofportfolio is exposed, it has a high turnover rate. As thename implies, conservative portfolios invest with an eyeon yield and long-term stability.In any financial market analysis, if the objective of theanalysis involves determination of stocks to buy and atwhat price, there are two basic methodologies:Fundamental analysis, which maintains that markets maymisprice a security in the short run but that the "correct"price will eventually be reached. Profits can be made bytrading the mispriced security and then waiting for themarket to recognize its "mistake" and re-price thesecurity. Technical analysis, maintains that all informationis reflected already in the stock price. Trends 'are yourfriend' and sentiment changes predate and predict trendchanges. Investors' emotional responses to pricemovements lead to recognizable price chart patterns.Technical analysis does not care what the 'value' of astock is. Their price predictions are only extrapolationsfrom historical price patterns.Fundamental analysis of a business involves analysingits financial statements and forexhealth, its managementand competitive advantages, and its competitors andmarkets. When applied to futures and, it focuses on theoverall state of the economy, interest rates, production,earnings, and management.Fundamental analysis is performed on historical andpresent data, but with the goal of making financialforecasts. There are several possible objectives:1. To conduct a company stock valuation and predict itsprobable price evolution2. To make a projection on its business performance3. To evaluate its management and make internalbusiness decisions4. To calculate its credit riskWhile fundamental analysts examine earnings, dividends,new products, research and the like, technical analystsexamine what investors fear or thought of thesedevelopments and whether or not investors have theWhere withal to back up their opinions; these twoconcepts are called psych (psychology) and supply/demand.Technical analysts use market indicators of many sorts,some of which are mathematical transformations of price,often including up and down volume, advance/declinedata and other inputs. These indicators are used to helpassess whether an asset is trending, and if it is, theprobability of its direction and of continuation. Also,relationships between price/volume indices and marketindicators are sought. Examples include the relativestrength index, and MACD. Other avenues of studyinclude correlations between changes in options (impliedvolatility) and put/call ratios with price. Also, important aresentiment indicators such as put/call ratios, bull/bearratios, short interest, implied volatility, etc.There are many techniques in technical analysis.Adherents of different techniques (for example,Candlestick charting, Dow Theory, and Elliott Wavetheory), may ignore other approaches, yet many financialtraders combine elements from more than one technique.Some technical analysts use subjective judgment todecide which pattern(s) a particular instrument reflects ata given time and what the interpretation of that patternshould be. Others employ a strictly mechanical orsystematic approach to pattern identification andinterpretation.Technical analysis is frequently contrasted withfundamental analysis; the study of economic factors thatinfluence the way investors price financial markets.Technical analysis holds that prices already reflect allsuch trends before investors are aware of them.Uncovering those trends is what technical indicators aredesigned to do, imperfect as they may be. Fundamentalindicators are subject to the same limitations, naturally.Some traders use technical or fundamental analysisexclusively, while others use both types to make tradingdecisions.THE MODERN PORTFOLIO THEORY (MPT)Harry Markowitz 1991, an American economist in the1950s developed a theory of "portfolio choice," whichallows investors to analyse risk relative to their expectedreturn. For this work Markowitz, a professor at BaruchCollege at the City University of New York, shared the1990 Nobel Memorial Prize in Economic Sciences withWilliam Sharpe and Merton Miller.Markowitz’s theory is today known as the ModernPortfolio Theory, (MPT). The MPT is a theory ofinvestment which attempts to maximize portfolioexpected return for a given amount of portfolio risk, orequivalently minimize risk for a given level of expectedreturn, by carefully choosing the proportions of variousassets. Although the MPT is widely used in practice in thefinancial industry, in recent years, the basic assumptions
Omisore et al.210.01Capital Market Line (CML)0.0250.020.015Markowitz Efficient ure 1. Markouwitz Modern Portfolio theory.of the MPT have been widely challenged.The Modern Portfolio Theory, an improvement upontraditional investment models, is an important advance inthe mathematical modelling of finance. The theoryencourages asset diversification to hedge against marketrisk as well as risk that is unique to a specific company.The theory (MPT) is a sophisticated investmentdecision approach that aids an investor to classify,estimate, and control both the kind and the amount ofexpected risk and return; also called PortfolioManagement Theory. Essential to the portfolio theory areits quantification of the relationship between risk andreturn and the assumption that investors must becompensated for assuming risk. Portfolio theory departsfrom traditional security analysis in shifting emphasisfrom analysing the characteristics of individualinvestments to determining the statistical relationshipsamong the individual securities that comprise the overallportfolio (Edwin and Martins 1997).The MPT mathematically formulates the concept ofdiversification in investing, with the aim of selecting acollection of investment assets that has collectively lowerrisk than any individual asset. The possibility of this canbe seen intuitively because different types of assets oftenchange in value in opposite ways. But diversificationlowers risk even if assets' returns are not negativelycorrelated-indeed, even if they are positively correlated.More technically, the MPT models an asset return as anormally distributed function (or more generally as anelliptically distributed random variable), define risk as thestandard deviation of return, and models a portfolio as aweighted combination of assets so that the return of aportfolio is the weighted combination of the assets'returns. By combining different assets whose returns arenot perfectly positively correlated, MPT seeks to reducethe total variance of the portfolio return. MPT alsoassumes that investors are rational and markets areefficient.The fundamental concept behind the MPT is thatassets in an investment portfolio should not be selectedindividually, each on their own merits. Rather, it isimportant to consider how each asset changes in pricerelative to how every other asset in the portfolio changesin price.Investing is a trade-off between risk and expectedreturn as shown in Figure 1. Generally, assets with higherexpected returns are riskier (Taleb, 2007). For a givenamount of risk, the MPT describes how to select aportfolio with the highest possible expected return. Or, fora given expected return, the MPT explains how to selecta portfolio with the lowest possible risk (the targetedexpected return cannot be more than the highestreturning available security, of course, unless negativeholdings of assets are possible).Concept of risk and expected returnReturnReturn is the basic motivating force and the principalreward in any investment process. Returns may be
22J. Account. Taxationdefined in terms of realized return (that is, the returnwhich has been earned) and expected return (that is, thereturn which the investor anticipates to earn over somefuture investment period). The expected return is apredicted or estimated return and may or may not occur.The realized returns in the past allow an investor toestimate cash inflows in terms of dividends, interest,bonus, capital gains, etc., available to the holder of theinvestment. The return can be measured as the total gainor loss to the holder over a given period of time and maybe defined as a percentage return on the initial amountinvested. With reference to investment in equity shares,return is consisting of the dividends and the capital gainor loss at the time of sale of these shares.market risk. The market risk is represented by fluctuationin the market benchmark, that is, market index. Shareswhose β factor is more than 1 are considered less risky. Itmay be noted that β is a measure of systematic riskwhich cannot be diversified away.The total risk of an investment consists of twocomponents: diversifiable (unsystematic) risk and nondiversifiable (systematic) risk. The relationship betweentotal risk, diversifiable risk, and non-diversifiable risk canbe expressed by the following equation:Total risk Diversifiable risk Non diversifiable riskAssumptions of the modern portfolio theoryRiskRisk in investment analysis, is the unpredictability offuture returns from an investment. The concept of riskmay be defined as the possibility that the actual returnmay not be same as expected. In other words, risk refersto the chance that the actual outcome (return) from aninvestment will differ from an expected outcome. Withreference to a firm, risk may be defined as the possibilitythat the actual outcome of a financial decision may not besame as estimated. The risk may be considered as achance of variation in return. Investments having greaterchances of variations are considered more risky thanthose with lesser chances of variations.Risk should be differentiated from uncertainty; Risk isdefined as a situation where the possibility of happeningor non-happening of an event can be quantified andmeasured: while uncertainty is a situation where thispossibility cannot be measured. Thus, risk is a situationwhere probabilities can be assigned to an event on thebasis of facts and figures available regarding thedecision. Uncertainty, on the other hand, is a situationwhere either the facts and figures are not available, or theprobabilities cannot be assigned.Measurement of risk and the beta coefficientNo investor can predict with certainty whether the incomefrom an investment will increase or decrease and by howmuch. Statistical measures can be used to make precisemeasurement of risk about the estimated returns, togauge the extent to which the expected return and actualreturn are likely to differ. The expected return, standarddeviation and variance of outcomes can be used tomeasure risk.Beta coefficientThere is another measure of risk known as β whichmeasures the risk of one security/ portfolio relative toThe framework of the MPT makes many assumptionsabout investors and markets. Some are explicit in MPTequations; such as the use of Normal distributions tomodel returns. Others are implicit, such as the neglect oftaxes and transaction fees. None of these assumptionsare entirely true, and each of them compromises the MPTto some degree. Predominant among the MPT assumptions is the efficient market theory.The efficient market theoryThe efficient market theory is widely referred to as ahypothesis, and thus efficient market hypothesis (EMH)asserts that financial markets are "informationallyefficient". That is, one cannot consistently achieve returnsin excess of average market returns on a risk-adjustedbasis, given the information available at the time theinvestment is made.There are three major versions of the MPT hypothesis:"weak", "semi-strong", and "strong". The weak EMHasserts that prices of traded assets (for example, stocks,bonds, or property) already reflect all past publiclyavailable information. The semi-strong EMH opines thatprices reflect all publicly available information and thatprices change to reflect new public information. Thestrong EMH additionally claims that prices instantly reflecteven hidden or "insider" information. There is evidencefor and against the weak and semi-strong EMHs, whilethere is powerful evidence against the strong EMH(Andrei, 2000).Extensive researches have revealed signs ofinefficiency in financial markets. Critics have blamed thebelief in rational markets for much of the late-2000sglobal financial crisis. In response, proponents of thehypothesis have stated that market efficiency does notmean having no uncertainty about the future, rather themarket efficiency is a simplification of the world whichmay not always hold true, and that the market ispractically efficient for investment purposes for mostindividuals (Chambernan, 1983).
Omisore et al.Asset returns are (jointly) normally distributedrandom variablesDespite this assumption, evidence from frequentobservations shows that returns in equity and othermarkets are not normally distributed. Large swings (3 to 6standard deviations from the mean) occur in the marketfar more frequently than the normal distributionassumption would predict. While the model can also bejustified by assuming any return distribution which isjointly elliptical, all the joint elliptical distributions aresymmetrical whereas asset returns empirically are not.Correlations between assets are fixed and constantforeverCorrelations depend on systemic relationships betweenthe underlying assets, and change when these relationships change. During times of financial crisis, all assetstend to become positively correlated, because they allmove (down) together. In other words, the MPT fails tofunction when investors are most in need of protectionfrom risk.23expectations being biased, causing market prices to beinformationally inefficient. This possibility is studied in thefield of behavioural finance, which uses psychologicalassumptions to provide alternatives to the capital assetpricing model (Owen and Rabinovitch, 1983).There are no taxes or transaction costsReal financial products are subject both to taxes andtransaction costs (such as broker fees), and taking theseinto account will alter the composition of the optimumportfolio. These assumptions can be relaxed with morecomplicated versions of the model.All investors are price takersTheir actions do not influence prices. In reality,sufficiently large sales or purchases of individual assetscan shift market prices for that asset and others (viacross-elasticity of demand). An investor may not even beable to assemble the theoretically optimal portfolio if themarket moves too much while they are buying therequired securities.All investors aim to maximize economic utilityInvestors aim to maximize economic utility in order tomake as much money as possible, regardless of anyother considerations. This is a key assumption of theefficient market hypothesis, upon which the MPT relies.Any investor can lend and borrow an unlimitedamount at the risk free rate of interest.In reality, every investor has a credit limit.All securities can be divided into parcels of any sizeAll investors are rational and risk-averseThis is another assumption of the efficient markethypothesis, but we now know from behaviouraleconomics that market participants are not rational. Itdoes not allow for "herd behaviour" or investors who willaccept lower returns for higher risk. Even gamblersclearly pay for risk, and it is possible that some stocktraders will pay for risk as well.All investors have access to the same information atthe same timeThis also comes from the efficient market hypothesis. Infact, real markets contain information asymmetry, insidertrading, and those who are simply better informed thanothers.Investors have an accurate conception of possiblereturnsThe probability beliefs of investors match the truedistribution of returns. A different possibility is investors'In reality, fractional shares usually cannot be bought orsold, and some assets have minimum order sizes.More complex versions of the MPT take into account amore sophisticated model of the world (such as one withnon-normal distributions and taxes) but all mathematicalmodels of finance still rely on many unrealistic premisesas stated e MPT assumes that investors are risk averse,meaning that given two portfolios that offer the sameexpected return, investors will prefer the less risky one.Thus, an investor will take on increased risk only ifcompensated by higher expected returns. Conversely, aninvestor who wants higher expected returns must acceptmore risk. The exact trade-off will be the same for allinvestors, but different investors will evaluate the trade-offdifferently based on individual risk aversion characteristics. The implication is that a rational investor willnot invest in a portfolio if a second portfolio exists with
24J. Account. Taxationa more favourable risk-expected return profile – that is, iffor that level of risk an alternative portfolio exists whichhas better expected returns.The MPT is therefore a form of diversification. Undercertain assumptions and for specific quantitative definitions of risk and return, MPT explains how to find the bestpossible diversification strategy.Applying the theoryThe Portfolio theory (MPT) approach has four basicprocedures: Security valuation-describing a universe ofassets in terms of expected return and expected risk;asset allocation decision- determining how assets are tobe distributed among classes of investment, such asstocks or bonds; portfolio optimization-reconciling riskand return in selecting the securities to be included, suchas determining which portfolio of stocks offers the bestreturn for a given level of expected risk; and performancemeasurement-dividing each stock’s performance (risk)into market-related (systematic) and industry/securityrelated (residual) classifications (Brodie, 2009).Step one: Data collectionGet historical data for all the selected equities. Determinethe average weekly (or daily) returns and correspondingstandard deviation in weekly returns. Find the correlationbetween selected assets.rate per unit of risk for any available portfolio with riskyassets. Or, a portfolio with the maximum Sharpe ratio.Step four: Create the capital market lineAccording to capital market theory, investors who allocatetheir capital between a riskless security and the riskyportfolio (M) can expect a return equal to the risk-freerate plus compensation for the number of risk units theyaccept. In other words;Step five: The optimal portfolioFinally, we are ready for our optimal portfolio. Optimalportfolio is represented by the point of tangency betweenthe capital market line and the Markowitz efficient frontier.The theory is a mathematical model that uses standarddeviation of return as a proxy for risk, which is valid ifasset returns are jointly normally distributed or otherwiseelliptically distributed.Under the model:1. Portfolio return is the proportion-weighted combinationof the constituent assets' returns.2. Portfolio volatility is a function of the correlations ρij ofthe component assets, for all asset pairs (i, j).Expected return:Step two: Create a Markowitz efficient frontierThe portfolio standard deviation is provided by the followequation:Where Rp is the return on the portfolio, Ri is the return onasset i and wi is the weighting of component asset i (thatis, the share of asset i in the portfolio).σP sqrt ( square(wi). square(σi2) wi .wj.CovijPortfolio return variance:Construct different portfolios with given target returns(0.001, 0.002, etc.) and use the “solver” in excel to findweights such that the standard deviation for the portfolio(expressed previously) is minimized. Then, plot theseportfolios with return on y-axis and risk or standarddeviation on x-axis. The resulting envelope curve iscalled the “Markowitz efficient frontier“. All the portfolioson this frontier are efficient in the sense that any portfoliobeneath this line will not provide a better risk-returnalternative (either the portfolio will have lower return forgiven risk or higher risk for given return) (Markowitz,1959, 1952).Where ρij is the correlation coefficient between thereturns on assets i and j.Alternatively the expression can be written as:,Where ρij 1 for i j.Step three: Create the market portfolioMarket portfolio is defined as the portfolio with riskyassets that provide highest expected return over risk-freePortfolio return volatility (standard deviation):
Omisore et al.25Risk free rateFigure 2. Efficient frontier. The hyperbola is sometimes referred to as the 'Markowitz Bullet', and is theefficient frontier if no risk-free asset is available. With a risk-free asset, the straight line is the efficientfrontier.For a two asset portfolio, we have the following:Portfolio return:Portfolio variance:For a three asset portfolio, we have the following:Portfolio return:Portfolio variance:PRACTICAL APPLICATIONS OF THE THEORY ININVESTMENT DECISION MAKINGDiversificationAn investor can reduce portfolio risk simply by holdingcombinations of instruments which are not nt). In other words, investors can reducetheir exposure to individual asset risk by holding adiversified portfolio of assets. Diversification may allowfor the same portfolio expected return with reduced risk.If all the asset pairs have correlations of 0—they areperfectly uncorrelated—the portfolio's return variance isthe sum over all assets of the square of the fraction heldin the asset times the asset's return variance (and theportfolio standard deviation is the square root of this sum)(Koponen, 2003).The efficient frontier with no risk-free assetAs shown in Figure 2, every possible combination of therisky assets, without including any holdings of the riskfree asset, can be plotted in risk-expected return space,and the collection of all such possible portfolios defines aregion in this space. The left boundary of this region is ahyperbola, and the upper edge of this region is theefficient frontier in the absence of a risk-free asset(sometimes called "the Markowitz bullet"). Combinationsalong this upper edge represent portfolios (including noholdings of the risk-free asset) for which there is lowestrisk for a given level of expected return. Equivalently, aportfolio laying on the efficient frontier represents thecombination offering the best possible expected return forgiven risk level (Kent et al., 2001).Matrices are preferred for calculations of the efficientfrontier. In matrix form, for a given "risk tolerance", the efficient frontier is found by minimizingthe following expression:wTΣw q * RTw
26J. Account. TaxationWhere, w is a vector of portfolio weights and i wi 1(The weights can be negative, which means investorscan short a security); Σ is the covariance matrix for thethe loss of the sunk costs (that is, there is little or norecovery/salvage value of a half-complete project).returns on the assets in the portfolio;is a "risktolerance" factor, where 0 results in the portfolio withminimal risk andresults in the portfolio infinitely far outon the frontier with both expected return and riskTunbounded; and Ris a vector of expected returns. w ΣwisTthe variance of portfolio return. R w is the expectedreturn on the portfolio.The foregoing optimization finds the point on thefrontier at which the inverse of the slope of the frontierwould be q if portfolio return variance instead of standarddeviation were plotted horizontally. The frontier in itsentirety is parametric on q.Many software packages, including Microsoft Excel,MATLAB, Mathematical and R, provide optimizationroutines suitable for the foregoing problem.An alternative approach to specifying the efficientfrontier is to do so parametrically on expected portfolioreturn RTw. This version of the problem requires that weminimize;wTΣwHowever, that neither of these cited differencesnecessarily eliminates the possibility of using the MPTand such portfolios. They simply indicate the need to runthe optimization with an additional set of mathematicallyexpressed constraints that would not normally apply tofinancial portfolios.Furthermore, some of the simplest elements of themodern portfolio theory are applicable to virtually anykind of portfolio. The concept of capturing the risktolerance of an investor by documenting how much risk isacceptable for a given return may be applied to a varietyof decision analysis problems. MPT uses historicalvariance as a measure of risk, but portfolios of assets likemajor projects do not have a well-defined "historicalvariance". In this case, the MPT investment boundary canbe expressed in more general terms like "chance of areturn on investment (ROI) less than cost of capital" or"chance of losing more than half of the investment".When risk is put in terms of uncertainty about forecastsand possible losses then the concept is transferable tovarious types of investment.Subject to:Application to other disciplinesTR w μFor parameter μ. This problem is easily solved using aLagrange Multiplier (Merton, 1972).Applications to project portfolios and other "nonfinancial" assetsThe MPT is gradually being applied to portfolios ofprojects and other assets besides financial instruments.When applied beyond traditional financial portfolios,some fundamental differences between the differenttypes of portfolios must be considered:1. The assets in financial portfolios are, for practicalpurposes, continuously divisible while portfolios ofprojects are "lumpy". For example, while we can computethat the optimal portfolio position for 3 stocks is, say, 47,30 and 23%, the optimal position for a project portfoliomay not allow us to simply change the amount spent
expected risk and return; also called Portfolio Management Theory. Essential to the portfolio theory are its quantification of the relationship between risk and return and the assumption that investors must be compensated for assuming risk. Portfolio theory departs from traditiona
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