*Markowitz Risk Return OptimizationTHE PORTFOLIO SELECTION PROBLEM

*INTRODUCTIONTHE BASIC PROBLEM:given uncertain outcomes, what risky securities should an investor own?

*INTRODUCTIONTHE BASIC PROBLEM:The Markowitz Approachassume an initial wealtha specific holding period (one period)a terminal wealthdiversify

*INTRODUCTIONInitial and Terminal Wealthrecall one period rate of return

where rt = the one period rate of returnwb = the beginning of period wealth we= the end of period wealth

*INITIAL AND TERMINAL WEALTHDETERMINING THE PORTFOLIO RATE OF RETURNsimilar to calculating the return on a securityFORMULA

*INITIAL AND TERMINAL WEALTHDETERMINING THE PORTFOLIO RATE OF RETURNFormula:

where w0 = the aggregate purchase price at time t=0 w1 = aggregate market value at time t=1

*INITIAL AND TERMINAL WEALTHOR USING INITIAL AND TERMINAL WEALTH

where w0 =the initial wealth w1 =the terminal wealth

*THE MARKOWITZ APPROACHMARKOWITZ PORTFOLIO RETURNportfolio return (rp) is a random variable

*THE MARKOWITZ APPROACHMARKOWITZ PORTFOLIO RETURNdefined by the first and second moments of the distributionexpected returnstandard deviation

*THE MARKOWITZ APPROACHMARKOWITZ PORTFOLIO RETURNFirst Assumption:nonsatiation: investor always prefers a higher rate of portfolio return

*THE MARKOWITZ APPROACHMARKOWITZ PORTFOLIO RETURNSecond Assumptionassume a risk-averse investor will choose a portfolio with a smaller standard deviationin other words, these investors when given a fair bet (odds 50:50) will not take the bet

*THE MARKOWITZ APPROACHMARKOWITZ PORTFOLIO RETURNINVESTOR UTILITYDEFINITION: is the relative satisfaction derived by the investor from the economic activity.It depends upon individual tastes and preferencesIt assumes rationality, i.e. people will seek to maximize their utility

*THE MARKOWITZ APPROACHMARGINAL UTILITYeach investor has a unique utility-of-wealth functionincremental or marginal utility differs by individual investor

*THE MARKOWITZ APPROACHMARGINAL UTILITYAssumesdiminishing characteristicnonsatiationConcave utility-of-wealth function

*THE MARKOWITZ APPROACHUTILITY OF WEALTH FUNCTIONWealthUtilityUtility of Wealth

*INDIFFERENCE CURVE ANALYSISINDIFFERENCE CURVE ANALYSISDEFINITION OF INDIFFERENCE CURVES: a graphical representation of a set of various risk and expected return combinations that provide the same level of utility

*INDIFFERENCE CURVE ANALYSISINDIFFERENCE CURVE ANALYSISFeatures of Indifference Curves:no intersection by another curvefurther northwest is more desirable giving greater utilityinvestors possess infinite numbers of indifference curvesthe slope of the curve is the marginal rate of substitution which represents the nonsatiation and risk averse Markowitz assumptions

*PORTFOLIO RETURNCALCULATING PORTFOLIO RETURNExpected returnsMarkowitz Approach focuses on terminal wealth (W1), that is, the effect various portfolios have on W1

measured by expected returns and standard deviation

*PORTFOLIO RETURNCALCULATING PORTFOLIO RETURNExpected returns:Method One:

rP = w1 - w0/ w0

*PORTFOLIO RETURNExpected returns:Method Two:

where rP = the expected return of the portfolioXi = the proportion of the portfolios initial value invested in security iri = the expected return of security iN = the number of securities in the portfolio

*PORTFOLIO RISKCALCULATING PORTFOLIO RISKPortfolio Risk:DEFINITION: a measure that estimates the extent to which the actual outcome is likely to diverge from the expected outcome

*PORTFOLIO RISKCALCULATING PORTFOLIO RISKPortfolio Risk:

where sij = the covariance of returns between security i and security j

*PORTFOLIO RISKCALCULATING PORTFOLIO RISKPortfolio Risk:COVARIANCEDEFINITION: a measure of the relationship between two random variablespossible values:positive: variables move togetherzero: no relationshipnegative: variables move in opposite directions

*PORTFOLIO RISKCORRELATION COEFFICIENTrescales covariance to a range of +1 to -1

where