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Prof. Dr. Roland Kirstein Economics of Business and Law Faculty of Economics and Management http://econbizlaw.de Slope of the upper part of the Markowitz curve (=efficient frontier, until the bliss point). The curve can be described as a function r(σ). x is the share of the high risk asset => 1-x of the low risk asset. Starting at x=1 (only high risk), decrease x (moving on the efficient frontier: both r and decrease (they are functions of x) Hence, the efficient frontier can be written as r[σ(x)]. Then dr/dx = [dr/dσ]*[dσ/dx] <=> dr/dσ = [dr/dx] / [dσ/dx] Recall that r(x)=xrH+(1-x) rL and σ2(x)= x2σ2H+(1-x)2 σ2L+2x(1-x) σLH from which dr/dx and dσ/dx can easily be derived: dr/dσ =[ xrH+(1-x) rL] *[x2σ2H+(1-x)2 σ2L+2x(1-x) σLH](-0.5) please do not learn this by heart; it’s sufficient if you understand it 100 Prof. Dr. Roland Kirstein Economics of Business and Law Faculty of Economics and Management http://econbizlaw.de Portfolio Risk While the expected return of a portfolio equals the weighted average of its components: Expected Portfolio Return = (x1 r1 ) + ( x 2 r2 ) the portfolio variance requires a correction term which is influenced by the correlation coefficient: Portfolio Variance = x12 σ12 + x 22 σ 22 + 2( x1x 2ρ12 σ1σ 2 ) Covariance σ12 101 Prof. Dr. Roland Kirstein Economics of Business and Law Faculty of Economics and Management http://econbizlaw.de Risk Minimizing Portfolio (of two risky assets) Derivation of the risk-minimizing portfolio composition (if σ1, σ2 >0 and ρ12 are given) Remember that x2=1-x1, thus portfolio variance is given by: x12 σ12 + (1 - x1 ) 2 σ 22 + 2 x1 (1 - x1 )σ12 FOC: 2x1 σ12 − 2(1 - x1 )σ 22 + 2(1 - 2x1 )σ12 = 0 Solution: 2 σ * 2 − σ 12 x1 = 2 σ 1 + σ 22 − 2σ 12 SOC: σ12+σ22-2σ12>0 (=>minimum) 102 Prof. Dr. Roland Kirstein Economics of Business and Law Faculty of Economics and Management http://econbizlaw.de σ − σ 12 x = 2 σ 1 + σ 22 − 2σ 12 * 1 2 2 Application of the binomial formula shows that the denominator is always positive, hence we only have to check whether the nominator is between 0 and 1. Closer inspection of x* reveals the two conditions for an internal solution: x1* > 0 ⇔σ22 > σ12 ⇔ σ2 > σ1ρ12 x1* <1 ⇔σ22 −σ12 < σ12 +σ22 − 2σ12 ⇔ σ2 < σ1 / ρ12 If both conditions hold, then 0<x1*<1 => 0<x2*<1. If, however, σ2 ≤ σ1ρ12 then x1*= 0 => x2*=1 if σ2 ≥ σ1/ ρ12 then x1*= 1=> x2*=0. 103 Prof. Dr. Roland Kirstein Economics of Business and an Law Faculty of Economics and Management http://econbizlaw.de With a positive correlation coeff., interior solutions exist if the individual SD are „similar enough, “otherwise, the less risky asset is preferred. With a negative correlation coeff. only interior solutions exist. 104 Prof. Dr. Roland Kirstein Economics of Business and Law Faculty of Economics and Management http://econbizlaw.de Portfolio risk with one risk-free asset Assume that 1 is a risky asset, characterized by (R>r,σ>0), and 2 is a risk free asset (r>0,0). x denotes the share of asset 1. Then ExpectedPortfolioReturn = xR + (1 − x)r = x( R − r ) + r and Portfolio Variance = x 2σ2 + (1− x)20 + 2x(1− x)0 = x 2σ2 => Portfolio SD =x1σ1 Both E and SD of the portfolio are linear in x: dE/dx=(R-r), dSD/dx=σ. The slope of the efficient frontier equals the “Sharpe ratio” of the risky asset (excess return over SD): dE/dSP = [dE/dx]/[dSD/dx]= (R-r)/σ. 105 Prof. Dr. Roland Kirstein Economics of Business and an Law Faculty of Economics and Management http://econbizlaw.de (R-r)/σ 106 Prof. Dr. Roland Kirstein Economics of Business and Law Faculty of Economics and Management http://econbizlaw.de The same is true if the risky asset is, indeed, a portfolio of risky assets => If a portfolio of risky assets can be combined with a risk-free asset, then the attainable risk-return combinations are linear combinations of the r-σ-parameters of the risky portfolio and the risk-free asset. When creating a portfolio by buying a linear combination of a riskless and a risky asset, the efficient frontier in the r-σ-diagram is a straight line starting in (r,0) through (R,σ) => slope = (R-r)/σ. Two scenarios exists in which the portfolio SD is a linear combination (weighted average) of the individual SDs: • if the correlation coefficient is +1, or • when mixing a risk-free asset with a risky one. 107 Prof. Dr. Roland Kirstein Economics of Business and Law Faculty of Economics and Management http://econbizlaw.de We now distinguish four variations of the portfolio model Two risky assets One risk-free asset Markowitz: available 1st model: bulky 2nd model: fund is invested into efficient frontier straight line Zero investment 4th model: next model: strategy: short sale of flatter straight line steeper straight line two assets one asset is used to through the origin fund the investment into the other (risky) asset 108 Prof. Dr. Roland Kirstein Economics of Business and Law Faculty of Economics and Management http://econbizlaw.de Zero investment strategy (with one risk-free asset) Above: no short sales (x1,x2≥0) Zero investment strategy: invest x into one risky asset (r1=r, σ1=σ>0), fully funded by credit (i.e., short sale of riskless asset, r2=f with 0<f<r, σ2=0) => x1=-x2=x (the portfolio is characterized by the amount of asset 1) Since 2 is risk-free: σ12=0=ρ12 Portfolio return = x1r1+x2r2 = x1r-x1f = x(r-f) Portfolio variance = x12σ12+ x22σ22+ 2x1x2σ12= x2σ2 => Portfolio SD = xσ Observation: Portfolio risk (and return) is strictly increasing in x (for r>f and σ>0) 109 Prof. Dr. Roland Kirstein Economics of Business and an Law Faculty of Economics and Management http://econbizlaw.de Sharpe ratio: risky asset’s excess return over its SD (r-f)/σ is the slope of the (green) line connecting the riskless and the risky asset in a r-σ-diagram. The blue line (same slope) symbolizes for x≥0 the set of efficient r-σ-combinations a zero-investment strategy (ZIS) can bring about (r-σ-combinations below this line are feasible). 110 Prof. Dr. Roland Kirstein Economics of Business and Law Faculty of Economics and Management http://econbizlaw.de Zero investment strategy (with two risky assets) Investment into a risky asset (r1, σ1>0), funded by short sale of another (less) risky asset (r2, σ2>0) with σ1>σ2 => x1=-x2=x with -1≤ρ12≤1 Portfolio return = x(r1-r2) Portfolio variance = x2σ12+(-x)2σ22+ 2x(-x) σ12 = x2(σ12+σ22)- 2x2σ12 = x2(σ12+σ22- 2σ12) and remember that σ12+σ22- 2σ12>0 1st observation: portfolio risk is increasing in x, with a minimum at x=0. Markowitz: portfolio risk is independent of portfolio size. 111 Prof. Dr. Roland Kirstein Economics of Business and Law Faculty of Economics and Management http://econbizlaw.de ZIS, portfolio variance for ρ12=+1: x2(σ12+σ22-2σ1σ2) = x2(σ1-σ2)2 => σP = x (σ1-σ2) ρ12=0: x2(σ12+σ22) ρ12=-1: x2(σ12+σ22+2σ1σ2) = x2(σ1+σ2)2 => σP = x (σ1+σ2) => σP = x√(σ12+σ22) 2nd observation: for x>0, portfolio risk is decreasing in ρ12 Markowitz: risk is increasing in ρ12 If investment into one risky asset is financed by short selling another risky asset, it curbs the portfolio risk if both assets’ returns are highly correlated (perfect correlation would fully eliminate risk if σ1=σ2). 3rd observation: for some x>0 (1 is suff./not necc.), the portfolio SD always exceeds the lower individual SD => no diversification effect. 112 Prof. Dr. Roland Kirstein Economics of Business and Law Faculty of Economics and Management http://econbizlaw.de Sharpe ratio for ZIS with two risky assets: (r1-r2)/(σ1-σ2) Efficient r-σ-combinations: rP= x(r1-r2) => dr/dx=r1-r2 σP= x√(σ12+σ22- 2σ12) => dσ/dx=√(σ12+σ22- 2σ12) dr/dσ >0 <=> r1>r2 (leverage effect) If (as) r1> r2, σ12, σ22, and σ12 are exogenously given, the efficient frontier in the ZIS model is a straight line through the origin with positive slope: drP/dσP = [drP/dx]/[ dσP/dx] = (r1-r2)/ √(σ12+σ22- 2σ12) 113 Prof. Dr. Roland Kirstein Economics of Business and Law Faculty of Economics and Management http://econbizlaw.de Efficient frontier in the Markowitz model (x2=1-x1) • two risky assets,perfect correlation: connecting line • two risky assets, imperfect correlation: bulked line • two risky assets, perf. neg. corr.: triangle • riskless asset and risky asset/portfolio: line through riskless asset; slope=Sharpe ratio of risky asset. =>with low correlation, diversification possible =>adding riskless asset increases feasible set beyond Markowitz curve Efficient frontier in Zero-Investment Strategy (x2=-x1) • One risky, one riskless asset: line through origin (x=0), slope equals Sharpe ratio of risky asset. • Two risky assets: line through origin, slope (r1-r2)/ √(σ12+σ22- 2σ12) => no diversification; riskless fund increases slope. 114 Prof. Dr. Roland Kirstein Economics of Business and an Law Faculty of Economics and Management http://econbizlaw.de Portfolio Choice + Risk Aversion 115 Prof. Dr. Roland Kirstein Economics of Business and Law Faculty of Economics and Management http://econbizlaw.de µ-σ-utility If a decision-maker requests some compensation in terms of expected return for accepting higher risk (i.e., a positive risk premium), she can be described as a risk-averse µ-σ-utility maximizer; her indifference curves (IC) are upwards sloping. Example: U(µ, σ)= µ-cσ2, c>0, then ∂U/∂µ=1>0>∂U/∂σ=-2cσ and ∂2U/∂µ2=0, ∂2U/∂σ2=-2<0, ∂2U/∂µ∂σ=0 On an IC, the utility is constant =>dU=0 116 Prof. Dr. Roland Kirstein Economics of Business and Law Faculty of Economics and Management http://econbizlaw.de Total differential of U(µ, σ): dU=dµ∂U/∂µ+dσ∂U/∂σ = 0 <=> dµ/dσ = - [∂U/∂σ] / [∂U/∂µ] or: slope of IC = MRS between µ and σ Inspection of second derivative of TD: positive => in the µ-σ-diagram, the IC for U(µ, σ)= µ-cσ2 are convex/upwards sloping (In a µ-σ2-diagram, the IC for U(µ, σ)= µ-cσ2 are linear/upwards sloping) 117