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)
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