The document summarizes the efficient set and how to construct efficient portfolios from two risky assets with different combinations of expected returns and standard deviations. It examines four cases: 1) Perfect positive correlation, where there is no reduction in risk. Minimum risk is achieved at 100% allocation to the lower risk asset. 2) Perfect negative correlation, where minimum risk is achieved at a 1/3 allocation to the higher risk asset. 3) No correlation, where minimum risk is achieved at a 20% allocation to the higher risk asset. 4) Intermediate correlation, where minimum risk allocation is between the no and perfect correlation cases. The document also introduces the risk-free asset and shows the

1. DELLINEATING EFFICIENT PORTFOLIOS
THE EFFICIENT SET
The portfolios that have the same return at the lower risks, or the same risk at
higher returns.
COMBINATION OF TWO RISKY ASSETS: SHORT SALES NOT ALLOWED
1. Expected Return of Two Assets:
)
R
X
(
)
R
X
(
R B
B
A
A
p +
=
Since:
XA + XB = 1 → X
B
BB = 1- XA
Thus:
B
A
A
A
p R
)
X
1
(
)
R
X
(
R −
+
=
2. Standard Deviation
σ2
p = XA
2
σ2
A + XB
2
σ2
B + 2XAXBσ
B
AB
σp = (XA
2
σ2
A+ X2
Bσ2
B + 2XAXBσAB)1/2
Recall that:
B
A
AB
AB
σ
σ
σ
=
ρ
Thus: σAB = ρABσAσB
The equation becomes:
σp = [X2
Aσ2
A+ (1 – XA)2
σ2
B + 2XA(1 – XA)ρABσAσB]1/2
Efficient port/Ivan 1

2. Example:
Expected Return Standard Deviation
Colonel Motor (C) 14% 6%
Separated Edison (S) 8% 3%
Case 1: Perfect Positive Correlation (ρ = +1)
• Standard Deviation
σp = [X2
Cσ2
C + (1-XC)2
σ2
S + 2XC(1-XC)σCσS]1/2
The square of the bracket has the form:
X2
+ 2XY + Y2
→ (X + Y)2
Can be written as: [XCσC + (1-XC)σS]2
Since σ = √σ2
, so that:
σp = XCσC + (1-XC)σS
σp = 6XC + 3(1-XC)= 3 + XC
• Expected return
S
C
C
C
p R
)
X
1
(
)
R
X
(
R −
+
=
C
C
C
p X
6
8
)
X
1
(
8
)
X
14
(
R +
=
−
+
=
Table for ρ = +1
XC 0 0.2 0.4 0.5 0.6 0.8 1.0
Rp 8.0 9.2 10.4 11.0 11.6 12.8 14.0
σp 3.0 3.6 4.2 4.5 4.8 5.4 6.0
• Conclusion
At ρ = +1, there is no reduction in risk for portfolio
Efficient port/Ivan 2

3. Case 2: Perfect Negative Correlation (ρ = -1)
• Standard Deviation
σp = [X2
Cσ2
C + (1-XC)2
σ2
S + 2XC(1-XC)ρCSσCσS]1/2
σp = [X2
Cσ2
C + (1-XC) 2
σ2
S - 2XC(1-XC)σCσS]1/2
The square of the bracket is equivalent:
[XCσC - (1-XC)σS]2
or [-XCσC + (1-XC)σS]2
Thus, σp is either:
σp = XCσC - (1-XC)σS …… (a) or
σp = -XCσC + (1-XC)σS ……(b)
Since σ = √σ2
and square roots of a negative number is imaginary, the
equation is valid when the right hand side is positive:
σp = -XCσC + (1-XC)σS
σp = 6XC - 3(1-XC) or
σp = -6XC + 3(1-XC)
Table for ρ = -1
XC 0 0.2 0.4 0.6 0.8 1.0
σp 3.0 1.2 0.6 2.4 4.2 6.0
• Minimum Risk
Set equation (a) or (b) equal to zero
-XCσC + (1-XC)σS = 0
-XCσC - XCσS + σS = 0
XCσC + XCσS = σS
XC(σC + σS ) = σS → XC = σS/(σC + σS )
Minimum risk: XC = 3/(6 + 3) = 1/3
Efficient port/Ivan 3

4. Case 3: No Relationship (ρ = 0)
σp = [X2
Cσ2
C + (1-XC)2
σ2
S]1/2
σp = [62
X2
C + (32
)(1-XC) 2
]1/2
= [45X2
C - 18XC + 9]1/2
Table at ρ = 0
XC 0 0.2 0.4 0.6 0.8 1.0
σp 3.0 2.68 3.00 3.79 4.84 6.0
• Minimum Risk
XC = σ2
S/(σ2
C + σ2
S )
XC = 9/(9 + 36)= 1/5 = 0.20
Case 4 : Intermediate Risk (ρ = 0.5)
• Standard Deviation
σp = [X2
Cσ2
C+ (1-XC)2
σ2
S+2XC(1-XC)ρCSσCσS]1/2
σp = [(62
)X2
C + (32
)(1-XC)2
+ 2XC(1-XC)(3)(6)(1/2)]1/2
= [27X2
C + 9]1/2
Table at ρ = 0.5
XC 0 0.2 0.4 0.6 0.8 1.0
σp 3.0 3.17 3.65 4.33 5.13 6.0
• Minimum Risk
CS
S
C
S
2
C
2
CS
S
C
S
2
C
2
X
ρ
σ
σ
−
σ
+
σ
ρ
σ
σ
−
σ
=
Efficient port/Ivan 4

5. INTRODUCING THE RISK-FREE RATE (RF)
• Characteristic
Certain return and zero standard deviation
• Expected return
A
F
C R
X
R
)
X
1
(
R +
−
=
• Standard Deviation
σC = [(1-X)2
σ2
F + X2
σ2
A+ 2X(1-X)ρFAσAσF]1/2
Since σF = 0, so that:
σC = (X 2
σ2
A)1/2
= XσA
Solving expression for X:
X = σC/σA
Substituting for expected return:
C
C
C
C
C
)
R
R
(
R
R
R
R
)
1
(
R
A
F
A
F
A
A
F
A
σ
σ
−
=
σ
σ
+
σ
σ
−
=
The equation is the equation of straight line.
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Efficient port/Ivan 5