The document provides steps for determining the efficient frontier using the single-index model. It outlines calculating the excess return to beta ratio for stocks and ranking them, setting a cut-off rate to include higher-ranking stocks in the optimal portfolio, and constructing the portfolio to maximize return for a given level of risk. An example is shown for 10 stocks, where the excess return to beta is calculated, the cut-off rate is determined as 5.45, and the optimal portfolio composition is 23.5% in stock 1, 24.6% in stock 2, etc., with an expected variance return.

1. Ivan/Simple Tech. 1
SIMPLE TECHNIQUES FOR DETERMINING
THE EFFICIENT FRONTIER
 ASSUMPTION
The single-index model is accepted as the criteria
 THE STEPS
1. Find excess return to beta and rank from highest to lowest:
2. Set the cut-off rate, all stocks with higher ratio will be included,
and otherwise excluded:
3. The optimum portfolio consists of all stocks for which:
4. Constructing the optimal portfolio
Xi = fraction to invest in each security
Zi = proportional of that fraction
i
R
R F
i








 i
1
j ej
2
j
2
m
2
i
1
j ej
2
j
F
i
m
2
i
)
(σ
)
(β
(
σ
1
σ
)β
R
R
(
σ
C














*
C
β
R
R
σ
β
Z
:
where
Z
Z
X
i
F
i
ei
2
i
i
N
1
j
j
i
i
*
C
βi
R
R F
i



2. Ivan/Simple Tech. 2
 SIMPLE TECHNIQUE: AN EXAMPLE
Table 1. Assume RF=5%
1
Stock No.
2
Mean Return
Ri
3
Excess Return
Ri - RF
4
Beta
i
5
Unsystematic
Risk
2
ei
6
Excess Return
to Beta
(Ri - RF)/i
1 15 10 1.0 50 10
2 17 12 1.5 40 8
3 12 7 1.0 20 7
4 17 12 2.0 10 6
5 11 6 1.0 40 6
6 11 6 1.5 30 4
7 11 6 2.0 40 3
8 7 2 0.8 16 2.5
9 7 2 1.0 20 2
10 5.6 0.6 0.6 6 1
Table 2. Determining Cut-off rate with 2
m = 10
1
Stock
No.
2
(Ri – RF)
i
3
[(Ri – RF)i]
2
ei
4
2
i/2
ei
5
[(Rj – RF)j]
2
ej
6
[2
j/2
ej]
7
Ci
1 10 2/10 2/100 2/10 2/100 1.67
2 8 4.5/10 5.625/100 6.5/10 7.625/100 3.69
3 7 3.5/10 5/100 10/10 12.625/100 4.42
4 6 24/10 40/100 34/10 52.625/100 5.43
5 6 1.5/10 2.5/100 35.5/10 55.125/100 5.45
6 4 3/10 7.5/100 38.5/10 62.625/100 5.30
7 3 3/10 10/100 41./10 72.625/100 5.02
8 2.5 1/10 4/100 42.5/10 76.625/100 4.91
9 2.0 1/10 5/100 43.5/10 81.625/100 4.75
10 1.0 0.6/10 6/100 44.1/10 87.625/100 4.52
Constructing Optimal Portfolio:
Z1 = (2/100)(10 – 5.45) = 0,091
Z2 = (3.75/100)(8 – 5.45) = 0.095625
Z3 = (5/100)(7 – 5.45) = 0.0775
Z4 = (20/100)(6 – 5.45) = 0.110
Z5 = (2.5/100)(6 – 5.45) = 0.0101375
Zi = 0.387875
Dividing each Zi by Zi, thus, invest 23.5% in 1, 24.6% in 2, 2.20% in 3, 28.4%
in 4, and 3.5% in 5. Portfolio and variance return?
---…why so sad … Alicia Key---


3. Ivan/Simple Tech. 3
Notes From Table 2:
1. Column 3:
Example for stock i= 1
[(Ri – RF)i]/2
ej = [(15-5)1]/50 = 2/10
2. Column 5 the cumulative total of column 3.
3. Column 4, for i = 1
2
1/2
e1 = (12
)/50 = 2/100.
4. Column 6 the cumulative total of column 4.
5. Finding Ci, remember that 2
m = 10






 i
1
j ej
2
j
2
m
2
i
1
j ej
2
j
F
i
m
2
i
)
(σ
)
(β
(
σ
1
σ
)β
R
R
(
σ
C
67
.
1
100
2
10
1
10
2
10
(
1
(
σ
C
m
2
1 

















6)
column
5)
column
m
2
--- 99 by Toto ---