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Chapter 11
Performance-Measure
Approaches for selecting
Optimum Portfolios
By
Cheng Few Lee
Joseph Finnerty
John Lee
Alice C Lee
Donald Wort
Chapter Outline
•
11.1 SHARPE PERFORMANCE-MEASURE APPROACH WITH SHORT
SALES ALLOWED
•
11.2 TREYNOR-MEASURE APPROACH WITH SHORT SALES ALLOWED
•
11.3 TREYNOR-MEASURE APPROACH WITH SHORT SALES NOT
ALLOWED
•
11.4 IMPACT OF SHORT SALES ON OPTIMAL-WEIGHT DETERMINATION
•
11.5 ECONOMIC RATIONALE OF THE TREYNOR PERFORMANCEMEASURE METHOD
•
2
11.6 SUMMARY
•
•
•
•
3
This chapter assumes the existence of a risk-free borrowing and lending rate
and advances one step further to simplify the calculation of the optimal
weights of a portfolio and the efficient frontier.
First discussed are Lintner’s (1965) and Elton et al.’s (1976) Sharpe
performance-measure approaches for determining the efficient frontier with
short sales allowed.
This is followed by a discussion of the Treynor performance-measure
approach for determining the efficient frontier with short sales allowed.
The Treynor-measure approach is then analyzed for determining the efficient
frontier with short sales not allowed.
11.1 Sharpe Performance-Measure
Approach With Short Sales Allowed
•
Following previous chapters, the objective function for portfolio selection can
be expressed:
12

n
n


 n



Max L   Wi Ri  1    w i w j Cov  Ri , R j     p   2   Wi  1
i 1
 i 1




  j 1 i 1

n
where:
Ri = average rates of return for security i;
Wi (orW j ) = the optimal weight for ith (or jth) security;
Cov  Ri , R j  = the covariance between
Ri and R j ;
 p = the standard deviation of a portfolio; and
1 , 2 = Lagrangian multipliers.
•
4
Equation (11.1) maximizes the expected rates of return given targeted
standard deviation.
(11.1)
•
If a constant risk-free borrowing and lending rate R f is subtracted from EQ(11.1) :
12

n
n


 n




Max L   Wi Ri  R f   1    WW
  p   2  Wi  1
i
j Cov  Ri , R j  
i 1
 i 1




  i 1 j 1

n
(11.2a)
•
•
5
Equations (11.1) and (11.2a), both formulated as a constrained maximization
, n .
problem, can be used to obtain optimum portfolio weights Wi  i  1, 2,
Since R f is a constant, the optimum weights obtained from Equation (11.1) will be
equal to those obtained for Equation (11.2a).
•
Previous chapters used the methodology of Lagrangian multipliers; it can be
shown that Equation (11.2a) can be replaced by a nonconstrained maximization
method as follows.
n
•
Incorporating the constant
W
i 1
i
 1 into the objective function by substituting
n
 n

R f  1 R f   Wi  R f  Wi R f
i 1
 i 1 
into Equation (11.2a):
12
 n n


Max L   Wi  Ri  R f   1   WW
 p 
i
j Cov  Ri , R j  
 i 1 j 1

i 1

n
•
6
A two-Lagrangian multiplier problem has been reduced to a one-Lagrangian
problem as indicated in Equation (11.2b).
(11.2b)
 w  R  R  and
n
•
By using a special property of the relationship between
i 1
12
 n n

WW
Cov
R
,
R


  i j
i
j  the
 i 1 j 1

i
i
f
constrained optimization of Equation (18.2A) can be
reduced to an unconstrained optimization problem, as indicated in Equation (11.3).
W  R  R 
n
Max L 

i 1
i
i
f
12
n
n
 n

2 2
  Wi  i   WW
i
j ij 
i 1 j 1
 i 1

i  j 
(11.3)

Where  ij  Cov Ri , R j .
•
Alternatively, the objective function of Equation (11.3) can be developed as
follows.
•
This ratio L is equal to excess average rates of return for the ith portfolio divided
by the standard deviation of the ith portfolio.
•
7
This is a Sharpe performance measure.
Figure 11.1 Linear Efficient Frontier
•
•
•
8
Following Sharpe (1964) and Lintner (1965), if there is a risk-free lending and
borrowing rate  R f  and short sales are allowed, then the efficient frontier
(efficient set) will be linear, as discussed in previous chapters.
In terms of return  R p  standard-deviation  p space, this linear efficient frontier
is indicated as line R f E in Figure 11.1.
AEC represents a feasible investment opportunity in terms of existing securities to
be included in the portfolio when there is no risk-free lending and borrowing rate.
•
•
•
If there is a risk-free lending and borrowing rate, then the efficient frontier
becomes R f E.
An infinite number of linear lines represent the combination of a riskless asset
and risky portfolio, such as R f A, R f B, and R f E .
It is obvious that line has the highest slope, as represented by

Rp  R f
(11.4)
p
n
•
in which R p   Wi Ri , R f , and  pare defined as in Equation (11.2a).
i 1
•
Thus the efficient set is obtained by maximizing
n
•
By imposing the constraint
W
i 1
 
9
i
Rp  R f
p
.
 1 , Equation (11.4) is expressed:
 n

   Wi  1
 i 1

(11.5a)
•
By using the procedure of deriving Equation (11.2b), Equation (11.5a) becomes
W  R  R 
n
 
i
i 1
i
n
f
n
W    
2
i 1
i
2
i
i  j 
n
i 1 i 1
(11.5b)
ij
•
Following the maximization procedure discussed earlier in previous chapters, it
is clear that there are n unknowns to be solved in either Equation (11.3) or
Equation (11.5b).
•
Therefore, calculus must be employed to compute n first-order conditions to
formulate a system of n simultaneous equations:
10
1
dL
0
dW1
2
dL
0
dW2
n
dL
0
dWn
•
From Appendix 11A, the n simultaneous equations used to solve H i are
where Hi  kWi
R1  R f  H1 12  H 2 12  H 3 13 
 H n 1n
R2  R f  H1 12  H 2 22  H 3 23 
 H n 2 n
Rn  R f  H1 1n  H 2 2 n  H 3 3n 
 H n n2
i  1, 2,
, n  ; and
k 
•
11
(11.6a)
Rp  R f
 p2
The H S is proportional to the optimum portfolio weight Wi  i  1, 2,
constant factor K.
, n  by a
•
•
To determine the optimum weight Wi , H i is first solved from the set of equations
indicated in Equation (11.6).
Having d so the Hi must be called to calculate Wi, as indicated in Equation (11.7).
Wi 
Hi
n
H
i 1
•
(11.7)
i
If there are only three securities, then Equation (11.6a) reduces to:
R1  R f  H1 12  H 2 12  H 3 13
R2  R f  H1 12  H 2 22  H 3 23
R3  R f  H1 13  H 2 23  H 3 32
12
(11.6b)
Sample Problem 11.1
Let
R1  15%
R2  12% R3  20%
 1  8%
 2  7%
 3  9%
r12  0.5
r13  0.4
r23  0.2
R f  8%
Substituting this information into Equation (l1.6b):
15  8  64 H1   0.5 8  7  H 2   0.4 8  9  H 3
12  8   0.5  8  7  H1  49 H 2   0.2  7  9  H 3
20  8   0.4 8  9  H1   0.2  7  9  H 2  81H 3
Simplifying:
7  64 H1  28H 2  28.8H 3
4  28H1  49 H 2  12.6 H 3
12  28.8H1  12.6 H 2  81H 3
(11.6c)
Using Cramer’s rule, H1, H2, H3 and can be solved for as follows:
H1 
7
28
28.8
4
49
12.6
12 12.6
64
28
81
28.8
28
12.6
49
28.8 12.6

81
 4233.6  1451.5  27783  1111.3  9072  16934.4 
10160.6  10160.6  254016   10160.6  63504  40642.6 
64
28
33648.1  27.1177 6350.4


 3.97%
160030
160030
64
28
H2 
7
4
81
28.8
28
12.6
28.8 12.6


14
28.8
12.6
28.8 12
64
28
49
H3 
28
49
28.8 12.6 81
64
28 28.8
28
49
28.8 12.6

81
 2540.2  9676.6  20736    9676.8  15876  3317.8
160030
32952.8  28870.6
 2.55%
160030

28.8
12.6
12.6
81
 3225.6  2469.6  37632    3225.6  9480  9878.4 
160030
20815.2
 13.01%
160030
•
Using Equation (11.7), W1, W2, W3 are obtained:
W1 
H1

3
H
i 1
3.97
3.97

3.97  2.55  13.01 18.53
i
 20.33%
W2 
H2

3
H
i 1
2.55
19.53
i
 13.06%
W3 
H3

3
H
i 1
i
 66.61%
15
13.01
19.53
•
Here Rp and  p2 can be calculated by employing these weights:
Rp  15  0.2033  12  0.1306    20  0.0061
 3.049  1.5672  13.322
 17.9382%
3
3
3
   Wi    WW
i
j rij i j
2
p
2
i 1
2
i
i j
i 1 j 1
  0.2033  64    0.1306   49    0.6661 81
2
2
2
 2  0.2033 0.1306  0.5 8  7 
 2  0.2033 0.6661 0.4 8  9 
 2  0.2  0.1306  0.6661 7  9 
 2.645  0.836  35.939  1.487  7.8  2.192
 50.899%
16
Figure 11.2 Efficient Frontier for Example 11.1
17
•
The efficient frontier for this example is shown in Figure 11.2.
•
Here A represents an efficient portfolio with Rp  17.94% and  p2  50.90% .
•
In addition to Cramer’s rule method used in this example, we can also use the
matrix inversion method to solve this question.
•
Equation (11.6b) can be written in the matrix form as following:
•
Then Equation (11.6c) can be written as following:
H1
•
By using inverse matrix method in Appendix 10C of Chapter 10, we can obtain
H1 , H 2 ,and H3 .
18
11.2 Treynor-Measure Approach With
Short Sales Allowed
•Using single index market model discussed in Chapter 10, Elton et al (1976)
12
define:
n
 n 2 2 2 n n

2
2 2
 p    Wi i  m   WW
i
j  i  j m   Wi  i 
i 1 j 1
i 1
 i 1

ji
•Substituting of this value of  p into Equation (11.3)
W  R
n
L
i 1
i
i
 Rf

12

n
n
n
n


i 1
i 1
j 1
i 1

2
2
2
 p    Wi 2 i2 m2    WW
i
j  i  j m   Wi  i 
19
ji
(11.8)
•
In order to find the set of Wi s that maximize L, take the partial derivatives of
above equation with respect to each Wi and some manipulation.
•
Then we can obtain
Wi 
Hi
n
H
i 1
(11.9)
i
n

Rj  Rf

2
 m
R

R

 2i
i
f 

j 1
Hi 

2
i 2
 i
2
 1   m  2j

j 1  ej

where
 
2
  i 
  2 
  i 


(11.10)
•
The procedure of deriving Equations (11.9) and (11.10) can be found in Appendix
11B.
•
The H i s must be calculated for all the stocks in the portfolio.
•
By the Treynor measure approach, if H i is a positive value, this indicates the
stock will be held long, whereas a negative value indicates that the stock should be
sold short.
20
•
One method follows the standard definition of short sales, which presumes that a
short sale of stock is a source of funds to the investor; it is called the standard
method of short sales.
•
This standard scaling method is indicated in Equation (11.10). In Equation
(11.10), H i can be positive or negative.
•
This scaling factor includes a definition of short sales and the constraint:
n
W
i 1
•
21
i
1
A second method (Lintner’s (1965) method of short sales) assumes that the
proceeds of short sales are not available to the investor and that the investor
must put up an amount of funds equal to the proceeds of the short sale.
•
The additional amount of funds serves as collateral to protect against adverse
price movements.
•
Under these assumptions, the constraints on the Wi s can be expressed as
n
W
i 1
•
i
1
And the scaling factor is expressed as
Wi 
Hi
n
H
i 1
22
(11.11)
i
Sample Problem 11.2
•
•
•
The following example shows the differences in security weights in the
optimal portfolio due to the differing short-sale assumptions.
Data associated with regressions of the single-index model are presented in
the Table 11.1.
The mean return, R , the excess return Ri  R f , the beta coefficient i , and
2
the variance of the error term  i are presented from columns 2 through 5.
Table 11.1 Data Associated with Regressions of the Single-Index Model
23
•
•
From the information in Table 11-1, using Equations (11.9) and (11.10).
Hi (i  1, 2, ,5) can be calculated as
 1   15  5 
H1    
  3.067  0.2311
30
1
 

 2   13  5 
H2   
  3.067  0.0373
 50   2 
 1.43   10  5 
H3  

  3.067  0.0307
20
1.43



 1.33   9  5 
H4  

  3.067  0.0079
 10   1.33 
 1  7  5 
H5    
  3.067  0.0356
30
1
 

•
W1 
0.2311
 0.9041
0.2556
W2 
0.0373
 0.1459
0.2556
W3 
0.0307
 0.1201
0.2556
W4 
0.0079
 0.0309
0.2556
W5 
0.0356
 0.1393
0.2556
If this same example is scaled using the standard
5
definition of short sales (
 H ) , which provides
i 1
funds to the investor:
i
5
H
i 1
i
 0.2556
•
According to Lintner’s method:
5
H
i 1
•
•
•
i
 0.3426
W1 
0.2311
 0.6745
0.3426
Now to scale the H i values into an optimum portfolio,
0.0373
W2 
 0.1089
0.3426
apply Equation (11.11):
W3 
0.0307
 0.0896
0.3426
W4 
0.0079
 0.0231
0.3426
W5 
0.0356
 0.1039
0.3426
The difference between Lintner’s method and the standard method are due to the
different definitions of short selling discussed earlier.
The standard method assumes that the investor has the proceeds of the short sale,
while Lintner’s method assumes that the short seller does not receive the
proceeds and must provide funds as collateral.
11.3 Treynor-Measure Approach With
Short Sales Not Allowed
•
Equation (11.10) can be modified to

i  Ri  R f
*
H i  2 
 C 
 i  i

in which
 R  R  is the Treynor performance measure and C*can be
i
f
i
defined as

C 
*
26
(11.12)
i
2
m

R  R 
i
f
2

j 1
j
i 2
1   m2  2j
j 1   j
i
(11.13)
•
Elton et al. (1976) also derive a Treynor-measure approach with short sales not
allowed.
•
From Appendix 11C, Equation (11.13) should be modified to

i  Ri  R f
*
H i  2 
 C   i
 i  i

(11.14)
where
Hi  0, i  0, and Hi i  0
then

C 
*
d
2
m

j 1
Rj  Rf

2
j
j
2

1   m2  2j
j 1   j
d
where d is a set which contains all stocks with positive H i
27
(11.15)
If all securities have positivei s, the following three step procedure from Elton et
al. can be used to choose securities to be included in the optimum portfolio.
1) Use the Treynor performance measure( Ri  R f ) / i to rank the securities in
descending order.
2) Use equation (11.16) to calculate C* for first ith securities.
3) Include i securities for which ( Ri  R f ) / iis larger than Ci*. Then C*is equal to Ci.*
4) Use Equation (11.5) to calculate optimum weights for i securities.
Sample Problem 18.3
•
The Center for Research in Security Prices tape was the source of five years of
monthly return data, from January 2006 through December 2010, for the 30
stocks in the Dow-Jones Industrial Averages (DJIAs).
•
The value-weighted average of the S&P 500 index was used as the market while
three-month Treasury-bill rates were used as the risk-free rate.
•
The single-index model was used with an ordinary least-squares regression
procedure to determine each stock’s beta.
•
All data are listed in the worksheet, which lists the companies in descending
order of Treynor performance measure.
29
•
To calculate the Ci* as defined in Equation (11.16)
R
 
i
•
j 1
2
j
.j  ,
j  Rf  
2
j
2
2
2


R

R


,
and





j
f
j
j 
j
j .

j 1
i
2 j  are calculated and presented in the worksheet.
i
•


i

Substituting   0.00207,   R j  R f  j  , and   2j 2 j


j 1
j 1
2
m
2
j

*
into Equation (11.16) produces Ci for every firm as listed in the last column in
the worksheet.
30
Worksheet for DJIAs (pg.413)
Table 11.2 Positive Optimum Weight for Three Securities
•
Using company VZ as an example:
*
CVZ

•
•
32
 0.00294  7.35  0.0109
1   0.00294  337.42 
*
From Ci of the worksheet, it is clear that there are three securities that should
be included in the portfolio.
The estimated i  2i ,  Ri  R f
in Table 11.2.

i , and Ci* of these three securities are listed
•
Substituting this information into Equation (11.15) produces Hi for all three
securities.
•
Using security MCD as an example:
H MCD  (334.4223)(0.0318  0.0112)  6.8729
•
Using Equation (11.12), the optimum weights can be estimated for all three
securities, as indicated in Table 11.2.
•
In other words, 90.01% of our fund should be invested in security MCD, 4.73%
in security VZ, 5.26% in security KO.
•
Based upon the optimal weights, the average rate for the portfolio Rp is
calculated as 1.65%, as presented in the last column of Table 11.2.
33
11.4 Impact of Short Sales on OptimalWeight Determination
34
•
In both the Markowitz and Sharpe models, the analysis is facilitated by the
presence of short selling.
•
This chapter discusses a method proposed by Elton and Gruber for the
selection of optimal portfolios.
•
Their method involves ranking securities based on their excess return to beta
ratio, and choosing all securities with a ratio greater than some particular cutoff
level C*.
•
It is interesting to note that while the presence of short selling facilitated the
selection of the optimum portfolio in both the Markowitz and Sharpe models, it
complicates the analysis when we use the Elton and Gruber approach.
11.5 Economic Rationale of the Treynor
Performance-Measure Method
Cheung and Kwan (1988) have derived an alternative simple rate of optimal
portfolio selection in terms of the single-index model.
where:
C*
i 
 m i
(18.16)
i   im  i m ;
 im  covariance between Ri and Rm ;
i   Ri  R f
  , the Sharpe performance measure associated
i
with the ith portfolio; and
 i and  m  standard deviation for ith portfolio and market portfolio,
respectively.
•
•
Based upon the single-index model and the risk decomposition discussed
in Chapter 7, the following relationships can be defined:
 im
1.
i 
 i m
2.
 im  i m2
3.
 i2  i2 m2   2i
(11.17)
From Equation (11.17), Cheung and Kwan define i in terms of  i2 ,  m2 , and  i .
i   
2
i
2
m
 
2
i
 2i
1 2
i
in which  i is the nonsystematic risk for the ith portfolio.
• They use both i and i to select securities for an optimum portfolio, and they
conclude that i can be used to replace i in selecting securities for an optimum
portfolio.
• But i information is still needed to calculate the weights for each security.
2
11.6 SUMMARY
•
•
•
Following Elton et al. (1976) and Elton and Gruber (1987) we have discussed the
performance-measure approaches to selecting optimal portfolios.
We have shown that the performance-measure approaches for optimal portfolio
selection are complementary to the Markowitz full variance–covariance method
and the Sharpe index-model method.
These performance-measure approaches are thus worthwhile for students of
finance to study following an investigation of the Markowitz variance–covariance
method and Sharpe’s index approach.