2. 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 PERFORMANCE-
MEASURE METHOD
• 11.6 SUMMARY
2
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.
3
4. • Following previous chapters, the objective function for portfolio selection can
be expressed:
where:
= average rates of return for security i;
= the optimal weight for ith (or jth) security;
= the covariance between and ;
= the standard deviation of a portfolio; and
= Lagrangian multipliers.
• Equation (11.1) maximizes the expected rates of return given targeted
standard deviation.
11.1 Sharpe Performance-Measure
Approach With Short Sales Allowed
1 2
1 2
1 1 1 1
Max w w Cov , 1
n n n n
i i i j i j p i
i j i i
L W R R R W
iR
(or )i jW W
Cov ,i jR R iR jR
p
1 2,
(11.1)
4
5. • If a constant risk-free borrowing and lending rate is subtracted from EQ(11.1) :
• Equations (11.1) and (11.2a), both formulated as a constrained maximization
problem, can be used to obtain optimum portfolio weights .
• Since is a constant, the optimum weights obtained from Equation (11.1) will be
equal to those obtained for Equation (11.2a).
1 2
1 2
1 1 1 1
Max Cov , 1
n n n n
i i f i j i j p i
i i j i
L W R R WW R R W
1, 2, ,iW i n
fR
(11.2a)
fR
5
6. • 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.
• Incorporating the constant into the objective function by substituting
into Equation (11.2a):
• A two-Lagrangian multiplier problem has been reduced to a one-Lagrangian
problem as indicated in Equation (11.2b).
1 1
1
n n
f f i f i f
i i
R R W R W R
1
1
n
i
i
W
1 2
1
1 1 1
Max Cov ,
n n n
i i f i j i j p
i i j
L W R R WW R R
(11.2b)
6
7. • By using a special property of the relationship between and
the constrained optimization of Equation (18.2A) can be
reduced to an unconstrained optimization problem, as indicated in Equation (11.3).
Where .
• 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.
• This is a Sharpe performance measure.
1
1 2
2 2
1 1 1
Max
n
i i f
i
n n n
i i i j ij
i i j
W R R
L i j
W WW
1
n
i i f
i
w R R
1 2
1 1
Cov ,
n n
i j i j
i j
WW R R
(11.3)
Cov ,ij i jR R
7
8. Figure 11.1 Linear Efficient Frontier
• Following Sharpe (1964) and Lintner (1965), if there is a risk-free lending and
borrowing rate and short sales are allowed, then the efficient frontier
(efficient set) will be linear, as discussed in previous chapters.
• In terms of return standard-deviation space, this linear efficient frontier
is indicated as line 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.
fR
pR p
fR E
8
9. • If there is a risk-free lending and borrowing rate, then the efficient frontier
becomes .
• An infinite number of linear lines represent the combination of a riskless asset
and risky portfolio, such as .
• It is obvious that line has the highest slope, as represented by
• in which are defined as in Equation (11.2a).
• Thus the efficient set is obtained by maximizing .
• By imposing the constraint , Equation (11.4) is expressed:
fR E
, , andf f fR A R B R E
p f
p
R R
(11.4)
1
1
n
p f
i
ip
R R
W
1
, , and
n
p i i f p
i
R W R R
1
1
n
i
i
W
(11.5a)
9
10. • By using the procedure of deriving Equation (11.2b), Equation (11.5a) becomes
• 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:
1
2 2
1 1 1
n
i i f
i
n n n
i i ij
i i i
W R R
i j
W
(11.5b)
1
2
1 0
2 0
0
n
dL
dW
dL
dW
dL
n
dW
10
11. • From Appendix 11A, the n simultaneous equations used to solve are
where
• The is proportional to the optimum portfolio weight by a
constant factor K.
2
1 1 1 2 12 3 13 1
2
2 1 12 2 2 3 23 2
2
1 1 2 2 3 3
f n n
f n n
n f n n n n n
R R H H H H
R R H H H H
R R H H H H
1,2, , ; andi iH kW i n
2
p f
p
R R
k
iH
(11.6a)
SH 1,2, ,iW i n
11
12. • To determine the optimum weight 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).
• If there are only three securities, then Equation (11.6a) reduces to:
1
i
i n
i
i
H
W
H
2
1 1 1 2 12 3 13
2
2 1 12 2 2 3 23
2
3 1 13 2 23 3 3
f
f
f
R R H H H
R R H H H
R R H H H
,i iW H
(11.7)
(11.6b)
12
13. Sample Problem 11.1
Let
Substituting this information into Equation (l1.6b):
Simplifying:
1 2 3
1 2 3
12 13 23
15% 12% 20%
8% 7% 9%
0.5 0.4 0.2
8%f
R R R
r r r
R
1 2 3
1 2 3
1 2 3
15 8 64 0.5 8 7 0.4 8 9
12 8 0.5 8 7 49 0.2 7 9
20 8 0.4 8 9 0.2 7 9 81
H H H
H H H
H H H
1 2 3
1 2 3
1 2 3
7 64 28 28.8
4 28 49 12.6
12 28.8 12.6 81
H H H
H H H
H H H
(11.6c)
15. • Using Equation (11.7), W1, W2, W3 are obtained:
1
1 3
1
2
2 3
1
3
3 3
1
3.97 3.97
3.97 2.55 13.01 18.53
20.33%
2.55
19.53
13.06%
13.01
19.53
66.61%
i
i
i
i
i
i
H
W
H
H
W
H
H
W
H
15
16. • Here can be calculated by employing these weights:2
andp pR
15 0.2033 12 0.1306 20 0.0061
3.049 1.5672 13.322
17.9382%
pR
3 3 3
2 2 2
1 1 1
2 2 2
0.2033 64 0.1306 49 0.6661 81
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
p i i i j ij i j
i i j
W WW r i j
2.192
50.899%
16
17. • The efficient frontier for this example is shown in Figure 11.2.
• Here A represents an efficient portfolio with .2
17.94% and 50.90%p pR
Figure 11.2 Efficient Frontier for Example 11.1
17
18. • 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:
• By using inverse matrix method in Appendix 10C of Chapter 10, we can obtain
1H
1 2 3, ,and .H H H
18
19. p
11.2 Treynor-Measure Approach With
Short Sales Allowed
1 2
2 2 2 2 2 2
1 1 1 1
n n n n
p i i m i j i j m i i
i i j i
W WW W
j i
1
1 2
2 2 2 2 2 2
1 1 1 1
(11.8)
n
i i f
i
n n n n
p i i m i j i j m i i
i i j i
W R R
L
W WW W
j i
•Using single index market model discussed in Chapter 10, Elton et al (1976)
define:
•Substituting of this value of into Equation (11.3)
19
20. • In order to find the set of that maximize L, take the partial derivatives of
above equation with respect to each and some manipulation.
• Then we can obtain
where
• The procedure of deriving Equations (11.9) and (11.10) can be found in Appendix
11B.
• The must be calculated for all the stocks in the portfolio.
• By the Treynor measure approach, if is a positive value, this indicates the
stock will be held long, whereas a negative value indicates that the stock should be
sold short.
1
i
i n
i
i
H
W
H
iW s
iW
(11.9)
2
2 2
1
22 2
2
2
1
1
n
j f
m
i f j i i
i i
ji i
m
j ej
R R
R R
H
(11.10)
iH s
iH
20
21. • 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), can be positive or negative.
• This scaling factor includes a definition of short sales and the constraint:
• 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.
iH
1
1
n
i
i
W
21
22. • The additional amount of funds serves as collateral to protect against adverse
price movements.
• Under these assumptions, the constraints on the can be expressed as
• And the scaling factor is expressed as
iW s
1
1
n
i
i
W
1
i
i n
i
i
H
W
H
(11.11)
22
23. 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, , the excess return , the beta coefficient , and
the variance of the error term are presented from columns 2 through 5.
i fR R i
2
i
R
Table 11.1 Data Associated with Regressions of the Single-Index Model
23
24. • From the information in Table 11-1, using Equations (11.9) and (11.10).
• can be calculated as
• If this same example is scaled using the standard
definition of short sales , which provides
funds to the investor:
( 1,2, ,5)iH i
1
2
3
4
5
1 15 5
3.067 0.2311
30 1
2 13 5
3.067 0.0373
50 2
1.43 10 5
3.067 0.0307
20 1.43
1.33 9 5
3.067 0.0079
10 1.33
1 7 5
3.067 0.0356
30 1
H
H
H
H
H
5
1
0.2556i
i
H
1
2
3
4
5
0.2311
0.9041
0.2556
0.0373
0.1459
0.2556
0.0307
0.1201
0.2556
0.0079
0.0309
0.2556
0.0356
0.1393
0.2556
W
W
W
W
W
5
1
( )i
i
H
25. • According to Lintner’s method:
• Now to scale the values into an optimum portfolio,
apply Equation (11.11):
• 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.
5
1
0.3426i
i
H
1
2
3
4
5
0.2311
0.6745
0.3426
0.0373
0.1089
0.3426
0.0307
0.0896
0.3426
0.0079
0.0231
0.3426
0.0356
0.1039
0.3426
W
W
W
W
W
iH
26. • Equation (11.10) can be modified to
in which is the Treynor performance measure and C*can be
defined as
11.3 Treynor-Measure Approach With
Short Sales Not Allowed
*
2
i fi
i
i i
R R
H C
2
2
1*
2
2
2
1
1
i
i f i
m
j j
i
j
m
j j
R R
C
i f
i
R R
(11.12)
(11.13)
26
27. • 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
where
then
where d is a set which contains all stocks with positive
*
2
i fi
i i
i i
R R
H C
0, 0, and 0i i i iH H
2
2
1*
2
2
2
1
1
d
j f
m j
j j
d
j
m
j j
R R
C
iH
(11.15)
(11.14)
27
28. If all securities have positive , 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 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 is larger than . Then C*is equal to .
4) Use Equation (11.5) to calculate optimum weights for i securities.
*
iC( )/i f iR R
i s
( ) /i f iR R
*
iC
29. • 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.
Sample Problem 18.3
29
30. • To calculate the as defined in Equation (11.16)
.
• are calculated and presented in the worksheet.
• Substituting
into Equation (11.16) produces for every firm as listed in the last column in
the worksheet.
*
iC
2 2 2 2
1
, , and .
i
j f j j j f j j j j
j
R R R R
2 2 2 2
1 1
0.00207, , and
i i
m j f j j j j
j j
R R
*
iC
2 2
1
i
j j
j
30
32. Table 11.2 Positive Optimum Weight for Three Securities
• Using company VZ as an example:
• From of the worksheet, it is clear that there are three securities that should
be included in the portfolio.
• The estimated of these three securities are listed
in Table 11.2.
* 0.00294 7.35
0.0109
1 0.00294 337.42
VZC
2 *
, , andi i i f i iR R C
*
iC
32
33. • Substituting this information into Equation (11.15) produces for all three
securities.
• Using security MCD as an example:
• 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 is
calculated as 1.65%, as presented in the last column of Table 11.2.
(334.4223)(0.0318 0.0112) 6.8729MCDH
iH
pR
33
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.
11.4 Impact of Short Sales on Optimal-
Weight Determination
34
35. Cheung and Kwan (1988) have derived an alternative simple rate of optimal
portfolio selection in terms of the single-index model.
where:
*
(18.16)i
m i
C
;
covariance between and ;
, the Sharpe performance measure associated
with the th portfolio; and
and
i im i m
im i m
i i f i
i
R R
R R
i
standard deviation for ith portfolio and market portfolio,
respectively.
m
11.5 Economic Rationale of the Treynor
Performance-Measure Method
36. • Based upon the single-index model and the risk decomposition discussed
in Chapter 7, the following relationships can be defined:
• From Equation (11.17), Cheung and Kwan define in terms of .
in which is the nonsystematic risk for the ith portfolio.
• They use both to select securities for an optimum portfolio, and they
conclude that can be used to replace in selecting securities for an optimum
portfolio.
• But information is still needed to calculate the weights for each security.
i
2
2 2 2 2
1.
2. (11.17)
3.
im
i
i m
im i m
i i m i
2
2 2 2
2
1 i
i i m i
i
2
i
andi i
ii
2 2
, , andi m i
i
37. 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.
