Appendix 11A:
Derivation of
Equation (11.6a)
(11.1)
The objective function L as defined in Equation (18.3) can be rewritten:



2 2
Max L  Wi  Ri  R f  Wi  i  WW
i j ij 
i 1 j 1
 i 1
  i 1

n
n
n
n
1 2
Then, following the product and chain rule, we have:
n
n

dL
d  n
 n
2

W
R

R
W


W







i
i
f 
i i
j ij 
dWi dWi  i 1
i 1 j 1
  i 1

 n 2 2 n n

   Wi  i   WW

i
j ij 
i 1 j 1
 i 1

1 2
1 2
d  n


Wi  Ri  R f  


dWi  i 1

n
n

 n
 d  n
2
   Wi  Ri  R f  
W


W

  i i  j ij 
i 1 j 1
 i 1
 dWi  i 1

n
n
 n

   Wi 2 i2   WW
i
j ij 
i 1 j 1
 i 1

1 2
(11A.1)
1 2
 Ri  R f 
n
n

 n
 n
 12   Wi  Ri  R f     Wi 2 i2   WW
i
j ij 
i 1 j 1
 i 1
  i 1

n


2
  2Wi i  2 W j ij 
j 1


0
 i  1, 2, , n 
i j
3 2
i j
1 2
n
n
 n

2
2
,
W


WW

 i

i
i
j
ij  ,
i 1 j 1
 i 1

Multiplying Equation (11A.1) by
and
rearranging yields:

n

Wi  Ri  R f 


Ri  R f   n i 1
n
n

2 2
Wi  i  WW
i
j ij

j 1 j 1
 i 1
i j




n

2
  Wi i  W j ij 

j 1




Defining
W  R
n
k 
i 1
n
W
i 1
i
i
i
 Rf
n
n
i 1
j 1

 i2    WiW j ij
2
i j
(11A.2)
i j
Yields
n


k  Wi i2  Wi ij    Ri  R f   0
i 1


ji
Therefore:
dL
  (kW1 1i  kW2 2i 
dWi
0
 kWi i2 
 kWn ni )  ( Ri  R f )
Define Hi=kWi, where the Wi are the fractions to invest in each security and
the Hi are proportional to this fraction. Substituting Hi for Wi:
Ri  R f  Hi1i  H 2 2i 
 Hi i2 
There is one equation like this for each value of i.
R1  R f  H1 12  H 2 12 
 H n 1n
R2  R f  H1 21  H 2 22 
 H n 2 n
Rn  R f  H1 n1  H 2 n 2 
 H n n2
This is Equation (11.6a) in the text.
 Hn ni
Appendix 11B:
Derivation of
Equation 11.10
(11.2)
Following the optimization procedure for deriving Equation (18A.2) in
Appendix 18A:
n

dL
  Ri  R f   i 1
dWi
0
Wi  Ri  R f
 p2

2
2 
W




W



W

 i
i
j j m
i i 
j 1


n
2
i
2
m
ji
Let H i   Rp  R f   p2  Wi and, solving for any H i ,


n
Hi 
Ri  R f
 2i

i m2  H j  j
j 1
2
i

(11B.1)
Multiplying both sides of the equation by i:
H i i 
 Ri  R f  i
 2i
i

n
2
m
H 
j 1
2
i
j
2
j
（11B.2）

Adding together the n equation of this form yields:
n
n
H 
j 1
j
j


R
j
 Rf   j
（11B.3）
 2i
j 1
 j2
1   2
j 1  i
n
2
i
By substituting Equation (11B.3) into Equation (11B.1), Equation (11.10) is
obtained.
n

 Rj  Rf
2
 m
Ri  R f  

 2i
j 1
Hi 

i 2
 2i
 1   m2  2j

j 1  ej

 
2
  i 
  2 
  i 


Appendix 11C:
Derivation of
Equation (11.15)
(11.3)
This appendix discusses the use of performance measure to examine the optimal
portfolio with short sales not allowed. Therefore, Equation (11B.1) from
Appendix 11B must be modified
Hi 
Ri  R f

2
i
i m2
 2
 i
n
 H
j 1
j
j
 i
(11C.1)
where Hi  0, i  0 and i H i  0 for all i.
The justification of this equation can be found in Elton et al. (1976). Assuming all
stocks that would be in an optimal portfolio (called d) can be found, and then
arranging these stocks as i=1, 2, ….. , d, for the subpopulation of stocks that
make up the optimal portfolio:
Hi 
Ri  R f
 2i
i m2
 2
 i
d
 H j  j and i  0
j 1
(11C.2)
Multiplying both sides by  j , summing over all stocks in d, and rearranging yields:
 Rj  Rf 
j



2


d
 j 
j 1 
H j j 

d 
2
j 1
1   m  2j
d
j 1
 j
Notice since the set d contains all stocks with positive Hi:
n
d
H   H 
j 1
j
j
j 1
j
j
(11C.3)
d
And Let:
C *   m2

j 1
Rj  Rf
 2 j
j
2

1   m2  2j
j 1   j
n
(11C.4)
Using (11C.4), the following equation for is obtained after substitution and
rearranging from Equation (11C.1):

i  Ri  R f
*
H i  2 
 C   i
 i  i

(11C.5)
Since i  0, the inclusion of i ; can only increase the value of Hi. Therefore, if Hi is
positive with i  0 can never make it zero and the security should be included, If
Hi<0 when i  0, positive values of i can increase Hi . However, because the
product of i and Hi must equal zero, as indicated in Equation (11C.1), positive values
i
of imply Hi=0. Therefore, any security Hi<0 when i  0 must be rejected.
Therefore, Equation (11.16) in the text can be used to estimate the optimal weight of a
portfolio.
Appendix 11D:
Quadratic
Programming and
Kuhn-Tucker
Conditions
•
The quadratic programming algorithm is based on a technique from advanced
calculus called Kuhn–Tucker conditions.
•
This technique can solve the optimal portfolio with short sales not allowed.
•
The problem can be stated as
Max{Wi ,i 1,2,...n}F (W ) 
•
Subject to
Rp  R f
p
n
Wi  1
i 1
Wi  0 , i  1, 2,...n
where
n
R p  Wi Ri
i 1
12
n n
n

2
2
 p  Wi  i    WiW j ij 
i 1

i 1 j 1


•
To find the maximum value of F(W), there are two case of the optimal weights
W=(W1, W2,…Wn) in Figure 11.D.1
F(W)
F(W)
P
P
P’
Wi
(a)
Wi
(b)
Figure 11.D.1 Value of the Function F(W) as Wi Changes
•
•
If F(W) is a function of Wi in Figure 11.D.1(a), then the maximum value of
dF (W )
F(W) occurs at P, where
at positive optimal weight (Wi > 0).
0
dWi
However, in Figure 11.D.1(b), the maximum feasible value of F(W) occurs at Pʹ
instead of P because of short sell not allowed, then dF (W )  0 at optimal weight
dWi
(Wi = 0).
•
Therefore, in general, we can obtain the optimal weights under conditions
dF (W )
0
dWi
dF (W )
Wi  0
dWi
•
We can rewrite the conditions as five Kuhn–Tucker conditions:
dF (W )
 Ai  0
dWi
Wi Ai  0
Wi  0
Ai  0
n
Wi  1
i 1
•
When maximum feasible value occurs on
to zero.
•
If maximum feasible value occurs on
positive.
dF (W )
0
dWi
dF (W )
0,
dWi
, Ai is positive and Wi is equal
then Ai is equal to zero and Wi is
•
•
In the following part, we will solve the optimal weights under Kuhn–Tucker
conditions by Excel.
dF (W )
dWi
Given an example in Figure 11.D.2: initial weights W = (0.1, 0.2, 0.7),
can be calculated by the Equation (11A.2) in Appendix 11A and the value of the
excess return and covariance matrix as below.
Figure 11.D.2 Solver Function in Excel
•
Then we use “Solver” function in Excel to find the optimal weights W.
•
Set target cell B5 equal to the value of 1 (the fifth Kuhn–Tucker condition), select
the range from B2 to C4 as change cells, and select the first and second Kuhn–
Tucker conditions as the constraints.
Figure 11.D.2 Solver Function in Excel (Continued)
•
For the third and fourth Kuhn–Tucker conditions, press “Options” into Solver
Options, select “Assume Non-Negative” and press “OK.”
•
Then go back to Solver Parameters and press “Solve.”.
Figure 11.D.2 Solver Function in Excel (Continued)
Figure 11.D.3 Optimal Weights by Solver Function
•
If first solve cannot find a solution, use “Solver” function again until the Solver
find a solution as Figure11.D.3.
Figure 11.D.3 Optimal Weights by Solver Function
Appendix 11E:
Portfolio
Optimization with
Short-Selling
Constraints
•
The traditional mean-variance efficient portfolios generate appealing
characteristics.
•
However, it is frequently questioned by financial professionals due to the
propensity of corner solutions in portfolio weights.
•
Therefore, adding portfolio weighting limits help portfolio managers to fashion
realistic asset allocation strategies.
•
Suppose risky asset investments can be characterized as a vector of multivariate
returns of N securities, RT.
•
The expected risk premiums and variance-covariance of asset returns can be
expressed as a vector  and a positive definite matrix V, respectively.
•
Let Ω be the set of all real vectors w that define the weights of assets such that wT
1=1, where 1 is an N-vector of ones.
•
•
The expected return of the portfolio is p =wT  and the variance of the portfolio
is p = wTVw.
Considering all constraints given the objective to minimize the portfolio’s risk,
the efficient frontier can be then expressed as a Lagrangian function:
min {w , , }  
•
1 T
w Vw   (  p  w T )  (1  w T 1)
2
(11E.1)
As described in the previous section, the optimal portfolio weights are a function
of means, variances, and covariances of asset returns.
•
Consider a generalized case with N assets, let
A=1TV-1R, B=RTV-1R, X=1TV-11, and =BX-A2,
the solution of the above quadratic function is (see Pennacchi (2008))
wp=
1
[(X p – A)(V-1R) + (B – A p )(V-11)]
Δ
(11E.2)
•
The short-selling constraints are further considered in Markowitz model.
•
The inequalities that represent non-negative portfolio weights are
w1  0, w2  0, ……, wN  0 , where w=[w1, w2,……, wN].
•
(11E.3)
The solution of this constrained diversification is to incorporate Equation (11E.3)
in Equation (11E.1).
A Numerical Example
•
Considering a three-asset case exemplified in the previous section, we have the
following objective and constraint functions:
3
Min
 p2  
j 1
3
W W Cov( R , R )
i 1
j
i
j
i
3
s.t.
E ( R p )   Wi E ( Ri )
(11E.4)
i 1
3
W
i 1
i
1
W1  0, W2  0, W3  0.
•
Replacing the variables in Equation (11E.4), we can rewrite Equation (11E.1) as
Min   [W12 11  W22 22  W32 33 )  2(WW
1 2 12  WW
1 3 13  W2W3 23 )
3
3
i 1
i 1
  ( E ( R p )  Wi E ( Ri ))   ( Wi  1)  1W1  2W2  3W3 ]
•
where  ,  , and i i=1, 2, 3, are Lagrange multipliers (LMs).
(11E.5)
•
The corresponding complementary slackness conditions are
3
 ( E ( R p )   Wi E ( Ri ))  0
i 1
3
 ( Wi  1)  0
(11E.6)
i 1
1W1  0
2W2  0
3W3  0
•
•
The above complementary slackness conditions indicate that either inequality
constraints should be active at a local optimum or the corresponding Lagarange
variable should equal zero.
For differentiable nonlinear program, solutions Wi,  ,  , i , i=1, 2, 3, satisfy the
Kuhn–Tucker conditions if they fulfill complementary slackness conditions, primal
constraints, and gradient equation

•
 p2
Wi
  Constraint (W1 ,W2 ,W3 )  LM
(11E.7)
Any combination of Wi, i = 1, 2, 3, for which there exist a corresponding LMs
satisfying these conditions is called a Kuhn–Tucker point.
•
Our portfolio model has the following objective function gradient
 0.91W1  0.0018W2  0.0008W3 




  0.0018W1  0.1228W2  0.002W3 
Wi 

 0.0008W1  0.002W2  0.105W3 
2
p
(11E.8)
and those of the five linear constraints are
Constraint1 (W1 , W2 , W3 )  (0.0053, 0.0055, 0.0126)
Constraint 2 (W1 , W2 , W3 )  (1,1,1)
Constraint 3 (W1 , W2 , W3 )  (1, 0, 0)
(11E.9)
Constraint 4 (W1 , W2 , W3 )  (0,1, 0)
Constraint 5 (W1 , W2 , W3 )  (0, 0,1)
•
Therefore the gradient equation part of Kuhn–Tucker conditions is
0.0053  1  11  0.91W1  0.0018W2  0.0008W3
0.0055  1  12  0.0018W1  0.1228W2  0.002W3
0.0126  1  13  0.0008W1  0.002W2  0.105W3
(11E.10)
plus the primal constraints as part of the conditions
3
E ( R p )   Wi E ( Ri )
i 1
3
W
i 1
i
1
W1  0
W2  0
W3  0
and complementary slackness conditions in Equation (11E.6).
(11E.11)
•
Notice that the weights are functions of five LMs and bounded by inequality
constraints.
•
Since the functions of solutions of Wi are multidimensional, we cannot show their
relation on a graph.
•
One may start from any feasible point and then search an improving feasible
direction (W1 ,W2 ,W3 ) chased by implementations of feasible and small steps of
LMs.
•
The stop of an improving feasible search not necessarily represents the current
Kuhn–Tucker point is the global optimum but suggests it is a local optimum.
•
Since there is no close-form solution function for each weight, one may need to
continue the search until no improvement can be found. In our case, if

•
 p2
Wi
  (W1 , W2 , W3 )  0
there is an improvement in objective function.
(11E.12)
•
We use the same data set to construct the nonconstrained (NC) efficient frontier
and short-selling constrained (SS) efficient frontier, which is shown in Figure
11E.1.
•
In Figure 11E.1, the SS optimal diversification is a subset of the NC portfolio.
3.0%
No Constraint
2.5%
Short-Selling Constraint
2.0%
1.5%
Short-selling Constrained Efficient Frontier
1.0%
0.5%
0.0%
0%
20%
40%
60%
Figure 11E.1 Efficient Frontiers
80%
100%
•
The global minimum variance (MV) for the two kinds of portfolio is identical in
this case.
•
The information is listed in Table 11E.1.
•
The conclusion of the same MV under different constraints does not always
happen when the coefficients of correlation among securities and the relative
magnitudes of return among assets change.
•
Note that the optimal diversification strategies are sensitive to the variation in
the first two moments of asset returns in the portfolio.
E(R)(%)  %
0.78
13.39
Portfolio
Weight
w1(%)
w2(%)
w3(%)
38.57
28.10
33.33
Table 11E.1 Minimum-Variance Portfolio
•
As the portfolios on efficient frontiers shown in Table 11E.2, assuming the longterm annual risk-free interest rate is 4%, the portfolios of corner solutions are less
mean-variance efficient than the ones without negative and extremely positive
weights.
E(R) (%)
0.78
1.04
1.24
2.63
3.38
 %
13.39
16.87
22.38
72.96
101.30
Sharpe Ratio
0.0333
0.0422
0.0406
0.0315
0.0300
Portfolio
Weight
w1 (%)
38.57
16.86
−25.82
−100.53
−366.73
Table 11E.2 Portfolios on Efficient Frontier
w2 (%)
28.10
13.04
−16.90
−199.47
66.73
w3 (%)
33.33
70.10
142.72
400.00
400.00