Appendix 11A:

Derivation of Equation (11.6a)

(11.1)

The objective function L as defined in Equation (18.3) can be rewritten:

Then, following the product and chain rule, we have:

(11A.1)

1 2

2 2

1 1 1 1

Max n n n n

i i f i i i j iji i i j

L W R R W WW i j

1 2

2

1 1 1 1

1 2

2 2

1 1 1 1

2

1 1 1 1

n n n n

i i f i i j iji i i ji i

n n n n

i i i j ij i i fi i j ii

n n n n

i i f i i j iji i i ji

dL dW R R W W i j

dW dW

dW WW W R R

dW

dW R R W W

dW

1 2

1 2

2 2

1 1 1

3 2

2 212

1 1 1 1

2

1

2 2

0 1, 2, ,

n n n

i i i j ij i fi i j

n n n n

i i f i i i j iji i i j

n

i i j ijj

W WW R R

W R R W WW

W W

i n

Multiplying Equation (11A.1) by , , and

rearranging yields:

(11A.2)

Defining

1 2

2 2

1 1 1

n n n

i i i j iji i j

W WW

i j

21

2 2 1

1 1 1

n

i i f ni

i f i i j ijn n nj

i i i j iji j j

W R RR R W W

W WW

i j

1

2 2

1 1 1

n

i i fi

n n n

i i i j iji i j

W R Rk i j

W WW

Yields

Therefore:

2

1

0 n

i i i ij i fi

k W W R R j i

21 1 2 2 ( ) ( )

0

i i i i n ni i fi

dLkW kW kW kW R R

dW

21 2 2i f i i i i i n niR R H H H H

21 1 1 2 12 1

22 1 21 2 2 2

21 1 2 2

f n n

f n n

n f n n n n

R R H H H

R R H H H

R R H H H

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:

There is one equation like this for each value of i.

This is Equation (11.6a) in the text.

Appendix 11B:

Derivation of Equation 11.10

(11.2)

Following the optimization procedure for deriving Equation (18A.2) in Appendix 18A:

Let and, solving for any ,

(11B.1)

2 2 2 212

1

0

n

i i f ni

i f i i m i j j m i iji p

W R RdL

R R W W WdW

j i

2i p f p iH R R W iH

2

1

2 2

n

i m j ji f j

ii i

HR R

H

Multiplying both sides of the equation by :

（ 11B.2 ）

Adding together the n equation of this form yields:

（ 11B.3 ）

By substituting Equation (11B.3) into Equation (11B.1), Equation (11.10) is obtained.

2 2

1

2 2

n

i m j ji f i j

i ii i

HR R

H

i

2

1

21 2

21

1

nj f j

nj i

j j nj j

ij i

R R

H

2

2 21

22 22

21

1

nj f

mi f j i i

i iji i

mj ej

R R

R RH

Appendix 11C:

Derivation of Equation (11.15)

(11.3)

2

2 21

ni f i m

i j j iji i

R RH H

2

2 21

and 0d

i f i mi j j i

ji i

R RH H

(11C.1)

where and for all i.0, 0i iH 0i iH

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:

(11C.2)

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

Multiplying both sides by , summing over all stocks in d, and rearranging yields:

1 1

n d

j j j jj j

H H

21

212

1

1

dj f

jd

j j

j j djj

mj j

R R

H

(11C.3)

j

Notice since the set d contains all stocks with positive Hi:

Using (11C.4), the following equation for is obtained after substitution and rearranging from Equation (11C.1):

21* 2

22

21

1

dj f

jj j

m nj

mj j

R R

C

*2

i fii i

i i

R RH C

And Let:

Since , the inclusion of ; can only increase the value of Hi. Therefore, if Hi is positive with can never make it zero and the security should be included, If Hi<0 when , positive values of can increase Hi . However, because the product of and Hi must equal zero, as indicated in Equation (11C.1), positive values of imply Hi=0. Therefore, any security Hi<0 when must be rejected. Therefore, Equation (11.16) in the text can be used to estimate the optimal weight of a portfolio.

(11C.4)

(11C.5)

0i i

0i 0i i

ii 0i

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

• Subject to

where

• 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

{ , 1,2,... } ( )i

p fW i n

p

R RMax F W

1

1

0 , 1, 2,...

n

ii

i

W

W i n

1

1 2

2 2

1 1 1

n

p i ii

n n n

p i i i j iji i j

R W R

W W W

• If F(W) is a function of Wi in Figure 11.D.1(a), then the maximum value of F(W) occurs at P, where at positive optimal weight (Wi > 0).

• 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 at optimal weight (Wi = 0).

(a) (b)

F(W) F(W)

Wi Wi

P P

P’

Figure 11.D.1 Value of the Function F(W) as Wi Changes

( )0

i

dF W

dW

( )0

i

dF W

dW

• Therefore, in general, we can obtain the optimal weights under conditions

• We can rewrite the conditions as five Kuhn–Tucker conditions:

• When maximum feasible value occurs on , Ai is positive and Wi is equal

to zero. • If maximum feasible value occurs on , then Ai is equal to zero and Wi is

positive.

( )0

( )0

i

ii

dF W

dW

dF WW

dW

1

( )0

0

0

0

1

ii

i i

i

in

ii

dF WA

dW

W A

W

A

W

( )0

i

dF W

dW

( )0

i

dF W

dW

• In the following part, we will solve the optimal weights under Kuhn–Tucker conditions by Excel.

• 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.

( )

i

dF W

dW

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)

• 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

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:

• As described in the previous section, the optimal portfolio weights are a function of means, variances, and covariances of asset returns.

m in ( ) ( ){ , , }w w V w w w 1 1

21T T T

p (11E.1)

• 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= [(X p – A)(V-1R) + (B – A p )(V-11)]

• 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].

• The solution of this constrained diversification is to incorporate Equation (11E.3) in Equation (11E.1).

Δ1

(11E.2)

(11E.3)

• Considering a three-asset case exemplified in the previous section, we have the following objective and constraint functions:

• Replacing the variables in Equation (11E.4), we can rewrite Equation (11E.1) as

• where i=1, 2, 3, are Lagrange multipliers (LMs).

A Numerical Example

3 32

1 1

3

1

3

1

1 2 3

( , )

s.t. ( ) ( )

1

0, 0, 0.

p j i j ij i

p i ii

ii

Min W W Cov R R

E R W E R

W

W W W

(11E.4)

2 2 21 11 2 22 3 33 1 2 12 1 3 13 2 3 23

3 3

1 1 2 2 3 31 1

Min [ ) 2( )

( ( ) ( )) ( 1) ]p i i ii i

W W W WW WW W W

E R W E R W W W W

(11E.5)

, , and i

• The corresponding complementary slackness conditions are

• 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=1, 2, 3, satisfy the

Kuhn–Tucker conditions if they fulfill complementary slackness conditions, primal constraints, and gradient equation

• Any combination of Wi, i = 1, 2, 3, for which there exist a corresponding LMs

satisfying these conditions is called a Kuhn–Tucker point.

3

1

3

1

1 1

2 2

3 3

( ( ) ( )) 0

( 1) 0

0

0

0

p i ii

ii

E R W E R

W

W

W

W

(11E.6)

, , ,i

LM),,(Constraint 321

2

WWWWi

p(11E.7)

• Our portfolio model has the following objective function gradient

and those of the five linear constraints are

• Therefore the gradient equation part of Kuhn–Tucker conditions is

321

321

3212

105.0002.00008.0

002.01228.00018.0

0008.00018.091.0

WWW

WWW

WWW

Wi

p

1 1 2 3

2 1 2 3

3 1 2 3

4 1 2 3

5 1 2 3

Constraint ( , , ) (0.0053,0.0055,0.0126)

Constraint ( , , ) (1,1,1)

Constraint ( , , ) (1,0,0)

Constraint ( , , ) (0,1,0)

Constraint ( , , ) (0,0,1)

W W W

W W W

W W W

W W W

W W W

(11E.8)

(11E.9)

1 1 2 3

2 1 2 3

3 1 2 3

0.0053 1 1 0.91 0.0018 0.0008

0.0055 1 1 0.0018 0.1228 0.002

0.0126 1 1 0.0008 0.002 0.105

W W W

W W W

W W W

(11E.10)

plus the primal constraints as part of the conditions

and complementary slackness conditions in Equation (11E.6).

3

1

3

1

1

2

3

( ) ( )

1

0

0

0

p i ii

ii

E R W E R

W

W

W

W

(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 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

• there is an improvement in objective function.

),,( 321 WWW

2

1 2 3( , , ) 0p

i

W W WW

(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.

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

0% 20% 40% 60% 80% 100%

No Constraint

Short-Selling Constraint

Short-sellingConstrained Efficient Frontier

Figure 11E.1 Efficient Frontiers

• 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)(%) s (%)Portfolio Weight w1(%) w2(%) w3(%)

0.78 13.39 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 long-term 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) (%) s (%) Sharpe RatioPortfolio Weight w1 (%) w2 (%) w3 (%)

0.78 13.39 0.0333 38.57 28.10 33.331.04 16.87 0.0422 16.86 13.04 70.101.24 22.38 0.0406 −25.82 −16.90 142.722.63 72.96 0.0315 −100.53 −199.47 400.003.38 101.30 0.0300 −366.73 66.73 400.00

Table 11E.2 Portfolios on Efficient Frontier