Conclusions and Recommendations - 共整合向量模型之平均數-變異數動態投資組合方法探討

We draw conclusions from the empirical results in chapter 4 and make

recommendations of future research on this topic.

Figure 1. Framework

Motivation and Purpose

The Cointegration Model

Research Method

One-stage method:

Complete the optimal asset allocation for both phases at the beginning of the first phase

based on the initial price.

Two-stage method:

Complete the optimal asset allocation for each phase at the

beginning of each phase based on prices at that time.

Discuss dynamic asset allocation using the mean-variance optimization method.

Comparison of Two Methods

Numerical Illustrations

Conclusions and Recommendations

Forecast asset price using PCB Cointegration Model.

, 2 1 , 2 2 , 2

, 1 , 2 1

, 1













+ +

+ Ξ +

− t

t t

t t T t

b x x

θ t =1,2,...,n; ~ (0, )

, 2

1  Σ







= N b

b b

iid

t t t

2 The Cointegration Model

In this chapter, we will show Cointegration Model which will be used to organize

the sequence information of the price of the financial commodity in this paper.

Before being engaged in each type of statistical inference of the time series, we

should first examine whether the sequence is in stationary state. When the time series is

stationary, its data follows a stochastic process, but the probability distribution of this

stochastic process does not change along with time; otherwise, this time series is called

non-stationary time series.

Elam & Dixon (1988) pointed out that the price series of financial products usually

show the characteristic of non-stationary process. That means the price sequence is a

random walk type in which the price sequence of current period is not affected by the

price sequence of the previous period. If a time series is non-stationary, the commonly

used method to convert it to stationary is to take the difference on the variable or

eliminate the tendency item. However, the conversion could potentially eliminate the

implicit long-term message from the data, so to avoid this potential problem, we could

adopt the cointegration concept proposed by Granger. According to the cointegration

concept, a stable long-term balanced relation might exist between unstable variables,

and this relation might cause synchronized tendency on the variables. Therefore, we use

formula(3) transformed from PCB Cointegration Model to model price sequence of

finance commodity in this paper.

PCB Cointegration model

Next, rewrite Eq.(2) in vector andmatrix forms.





3.1 Notations

x denotes the random vector of asset prices taken logarithm at time t. According to t

formula(3), x_t =β +Πx_t₋1+u_t^{. When}t≥², thex can be rewritten as: _t we allocate at stage t. When we know the price of assetsx , we should know all the _t

weights that are less than or equal to that at staget+1. For example, when we

know the price of assetsx₁, we should know the price of assetsx , then we can ₀

know the weightsw₁andw₂. In addition, positive weight represents buying of

stocks; negative weight means selling or short selling of stocks. Prices of financial

products do not always go up; they may go down as well. Short selling is the

trading option to maximize profits during down time, thus we allow short selling in

this paper. Because our assumption allows short selling of stocks, the sum of the

weights in each stage does not need to be equal to one.

Λ denotes the fees charged for different trading methods, such as buying, selling, and

short selling of stocks. It is determined based on the weight. For the convenience of

problem solving and calculation, we represent it using a fixed ratio. And we assume

that Λis a symmetric positive definite matrix. The model of the transaction cost in

this paper is w^TΛw 2

1 . The reason we use quadratic form to present the model of

transaction costs in this paper is to avoid the transaction costs becoming negative.

Because we allow negative weight, we must use the quadratic form to ensure that

transaction cost is positive.

Since we apply two-period dynamic asset allocation, the maximum number of t =2.

3.2 Objective functions

Markowitz's mean-variance portfolio optimization methods provide two

approaches: (1) under fixed risk, find the maximum return or (2) under fixed

remuneration, minimize the risk. The method that we chose in this paper is the former,

which is to find maximum return under fixed risk. In the special case of the following

section, we will verify that both methods give the same return when undertaking a unit

risk.

This paper uses two-stage mean-variance portfolio optimization framework, and

the objective function is as follows:

( )

1 1 2 2 0 1 1

(

2 1

) (

2 1

)

2 ,

1 2

max 1 ,

2 1

x w w w

w w w E x

r w r w E w

f ^T ^T ^T ^T

C = w + − Λ + − Λ − (10)

Restriction is as follow:

Var w¹^Tr¹+w²^Tr² x⁰ =σ⁰²

That means we want to optimize the net expected return of two-period, which is

deducting transaction cost under the constraint of restricting the variance of total return

toσ₀²at the end.

The objective function (10) and its restriction in this case which use the Lagrange

multipliers method can be rewritten as:

( )

1 1 2 2 0 1 1

(

2 1

) (

2 1

)

2 ,

1 2

max 1 ,

2 1

x w w w

w w w E x

r w r w E w

f ^T ^T ^T ^T

C = w + − Λ + − Λ −

[

¹ ¹ ² ² ⁰ ⁰²

]

2 σ

λ ₊ ₋

− ^C Var w^Tr w^Tr x (11)

where λ_C is a Lagrange multiplier.

Below we will discuss one-stage method and two-stage method respectively.

3.3 Dynamic portfolio selection methods

3.3.1 One-stage method

At statex , we optimize asset allocation for two periods according to asset prices ₀

at statex .₀

First, the objective function sort out from (11) is as follow:

( )

^w ⁼^E ^w ^r ^x⁰ ⁻¹₂^w ^Λ^∗^w⁻

^λ

₂¹

[

^Var ^w ^r ^x⁰ ⁻

^σ

⁰²

]

The vectors andmatrices represent respectively:



Then to optimize the objective function, we perform the first order differential equal to

zero on it.

According to 3.1 (8)、(9), the results are derived as follows:

Therefore, our approach is to get a starting estimate λ₁^∗ by using Taylor expansion first.

(The detailed derivations are in Appendix1.1.1.)

(The detailed derivations are in Appendix1.1.2.)

(15), and then we can get a new estimated value from the right side of the equation. We

then confirm if the difference between the initial estimated value and the new estimated

value is less than the estimated error of our setting. If not, we take the new estimated

value and substitute it into the left side of the formula until the values of two sides are

very close, expressing convergence, then we stop. If so, we can get an answer directly,

which results in a more accurate estimated valueλ₁^S^.

The optimal weight of one stage method:

w_CO^∗ =Ψ^∗⁻¹A where ^Ψ^∗ ⁼

(

^Λ^∗ ⁺^λ¹^S^B

)

(16) Expected return after deducting transaction cost of one-stage method as follows:

^∗ − ^∗ Λ^∗ ^∗ = ^∗ − ^∗ Λ^∗ ^∗CO

T CO T

CO CO T CO T

CO r x w w w A w w

E 2

1 2

= A^T ⋅Ψ^∗⁻¹⋅A− A^T ⋅Ψ^∗⁻¹⋅Λ^∗⋅Ψ^∗⁻¹⋅A 2

⁼ ^A^T ^⋅^Ψ^∗⁻¹^⋅^A⁻₂¹

(

^A^T ^⋅^Ψ^∗⁻¹^⋅^A⁻^λ¹^S ^⋅^σ⁰²

)

⁼ ¹₂

(

^A^T ^⋅^Ψ^∗⁻¹^⋅^A⁺^λ¹^S ^⋅^σ⁰²

)

3.3.2 Two-stage method

At beginning of the first period, we will allocate the weight of asset based on the

initial price. We will then re-allocate the weight of asset again based on the asset price at

beginning of the second period.

In this method, we use the backward way to solve it. In other words, we get the

optimal weight of the second phase by using the objective function of the second phase.

Then we apply the optimal weight of the second stage to the overall objective function

to get the optimal weight of the first phase.

In this paper, we don’t consider the transaction cost in the objective function of the

second stage. Instead, we use an undetermined coefficient a₁ to adjust the change.

Step1.

order differential equal to zero on it.

According to 3.1 (8)、(9), the results are derived as follows:

The deduced result is obtained:

Step2.

Next, we must take the optimal weight of the second phase (17), which multiplies

an undetermined coefficienta₁, to substitute it into the optimal objective function of two

periods (11), as shown below:

(

¹ ¹

)

¹ ¹ ¹ ² ² ⁰ ¹ ¹

(

¹ ² ¹

) (

¹ ² ¹

)

⁰

The objective function is sorted out as following:

( )

^V ⁼^E ^V ^Z ^x⁰ ⁻₂¹^E ^V ^CV ^x⁰ ⁻^λ₂²

[

^Var ^V ^Z ^x⁰ ⁻^σ⁰²

]

The vectors andmatrices represent respectively:



After the first order differential, we get:

2 =0

−

−HV GV

F λ

V^∗ =Ω⁻¹ where ^Ω⁼

(

^H ⁺^λ²^G

)

(19) According to 3.1 (8)、(9), the results are derived as follows:

(The detailed derivations are in Appendix2.1.1.)

(

1 0 1

)

1 0

also get a starting estimateλ^∗₂ by Taylor expansion.

(The detailed derivations are in Appendix2.1.2.)

(The detailed derivations are in Appendix2.1.3.)

2 0 2 1

1⋅ − ⋅Ω ⋅ ⋅Ω ⋅ =λ ⋅σ

Ω

⋅ ⁻ F F ⁻ H ⁻ F

F^T ^T (21)

Next, we take the initial estimated value λ^∗₂ into the function Ω⁻¹of the left side of

Eq.(21), so that we can get a new estimated value of the right side of the equation. We

then confirm if the difference of the initial estimated valueλ^∗₂and the new estimated

value is less than the estimated error of our setting. If not, we take the new estimated

value and substitute it into the left side of the formula until the value of two sides are

very close, expressing convergence, then we stop; If so, we can get an answer directly,

which would result in a more accurate estimated valueλ^S₂^.

The two-stage method obtains a vectorV_λ^∗which is combination of the optimal weight of

the first stage and a scale factor. V_λ^∗ =Ω^∗⁻¹F where ^Ω^∗ ⁼

(

^H⁺^λ^S²^G

)

(22) Expected return of two-stage method minus transition cost is represented as follows:

0 2

1E V CV x

x Z V

E _λ^∗^T − _λ^∗^T _λ^∗

∗

∗ −

=V_λ^TF V_λ^THV_λ 2

= F^T ⋅Ω^∗⁻¹⋅F − F^T ⋅Ω^∗⁻¹⋅H⋅Ω^∗⁻¹⋅F 2

⁼ ^F^T ^⋅^Ω^∗⁻¹^⋅^F ⁻¹₂

(

^F^T ^⋅^Ω^∗⁻¹^⋅^H ⁻^λ²^S ^⋅^σ⁰²

)

⁼ ¹₂

(

^F^T ^⋅^Ω^∗⁻¹^⋅^F ⁺^λ²^S ^⋅^σ⁰²

)

3.4 Special Cases

In this section, we will show four special cases.

Case #1: The first example is the typical Markowitz mean-variance portfolio

optimization approach in which asset allocation decisions are made at the beginning of

the investment period according to the price at the time regardless of the length of the

investment period. In other words, the asset allocation weight will not be changed

through time. This method is also known as static method. In this paper, we will

transform the PCB Cointegration Model to formula(3), and then incorporate the new

model together with the transaction cost model into the static method.

The objective function of the static method is as follows:

^f^C

( )

^w ⁼^max_w ^E ^w^T^r ^x⁰ ⁻¹₂^w^T^Λ^w⁻

^λ

₂

[

^Var ^w^T^r ^x⁰ ⁻

^σ

⁰²

]

[

⁰²

]

2 2

1 _Λ ₋λ ₋σ

−

=w^TA_S w^T w w^TB_Sw

According to the definition of return in 3.1 (8)、(9), the return random vector is as

follows:

(

2 1

) (

1 0

)

2 1 1 1 2 1 0 1

2 x x x x x r r u u x u

r = − = − + − = + =Πβ+Π + +β +Π +

=Π1x0 +

(

Π+I

) (

β + Π1+I

)

u1+u2 =Π1x0 +

(

Π+I

)

β+Πu1+u2 (23) After taking the first order differential, we get:

S A

w =Μ⁻¹ where Μ =

(

Λ+λBS

)

(24)

Result derived according to Eq.(23) is as follows:

(

)

β ^u ^u ^x ^x

(

)

x E x r

E 0 = Π1 0 + Π+ +Π 1 + 2 0 =Π1 0 + Π+

Var r x0 =Var Π1x0 +

(

Π+I

)

β +Πu1+u2 x0 =ΠΣ^∗Π^T +Σ^∗ Then, with the same approach used previously to estimate valueλ, we get a more

accurate estimateλ_S, thus the expected return after deducting transaction cost of static

method is as follows:

^E ^w^∗^S^T^r ^x⁰ ⁻¹₂^w^∗^S^T^Λ^w^S^∗ ⁼¹₂

(

^A^S^T ^⋅^Μ^∗⁻¹^⋅^A^S ⁺^λ^S ^⋅^σ⁰²

)

Case #2: Given the conditionΛ≠0;β =0, compare the two dynamic methods to

determine which method gives higher expected return after deducting transaction cost.

(See Appendix 1.2、2.2 for the detailed derivations)

The objective function is as follows:

( )

1 1 2 2 0 1 1

(

2 1

) (

2 1

)

2 ,

1 2

max 1 ,

2 1

x w w w

w w w E x

r w r w E w

f ^T ^T ^T ^T

C = w + − Λ + − Λ −

⁻^λ₂^C

[

^Var ^w¹^T^r¹⁺^w²^T^r² ^x⁰ ⁻^σ⁰²

]

Expected return after deducting transaction cost of one-stage method is as follows:

^E ^w^∗^CO^T^r ^x⁰ ⁻¹₂^w^CO^∗ ^T^Λ^∗^w^CO^∗ ⁼ ₂¹

(

^A⁰^T ^⋅^Ψ^∗⁻¹^⋅^A⁰ ⁺^λ³^S ^⋅^σ⁰²

)

Expected return after deducting transaction cost of two-stage method is as follows:

^E ^V^λ^∗^T^Z ^x⁰ ⁻₂¹^E ^V^λ^∗^T^CV^λ^∗ ^x⁰ ⁼ ¹₂

(

^F⁰^T ^⋅^Ω⁰^∗⁻¹^⋅^F⁰^T ⁺^λ⁴^S ^⋅^σ⁰²

)

Case #3: Given the conditionΛ=0;β ≠0, compare the two dynamic methods to

determine which method gives higher expected return after deducting transaction cost.

(See Appendix 1.3.1、2.3.1 for the detailed derivations)

The objective function is as follows:

(

¹^, ²

)

^max_, ¹ ¹ ² ² ⁰ ₂

[

¹ ¹ ² ² ⁰ ⁰²

]

2 1

λ

₊ ₋

σ

− +

= E w r w r x Var w r w r x w

f ^T ^T ^T ^T

Expected return of one-stage method is as follows:

E w_O^∗^Tr x₀ =σ_O A^TB⁻¹A

Expected return of two-stage method is as follows:

E Vλ^∗^TZ x₀ =σ_O F^TG⁻¹F

We will have further discussion on this case. In previous discussion, we talk about

seeking maximum return while undertaking the fixed risk. We want to verify if the

return per unit of risk remains the same when seeking for a minimum risk for a fixed

return. We decided to use this case for discussion because the λ values are estimated

in the cases involving transaction costs, which could affect the verification results. In

addition, if the verification results are the same in this case then the case where the

intercept item equals to zero will also have the same result.

(See Appendix 1.3.2、2.3.2 for the detailed derivations)

The objective function is as following:

(

^w ^w

)

_w _w ^Var ^w^T^r ^w^T^r ^x

[

^E ^w^T^r ^w^T^r ^x ^O

]

f ₌ ₊ ₋

λ

₊ ₋

µ

0 2 2 1 1 0

2 2 1 , 1

1, min 2

The minimum variance of one-stage method is:

A B x A

r w

Var _O _T ^O₁

0 T

−

∗ = µ

The minimum variance of two-stage method is:

− ∗

∗

∗ =

F G F x Z V

Var ^T _T ^O₁

λ µ

Before comparing the minimum variance method and the maximum return method,

we need to defineδ , which denotes the return per unit of risk.

The comparison of the two methods using one-stage method:

The minimum variance method:

A B A A B A

T O O ov

1 2

−

= µ

δ µ

The maximum return method:

A B A A

A _T

O T O or

1 −

− =

= σ

δ σ

Above shows that the return per unit of risk remains the same when seeking for a

minimum risk for a fixed return using one-stage method.

The comparison of the two methods using two-stage method:

The minimum variance method:

− ∗

∗

− ∗

∗

= F G F

F G F

T O O tv

µ2

δ µ

The maximum return method:

F G F F

F _T

O T O or

1 −

− =

= σ

δ σ

This part can not be proved by the analytical way; we can only verify it with numerical

method in the next chapter.

Case #4: Given the conditionΛ=0;β =0, compare the two dynamic methods to

determine which method can give higher expected return after deducting transaction

costs. (See Appendix 1.4 、2.4 the detailed derivations)

The objective function is as following:

(

¹^, ²

)

^max_, ¹ ¹ ² ² ⁰ ₂

[

¹ ¹ ² ² ⁰ ⁰²

]

2 1

λ

₊ ₋

σ

− +

= E w r w r x Var w r w r x w

f ^T ^T ^T ^T

Expected return of one-stage method is as follows:

E w_O^∗^Tr x₀ =σ_O A₀^TB⁻¹A₀

Expected return of two-stage method is as follows:

E Vλ^∗^TZ x₀ =σ_O F₀^TG₀⁻¹F₀

From section 3.3 and 3.4, we observe that when transaction costs are involved, we

can only get estimated λ value whether there is intercept item or not. In contrast,

when there is no transaction cost, regardless of the intercept item, λ can always be solved. In addition, in the caseΛ=0;β ≠0, we can prove that the two methods of

Markowitz mean-variance portfolio optimization approach using one-stage method have

the same δ value. We will use the numerical methods to verify if the return per unit of

risk remains the same when seeking for a minimum risk for a fixed return using

two-stage method in the next section.

4 Numerical Illustrations

The data in this paper is collected from Taiwan Economic Journal Data Bank,

consisting 253 records of each kind of the Taiwan sector index during 08/18/2009 and

08/18/2010.

4.1 Unit root test and cointegration test

The statistical analysis software Eviews is used to analyze the 19 sector indices,

examine if they are stationary, and find the cointegration relation. 5 out of the 19 sector

indices from the Taiwan market are chosen for asset allocation, including (1) Building

Construction, (2) Financial and Insurance, (3) Steel and Iron, (4) Electronic and

Electrical, and (5) Biotech. Data of these 5 sectors are I(1) series. I(1) series, a sequence

that can become a stable series after the first difference. Among which (1) Building

Construction and (2) Financial and Insurance indices are cointegrated; the other 3 sector

indices are not co-integrated with these 2 sectors.

4.2 Parameter estimation

The estimated values for parameters θ₁^、θ2^、Ξ are derived using formula(1):

With formula(1) and the estimated θ^、Ξ values, use Eviews to estimate the covariance

matrix for formula(1)

From formula(5) we get:

From formula(6) we get:

4.3 Transaction costs

A 0.1425% transaction fee is charged for buying, selling and short selling stocks,

and a trading tax of 0.3% is imposed when selling and short selling stocks. Lending fee

for short selling is about 0.08%. The above figures do not include any discount. So, in

short, the cost of buying stocks is the transaction fee while the cost for selling stock is

the trading tax plus the transaction fee; and the cost for short selling is the sum of

transaction fee, trading tax, and lending fee. To simplify, a fixed ratio is used as

transaction cost for buying, selling and short selling:

(Securities transaction tax + borrowing cost) / 2 + transaction fee = 0.3325%

The transaction cost matrix is as below:

4.4 Numerical results

In this section, the real data is applied to the model, and the values of the vectors,

matrix, and parameters of the two dynamic programming methods are examined.

4.4.1 Results of one-stage method

The vector and matrix of formula (12) in this paper are:

Expected return matrix A

Transaction cost matrix Λ^∗

4.4.2 Results of two-stage method

The vector and matrix of formula (18) are:

Expected return matrix F

Expected transaction cost matrix H

Covariance matrix G

4.5 Methods Comparison

In order to compare the one-stage method with the two-stage method and

determine which method can give higher expected return under the same risk, we apply the estimated parameters and the actual asset price datax to the formula of the expected ₀

return and transaction cost derived in this paper.

For the empirical research, we found that the size ofΣwill affect the comparison

results. We first examine the results by using differentΣvalues. We will sum the

covariance matrix estimated by the 5 sector indices and that the identity matrix

multiplied with different coefficients, respectively. And then we observe the changes

under the two cases: (1)Λ≠0;

β

≠0⁽²⁾Λ=⁰^;

β

≠⁰. We only discuss these two cases

because the information used in this paper has an intercept item.

According to the first table, the net expected returns of the two-stage method are

much better than those of the one-stage method. However, the expected returns of the

one-stage method are more sensitive to the change of the covariance matrix. So with the

reduction of the diagonal covariance matrix, the net expected returns of the two-stage

method will be closer to that of the one-stage method.

Table 1. Comparison of expected net return Expect net return

Λ ≠ 0 ; β ≠ 0 Λ = ⁰ ^; β ≠ ⁰

Σ+5* identity (0.0014, 0.0408) (0.0015, 0.0408) Σ+1* identity (0.0032, 0.0409) (0.0033, 0.0409) Σ+0.5* identity (0.0045, 0.0409) (0.0046, 0.0410) Σ+0.1* identity (0.0097, 0.0414) (0.0103, 0.0415) Σ+0.05* identity (0.0133, 0.0419) (0.0146, 0.0421) Σ+0.01* identity (0.0265, 0.0454) (0.0323, 0.0468) Σ+0.005* identity (0.0345, 0.0483) (0.0453, 0.0519) Σ+0.001* identity (0.0575, 0.0583) (0.0575, 0.0583)

Note 1. The results of this table is under σ ₌_0.05.

Note 2. The numbers in the table are in the following order: one-stage method, two-stage method.

The second table shows that when the diagonal of covariance matrix is smaller

than10^-4, the one-stage method performs better than the two-stage method. The

covariance matrix of this paper is about10^-4, so obtained numerical result of the one-stage method is better than that of the two-stage method. In addition, we also

observe that when the diagonal of covariance matrix is reduced to10^-4from10^-3, the net

expected returns of the above two cases increase significantly.

Table 2. Comparison of expected net return

Expect net return

Λ ≠ 0 ; β ≠ 0 Λ = ⁰ ^; β ≠ ⁰

Σ+0.0001* identity (0.1017, 0.0858) (0.1940, 0.1432)

Σ (0.1630, 0.1249) (0.2637, 0.1916)

Note 1. The results of this table is under σ ₌_0.05.

Note 2. The numbers in the table are in the following order: one-stage method, two-stage method.

From the above two tables, we observe that the expected net returns increase with

the reduction of the covariance matrix. Relatively, the expected net return of the

one-stage method is more sensitive to the change of the covariance matrix than the

expected net return of the two-stage method. A slight change to a small covariance

matrix can cause significant changes in expected net returns. In addition, when the

return rate is lower, there is almost no difference in the expected net returns whether

transaction cost is taken into account. When the return rate is higher, the difference

between the case of transaction cost and that of no transaction cost is relatively large.

However, that won’t affect the comparison results of the two methods.

Now we understand the impact of the change of the covariance matrix, let us look

at the comparison of two dynamic methods under different standard deviations.

Table 3 shows that the expected net return of the one-stage method is still better

than that of the two-stage method under a different standard deviation.

Table 3. Comparison of expected net return

Expect net return σ0=⁰^.⁰⁵ σ₀=0.1 σ0=⁰^.¹⁵

;

0 ≠

≠

Λ β

;

0 ≠

Λ β

(0.1630, 0.1249) (0.2637, 0.1916)

(0.2404, 0.1765) (0.5274, 0.3833)

(0.2832, 0.1816) (0.7911, 0.5749)

Note. The numbers in the table are in the following order: one-stage method, two-stage method.

In addition, we can see from Table 4 that higher risk gives higher return, but the

return per risk unit degrades.

Table 4. Comparison of net return and the return per risk unit under 0

;

0 ≠

≠

Λ β

05 .

0=0

σ σ₀=0.1 σ0=⁰^.¹⁵ Expect net return

Delta

(0.1630, 0.1249) (3.26, 2.498)

(0.2404, 0.1765) (2.404, 1.916)

(0.2832, 0.1816) (1.888, 1.211)

Note: Delta represents the return per risk unit.

Finally, we used the numerical method and verified that the return per unit of risk

remains the same when seeking for a minimum risk for a fixed return using two-stage.

5 Conclusions and Recommendations

In the past, many studies conducted in Taiwan and abroad had confirmed the

importance of asset allocation. The topic of asset allocation has attracted great attention

from investors. Today, with the availability of many investment products, investors can

use portfolio to diversify investment risk and apply research results to maximize the

return profits.

The shortcomings of the approach of traditional asset allocation are: regardless of

the length of the investment, asset allocation decision is made at the beginning of the

investment period using the price at that time. The allocation weight will not change

over the period of time. The approach of traditional asset allocation does not consider

在文檔中共整合向量模型之平均數-變異數動態投資組合方法探討 (頁 12-61)