IJRRAS 11 (1) ● April 2012
www.arpapress.com/Volumes/Vol11Issue1/IJRRAS_11_1_06.pdf
TWO-STAGE DYNAMIC MULTI-PERIOD PORTFOLIO
OPTIMIZATIONS
Shu-Ing Liu
Professor, Department of Finance, Shih Hsin University, No. 111, Mu-Cha Road, Sec 1, Taipei, Taiwan, 11604
ABSTRACT
This paper extends the traditional Markowitz’s mean-variance optimization to a two-stage dynamic multi-period
portfolio optimization. The underlying assets time series data are supposed to follow a discrete time triangular
cointegrated vector model, and in addition random quadratic transaction costs are taken into consideration. A twostage dynamic multi-period approach is proposed, and the optimal solution under the discussed model is analytically
derived. Also, comparisons between a standard one-stage static approach and some two-stage approaches will be
numerically examined under constructed data. The results indicate that the proposed two-stage dynamic method
performs quite efficiently, and that higher net returns per risk can be expected.
Keywords: Cointegration, Dynamic asset allocation, Markowitz optimization, Portfolio weight vector.
1. INTRODUCTION
The modern asset allocation theory was originated from the mean-variance portfolio model introduced by
Markowitz [1] in 1952. The original Markowitz model simply dealt with a static single period asset allocation
problem, calculating the tradeoff between risks and returns. A comprehensive theoretical framework was introduced,
adapting the diversification concepts, that investors selected the ideal portfolio which gave the best return for a
given risk. The proposals initiated the research foundation on current portfolio selection discussions. By applying
the Bayeaisn analysis technique, Black and Litterman [2] introduced the concept that investor view has an input into
the asset allocation problem. They believed that equilibrium exists in the financial market, thus the existing
capitalization weights could serve as the basis for establishing an optimal allocation. In the investment industry, the
model has significantly been applied by Goldman Sachs since 1990 though it is not so popular in academic research.
Morton and Pliska [3] considered some correlated geometric Brownian motion stocks and a riskless bond as the
underlying assets with numerical results showing that transaction costs were significantly influential in designing the
optimal trading strategy. The impact of joining with transaction costs was discussed by Lobo, Fazel and Boyd [4],
and Borkovec, Domowitz, Kiernan and Seibin [5], and the empirical results indicated that model with transaction
costs lead to improvement in realized returns, and better alignment of return with risk. Gârleanu and Pedersen [6]
analytically investigated an optimal dynamic portfolio policy with quadratic transaction costs and predictable returns
under different mean-reversion speeds. Numerical results revealed that the proposed optimal dynamic strategy
significantly performed better than the optimal static strategy.
An extension of the mean-variance formula in multi-period portfolio selection with analytical optimal solution was
discussed by Li and Ng [7]. The explicit solution could provide investors with the optimal strategy to follow in a
dynamic investment environment. A look back straddle approach was applied by Darius et al. [8] for evaluating the
return characteristics of a trend-following strategy via a multi-period dynamic portfolio model. Empirical results
showed that the advantages of the discussed strategy for investors were at the top end of the multi-period efficient
frontier. Moreover, in order to obtain a more accurate estimation of the covariance matrix of underlying assets, the
shrinkage technique popularly used in Bayesian analysis, was applied by Ledoit and Wolf [9]. The proposed
shrinkage estimator could select portfolios with significantly lower out-of-sample variance demonstrated by some
numerical examples.
It is popularly known that the cointegration approach, introduced by Granger [10], could represent the relationship
of the long-term mobile trend between some non-stationary time series. Empirical results significantly showed that
for most financial economic data, the log-prices of the underlying assets were non-stationary, and the cointegration
structure helped to model non-stationary data without taking differencing transformation. A naive two-stage meanvariance portfolio selection approach, discussed by Rudoy and Rohrs [11], assumed that the underlying assets follow
a discrete-time cointegrated vector autoregressive model. Some promising numerical results were obtained by
comparing this with other trading strategies.
Extending Rudoy and Rohrs’ work [11] a two-stage multi-period portfolio selection with random portfolio weights
was constructed by Liu [12]. The optimal solutions were derived in closed forms and equivalence comparisons
between different formulations were shown analytically. In this paper, attention will be focused on the optimal mean
return under a prescribed risk portfolio selection approach. A more efficient and informative two-stage multi-period
53
IJRRAS 11 (1) ● April 2012
Liu ● Two-Stage Dynamic Multi-Period Portfolio Optimizations
optimal solution, an extension of Liu’s work [12], will be analytically discussed and numerically investigated.
This paper is organized as follows. In Section 2, the discussed discrete-time cointegrated vector autoregressive
model is briefly reviewed. Under the proposed cointegrated model, multi-period portfolio selection formulation with
quadratic transaction cost, and discussion of a traditional one-stage optimal portfolio weight, are included in Section
3. In Section 4 an innovative extension of Liu’s work [12] is proposed with a n-stage multi-period portfolio selection
model that includes a random portfolio vector and the optimal portfolio allocation presented in closed form.
Numerical comparisons between the one-stage and the two-stage approaches, one proposed by Liu [12] and the
other discussed in this paper, under some special cases are demonstrated in Section 5. Finally, the conclusions are
given in Section 6.
2. DISCRETE TIME SERIES MODEL FOR ASSET RETURNS
Granger [10] suggested that in a vector time series when all of the components are stationary after taking the first
difference, there may exist stationary linear combinations. This caused the study of a financial cointegrated time
series model which is a common approach to eliminate illogical correlation and still keep the long-term equilibrium
between individual time series data. Suppose a capital market with k assets is considered and the investor will
allocate his wealth among the k assets, and re-allocate at the beginning of each of the following n consecutive
periods.
Let y t be a k  1 random vector time series, representing the log-prices of each asset. The portfolio selection
problem for time series data
yt  with model structure discussed by Liu’s [12] approach is adopted. A brief review
is stated as follows: Suppose that each entry of
yt has a unit root and a triangle cointegrated model with rank k1 is
defined as follows:
 y1,t   1  T y2,t  a1,t 
,

yt  
   y

y

a
2
,
t
2 , t 1
2,t 

  2
for
(1)
t  1,2,, n . Both yit ,  i and ait are ki  1 vectors, with k1  k2  k and  is a k 2  k1 matrix; and the


random vector, atT  a1Tt , a2Tt , is supposed to satisfy a VAR ( p) model. After algebraic computations, the a’s could
be assumed to be multivariate normally distributed, with mean 0 and some variance-covariance matrix, 1 , denoted
by MN 0, 1  . The elements of 1 are function of parameters involved in the VAR ( p) model.
Denote I j as a
 0k
1   1
 0


 I k1
 0

j  j identity matrix, 0 j as a j  j zero matrix, 0 as a k2  k1 zero matrix, 0  
T 
,
I k 2 
T 
 ,    1  ,   0 , and et  0 at , then model (1) is re-written as:
 
I k 2 
 2
yt    1 yt 1  et .

T T
n
Here e1T , e2T ,, e
(2)
~ MN 0, ( n )  , with ( n )  I n  0 1 I n  0T  , and  denotes the Kronecker product.
For detailed derivations please refer to Liu [12].
x0 denote the investor initial wealth, and y0 is the natural logarithm of x0 ; and 2  1  I k and
 t  2et 1  et , then for t  2
rt  yt  yt 1    2 yt 1  et    2   1 yt 2  et 1   et  1  2 et 1  et  1   t .
Let
Therefore,
  2 y0  e1 , for t  1,
rt  yt  yt 1  
for t  2.
1   t ,

Moreover, define r( n )  r1T , r2T , , rnT

T
and
 ( n )  e1T ,  2T ,  3T ,,  nT  , then given an initial log-asset value, y0
T
, the overall n-period joint return resulted as:
r( n )  ( n )   ( n ) ,
(3)
54
IJRRAS 11 (1) ● April 2012
where
( n )  E ( r( n ) | y0 )  1T , 2T ,, nT  ,
T
  2 y0 , if
if
1 ,
i  
Liu ● Two-Stage Dynamic Multi-Period Portfolio Optimizations
(4)
i  1,
i  2,
Var ( r( n ) | y0 )  ( n )  ( n )( n ) (Tn ) ,
With
3.
( n )
(5)
0 
 Ik 0 0


0 
 2 I k 0
. Also,  ( n ) ~ MN 0, ( n ) and as  2  0 , this implies that 1  0 .
  0 2 I k
0 


0 0 

0 0   I 
2
k



MULTI-PERIOD MEAN-VARIANCE FORMULATIONS
3.1 The framework
The investor plans to allocate his wealth among some k assets by deciding the portfolio weights at time t denoted by
wt . Positive entries of wt denote purchasing positions, while negative entries of wi denote shorting positions. In
the following discussion no restrictions on the sign of entries of
wt are attached. For convenience, when asset
prices are mentioned, they mean log-asset prices, instead of the actual asset prices. At the beginning time, suppose
the investor has initial asset value, y0 , which is known in advance. The goal is to design an optimal n-period asset

allocation by deciding portfolio weights denoted by w( n )  w1T , w2T ,, wnT

T
.
Let the one-period transaction costs associated with trading wt  wt 1 shares, may change the price by the amount,
say, 1 t wt  wt 1  . The coefficient, one-half, is set for computational convenience, and t , is a symmetric matrix,
2
measuring the level of trading costs. Then, the one-period convex transaction costs are defined as
1
wt  wt 1 T t wt  wt 1  , and the entire n-period transaction costs are thus
2
1 n
wt  wt 1 T t wt  wt 1   1 w(Tn )  ( n ) w( n ) .

2 t 1
2
For convenience, w0 is defined as 0, and  (n ) is a nk  nk symmetric positive-definite matrix expressed as:
 1  2  2



2  3   3
0
  2


.
3  4   4
 3

(n)  




0
 n 1 n 1  n   n 







n
n

In this paper the discussed mean-variance algorithm will be focused on the one that maximizes the overall n-period
net expected returns, under a prescribed overall risk level, under the given initial asset value y0 : That is
1


max  E w(Tn ) r( n ) | y0   E w(Tn )  ( n ) w( n ) | y0  , subject to Var w(Tn ) r( n ) | y0    02 .
w
2


The future portfolio weights will be formulated at the beginning time: Two approaches will be considered,
depending upon the structure of the portfolio weight vector, w(n ) : One, treated w(n ) as deterministic, is called a
standard one-stage method hereafter. The future portfolio weights are explicitly determined at the beginning time;
the other is designed as a random vector constructed by a two-stage procedure, called a two-stage method. The
future portfolio weights conditionally determined at the beginning time, are updated period by period. These two
55
IJRRAS 11 (1) ● April 2012
Liu ● Two-Stage Dynamic Multi-Period Portfolio Optimizations
approaches will be summarized in the following subsection.
3.2 Standard One-Stage Multi-period Approach
Firstly, we discuss the case where the portfolio weight w(n ) is deterministic, a standard one-stage method: By using
the Lagrange multiplier method, the optimal weight is straightly obtained from maximizing the following objective
function:
1

f s ( w( n ) | y0 )  E w(Tn ) r( n ) | y0   w(Tn )  ( n ) w( n )  s Var w(Tn ) r( n ) | y0    02 .
2
2

Here
s

is the Lagrange multiplier. Substituting the model structure stated in Section 2, into the above mentioned
formula, the objective function is re-written as:
1

f s ( w( n ) | y0 )  w(Tn )( n )  w(Tn )  ( n )  s ( n ) w( n )  s  02 .
2
2
Thus the optimal allocations under the standard one-stage algorithm, will be obtained by solving the equation,
( n )  ( n )  s( n ) w( n )  0 . The optimal solution resulted


wˆ ( n )   ( n ) ,
1
(n)

The optimal Lagrange multiplier

T
(n)

s
can be solved from the following equation:
s( n )  ( n ) ( n )  s( n )  ( n )   02 .
1
(n)

where ( n )   ( n )  s( n ) .
1
(6)
Then the optimal returns under the constant weight approach is
1
1
ˆ
f s wˆ ( n ) | y0   (Tn )  ( n )  ˆs ( n ) ( n )  s  02 .
2
2


And the overall net expected return per risk is defined as
Ratio n , S 

E wˆ (Tn ) r( n ) | y0  



1  1
1

1 T
T
wˆ ( n )  ( n ) wˆ ( n ) ( n )  ( n )  ˆs ( n )   ( n )  ˆs ( n )   ( n )  ˆs ( n ) ( n )
2

2
.

T

Var wˆ ( n ) r( n ) | y0 
0
4. TWO-STAGE DYNAMIC MULTI-PERIOD APPROACH
In this section, an innovative optimal portfolio problem, with a random portfolio vector and random transaction
costs are investigated. At each time epoch t , the current portfolio weight is determined conditional on the
information up to a given time epoch t  1 : At the first stage, starting from the beginning time epoch, a sequence of
optimal portfolio weight vector is period by period consecutively constructed. Following this an overall optimal
portfolio weight vector will be suitably combined together at the second stage to constitute a random portfolio
weights vector. This approach is denoted as a two-stage method. A detailed construction is discussed as follows:
4.1 Background of the Proposed Two-stage Approach
From model (3), the n-period returns r(n ) follow a multivariate normal distribution, say, r( n ) ~ MN ( n ) , ( n ) ,


where  (n ) are  (n ) defined by equation (4) and (5) respectively. In this subsection, instead of treating the portfolio
weight w(n ) as a deterministic vector, it is regarded as a random vector. By the conditional property of a
multivariate normal distribution, it turns out that, for i  1 ,
ri1 | r(i ) , y0 ~ MN i1.i , i1.i  , and wiT1ri1 | r(i ) , y0 ~ MN wiT1i1.i , wiT1i1.i wi1 , where
 i 1.i  i1   i1,(i ) 
1
( i ), ( i )
r
(i )

  (i ) , i 1.i  i 1,i 1  i 1,(i )

1
( i ), ( i )
(i ), i 1 ,  i , j  Covri , rj  ,
(i ), j  Covr(i ) , rj  , i , ( j )  Covri , r( j ) , and (i ), ( j )  Covr(i ) , r( j )  . And the initial conditional distribution is:


r1 | y0 ~ MN 1.0 , 1.0  , and w1T r1 ~ MN w1T 1.0 , w1T 1.0 w1 , where  1.0 1 and 1.0  ( n ) .
At the first stage, the current portfolio weight is constructed depending upon returns of all the preceding periods.
56
IJRRAS 11 (1) ● April 2012
Liu ● Two-Stage Dynamic Multi-Period Portfolio Optimizations
Temporarily define the objective function at the (i+1)-th period, for giving the previous returns r(i ) , as




1
1
hi 1 (wi 1 | r(i ) , y0 )  E wiT1ri 1 | r(i ) , y0  Var wiT1ri 1 | r(i ) , y0  wiT1i 1.i  wiT1i 1.i wi 1 , for i  1 .
2
2
Then the preliminary optimal solution of wi 1 is
~   1 
w
 a  AT r    , for i  1,2,, n  1 ,
i 1
i 1.i i 1.i
i 1
i 1
(i )
(i )
i 1 , Ai 1   (i ),i 1 , and i 1.i  i 1,i 1  i 1,(i )(i1), (i )(i ), i 1 . It is worthy to note that
~ is random; while after r is observed, w
~ is deterministic.
before r(i ) is observed, w
i 1
i 1
(i )
~ ,w
~ , , w
~ , the final
After the portfolio weight vectors for each period are temporarily obtained, denoted by, w
1
i 1.i
here ai 1  
1
( i ), ( i )
1
i 1.i
2
3
n
portfolio weight vector for the whole future period are constructed by a rolling scheme: Starting from the beginning
period, the portfolio weight denoted by w1 is assumed to be deterministic; Then the portfolio weights for the
~ ’s, for i  2 , are
~ , w
~ , ,  w
~ , consecutively. Thus at the beginning time, w
remaining periods are  2 w
i
2
3 3
n n
random, instead of deterministic. Now, according to the strategy, innovative optimal portfolio weights are obtained
by finding suitable w1 and  i , which maximize the following objective function:
~T r     w
~T
E w1T r1   2 w
2 2
n n rn | y0  
b

1 n
~  w
~ T
~
~
E  t w

t
t 1 t 1  t  t wt   t 1wt 1  | y0
2 t 1



2
~T r     w
~T
,
Var w1T r1   2 w
2 2
n n rn | y0    0
2
~ w , w
~  0 ,   1 ,   0 and  is the Lagrange multiplier.
where w
1
1
0
1
0
b

In order to obtain the optimal solution analytically, it is convenient to re-represent the objective function. Define
w( n ), b
 r1 
 w1 
 Ik

 w1 
 T 
 


 ~ 
~
~
w
r



w2
0 
2 2
 2

  2 w2 
~ T r  , and U  
~
~ , u    ,   w
,
   3w
w
(n)
3
3



 (n)  3  (n)  3 3 



 
 0



 ~T 
 

 w
~
~ 
 wn rn 
wn 
 n
 n n



then w( n ), b  U( n )u( n ) , w(Tn ), b r( n )  u(Tn ) ( n ) , w(Tn ), b  ( n ) w( n ), b  u(Tn )U(Tn )  ( n )U( n )u( n ) . Let
( n )  Var ( ( n ) | y0 ) and C( n )  E U(Tn )  ( n )U( n ) | y0  ; in detail:


~T r | y ,, E w
~T r | y  ,
( n )  E  ( n ) | y0   E r1 | y0 T , E w
2 2
0
n n
0
( n )
T
~T r | y  
~T r | y  
 Cov r1 , r1 | y0 
Cov r1 , w
Cov r1 , w
2 2
0
n n
0


T
T
T
T
T
~
~
~
~
~
 Cov w2 r2 , r1 | y0  Cov w2 r2 , w2 r2 | y0   Cov w2 r2 , wn rn | y0 

,



 Cov w
~T r , r | y  Cov w
~T r , w
~T r | y   Cov w
~T r , w
~T r | y 
n n 1
0
n n
2 2
0
n n
n n
0 

and
57
( n )  E ( ( n ) | y0 ) ,
IJRRAS 11 (1) ● April 2012
C( n )
 1  2

~T | y 
  E w
2
0
2




0



Liu ● Two-Stage Dynamic Multi-Period Portfolio Optimizations
~ |y 
 2 E w
2
0
~T    w
~ |y
~T  w
~
Ew
 E w
2
2
3
2
0
2
3 3 | y0 
~T  w
~
~T    w
~ |y
 E w
Ew
3
3 2 | y0 
3
3
4
3
0




T
~
~ |y 
 E wn 1n 1w
n2
0


0


T
~
~
 E w3 4 w4 | y0 
.


~T    w
~ |y
~T  w
~
Ew
 E w
n 1
n 1
n
n 1
0
n 1 n n | y0 

~T  w
~
~T  w
~
 E w
E w
n
n n 1 | y0 
n
n n | y0 




4.2 Two-stage Optimal Portfolio Solution
The objective function is re-written by notations stated in subsection 4.1, as:
1

hb (u( n ) | y0 )  E u(Tn ) ( n ) | y0   E u(Tn )U (Tn )  ( n )U ( n )u( n ) | y0   b Var u(Tn ) ( n ) | y0    02
2
2
1

 u(Tn )( n )  u(Tn )C( n )u( n )  b u(Tn )( n )u( n )   02  .
2
2
1
~
~
The optimal solution is u( n )  C( n )  b( n ) ( n ) , where ̂b is determined by the equation
1
1
~
~
(7)
(Tn ) C( n )  b( n ) ( n ) C( n )  b( n ) ( n )   02 .








Finally, the optimal return under the two-stage approach is summarized as:
~
1
1

~
hb (u~( n ) | y0 )  (Tn ) C( n )  b( n ) ( n )  b  02 .
2
2


And the overall net expected return per risk is defined as:
Ratio n , B


1
E u~(Tn ) ( n ) | y0   E u~(Tn )U (Tn )  ( n )U ( n )u~( n ) | y0 
2

Var u~(Tn ) ( n ) | y0 
~

1
1
2
~




~
1
(Tn ) C( n )  b( n )  C( n )  b( n )  C( n )  b( n ) ( n )
As the optimal
0
.
u~( n ) is determined, the optimal portfolio weight vector
~
is obtained, say
w
( n ), 
~ ~T T . The two-stage approach decides the portfolio weight vector period
~
~
~T ~ ~T ~ ~T
w
( n ),b  U ( n ) u( n )  w1 ,  2 w2 ,  3 w3 ,,  n wn 
~ is first decided, then once the
by period: Starting from the beginning period the optimal portfolio weight vector w
1
~ .
first period return r1 is observed, the optimal weight vector for the second period is determined by ~2 w
2
~ , which depends upon all the previous (n  1) returns
Analogically, the last optimal weight vector is decided by ~n w
n
r( n 1) . Finally, an adaptive approach of the optimal portfolio weight vector is thus consecutively established. Before
~ is random; whereas after r is observed, w
~ is deterministic. Therefore, at the beginning
r is observed, w
(i )
i 1
i 1
(i )
time, the portfolio weights for each future period are random, except for the first period weights.
~ depending upon the initial wealth
It is worthy to note that the discussed dynamic weight, w
i 1
previous returns r j , for
y0 and all the
j  i , instead of only the recently observed return ri , discussed by Liu [12]. To complete
~
the proposed allocation strategy, values of the Lagrange multiplier coefficients, b , and  (n ) ,  (n ) and C(n ) should
be explicitly expressed. The computing procedures are similar to those stated in the Appendix of Liu [12]. Some
results are sketched in the Appendix A.
Lemma 1. Computation of  (n ) :
~T r | y   aT   tr A 
E w
i 1 i 1
0
i 1 i 1
i 1 i 1, ( i ) , for i  1,2,, n  1 .
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IJRRAS 11 (1) ● April 2012
Liu ● Two-Stage Dynamic Multi-Period Portfolio Optimizations
Lemma 2. Computation of  (n ) :
Embedding matrix A j into a suitable
A~ 
*
j s ,l
~ ~
nk  nk matrix, say Aj , for j  2 , where Aj  A
, and
j  A j 
T

Aj s ,l  ( j 1) k , if 1  s  ( j  1)k & ( j  1)k  1  l  jk ,


otherwise.
0,
Aj is obviously a symmetric matrix, and satisfying the following properties:
1
T
T
(2.1) r( j 1)   ( j 1)  Aj rj   j   r    A*j r    .
2
T
~T r , w
~T r | y   a T  a  a T 
(2.2) Covw
i i
j j
0
i
i, j j
i
i , ( j 1) Aj j  a j  j , ( i 1) Aii
The re-constructed matrix
1
 iT AiT  ( i 1), ( j 1) Aj j  tr Ai*A*j   , for 2  i, j  n .
2
(2.3) Covr1, r1 | y0   Cov y1  y0 , y1  y0 | y0   Cove1, e1  .
~T r | y      1 
(2.4) Cov r , w
a , for i  1,2,, n  1 .

1
i 1 i 1
0
 
1,i 1
1, ( i )
( i ), ( i )
( i ), i 1

i 1
Lemma 3. Computation of C (n ) :
T
into a suitable nk  nk matrix, say C(i , j ) , for i  2 and J  2 , where
A
i Aj
Embedding matrix



~
~
~ T
C(i , j )  C(i , j )  C(i , j ) , and C(*i , j )

s,l
T

Ai Aj s ,l , if 1  s  (i  1)k & 1  l  ( j  1)k ,


otherwise.
0,
Similarly, the re-constructed matrix C(i , j ) is a symmetric matrix, and satisfying the following properties:

(3.1) r(i 1)   (i 1)

T
Ai ATj r( j 1)   ( j 1)  
~ | y  a .
(3.2) E w
2
0
2



1
r   T C(*i , j ) r    .
2

~T w
~ | y  aT a  1 tr C *  .
(3.3) E w
i
j
0
i
j
(i , j ) ( n )
2
The one-stage approach deterministically decides the n-period optimal portfolio weights at the beginning; while the
two-stage approach constructs the portfolio weight vector period by period: At the beginning of the i  1 -th period,
before previous information r(i ) was completely observed,
~ is random; whereas after r was observed, then
w
i 1
(i )
~ is deterministic. Comparisons between the two approaches, the standard one-stage and the proposed two-stage
w
i 1
will be discussed numerically in the next section. Before this is done however, one special case can first be
analytically investigated below.
Lemma 4: Suppose in model (1), y0  0 ,   0 , and without transaction costs, are assumed, then the expected
return to risk ratio for the proposed two-stage, is always greater than that under the one-stage approach.
Pf): Since y0  0 and   0 , this implies that  ( n )  0 ; moreover, by no transaction costs assumption, the two
objective functions are reduced as follows:


1
1
ˆ
f s wˆ ( n ) | y0   (Tn ) ˆs  ( n ) ( n )  s  02  ˆs 02  (Tn ) (n1)( n )  0  0
2
2
and
~
1
1

~
~
hb (u~( n ) | y0 )  (Tn ) b( n ) ( n )  b  02  b 02  (Tn )(n1)( n )  0  0
2
2


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IJRRAS 11 (1) ● April 2012
Liu ● Two-Stage Dynamic Multi-Period Portfolio Optimizations
5. NUMERICAL ILLUSTRATIONS
In this section, performing comparisons in terms of the overall net expected return per risk, among three approaches
the one-stage standard method and two two-stage dynamic methods, are numerically demonstrated: One of the two
two-stage methods is the one discussed by Liu [12] and the other is introduced in Section 4. In the following
numerical investigations, the former is referred as the two-stage method A, and the latter denoted as the two-stage
method B. A brief sketch of the development of the two-stage method A is given in Appendix B. The major
differences between the two-stage methods rely on the preliminary estimations of portfolio weight, wi 1 , at the
wi 1 under method A is conditioned only on information ri , while that under method B is
conditioned on information r(i ) , extracting more information.
(i+1)-stage; the defining
In detail, the preliminary (i+1)-th period objective function for the two-stage method A giving the previous returns
ri is defined as




1
g i 1 ( wi 1 | ri , y0 )  E wiT1ri1 | ri , y0  Var wiT1ri 1 | ri , y0 .
2
wi for the two-stage method A is to maximize the stated objective function (8).
While the preliminary estimate of

Since r( i )  r1T , r2T ,, ri
,
T T
(8)
ri is an element of r(i ) , it seems that method B contains more information than
method A; therefore we may expect that method B should be more efficient than method A.
In order to provide the scenario of a real financial market, the required parameters were set up by using historical
market data collected from the Taiwan Economic Journal Data Bank. Data were screened from the 19 Taiwan
Industrial Index, with only five of them included, denoted as k=5 in the following discussion: The five discussed
industrial indexes are, Building Construction, Financial and Insurance, Steel and Iron, Electronic and Electrical, and
Biotech.
5.1 Parameters Setup
To establish a suitable cointegration model like model (1), we data vector as the logarithm of the five Taiwan
industry index. k1  2 and k2  3 , and values of parameter are set as follows:
y0  1.2,1.5,0.8,0.5,0.9
T
 0.03 - 0.34 
 0.01 


 0.62 


 ,  2   0.05  and    - 0.52 - 0.63  .
1  
 1.50 
 1.37
 0.02 
1.64 



Furthermore, after numerical algebra, parameters in model (2) are obtained as:
0
 0.622 



0
1.498 
   0.010  , 1   0


0
 0.050 
0
 0.020 



For
simplicity,
0
0
0.03
- 0.34
0
0
1
0
0
0
1  I n  
- 0.52 1.37 
0 
 I5 0 0



- 0.63 1.64 
0 
 2 I 5 0
0
0  , ( n )   0 2 I 5
0  and 2  1  I 5 ,



1
0 
0 0 




0
1 
 0 0  2 I 5 
is assumed
for
some
symmetric
positive
definite
matrix
 . Since
( n )  I n  0 1 I n    , therefore ( n)  I n  W  . Here W is a 5 5 symmetric positive definite matrix

T
0
defined as:
 0.25

 0.86

W   0.20

 0.27
 0.38

0.86
0.82
0.20
0.24
0.24
0.32
0.30
0.18
0.48 0.20
0.27 0.38 

0.32 0.48 
0.18 0.20  .

0.40 0.30 
0.30 0.48 
After  (n ) is defined, then  (n ) is obtained from the relationship, ( n )  ( n )( n ) (Tn ) . In order to investigate the
60
IJRRAS 11 (1) ● April 2012
Liu ● Two-Stage Dynamic Multi-Period Portfolio Optimizations

impact of the covariance matrix  (n ) , different versions of W will be introduced. In the following discussions, for
simplicity, we set
( n )  I n  W , where W  W     I k .
The impacts of different values of “  ” will be numerically investigated.
The transaction cost in the Taiwanese stock market, includes a transaction fee of 0.1425% for buying, selling and
short selling stocks, a 0.3% of selling trading tax, and also a 0.08% for short selling fee. For simplicity, a fixed ratio
is used as a transaction cost for buying, selling and short selling, expressed as,
Transaction fee + (securities transaction tax + borrowing cost) / 2 = 0.3325%
For simplicity, the transaction matrix  (i ) is set to be equal, say i   , where  is a diagonal matrix with the
diagonal entry taking value 0.3325%. Thus, the overall transaction cost matrix  (n ) is simplified as below:
 (n)
 2  



   2  

0

.





  2   
 0

   

5.2 Numerical Results
Comparisons are focused on the optimal overall net expected return per risk, for convenience, key expression for
each method is summarized as follows: For the standard one-stage method, the net expected return per risk is:
Ratio n , S 
where
̂s




1
1
1
(Tn )  ( n )  ˆs ( n )  ˆs ( n )   ( n )   ( n )  ˆs ( n ) ( n )
is determined by


2
0


,

1

1
(Tn )  ( n )  ˆs( n ) ( n )  ( n )  ˆs( n ) ( n )   02 .
For the two-stage method A, proposed by Liu [12], the net expected return per risk is:
Ratio n, A 
where
̂a

 
1


1
2


1
 (Tn ) D( n )  ˆa ( n )  ˆa ( n )  D( n )  D( n )  ˆa ( n )  ( n )
0

,
is decided by the equation  (Tn ) D( n )  ˆa ( n )

1


1
( n ) D( n )  ˆa ( n )  ( n )   02 .
 ( n )  E ( ( n ) | y0 ) , ( n )  Var ( ( n ) | y0 ) ,  ( n )  r1T , wˆ 2T r2 , wˆ 3T r3 ,, wˆ nT rn  , and wˆ i 1 is obtained by
T
Here
gi 1 (wi 1 | ri , y0 ) defined by equation (8). D( n )  E V(Tn )  ( n )V( n ) | y0  , where V(n ) has a structure
~ by ŵ . For more detailed expressions about  , D and  refer to
similar to U (n ) , just replacing w
i
i
(n )
(n )
(n )
maximizing
Appendix B.
For the two-stage method B, the net expected return per risk is:
Ratio n , B 
where




1 ~
1
~
~
1
(Tn ) C( n )  b( n )  b( n )  C( n )  C( n )  b( n ) ( n )

2
0

,
1
1
~
~
~
b is decided by the equation (Tn ) C( n )  b( n )  ( n ) C( n )  b( n )  ( n )   02 .
For convenience, we set the two-stage method B as the benchmark for comparisons. Define the two ratios,
Ratio n , B
and Ratio _ 2  Ratio n , B
;
Ratio _ 1 
Ratio n , A
Ratio n , S
the former ratio explores the efficiency between the two-stage method B and the one-stage method; while the latter
ratio provides impacts on the two two-stage methods, which are constructed under different conditional information.
The numerical results indicate that the performance of the two-stage method B dominates the other two methods, in
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IJRRAS 11 (1) ● April 2012
Liu ● Two-Stage Dynamic Multi-Period Portfolio Optimizations
particular, as the time period n gets longer. When values of  become larger, volatilities enhance, and the net
expected returns for either the one-stage method or the two-stage method B decrease; however, the former declines
faster than the latter. Referring to the two-stage method A, the results seem not as stable as comparing to that of the
two-stage method B. In the meanwhile, both two-stage methods could obtain larger net expected returns than the
one-stage method, as data volatilities increase.
As the time horizon gets longer, the two-stage method B outperforms the one–stage method: Since the two-stage
method B obtains more previous information therefore net expected returns significantly increase. In comparisons
between the two-stage, method B and method A, the numerical results showed that method B is always more
efficient that method A. At each current time, method B determines the portfolio weights based on all the previous
information, as opposed to method A, which only utilizes information from one period ahead. Thus more
information may introduce higher net expected returns. In particular, as the discussed period gets longer, the
efficiency becomes more significant. However, when the time horizon is shorter, as volatilities increase, the
superiority of method B vanishes gradually and the performances of methods A and B tend to be close. In summary,
the two-stage dynamic approaches are more efficient that the static approach and suitably extracting useful
information will be helpful to construct more effective portfolio selection procedure. All the discussed results are
exhibited in Table 1.
n
2
3
4
6
8
10
20
0.0
0.5
1.0
5.0
10.0
0.0
0.5
1.0
5.0
10.0
0.0
0.5
1.0
5.0
10.0
0.0
0.5
1.0
5.0
10.0
Table 1. Comparisons of Net Expected Return per Risk
One-stage
Two-stage
standard
method A method B Ratio_1 Ratio_2
0.9808
1.0753
1.0753
1.0963 1.0000
0.8231
1.0157
1.0157
1.2340 1.0000
0.7464
0.9807
0.9807
1.3139 1.0000
0.5029
0.8744
0.8744
1.7387 1.0000
0.3887
0.8372 0.8372
2.1540 1.0000
1.1961
1.1856
1.3609
1.1378 1.1479
1.0085
1.1852
1.3166
1.3055 1.1109
0.9121
1.1750
1.2853
1.4092 1.0939
0.6121
1.1190
1.1860
1.9376 1.0599
0.4769
1.0928 1.1528
2.4137
1.0549
1.3791
1.3213
1.5935
1.1555 1.2060
1.1590
1.3503
1.5614
1.3472 1.1563
1.0516
1.3539
1.5292
1.4542 1.1295
0.7058
1.3201
1.4307
2.0271 1.0838
0.5501
1.3033 1.3982
2.5417 1.0728
1.6820
1.5490
1.9825
1.1787 1.2799
1.4156
1.6287
1.9585
1.3835 1.2025
1.2856
1.6533
1.9255
1.4977 1.1646
0.8649
1.6534
1.8231
2.1079 1.1027
0.6712
1.6419 1.7902
2.6667 1.0903
0.0
0.5
1.0
5.0
10.0
1.9366
1.6327
1.4814
0.9972
0.7760
1.7467
1.8655
1.9032
1.9285
1.9241
2.3052
2.2889
2.2561
2.1460
2.1105
1.1903
1.4019
1.5230
2.1520
2.7197
1.3198
1.2270
1.1854
1.1128
1.0969
0.0
0.5
1.0
5.0
10.0
0.0
0.5
1.0
5.0
10.0
2.1615
1.8203
1.6526
1.1140
0.8658
3.0285
2.5545
2.3189
1.5657
1.2188
1.9261
2.0736
2.1260
2.1715
2.1680
2.6413
2.9054
3.0076
3.2248
3.2415
2.5877
2.5755
2.5422
2.4264
2.3894
4.5480
4.5471
4.4524
4.1004
3.9971
1.1972
1.4149
1.5383
2.1781
2.7596
1.5017
1.7800
1.9208
2.6189
3.2794
1.3435
1.2420
1.1958
1.1174
1.1021
1.7219
1.5650
1.4804
1.2715
1.2331

Note:
1. ( n )  I n  W , W  W     I k .
2. Ratio_1=two-stage method B/one-stage standard method.
3. Ratio_2=two-stage method B/ two-stage method A.
62
IJRRAS 11 (1) ● April 2012
Liu ● Two-Stage Dynamic Multi-Period Portfolio Optimizations
6. CONCLUSIONS
This paper provides an innovative two-stage dynamic multi-period portfolio selection approach with random
portfolio weights and random quadratic transaction costs. The underlying assets time series data are supposed to
follow a discrete-time triangle cointegrated model with vector autoregressive noise structure. The utilized meanvariance algorithm is focused on the one that maximizes the overall multi-period net expected returns, under a prescribed overall risk level, and a given initial asset value. Based on the proposed two-stage approach, the optimal
portfolio allocation is analytically derived, expressed in a closed form. Moreover, some numerical results show that
the proposed algorithm is tremendously more efficient than the static approach in particular, as the discussed time
horizon becomes longer or data volatilities increase, higher net expected returns are indicated. Moreover, the
performances between the two two-stage dynamic approaches exhibit that the one proposed in this article, which
applies more previous information outperforms the other approach, which utilizes less information.
ACKNOWLEDGEMENTS
The author wishes to thank Miss. Meng-Yu Chan for her assistance in carrying out some data collection and
parameter estimations.
APPENDIX A
Proofs of Lemma 1-3
The conditional property of a multivariate normal distribution: for i  1 ,
ri 1 | r(i ) , y0 ~ MN i 1.i , i 1.i  , and wiT1ri1 | r(i ) , y0 ~ MN wiT1i1.i , wiT1i 1.i wi 1 , where
 i 1.i  i 1   i 1,(i ) 
1
( i ), ( i )

r
(i )
  (i ) , i 1.i  i 1,i 1  i 1,(i )




1
( i ), ( i )
(i ), i 1 ,  i , j  Covri , rj  , (i ), j  Covr(i ) , rj  ,
i , ( j )  Covri , r( j ) , and (i ), ( j )  Cov r(i ) , r( j ) . Therefore, the preliminary optimal weight is,
T
~   1 
, here ai 1  i11.ii 1 , Ai1   (i1),(i )  (i ),i1i11.i , and
w
i 1
i 1.i i 1.i  ai 1  Ai 1 r( i )   ( i )

i 1.i  i 1,i 1  i 1,(i )(i1), (i )(i ), i 1 . Before developing the proofs of Lemma 1-3, for convenience, some
preliminary results are stated without proofs as follows:
Property A1: Let Y be a random vector and normally distributed, with mean  and covariance matrix V , and
qi  Y T DiY , here Di is a symmetric matrix, then
(A1) E qi   tr DiV    T Di 
(A2) Var qi   2tr DiVDiV   4 T DiVDi 




(A3) Cov qi , q j  2tr DiVD jV  4 T DiVD j 
(A4) CovY , qi   2VDi 
Two more properties are briefly shown as follows, for more detailed derivations please refer to Liu [12]:


Property A2: Cov ri  i , r( j )  ( j )


Pf): Cov ri   i , r( j )   ( j )


T

T

Aj 1rj 1 | y0  i ,( j ) Aj 1 j 1
Aj 1rj 1 | y0



 Cov ri  i , r( j )  ( j )  Aj 1 rj 1   j 1  | y0  Cov ri  i , r( j )  ( j )  Aj 1 j 1 | y0
T


T

1
T
 Cov ri  i , r    A*j 1 r    | y0  i ,( j ) Aj 1 j 1  i ,( j ) Aj 1 j 1
2
1
T
T
Property A3: Cov r( i 1)  ( i 1)  Ai ri , r( j 1)  ( j 1)  Aj rj | y0  tr Ai*A*j    iT AiT  (i 1),( j 1) Aj j
2
T
T
Pf): Cov r( i 1)  ( i 1)  Ai ri , r( j 1)  ( j 1)  Aj rj | y0









1
1
T
T
T
T
 Cov r    Ai* r   , r( j 1)  ( j 1)  Aj j | y0  Cov r( i 1)  ( i 1)  Aii , r    A*j r    | y0
2
2




1
T
T
T
T
 Cov r(i 1)  (i 1)  Aii , r( j 1)  ( j 1)  Aj j | y0  Cov r    Ai* r   , r    A*j r    | y0
4
63

IJRRAS 11 (1) ● April 2012
Liu ● Two-Stage Dynamic Multi-Period Portfolio Optimizations



1
T
T
 Cov r(i1)   (i1)  Ai i , r( j 1)   ( j 1)  Aj j | y0  Cov r   T Ai* r   , r   T A*j r    | y0
4
1
  iT AiT  (i 1),( j 1) A j j  tr Ai*A*j  
2
Proof of Lemma 1:
 

~T r | y  E aT r  r   T A r | y
Ew
i 1 i 1
0
i 1 i 1
(i )
(i )
i 1 i 1
0


 a  i1  E r(i )   (i )  Ai1i1,(i )  (i1),(i ) r(i )   (i )  | y0
T
i 1
T
 aiT1 i1  tr Ai1i1,(i )  , for i  1 .
Proof of Lemma 2:



T
~T r , w
~T r | y   Cov aT r  r   T A r , aT r  r
Covw
i i
j j
0
i i
( i 1)
( i 1)
i i
j j
( j 1)  ( j 1)  Aj rl | y0

1
 aiT  i , j a j  aiT  i ,( j 1) Aj j  a Tj  j ,( i 1) Aii  iT AiT  ( i 1),( j 1) Aj j  tr AiAj  ,
2

~T r | y   Covr   , aT r | y   Cov r   , r   T A r | y
Covr1, w
i 1 i 1
0
1
1
i 1 i 1
0
1
1
(i )
(i )
i 1 i 1
0
 1,i 1ai 1  1,(i ) Ai 1i 1  1,i 1  1,(i )
Proof of Lemma 3:



 a a j
 aiT a j
(i ), i 1 ai 1
 a  A r   | y 
| y 
   A A r

1
1
 E r    C r    | y   a a  tr C
2
2

 E r
~T w
~ | y  E a  AT r  
(3.3) E w
i
j
0
i
i
( i 1)
( i 1)
T
i
1
( i ), ( i )
T
T
( i 1)
( i 1)
T
T
j
j
i
T
j
*
(i, j )
( j 1)
0
( j 1)
( j 1)
( j 1)
T
i

0
0
j
*
(i, j )
( n ) 
APPENDIX B
Reviews of the Two-stage Method A
The following results are summarized from Liu [12]. Firstly, the preliminary objective function for the (i+1)-th
period, giving previous return ri , was defined as
1
1
gi 1 ( wi 1 | ri , y0 )  E wiT1ri 1 | ri , y0   Var wiT1ri 1 | ri , y0   wiT1 i 1.i  wiT1i 1.i wi 1 ,
2
2
here  i 1.i i 1  i 1,i i,1i ri  i  , i 1.i  i 1,i 1  i 1,i i,1i i ,i 1 ,  i , j  Covri , rj  . Thus the preliminary
optimal solution of

1
i 1.i i 1.i
wˆ i 1  
wi 1 , denoted by wˆ i 1 , is derived as:
 bi 1  BiT1 ri  i  , where bi 1  i 11.ii 1 and Bi 1  i,1i i ,i 1i 11.i .
Next, the overall optimal portfolio weights are obtained, by finding suitable
following objective function:
E w1T r1   2 wˆ 2T r2     n wˆ nT rn | y0  

a
2

1 n
T
E t wˆ t  t 1wˆ t 1  t t wˆ t  t 1wˆ t 1  | y0

2 t 1
Var w r   wˆ r     wˆ r | y    ,
T
1 1
2
T
2 2
n
T
n n
w1 and  i , which maximize the
0
2
0
where w
ˆ 1  w1 , wˆ 0  0 , 1  1 ,  0  0 and a is the Lagrange multiplier.
To re-organize the objective function, the following notations are introduced:
64

IJRRAS 11 (1) ● April 2012
w( n ), a
Liu ● Two-Stage Dynamic Multi-Period Portfolio Optimizations
 w1 
 w1 
 Ik



 
 r1 


 T 
ˆ

w
 2 2
 2 
wˆ 2
0 

wˆ 2 r2 

   3wˆ 3  , v( n )    3  ,
.

 , and V  
wˆ 3
(n)


   ( n )   wˆ 3T r3 




 



 0

 T 
  wˆ 
 

 wˆ n rn 
wˆ n 
 n n
 n



Then w( n ), a  V( n ) v( n ) , w(Tn ), a r( n )  w1T r1   2 w
ˆ 2T r2    n w
ˆ nT rn  v(Tn ) ( n ) , and
w(Tn ), a ( n ) w( n ), a  v(Tn )V(Tn ) ( n )V( n )v( n ) , and the overall objective function is re-written as:
1

g a (v( n ) | y0 )  v(Tn ) ( n )  v(Tn ) D( n ) v( n )  a v(Tn ) ( n )v( n )   02 ,
2
2
here  ( n )  E ( ( n ) | y0 ) , ( n )  Var ( ( n ) | y0 ) and D( n )  E V(Tn )  ( n )V( n ) | y0  .



D
The optimal solution is vˆ( n )  D( n )  ˆa ( n )


1
 (Tn ) D( n )  ˆa ( n ) ( n )
1
(n)
(n)

, where

̂a
is decided by solving the equation
1
 ˆa ( n )  ( n )   02 .
Finally, the optimal net returns under the two-stage method A is:


1
1
ˆ
g a (vˆ( n ) | y0 )   (Tn ) D( n )  ˆa ( n )  ( n )  a  02 .
2
2
And the overall net expected return per risk is defined as:
1 1
1
 (Tn ) D( n )  ˆa ( n )  D( n )  ˆa ( n )  D( n )  ˆa ( n )  ( n )
2

,
Ratio n , A 




0
The detailed representations of
(B1)

 (n ) , (n ) , and D(n )
are summarized as follows:

 (Tn )  E  ( n ) | y0 T  E r1 | y0 T , E wˆ 2T r2 | y0 ,, E wˆ nT rn | y0  , where


ˆ iT1ri 1 | y0  biT1i 1  tr Bi 1i 1,i  , for i  1,2,, n  1 .
E r1 | y0   1 and E w
(B2) ( n )
 Covr1 , r1 | y0 
Cov r1 , wˆ 2T r2 | y0  Cov r1 , wˆ nT rn | y0  


 Cov wˆ 2T r2 , r1 | y0  Cov wˆ 2T r2 , wˆ 2T r2 | y0 Cov wˆ 2T r2 , wˆ nT rn | y0 

 , where



 Cov wˆ T r , r | y  Cov wˆ T r , wˆ T r | y Cov wˆ T r , wˆ T r | y 
n n 1
0
n n
2 2
0
n n
n n
0 

Covr1, wˆ iT1ri 1 | y0   1,i 1  1,i i,1i i ,i 1 bi 1 , for i  1,2,, n  1 ; and
Covwˆ iT ri , wˆ Tj rj | y0   biT i , jb j  biT i , j 1B j j  bTj  j ,i 1Bii
1
 iT BiT i 1, j 1B j j  tr Bi*B*j   , for 2  i, j  n .
2
T


ˆ
Here, for j  2 , B j  B j  Bˆ j , and
 
Bˆ 
*
j s ,l

B j s  ( j  2 ) k ,l  ( j 1) k , if ( j  2)k  1  s  ( j  1)k & ( j  1)k  1  l  jk ,


otherwise.
0,
65
IJRRAS 11 (1) ● April 2012
Liu ● Two-Stage Dynamic Multi-Period Portfolio Optimizations
2
 E wˆ 2 | y0 




T
T
T
0
  E wˆ 2 | y0  2 E wˆ 2 wˆ 2 | y0   E wˆ 2 wˆ 3 | y0 



T
T
T
 E wˆ 3 wˆ 2 | y0  2 E wˆ 3 wˆ 3 | y0  E wˆ 3 wˆ 4 | y0 
(B3)
,
D( n )  





0
 E wˆ nT1wˆ n  2 | y0  2 E wˆ nT1wˆ n 1 | y0  E wˆ nT1wˆ n | y0 



 E wˆ nT wˆ n 1 | y0 
E wˆ nT wˆ n | y0  

ˆ 2 | y0   b2 ; for i  2 & j  2 , E wˆ iT wˆ j | y0   biT b j  1 tr D( i , j )  , with
here E w
2
T
 Bi B j s(i2) k ,l ( j 2) k , if (i  2)k  1  s  (i  1)k

T
D( i , j )  Dˆ ( i , j )  Dˆ ( i , j ) , and Dˆ (i , j )  
& ( j  2)k  1  l  ( j  1)k ,

s ,l
0,
otherwise.










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