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Citation for published version
Zhou, Zhongbao and Zeng, Ximei and Xiao, Helu and Ren, Tiantian and Liu, Wenbin (2018)
Multiperiod portfolio optimization for asset-liability management with quadratic transaction costs.
Journal of Industrial & Management Optimization, 15 (3). pp. 1493-1515. ISSN 1553-166X.
DOI
https://doi.org/10.3934/jimo.2018106
Link to record in KAR
https://kar.kent.ac.uk/73737/
Document Version
Multiperiod portfolio optimization for asset–liability management
with quadratic transaction costs
Zhongbao ZHOUa一
Ximei ZENGa Helu XIAOb,a Tiantian RENa Wenbin LIUc,a
a. School of Business Administration, Hunan University, Changsha 410082, China b. Business School, Hunan Normal University, Changsha 410081, China
c. Kent Business School, University of Kent, Canterbury, CT2 7PE, UK
Abstract:
T
his paper investigates the multiperiod asset-liability management problem withquadratic transaction costs. Under the mean-variance criteria, we construct tractability models with/without the riskless asset and obtain the pre-commitment and time-consistent investment strategies through the application of embedding scheme and backward induction approach, respectively. In addition, some conclusions in the existing literatures can be regarded as the degenerated cases under our setting. Finally, the numerical simulations are given to show the difference of frontiers derived by different strategies. Also, some interesting findings on the impact of quadratic transaction cost parameters on efficient frontiers are discussed.
Keywords: Asset-liability management; Multiperiod portfolio optimization; Quadratic
transaction costs; Pre-commitment strategies; Time-consistent strategies
1 Introduction
Asset-liability management (ALM) is a general risk management problem for financial services companies, such as pension funds and insurance companies. Typically, ALM involves the management of assets in such a way as to earn adequate returns while maintaining a comfortable surplus of assets over existing and future liabilities. Since the seminal work of Sharpe (1990), many attempts have been made to solve the ALM problem, among which the mean-variance criteria presented by Markowitz (1952) is of great importance. Actually, the mean-variance asset-liability management (MVALM) problem is a portfolio optimization problem, so as to realize the trade-off between the expectation of the terminal surplus maximization and minimum risk measured by the variance of the terminal surplus. Keel and Müller (1995) studied the asset-liability management problem in a single period setting, and validated the significant effect of liability on efficient frontier. However, the multiperiod MVALM problem faces with the difficulties in solving the analytical solution
一
due to non-separability of variance. In this sense, the multiperiod mean-variance ALM problem cannot be directly solved by the dynamic programming approach.
Up to now, there are two mainstream approaches are applied to deal with this time-inconsistent problem. One is the embedding method initiated by Li and Ng (2000) and Zhou and Li (2000) in multiperiod and continuous-time portfolio optimization respectively, and the corresponding optimal investment strategy is called the pre-commitment strategy. Leippold et al. (2004) firstly applied the embedding method in multiperiod asset-liability management to acquire the closed form of efficient frontier. Subsequently, Xie et al. (2008) modeled the uncontrollable liability under the framework of Zhou and Li (2000) in continuous-time setting. Chang (2015) considered the issue of asset-liability management and derived effective strategy through the dynamic programming and Lagrangian duality theory. And he further validated the impact of liability on investment strategies. More studies can be found in Chen et al. (2008), Chen and Yang (2011) and Bensoussan et al. (2013). The other is the game approach, which is firstly developed by Bjork and Murgoci (2010). In this case, this optimization problem is treated as a non-cooperative game, in which the strategies at different period are determined by different players aiming at optimizing their own target functions. Nash equilibrium of these strategies was then utilized to define as the time-consistent strategy for the agent of the original problem. Wei et al. (2013) provided the first study in the time-consistent solution of the mean-variance asset-liability management. And the time-consistent strategy is derived in continuous-time setting. For more researches regarding the time-consistent strategy of asset-liability management, readers may refer to Li et al. (2012), Chen et al. (2013) and Long and Zeng (2016). Besides, some scholars have applied
genetic algorithms
to portfolio optimization for numerical solutions, such as Guo (2016), Li(2015), Yu et al. (2012, 2009, 2008).
However, these studies do not take into account market frictions, such as transaction costs. It is generated by investors to aggressively adjust their portfolio for the goal of the maximum profit and risk minimization. For the institutional investors engaged in bulk trading, transaction costs are particularly high. Thus, how to effectively allocate financial assets in the presence of transaction costs is a key problem to be solved. Further, Arnott (1990) found that ignorance of transaction costs would lead to invalid portfolios through empirical study.
Yoshimoto (1996) once again proved this judgment. Nevertheless, portfolio optimization with transaction costs has been an insurmountable problem. In order to obtain analytical solutions, Fu et al. (2015) represented a two-stage portfolio including a risky asset and a riskless asset, and deduced an analytical expression of the investment strategy when considering the proportional transaction cost. However, this approach is limited to one or two investment stages, and also the investor can only invest one riskless asset and one risky asset. To deal with this dilemma, Gârleanu and Pedersen (2013) promoted the optimal feedback solution for dynamic portfolios with a quadratic transaction cost which was followed by some researchers, such as Boyd et al. (2014), DeMiguel et al. (2015) and Zhang et al. (2017). As far as we know, there is very little research focus on ALM problem with transaction costs. Papi et al. (2006) considered the proportional transaction cost in ALM problem and proposed an approximation method based on the classical dynamic programming algorithm. Though the method reduces the computational and storage requirement of algorithm, it fails to acquire the analytical solutions.
Motivated by the difficulties for multiperiod asset-liability management problem with transaction costs, we provide the tractability framework to obtain the analytic solutions, which considers the quadratic transaction costs adopted by Gârleanu and Pedersen (2013). Since investors tend to pursue the goal of maximizing ultimate surplus, not the wealth of a particular period. We take the ultimate surplus of investment as the optimization target and cover the wealth accumulation of investment process, which is different from Gârleanu and Pedersen (2013). We then derive the pre-commitment and time-consistent investment strategies by applying the embedding scheme and backward induction approach, respectively. Also, we obtain the analytical expressions for the optimal investment strategies, the corresponding expectation and variance of surplus and the expected transaction costs. What is more, we study two cases, namely, the market containing a riskless asset and the investment without riskless assets. Finally, some numerical simulations are presented to compare the frontiers from different strategies and further verify the formulations derived in this paper.
The rest of this paper is organized as follows. In Section 2, we formulate the multiperiod MVALM problem containing a riskless asset with quadratic transaction costs. The pre-commitment strategy and time-consistent strategy are solved in Section 3. In Section 4,
we consider the portfolio without riskless assets and derive the pre-commitment and time-consistent strategies. In Section 5, some numerical simulations are presented to show our findings for different strategies. Section 6 concludes this whole paper.
2 Problem formulation
Consider a capital market with n1 assets and an investment process for T periods. Here, asset 0 is a riskless asset with a constant return rate 0
t
r while asset i is a risky asset with a random return rate i
t
e
at period t for i1,2,...,n and t0,1,...,T 1. It is assumed that the vector [ 1, 2, , n]t t t
t e e e
e , are statistically independent and return
e
t has a known mean ( )[ ( 1), ( 2), , ( n)] t t t t E e E e E e eE and a known covariance matrix
)
,
(
)
,
(
)
,
(
)
,
(
)
,
(
)
,
(
)
,
(
)
,
(
)
,
(
2 1 2 2 2 1 2 1 2 1 1 1 n t n t t n t t n t n t t t t t t n t t t t t t te
e
Cov
e
e
Cov
e
e
Cov
e
e
Cov
e
e
Cov
e
e
Cov
e
e
Cov
e
e
Cov
e
e
Cov
, (2.1)which is supposed to be positive definite. The investor allocates his initial wealth W0
among all the securities in the market at initial time, along with the accumulation of wealth, and then adjust the amount of investment for each asset at period t . In order to better describe the investment process, we define i
t
v , i1,2,...,n, as the investment amount on risky asset i which is allowed short selling and { 1 , 2,..., n}
t t t t v v v v as the
adjustment amount on risky asset i at period t . Therefore, the investment amount on riskless asset is vt0 Wt1Ivt , and the adjustment amount on riskless asset vt0 is equal to 1vt based on the self-financing assumption.
In addition, suppose that the investor has an exogenous liability. The initial liability is
0
L . Let qt be the return of liability at the t -th investment period, where (et,qt) is statistically independent. We diagonalize the co-variance vector about the liability and risky assets, denoted as 0 ( ( , 1), , ( , n))
t t t
t
t diag Cov q e Cov q e
. Therefore, we have
t t
t q L
and the surplus at period t can be expressed as St WtLt.
We follow the quadratic transaction costs adopted in Gârleanu and Pedersen (2013). Under this setting, the transaction cost (TC) associated with trading volumes vt is given by
1 0 1 0 T t t t T t t T C v v TC . (2.3)where is a symmetric positive-definite matrix measuring the level of total trading costs. Note that the transaction cost Ct depicts the expense arising from changes on investment amounts at period t rather than trading shares shown in Gârleanu and Pedersen (2013). Trading volume vt moves the average price by vt , and it leads to a total transaction costs according to T period, which can be denote as TCT. More importantly, we assume that the transaction cost is paid beyond the amount of investment wealth Wt, that is, the transaction cost is independent of Wt. Obviously, the transaction cost is regarded as an undesirable payment, and it should be minimized in the objective function.
Let v(t){vt,vt1,...,vT1} be the strategy at period t, and then the multiperiod asset-liability management problem with quadratic transaction costs can be expressed as:
) ( ) ( ) ( max )) ( , ( ( ) ( ) ( ) t t v T t t v T t t v T t t J v t E S J Var S J E TC J F
, (2.4) where v(t) T S and v(t) TTC denote the terminal surplus and the total transaction cost
corresponding to the investment strategy v(t), respectively. In addition, Jt denotes the information at period, and
(0,) is risk-aversion coefficient,
(0,) is cost-aversion coefficient.The difficulty in solving the problem Ft(Jt,v(t)) is cause by the non-separability of variance. That is, it does not satisfy the Bellman optimality principle. Therefore it can not be directly solved by dynamic programming approach. In the following, we will adopt embedding scheme and backward induction to solve this problem. According to the idea of Li
and Ng (2000), for the pre-commitment strategy, we embed it into a separable auxiliary problem which can be solved by dynamic programming. Then the solution of the original problem can be obtained by the following theorem.
Theorem 2.1. If ( ) { , ,..., * } 1 * 1 * *
v t vt vt vT is the optimal strategy for the auxiliary problem ) ( ) ) (( ) ( max )) ( , ( ~ ( ) () 2 ( ) t t v T t t v T t t v T t t J v t E S J E S J E TC J F
, (2.5)then v*(t) is also the optimal strategy for the problem F(J , v(t)) t t for ) ( 2 1 * T t S E
. ( *) T t SE denotes the expectation of the investors’ final surplus when he invests according to the optimal strategy v*(t) . The proof of theorem 2.1 is detailed in Appendix A.
According to Theorem 2.1, the pre-commitment strategy can be obtained by the following steps:
1) We first construct the auxiliary problem
) ( ) ) (( ) ( max () () 2 () t t v T t t v T t t v T J E S J E TC J S E
, (2.6)which is a separable structure in the sense of dynamic programming.
2) Through the idea of dynamic programming, we obtain the solution vˆ t( ) of the auxiliary problem, and vˆ t( ) is a function of
.3) By iterating each period of the strategy vˆ t( ) with the state transition equation of surplus, it is easy to find the expected final surplus Et(ST) , which is a function of
. Then, by using equation
12
Et(ST) and the expression of Et(ST) for
, the pre-commitment strategy v*(t) for the problem F(J , v(t))t
t is solved.
Also, the problem Ft(Jt,v(t)) can be solved by the time-consistent strategy. Bjork and Murgoci (2010), from a mathematical point of view, proves the application of Nash equilibrium strategy to solving time-inconsistent problems. Then Wu (2013) investigates the
time-consistent Nash equilibrium strategies for a multiperiod mean-variance portfolio selection problem. Mathematically, the time-consistent strategy can be defined as follows.
Definition 2.1. Let v~ be a fixed control law. For an arbitrary point
(
01,,...,T 1),one selects an arbitrary control value v and define the strategy } ~ ,..., ~ , { ) ( 1 1
v
v v vT . Then v~ is call as the time-consistent strategy if for allT
, it satisfies )) ( ~ ; ( )) ( ; ( max ) ( F J v
F J v
v (2.7)Let v~ t( ) be the time-consistent strategy at period t, Definition 2.1 makes it possible to solve the problem by the following procedures:
1) )] ( ) ( ) ( [ max arg ~ ) 1 ( ~ 1 ) 1 ( 1 ) 1 ( 1 ) 1 ( v 1 1 -T T T v T T T v T T T v T T J TC E J S Var J S E v T v
(2.8)2) Given that the decision maker T 1 will use v~T1, ~vT2 is the optimal strategy
by optimizing objective function FT2(JT2;(vT-2,v~T-1));
3) Generally, given that the forthcoming decision makers t1,...,T1 choose the strategy v~(t1)(~vt1,...,v~T1), v~t is obtained by letting decision maker t choose
t v to maximize Ft. That is )) ~ ,..., ~ , ( ; ( max arg ~ 1 1 vt v Ft Jt vt vt vT t (2.9) For a mean-variance investor, the pre-commitment as well as time-consistent strategies are available. We will show them in the following sections.
3 Analytical solutions of multiperiod MVALM problem with a riskless asset
In this section, we consider the market with a riskless asset and derive the analytical solutions which contain pre-commitment strategy and time-consistent strategy. The corresponding investment strategies, the expectation and variance of surplus and the expected transaction costs are showed in this section.
To sum up, the formulation for the market with a riskless asset can be expressed by the following model:
1
,...,
1
,
0
,
'
,...,
1
,
0
,
1
,...,
1
,
0
,
1
,...,
1
,
0
,
1
,...,
1
,
0
),
(
1
,...,
1
,
0
),
(
.
.
)
(
)
(
ar
)
(
max
))
,
(
(
1 0 1 0 1 1 1 0 0 0 1 1 ) (T
t
v
v
TC
T
t
L
W
S
T
t
L
q
L
T
t
v
v
I
W
T
t
Δv
I
v
r
v
T
t
v
v
G
v
t
s
TC
E
S
V
S
E
P
T t t t T t t t t t t t t t t t t t t t t t T t T t T t t v
(3.1)where
(0,) is the risk-aversion coefficient,
(0,) is the cost-aversion coefficient. For a specific investor, and are constant.3.1 Pre-commitment strategy for problem (P(
,
))As the non-separability of variance in problem (P(
,
)), the objective function does not meet the requirement of dynamic programming approach. Thus, according to the Theorem 2.1, we first construct the auxiliary problem (A(
,
,
)) and solve the problem (P(
,
)) based on solutions of problem (A(
,
,
)).
1 ,..., 1 , 0 , ' ,..., 1 , 0 , 1 ,..., 1 , 0 , 1 ,..., 1 , 0 , 1 ,..., 1 , 0 ), ( 1 ,..., 1 , 0 ), ( . . ) ( ) ( ) ( max )) , , ( ( 1 0 1 0 1 0 0 0 1 1 2 ) ( T t v v TC T t L W S T t L q L T t v v I W T t Δv I v r v T t v v G v t s TC E S E S E A T t t t T t t t t t t t t t t t t t t t t t T t T t T t t v
(3.2)Obviously, (A(
,
,
)) is a separable structure in the sense of dynamic programming. According to Theorem 2.1, we can obtain the optimal asset allocation and the optimal value of objective function by solving the analytical solution of auxiliary problem (A(
,
,
)). For convenience, we list the notations of this section as following.0 ~ , ~ ~ , ~ , 2 ~ , 2 ~ ~ , ~ ~ , ~ T T T T T T T T T T II N O P Q I R X I Y Z K M
1 0 2 0 1 ), ~ ( ) , ( )~ ~ ( t t t t t 1 t t t t t t E GM G
N r
r E G P
, t 0,1,2,...,T 1 ] ' ' ' [ 2 t t t t t
II
I I
, t 0,1,2,...,T 1 0 1 0 1 1, ~ ( ), ~ ~ ) ( t t t t t t t t t t t E qG Q
Rr E q
Yr
, t0,1,2,...,T1 ] ) ( [ ~ 1 1 E G x I At
t t t
t , t0,1,2,...,T 1 ] [ ~ 1 I Bt
t
t
t , t0,1,2,...,T 1 ] 2 [ ~ 1 t t t t I C
, t0,1,2,...,T 1 ]' 2 [ ~ 1 t t t t I D
, t0,1,2,...,T1 1 ,..., 2 , 1, 0 , ~ ~ ~ , ~ ~ ~ , ~ ~ ~ M1 M2 N N1 N2 O O1 O2 t T Mt t
t t t
t t t
t 2 1 2 1 2 1 ~ , ~ ~ ~ , ~ ~ ~ ~ ~ t t t t t t t t t P P Q Q Q R R R P
, t0,1,2,...,T1
~ ~ , ~ ~ / ~ ~ 1 2 3 t t t t t t X X X x X X , t0,1,2,...,T 1
~ ~ , ~ ~/ ~ ~ 1 2 3 t t t t t t Y Y Y y Y Y , t0,1,2,...,T 1 3 2 1 ~ ~ ~ ~ t t t t Z Z Z Z
, t0,1,2,...,T 1 3 2 1 ~ ~ ~ ~ t t t t K K K K
, t0,1,2,...,T 1 0 ~ , ~ ~ , ~3 3 3 3 T T T T I Y Z K X
0 3 1 3 3 1 3 ( )~ , ~ t t t t t t E G X
Yr
, t0,1,2,...,T 1. I D D X~t3 (
~t)'
t3
t3~'t , t0,1,2,...,T 1. ) ~ ' 1 ( ~ ' ~3 3 3 t t t t t C I C Y
, t0,1,2,...,T 1. ) ( ~ ~ ' ~ ' ~ 3 1 3 3 3 t t t t t t t B I B Z E q Z
, t0,1,2,...,T 1. t t t t t t K A I A K~ ~3 3'~ 3 '~ 1 3
, t0,1,2,...,T 1. 0 ~ ~ ~ ~ , 2 ~ , 2 ~ ~ , ~ ~ , ~1 1 1 1 1 1 1 1 1 1 T T T T T T T T T T II N O P Q I R X Y Z K M
1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 ~' ~ ,~ ~' ~ , ~ ~' ~ ,~ ~' ~ ~ T T T T T T T T T T T T D B R B C X D A Y A C Q1 1 2 1 1 1 2 1 ~' ~ , ~ ~' ~ ~ T T T T T T A B K A A Z
The following notations are defined for t0,1,2,...,Ti and i1,2: i t t t i t t i t i t t i t t i t E(GM~ 1G)(i1),
N~1(r0)2,
r0E(G)P~1
) ( ~ , ~ ) ( 0 1 1 ti ti t t i t t t i t E qG Q
Rr E q
t i t t t t i t t i t t i t D D D II D D I D M~ (
~)
(
~)
~' ' ~ (
~)'
'~ ) ~ ' 1 ( ~ ' ) ~ ' 1 ( ~ ' ~ ~ 2 t t i t t i t t i t t i t C C I C C I C N
t i t t i t t t i t t i t t t i t t i t t i t B B B II B O E q BI B B I B O~ ~' ~ ~' '~ ~ ( 2) '~ '~ '~ '~ 1
i t t t t t t t i t t i t t i t t i t D C C I C D I I C D D IC P~ 2~'
~ 2
~ 2
(1 '~)~' [(1 '~)(
~)'~' ~' ]
I D D B I D D I B B II D B B D Q~ti 2~'t
ti~t2
ti~t2
ti~'t '~t[~'t (
~t)'~'t ~t]
ti (
~t)'
ti
ti~'t ) ~ ' 1 ( ~ ' ] ~ ' ~ ) ~ ' 1 ( ~ [' ] ~ ' ~ ) ~ ' 1 ( ' ~ ~ [ 2 ~ 0 1t t t t ti t ti t t t t ti t ti t i t i t N r B I I C B C B I C CI B C I C R
} ] ' ~ ' ~ )' ~ ( ' ~ [ ] ~ ' ~ ~ ~ ' ~ [ 2 { ~ i t t t t t t t i t t i t t i t t i t D A A DII A A I D D IA X
]} ~ ' ~ ) ~ ' 1 ( ~ [' ) ~ ' 1 ( ' ~ 2 ~ ' ~ 2 { ~ t t t t i t t t i t t i t t i t A C A I I C A I C CI A Y
} ~ ' ~ ' ] ~ ' ~ ~ ' ~ [' ~ )' ( ' ~ 2 { ~ t i t t i t t t t t i t t i t i t t i t A II B AI B BI A A I A Z
] ~ )' ' ' ( ' ~ [ ~ ~ 2 1 t ti t ti t i t i t K A II I A K
.where I is the n-dimensional column vector of element 1, and is a unit matrix.
By using the procedure on the pre-commitment strategy in Section 2, the corresponding investment strategy for problem (A(
,
,
)) can be given in the following Theorem 3.1.Theorem 3.1. The optimal strategy vt and optimal value function ( , 0, )
t t t t v v L
f at the
period t for problem(A(
,
,
)), respectively, is t t t t t t t tA
B
L
C
v
D
v
v
~
~
~
0
~
，t0,1,2,...,T 1 (3.3) 0 0 2 2 0 1 0,
)
'
~
~
(
)
~
~
'
~
'
~
,
(
t t t t t t t t t t t t t t t t t t t tv
v
L
v
M
v
N
v
O
L
P
v
v
Q
v
L
R
L
v
f
1 ,..., 2 , 1 , 0 , ~ ~ ~ ' ~ 0 X tvt Ytvt ZtLt Kt t T . (3.4)And then, in accordance with Theorem 2.1, we can obtain 3 0 3 0 3 0 3 0 * ~ 2 1 ) ~ ~ ~ ( 2 1 ~ K Z Y X
. (3.5)Theorem 3.2. The optimal investment strategy of problem (P(
,
)), the correspondingexpectation and variance of surplus and expected transaction cost for t0,1,2,...,T 1 is, respectively, as follows t t t t t t t t A BL Cv Dv v* *~ ~ ~ 0 ~
, (3.6) 3 3 3 3 * *) ~ ~ ~ ~ ( T t t t t t S K X Y Z E
(3.7) 0 1 1 0 1 2 1 2 0 1 1 *) ' ~ ~ ( ) ~ ~' ~ ' ~ ( T t t t t t t t t t t t t t t t t t S v M v N v O L P vv Q vL R Lv Var 2 * 1 1 * 0 1 * 1 *~ ~ ~ ~ [ ( )] T t t t t t t t t v y v z L K E S x
(3.8) t t t t t t t t t t t t t T t TC v M v N v O L P vv Q vL E( *) ' ~2 ~2( 0)2 ~2 2 ~2' 0 ~2' 2 2 0 2 2 0 2 ~ ' ~ ~ ~ ~ t t t t t t t t t t Lv X v Y v Z L K R . (3.9)Remark 3.1. When the investor has no liability, that is Lt 0, then the pre-commitment
strategy reduces to t t t t t t A Cv Dv v* *~ ~ 0 ~
(3.10)and the expectation and variance of surplus and the expected transaction costs for
T t0,1,2,..., , respectively, is： 3 3 3 * *) ~ ~ ~ ( T t t t t S K X Y E
(3.11) 2 * 1 0 1 * 1 * 0 1 2 0 1 1 *) ' ~ ~ ( ) ~ ' ~ ' ~ ~ [ ( )] ( T t t t t t t t t t t t t t t T t S v M v N v P vv x v y v K E S Var
(3.12) 2 0 2 2 0 2 2 0 2 2 *) ' ~ ~ ( ) ~ ' ~ ' ~ ~ ( T t t t t t t t t t t t t t t TC v M v N v P vv X v Y v K E . (3.13)Remark 3.2. When ignoring the transaction cost, that is Ct 0,t 01,,...,T1 , the
pre-commitment strategy can be acquired by setting
0 in equations (3.10). In addition, if the liability Lt 0 ( t0,1,...,T 1 ) at the same time, the pre-commitment optimal strategy and the frontier is equivalent to those in Li and Ng (2000).In summary, Theorem 3.1 generally includes a portfolio optimization strategy and the corresponding frontier that does not contain transaction costs or liabilities, or both.
3.2 Time-consistent strategy for problem (P(
,
))Here, we show the time-consistent strategy for multiperiod MVALM problem with quadratic transaction cost. The backwards induction is applied to solve the time-consistent strategy containing a riskless asset.
For t0,1,...,T1 and
0 if t T1,
1 if t T1, we define: 1 ~ T x,
~ yT 1,
~ zT 1 t t t t t t t E GM G x x ~
( ~ 1 ) ~1~'1 2 ,..., 1 , 0 , ~ ~ ' ~ )' ( 1 , 2 ~ 1 1 0 1 0 1 0 T t y x o q G E T t I t t t t t t T t ) ~ ( ~ 1 t t t t E G d G
1 ] ~ 2 ~ 2 [ ~ t
t
t )) ~ ( 2 ~ 2 ( ~ ~ t t t t t t E Gd G a
)] ~ ~ )( ( ~ ~ ) ( [ ~ ~ 1 1 0 1 1 t t t t t t t t t E G x z r I E G p k c
1 0 ( )~ ~ ( ~ ~ t t t t t t E Gq j b
t t t t t t t a E G x z r I a x ( ~)' ( )~ ~ '~ ~ 0 1 1
t t t t t t t t t x E G b y E q z r I b y ~' ( )~ ~ ( ) ~ '~ ~ 0 1 1 1 0 1 ~ ~ t t t z r z t t t t t t t ~ [x~ E(G) ~z r I ]'c~ ~ 0 1 1 1
) ( ~ )' ( ~ t t t t a a m
) ( ~ ~ ' ~ ~ ~ ' ~ ~ 2 1 0 t t t t t t t t t b b b n E q n
) ~ ~ ' ~ 2 ( )' ~ ( ~ 0 t t t t t a b o
] ~ ) ( ~ ' ~ 2 [ )' ~ ( ~ 1 1 T t t t t t a c E G p p
] ~ ) ( ' ~ ~ ) ( [ ~ ' ~ ~ ~ ' ~ 2 ~ 1 1 0 t t t t t t t t t t t b c c
E q
p E G b
] ~ ~ ) ( ' ~ [ ~ ˆ ' ~ ~ 1 t 1 t
ct tct pt E Gt ct) ~ )( ~ ( )' ~ ( ~ ' ~ ~ 1 t t t t t t t t a a a E Gd G a d
] ~ ) ( ' ~ ) ( ~ [ ~ ' ~ ~ 1 2 1 t t t t t t t t t b b h E q j E Gq b h
)] ~ ) ( ~ ) ~ ( 2 )' ~ [( ~ ' ~ 2 ~ 1 1 t t t t t t t t t t t a b a E Gd G b E Gq j j
（ ] ~ ) ( ~ ) ~ ( 2 )' ~ [( ' 2 ~ 1 1 t t t t t t t t t t t a c a E Gd G c E Gq k k
)] ( ~ ~ ) ( ' ~ ~ ) ( ' ~ [ ~ ~ ' ~ 2 ~ 1 1 1 t t t t t t t t t t t t t b c j E Gq c k E G b u E q u
] ~ ) ( ' ~ ~ [ ~ ~ ' ~ ~ 1 1 t t t t t t t t c c
k E G c
.By applying Bellman's principle of optimality, the time-consistent investment strategy of problem (P(
,
)) is given in the following theorem.Theorem 3.2. The time-consistent investment strategy of problem (P(
,
)) for1 ,..., 1 , 0 T t is given by t t t t t t av bL c v ~ ~ ~ , (3.14)
and the expectation of surplus is
t t t t t t t T t S x v yL zv E( )~' ~ ~ 0
~ (3.15)the variance of surplus is
t t t t t t t t t t t t t T t S v mv nL o vL p v L Var( ) ' ~ ~ 2 ~' ~'
~
~ (3.16)the expected transaction costs is
t t t t t t t t t t t t t T t TC v dv hL j vL k v uL E( ) ' ~ ~ 2~' ~' ~
~. (3.17)Proof. See Appendix C.
Remark 3.3. If the investor have no liability, that is Lt 0 for t0,1,...,T1, then the
time-consistent strategy reduces to
t t t t av c v ~ ~ (3.18)
and the expectation and variance of surplus and the expected transaction costs, respectively, is： t t t t t T t S x v zv E( )~' ~ 0
~ (3.19) t t t t t t T t S v mv p v Var( ) ' ~ ~'
~ (3.20) t t t t t t T t TC v dv k v E( ) ' ~ ~'
~. (3.21)Furthermore, the Theorem 3.2 still generalizes the situation without transaction cost when 0
, and the situation without liability and transaction cost.Remark 3.4. When ignoring the transaction cost, that is Ct 0,t01,,...,T 1, then the
time-consistent strategy can be acquired by setting
0 in the equations (3.18). And the expectation and variance of surplus and the expected transaction costs at period t can be obtained in the same way.Similarly, Theorem 3.2 generally includes a portfolio optimization strategy and the corresponding frontier that does not contain transaction costs or liabilities, or both.
4 Analytical optimal solutions of multiperiod MVALM problem without riskless assets
To our best knowledge, most existing literatures about portfolio selection only concern the market with a riskless asset and risky assets. However, Yao et al. (2014) pointed out that, in some real investments, the riskless asset does not exist due to the stochastic nature of real interest rates and the inflation risk. In addition, Viceira (2012) held that the expected return on riskless asset is time-varying especially in multiperiod investment. Ma et al. (2013),Gülpınar
et al.(2016) and Chiuet al.(2017) alsostudied the market without riskless assets. Therefore, it is necessary to take an economy with only risky assets into account for the multiperiod asset allocation.
Here, we consider a market consisting of only n risky assets presented in Section 2. In this setting, this portfolio optimization problem can be written as follows:
1
,...,
1
,
0
,
'
,...,
1,
0
,
1
,...,
1
,
0
,
1
,...,
1
,
0
,
0
1
,...,
1,
0
),
(
1
,...,
1
,
0
),
(
.
.
)
(
)
(
ar
)
(
max
))
,
(
ˆ
(
1 0 1 1 1 ) (T
t
v
v
TC
T
t
L
W
S
T
t
L
q
L
T
t
v
I
T
t
v
v
G
v
T
t
v
v
e
W
t
s
TC
E
S
V
S
E
P
T t t t T t t t t t t t t t t t t t t t T t T t T t t v
(4.1)Obviously, the solving of multiperiod portfolio model without riskless assets is similar to that of the model with a riskless asset. Thus, we omit the proving process and only show the results in this section.
4.1 Pre-commitment strategy for problem (Pˆ(
,
))From a mathematical point of view, the nature of the problem (Pˆ(
,
))is similar to )), (
(P
. And the difference between problem(Pˆ(
,
)) and problem (P(
,
)) is only the wealth equation. Therefore, it essentially has the same non-separable structure in the sense dynamic programming. Thus, we solve it by the Theorem 2.1. Similarly, we first construct auxiliary problem (Aˆ(
,
,
)) and then solve problem (Pˆ(
,
)) through the relationship between them. The auxiliary problem is showed below:
1 ,..., 1 , 0 , ' ,..., 1 , 0 , 1 ,..., 1 , 0 , 1 ,..., 1 , 0 , 0 1 ,..., 1 , 0 ), ( 1 ,..., 1 , 0 , ) ( . . ) ( ) ( ) ( max )) , , ( ˆ ( 1 0 1 1 1 2 ) ( T t v v TC T t L W S T t L q L T t v I T t v v G v T t v v e W t s TC E S E S E A T t t t T t t t t t t t t t t t t t t t T t T t T t t v
(4.2)The analytical solution and the optimal value of objective function to problem ))
, , ( ˆ
(A
are derived by dynamic programming approach. Define:
T T T T T II N I P I Q R M , , 2 , , t t t t t t t tM G E GM G G E( 1 )
, ( 1 )
,t0,1,2,...,T 1 1 1 1 1 ) ( ) ' ' ( 2 1 t t t t t t III I E G n A
, t0,1,2,...,T1 1 1 1 1 ) ( ) ' ' ( 2 1 t t t t t t t III I E qG P B
, t0,1,2,...,T1 t t t t t III I C
)
' ' ( 11 1 , t0,1,2,...,T 1
2, 1 2 3, / 1 t t t t t t t t t M M N N N N n N M , t0,1,2,...,T1 2 1 2 1 , t t t t t t P P Q Q Q P
, t0,1,2,...,T 13 2 1 3 2 1 , t t t t t t t t R R R O O O O R
, t0,1,2,...,T 1 0 ,1 , 3 3 3 T T T I R O N 3 1 3 ( ) ( ) t t t t C E G N N
, t0,1,2,...,T1 ) ( ) ( 3 1 3 1 3 1 t t t t t t N E G B R E q R , t0,1,2,...,T1 t t t t t O N E G A O 3 ( ) 1 3 1 3 ,t0,1,2,...,T1. 0 , , 0 , 2 , 0 , 1 1 1 1 1 1 T T T T T T II N P I R Q O M
1 1 2 1 1 1 2 1 1 1 2 1 , 2 , 2 T T T T T T T T T C C N C A P C B M 1 1 2 1 1 1 2 1 1 1 2 1 , 2 , T T T T T T T T T B B R A B O A A Q .The following notations are defined for t0,1,2,...,Ti and i1,2: t t t i t t i t C C i C C M (
)'
(
)( 1) ' , ( i1 t) t t i t E GMG
] ' ) 1 ( ) ' [( 2 i t t t t t i t C A i C A N
t t i t t t t t i t t i t C B C E qG P i C B P 2(
)
(
)' ( ) 12( 1) ' ) ( ) ( ] ) 1 ( [' 2 1 1 t t t ti t i t t i t t i t B i B P E qG B Q E q Q
t t t i t t i t t i t A i B P E qG A R 2
['
( 1)]
1 ( ) t i t t i t i t O A i A O 2 [' ( 1) ] 1
.where I is the n-dimensional column vector of element 1, and is a unit matrix.
Theorem 4.1. The optimal strategy vt of problem (Pˆ(
,
)) for t0,1,...,T 1 isspecified by: t t t t t t A BL Cv v *
* . (4.3)And the expectation and variance of surplus and the expected transaction costs, respectively, is： 3 3 3 * *) ( T t t t t S O N R E
(4.4) 2 * 1 1 2 1 1 1 1 *) ' ' ' [ ( )] ( T t t t t t t t t t t t t t t T t S v M v N v P L v QL RL O E S Var (4.5)2 2 2 2 2 2 2 *) ' ( T t t t t t t t t t t t t t t TC v M v N v P Lv Q L R L O E , (4.6) where 3 0 3 0 3 0 * 2 1 ) ( 2 1 O R N
and t0,1,2,...,T .Remark 4.1. When the investor has no liability, that is Lt 0 for t0,1,...,T1, the
above optimal strategy remains valid. Under this situation, the optimal strategy vt is specified by t t t t A Cv v *
* ，t0,1,2,...,T 1. (4.7) And the expressions of expectation and variance of surplus and the expected transaction costs for t0,1,...,T, respectively, is 3 3 * *) ( T t t t S O N E
(4.8) 2 * 1 1 1 *) ' ' [ ( )] ( T t t t t t t t T t S v M v N v O E S Var (4.9) 2 2 2 *) ' ' ( T t t t t t t t TC v M v N v O E . (4.10)This implies that the Theorem 4.1 can generalize the situation without liability.
Remark 4.2. When ignoring the transaction cost, that is Ct 0,t 01,,...,T1 , the
pre-commitment strategy can be acquired by setting
0 in equations (4.7). If the liability 0 t
L (t0,1,...,T1) and cost-aversion coefficient
0that ignores the transaction cost, the pre-commitment optimal strategy is equivalent to that in Li and Ng (2000). Therefore, Theorem 4.1 generally includes three situations just like Theorem 3.1.4.2 Time-consistent strategy for problem (Pˆ(
,
))It is not difficult to find the problem (Pˆ(
,
)) can be solved by the time consistent strategy. By using backwards induction, we derive the time-consistency strategy for multiperiod MVALM problem without riskless assets by using the procedures presented in Section 2.Define:
1 T x
